
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00098464/00001
Material Information
 Title:
 An Investigation of production scheduling problems motivated by semiconductor manufacturing
 Creator:
 Herrmann, Jeffrey William ( Dissertant )
Lee, ChungYee ( Thesis advisor )
Elzinga, Jack ( Reviewer )
Bai, Sherman ( Reviewer )
Koehler, Gary ( Reviewer )
Erengne, Seleuk
Phillips, Winfred M. ( Degree grantor )
Holbrook, Karen A. ( Degree grantor )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1993
 Copyright Date:
 1993
 Language:
 English
 Physical Description:
 viii, 219 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Deadlines ( jstor )
Dispatching ( jstor ) Genetic algorithms ( jstor ) Heuristics ( jstor ) Job hunting ( jstor ) Job shops ( jstor ) Operations research ( jstor ) Scheduling ( jstor ) Semiconductor wafers ( jstor ) Semiconductors ( jstor ) Dissertations, Academic  Industrial and Systems Engineering  UF Genetic algorithms ( lcsh ) Industrial and Systems Engineering thesis Ph.D Production scheduling  Mathematical models ( lcsh ) Semiconductor industry ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Abstract:
 Manufacturing and service organizations frequently face the challenges of making highquality products quickly and of delivering those products to their customers ontime. Improvements in the scheduling of their operations can often contribute to their success in meeting these goals. However, as manufacturing processes become more complex, the difficulty of finding good production schedules increases. This dissertation addresses dynamic deterministic job shop scheduling, a problem that occurs in many manufacturing environments. The problem is among the most difficult scheduling problems, and few solution procedures have been implemented. The approach in this research is to consider specific subproblems that are motivated by semiconductor test operations and the develop genetic algorithms that exploit alternative search spaces. The research includes new analytical and empirical results for previously unstudied onemachine class scheduling problems and threemachine lookahead problems. The onemachine problems include sequencedependent setup times. In the threemachine problems, two groups of jobs are processed on separate second stage machines. Testing shows that a new type of genetic algorithm can find good schedules for the onemachine problems by adjusting the problem data while using an appropriate heuristic. An approximation algorithm for the threemachine problem is able to find nearoptimal schedules. Moreover, this dissertation describes the development and application of a global job shop scheduling system for the semiconductor test area. This system uses a detailed deterministic simulation model of the shop floor, data about the current status of the shop, and a genetic algorithm to search over combinations of dispatching rules in order to create a good shift schedule. These rules include these motivated by this research into the onemachine and threemachine problems. The scheduling system is able to adapt to changing conditions each shift. The benefits of this work consist of the identification of dominance properties for the onemachine class scheduling and threemachine lookahead problems, the development of a problem space genetic algorithm, the definition of new lookahead heuristics, the creation of a new genetic algorithm for global scheduling, and the implementation of these results to the actual semiconductor test floor that is the motivation for this work.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1993.
 Bibliography:
 Includes bibliographical references (leaves 206218).
 Additional Physical Form:
 Also available in electronic format.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Jeffrey William Herrmann.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 030418663 ( alephbibnum )
31182006 ( oclc ) AKC5081 ( notis )

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Full Text 
AN INVESTIGATION OF PRODUCTION SCHEDULING PROBLEMS MOTIVATED BY
SEMICONDUCTOR MANUFACTURING
By
JEFFREY WILLIAM HERRMANN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
ACKNOWLEDGMENTS
Years of learning are not sufficient for one who wishes to prepare a dissertation; also
necessary is an environment which encourages one to develop the talents which God has given
and to become an intelligent person, a responsible person, a kind person, a successful person. For
providing such a setting I thank my family.
To many others am I indebted; to the chairman of my supervisory committee, Dr. Chung
Yee Lee, who has been an exceptional influence on my development as a researcher, teacher, and
scholar; to department chairman Dr. Jack Elzinga, Dr. Boghos Sivazlian, and the members of my
committee; to all who were involved in the Harris project and whose effort was critical to this
work; and to the quality friends, classmates, and teachers whom I have known and who have all
added something valuable to my life.
DISCLAIMER
This material is based upon work supported under a National Science Foundation Graduate
Research Fellowship. Any opinions, findings, conclusions or recommendations expressed in this
dissertation arc those of the author and do not necessarily reflect the views of the National
Science Foundation.
u
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ii
ABSTRACT... vii
CHAPTERS
1. INTRODUCTION 1
1.1 Semiconductor Manufacturing ..... 2
1.2 Job Shop Scheduling ...... 4
1.3 Lookahead and Lookbchind Scheduling ... 4
1.4 Setups. ........ 5
1.5 Smartandlucky Searches . ..... 6
1.6 Problem Space and Heuristic Space .... 8
1.7 Objective Functions. ...... 8
1.8 Overview of Research ...... 9
1.9 Plan of Dissertation . ...... 10
2. BACKGROUND 13
2.1 Semiconductor Test Operations ..... 13
2.2 Semiconductor Scheduling. ..... 16
2.2.1 Production Planning ..... 17
2.2.2 Shop Floor Control ...... 19
2.2.3 Performance Evaluation ..... 32
2.2.4 Summary ....... 35
2.3 Job Shop Scheduling ...... 36
2.3.1 Scheduling Notation ..... 36
2.3.2 Shifting Bottleneck ...... 37
2.3.3 Dispatching Rules ...... 39
2.3.4 Summary ....... 45
2.4 Flow Shop Scheduling ...... 45
2.4.1 Makespan ....... 46
2.4.2 Total Flowtime ...... 46
2.4.3 Maximum Lateness (Lmax) .... 50
2.4.4 Number of Tardy Jobs ..... 51
iii
2.4.5 General Topics ...... 52
2.4.6 Summary ....... 55
2.5 Lookahead and Lookbchind Scheduling ... 55
2.6 Class Scheduling ....... 57
2.7 Some Onemachine Problems ..... 60
2.7.1 Constrained Flowtime ..... 61
2.7.2 Release and Due Dates . . . . 61
2.7.3 Flowtimc and Release Dates .... 62
2.8 Smartandlucky Searches ...... 64
2.8.1 Introduction ....... 64
2.8.2 Simulated Annealing ...... 64
2.8.3 Tabu Search ....... 67
2.8.4 Genetic Algorithms ...... 70
2.8.5 Summary ....... 75
2.9 Problem and Heuristic Space ..... 75
2.10 NPComplctencss ....... 77
2.11 Chapter Summary ....... 78
3. ONEMACHINE CLASS SCHEDULING PROBLEMS . 80
3.1 Introduction ........ 80
3.2 Constrained Flowtimc with Setups .... 81
3.2.1 Introduction . . . . . 81
3.2.2 Literature Review ...... 84
3.2.3 Notation and an Optimal Properly. ... 85
3.2.4 The Heuristic ...... 87
3.2.5 The Genetic Algorithm ..... 91
3.2.6 Empirical Testing ...... 100
3.2.7 Conclusions ....... 104
3.3 Class Scheduling with Release and Due Dates . . . 104
3.3.1 Introduction ....... 105
3.3.2 Literature Review ...... 105
3.3.3 Notation and Problem Formulation . . . 106
3.3.4 Heuristics ....... 106
3.3.5 Analysis of the Heuristic ..... 109
3.3.6 The Genetic Algorithm . . . . 113
3.3.7 Empirical Tests and Results . . .115
3.3.8 Extension to Minimizing Tardiness . 120
3.3.9 Conclusions . . . . . .123
3.4 Flowtimc with Setups and Release Dates . . 124
3.4.1 Introduction ....... 124
3.4.2 Notation and Problem Formulation . . .125
3.4.3 Background . . . . .125
3.4.4 Solution Techniques . . . . .126
IV
3.4.5 Empirical Testing ...... 134
3.4.6 Special Case . . . . . . .136
3.4.7 Conclusions ....... 140
3.5Chapter Summary . . . . .141
4. LOOKAHEAD SCHEDULING PROBLEMS . ... 142
4.1 Introduction ........ 142
4.2 Minimizing the Makespan ...... 145
4.2.1 Notation ....... 145
4.2.2 Johnson's Algorithm ..... 146
4.2.3 Permutation Schedules ..... 147
4.2.4 NPCompleieness ...... 149
4.2.5 Makespan Optimality Conditions and
PolynomiallySolvable Cases . . 150
4.2.6 BranchandBound Algorithm .... 158
4.2.7 Heuristics ....... 160
4.2.8 Empirical Results ...... 160
4.2.9 Heuristic Error Bounds . . . . .162
4.3 Minimizing the Total Flowlime . . .164
4.3.1 Total Enumeration ...... 165
4.3.2 Lower Bounds . . . . .167
4.3.3 Special Case . . . . . . .168
4.3.4 Empirical Testing ...... 169
4.4 Minimizing the Number of Tardy Jobs .... 170
4.4.1 Problem Introduction . . . .170
4.4.2 Lower Bound and Special Case .... 171
4.4.3 Heuristics ....... 172
4.4.4 Results . . . . . . .173
4.5 Application to Job Shop Scheduling . . .174
4.6 Chapter Summary ....... 175
5. GLOBAL JOB SHOP SCHEDULING 177
5.1 Introduction ........ 177
5.2 Job Shop Scheduling ...... 179
5.3 A Genetic Algorithm for Job Shop Scheduling . . 181
5.3.1 The Heuristic Space. ..... 182
5.3.2 A Genetic Algorithm for Global Scheduling . . 185
5.4 Global Job Shop Scheduling . . .187
5.4.1 The Semiconductor Test Process .... 187
5.4.2 The Previous Scheduling System . . . 190
5.4.3 Scheduling Needs ...... 191
5.4.4 Scheduling System Design. .... 192
5.4.5 Information Requirements . .... 193
v
5.4.6 Implementation of Global Scheduling 195
5.4.7 Implementation Issues ..... 197
5.4.8 Contributions of Global Scheduling . 199
5.5 Chapter Summary ....... 200
6. SUMMARY AND CONCLUSIONS .... 201
6.1 Onemachine Class Scheduling Problems .... 201
6.2 Lookahead Scheduling ...... 202
6.3 Searching for Job Shop Schedules ..... 202
6.4 Conclusions ........ 203
REFERENCES 206
BIOGRAPHICAL SKETCH 219
vi
Abstract of Dissertation Presented to the Graduate Sehuol of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doetor of Philosophy
AN INVESTIGATION OF PRODUCTION SCHEDULING PROBLEMS MOTIVATED BY
SEMICONDUCTOR MANUFACTURING
By
Jeffrey William Herrmann
December, 1993
Chairman: Dr. ChungYee Lee
Major Department: Industrial and Systems Engineering
Manufacturing and service organizations frequently face the challenges of making high
quality products quickly and of delivering those products to their customers ontime.
Improvements in the scheduling of their operations can often contribute to their success in
meeting these goals. However, as manufacturing processes become more complex, the difficulty
of finding good production schedules increases.
This dissertation addresses dynamic deterministic job shop scheduling, a problem that
occurs in many manufacturing environments. The problem is among the most difficult
scheduling problems, and few solution procedures have been implemented. The approach in this
research is to consider specific subproblems that are motivated by semiconductor test operations
and to develop genetic algorithms that exploit alternative search spaces.
The research includes new analytical and empirical results for previously unstudied one
machine class scheduling problems and threemachine lookahead problems. The onemachine
problems include sequencedependent setup times. In the threemachine problems, two groups of
jobs are processed on separate secondstage machines. Testing shows that a new type of genetic
algorithm can find good schedules for the onemachine problems by adjusting the problem data
while using an appropriate heuristic. An approximation algorithm for the threemachine problem
is able to find nearoptimal schedules.
vii
Moreover, this dissertation describes the development and application of a global job shop
scheduling system for the semiconductor test area. This system uses a detailed deterministic
simulation model of the shop floor, data about the current status of the shop, and a genetic
algorithm to search over combinations of dispatching rules in order to create a good shift
schedule. These rules include those motivated by this research into the onemachine and three
machine problems. The scheduling system is able to adapt to changing conditions each shift.
The benefits of this work consist of the identification of dominance properties for the one
machine class scheduling and threemachine lookahead problems, the development of a problem
space genetic algorithm, the definition of new lookahead heuristics, the creation of a new genetic
algorithm for global scheduling, and the implementation of these results to the actual
semiconductor test floor that is the motivation for this work.
viii
CHAPTER 1
INTRODUCTION
Why investigate production seheduling problems motivated by semieonductor
manufacturing? Because scheduling works. Scheduling, the term, refers to the process of
assigning tasks to resources and determining when each task will be done. Scheduling is not,
however, limited to manufacturing, since this type of problem occurs in other activities.
Scheduling, the science, embodies knowledge about models, techniques, and insights related to
actual systems (Baker, 1992). This field is an important part of operations research. In order to
compete effectively in the marketplace, firms have used operations research as a tool for
understanding and improving their manufacturing systems.
This is especially true in the manufacturing of hightechnology products, where the new,
complicated processes lead to production scheduling problems that are not wellsolved. The
postassembly testing of semiconductors, for instance, is a complicated job shop seheduling
problem; among the difficulties is the presence of sequencedependent setups at certain
operations. Moreover, scheduling, which has been studied from the earliest days of operations
research, has received new attention due to the successful implementation of justintime systems
that emphasize the close coordination of resources and the maintenance of low workinprocess
inventory levels.
Although the research in production scheduling has yielded a large body of knowledge
which has been successfully applied to a number of areas, there still remain problems to be
solved. The problems previously studied have always simplified reality in some way. Adding
realism creates problems that resist simple solution techniques. Further research into these types
of problems is necessary in order to increase our ability to control manufacturing processes
effectively.
1
2
The research described in this dissertation falls into this category. It includes factors that
model reality more closely and studies a number of different problems in an effort to gain insight
into how improved job shop scheduling may be achieved. This insight will be applied to the
semiconductor test area being studied.
The problems that this research investigates arc motivated by a hightechnology field where
much time has been spent investigating manufacturing systems, namely the semiconductor
industry. The research is particularly motivated by the operations of semiconductor testing,
although the problems can also be found in the processes of other manufacturing industries.
This introduction includes a number of sections about topics that will arise in the
scheduling of semiconductor test operations. These topics are introduced here in order to
describe how they are related to this research. The first of these is an overview of semiconductor
manufacturing.
U Semiconductor Manufacturing
In order to give some context to the problems being studied, this section describes the
general flow of semiconductor manufacturing.
The manufacturing of semiconductors consists of many complex steps. The first process is
that of wafer fabrication, done in a superclean environment in order to protect the delicate
structures on the wafers from contaminants. The wafers of silicon undergo repeated applications
of the cycle of photolithography, etching, and diffusion in order to build, layer by layer, the
different electronic devices. A wafer will contain a number of identical circuits laid out in a grid
pattern on the wafer. In the next process, probe, these individual circuits are inspected with a
wafer probe and marked if defective. In the assembly process, another clean area, the wafers arc
cut into the separate circuits. Good circuits are placed into the packages that protect them from
the environment and that provide electrical connectioas to allow them to interact with the world.
The identical devices from a lot of identical wafers become a lot that then moves into the test
3
facility, an area where environmental safeguards protect the devices from static electricity that
could still damage die circuits.
The test process consists of initial testing, bumin, and final testing. The testing includes
electrical testing of each device and other testing of the device packages. Bumin consists of
subjecting the semiconductor chips to extreme thermal stresses. Other nontest operations, such
as serialization and brand, take place during the process.
Not all of the lots follow the same route through the test floor. Different products arc tested
on different machines and undergo a different set of tests. Thus, the test area is a job shop, and
the scheduling problem is a job shop scheduling problem. This job shop of semiconductor test
operations has sequencedependent setup times, reentrant flows, batch processing, routes that
temporarily leave the test area, and other complicadons that lead to many interesting scheduling
questions.
The primary management objectives for diis area are customer satisfaction and making
profit. Because testing is the last process in the manufacturing flow, the output of this area
affects how well the company can get orders to its customers on time and how much product the
company can ship to generate revenue.
The particular semiconductor test area being studied has implemented a decision support
system to assist in the scheduling of lots through the area. The programs in the system can order
the lots waiting for an workstation with priority rules that include information about lateness,
priority, and setups. As part of the facilitywide computerintegrated manufacturing system, it is
able to access realtime information about the lots. The system is used to create schedules for a
short time period (two to four hours), since dispatching more often would require excessive
computer resources. Lots that arc tardy and lots that have some externallyimposed priority are
expedited whenever possible.
4
L2 Job Shoo Scheduling
The semiconductor test area is a job shop, and optimizing the scheduling of the area is a job
shop scheduling problem. This section discusses job shop scheduling.
Job shop scheduling consists of those scheduling problems in which different jobs may
follow different routes through the shop. A job consists of a number of operations or tasks, which
must be processed in the given order. There exists a specified machine (or workstation) that can
perform each operation. These problems are generally the hardest to solve optimally, since few
properties of optimal schedules arc known and the number of possible solutions explodes as the
problems increase in size. However, in many cases, the job shop is a better model of reality than
onemachine, parallelmachine, or flow shop problems.
Because of the complexity of job shop scheduling, algorithms to find the optimal solution
(in a reasonable amount of time) for any arbitrary objective function do not exist. Recent
research has shown that the shifting bottleneck heuristic is successful at finding good solutions for
a simple job shop scheduling problem. Traditionally, however, researchers have studied and
schedulers have used dispatching rules to order the jobs waiting for processing at a machine.
When a machine becomes available, it chooses from among the jobs in its queue by using a
dispatching rule. Common dispatching rules employ processing times and due dates in both
simple rules and complex combinations. These dispatching rules arc often extensions of the
algorithms used to solve onemachine problems.
In any shop, there may exist a bottleneck machine whose throughput is a limiting factor on
the capacity of the shop. If so, the improved scheduling of this one machine becomes an
important objective.
1.3 Lookahead and Lookbehind Scheduling
This section addresses a weakness of the traditional dispatching rule approach to
scheduling job shops and defines two t\pes of rules that overcome this weakness.
5
Traditional dispatching rules are myopic; they are concerned only with the machine to be
scheduled and the jobs waiting for that machine. Improved scheduling may be realized by using
rules that can consider more information, in order to see that critical lots are arriving soon or that
a certain machine has an excessive queue. Lookahead and lookbehind scheduling includes
procedures that look around the shop for more information to use in making a scheduling
decision. With this additional information, they can hopefully produce better decisions. Look
ahead models consider the machines where the jobs will be headed after this stage. Lookbehind
models consider the jobs that will be arriving at this machine soon.
The terms lookahead and lookbehind are used to designate scheduling procedures that do
more than consider just the state of one machine. Both types of models look into the future
(where jobs will go and what jobs will be arriving). The difference is what part of the future they
consider.
1.4 Setups
One of the complications of scheduling the operations in semiconductor testing is the
presence of different product types that require different configurations on the same machine.
The task of configuring a machine in order to process a job is a setup.
If the setup for a job is independent of the job that was scheduled before it on the same
machine, the setup time can be included in the processing requirements of that job. If the setup
for a job is sequencedependent, that is, the time of the setup depends upon the immediate
predecessor job, the scheduling problem becomes quite difficult.
Because the problems with sequencedependent setups are quite difficult, researchers have
examined special cases of the general problem. The most common problem is the class
scheduling problem, where the jobs to be processed are grouped into a number of job classes.
There exists no sequencedependent setup between jobs from the same class, although there does
exist a special setup, the class setup, when a job from one class is processed after one from
another class. There may also exist a class setup for the first job processed. These class setups
6
may be sequencedependent in that they depend upon the class that was last processed. Class
scheduling is still difficult, though: the class scheduling extensions of onemachine problems that
are easy to solve are usually NPcomplete problems.
An example of class setups occurs in the electrical testing of assembled semiconductor
devices on a machine which can test a number of different types of semiconductors. If a machine
is scheduled to test a lot consisting of devices that are different from the devices tested in the
previous lot, various setup tasks are required. These tasks may include changing a handler and
load board that can process only certain types of semiconductor packages or loading a new test
program on the tester. However, if the new lot consists of devices that are the same as the
previous type, none of this setup is required. This is a class scheduling problem.
1.5 Smartandluckv Searches
As mentioned before, algorithms to solve the job shop scheduling problem optimally in
reasonable time do not exist. In addition to the shifting bottleneck heuristic and the use of
dispatching rules, one way to find good solutions is to search for them. This section will discuss
standard local searches and new, more sophisticated heuristic searches.
One method of finding good solutions to hard optimization problems has been local search.
A local search begins with some initial solution and moves from an incumbent solution to a better
neighboring solution, ending when no improvement can be found. At this point, the search has
reached a local optimum. Two common local searches arc hillclimbing and steepest descent. In a
hillclimbing search, the neighbors are chosen at random, and the first better neighbor found is
chosen as the new solution. In a steepest descent search, the entire neighborhood of the
incumbent solution is searched and the neighbor that has the best performance improvement is
selected. In order to overcome the fact that these searches converge only to local optima, new
heuristic searches have been developed. These include simulated annealing, tabu search, and
genetic algorithms.
7
These heuristic searches are complex searches that arc .smart enough to escape from most
local optima and are lucky enough to generally find good solutions.
Simulated annealing (SA) is a variant of the hillclimbing algorithm. Simulated annealing is
so called because the algorithm views optimization as a process analogous to physical annealing,
the cooling of a system until it reaches a lowenergy state. In the same way that a system may
pass through higherenergy states during the cooling process, the simulated annealing search
occasionally moves to worse neighbors before settling into a good solution.
Tabu search (TS) is a variant of steepest descent. Given an incumbent solution, a TS
canvasses the neighborhood of this solution in order to find the best allowable move. If the
search is at a local optimum, it is forced to move away, and the move that would return to the
previous solution is temporarily prohibited (tabu) by adding it to the tabu list for a number of
moves. In this way, TS may continue to move away from a local optimum and find an area of the
space that leads to another local optimum. An aspiration level insures that the search will make a
move that leads to a solution better than any yet found, even if tabu. Thus, a tabu search works
because the tabu list forces the search to explore new areas of the solution space. The shortterm
aspect of the memory and the aspiration level allow, however, the search to get to a global
optimum.
A genetic algorithm (GA) is a procedure that mimics the adaptation that nature uses to find
an optimal state. In genetic algorithms, solutions are represented as strings (chromosomes) of
alleles. An allele is a bit of information about the solution. The search performs operations on
the population of solutions. These operations are 1) the evaluation of individual fitness, 2) the
formation of a gene pool, and 3) the recombination and mutation of genes to form a new
population. After a period of time, good strings dominate the population, providing a good
solution.
When applied to scheduling problems, simulated annealing and tabu search easily search
the complex solution spaces with simple types of moves, generally find good solutions fast, and
are smart and lucky variations of standard searches such as hillclimbing and steepest descent.
8
Genetic algorithms arc something completely different, work well by using good strings and
implicit parallelism, and arc harder to implement.
1,6 Problem Space and Heuristic Space
Typical problemsolving searches like those described in the above section have examined
the solution space. A solution can be a vector of values or a sequence of objects. Applying a
heuristic to a problem yields a point in the solution space. This implies that a search over all of
the possible heuristics would find a solution that gives the optimal objective function value.
Moreover, applying a heuristic to another, similar problem also generates a point in the
solution space which can be evaluated for the original problem using the objective function.
Thus, a search over the new problem space provides a way to solve the problem.
An example of a heuristic space is the space of vectors of m dispatching rules for a Â«job
and mmachinc job shop scheduling problem, where each dispatching rule is applied to a different
machine. Searching over the vectors of dispatching rules is similar to machine learning. For a
complicated onemachine problem, a sample problem space would be the space of Â«element
vectors, where each element corresponds to an alternative due date for a job. Sorting the jobs by
these alternative due dates creates a new solution.
1.7 Objective Functions
The management of a manufacturing system is concerned with many different performance
measures, including customer service, inventory holding costs, throughput, and machine
utilization. This section describe how the objectives used in scheduling mirror these concerns.
Scheduling problems include many different objective functions that attempt to model the
concerns of those who are running the system. Most objectives are functions of the completion
times of the jobs to be scheduled. The makespan is the maximum completion time, which is
when all of the jobs are finished, and is some measure of the throughput of the system. The total
flowtime is the total time that the jobs spend in the system and is a measure of the workin
9
process inventory held while processing die jobs. Other objective functions include due dates for
each job, where a job is tardy if it completes after its due date. These objective functions arc
concerned with customer satisfaction, and include maximum lateness, which is the maximum
difference between a completion time and a due date, the number of tardy jobs, and the total
tardiness. Another interesting objective function is earliness, which is a measure of the holding
cost incurred by storing finished product until it is delivered at its due date.
While all of these objective functions (and many more) have been studied for onemachine
problems, most analysis of flow shop and job shop scheduling has focused on the minimization of
makespan. Since this objective is not appropriate in semiconductor manufacturing, this research
is especially interested in solving job shop scheduling problems with other objectives.
1 .X Overview of Research
There exist different approaches to attacking the job shop scheduling problem. One can use
the shifting bottleneck algorithm to minimize the makespan, use good dispatching rules in a
dynamic environment, or search for good global schedules. This research is concerned with the
last two methods (see Figure 1.1). This research investigates the use of smartandlucky searches
over the new search spaces to find good solutions to the job shop scheduling problem. Also
investigated are a number of interesting onemachine class scheduling problems and lookahead
models in order to devise techniques that can be used as good dispatching rules. Finally, this
research studies how these techniques may be applied to the actual semiconductor test floor that
is the motivation for this work.
The benefits of this work include three areas: 1. The addition of results to the body of
knowledge about specific scheduling problems, including the difficult class scheduling problems
and littlestudied lookahead models. 2. The construction of a robust genetic algorithm for one
machine class scheduling problems. 3. The development and implementation of a genetic
algorithm for global job shop scheduling. Moreover, this work continues the investigation of
scheduling semiconductor test operations that has only recently begun.
10
Figure 1.1. Relationship of dissertation topics.
1.9 Plan of Dissertation
This research investigates both onemachine and multimachine problems. The one
machine problems studied are class scheduling problems that model the complicating factor of
machine setups in the manufacturing process. This work also considers some threemachine
general flow shop problems that are attempts to model how scheduling can be improved by
considering the states of other machines. The research also investigates a general job shop
scheduling problem in an attempt to see how new search spaces may be used for global job shop
scheduling.
For these problems, the research has focused on finding good lower bounds, developing
heuristics to find good solutions for the problems, and using genetic algorithms to find better
solutions. Empirical testing of these bounds and heuristics serves as a way of evaluating the
11
heuristics and the searches. Also, good heuristics for the subproblems can be implemented as
advanced dispatching rules for solving the job shop scheduling problem.
In addition, analytical results such as NPcomplctcncss, lower bounds, error bounds, and
optimality conditions arc presented for the problems being studied.
The machine scheduling problems under investigation arc as follows:
1. Constrained Flowtimc with Setups (CFTS)
2. Class Scheduling with Release and Due Dates (CSRDD)
3. Flowtimc with Setups and Release Dates (FTSRD)
4. ThreeMachine LookAhead Scheduling: Makespan (3MLAMS)
5. ThreeMachine LookAhead Scheduling: Flowtime (3MLAFT)
6. ThreeMachine LookAhead Scheduling: Number of Tardy Jobs (3MNT)
7. Job shop scheduling
The first problem is the class scheduling extension of the onemachine problem of
minimizing the total flowtimc of a set of jobs that have deadlines. The second problem is the
class scheduling problem where each job has a release date and a due date. The objective is to
minimize the number of tardy jobs. The objective in the third problem is to minimize the total
flowtime where each job has a release date.
The research next considers the threemachine lookahead problems with the objectives of
makespan, total flowtimc, and number of tardy jobs.
Finally, this research investigates the general job shop scheduling problem and a heuristic
space approach to finding good solutions with a genetic algorithm.
The dissertation consists of the following five chapters: the background; the research on
onemachine class scheduling problems, on threemachine lookahead problems, and on job shop
scheduling; and the conclusions.
The background material contains detailed information about the problems and methods
under investigation and summaries of a number of papers that arc related to this work. The topics
include semiconductor test operations and semiconductor scheduling, job shop scheduling
12
techniques, class scheduling, some oncmachinc problems, smartandlucky searches, problem
and heuristic space, lookahead and flow shop scheduling, bumin scheduling, and NP
complctcncss.
The discussion of the research contains reports on the problems that have been investigated:
three onemachinc class scheduling problems, three threemachine general flow shop problems,
and a heuristic search for use on the general job shop scheduling problem. In the conclusions we
summarize this research and identify some directions for future research.
CHAPTER 2
BACKGROUND
This chapter of the dissertation performs two functions: it elaborates on the topics in the
introduction and provides a review of the relevant literature. The discussion in this chapter is
both wideranging and descriptive. In Chapters 3,4, and 5, discussions of previous research will
be problemspecific and less explicit.
The first section will describe in more detail the particular operations associated with the
testing of semiconductor devices. Remaining sections will discuss papers on semiconductor
manufacturing, job shop scheduling techniques, and flow shop scheduling problems. Also
included in this chapter are sections on lookahead and lookbchind scheduling, the scheduling of
the bumin operation, class scheduling, some onemachine problems, smartandlueky searches,
problem and heuristic space, and NPcompleteness.
2,1 Semiconductor Test Operations
The manufacturing of semiconductors consists of many complex steps. As described in the
introduction, this includes wafer fabrication, probe, assembly, and test. This research is
concerned with this last facility. Although the routes of lots through the test process vary
significantly over the many different types of products, general trends can be described. Figure
2.1 shows what these routes look like for seventeen representative products of a lest area. The
numbers on each arc arc the number of products that follow that path.
Presented here is a description of the operations in a typical product route. The terms
semiconductor and device describe the same thing.
13
14
Figure 2.1. Product Routes in ihe Test Area.
15
The lot arrives at the lotform area with an assembly traveler that describes the history of the
lot up to that point. A lot is started through the test lloor when it is needed to meet back ordered
demand or when the area planner includes it on the weekly release plan. The test traveler for this
product is printed, and engineering information is extracted from the database and attached to the
traveler. The tubes of deviees are placed into a box, and the box is moved to a waiting area.
The lot is then serialized. Done in the brand area, serialization is the process of giving each
device a unique serial number. This process is laborintensive as someone must stamp each
device with a branding machine.
The lot then undergoes Xray testing. This operation includes six tasks: loading the devices
into racks, shooting the Xrays, unloading the devices, developing the Xrays, reading the Xrays
for internal defects in the package assembly, and culling the rejects. The lot may experience
significant waiting before and after the reading of the Xrays. Three people handle the lot: one
who loads and shoots, one who reads, and one who culls.
Electrical testing, the next step, is an important, machineintensive operation. During an
electrical test, the lot is processed by a handler machine that positions each device over a contact
with the tester, a computer that performs a number of tests by subjecting the device to a number
of different electrical inputs. After the test of that semiconductor is done, the device is fed to an
output bin. The test floor contains a number of different types of testers and different handlers.
Since each product requires a different combination of handler and tester, significant setup time
can be incurred if the operator must find and install a handler to process the lot.
After the initial test, the lot must be bumedin. The devices in the lot are loaded onto
boards that hold from 10 to 100 devices each. These boards are placed into ovens, where the
devices are subjected to thermal stresses and electrical inputs in order to cause infant mortality.
The bumin period lasts a minimum of 24 hours, although no penalty is associated with keeping
the devices in the oven for longer than this time. After the boards are removed from the oven, the
devices are unloaded from the boards.
16
After the bumin period, the lot returns to a tester and undergoes interim test. Interim test
consists of testing at room, low, and high temperatures. For the low temperature test, a bottle of
liquid nitrogen must be attached to the handler and the handler must chill the testing environment
to the desired temperature. In a test at high temperature, the handler warms the test chamber to
the desired temperature.
After the last bumin and interim test, a final test is conducted. The lot then goes to the
assembly facility, where fine and gross leak testing of the package seal is performed. During this
time, the lot is out of the test facility's area of control.
The lot returns and moves to brand, where each device is stamped with the company logo.
The devices arc placed into pans and then sit through a threehour bake in a kiln. After this step,
testing is performed to verify that no damage was done to the devices during brand.
After another trip to assembly, the lot undergoes a visual test, where each device is
inspected under magnification for obvious defects in the packaging. The lot enters document
review next. At this time, all of the paperwork associated with the lot must be reviewed for
completeness.
The lot then undergoes customer source inspection, where a representative of the customer
personally supervises the testing of a sample of the lot. In the shipping operation, the devices are
placed in boxes and the boxes arc addressed, ready for delivery to the customer.
2,2 Semiconductor Scheduling
The manufacturing of semiconductors has received much attention form production
planning and scheduling researchers. This section reviews a number of papers that address the
issue of semiconductor production planning and scheduling directly or indirectly. As in Uzsoy,
Lee, and MartinVega (1992a, 1993), the papers are classified into the following topics:
production planning; shop floor scheduling: dispatching rules and work release, deterministic
scheduling, batch processing, controltheoretic approaches, knowledgebased approaches, and
simulation; and performance evaluation: queueing models, and simulation. This review is
17
intended to display the numerous methods that have been used in the research of production and
scheduling issues of semiconductor manufacturing.
2.2.1 Production Planning
A number of authors have addressed the largescale problem of production planning for the
semiconductor industry. The most common approach has been a hierarchical decomposition of
the problem.
Leachman (1986) describes a corporatelevel production planning system. This system
divides the manufacturing process into the stages of wafer fab, probe, assembly and test, linked
by inventories. Its model may include multiple facilities. The production processes in a plant are
treated as a single entity, and problem complexity is reduced by the aggregation of products that
are in the same process at each stage.
Information from corporate databases is used in a linear program with dynamic production
functions that capture the relationships between the processes. This yields a production plan.
Through a set of linear programs, the aggregate plan is then decomposed into a capacityfeasible
weekly start schedule for various facilities. In a later work, this system was implemented by
Harris Semiconductor.
Hadavi and Voigt (1987) describe the planning system for a Siemens development wafer
wafer fab. In their approach, they create different levels of abstraction for different levels of
planning and localize rescheduling while making minimal resource constraints.
The system architecture is a hierarchy of constraint sets that represent different time
windows. Quarterly requirements are divided into months, weeks, and days. Arriving orders
start feasibility analysis (can it be done?) and then a scheduling heuristic (when will it be done?)
that tries to satisfies the constraint sets (based on the hierarchy). Scheduled orders are sequenced
by an algorithm that maximizes throughput. A rulebased expert system recovers from
disturbances by local rescheduling.
18
Harrison, Holloway, and Palcll (1989) discuss the production planning question and present
a ease study of National Semiconductor Corporation, 'llicy list the following programs that can
improve customer service: management information systems; logic and algorithms used to
generate delivery quotations and to schedule production; and performance evaluations and
incentive systems based on measures of delivery performance. They place the emphasis on
operational decisionmaking, from booking orders to scheduling lots on machines.
The authors also consider performance measurement, which should give the upper
management information about the different groups, all of whom arc acting to improve their
performance on these scales. These measures should encourage mangers to act optimally for the
company.
The authors make the following comments about scheduling. Marketing wants
commitments to be inviolate; production planners thus want control over factory loading.
Managers view scheduling as a MIS problem that provides reliable routine execution. The
company must get the right information to the right people and then coerce them to do the right
thing.
Golovin (1986) describes various attempts to solve the problem of production planning and
factory scheduling. Mathematically, an integer programming problem of the factory scheduling
problem considers all costs. The solution philosophies arc diverse and involve certain tradeoffs.
Mathematical optimization gives the best solution but is computationally costly in data and time
and ignores local conditions. Dispatching rules are concerned only with the present. They may
make poor scheduling decisions by ignoring global conditions, but they work quickly and
cheaply. JustInTime maintains a balance of work in the factory and attempts to maintain a high
level of quality.
The authors find most promising hierarchical systems that decompose the problem into sets
of decisions made at different times: capacity planning, release planning, and lot dispatching.
This approach may yield suboptimal policies but it gives control of decisions to the users who are
responsible for the results. Scheduling is done under the assumption that sufficient capacity
19
exists. Dealing with uncertainty in yields and equipment calls for keeping safety stock of
standard product and buffers in front of each machine to prevent starvation.
2,2,2 Shop Floor Control
Methods used to control the shop floor vary widely in their approaches to solving the
problems encountered there. The basic areas include lot release policies, dispatching rules,
deterministic scheduling, controltheoretic approaches, knowledgebased approaches, and
simulation.
The production of detailed shop schedules for planning and shop floor control has also been
considered by researchers looking for some way to go beyond materials requirements planning,
which does not consider capacity constraints, and to search for schedules better than those found
by dispatching rules.
Vollmann, Berry, and Whybark (Chapter 5, 1988) discuss three simple ways in which shop
floor control can be done: Gantt charts (often created backwards from the job due dates), priority
sequencing (dispatching) rules, and finite loading schemes, which create a schedule for the time
horizon by simulating the operation of the shop.
Bai and Gershwin (1989) initiate a discussion of all important phenomena in semiconductor
fabrication and characterization of all events and scheduling objectives and factory types. They
start with the observation that two types of events exist: controllable and uncontrollable.
Controllable events are those that the person in charge of scheduling the shop can start. The state
of the system is a complete description of the production variables, such as the status of each
machine and each worker. The variables in the state are chosen by the scheduler for his purposes,
and he may ignore certain inconsequential quantities. Events change the state of the system, and
the current state limits the options of the scheduler, who must make a decision based upon the
state.
The authors note that schedules in the real world are subject to disruption by uncontrollable
events and that schedulers have three weapons to reduce the effects of disruption: realtime
20
scheduling systems, prediction, and inhcrcntlyrobust schedules. The first is the most powerful,
but the computational effort makes combinatorial optimization impractical; thus, scheduling
heuristics arc necessary.
The paper includes a summary of the semiconductor manufacturing processes and a
detailed description of wafer fabrication. This description includes an explanation of the different
machines, workers, and events in the fabrication process. The uncontrollable events arc classified
into those that are predictable and unpredictable. The latter include machine failure and defective
wafers and arc the ones that make life difficult for schedulers. The authors also include the
constraints on scheduling and the objectives of scheduling, which arc the minimization of WIP,
throughput, variability in throughput, and costs and the meeting of demand.
Lot release policies and dispatching rules. A number of researchers have realized that the
release of work into the shop has large ramifications on the performance of that shop.
Dispatching rules are in use in many shops, and some work has been devoted to developing better
rules.
Wein (1988) considers the problem of minimizing cycle time in a semiconductor wafer
fabrication. His approach is to study the input regulation policy. He studies four alternatives: no
control (Poisson arrivals); uniform start policy (constant release rate); constant WIP (closed
loop); and workload regulating. This last rule focuses on the heavilyutilized workstations and
uses a Brownian network model to approximate a muliiclass queueing model with dynamic
control capability. A lot is released when the work in the system for a bottleneck machine falls
below a certain level. Bottleneck machines use various dispatching rules while other machines
use FIFO. The workload regulation rule was found through simulation studies to reduce the
mean and variance of cycle time, and the effects of dispatching rules were less significant and
varied by system and input type. The study uses data gathered at HewlettPackard Technology
Research Center in Palo Alto.
Glassey and Resende (1988 a,b) examine a wafer wafer fab with a single bottleneck
workstation, single product, and constant demand. Their approach is to control input regulation
21
by considering global information. The measure of performance is the cyclelime versus
throughput curve. Their answer is to release jobs so that they arrive at the bottleneck just in time
to avoid starving that machine (starvation avoidance). This requires the determination of the
virtual inventory, the amount of work for the bottleneck machine that is there now or will be soon
(within the lead time it takes a released job to reach the machine). If this inventory is less than
the lead time, release a job. This is similar to a safetystock inventory policy.
Simulation studies reveal that starvation avoidance results in lower cycletime vs.
throughput curves than other release rules: uniform, fixedWIP, and workload regulation.
Solorzano (1989) describes the implementation at Harris Semiconductor.
Glassey and Petrakian (1989) consider the problem of minimizing the queue of a bottleneck
machine in a shop using starvation avoidance input regulation. They do so by using queue
predictions and dispatching rules that give higher priority to lots that should encounter a shorter
queue on their next visit to the bottleneck. This extra queue prediction computation is intensive,
although certain simplifying assumptions are made. Objectoriented simulation studies show
good behavior (better than dispatching rules such as SPT and FIFO) in both a oneproduct wafer
fab and a twoproduct wafer fab.
Wein and Chevalier (1992) take a broader view of the jobshop scheduling problem by
considering three dynamic decisions: assigning duedates to arriving jobs, releasing jobs from
backlog to shop floor, sequencing jobs at each workstation. Their objective is to minimize WIP
and duedate lead time, subject to an upper bound constraint on fraction tardy. They take a two
step approach: (1) release and sequence jobs to minimize WIP subject to throughput rate; and (2)
set due dates that minimize duedate lead time.
The authors propose three principles: (1) while maintaining fraction tardy, average due date
lead times can be reduced by dynamically basing duedates on the status of the backlog and shop
floor, the type of arriving job, and the job release and sequencing policies used; (2) without
affecting the throughput, WIP can be reduced by regulating the amount of work on the shop floor
22
tor bottleneck stations; and (3) better duedate performance can be achieved by focusing on
efficient system performance and ignoring due dates when sequencing.
The proposed job release policy is to inject a customer into the shop whenever the
workload at the bottlenecks is at a certain level, determining the customer by a workload
balancing input heuristic. Priority sequencing uses dynamic reduced costs from an LP. In step
two, due dates are set using rough approximations that follow the spirit of principle 1. Simulation
experiments considered a twomachine, twoproduct shop. The proposed policies beat standard
policies.
Lee et al. (1993) describe a decision support system for shopfloor control in a test facility.
The system uses the ShortInterval Scheduler (SIS) module of COMETS to perform dispatching.
The main contribution of the work is the development and implementation of a mechanism that
considers sequencedependent setups while making despatching decisions. This is done by
classifying the setups and assigning each operation a setup code representing the setup
configuration (determined by handler, temperature, and package type). This allows the operator
to select operations with desirable setup characteristics in addition to the due date or operation
type allowed by COMETS. This system has been implemented in a test facility and has been
running successfully for over two years.
Deterministic scheduling. Under certain conditions, or using simplifying assumptions,
controlling the shop floor can be represented by a scheduling problem that can be solved
deterministically. The solution to this problem gives a schedule that can be used on the shop
floor.
The papers by Uzsoy, Lee, and MartinVega (1992b) and Uzsoy et al. (1991a, 1991b)
consider the scheduling of a semiconductor test facility and the associated singlemachine
problems. This work addresses backend and duedate issues. Their first approach is the use of
the shifting bottleneck algorithm, which iteratively schedules workcenters based on some
measure of criticality.
23
In another result, they use dynamic programming algorithms and heuristics to minimize
maximum lateness and number tardy on a singlemachine problem with sequencedependent
setup times. Finally, they minimize maximum lateness on a single tester with a branch and bound
algorithm and local search improvements to heuristics. They also describe the prototype
implementation of an approximation scheme for an entire test facility (Harris Semiconductor),
which may be a practical alternative to dispatching rules.
Graves et al. (1983) study the problem of scheduling a production facility with reentrant
product flows for identical products. Their objective is to minimize average cycle time while
meeting a target production rate. They develop cyclic schedules that process each operation of a
job once every cycle, where the length of the cycle is the reciprocal of the production rate. Thus,
in each cycle, one job is started and one job is finished. The scheduling of tasks on machines in
the cycle is done with a greedy heuristic that maintains feasibility. The machines in the problem
may be multichannel or batch facilities. The authors compare their rule to the simple FIFO
dispatching rule.
Kubiak, Lou, and Wang (1990) consider a reentrant job shop with a hub machine, the
machine to which jobs return repeatedly. They wish to minimize total completion time under the
following assumptions: (1) the shortest operation on the bottleneck is longer than any other
operation (allowing the singlemachine simplification); and (2) the jobs possess a hereditary
order, where a smaller total processing time implies a shorter processing time for any operation
on hub machine. An optimal wellordered schedule sequences jobs at the same operation for the
hub machine by SPT. They develop a dynamic program and present a heuristic that develops
clustered schedules, where groups of jobs scheduled together, finishing one cluster before moving
to the next.
Lee, Uzsoy, and MartinVega (1992) study bumin as a singlemachine problem. They
assume that each lot of devices has been loaded onto a number of boards, and each of these
boards forms a job for a bumin oven. If a batch of jobs is placed into the oven, the entire batch
must remain in the oven until all of the jobs have been processed long enough. Thus, the
24
processing time of the hatch equals the maximum processing time of the jobs in the batch. The
authors examine the performance measures of maximum tardiness, number of tardy jobs,
maximum lateness, and makespan. In this mathematical paper, the authors present dynamic
programming algorithms to solve batch scheduling problems with release dates and parallel batch
scheduling problems.
Bitran and Tirupali (1988 a,b) consider the problem of epitaxial wafer manufacturing.
They identify the bottleneck as the epitaxial growth operation, which takes place in a number of
different reactors. Their model is a singlestage parallel unrelated machine problem, with the
objectives of makespan and total tardiness. They develop static scheduling heuristics that create
schedules in two phases: product group priorities (by workload and due date measures) and then
job priorities (by due date) within each group.
They also address the planning problem of assigning reactors to product groups in order to
decompose the problem into independent shops. This approach reduces the complexity of the
problem and results in the following observation: the choice of heuristic to solve the problem
should be guided by the homogeneity of product set and the objective function. An implemented
scheduling system provides shop floor schedule for each reactor, job status, leadtime estimates,
and reactor load profile. Periodic resolving of model after planning preprocessing creates
uniform reactor loads and homogeneous product mixes.
Controltheoretic approaches. In an attempt to find policies that perform better than
standard rules, some researchers have created controltheoretic models. The solutions to these
models are then used to manage the shop floor.
Bai, Srivatsan, and Gershwin (1990) consider a hierarchical production planning and
scheduling system for a semiconductor wafer fabrication facility. They attempt to meet
throughput goals while treating random disruptions explicitly. They integrate the scheduler with
the system data base. Events in the wafer wafer fab arc classified by their frequency and whether
they are controllable (starting a lot vs. machine breakdown). Planning hierarchy is organized by
these frequencies. Each level has events with the same magnitude of frequency, capacity
25
constraints, and objectives passed to lower levels. When something with a low frequency
happens, the target frequencies of higherfrequency events arc recalculated. The realtime
scheduling is derived from control theory and treats random events as part of the system. This
decomposition allows small but very detailed models. The approach is implemented at the MIT
Integrated Circuit Lab. Recent results on this work are covered in Bai and Gershwin (1992).
Bai and Gershwin (1990) cover previous work on scheduling singlepart, multiplepart, and
reentrant flow systems using a realtime feedback control algorithm. For singlepart systems, the
linear programming problem for the feedback control law divides the surplus space into regions
where the optimal production rate is constant. There also exists a spot called the hedging point
where the surplus is enough to compensate for any disruptions. The feedback controller attempts
to drive the system to this point.
The authors measure the system's performance in keeping the amount of surplus close to
zero with low buffer inventories. They also attempt to improve behavior by separating the
machines from each other to reduce effect of machine failure. Solving a nonlinear program
yields an optimal hedging point (minimizes buffer inventory and buffer sizes). The current buffer
inventories and the hedging point are then used to calculate the desired production rate for the
system. Loading times for machines are calculated hcuristically to be close to the optimal
production rate.
For multiplepart systems, machines are divided into singlepart subsystems with the same
failure and repair rates and capacities proportional to the demand for that part. In reentrant
systems, machines are again divided by part and also by operation, with appropriate capacities
calculated.
In Lou and Kager (1989), the authors consider VLSI wafer fabrication with the goal of
reducing WIP while following target production and observing machine capacity. Their approach
is to use lot release and lot dispatching control rules based on flow rate control, a stochastic
optimal control problem. Assuming a continuous flow, the authors divide the shop into
workstations and determine a production rate for each according to control rules that consider the
26
inventories across the floor and predetermined hedging points. The advantages of this model
include reducing the dimension tby ignoring machines) and providing dynamic feedback control
(by responding to surpluses, down machines). The authors compare their policy to an event
driven simulation using uniformloading by costs of inventory at all stages, and claim that the
flowrate control reduces costs by 50%.
Gong and Matsuo (1990a) examine a multiperiod production system with random yield,
with the objective of minimizing fluctuation of WIP inventory, leading to more predictable
system performance. Their approach is to consider control rules for starting product into each
stage and raw material. They report that intuitive control rules can be destabilizing. Their new
control rule  Minimum Weighted Variance  is the optimal control policy for a stochastic
dynamic program with a quadratic objective function that penalizes WIP deviation from targets
and infinite time horizon. Rule performs well in systems close to capacity.
Gong and Matsuo (1990b) also consider a problem with multiple products, limited
workccnter capacity, rework, and reentry. Again, they wish to minimize weighted WIP
variance, where the weights in the objective arc determined by a nonlinear program that
minimizes total expected WIP. With sufficient conditions on the variance of yield and rework
fractions, the authors find an optimal control policy for the associated dynamic program. An
important conclusion is that the development of stable control policies in uncertain environments
is challenging and depends upon the yield and rework distributions.
In Ou and Wein (1991), the wafer fabrication system has a single bottleneck, multiple
processes with reentry, and byproducts. The authors use for their model a singleserver queue
with job classes that correspond to different operations. The model's objective is to minimize
total cost: the sum of holding costs for WIP (jobs waiting for reentry) and finished goods and
backorder costs. Their approach is to develop a control problem approximation involving
Brownian motion. The optimal control policy can be interpreted as the optimal schedule. The
authors compare their policy to two statedependent heuristic policies with a simulation and find
that their policy reduces costs.
27
Know led cc based approaches. Many researchers have attempted to give managers better
contad of the shop floor with decision support and expert systems, which incorporate knowledge
that can be used to set control policies or schedule machines. Some expert systems can even
perform scheduling events themselves.
Adachi, Moodie, and Talavagc (1988) considera production system with reentrant product
flows. They use a simulation model to examine the effect of management decision variables on
system performance. The variables arc lot si/.c, dispatching rules, start rate, frequency of urgent
jobs, and frequency of machine breakdowns. The measures are product mix, throughput, cycle
time, and WIP.
The authors report that the start rate was a more important factor than the dispatching rules.
The lot sizes also had significant effects. The dispatching rules were important policies only in
overcapacity shops. The authors then develop a pattern recognition DSS that, through an
iterative procedure, helps the user find a system that matches the userdefined goals. A prototype
is implemented on the printed circuit board fabrication line at NEC Corporation.
In Adachi, Moodie, and Talavage (1989), the authors extend their work on production
sy stems with reentrant product flows. This time, the decision support system includes a rule
based component and simulation model. The simulation model comes from the previous work
and evaluates the effect of control variables on measures of inventory, product mix, cycle time,
throughput. The results are used to obtain regression coefficients, which are stored in a database
for the rulebased component, which organizes the control variables into the hierarchy of start
rate, lot size, and priority rule. This DSS was implemented for a printed circuit board fabrication
line, resulting in superior control policies to the previous pattern recognition DSS.
Adclsbcrgcr and Kanet (1991) report on a new tool in computeraided manufacturing
scheduling: the leitstand, a decision support system with a computeraided graphical interface.
Leitstands include the following components: the graphics component is a electronic Gantt chart;
a schedule editor allows a user to easily change an existing production schedule; the data base
manager incorporates data from production planning system, engineering, and shop floor and
28
uses specific knowledge; an evaluation component measures the schedules on objective functions
and creates reports; and an advanced automatic schedule generator produces feasible schedules.
Saveli, Perez, and Koh (1989) describe scheduling semiconductor wafer production with an
expert system that takes advantage of the modularity of workccntcrs and sophisticated data
analysis. The wafer fab is divided into into thirty cells. The expert system, which is an offthe
shelf PC package with some external routines for data manipulation, creates daily schedules. It
includes two modules: (1) the priority assignment module uses information from CAM about the
operational slack of each lot and knowledge about special lots; and (2) the equipment scheduling
module for each cell uses specific knowledge about the cell and the lot priorities to create a
schedule for the lots in the cell and those arriving at the cell within a certain time frame. The
expert system is implemented in two cells (PDiffusion and Aluminum Deposition) at Harris
Semiconductor.
Sullivan and Fordyce (1990) report on IBM Burlington's Logistics Management System for
wafer fabrication. The main function is the shop floor dispatching of lots. It replaces slack lead
times with information to handle the coupling of strategic and operational decisions.
The LMS includes realtime lot and machine status and proactive intervention, with an
expert system that has knowledge about generating alerts in certain conditions and responding to
some of these by making dispatch decisions that consider five conflicting objectives: lot priority,
ontime delivery, flow requirements, increasing throughput, meeting engineering specifications.
The LMS is implemented in various areas and realizes the importance of accurate data and
continual updating to reflect changing environment.
Fordyce et al. (1992) discuss the current version of the Logistics Management System
(LMS) in place at the IBM Burlington Semiconductor manufacturing site. The emphasis of this
paper is on the last of the four tiers of the scheduling decision hierarchy. This is dispatch or
shortinterval scheduling for periods from one hour to two weeks. This tier contains the decisions
concerning the actual manufacturing flow, including the scheduling of an operation. The LMS
contains a dispatcher/shortintcrvalschcdulcr, that creates zones of control around the bottleneck
29
points, which may a sequence of operauons that lots must visit repeatedly. Kanbans within these
zones monitor the WIP level for the different passes through the zone. The dispatching is done
by something called a Judge. The Judge receives information from goal advocates in order to
make its decision, and the different advocates have different goals, including meeting due dates,
meeting the daily plan, maintaining low WIP (this is the goal of the kanbans), and increasing
machine utilization (this includes minimizing setups).
Hadavi etal. (1991) present a distributed architecture for realtime scheduling and describe
its implementation in a wafer fabrication factory. The system (ReDS) works to meet
management objectives of meeting due dates, reducing WIP and finished goods inventory,
reducing cycle times, and maximizing machine utilization. The system abstracts constraints and
time into a tree with nodes that correspond to an interval of time; it also abstracts an order into an
"essence function" that describes its critical resources. The release policy uses "continuity
indices" to reduce cycle times. The modules in the system include a preprocessor to abstract the
constraints and orders and a feasibility analysis to release orders. Also in the system are a
detailed scheduling module that uses least commitment planning in determining a daily or shift
schedule. A sequencer then dispatches the operations scheduled for a time period by using a
dynamic sequencing rule that responds in real time to a changing floor.
Rao and Lingaraj (1988) review a number of expert systems for production and operations
management decision making. They classify the systems along two dimensions: strategic
decisions versus tactical decisions and operations orientation versus resources orientation.
Scheduling systems arc classified as tactical, operationsoriented applications. They review a
number of systems in scheduling, capacity planning, facility layout, process & product design,
quality control, aggregate planning, inventory control, and maintenance & reliability. They
conclude that expert systems should combine technological knowledge and logistical data in
order to be effective.
3U
ISIS (.Fox and Smith, 1984) is an expert system tor scheduling that uses a hierarchical
approach to scheduling: job selection, capacity analysis, resource analysis, and reservation
selection. It uses decisions made at each level as constraints for the lower levels.
Opportunistic scheduling is a knowledgebased approach introduced in the OP1S system
(Smith, Fox, and Ow, 1986, and Ow and Smith, 1988). In this system, the authors extend the job
oriented scheduler of ISIS with a machineoriented scheduling procedure. After identifying an
initial bottleneck, the search for a good schedule proceeds by directing activity towards the
bottleneck subproblcms of the job shop scheduling problem. The characteristics of this system
include alternative problem decompositions, multiple scheduling heuristics, and multiple problem
abstraction.
Sadch (1991) discusses a system called MICROBOSS, another opportunistic scheduler
that identifies an initial bottleneck and revises its strategy as new bottlenecks emerge during the
construction of the schedule. This system is a more flexible procedure, however, for it can revise
its scheduling strategy after each operation. Thus, it avoids having to scheduling large
subproblems for a machine or a job.
Bcnsana, Bel, and Dubois (1988) describe a system called OPAL, a job shop scheduling
software that combines three types of knowledge: theoretical knowledge about scheduling
problems, empirical knowledge about scheduling heuristics, and practical knowledge of the
scheduling environment. They claim that combining artificial intelligence techniques with
operations research should be an effective approach to scheduling problems.
OFT and other approaches. OPT (Optimized Production Technology) is a proprietary
scheduling system that focuses on the scheduling of the bottleneck resource. The system has
been reviewed by a number of authors, including Jacobs (1984), Melcton (1986), Lundrigan
(1986), and Vollman (1986). OPT uses a forward finiteloading scheduling procedure to
schedule the identified bottleneck. The remaining, noncritical operations are scheduled using an
backward infiniteloading procedure. According to Morton (1992), the advantages of OPT arc its
good solutions to large scheduling problems and its focus on the bottlenecks. Disadvantages
31
include the inability to do rcaetive scheduling without resolving the whole problem. Details
about OPT ean be found in the above articles and in Vollmann, Berry and Whybark (Chapter 20,
1988).
Faaland and Schmitt (1993) use a costbased system to enhance the planning function of
MRP in an assembly shop. They developed a system that can create a detailed schedule of jobs,
workers, and work centers for a maker of aircraft audio, power, and light systems. They include
inventory holding costs associated with finishing early, opportunity costs associated with late
delivery, and payroll costs, since the model includes the crosstraining of workers.
Morton (Chapter 16, 1992) describes an early version of bottleneck dynamics called
SCHEDSTAR (initially reported in Morton et alâ€ž 1988). This system iterated lead times and
prices over the bottleneck and considered an objective function that included revenue, tardiness,
direct completion costs, and holding costs. A number of release and dispatch heuristics were
studied on a variety of shop situations, and the authors conclude that bottleneck dynamics (and
iterated pricing and lead times) leads to better schedules.
Simulation scheduling. Primarily, simulation models arc used to predict the performance of
certain policies. However, once the policies arc set, the simulation model can be used to create a
feasible schedule for die shop floor.
Atherton (1988) states that simulation can be used on all levels of planning. For shortterm
scheduling, models may perform shortinterval and shopfloor scheduling. This form of
scheduling considers factory capacity (dispatching rules don't). A validated simulation model,
provided with current information on the system status, can use rules to determine what will
happen in the short tenm, providing a shop floor schedule. If the projected lot completions do not
meet production requirements, further simulations may be necessary to find a schedule that is
close to the goal.
Leachman and Sohoni attack the problem of semiconductor manufacturing using an
automated scheduling system and teamworkfostering management. For every shift, a target is
set that considers the real floor. The entire staff meets to identify problems and propose
32
solutions. The targets arc generated by the scheduling system: a simulation model of the wafer
fab in BLOCS that is linked to the WIP tracking system, providing it with realtime data. The
simulation considers availability but assumes perfect execution. Thus, it is fair and accurate. To
make the schedule, it uses logic accepted by the staff, including least slack, FIFO, and starvation
avoidance rules and furnace waiting cost analysis.
The authors discuss how a shift proceeds: meetings before and after the shift with explicit
incentives for meeting the target. They also review the organizational effectiveness of their
approach in terms of motivation, satisfaction, communication and coordination, problem solving
capability, acceptance of change.
Najmi and Lozinski (1989) discuss the implementation of the system in two wafer fabs at
NCR, Inc. The simulation model is written in BLOCS, an objectoriented language where each
physical object or collection of data is a object (composed of data and procedures), and the
objects communicate to each other with messages that represent the interaction of objects.
2,2,3 Performance Evaluation
The complexity of semiconductor fabrication facilities makes direct analytical evaluation of
them difficult at best. In many cases, researchers use queueing models or simulation models in
order to gain some insight on how the system performs in the current or some proposed
environment.
Burman et al. (1986) discuss the ways that OR tools and techniques are used to analyze IC
manufacturing lines: simulation, queueing analysis, and deterministic capacity models. The
authors describe a simulation study for direct step on wafer printers in photolithography. The
study determined the effect of lot size, number of products, and setup time on capacity and WIP.
It also used largescale simulations to obtain arrival distributions to this area. A deterministic
capacity model of a clean room uses the mean number of steps and the mean process time. The
analysis took much less computing time than a simulation. The model was useful for estimating a
room's capacity and determining the minimal number of machines required for a proposed
33
program. They conclude that simulation models have the greatest flexibility and best success but
take considerable time to develop and run. The other techniques are quicker and, if not quite
accurate, do provide estimates that may be verified in simulation.
Queueing models. Queueing models attempt to model the shop floor as a set of servers
with service time distributions and possible stochastic routings. The system yields a set of
equations that describe its behavior. Because the success of solving these equations is usually
undermined by the complexity of the shop being modelled, researchers arc forced to make
simplifying assumptions.
Chen et al. (1988) develop a simple queueing network model to predict performance
measures in an research and development wafer fab. The HewlettPackard Technology Research
Center Silicon Wafer fab serves as the model. The model is a classic job shop, and the
performance measures arc throughput and cycle time. Machine breakdown rates added to actual
service time to create an effective service time. The naive queueing network model includes
different types of customers that have different routing distributions. Straightforward formulas
from previous researchers yield performance measures. The authors report that the model yielded
performance measures that were within 10% of the actual numbers.
Wein (1991) investigates a wafer wafer fab with timedependent stochastic defects, using a
simple queueing theoretic model to determine the relationship of yield and cycle time to
throughput. The model is a singleserver queueing system with exponential arrival and service
times. The author derives a closed form relationship between the mean cycle time and the
throughput of good die. In standard models, as the start rate is increased, the throughput
increases and asymptotically approaches some upper bound as cycle times increase to infinity. In
this model, the cycle times increase without bound but throughput reaches some maximum and
then decreases as the increased cycle time leads to a higher defect rate (and thus less yield).
Simulation. Simulation models remain as a popular approach to measuring the
performance of manufacturing system, and many packages have been developed. The state of the
34
art is probably the BLOCS package developed at CalitomiaBerkeley and partially described in
Najmi and Lozinski (1989). The following papers report on some uses of simulation.
In two papers, Dayhoff and Atherton (1986 a,b) develop simulation models of wafer fabs.
fhey model wafer fab as a queueing network with components for each lot and process flow.
Models are used to analyze system performance, measured by its signature: the plots of cycle
time, inventory level, and throughput versus start rate. The signature provides a way to gain
information from the large quantity of data provided by a simulation by displaying the
relationships among different variables. As a specific example, they model a bipolar wafer fab.
They compare the signatures for a number of dispatch schemes for a bipolar wafer fab
where products visit the masking station seven times. In the wafer fab, there exist three levels of
dispatching: (1) among lots of one product waiting for one process, (2) among process steps of
the same product, (3) among products. This paper works at level two; the first level is done by
FIFO, third by product priority. The study compares the following dispatch schemes for
masking: earlier steps first (a form of longest remaining processing time), later steps first (a form
of shortest remaining processing time), roundrobin (changing priority). Qualitative analysis
shows that earlier steps first failed at higher start rates by building inventory and that the later
steps first was better by getting higher throughput with no increase in inventory.
Dayhoff and Atherton (1987) provide definitions of the elements of wafer fabrication and
their interactions. They identify the important parameters that govern wafer wafer fab and result
from wafer fab operations. The elements defined include work station, process flow, products,
wafer lot, process step, batch, batching down, service, dispatch system, rework, and wafer fab
graph. They discuss the following types of simulation results: average wait times, queue lengths,
inventory, cycle time, equipment utilization, yield, throughput.
Atherton and Pool (1989) discuss the ACHILLES simulation model in a wafer wafer fab of
Silicon Systems, Inc. The describe the model and its initialization, calibration, and validation by
comparison to actual measures of factory performance. The discuss the use of the model to
predict the effect of reducing inventory levels on cycletimes.
35
Spence and Welter (1987) discuss a project by Stanford University and Motorola Inc. that
attempts to improve the performance of a semiconductor wafer fabrication line. 'Hie goal is to
increase the throughput of the line without sacrificing cycle time. The analysis was done with a
Monte Carlo simulation to determine capacity, defined as throughput rate, cycle time, and WIP.
The performance was measured on the steadystate cycletime vs throughput tradeoff curve,
instead of using transient signatures (c.f. Dayhoff and Atherton). The photolithography cell was
studied for its analytical difficulty, the build up of WIP in this cell, and the proposal of system
changes. The authors report that adding resources (operators, aligner equipment) reduced cycle
lime at higher throughput. Reducing reworks, setup times, and the time to wait for repair reduced
cycle times at all levels. Larger lot sizes were preferable at higher throughputs.
2.2,4 Summary
From this review, it can be seen that many different approaches have been applied to the
problems of production planning and scheduling. However, most of this work addresses the
questions raised in wafer fabrication. The primary exceptions arc Uzsoy, Lee, and MartinVega
(1992b), Uzsoy et al. (1991a, 1991b), Lee, Uzsoy, and MartinVega (1992), and Lee et al. (1993)
which do explicitly consider the problems of semiconductor test. However, this work does not
make use of the new, sophisticated heuristic searches and is not concerned with the ideas of class
scheduling and lookahead and lookbchind scheduling.
Although Saveli, Perez, and Koh (1989) do develop a system that is lookbehind, it is
implemented in an offline system in a wafer fabrication cell. The research in this dissertation,
however, investigates the application of both lookbehind and lookahead rules into the current
realtime dispatching system of a semiconductor test area.
36
2 3 Job Shop Scheduling
Before beginning the analysis of scheduling problems, it is useful to review the notation to
be used and the basics of scheduling problems. Because even the simplest job shop scheduling
problems are NPcomplete, the literature consists of different heuristics.
Job shop scheduling, as one of the most difficult scheduling problems, has attracted a lot of
attention from researchers. Techniques such as the shifting bottleneck algorithm (Adams, Balas,
and Zawack, 1988) or bottleneck dynamics (described in Morton, 1992) concentrate on solving
the problem at one machine at a time. Other researchers have studied how well different
dispatching rules perform in minimizing makespan and other objective functions. Panwalkar and
Iskander (1977) present a list of over 100 rules. Recent studies include Fry, Philipoom, and
Blackstone (1988) and Vepsalainen and Morton (1988). Various scheduling systems for shop
floor control are reviewed in Section 2.2. More sophisticated lookahead and lookbchind rules
have also been introduced; see Section 2.5 for a discussion of these ideas. In this section we
review scheduling notation, the shifting bottleneck algorithm, and dispatching rules.
2.3.1 Scheduling .Notation
In the general job shop scheduling problem, there exists a set of jobs Jj,j = 1,..., n, and set
of machine MÂ¡, 1=1,..., m. Each job Jj has rij operations (or tasks) Oy, i= 1,..., nj, where Oy =
k if the tth operation of Jj is to be processed on machine M^. In a flow shop, all nj = m and
Oy = i for all i. In general, each operation has a processing time py > 0. A job can have release
dates rj and due dates dj and deadlines Dj.
A feasible schedule a is a plan that determines when each operation is processed on each
machine subject to the following constraints: the operations of each job must be performed in
order, and no machine can process more than one operation at at time.
For a schedule, certain performance measures can be associated with a job Jj. the
completion time Cj, the lateness Lj = Cj  dj, the tardiness Tj = max [Lj, 0}, and whether or not
37
ihc job is tardy, Uj = 1 if 7y > 0 and 0 otherwise. Objective functions that measure the
performance of the schedule include the makespan, Crnax  max {Cy}, the total flowtime X Cy,
and the number of tardy jobs X Uj.
The scheduling problem is to find a feasible schedule that minimizes the objective function.
For the regular performance measures mentioned above, the set of schedules that need to be
considered is the set of active schedules, where no operation can be started earlier without
causing another operation to be delayed. In this set, there exists a onetoone correspondence
between a schedule and its representation by a sequence of operations for each machine. Thus, a
sequence that orders the jobs on each machine can be considered a solution to the problem.
Standard scheduling problems can be classified in a concise way by using the following
threeelement description: x /y / z. The field x describes the machine environment as one
machine, parallelmachine, or shop. The second field (y) describes any constraints or special
characteristics of the problem. The third field (z) describes the objective function. Consider the
following examples:
I / ry / X Uj is a onemachine problem where the jobs have release dates and the objective is to
minimize the maximum lateness.
II Djl X Cj is the onemachine problem where the jobs have deadlines and the objective is to
minimize the total flowtime.
F2 // Cftiax is ^ twomachine flowshop problem of minimizing the makespan.
ill Cmax is the general job shop scheduling problem of minimizing makespan.
2,3.2 Shifting Bottleneck
Adams, Balas, and Zawack (1988) introduce the Shifting Bottleneck procedure to minimize
the makespan of job shop scheduling. This algorithm sequences the machines in a job shop
successively by identifying the machine that is a bottleneck among the machines not yet
sequenced. After the scheduling of the new bottleneck, all of the previouslyconsidered machines
38
have their schedules readjusted. The individual steps involve finding the solution to onemachine
problems.
The authors lean heavily on the disjunctive graph associated with the job shop scheduling
problem. In this graph, there exists a node for each operation to be performed as well as dummy
start and termination nodes. For each job there exist directed arcs from each operation to its
successor and an arc from the start to the first operation and an arc from the last operation to the
termination. For each machine, there exist a pair of disjunctive arcs between each pair of
operations on this machine. All of the arcs have a length equal to the processing time of the head
operation.
If a selection is made for the disjunctive arcs of each machine and the corresponding graph
is noncyclic, a schedule for the entire shop is made by scheduling each job as soon as possible
under the constraints imposed by the selections and the job operation sequence. The makespan of
the schedule is equal to the longest path from the start node to the termination node.
The Shifting Bottleneck procedure iteratively selects a bottleneck machine, schedules this
machine, and readjusts all previouslyscheduled machines. Given that a set of machines has been
scheduled, procedure determines the bottleneck by considering each unscheduled machine
individually. This yields a onemachine problem where the minimization of the makespan for the
shop is equivalent to minimizing the maximum lateness of the operations on this machine. The
minimization problem is NPcomplete, but the authors use the algorithm of Carlier, a branchand
bound procedure that gives excellent results. The machine that has the largest such maximum
lateness is the bottleneck, and that machine is added to the set of scheduled machines.
The local reoptimization procedure considers the set of scheduled machines and performs a
number of cycles. Each cycle consists of solving the one machine problem for each machine in
turn, using the selections found for all of the other machines.
The authors report that the Shifting Bottleneck procedure performs exceptionally well,
including finding the optimal makespan for the notorious 10job and 10machine problem of
Gi filer and Thompson.
39
2.3.3 Disnatchint; Rules
Because of the complexity of job shop scheduling, algorithms to find the optimal solution
for any arbitrary objective function do not exist. Thus, researchers have studied and schedulers
have used dispatching rules to order the jobs waiting for processing at a machine. Most of the
research in this area has examined the performance of various dispatching rules, which sequence
the jobs that are waiting for a machine according to some statistic that is a function of the jobs'
characteristics.
When a machine becomes available, it chooses from among the jobs in its queue by using a
rule that sorts the jobs by some criteria. Common dispatching rules employ processing times and
due dates in simple rules and complex combinations.
These dispatching rules arc sometimes extensions from simple onemachine problems. For
instance, the Shortest Processing Time (SPT) algorithm is known to minimize the total flowtime
of jobs processed on one machine. The SPT dispatching rule sorts jobs waiting for a machine by
the amount of processing time they require on the machine. Also, the Earliest Due Date (EDD)
algorithm is known to minimize the maximum lateness of a set of jobs being processed on one
machine. The EDD dispatching rule is used in job shop scheduling in an attempt to reduce
maximum lateness. While the list of known dispatching rules includes over 100 items, only a
handful arc commonly used. And most of the rest are combinations of the most common rules.
Day and Hottenstcin (1970) present a review of sequencing research, in which they discuss
Jackson's Decomposition Principle (1967), which assumes that the arrival times for each job
arriving from outside the system are exponentially distributed, the processing times at each
machine are exponentially distributed, the jobs arc routed to a machine by a fixed probability
transition matrix, and the priority rule at each machine is FirstComeFirstServed (FCFS).
They also discuss several due date assignment schemes presented by Conway (1965).
These arc CON (constant from the order to the due date), RAN (random: due date chosen by
customer and accepted by salesman), TWK (Total Work Content: the allowable shop time is
40
proportional to the sum of the processing times of the operations of the job), and NOP (Number
of Operations: the allowable shop time is proportional to the number of operations).
The authors also review the previous research on dispatching rules by Conway (1965), who
compared SI to 31 other rules and found that it dominated the set of rules tested. It is simpler and
easier to implement. However, the biggest objection to SI is the following fact: if the mean time
which jobs spend in the system is small, individual jobs (those with long operations) may be
intolerably delayed. Thus the variance of the lateness distribution is the basic disadvantage of the
SI rule. Conway tried three methods to reduce the variance: (i) Alternating the SI rule with a low
variance rule (with respect to flowtimc) to periodically clean out the shop, (ii) forcibly truncating
the SI rule by imposing a limit on the delay that individual jobs will tolerate; and (iii) dividing
jobs into two classes, preferred and regular, as the primary criteria, leaving SI as a secondary
criteria.
The authors then move to the COVERT dispatching rule. Buffa (1968) retained the
performance of the SI rule and tended to minimize the extreme completion delays of a few orders.
Trilling (1966) used a ratio of delay cost rate of a job to the processing time of that job on the
machine in question. The job with the highest ratio is dispatched first.
The authors also report on the priority rules employed in industry, where job lateness is a
primary concern. Henee, EDD and least slack rules are most used. However, LPT becomes a
popular rule because schedulers rank jobs by the index of importance and it is reasonable to
expect some positive correlation between importance and processing time.
Holloway and Nelson (1974) study the problem of job shop scheduling with stochastic
processing times. Their performance measures are the mean, variance, and maximum of
tardiness. For dispatching, the authors use HSP, a multipass heuristic that produces delay
schedules, HSPNDT (a nondelay version), DDATE, SPT, SLACK, and a SPTSLACK
combination.
The authors study three problems of different size, tightness, and utilization, where the
processing times were constant or from one of three distributions. They report that HSP did well
41
on lowvariance problems and on maximum tardiness. HSPNDT beat HSP in some situations.
Of the nondelay rules, the deterministic problems were a good predictor of relative performance
on problems with variability. DDATE improved its relative performance as variability of
processing times increased.
Panwalkar and Iskander (1977) list 100 dispatching rules. They categorize these rules into
five classes: simple dispatching rules, combinations of simple rules, weighted priority indexes,
heuristic scheduling rules, and other rules. The simple dispatching rules are organized into
processing time, due date, number of operations, cost, setup time, arrival time, and machine rulÃ©s.
Combination rules sequence jobs by considering first one characteristic and then another.
Weighted priority indexes sequence jobs by combining values from different job characteristics
(by adding or dividing).
Blackstone, Phillips, and Hogg (1982) state that the best measure of performance is cost
effectiveness. Their work covers this objective function and others: tardiness, lateness, flowtime,
inventory. They note that analytic measures depend upon Jackson's decomposition, which
assumes a FIFO dispatching rule. Other rules lead to interrelated queues and thus researchers use
simulation.
For the singleserver model, it is known that SI (shortest imminent processing time)
minimizes mean flowtime and mean lateness and EDD (earliest due date) minimizes maximum
lateness and maximum tardiness. Mean tardiness cannot be optimized by any dispatching rule;
thus, the authors conclude that "no single dispatching rule yet developed will optimize delay costs
in the job shop environment."
The authors consider a number of dispatching rules. Their first is SI. They report that SI is
not affected much by imperfect data, it performed best on mean flowtime, and it minimized the
number of tardy jobs and mean lateness for exogenous (constant and random) due date
assignment. It also performed better when internal due dates are less than seven times processing
time and utilization is high. Modifications to SI to clear jobs that have been waiting include
42
alternating rules and truncation. Truncation versions seem useful for shops having control over
due dates and concerned about very late jobs.
The authors also consider a number of due date rules: EDD, Slack, and Slackpcropcration.
Slackpcropcration is the best and, compared to SI or FIFO, produces a smaller variance of job
lateness independent of due date assignment. Compared to EDD and slack, it performs best on
lateness variance, costperorder, job tardiness, number of tardy jobs. Slack may be defined in
static or dynamic terms, although rules that use the latter arc better. The authors also report that
critical ratio rules are also in use.
Among rules that used neither processing time nor due date information are FIFO and NOP
(number of operations). The authors report that both perform worse than other good rules. They
also conclude that the look aheadrules NINQ (number in next queue) and WINQ (work in next
queue) are not as good as SI in fiowtime and inventory criteria and that a valuebased rule usually
becomes longest processing time, which generally behaves poorly.
The authors report that weighted combinatorial rules arc not better than any single rule,
although COVERT (delay cost over time remaining) may be useful in shops willing to estimate
delay costs. The authors also discuss dispatching heuristics. The lookahead heuristic LAH
allows insertion of idle time in order to process a critical job. Heuristics improve the
performance of dispatching rules but their implementation may not be cost effective; a complete
study has not been done.
Green and Appel (1981) examine the problem of job shop scheduling by asking the
following questions: What traditional dispatching rules do experienced schedulers select? Would
dispatch rule selection be influenced by urgency? Would schedulers select a dispatch order based
on organizational influence and/or peer pressure? The authors asked schedulers in a number of
plants to denote which of the following rules they used: Due Date, Slack, Operations Due Date,
Slack per Operations, SPT, FCFS, COVERT, Program in Greatest Trouble (PGT), or Friend
Needs a Favor (FNF). The authors report that influence systems affect scheduling. PGT (a
43
coalition rule) was highly valued, but FNF (an individual rule) was rejected. Traditional and
theoretical rules were not highly valued.
Kanet and Hayya (1982) also consider the problem of job shop scheduling. Their paper
tries to determine if operation due dates are better. Their comparison is done by running
controlled simulation experiments using the following dispatching rules: Earliest Due Date:
DDATE and OPNDD; Smallest Job Slack: SLACK and OPSLK; Critical ratio: CR and OPCR.
The slack is static; the critical ratio is dynamic: time until duedate over remaining allowance or
operation allowance. For critical ratio, each operation must be assigned an allowance. This
allowance is proportional to the processing time. The allowance multiple controls the difficulty
of meeting due dates.
The authors consider the performance measures of lateness (mean and standard deviation),
fraction tardy, conditional mean tardiness, flowtime (mean and standard deviation), and
maximum tardiness. The authors report that all rules were outperformed by their operation
counterparts on all measures. Introducing operation due dates shifts distribution of job lateness to
the left Oess) and compresses it. One unexpected result was that OPNDD (due date)
outperformed OPSLK (slack) and always minimized maximum tardiness. The CR rule gives
lower lateness variances, but the OPCR rule was even better. The authors note that, as the
allowance multiple is increased, due dates become larger and DDATE and SLACK rules act more
like SPT, minimizing fiowtime. For the critical ratio rules, a larger allowance means longer jobs
get lower ratios, causing CR to act like LPT.
Baker and Bertrand (1982) take up the problem of singlemachine dynamic scheduling and
study different due date assignment methods and dispatching rules. They arc concerned with
average tardiness. The authors introduce a new dispatching rule: the modified due date rule
(MDD), where the modified due date is defined as the maximum of the due date and earliest
finish date. This rule is a combination of EDD and SPT that implicitly responds to changes in the
amount of slack. The authors report that MDD dominates both SPT and EDD on average
tardiness.
44
Baker and Kanet (1983) extend the MDD rule to attack the problem of job shop scheduling.
They consider the performance measures of mean job tardiness and proportion of tardy jobs.
They introduce the modified operation due date rule (MOD), where the modified operation due
date is the maximum of operation due date and earliest operation finish time. The authors also
consider other dynamic rules: Slack per operation, Critical ratio, and COVERT under different
allowances and utilization levels.
The authors report the following results: MOD performs better than MDD and SPT and is
sometimes better than operation due date, critical ratio, and slack per operation rules. The
authors thus conclude that MOD is an important dispatching mechanism.
Baker (1984) considers the job shop scheduling problem and attempts to clarify some
conflicting results between different rules of dispatching and duedate assignments. He points
out that two factors are of primary interest: flowtime and duedate performance. He reports that
SPT is the best tactic for reducing mean flowtime. However, no single priority rule dominates
performance comparisons. He reports that the critical ratio is best for minimizing conditional
mean tardiness (that is, the average tardiness of tardy jobs); SPT is best for number of tardy jobs;
however, for mean tardiness, the results are mixed.
Fry, Philipoom, and Blackstone (1988) consider the problem of job shop scheduling with
90% utilization. For the due date assignment, they use the total work method with two different
allowances. They consider the following performance measures: mean flowtime, mean tardiness,
and root mean square tardiness (designed to punish large tardiness).
They study truncated and alternating versions of the SPT dispatching rule. The authors
define the critical percentage (%C) as the percentage of jobs that enter a higher priority queue
because they have been waiting a long time. They report that the best rule depends upon the this
critical value.
Vcpsalainen and Morton (1987, 1988) consider the problem of minimizing weighted
tardiness in job shop scheduling. They use the following dispatching rules: FCFS, EDD, Slack
per remaining processing time (S/RPT), WSPT, weighted COVERT, and Apparent Tardiness
45
Cost (ATC). A lead time estimation is neecssary for last two rules. Lead time estimates are
computed as a fixed multiple of processing times, derived from rough waitingline analysis and
job priority indices, or found from iterating simulations.
The authors first studied the flow shop problem, where ATC and COVERT dominated
using any leadtime estimate. Using the iteration estimate resulted in better weighted tardiness.
ATC did better on fraction of jobs tardy and inventory measures. In the job shop, under different
loads and due date tightnesses, the authors report that the ATC rule did better, followed closely
by COVERT. The use of the priority estimates yielded a smaller fraction of jobs tardy.
2,3.4 Summary
In addition to introducing some notation, this section covered two important methods of job
shop scheduling: the shifting bottleneck procedure and dispatching rules. Both methods have
limitations, however. The shifting bottleneck procedure searches for a schedule and continuously
improves upon it. Thus, it is only a local search technique. The dispatching rules that have been
studied are mainly shortsighted techniques that do not consider other machines. The research in
this dissertation extends this work by considering smartandlucky searches and investigating
lookahead and lookbchind dispatching rules.
2,4 How Shop Scheduling
This dissertation considers a threemachine problem where all jobs are processed on a
certain machine and then go to one of two secondstage machines. This problem is similar to a
flow shop, and it was useful to review the flow shop scheduling problems that have been
previously studied.
The flow shop problem is actually a collection of problems that deal with the minimization
of some regular objective function for a set of jobs in a flow shop, where each job consists of a
number of different operations that must be processed on a set of machines. The primary feature
46
of a flow shop is that the sequence of operations is the same for each job. That is, each follows
the same path through the shop.
All How shop analysis starts with Johnson (1954) who studied the minimization of
makespan for twomachine flow shop problems and for some special threemachine flow shop
problems. His algorithm starts jobs with the smallest tasks on machine 1 as soon as possible and
jobs with the smallest tasks on machine 2 as late as possible. For the two and three machine
cases, it can be shown that only permutation schedules need be considered. Permutation
schedules are schedules for the machines in which the sequence of jobs is the same for each
machine. Johnson showed that for four machines, passing may be necessary for optimality.
Although most analytical research in shop scheduling has dealt with the makespan
objective function, this research is interested in the minimization of total flowtime, also known as
the mean completion time, the sum of completion times, the mean time in system, or the total
time in system. Therefore, this section concentrates on a number of papers on the minimization
of total flowtime, a lcsscommonly studied objective, and then moves on to other objectives,
including maximum lateness, and the number of tardy jobs. Also included in this section are
reports on some NPcomplctcness results and problems with separated setup times.
2.4.1 Makespan
Special cases of the flow shop makespan problem have been studied by a number of
researchers, including Mitten (1958), Conway, Maxwell, and Miller (1967), Bums and Rooker
(1975), and Szwarc (1977). Garey, Johnson, and Sethi (1976) proved that the general three
machine problem was NPcomplete. Problems with release dates, preemption, precedence
constraints, or more than three machines have also been studied.
2.4.2 Total Flowtime
Ignall and Schrage (1965) describe a branchandbound algorithm for F2 // X Cj and for
F3 / / Cmax. For the total flowtime problem, the authors consider two values: the sum of
47
completion times it the first machine is the only capacity constraint and the sum it the second
machine is the capacity constraint. The larger ot these two values is the bound at each node.
Krone and Sleiglitz (1974) consider the static (low shop scheduling problem with the
objective of minimizing the mean llowtime of Â«jobs that must visit m machines. They present a
heuristic method. The authors note that two previous researchers performed separate sampling
experiments of the flow shop scheduling problem. Heller concluded that permutation schedules
should be investigated while Nugent found that good schedules tended to be permutation
schedules that had allowed some jobs to pass others (a local reversal).
For the flow shop problem, the authors consider semiactive schedules (where no operation
can be shifted forward in time) and define a schedule as an array S that gives a sequence for each
machine. SÂ¡j is the job that isy'th on the ith machine. The authors define a twophase heuristic.
In the first phase, the search considers only permutation schedules. In the second phase, the
search takes an initial permutation schedule and allows deviatioas from the uniform ordering.
Each phase is a local search, although the neighborhood structures are different for each phase.
Kohler and Steiglitz (1975) study the twomachine flow shop problem with the objective of
minimizing the mean llowtime. They present algorithms that they combine with lower bounds
that guarantee the accuracy of the heuristics. The authors use the lower bound of Ignall and
Schrage. They compute an initial solution by moving down a branchandbound tree without
backtracking, selecting the node with the lowest lower bound at each stage. They consider
different local searches, with either a random start or the above initialization, hillclimbing (first
improvement), and one of four neighborhood structures: backward insertion, forward insertion, a
double adjacent pair interchange (switches two pairs), and finally the union of the first and third.
The authors report that the good initialization helped the local search perform better than
the random starting search.
The authors then present three different branchandbound algorithms that differ in the way
the nodes are eliminated. Since some of the algorithms exceed computational limits, they
determine a bracket that is the relative difference between the final upper bound and the greatest
48
lower bound. 11 the goal is to find a solution that is within a certain amount of the optimal, this
bracketing idea allows great speedup since a good (sometimes optimal) initial upper bound was
used and the lower bound is very close. For a given r < 1, the algorithm stops when the lower
bound L is greater than or equal to r times the best upper bound U, i.c., rU
Miyazaki, Nishiyama, and Hashimoto (1978) present a heuristic algorithm for F // X Cj.
Given a sequence, the authors interchange two adjacent jobs and determine the change in
flowtime. They then derive a number of conditions that arc sufficient to say that the second job
should never directly precede the first. Using these different conditions they create temporary
sequences and assign each job an ordinal number corresponding to its place in these sequences.
The jobs are then sorted by the sum of these values, (more here)
Miyazaki and Nishiyama (1980) extend this work by creating permutation schedules for the
weighted mean flowtime flow shop problem. The authors derive some precedence conditions and
create an algorithm to generate good solutions. The first analysis considers the effect of
interchanging two jobs in the permutation sequence to create. If this switch causes no decrease in
the weighted flowtime, then the second job cannot directly precede the first in an optimal
sequence. The authors then embark on finding a number of relations for two jobs that are
independent of all other jobs and that will determine if a job should precede another.
For their algorithm, the authors evaluate these various relations for each job. If each of the
relations yields the same sequence, this sequence is optimal. If not, each job is given a value for
each relation that is its performance on the relation (l=best,...). These values are added over the
relations and the jobs are ordered by these sums.
Szwarc (1983) discusses the mean flowtime flow shop problem and a number of special
cases. The author develops two properties that create a precedence relation between two jobs, the
first for a fixed prescqucncc, and the second for an arbitrary prescquences. Using these
properties, the author presents a special case of a 2job problem, and a case where the final
completion times depend only on the completion times on some earlier machine. This last case
49
holds for die twomachine case where die t/r > t2r for all jobs Jr and the case where the columas
and rows form a perfect order under dominance.
Ahmadi et al. (1989) study the twomachine flow shop subproblcm with one or two batch
processors as a segment of a larger job shop. They study the objecdve functions of makespan and
total flowtime. They consider two types of processors. The first is a batch processor that has a
fixed capacity and the time necessary to process the jobs in a batch is a constant. The other type
of processor is called a discrete processor, which refers to a standard singlemachine processor.
The authors consider the six types twomachinc problems that can be formed.
To minimize the makespan of a batchdiscrete shop, full batches are optimal, permutation
schedules dominate nonpermutation schedules, and the opdmal solution is to sequence jobs by
LPT and fill each batch. For the discretebatch shop, the analysis is similar, but in this case the
jobs should be ordered by SPT. For twobatchmachineshop, all of the jobs are equivalent. The
optimal policy is to completely fill the batches on machine one and to use a dynamic program to
fill the batches on machine 2, using the completion times on machine 1 as release dates.
To minimize the total flowtime of a discretebatch shop, SPT should sequence the jobs on
the discrete machine 1. A dynamic program can be used to perform the batch dispatching on
machine 2. For the twobatchmachineshop, the batches on machine 1 should be completely
filled and the completion times from this can be used as release dates to batch machine 2, which
is dispatched using the same dynamic program. For the problem where a discrete processor is fed
by a batch machine, the batches should be completely filled, but the authors prove that this case is
strongly NPcompIete. They do consider special cases relating the processing times on one
machine to another and derive two optimal algorithms.
For the NPcomplete problem, the authors propose a heuristic determines the number of
shortest batches that arc shorter than the batch processing time. To these batches the shortest jobs
are dealt. The remaining jobs are sequenced by SPT to form batches. The authors show that the
error tends to 1/2 as the number of batches tends to infinity.
50
Ahmadi and Bagchi (1990) present a lower bound on the total flowtimc. Given a partial
schedule Jr of r jobs and fixing a machine j, the authors bound the final completion times based
on the completion times on machine j. Then, they search over sequences on machine j,
considering the sequence constraints and the machine j release dates for each job. This is the NP
hard problem of solving 1/ryX Cj. The authors solve the problem by allowing preemption. The
lower bound is the greatest found by performing this procedure over each of the machines.
Van de Velde (1990) studies the twomachine flow shop problem and minimizes the sum of
completion times by using a Lagrangcan relaxation. The author decides to relax the precedence
constraints between the two operations of a job (instead of the capacity constraints on each
machine). Let the vector of multipliers be X = (A.,, X2,.... Xn). The relaxed problem is one of
minimizing the weighted sum of operation completion times, solved by SWPT on each machine
separately. The author, however, wants to restrict the solution to permutation schedules.
Thus, the author reformulates the problem into a linear ordering problem, which is
polynomially solvable in certain cases. The author lets all XÂ¡ = c, where 0 < c < 1, and if c  0 or
c = 1, the problem yields the bounds of Ignall and Schrage. Thus, for c, the relaxation can be
solved optimally with a permutation schedule. The author claims that the lower bound as a
function of c is continuous, concave, piecewiselinear function, and the maximum can be found in
polynomial time.
One partial sequence dominates another if the sum of completion times is smaller and the
makespan is also smaller. The author develops a number of sufficient conditions for dominance.
Also, if P21 = P2j and Pa ^ pÂ¡j, then the author proves that there exists an optimal sequence
where job i precedes job j.
2.4.3 Maximum Lateness (Lmax)
Masuda, lshii, and Nishida (1983) find a solvable case of F2 / / Lmax and present an
algorithm for the general problem with a worst case bound. The authors first prove that the EDD
sequence is an optimal schedule if for i and j :dÂ¡< dj if and only if min [ap bj\ < min {aj, bÂ¿}.
51
This says that the EDD and Johnson sequence are the same. However, this property does not
provide any precedence for the general ease, but the authors schedule the jobs according to EDD
and give an error bound that is tight asymptotically.
Grabowski, Skubalska, and Smutnicki (1983) the flow shop problem of minimizing
maximum lateness with release dates. They analyze two lower bounds, one with one bottleneck
machine and a nonbottleneck preceding it and the other with two bottleneck machines and a nonÂ¬
bottleneck between them.
2,4,4 Number of Tard\ Jobs
Hariri and Potts (1989) consider a permutation flow shop and develop a lower bound and a
branchandbound procedure to minimize the number of tardy jobs. This problem is NPhard.
For a given partial sequence and a machine j, wc can consider the time this machine is available,
disregard the processing before machine j and suppose that the machines after machine j have
infinite capacity. The additional processing can be subtracted from the due date and The Moore
Hodgson algorithm applied to find a minimum number of late jobs. Applying this procedure to
each of the machines yields a lower bound.
The authors derive another bound using the consistent early set, a set of jobs that is feasibly
early for each of the machine subproblems. The size of the smallest consistent late set (a set
whose complement is a consistent early set) is the lower bound. The authors extend this to the
idea of a consistent feasible early sequence for singlemachine and rmachinc problems.
In generating problems to test their bounds, the authors set due dates that were uniformly
distributed in a range proportional to an estimate of the total processing time. The fractions
ranged from 0.2 to 1.0. The branchandbound algorithm with the simple bound was good
enough for the small problems, but the use of consistent sequences yielded the most efficient
procedure on the larger problems. The due date range of 0.4 to 0.6 yielded the hardest problems
since the range was not great enough to guide sequencing and the due dates were tight.
52
2,4,5 General Topics
Garey, Johnson, and Sethi (1976) prove a number of NPcompleteness proofs tor
scheduling problems. They do allow zero processing times. The authors begin by discussing
some of the terms of NPtheory. All of the proofs will be by transformation from 3partition,
which is NPcomplete in the strong sense.
The first proof is that of the F3 // Cmax problem, where dummy jobs are created to give
slots on the second machine that jobs corresponding the elements of A must fit. The next proof is
for the F2 / / X Cj problem. Dummy jobs create slots on the second machine that dummy jobs of
intermediate length and the jobs corresponding to the partition problem must fill. Long tasks arc
added to the end to insure that the spacers are completed as soon as possible. Lastly, they prove
that J2 / / Cmax is NPcomplete, although this depends upon a job that is reentrant n times to
machines 1 and 2.
Gonzalez and Sahni (1978) consider a number of flow shop and job shop problems, with
and without preemption, minimizing makespan and flowtime. The authors extend the makespan
results to the preemptive problems.
F3/pmtn/Cmax is NPcomplete even if no job has more than 2 operations. They have jobs
with 12, 13, and 23 flows. They transform Partition into this, and the result and proof arc the
same for nonpreemptive jobs. F3/pmtn/Cmax is strongly NPcomplete, including jobs with
operations on all three machines.
J2/pmtn/Cmax is NPcomplete if all but two jobs have only two operations. The two other
jobs go 121 and 212. The result also holds for nonpreemptive jobs. J2/pmtn/Cwa;c is
strongly NPcomplete with one job that visits each of the two machines n times.
The authors move to some approximation algorithms and bounds. For the total flowtime
job shop problem, the bound on the ratio of the flowtimes for any busy schedule to the optimal is
m. And this holds if the busy schedule is SPT. The ratio for makespan is also bounded by m. A
variation of Johnson's algorithm that considers pairs of machines has an error bound of m/2.
53
Lageweg, Lenstra, and Rinnooy Kan (1978) report on a number of different lower bounds
and a classification of them as well as showing that a new lower bound for the makespan problem
is better than the rest.
They present two dominance criteria and then move to the lower bounds, which relax the
problem by allowing some machines to have infinite capacity. The unitcapacity machines are
called bottlenecks and the infinitecapacity one are nonbottlenecks. The authors limit
consideration to at most two bottleneck machines Mu and Mv and combine consecutive non
bottlenccks into one by summing the processing times for each job. Additionally, a non
bottleneck can be eliminated by adding the minimum processing requirement on this machine to
the lower bound. This leads to nine different schemes (excluding symmetric ones). The authors
describe each of the nine and analyze the effort required for each. They describe some upper
bound calculations.
They test all of the lower bounds on problems with six jobs and three, five, or eight
machines. The better ones were tested on larger problems with 10 to 50 jobs and 3 to 5 machines.
The best results were obtained using elimination criteria and the two bottleneck machine bound
with a nonbottleneck between them and the head and tail nonbottleneck machines removed.
Some researchers have looked at the flexible flowshop problem, where there may exist two
or more parallel machines at a stage of the flow. Wittrock (1988) attempts to minimize the
makespan and the queueing time. He creates three subproblcms: machine allocation, sequencing,
and timing. In order to simultaneously minimize makespan and queueing, he develops a heuristic
procedure to greedily sequence the jobs given an allocation of the jobs to machines. He applies
the LPT heuristic to do this allocation at each stage. Timing consists ofloading the parts into the
system as late as possible while not delaying any subsequent operation. He also considers buffer
limits.
Sriskandarajah and Sethi (1989) consider a number of simple heuristics for the flexible
flowshop problem. They derive worstease error bounds for list scheduling and a Johnsonlike
sequence on the twostage case where the first stage has only one machine. Their best relative
54
error bound is 2. They also examine the problem where each of the two stages has m machines,
rheir heuristic creates rn twomachine flowshops and uses list scheduling and LPT to allocate the
jobs to the different shops. Each shop is sequenced by Johnsonâ€™s algorithm.
Gupta (1988) and Gupta and Tunc (1991) study the special twostage eases where one of
the stages has only one machine. Both papers propose heuristics that sequence the jobs (by
Johnsonâ€™s or some algorithm rule) and then assign the jobs to the parallel machines. Lower
bounds on the makespan and the results of empirical testing are also discussed.
Lee, Cheng, and Lin (1993) study a threemachine problem where each job consists of two
tasks that are assembled in a third operation. Each of the two firststage tasks is processed on a
different machine and the tasks can be processed in parallel. The assembly task is performed on
the third machine and cannot begin until both firststage tasks are finished. They show that the
problem is NPcomplcte. The authors provide some restricted versions that are polynomially
complete. These include cases where one of the tasks dominates the other, where the assembly
tasks is dominated by both firststage tasks, and where the assembly tasks dominates the first
stage tasks. The authors present a branchandbound approach that uses the eases presented to
trim the search tree. The authors present heuristics (with error bounds) that create new firststage
tasks from the job data and scheduling according to Johnson's algorithm to minimize the
makespan of a twomachine flowshop. The best relative error bound is 1/3.
This threemachine problem is therefore closely related to problems previously studied, but
the special structure of the problem (the different flows) leads to interesting twists on these
results.
Lee and Herrmann (1993) look at the threemachine lookbehind problem with the
objective of minimizing the makespan. They derive results similar to the ones we derive for the
threemachine lookahead case.
55
2.4.b Summary
The papers covered in this section provide a number of ideas that may be useful in the
threemachine problems to be studied. The threemachine problem docs not fit into the flow shop
model, and thus work must be done on how to modify the above approaches. Also, the NP
completeness proofs do not apply, and the NPcomplctcncss of the threemachine problem must
be proved. The flexible flowshop models and the assembly flowshop model of Lee, Cheng, and
Lin (1993) may be provide some results on minimizing makespan that can be applied, although
the problem is fundamentally different for the flowtime objective.
2.5 Lookahead and Lookbehind Scheuuliiiii
This research includes in its investigation of the complicated job shop scheduling problem
the use of more sophisticated dispatching techniques that consider more information in their
decisionmaking. This research tries to be precise in its use of the terms lookahead and look
behind. The idea of looking at the other machines in the shop when dispatching is not well
researched, although the workinnextqueue and numberinnextqueue dispatching rules
mentioned by Panwalker and Iskander (1977) are simple lookahead rules. '
Lookbchind rules consider the jobs that will be arriving soon (called xdispatch by Morton
and Ramnath, 1992). Lookahead rules consider information about the machines downstream in
the flow. This includes the workinnextqueue and the numberinnextqueue rules of Panwalkar
and Iskander (1977), bottleneck starvation avoidance (Glasscy and Petrakian, 1989), bottleneck
dynamics (described in Morton, 1992), and lot release policies that lookahead to the bottleneck
(Wein, 1988; Glasscy and Resende, 1988a; and Lcachman, Solorzano, and Glasscy, 1988). A
threemachine lookbchind scheduling problems similar to problems studied in this dissertation is
considered in Lee and Herrmann (1993).
56
Also, Blackstone, Phillips, and Hogg (1982) report ori a heuristic that is called lookahead
but would be classified here as lookbchind, and Saveli, Pere/., and Koh (1989) did develop a
lookbchind system.
More specifically, lookahead models consider the states of the machines where the jobs
will be processed after being processed on the machine being scheduled. Information about the
workload of other machines may be useful in balancing the flow of product through the shop.
Lookbchind models consider the machines the precede the machine being scheduled in the flow
and the jobs being processed on these machines. Some of these machines will be processing jobs
that will next need processing at the machine being scheduled. If the times that these jobs will
complete is known, these times form release dates to the machine being scheduled, and the
scheduling decision can explicitly consider these imminent arrivals.
Other researchers have studied procedures that they called lookahead scheduling, but the
problem setting or interpretation is slightly different.
Vepsalaincn and Morton (1987) called their weighted COVERT and apparent tardiness cost
rules "lookahead" since they are concerned with the remaining waiting time of a job. They use,
however, average waiting times without looking at the current queues.
Koulamas and Smith (1988) arc concerned with the scheduling of jobs on machines that are
attended by a server that must unload a job from a machine and load another job onto the
machine. Interference results from one machine finishing while the server is busy at another
machine. This interference degrades the system performance by preventing the machine from
being maximally utilized.
The authors study a twomachine system where each machine has a distinct set of job types
arriving to it. The authors propose a lookahead rule for scheduling a machine that considers the
state of the other machine when deciding what job to sequence next. The rule attempts to
schedule a job whose completion will not interfere with the completion of the job on the other
machine.
57
This work is similar to the definition of lookahead in this paper, although its objective (to
minimize interference) is not related to the completiontime objectives being studied.
Zeestraten (1990) is concerned with minimizing makespan in a job shop with routing
flexibility. That is, some operations may be able to choose from more than one machine for their
processing. The author defines a lookahead rule as one that considers the entire state of the
system and all of the unscheduled operations and creates a partial schedule that specifies a few
operations for each machine. This schedule is followed until another scheduling decision is
made. At this point, the procedure is repeated, using the new information about state of the
system.
The lookahead rule is so called because it searches through the states that the system could
reach in a period that is approximately twice the average cycle time. That is, it looks ahead in
time without attempting to schedule all of the remaining operations. Thus, it falls somewhere
between fixed schedules and realtime dispatching.
This type of scheduling is actually more global in nature, since it considers the entire shop
when scheduling. It is not looking at specific machines.
In summary, although scattered lookahead and lookbehind techniques have been
considered, this research attempts to categorize these ideas, conduct analysis of lookahead and
lookbehind scheduling problems in order to derive good rules, and integrate the results into a job
shop scheduling environment.
2.6 Class Scheduling
Manufacturing often involves machines that process different product types, and this
phenomenon can be modelled as a class scheduling problem, a topic mentioned in the
introduction (Section 1.4). A number of researchers have studied the class scheduling case of
sequencedependent setup limes. Examples reported by Moruna and Potts (1989) include paint
production machines that are cleaned between the production of different colors, a computer
58
system that must load the appropriate compiler for a type of task, and a limited labor force with
workers switching between two or more machines.
Bruno and Downey (1978) study class scheduling problems with deadlines. Their problem
is the singlemachine scheduling of classes of tasks with deadlines and a setup time or
changeover cost between classes. They prove that, with setup times, the question of finding a
feasible schedule is NPcomplctc. With changeover costs, the problem of finding a minimal cost
feasible schedule is also NPcomplctc.
Monma and Potts (1989) present complexity results for class scheduling problems without
deadlines. The assume that the class setups satisfy the triangle property. They show that
minimizing makespan, maximum lateness, the number of tardy jobs, and the unweighted and
weighted flowtimc is NPcompletc, although they discuss a number of optimal properties and
dynamic programming algorithms that arc polynomial in number of jobs but exponential in the
number of batches. They also show that the corresponding parallel machine problems are NP
complete. The authors conclude that the design and analysis of heuristics for these problems is
important.
Ahn and Hyun (1990) study the problem of minimizing flowtime and present a dynamic
programming algorithm that is similar to those of Monma and Potts. This algorithm is
exponential in the number of classes, and the authors develop an iterative improvement heuristic
that finds nearoptimal schedules.
Sahney (1972) considers the problem of scheduling one worker to operate two machines in
order to minimize the flowtimc of jobs that need processing on one of the two machines. Sahney
derives a number of optimal properties and uses these to derive an intuitive branchandbound
algorithm for the problem.
Gupta (1984) also studies the twoclass scheduling problem and derives an O(n log n)
algorithm to minimize flowtime. Potts (1991) presents an example that Guptaâ€™s algorithm does
not solve and goes on to describe an 0(n2) dynamic program to minimize flowtime and an 0(n3)
algorithm to minimize weighted fiowtime.
59
Coffman, Nozan, and Yannakakis (1989) consider a class scheduling problem with two
subassembly part tjpes. A product consists ol two parts, one ol each type, both made on the
same machine, and the product cannot be delivered until both parts arc finished. Their objective
is to minimize the llowtime of delivered products, and using properties of an optimal schedules,
they authors create an 0(Vn) algorithm. The authors describe extensions to multiple copies,
limits on the number of changeovers, and limits on batch size as solvable cases without details.
Mason and Anderson (1991) study the oncmachine class scheduling problem with the
objective of minimizing weighted completion times under sequenceindependent setups that
fulfill the triangle inequality. By using properties of an optimal sequence and aggregating jobs
into composite jobs, the authors develop a brandandbound algorithm using their dominance
criteria and lower bound.
Dobson, Karmarkar, and Rummel (1987) consider a class scheduling problem with the
objective of minimizing flowtime under both the itemflow and batchflow delivery schedules.
The batch (or class) setups are sequenceindependent and the processing times are equal for all
parts (jobs) in one part type (class). They formulate integer programs for both problems, solving
the itemflow problem and singleproduct batchflow problems optimally. For the multiple
product batch flow, the authors can only provide some heuristics based on their other results.
Dobson, Karmarkar, and Rummel (1989) consider the above singleitem problems on
uniform parallel machines. They assume that the work is continuously divisible and determine
the amount of work to allocate to each machine by solving a convex programming problem. The
authors also apply their results to solve the itemflow problem.
Woodruff and Spearman (1992) consider an interesting class scheduling problem where the
objective is to maximize the profit of feasible schedules. That is, jobs must be completed by their
deadlines. The profit calculation includes the value of jobs that are not required to be processed
but add some revenue and two different costs: holding and setup. The authors search for good
solutions with a tabu search. Discussion of this search can be found in Section 2.8.3.
60
A more typical problem is studied by Ho (1992), who examines the problem of minimizing
die number of tardy jobs where die jobs fall into exactly two classes. He develops and efficient
branchandbound algorithm.
Gupta (1988) studies the class scheduling problem of minimizing the total flowtime. He
develops a heuristic that is based upon the standard shortest processing time rule. Empirical
testing shows that the heuristic produces good results on small problems.
A more sophisticated class scheduling problem is the minimizadon of the maximum
lateness of a set of jobs with nonzero release dates. This problem is considered by Schutten and
Zijm (1993), who develop a branchandbound algorithm and a tabu search over the sequences of
jobs. They report good results on problems with up to 50 jobs.
In summary, this body of research does not yet include much work on problems with
additional criteria, such as deadlines or release dates, or on shop problems. Moreover, this
research has concentrated on branchandbound techniques and optimal algorithms for problems
with just two classes. Little work has been done on heuristics for problems with an arbitrary
number of job classes or on the use of smartandlucky searches to solve class scheduling
problems.
2.7 Some Onemaehinc Problems
In order to gain insight into new dispatching rules that may be useful in the job shop
scheduling question, some oncmachine problems will be investigated. The three onemachine
problems reviewed in this section arc the problem of minimizing total flowtime with deadline
constraints, the problem of minimizing the number of tardy jobs subject to matching release
dates, and the problem of minimizing the total flowtime of jobs with release dates. These
questions will be the first to be studied as classscheduling problems.
61
2.7.1 Constrained How nine
The first one machine problem studied in the preliminary work is a class scheduling
extension of 1 / Dj/'Z Cj, the constrained flowtime problem first studied by Smith (1956).
Smith, in this paper, proves the optimality of ordering the jobs by Shortest Processing Time
to minimize the total flowtime problem 1 / / X Cy. He extends this to the problem where the jobs
all have deadlines that must be met.
The makespan of the jobs is known, and following the SPT property of minimizing total
flowtime, the algorithm tries to schedule the longest job to end at this time. However, the longest
job's deadline may be less than the makespan, in which case the job is not available to end at this
time. So, the algorithm searches for the longest job that can end at this time and schedules it.
The algorithm then moves to the start time of this scheduled job. This time is now the completion
time of the remaining unscheduled jobs, and the algorithm again searches for the longest
available job. This algorithm is hereafter referred to as Smithâ€™s algorithm.
The weighted problem is strongly NPcomplete (Lenstra, Rinnooy Kan, and Brucker,
1977), and different elimination criteria and branchandbound techniques have appeared. Potts
and van Wassenhove (1983) study the Lagrangean relaxation of the deadline constraints, leading
to the discovery of optimal solutions. Work by Posner (1985) and Bagchi and Ahmadi (1987)
present improved varieties of this bound. Many other problems have been studied; see
Herrmann, Lee, and Snowdon (1993) for a survey of dual criteria problems.
2.7.2 Release and Due Dates
The other onemachine problem in the preliminary results is the lookbehind problem
1 / rj / X Uj, which is a strongly NPcompletc problem in general (Lawler, 1982). The problem
has been solved optimally if all release dates are zero by the MooreHodgson algorithm (Moore,
1968). The problem has been solved optimally by Kise, Ibaraki, and Mine (1978) if the release
62
and due dates match (rj < rÂ¿ implies dj <, dÂ¿). They present an 0(/z2) algorithm (Kise's algorithm,
described below) in this case.
Kise's algorithm orders the jobs by their release and due dates (a nonambiguous ordering
since the dates must match). The algorithm is an exteasion of MoorcHodgson algorithm for
minimizing the number of late jobs. Each job is scheduled after the partial schedule of ontime
jobs while maintaining release availability. If the new job is tardy, the algorithm searches the on
time jobs for the job whose removal leaves the shortest schedule of ontime jobs. The removed
job is made tardy and will be processed with the other tardy jobs after the ontime jobs. In this
manner, the algorithm finds the largest subset of the jobs that can be delivered ontime. These
jobs are scheduled in order of their release and due dates.
Unlike the MooreHodgson algorithm, the subalgorithm that removes a job cannot just
choose the longest job, since the presence of release dates limits how the removal a single job
affects the completion times of later tasks. They present an efficient way to determine which job
should be removed.
2.7.3 Flowtime and Release Dates
A last onemachine problem that may yield some good results when studied as a class
scheduling problem is closely related to the constrained flowtime problem. This is the problem
of minimizing total flowtime subject to job release dates. Written as 1 / rj / Z Cy this problem is
a strongly NPcomplete question, as shown by Lenstra, Rinnooy Kan, and Brucker (1977).
Dessouky and Deogun (1981) present a branchandbound algorithm with a lower bound
and dominance properties used to prune the search tree. They list a number of dominance
properties that hold for a given partial sequence of the jobs. Their lower bound is derived from
the EFT schedule but starts the selected job at the first release time. They can solve problems
with up to 50 jobs. They also report that the EFT sequence generally finds good solutions (within
3c/c of optimal).
63
Bianco and Ricciardclli (1982) consider the weighted version of the problem. Let the
weighted processing time be the processing time divided by the weight of the job. They present
an improved branchandbound algorithm with further dominance properties for nodes with a
partial schedule. They compute a lower bound by allowing preemption. The hardest 10job
problems to solve were those with a larger range of weights and a maximum release date near the
expected sum of processing times (that is, the average intcrarrival and processing times were
nearly equal).
Hariri and Potts (1983) also attack the problem of minimizing weighted flowtime with
release dates. They use a Lagrangcan relaxation of the constraints Cj > rj + pj to find a lower
bound. They make use of previous dominance properties and add another that measures the
effect of interchanging two consecutive jobs. They solve problems with 10 to 50 jobs, and the
hardest problems (some of which remain unsolved) were those with a range of release dates
approximately the same as the expected sum of processing times.
Gazmuri (1985) studies the question probabilistically. He develops two eases. In the
undersaturated ease, where the expected processing time is strictly less than the expected time
between release dates, the author develops an algorithm that partitions the jobs into smaller sets
and schedules each of the sets optimally. The algorithm for the oversaturated case starts with the
optimal preemptive schedule and patches preempted jobs by shifting the delayed segments to the
left until the job is whole. For both eases, the heuristic is asymptotically optimal as the number
of jobs goes to infinity.
Liu and MacCarthy (1991) use both mixedinteger linear programming and branchand
bound techniques to solve the problem. They present a number of heuristics that are different
priority rules and report that the heuristics generally find nearoptimal solutions (within 1%) with
little computational effort until the problem sizes exceed 100 jobs. The MILP can solve problems
with only 10 jobs, the branchandbound procedure problems with 25 jobs.
Finally, Rinaldi and Sassano (1977) also report on a branchandbound technique for the
weighted problem.
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2A Snmnandluekv Searches
In this section we will discuss a class of searches that have been receiving much attention
from researchers recently. This class include tabu search, simulated annealing, and genetic
algorithms. We will use the last of these in the research into different scheduling problems.
2.8.1 Introduction
One approach to difficult scheduling problems such as job shop scheduling is local search,
an iterative procedure that moves from solution to solution until it finds a local optimum. Two
examples arc hillclimbing and steepest descent. The hillclimbing algorithm chooses a nearby
point at random and moves there if the point improves the value of the objective function. The
steepest descent procedure examines the entire neighborhood of an incumbent point and selects
the point (if any exist) that is most improving.
Heuristic searches (or probabilistic search heuristics) attempt to improve upon the primary
problem of these simple searches: convergence to local optima. We can use the term smartand
lucky to describe these more complex searches: they are smart enough to escape from most local
optima; they still must be lucky, however, in order to find the global optimum. Simulated
annealing, the most popular of the methods described in this work, was developed independently
by Kirkpatrick, Gelatt, and Vccchi (1983) and Cemy (1985). Glover claims that tabu search,
which is also widely used, goes back to a paper of his from 1977. Holland developed the ideas of
genetic algorithms, which have had the least success so far, in his 1975 book. This section
reviews the basic concepts of these searches and a number of scheduling applications.
2.8.2 Simulated Annealing
Simulated annealing (SA) is a variant of the hill climbing algorithm. As mentioned before,
both Kirkpatrick, Gelatt, and Vccchi (1983) and Cemy (1985) produced the initial papers on the
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simulated annealing algorithm, so called because the algorithm views optimization as a process
analogous to physical annealing, the cooling of a system until it reaches a lowenergy state.
In annealing, as a system cools, configurations occur with weight dependent upon their
energy and the system's temperature. In 1953 Metropolis ctal. developed a statistical simulation
of an annealing system. This algorithm randomly shifted the system from one configuration to
another, using the physical laws that governed behavior to guide the procedure. Kirkpatrick,
Gelatt, and Vecchi (and Cemy) modified his procedure to solve optimization problems. The
crucial element was the acceptance of a nonimproving move with a probability that was a
function of the difference D in objective function and some cooling constant T. The standard
formula was eorr. This acceptance of a nonimproving move is what differentiates a simulated
annealing from hillclimbing.
Kirkpatrick, Gelatt, and Vecchi (1983) list four ingredients for a simulated annealing
algorithm: a precise description of system (that is, a solution space), the random generation of
moves, a quantitative objective function, and an annealing schedule. Simulated annealing makes
a move by picking from a set of possible operations that can be applied to the incumbent solution.
Usually, the choice is made randomly, although the set may be ordered somehow. At a given
temperature, the algorithm may stop when some equilibrium or maximum number of moves is
reached. Then the algorithm reduces the temperature geometrically or linearly. The initialization
of the algorithm with a starting point may be random or may use a heuristic.
It can be proved that simulated annealing will converge to a global optimum. Aarts and van
Laarhoven (1985) use a homogeneous Markov chain that reaches equilibrium at each temperature
to prove this. They use a general form of acceptance probabilities, of which the standard
exponential is a special case. However, reaching equilibrium at a temperature may require an
exponential number of moves. They avoid the exponential chain length by defining quasiequilibÂ¬
rium conditions and conclude with a polynomial algorithm.
Kirkpatrick, Gelatt, and Vecchi (1983) studies a number of chip design problems and the
traveling salesman problem. In his example, he places the cities in nine clumps and examines the
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effect of temperature on the solutions found. At high temperature, the algorithm minimizes the
number oflong jumps; at a medium temperature, the coarse structure changes but stays minimal;
and at low temperatures, the procedure improves the solutioas at the small scale. Ccmy also
studies the traveling salesman problem.
Van Laarhovcn, Aarts, and Lenstra (1988) study the job shop scheduling problem by
making use of the disjunctive graph, where the makespan of a solution is the length of the longest
path. The disjunctive graph has an arc for each precedence among operations for a job and an
(undirected) edge connecting those operations on different jobs that use the same machine. A
schedule can be found by directing the edges to form an acyclic graph. Given a feasible solution,
any swap of a critical disjunctive edge will form another feasible solution. This becomes their
move, and they claim that their simulated annealing is as good as shifting bottleneck and simpler.
A different type of simulated annealing is included in the Matsuo, Suh, and Sullivan papers
(1988, 1989) on controlled search simulated annealing (CSSA). In this approach, the algorithm
uses a good initialization, independent acceptance probabilities, a low initial acceptance
probability, and a sequential search of smaller neighborhoods. The motivation for acceptance
probabilities independent of the change in objective is its use in empirical testing of different
cooling schedules. Their first paper addresses the singlemachine weighted tardiness problem.
The CSSA uses a fixed number of iterations and its move swaps an adjacent pair. The authors
compare different cooling schedules and claim that the CSSA is a better procedure than standard
SA.
In the paper on job shop scheduling, the authors use the shifting bottleneck algorithm for
initialization. The CSSA identifies the critical path and then switches two operations on the
critical path with lookback and lookahead options that swap other operations to improve
makespan. If a nonimproving move is not accepted, they perform a local search to find a
possible improving solution. They claim better makespans with the same effort as shifting
bottleneck.
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For How shops, Ogbu and Smith (1990) present a simulated annealing that they call
"probabilisticexhaustive" because it uses independent acceptance probabilities with a random
search of the entire neighborhood, moving to the last point accepted. They minimize makespan,
and their moves arc pair swaps and insertions. They get better processing time than the standard
SA and better results than repetitive local searches. Vakharia and Chang (1990) look at a fiow
shop with jobs in families and family setup limes. They also use independent acceptance
probabilities. Their moves swap jobs within a family or swap families. They use different
initializations and compare their search to other heuristics.
2.8.3 Tabu Search
Tabu search (TS) is a variant of steepest descent. Glover (1989, 1990) presents a good
discussion of tabu searches. Given an incumbent solution, a TS searches the neighborhood of this
solution, finding the best allowable move. A TS allows bad move away from local optima,
prohibits moves which lead backwards through shortterm memory (the tabu list) and has an
aspiration level to override the tabu list under certain conditions (usually best found). A tabu
search works because the tabu list forces the search to explore new areas of the solution space.
The shortterm aspect of the memory and the aspiration level allow the search to get to a global
optimum however.
Table 2.1. Outline of a Simple Tabu Search.
1. Pick initial x. let x'= x. T, the tabu list, is empty.
2. Let S(x) be the neighborhood of x. Take s' as best member of S(x) \ T.
3. Let jc = s'(x). If c(x) < c(x'), x'= x.
4. Check stopping criteria. Update T. Go to 2.
The best member is usually that which gives the best value of the objective function c(x).
The stopping criteria may be a total number of moves or a number of moves since the last best.
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Updating the tabu list T is the crucial part of a tabu search. Usually this means adding moves that
would reverse the step just taken and removing old tabu moves.
Laguna, Barnes, and Glover (1989) investigate a tabu search for a singlemachine
scheduling problem with linear delay penalties and setup cost dependencies. They use a heuristic
initialization and insertion and swap moves. They claim results much better than a branchand
bound.
Bamcs and Chambers (1991) investigate the use of a tabu search to solve the job shop
scheduling problem of minimizing makespan. The authors describe an improved approach that
uses an actual makespan change when evaluating new schedules instead of a projected change.
The authors compare their job shop scheduling approach to Applegate and Cook's shuffle
algorithm and the Shifting Bottleneck algorithm of Adams, Balas, and Zawack. They observe
that their tabu search achieves better makespans in most eases over a range of problem sizes.
Bamcs and Laguna (1992) examine the multiplemachine weighted flowtime problem with
a similar tabu search. This problem reduces to a partition problem, and their search docs prevent
nonimproving swaps: they claim that swaps are too small to move through the solution space
effectively. Again, they claim results that beat a branch and bound approach.
Widmer and Hertz (1989) take up the flow shop and use permutation schedules. They
define a distance between two jobs as the approximate increase in makespan and perform a TS
upon the open TSP. A move in the solution space is the swap of a pair. They get slightly better
makespans than six heuristics at the cost of terrible processing times.
An ambitious paper by Malek et al. (1989) examines parallel and serial tabu searches and
simulated annealing on the traveling salesman problem. For each search, a move was a 2opt
(subsequence reversal). The parallel runs communicate periodically, sharing good solutions.
Parallel SA perfonned a quick annealing on each processor and achieved superlinear speedup.
According to the authors, parallel TS performed best, but the simulated annealing was more
robust Oess sensitive to parameter changes).
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In an interesting article. Woodruff and Spearman (1992) study the problem of maximizing
the profit gained from scheduling a set of jobs on one machine. The authors claim that the onc
machine problem has significance for shops with a bottleneck operation, where the sequencing of
the bottleneck determines how the shop performs. These jobs include jobs that are required in
some sense, say to meet specific orders, and jobs that are filler, in that they are not required now,
but some profit is realized by producing them. The jobs have due dates, and the schedule should
be feasible with respect to these. The costs include the holding costs of finishing jobs too early
and setup costs incurred when the machine must be switched from processing jobs of one family
to those of another. Thus, this problem is a class scheduling problem, and it is an NPcomplete
question just to ask if there exists a feasible schedule.
The authors show that if the holding costs arc zero, it is optimal to order the jobs in each
family by EDD. If the holding costs arc positive, this EDD within a family is still used as a
heuristic in order to simplify the problem of finding a good schedule.
The authors use a tabu search to search the solution space for good sequences. The moves
of the tabu search are insertions of jobs while maintaining EDD within a family. Complications
are added by the presence of filler jobs that are not required to be in the schedule. The authors
use a diversification parameter for two reasons: first, the parameter diversifies the search by being
included in a modified cost function that provides a penalty for infeasible solutions while
allowing the search to use these as passes into new areas. Secondly, this parameter allows the
search to optimize its performance. This is done by initially performing tabu searches over a
number of different values of the diversification parameter and then continuing the search with
the best values.
Empirical testing of the algorithm on data motivated by an actual manufacturing
environment showed that the search is a useful method of finding good solutions for this problem.
Glover, Taillard, and dc Wcrra (1991) describe the main aspects of tabu search and discuss
various refinements that may lead to a new generation of search. These refinements occur on the
tactical, technical, and computational levels. Tactical improvements are concerned with
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improving the neighborhood structure and defined moves. This may include the use of learning
approaches, shifting penalty tactics, and strategic oscillation.
Technical improvements focus on the issues of neighborhood size, tabu list structures, and
aspiration conditions. Computational improvements include reducing the time required to
compute the objective function over the neighborhood and the parallelization of the search.
Laguna and Glover (1991) use target analysis to promote diversification in a tabu search
used to find good solutions to a single machine problem with linear delay and setup costs. The
tabu search uses both swap and insert moves.
The goal of using target analysis is to teach the tabu search to use good rules. The authors
describe five phases of target analysis. The first phase is to take some specific problems and find
very good solutions to those problems. In the second phase, with these very good solutions as
targets, the problems are solved again and information is gathered on how well the current
decision rules lead to the target. In the third phase, this information is integrated into a master
decision rule. The fourth phase finds good parameter values for this master. Finally, the master
is applied to the original problems to verify its merit.
Reeves (1993) examines how the definition of the neighborhood in a tabu search affects the
balance between exploration and exploitation (or diversification and intensification). Given the
proper balance, he finds that tabu search is a more efficient procedure than simulated annealing
for the permutation flowshop problem.
2,8.4 Genetic Algorithms
A genetic algorithm is a smartandlucky search that manipulates a population of points in
the effort to find the optimal solution. Each individual in the population is a string of genes,
where each gene describes some feature of the solution. Genetic algorithms mimic the processes
of natural evolution, including reproduction and mutation. The most powerful operator of a
genetic algorithm is the crossover operator: the recombination of the genes of two parents to
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create two offspring. This crossover allows the offspring to acquire the good characteristics of
different points in the search space.
Most genetic algorithms perform the following steps, stopping when a fixed number of
offspring has been created:
Step 0: Form an initial population.
Step 1: Evaluate the individuals in the population.
Step 2: Select individuals to become parents with probability based on their fitness.
Step 3: For each pair of parents, perform a crossover to form two offspring. Mutate each
offspring with some small probability.
Step 4: Place the offspring into the population. Return to Step 1.
Holland (1975), Davis (1987, 1991), and Goldberg (1989) provide good descriptions of
genetic algorithms. Holland's 1975 book introduced the genetic algorithm (GA), a procedure that
mimics the adaptation that nature uses to find an optimal state. In genetic algorithms, solutions
are represented as strings (chromosomes) of alleles, and the search performs operations on the
population of solutions. Liepins and Hilliard (1989) identify these operations as 1) the evaluation
of individual fitness, 2) the formation of a gene pool, and 3) the recombination and mutation of
genes to form a new population. After a period of time, good strings dominate the population,
providing an optimal (or nearoptimal) solution.
The solution strings may be a sequence of binary bits or a permutation. The fitness of a
solution is its objective function evaluation or ranking. In forming the gene pool, the algorithm
takes, through some random process, those solutions that are more fit. The recombination is
some type of crossover, onepoint or multipoint. It is this powerful crossover mechanism that
makes GAs special. The mutation operation, so important in SA, is a secondary operation here
and serves only to maintain diversity.
Why do GAs work? A schemata is a pattern possessed by individuals in population. The
fitness of a schema is the average fitness of those solutions that possess it. Genetic algorithms
work because good (short) schema survive exponentially, and the population of solutions
provides implicit parallelism.
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When do they fail? A genetic algorithm may fail if the strings are inappropriate
representations of solutions, the strings exclude important problem information (such as
constraints), or the algorithm converges to a local optimum. The first issue is the largest for
scheduling problems: representation is difficult for combinatorial problems because standard
crossovers may lead to infeasibility.
In an effort to address this problem for the traveling salesman problem, where a solution is
a permutation of cities, Oliver, Smith, and Holland (1987) propose a number of unusual
crossovers. The first is the Order crossover, which substitutes a subsequence from the first parent
into the second and then visits the remaining cities in the same relative order as before. The
crossover maintains short schema. Their PMX crossover compares subsequences from both
parents and then performs a swapping in the second parent. The Cycle crossover matches cities
to form independent tours. Then, for each tour, the crossover picks a parent to supply the cities.
After comparing the tree crossovers, they claim that the Order is better on TSP by preserving the
short schema, which are important for this problem.
The job shop scheduling problem is much more complicated problem. Davis (1985) solves
a simple job shop problem with a GA. Another method is an idea by Storer, Wu, and Vaccari
(1990, 1992): searching problem spaces and heuristic spaces, which are discussed in detail in
Section 2.9.
Storer, Wu, and Vaccari (1990) note that a solution is simply the application of a heuristic
to a problem. Changing the problem or the heuristic generates a new schedule and thus a new
solution. A problem is a vector of processing times, and a heuristic is a vector of dispatching
rules that can be used to create a nondelay or active schedule. These data structures create
simple strings and perform well under standard crossover as well as simulated annealing and tabu
search. The authors perform all three searches on both types of spaces. For the tabu search, the
tabu list was a set of tabu makespans. Otherwise, the SA and TS were standard. The authors also
tried shifting bottleneck, probabilistic dispatching, and random search over their new spaces.
They test their heuristics on a number of problems ranging in size from 10 to 50 jobs. The
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problems were also classified as easy or hard. The easy problems had completely random
routings, and the hard problems tended have greater competition for the machine resources at any
point in time. The genetic algorithms on problem space generally found higherquality solutions.
Fox and McMahon (1990) are interested in the job shop scheduling problem but study the
traveling salesman problem in order to gain insight into using genetic algorithms for sequencing
problems.
The authors consider the binary precedence matrix M where mÂ¡j = 1 if city i precedes city j.
mÂ¡Â¡ = 0 for all cities i. This matrix incorporates micro and macroinformation. The authors
consider row and column swap operators that exchange the successors or predecessors of two
cities.
The first new operator is the intersection operator, in which an offspring inherits the
precedences that exist in both parents. That is cÂ¡j = aÂ¡j and by, over all ij. This will yield a
matrix that is consistent (no cycles) but incomplete; in other words, a partial ordering. The
matrix is completed through an analysis of the row and column sums.
The second new operator is the union operator. First, the cities are partitioned into two
sets. This forms for each parent two precedence submatrices corresponding to the two subsets.
The operator then takes one submatrix from one parent and the submatrix for the other subset
from the other parent. Again, this leads to a partial ordering that must be completed. The authors
claim that this operator contains little microinformation from the parents, unless a Markov
process is used to partition the cities.
The authors compare different operators for the TSP, including a random operator used as a
benchmark. If the proposed operators arc no better than this, they should not be pursued. Their
city topologies included random distances, clustering, concentric circles, and a 30city problem
from Oliver, Smith, and Holland (1987). The population size was 900 with no mutation. The
also tried searches with and without elitism (the keeping of the best solution).
Among nonelite searches, the two new operators were great, even when considering
processing time. Their advantage disappeared when the searches got to use elitism.
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The authors claim that this work will be useful in scheduling problems, although execution
time remains a big problem.
Biegcl and Davcm (1990) begin with a general discussion of genetic algorithms. They then
describe how a genetic algorithm could play a part in the scheduling of a shop. They discuss
what data are essential and how the GA could form a schedule for a known group of jobs that
follow the same route.
The authors move to the singlemachine problem and consider how the analogous GA
would work. Moving to the 2 and wimachine flow shop, the authors present a simple discussion
of what the GA would do.
For the dynamic problem, the authors consider performing a rescheduling in realtime to
deal with arrivals and use a FIFO dispatching rules on all other machines. The authors then list a
number of areas for future research.
Cleveland and Smith (1989) use a genetic algorithm to schedule the release of jobs into a
manufacturing cell to minimize weighted tardiness. They consider both the sequencing problem
and the problem of determining the release times.
Nakano and Yantada (1991) introduce a binary representation to solve the job shop
scheduling problem. This representation denotes the relative ordering of each pair of jobs on
each machine. If an illegal chromosome is formed by a genetic operator, a nearby legal one
replaces it. The authors report that their algorithm finds good solutions on some standard
problems.
Whitley, Starkweather, and Shaner (1991) develop an edge recombination operator for the
traveling salesman problem and report that a genetic algorithm with this operator is able to find
very good schedules.
Starkweather et al. (1991) compare six sequencing operators for the traveling salesman
problem. They discover that the different operators stress different types of information; since
the TSP depends upon adjacency, the edge recombination operator is the best.
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In Syswerda (1991), the author discusses the scheduling of a fighter simulation lab.
Finding a feasible scheduling is made difficult by a number of constraints. A genetic algorithm
that manipulates sequences of jobs and employs a schedule builder to translate the sequence into
a legal schedule was able to find good schedules.
:,8.5 Summary
Simulated annealing and tabu search easily search complex spaces by lending themselves
to usually simple types of moves, generally find good solutions fast, and are smart variations of
standard searches such as hillclimbing and steepest descent. Genetic algorithms work in a
different manner. They work well due to the survival of good schema and their implicit
parallelism. They have been harder to implement on scheduling problems, however.
This research in this dissertation builds upon this work into smartandlucky searches by
looking into some alternative search spaces.
2.9 Problem and Heuristic Space
This dissertation includes the development of global job shop scheduling models, whose
solution will yield a schedule that can be followed for time period like an eighthour shift. As
mentioned in the last section, one way to find good schedules is to use a search. One recently
proposed idea is to search problem and heuristic spaces. This section will define mathematically
how these spaces can be used and will review the ideas of a few papers in this area.
Define a problem p as a set of data about which an optimization question can be asked. A
solution s is a point that is consistent with the problem structure, where the solution has some
performance z measured by applying the objective function/to the solution with the problem
data, z =f(p,s). If the problem is fixed, there is a performance function Cp over the solution
space such that z = Cp(s) = f(p,s). Solving a problem translates as finding the solution that gives
the minimum function value.
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Traditionally, problemsolving searches have occurred in the solution space. In the solution
space live points (for standard optimization) or sequences (for combinatorial optimization).
Heuristics arc also applied to problems. A heuristic h, applied to a problem p, yields a solution s,
s = G(h,p) orÃ = h(p). Thus, for a given problem p, there exists a performance function Dp(h)
over the heuristic space such that z = Dp(h) = Cp(h(p)). This implies that a search over different
heuristics could find a solution that gives the optimal objective function value.
Moreover, a solution can be generated in ways besides applying a heuristic to the original
problem. Applying a heuristic h to a problem y in another space can generate a solution s = h(y),
which can be evaluated using the objective function. Thus, given a problem p and the heuristic h,
there exists a performance function Ep rfy) over the other problem space, where z = Ep yÂ¿y) =
Cp(h(y)). And as before, a search over this new problem space provides a way to solve the
problem.
The idea of searching heuristic and problems spaces was investigated by Storcr, Wu, and
Vaccari (1990, 1992). In these papers, the authors are investigating the general job shop
scheduling problem. They define a heuristic space composed of vectors of dispatching rules.
The heuristic uses each rule in turn for a fixed number of dispatching decisions. For example, if
the problem is a tenjob by tenmachine problem, the vector might have five elements, where the
dispatching rule in the first clement is used for the first window of twenty decisions, the next rule
for the next window, and so on, until all one hundred operations have been processed.
For a problem space, they use a space of vectors of processing times, where each position
corresponds to a different operation. At each scheduling decision in the formation of the
schedule, the dispatching rule selects a job by considering these alternative processing times. The
operation of that job is scheduled using the actual job data in order to calculate the finish lime of
the operation, thus ensuring that the schedule produced will be feasible. Thus, a vector of
processing times yields a sequence of operations for each machine.
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The authors include a number of concepts related to searching these new spaces, including
the definitions of neighborhoods, the use of these spaces in genetic algorithms, the fact that the
spaces contain optimal solutions, and details of local search implementations.
The application to genetic algorithms merits special comment. Traditional genetic
algorithms have difficulty with scheduling problems since the components of the strings (the jobs
in the sequence) are not independent of each other. The problem space and heuristic space
described above, however, consist of vectors that have independent elements. That is, the
dispatching rule (or processing time) in the first element docs not affect what values the other
elements can have. Thus, a crossover operation that breaks two strings (vectors) and joins the
separate pieces yields offspring that arc valid points in the search space.
The authors extend this work in Wu, Storcr, and Chang (1993) to the onemachine
rescheduling problem. They use heuristics that adjust "artificial tails" to find schedules that are
efficient and deviate little from the original schedule.
Bean (1992) uses the idea of random keys to provide an alternative search space for a
number of problems: multiplemachine scheduling, resource allocation, and quadratic
assignment: the keys arc used to sequence the jobs (or other variables). Then some simple rule is
used to generate a solution from this sequence. Good empirical results arc reported.
Our research extends these ideas to onemachine class scheduling problems and
investigates a similar heuristic space for the job shop scheduling problem.
2.10 NPC'omplctcness
Job shop scheduling problems are usually quite difficult to solve, and they are among the
hardest optimization problems known. Combinatorial problems, as well as other types of
problems, can be classified by their complexity, or how difficult they are to solve. The hardest
problems arc called NPcomplctc. The primary guide to the theory and use of NPcomplctcncss
is the book by Garey and Johnson (1979).
78
Most ot the problems studied in operations research have been classified by their
complexity, and as researchers study new problems, their complexity is identified also. There are
three primar)â€™ classifications: polynomial, NPcomplctc, and strongly NPeomplctc. Problems in
the first set can be solved with algorithms that have a running time proportional to some
polynomial function of the problem size. Algorithms to solve problems in the second set require
effort that is a polynomial function of the problem size and the problem data. Such an algorithm
is called a pseudopolynomial algorithm. Algorithms to solve problems that are strongly NP
complete require an exponential amount of effort.
The complexity of a problem can be determined by identifying the problem as a more
difficult case of a hard problem, by developing a polynomial algorithm to solve the problem, or
by transforming a previously classified problem into the new problem. By being able to
determine the complexity of a problem, researchers arc able to know what types of approaches
may be possible in finding optimal solutions to the problem. A problem that is known to be NP
complete will not be solved optimally by any polynomialtime algorithm. Pseudopolynomial
dynamic programs will not solve strongly NPcomplcte problems. This docs leave the
possibility, however, that certain special cases may be more easily solved, that algorithms that
search for an optimal solution may have good average performance (e.g. the simplex method of
solving linear programs), and that polynomialtime heuristic algorithms may be able to find
generally highquality solutions.
This research is concerned with a number of different scheduling problems, which will
classified by their complexity as necessary. We will directly prove NPcompleteness for one of
the problems that wc study. Each of the other problems is a more difficult case of a previously
considered NPcomplete problem.
2.11 Chapter Summary
This chapter has attempted to cover a broad spectrum of topics and literature. Obviously,
much research has been done in the fields of job and flow shop scheduling and the field of
79
managing semiconductor manufacturing. However, no system to optimize the scheduling of such
a process has emerged, despite the use of many different and sophisticated procedures.
Still, new tools arc needed that may have good performance when applied to specific
settings. There do exist new ideas, such as class scheduling, lookahead and lookbchind
approaches, smartandlucky searches, and problem and heuristic spaces. The literature in these
areas reports on some initial research.
However, the full capabilities of the these methods, especially when joined to solve a
scheduling problem, have not been fully explored.
CHAPTER 3
ONEMACHINE CLASS SCHEDULING PROBLEMS
In Chapters 3, 4, and 5, we discuss the results of our research into a number of diflerunt
scheduling problems and the general job shop scheduling problem. The primary motivation is to
examine subproblems of the job shop scheduling problems in order to gain insight into the larger
problem. The subproblems that we will examine arc interesting scheduling problems that have
not been previously considered.
We will start with the onemachine class scheduling problems. Our approach in this
chapter is to develop analytical results, to test extended heuristics, and to show that a problem
space genetic algorithm can be a good procedure for a variety of scheduling problems.
3.1 Introduction
The three onemachine problems studied in this work are class scheduling problems that
model the complicating factor of machine setups in the manufacturing process. Most problems
with sequencedependent setup times are NPcompletc. Gass scheduling problems have a
special structure that makes them good candidates for further research: the jobs to be scheduled
form a number of disjoint job classes and setups occur whenever the machine processes
consecutive jobs from different classes. And although we will study a number of different
problems, we will sec that a problem space genetic algorithm will be a useful procedure for all of
them.
The onemachine class scheduling problems under investigation are as follows:
1. Constrained Flowtime with Setups (CFTS)
2. Class Scheduling with Release and Due Dates (CSRDD)
3. Flowtimc with Setups and Release Dates (FTSRD)
80
81
The first problem studied is the class scheduling extension of the onemachine problem of
minimizing the total fiowtime of a set of jobs that have deadlines. The research includes an
optimality properly for the jobs in the same class, a good heuristic, and the development of a
problem space genetic algorithm that can find better solutions.
The second problem is the class scheduling problem where each job has a release date and
a due date. The objective is to minimize the number of tardy jobs. We investigate a number of
heuristics and the use of a problem space genetic algorithm. We also look at a secondary criteria,
minimizing the total tardiness.
The objective in the third problem is to minimize the total flowtime where each job has a
release date. We consider a number of approaches to finding good solutions, including a problem
space genetic algorithm, and present a special case that can be solved with a pseudopolynomial
dynamic programming algorithm.
This chapter considers the research on each of these problems in turn. Research relevant to
these problems is also discussed in Chapter 2. See especially Sections 2.6 and 2.7.
2,2 Constrained Fiowtime with Setups
We will first consider the onemachine class scheduling problem of minimizing the total
flowtime subject to the constraint that each job must finish before its deadline. A new heuristic is
proposed for the problem. We investigate the use of a genetic algorithm to improve solution
quality by adjusting the inputs of the heuristic. We present experimental results that show that
the use of such a search can be a successful technique.
V2.1 Introduction
This research is motivated by the scheduling of semiconductor test operations. Assembled
semiconductor devices must undergo electrical testing on machines that can test a number of
different types of semiconductors. If a machine is scheduled to test a lot consisting of devices
that are different from the devices tested in the previous lot, various setup tasks are required.
82
These tasks may include changing a handler and load board that can process only certain types of
semiconductor packages or loading a new test program for the new pan. However, if the new lot
consists of circuits that are the same as the previous type, none of this setup is required. This type
of change is a sequencedependent setup that can be modelled by class scheduling.
Since postassembly testing is the last stage in semiconductor manufacturing, meeting a
job's due date is a very important objective for the manager of a test facility. A secondary
criterion is the minimization of total fiowtime (the sum of the job completion times), which
reflects the manager's desire to increase throughput and decrease inventory holding costs.
This is a dual criteria problem, in which the primary criterion is used as a constraint and the
secondary criterion is optimized under this restriction. The problem of minimizing the total
fiowtime subject to deadlines is an old problem. Smith (1956) provides an optimal solution
technique that repeatedly schedules the longest eligible job last. The class scheduling version of
the problem, however, is more difficult.
For our problem, finding a feasible schedule is an NPcomplctc problem. (A schedule is
feasible if every job finishes before or at its deadline.) Thus, there exist no exact algorithms to
minimize in polynomial time the total fiowtime subject to the deadline constraints. (For a
discussion of the theory of NPcompletcncss, see Section 2.10 and Garey and Johnson, 1979.)
Thus, we are motivated to try different heuristics. In this work we develop a multiplepass
heuristic that finds good solutions quickly. The first contribution of our investigation of this
problem is the extension of Smith's algorithm into a heuristic which considers the setup times
while sequencing the jobs by their deadlines and processing times.
We are also interested in using a genetic algorithm to improve the quality of our solutions.
A genetic algorithm is a heuristic search that has been used to find good solutions to a number of
different optimization problems, but genetic algorithms searching for good schedules must
overcome the difficulty of manipulating the sequences of jobs. We investigate the use of a
genetic algorithm to search a new type of space, the problem space. This type of approach was
introduced in Storer, Wu, and Vaecari (1992), who consider alternative search spaces for the
83
general job shop scheduling problem. In this work, we extend the idea to the problem of one
machine class scheduling.
Our search attempts to adjust the deadlines of the given problem so that our heuristic will
find even better solutions. The space of adjusted deadlines that we search forms a problem space
(we will return to this point in Section 3.2.5). The second contribution of this work is our use of
this method to improve the scheduling of a singlemachine problem.
If we use the principles of Davis (1991), then we can classify our genetic algorithm as a
t>pe of hybrid genetic algorithm. However, the only unusual characteristic of our algorithm is the
decoding (the bit string docs not describe a point in the solution space; instead it must be mapped
to a solution via the heuristic). Moreover, the range of hybrid genetic algorithms is so large (for
instance, Goldberg, 1989, describes hybrids differently) that our use of the term problem space
genetic algorithm is a more precise description of the search. Finally, this problem space exists
independently of the genetic algorithm, and the use of this new search space is not limited to our
search. Other searches (including steepest descent, simulated annealing, and tabu search) could
be used to explore the space. Therefore, we will continue to refer to our search space as a
problem space and to our search as a problem space genetic algorithm.
The next subsection summarizes some of the relevant literature on class scheduling
problems and the dual criteria objective under consideration. In Section 3.2.3, we discuss our
notation, an example instance of the problem, and a number of basic results. We discuss in
Section 3.2.4 the heuristic developed for the problem. Our genetic algorithm will employ this
heuristic. In Section 3.2.5, we present a problem space, introduce genetic algorithms, and
discuss the details of the genetic algorithm we developed to search the problem space. Section
3.2.6 describes the generation of the sample problems, the computational experiments, and the
results. Finally, in Section 3.2.7, we present our conclusions.
84
3.2.2 Literature Rcmcw
In this section we will briefly mention some of the relevant research on class scheduling
and on the dual criteria problem of minimizing total flowtime subject to job deadlines. This owrk
and die literature on genetic algorithms are discussed in more detail in Chapter 2.
One of the first papers on problems with class scheduling characteristics is Sahney (1972),
who considers the problem of scheduling one worker to operate two machines in order to
minimize the ilowtime of jobs that need processing on one of the two machines. Sahney derives
a number of optimal properties and uses these to derive an branchandbound algorithm for the
problem. Gupta (1984) defines the class scheduling problem, and Potts (1991), Coffman, Nozari,
and Yannakakis (1989), and Ho (1992) also study twoclass scheduling problems.
Bruno and Downey (1978) prove that, for more general class scheduling problems, the
question of finding a schedule with no tardy jobs is NPcomplete. Monma and Potts (1989) prove
that many class scheduling problems arc NPcomplete, including minimizing makespan,
maximum lateness, the number of tardy jobs, total flowtime, and weighted flowtime.
Dobson, Karmarkar, and Rummel (1987, 1989), Gupta (1988), Ahn and Hyun (1990), and
Mason and Anderson (1991) all study the class scheduling problem under different objective
functions. We will modify the procedure of Ahn and Hyun in order to use it as a comparative
heuristic. The only other dual criteria problem in this area is studied by Woodruff and Spearman
(1992); they consider a class scheduling problem with profit maximization and deadlines.
In the dual criteria literature, the problem of minimizing total flowtime subject to job
deadlines (a deadline is a constraint on the completion time) is among the oldest questions, being
first studied by Smith (1956). The problem of minimizing the weighted flowtime subject to job
deadlines is a strongly NPcompletc problem (Lcnstra, Rinnooy Kan, and Brucker, 1977), and a
number of researchers have examined branchandbound techniques.
3.2.3 Notation and an Optimal Property
In this section we introduce our problem and notation, give an example instance of the class
scheduling problem under consideration, and present some basic results.
Our class scheduling problem is the minimization of the total flowtimc of a set of jobs
where the jobs have deadlines on their completion. The problem can be formulated with the
following notation:
Jj
Job j,j= 1,... ,n
Pj
processing time of Jj
Dj
deadline of J;
Â¿i
job class i, i = 1 m
ni
number of jobs in GÂ¡
s0i
time of initial setup if first job is in GÂ¡
ski
time of setup between jobs in GÂ¿ and GÂ¡
CJ
completion lime of Jj
ICj
total flowtimc.
The problem is to find a sequence that minimizes the total flowtimc (X CJ) subject to the
deadline constraints (Cj < Dj for all Jp. We name this problem the Constrained Flowtime with
Setups problem (CFTS). Since job preemption or inserted idle time leads to a nonoptimal
solution, we will assume that schedules being considered have neither. Any schedule that is a
solution for CFTS will have a number of batches or runs that arc sets of jobs from one class
processed consecutively. Before each batch will be a class setup. The problem involves
determining the composition and order of batches from different classes.
CFTS is an extension of a onemachine problem studied by Smith (1956). In his problem,
which we name the Constrained Flowtime problem (CFT), there exist no sequencedependent
setup times.
An instance that we will use to illustrate our work is described in Example 3.1.
Example 3.1. The data in Table 3.1 form an instance of a class scheduling problem with five
jobs in two job classes. The first three jobs form one class, with the remaining two jobs in the
86
second class. Recall that no setup is required between jobs in the sanie class. However, a class
setup is necessary between jobs of different classes.
Table 3.1. Job and Class Data for Example 3.1.
j
12 3 4 5
Pj
12 2 3 2
Â°j
3 16 14 10 18
G,= (1,2,3)
n, = 3
S01 ~ S21  2
G2=(4,5)
n2 = 2
s02 = s12 = 1
CFTS is an NPcomplete problem since finding a feasible schedule is NPcomplete (Bruno
and Downey, 1978). Hence, it is unlikely that any polynomial algorithm to solve the problem
exists. We will study the use of heuristics to find good solutions.
We now describe Smith's rule for CFT and an optimal property for CFTS. Our new
heuristic extends Smith's rule by taking advantage of the optimal property, which we call Smithâ€™s
property.
In this and later sections, we will refer to each job that can complete at a given time without
violating the job's deadline as being eligible. In this problem, we are concerned with deadlines
that are constraints on the completion times, and we will create schedules backwards, starting
with the last position in the sequence. Thus, we say that a job Jj is eligible at a time t if t < Dj.
The job can feasibly complete at this time without violating the deadline constraint.
Smith's properly for CFT states that if a job is assigned the last position of an optimal
schedule, then it must be the longest eligible job. Smith's rule is derived from this property.
Algorithm 3.1 (Smith's rule for CFT). Let t = px + ...+ pn. Among the jobs that are eligible
at time t, that is, t < Dy choose the job Jj with the longest processing time pj. Schedule Jj to
complete at t, and solve the remaining problem in a similar manner.
87
The following property extends Smith's rule to caeh class in CFTS. This property will then
be extended to consider all classes in order to generate approximate solutions to CFTS.
Lemma 3.1 (Smith's property for CFTS). For each class in an optimal schedule for CFTS, the
only job that could be scheduled to complete at a time t is the longest eligible one.
Proof. It suffices to show that if two jobs JÂ¿ and Jj in die same class arc both eligible at time t
and JÂ¿ is longer than Jj (pÂ¡ > pj), then scheduling Jj to complete at time t leads to a nonoptimal
solution. Suppose we do. Then Cj = t, and JÂ¿ precedes Jj in the schedule formed. Create a new
schedule by interchanging the two jobs. Since Jj is moved to the left, it is still feasible, and the
new completion time is less than CÂ¿, the old completion time ofyÂ¿ (pÂ¿ > pj). The completion
times of any jobs between Jj and JÂ¡ arc decreased. Meanwhile, JÂ¿ completes when Jj did, but this
is feasible since / < DÂ¡. We have therefore created a feasible schedule with less total flowtimc,
and the original schedule cannot be optimal. QED.
3.2,4 The Heuristic
Quick methods of finding good solutions are sometimes effective ways to attack difficult
problems. In this section we describe a multiplepass heuristic that extends the idea of Smith's
rule. We illustrate how this heuristic works using Example 3.1.
Our heuristic finds solutions for CFTS by scheduling jobs in the spirit of Smith's rule,
working backwards from the end of the schedule. Since the makespan (the maximum completion
time) of the optimal solution is not known, the heuristic starts with a trial makespan. After
scheduling all of the jobs, we compute the actual makespan (by removing any idle time) and use
this makespan as the starting point for another iteration. We continue this process until some
limiting makespan is reached. At this point, another pass of the heuristic yields a schedule with
the same makespan or a schedule that is infeasible (because some job or setup starts before time
zero).
This heuristic constructs schedules that satisfy Smith's property for CFTS (Lemma 3.1).
While that lemma applies only to jobs in one class, our algorithm extends the idea of longest
88
eligible job by considering all of the job classes. We schedule the longest job with the minimum
wasted time. Wasted lime is time spent in a setup and idle time.
This (singlepass) Minimum Waste algorithm schedules an eligible job from the same class
as the previously scheduled job if one exists. Else, it selects a job from the class with the smallest
setup or selects the job with the latest deadline. It does this by measuring each jobâ€™s gap: the
wasted time incurred by selecting that job. Note that if no class setups exist, this algorithm is the
same as Smith's rule for CFT (Algorithm 3.1).
Algorithm 3.2 (Singlepass) Minimum Waste.
Step 0: Given a completion lime t, select for the last job the longest job eligible at this time Jj,
i.e. t < Dj. Schedule this job to end at t, and reduce r by pj.
Step 1 (a modification to Smith's rule): Suppose that at time r, a job from class GÂ¡ starts. Then,
for each unscheduled job Jj, define qj as the gap between the last possible completion time oUj
and r. If Jj is in class GÂ¿, qj = max (r  Dj, s^} (see Remarks below for an explanation of this
definition). Let q = min {qj over unscheduled Jj). Select the longest job Jj with qj= q and
schedule this job to end at t  q. Any necessary setup sÂ¡Â¿ can begin alt  q. Reduce t by q and p}.
Step 2: If there remain unscheduled jobs, return to Step 1.
Step 3: There are no more unscheduled jobs. If the first job in the schedule is in class GÂ¿, a setup
of length sqÂ¡ must end at r. Reduce t by this amount.
Step 4: If t < 0, the schedule created is infeasible. Else, compute the actual makespan of the jobs
and setups scheduled.
Remarks. In order to motivate the definition of qj, the gap, let us note that the setup aftcriy is
so the job may not complete after t  However, if the deadline Dj
qj= t  Dj, which is greater than s/,Â¿. This gap thus includes the setup and a period of inserted idle
time of length t  Dj  sÂ¡.Â¿. Note that if Jj is also in class GÂ¡, qj = 0 if and only if t < Dj. The
algorithm is a type of greedy heuristic, in that it attempts to minimize the setup time or idle time
in selecting jobs to be scheduled.
89
MultiplePass Minimum Waste Heuristic. To find a good solution forCFTS, wc can use the
following procedure that makes use of the singlepass Minimum Waste algorithm.
Step A: Let f' = max {Dj :j=\,...,n).
Step B: Let t = t'.
Step C: Perform one pass of the Minimum Waste algorithm (Algorithm 3.2) with completion
time t. This creates a trial schedule.
Step D: Let t' be the sum of processing times and setup limes of this schedule. If t' < t and the
trial schedule is feasible, go to Step B. (The smaller makespan may yield another schedule.)
Step E: If the trial schedule was infeasible or tâ€˜ = t, take the last feasible schedule created and
remove the inserted idle time, starting all jobs as soon as possible. This schedule is the result of
the heuristic. If an infeasible schedule was created on the first pass, then take the sequence of
jobs from the schedule and process the jobs in this order, starting at time zero. This will yield a
schedule with some violated deadline constraints.
Because the problem of finding a feasible schedule is NPcomplete, a single pass of the
Minimum Waste algorithm is not guaranteed to find one. Still, as wc shall sec, it is usually able
to find a feasible schedule if one exists. If a feasible schedule exists, it must finish by the
maximum deadline, which is the first trial makespan. Initially, the heuristic is concerned with
reducing the makespan. Eventually, as the makespan reaches a lower limit, the algorithm
concentrates on the flowtime objective through its use of Smith's property to select a job.
Example 3.2. Let us apply the MultiplePass Minimum Waste heuristic to Example 3.1. In the
first iteration, t = 18, the maximum deadline. The first pass of the Minimum Waste algorithm
performs the following calculations (sec Table 3.2 for complete algorithm): at time 18, no jobs
have been scheduled, and the only eligible job is J$. After choosing J$, t is reduced by p$ = 2 to
16. For J Â¡J 3, and Jj, the waste is the gap until the deadline. Thus, qÂ¡ = 16  DÂ¡ = 13, and
similarly for the other two jobs. For.^, however, Dj = 16, and the deadline gap is zero, but
because ^ is >n a different class than J5, Q2 = s 12 = * â€¢ Thus> J2 has the smallest waste and is
scheduled to end at time 15. After five steps, all of the jobs arc scheduled (sec Figure 3.1). There
90
arc two units of inserted idle time, however, so the actual makespan of the schedule can be
reduced to 16.
Table 3.2. Calculations of the first pass of the Minimum Waste algorithm. Initial makespan =
18.
Time:
Waste:
J1 h
J3
J4
J5
Schedule
Jj
cj
18
15 2
4
8
0
J5
18
16
13 1
2
6
J2
15
13
10
0
3
h
13
11
8
2
J4
9
6
3
h
3
Scheduled fiowtime: 58.
Reduced makespan: 16.
S01
Jl
s12
J4
S21
J3
h
s 12
J5
0 2 3 5 6 9 11 13 15 16 18
Figure 3.1. Schedule created on first pass of heuristic.
When the heuristic repeats the algorithm with the new makespan of 16 (see Table 3.3 for
calculations), jobs ^ and J$ are eligible at time 16. These jobs also have the same processing
time, but suppose J2 is chosen. Then, at time 14, J3 is eligible and has no gap, as it is in the same
class as ^ However, J $ has a gap q$ = sÂ¡2 = 2. The algorithm continues in this manner,
creating the schedule shown in Figure 3.2.
The MultiplePass Minimum Waste Heuristic again repeats the algorithm, which now
begins at the reduced makespan of 15, and a similar schedule can be found if J2 is chosen at time
15. This schedule has no idle time (see Figure 3.3), the reduced makespan is also 15, and so the
heuristic stops.
91
Table 3.3. The second pass of die Minimum Waste algorithm. Initial makespan = 16.
Figure 3.2. Second schedule.
S01
Jl
s12
J5
J4
S21
J3
J2
0 2 3 4 6 9 11 13 15
Figure 3.3. Third and final schedule.
3.2,5 The Genetic Algorithm
In this section we present an alternative search space, the problem space, provide a brief
introduction to genetic algorithms and some references to selected works, and discuss the details
of the problem space genetic algorithm we used to find good solutions for CFTS.
Problem space. The original work on problem and heuristic spaces is by Storcr, Wu, and
Vaccari (1990, 1992), who define some alternative search spaces for the job shop scheduling
problem: the problem space and the heuristic space. They note that a solution to a problem is the
result of applying a heuristic to the problem. Given a problem p, a heuristic h is a function that
creates a sequence corresponding to a solution s, i.e. h(p) = s. Thus, if one adjusts the heuristic,
one creates a different solution. The set of adjusted hcunstics is the heuristic space. Likewise, if
92
one adjusts the problem data that are used by the heuristic, one generates a different solution.
This set of adjusted problem data is the problem space. The idea is applied to the job shop
scheduling problem, and different heuristic searches over the spaces are performed, including hill
climbing, genetic algorithms, simulated annealing, and tabu search.
Our research extends this idea by defining a problem space for the onemachine class
scheduling problem. If we adjust the deadlines that arc inputs (along with the other problem data)
to a pass of the Minimum Waste algorithm, we will create a possibly different schedule. We will
use as a problem space for CFTS these adjusted deadlines, and we will use one pass of the
Minimum Waste algorithm to create a sequence of jobs using the adjusted deadlines. The
feasibility (against the actual deadlines) and total fiowtime of the sequence can be evaluated by
scheduling the jobs to start at time zero with no inserted idle time. The idea is to force jobs to be
done earlier or later by decreasing or increasing the deadlines. We will prove that every solution
for CFTS (including the optimal one) is in the range of h.
Theorem 3.1. For each solution to an instance of CFTS, there exists a vector of adjusted
deadlines that can be mapped to that solution using one pass of the Minimum Waste algorithm.
Proof. Suppose that a is a solution (a feasible schedule with no preemption or inserted idle
time) for an instance of CFTS. For each job, consider adjusting the deadline so that it equals the
job completion time. Then, if we use one pass of the Minimum Waste algorithm with the
adjusted deadlines, the job selected for the last position will be the job with the maximum
adjusted deadline. This job is the one with the maximum completion time CÂ¿ and thus was the
last job in a. It w ill be scheduled to complete at its adjusted deadline, which is CÂ¡.
Now we are at the start time t of a job./, and the job with the smallest gap is the
unscheduled job/y that immediately precedes JÂ¡ in o, since the adjusted deadline is Cy, and Cy < t.
Any setup necessary between Jj and JÂ¿ is already included in the difference between Cy and t.
Thus the gap cannot be larger for this job, and the gap for any other job is larger since CÂ¡, < Cy.
This job will be scheduled to complete at its adjusted deadline, which is Cj. If we continue in this
93
manner, all of the jobs will be sequenced in the same order as Uiey were in a, and we create the
same schedule. QED.
The more we adjust the deadlines, the more change we create in the schedule. For instance,
consider the following examples of applying one pass of the Minimum Waste algorithm to
vectors of adjusted deadlines where we have changed only die second and fifth deadlines: (Note
that the fourth schedule created is infeasible since J2 completes at time 18, which is greater than
the actual deadline: D2 = 16. The adjusted deadline of 19 was used only to sequence the jobs.)
Hcuristic(problcm) = solution:
Minimum Waste (3, 6, 14, 10, 20) = [ 1 24 35], flowtimc 46.
Minimum Waste (3, 16, 14, 10, 18) = [ 1 4 3 2 5], flowtime 50 (original deadlines).
Minimum Waste (3, 17, 14, 10, 16) = [ 1 5 4 3 2], flowtimc 46.
Minimum Waste (3, 19, 14, 10, 17) = [ 1 4 3 5 2 ], infeasible (Q> = 18).
In Figure 3.4 we show a graph that illustrates how adjusting just two of die five deadlines
of Example 3.1 can create a number of different schedules. The first, third, and fourth deadlines
were not adjusted. Each point in the plane (only nonnegative deadlines were considered)
corresponds to a pair of values for the second and fifth adjusted deadlines. The points in each
region of the plane arc mapped by a pass of the Minimum Waste algorithm to the job sequence
denoted by the fivedigit sequence shown in that region. The dot marks the point that
corresponds to the unadjusted deadlines (D2 = 16, D$ = 18). The best sequences achievable by
adjusting these deadlines are 12435 and 15432 (total flowtimc = 46), and the only other feasible
sequences are 14235 and 14325 (total flowtime = 50). The optimal solution (which cannot be
found by adjusting only the second and fifth deadlines) is 13452, with total flowtimc = 43.
Since the actual problem space consists of all of the problem data and there are numerous
heuristics that can be used, we can investigate other spaces and heuristics that might be useful.
Our first search was to adjust the job processing times and to use the Shortest Processing Time
(SPT) rule. However, it is difficult to find feasible solutions since SPT ignores the deadline
constraints entirely. We also tried the using the Earliest Due Date (EDD) rule while adjusting the
94
deadlines, but EDD docs not give enough attention to the flowtimc objective. Sequencing by
either SPT or EDD is a fairly naive heuristic, since neither makes use of the other available
information. The Minimum Waste algorithm, however, considers due dates, processing times,
and setups, and using it improves our searches. In addition, while it would be possible to use the
Minimum Waste algorithm while adjusting the processing or setup times, the effect of these
variables on the sequencing of jobs is more indirect than that of the deadlines.
The use of a heuristic space seems to be hard for this problem. Feasibility is a large
concern, and there arc very few heuristics we can use to find feasible schedules. Also, the
Minimum Waste algorithm has no parameters to adjust.
Job 5 Deadline
Job 2 Deadline
Figure 3.4. Graph of adjusted deadlines and schedules created.
95
A genetic algorithm for CITS. In tins section we discuss the details of the genetic
algorithm we developed to find better solutions for CFTS. As mentioned before, genetic
algorithms are heuristic searches that use a population of points in the effort to find the optimal
solution. The stronger members of the population survive, mate and produce offspring that may
undergo a mutation. These offspring form a new generation. Genetic algorithms have been used
on sequencing problems before, although they cannot use natural crossover techniques in
searching the solution space. The advantage of the problem space is that the genetic algorithm
can use standard techniques to create offspring.
Our genetic algorithm searches the space of adjusted deadlines. We use a binary coding for
the adjusted deadlines and a single pass of the Minimum Waste algorithm as the heuristic. For
this genetic algorithm, we use many of the ideas presented in Davis (1991), to which we refer
readers who wish to learn more about the issues discussed here.
In the problem space, each point is a vector of integers that are deadlines used as input for
one pass of the Minimum Waste algorithm. We will use a binary representation of the points in
problem space. In the population of the genetic algorithm, each individual is a string of bits.
Each successive sixbit substring represents a deadline for a specific job. The integer decoded
from this binary number ranges from zero to 63 and linearly maps to a real number in the range
from zero to the maximum deadline in the given problem data. This discretization reduces the
problem space but still allows the deadlines to vary significantly with respect to each other.
The adjusted deadlines are used as input to a single pass of the Minimum Waste algorithm,
which outputs a sequence of jobs. The algorithm uses the largest of the adjusted deadlines as the
initial makespan and schedules the jobs accordingly, using the actual job processing and class
setup limes where necessary but using the adjusted deadlines to determine when a job is eligible.
If necessary, the algorithm can start jobs before time zero.
Using the actual problem data and the sequence of jobs output from one application of the
Minimum Waste Algorithm, we can create a schedule of jobs that starts at time zero and has no
96
inserted idle time. We evaluate the original string of bits by computing the feasibility and the
total flowtime of this schedule.
Since we cannot guarantee that the algorithm will produce a schedule with no tardy jobs,
we use a penalty function to make undesirable those individuals in the population that yielded
infeasible schedules (with respect to the actual deadlines). This penalty function is O = X Tj2,
where Tj = max {0, Cj  Dj). Our objective function/is defined as/= Z Cj + r .
In order to encourage solutions with good total flowtime (regardless of feasibility) at the
beginning of the search and to encourage feasibility near the end of the search, we start the search
with the constant r small and increase it periodically.
The initial population includes one individual (the dummy, or seed) that we create by
dividing each actual deadline by the maximum deadline, multiplying by 63, rounding down to the
nearest integer, and converting this integer (which is in the range 0 to 63) to its binary
representation. The remaining individuals in the initial population arc constructed by mutating
the bits in the initial (dummy) chromosome. The mutation rate is set at fifty percent (0.5). Note
that using an initial mutation rate of 0.5 is equivalent to choosing a random chromosome from the
entire search space.
Let us illustrate this procedure using Example 3.1, the problem we introduced earlier (sec
Table 3.1 for problem data). Also, let us define LxJ as the greatest integer less than or equal to x.
Mapping the deadlines to the bit strings yields the dummy, shown in Table 3.4. Let us
create another member on the initial population. If two bits, the fourth of the fourth substring and
the first of the last substring, are flipped in the mutation, we have the point in problem space
shown in Table 3.5. Performing one pass of the Minimum Waste algorithm on the new deadlines
(see Table 3.6) yields the sequence [ 1 5 4 3 2], from which we create the feasible schedule
shown in Figure 3.5, with a makespan of 15 and a total fiowtime of 46.
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Table 3.4. Bit representation of the dummy.
L63 Dj/18j
10 56 49 35 63
Bits
ooioio mooo lioooi loooii min
Table 3.5. A point in the problem space.
Bits
ooioio mooo lioooi loom oimi
Integer
10 56 49 39 31
DJ
2.85 16 14 11.14 8.85
Table 3.6. Application of the Minimum Waste algorithm.
Time:
Waste:
h
h
J3
u
J5
Schedule:
Jj Cj
16
13.15
0
2
4.86
7.15
h
16
14
11.15
0
2.86
5.15
h
14
12
9.15
2
3.15
j4
10
7
4.15
0
J5
7
5
2.15
h
2.85
S01
h
s12
j4
S21
J3
h
0 2 3 4 6 9 11 13 15
Figure 3.5. Schedule corresponding to new bit string.
After some experimentation we decided to use a steadystate genetic algorithm that created
offspring by repeatedly (and randomly) selecting from a set of four genetic operators: onepoint
crossover, uniform crossover, and two types of mutation. Small mutation used a low probability
(two percent per bit) of flipping a bit in the string; large mutation used a higher probability (fifty
percent). The search used tournament selection for selecting the parents necessary for the
crossover or mutation, and duplicate bit strings were not allowed in the population. Tuning was
performed in order to determine some good settings for various algorithm parameters. (Sec Table
3.7. The problem sets used for tuning are described in Section 3.2.6.)
98
Table 3.7. Flowtime performance while tuning 2(K)0individual genetic algorithm on 30job
problems (Ncwprob4).
Population Operator Increase Frequency Mutation Average
Size Probabilities in r of Increase Rate Ratio
(in percent)
Tuning Increase in r and Frequency of Increase, Population Size = 10
10
25,25,25,25
+2
10
2%
0.8575
10
25,25,25,25
+ 10
10
2%
0.8666
10
25,25,25,25
+50
10
2%
0.8562
10
25,25,25,25
+2
50
2%
0.8646
10
25,25,25,25
+ 10
50
2%
0.8700
10
25,25,25,25
+50
50
2%
0.8624
10
25,25,25,25
+2
100
2%
0.8676
10
25,25,25,25
+ 10
100
2%
0.8755
10
25,25,25,25
+50
100
2%
0.8625
Tuning Increase in r and Frequency of Increase, Population Size 50
50
25,25,25,25
+2
10
2%
0.8597
50
25,25,25,25
+ 10
10
2%
0.8660
50
25,25,25,25
+50
10
2%
0.8623
50
25,25,25,25
+2
50
2%
0.8669
50
25,25,25,25
+ 10
50
2%
0.8687
50
25, 25, 25, 25
+50
50
2%
0.8654
50
25,25,25,25
+2
100
2%
0.8619
50
25,25,25,25
+ 10
100
2%
0.8626
50
25,25,25,25
+50
100
2%
0.8669
Tuning Increase in r and Frequency of Increase, Population Size 100
100
25,25,25,25
+2
10
2%
0.8800
100
25,25,25,25
+50
10
2%
0.8794
100
25,25,25,25
+2
50
2%
0.8738
100
25,25,25,25
+50
50
2%
0.8792
100
25,25,25,25
+2
100
2%
0.8751
100
25, 25,25,25
+50
100
2%
0.8716
Tuning Mutation Rate
10
25,25,25,25
+50
10
2%
0.8562
10
25,25,25,25
+50
10
10%
0.8990
10
25, 25,25,25
+50
10
50%
0.9962
10
25,25,25,25
+50
50
2%
0.8624
10
25,25,25,25
+50
50
10%
0.8953
10
25,25,25,25
+50
50
50%
0.9629
50
25,25,25,25
+50
10
2%
0.8623
50
25,25,25,25
+50
10
10%
0.8994
50
25,25,25,25
+50
10
50%
0.9236
99
Table 3.7. (Continued)
Population
Operator
Increase
Frequency
Mutation
Average
Size
Fitnesses
(in percent)
in r
of Increase
Rate
Ratio
Tuning Operator Fitness
10
75, 0, 0, 25
+50
10
2%
0.8664
10
50, 15, 10,25
+50
10
2%
0.8764
10
25,25,25,25
+50
10
2%
0.8562
10
15,50, 10, 25
+50
10
2%
0.8758
10
0, 75, 0, 25
+50
10
2%
0.8592
10
75, 0, 0, 25
+50
50
2%
0.8730
10
50, 15, 10, 25
+50
50
2%
0.8628
10
25,25,25,25
+50
50
2%
0.8624
10
15, 50, 10, 25
+50
50
2%
0.8649
10
0, 75, 0, 25
+50
50
2%
0.8688
50
75, 0, 0, 25
+50
10
2%
0.8574
50
50, 15,10,25
+50
10
2%
0.8732
50
25,25,25,25
+50
10
2%
0.8623
50
15,50, 10, 25
+50
10
2%
0.8689
50
0, 75, 0, 25
+50
10
2%
0.8720
Tuning 3000individual genetic algorithm on 50job problems (Prob50z).
10
25, 25,25,25
+50
50
2%
0.8880
50
25,25,25,25
+50
50
2%
0.8931
50
25,25,0, 50
+50
50
2%
0.8739
Notes: Performance is average ratio to solution found by the MultiplePass Minimum Waste
Heuristic. Operators are onepoint crossover, uniform crossover, large mutation, and small
mutation. Mutation rate is probability per bit.
These parameters included the population size, the mutation rate, the operator selection
probabilities, and the rate of change of the penalty coefficient r. The rate of increase in the
penalty coefficient affected the solution quality slightly. The mutation rale, population size, and
relative probabilities of operator selection affected solution quality more significantly, with a
smaller mutation rate (two percent, as mentioned earlier), smaller population sizes, and a higher
probability for selecting the small mutation yielding better solutions. These factors imply that the
search can easily find good neighborhoods (especially since a point corresponding to the original
problem data is included in the initial population) but needs to spend time hunting for a better
100
solution. Thus, a search that incorporates some kind of local search at the end of the genetic
algorithm may be useful.
3.2,6 Empirical Testing
In this section we describe a set of experiments performed in order to test the genetic
algorithm and the heuristics. We discuss the generation of sample problems and the
computational results.
Problem generation. In order to test the heuristics and the genetic algorithm described
above, it is necessary to create a set of test problems. We describe in this section how we can
create problems that have at least one feasible solution and problems where finding a feasible
solution is more difficult.
The problems in the first problem set have 30 jobs in four classes, with random processing
times in the range [1,20] and sequencedependent setup times in the range [0, 5], For this set, we
want to determine random deadlines in order to insure that some feasible schedule did exist.
We use the following procedure: after computing the random class setup times, each job is
given a random processing time, and an initial completion time is computed by scheduling it after
all previously constructed jobs. This firstgenerated, firstserved schedule yields a makespan that
becomes an upper bound for the deadlines, and each job is given a deadline determined by
sampling a random variable uniformly distributed between the job completion time (in this
schedule) and the makespan, i.e. the interval [ Cj, Cmax ]. Thus, the initial sequence is a feasible
solution.
In order to determine the performance of the genetic algorithm on minimizing the flowtimc
when feasible solutions arc harder to locate, a number of additional problem sets arc created. In
addition to the problem set described earlier, which includes problems that were known to have a
feasible solution, we generate 30job and 50job problems with tighter deadlines. The 30job
problems have four job classes and the 50job problems ten job classes. Tighter deadlines arc
achieved by extending the range of values that a random deadline could take. Let us define a
value k that can range from zero to one. The deadline for Jj is taken from the interval
101
Ik Cj, Cmatl, where the C: and Cmax are from the original generated schedule. It it = I, this plan
is the same as the original one, and the generated problem is guaranteed to have a feasible
schedule. If k = 0, all of the deadlines vary equally, and there may exist no feasible schedule. As
k decreases from one to zero, the problems we generate have a higher probability of having fewer
feasible schedules. We generated problems with k = 0, 0.2, and 1. Since the MultiplePass
Minimum Waste Heuristic cannot find feasible solutions for some of these problems, we will see
if the genetic algorithm can find solutions that arc feasible.
Results. In this section we discuss the results of our experiments with the solution
procedures on the generated problem sets. We summarize the findings and present tables of the
collected data. Since the optimal solutions arc not known (a branchandbound algorithm to find
optima requires excessive computational time) and no good lower bound can be determined, we
measure the performance of the solution procedures relative to each other.
Each procedure was run once on each of the problems in the problem sets. The procedures
include the MultiplePass Minimum Waste Heuristic and the problem space genetic algorithm.
For comparison purposes, we also implemented a version of the heuristic that Ahn and
Hyun (1990) use to reduce the total flowtime in class scheduling problems. They proposed an
iterative heuristic that starts with an initial feasible sequence where the jobs in each class are in
SPT order (since they arc not concerned with deadlines) and applies both a forward and backward
procedure to it, repeating the steps until no strict improvement is found. Each of the forward and
backward procedures interchanges different subschcdulcs where the second subschcdulc consists
of jobs from one class and the first subschcdule has no jobs from this class. If the interchange
reduces the total flowtimc, the subschcdulcs arc switched; this maintains the class SPT property.
Our version of this algorithm, called the Modified Ahn & Hyun heuristic, uses one pass of
the Minimum Waste algorithm to form the initial schedule. In addition, a potential swap of two
subschedules is performed only if the swap reduces the total flowtimc and maintains deadline
feasibility.
102
The results (see Table 3.8) show that the genetic algorithm can find solutions that arc much
better than those found by the MultiplePass Minimum Waste Heuristic and arc slightly better
than those that Modified Ahn & Hyun heuristic produces. The genetic algorithm needs more
time to find good solutions on the larger problems, although additional tuning may help improve
the performance of the search.
The searches for the 30job problems were for 2000 iterations using the following
parameter settings: population si/e of 10, all operator fitnesses equal, increase of 50 in r every 10
individuals. The searches for the 50job problems were for 3000 iterations using the same
population size, no large mutations, and an increase of 50 in r every 50 individuals.
Table 3.8. Total fiowtime performance of heuristics on problems where a feasible was
found.
Problem
Problems
Jobs
k
Performance*1 of Heuristics
Set
Genetic Algorithm
Modified Ahn
NewproM
10
30
1
0.8562
0.9130
Prob2a
4
30
0.2
0.9099
0.9346
Prob50z
10
50
1
0.8739
0.8755
Prob2c
8
50
0.2
0.8796
0.8914
Note: a: Performance is average ratio to solution found by the MultiplePass
Minimum Waste Heuristic.
We observed that if the MultiplePass Minimum Waste Heuristic is unable to generate a
feasible solution, the Modified Ahn & Hyun heuristic and the problem space genetic algorithm
cannot locate a feasible solution. Thus our results arc reported only for those problems where
feasible schedules were created.
The genetic algorithm is able to outperform the MultiplePass Minimum Waste heuristic on
total flowtime at the cost of increased computational time, as a full 2000individual run lasts 256
seconds on average. (All computations performed on a 386 personal computer.) This is due to
the effort of decoding the long bit strings into deadlines and the complexity of using one pass of
the Minimum Waste algorithm to evaluate the individual. This computational effort is substantial
103
compared to that of running the MultiplePass Minimum Waste and Modified Ahn & Hyun
heuristics. The heuristics need less than one second to find a good solution to a problem. A
faster computer, however, would be able to reduce the compulation time necessary for the genetic
algorithm. Still, these results show that genetic algorithms that search the problem space can find
very good solutions to scheduling problems.
The improvement in total flowtime of the solutions that the genetic algorithm can find is a
result of two things: the multiple sampling of the search space and the evolutionary process. This
leads to the following question: Arc the genetic characteristics of the search a significant factor?
We answer this question by changing the genetic algorithm so that each individual is a
completely new one. Instead of choosing parents and creating offspring, we create new
individuals by again mutating the dummy individual. This is a random sampling approach. (See
Table 3.9.) These results imply that even though the same number of individuals are evaluated,
the random sampling performs well but docs not generate the same quality of solutions that the
genetic algorithm docs. Thus, we feel that the evolutionary process docs contribute significantly
to the improvement in total fiowtime.
Table 3.9. Total flowtime performance of random sampling on problems where a feasible
was found.
Problem
Set
Problems
Jobs
k
Performance* of Random Sampling
NewproM
10
30
1
0.9435
Prob2a
4
30
0.2
0.9545
Prob50z
10
50
1
0.9728
Prob2c
8
50
0.2
1.0165
Note: a: Performance is average ratio to solution found by the MultiplePass
Minimum Waste Heuristic.
In conclusion, the genetic algorithm can find solutions with low total flowtimcs. This good
performance is due to the multiple sampling of the problem space (since more than one
neighborhood can be searched at once); the use of the Minimum Waste algorithm to create
KM
solutions from adjusted deadlines; and the ability of the genetic algorithm to combine the best
characteristics of the points in the initial population.
3.2.7 Conclusions
This portion of the work has two contributions: it introduces an extended heuristic for the
dual criteria class scheduling problem that we call CFTS, and it describes a problem space
genetic algorithm used to find good solutions. The problem is to minimize the total flowtime
subject to deadline constraints. In this section we present a multiplepass heuristic for finding
good solutions and discuss problem space and the genetic algorithm. Finally, we describe our
experimental results, in which we compared the genetic algorithm to some heuristic approaches.
From these results we make the following conclusions:
The MultiplePass Minimum Waste heuristic performs well at minimizing the total
flowtime of CFTS. Though not an exact procedure, it is usually able to find feasible, highquality
solutions.
A genetic algorithm that searches a problem space of the Minimum Waste algorithm for
CFTS can find solutions with lower total flowtime. This genetic algorithm includes a penalty
function for infeasible points that increases the cost of tardiness as the search progresses. In
addition, it produces slightly better solutions than another procedure modified for this problem.
3.3 Class Scheduling with Release and Due Dates
In this section, we study the onemachine class scheduling problem of minimizing the
number of tardy jobs. Moreover, some of the jobs have nonzero release dates. We describe an
extended heuristic developed for this problem and a genetic algorithm used to find good
solutions. We also discuss an extension of this problem to the question of minimizing tardiness
with minimum number of tardy jobs.
105
3.3.1 Introduction
The class scheduling problem studied in this section is to schedule a set of jobs, where
some jobs have nonzero release dates, in order to minimize the number of tardy jobs. This
problem is motivated by the semiconductor test area. Since postassembly testing is the last stage
in semiconductor manufacturing, meeting a job's due date is a very important objective for the
manager of a test facility. The consideration of release dates is an attempt to model the look
bchind situation that exists in the job shop, where the scheduling of a machine (the bottleneck, for
instance) may be improved by including information about the jobs that arc arriving soon.
This problem, like most class scheduling problems, is a difficult case. Since even finding a
schedule with no tardy jobs is an NPcomplctc problem, exact algorithms to solve our problem in
polynomial lime do not exist. Thus, we are motivated to try different heuristics and searches.
Our approach was to modify an existing algorithm to include class setups and sec how such
an algorithm performs on this problem. Since the heuristic was not guaranteed to find good
solutions, we also investigated a genetic algorithm. Thus, this research presents contributions in
the extension of class scheduling problems to include a problem that has not been previously
investigated and the use of both genetic algorithms and problem spaces to include the search for
good solutions to class scheduling problems.
3.3.2 Literature Review
In this section we will mention some of the most relevant research on class scheduling and
on the problem of minimizing the number of tardy jobs in the presence of release dates. A full
discussion can be found in Chapter 2.
Bruno and Downey (1978) prove that, for general class scheduling problems, the problem
of finding a schedule with no tardy jobs is NPcomplctc. Monma and Potts (1989) prove that
class scheduling to minimize the number of tardy jobs is an NPcomplcte problem.
106
As discussed in Chapter 2, the onemachine problem of minimizing the number of tardy
jobs when some have nonzero release dates (1 / rj/ X Up is a strongly NPcomplctc problem
(Lawler, 1982). A restricted version of the problem has been considered by Rise, Ibaraki, and
Mine (1978), who solve the problem optimally if the release and due dates match (rj < rÂ¿ implies
dj < dÂ¡Â¡). They present an 0(Â«2) algorithm (Kisc's algorithm, described in Section 3.3.4) for this
ease.
3.3.3 Notation and Problem Formulation
We will use the basic notation introduced in Section 3.2.3. For CSRDD, each job Jj has a
release date rj and a due date dj. For a given schedule, Uj = 1 if Cj > dj and 0 otherwise. The
problem is to find a sequence that minimizes X Uj subject to the constraint that Cj > rj + pj.
An NPcomplcte problem, CSRDD is unstudied in the literature on class scheduling. In
order to simplify the problem, it is assumed that the release and due dates match; that is, there
exists an ordering where the jobs are simultaneously in Earliest Release Date (ERD) order and in
Earliest Due Date (EDD) order. Our primary heuristic for CSRDD extends Riseâ€™s algorithm for
the problem without setups to form a heuristic for finding good solutions.
3.3.4 Heuristics
In this section we will describe a number of heuristics; Kisc's algorithm for the problem
without class setups, our extension of this algorithm, and other heuristics used for testing
purposes.
Rise's algorithm. Rise's algorithm orders the jobs by their release and due dates (a non
ambiguous ordering since the dates must match). The algorithm is an extension of the Moorc
Hodgson algorithm (Moore, 1968) for minimizing the number of late jobs. Each job is scheduled
after the partial schedule of ontime jobs while maintaining release date availability. If the new
job is tardy, the algorithm searches the ontime jobs for the job whose removal leaves the shortest
schedule of ontime jobs. The removed job is made tardy and will be processed with the other
107
tardy jobs after the feasible jobs. In this manner, the algorithm finds the largest subset of the jobs
that can be delivered ontime. These jobs are scheduled in order of their release and due dates.
The search subalgorithm has effort that is linear in the number of jobs in the partial schedule.
Since the subalgorithm may be performed up to n times, the total effort of Kisc's algorithm is
0(Â«2).
Kisc's heuristic is not optimal for CSRDD, although it can be modified to include setup
times. Take the following example:
Example 33.
j rj pj dj
10 5 6 1
204 13 2
3 5 5 14 2
462 15 1
^01 = ^02 = !â€¢ ^12 = 1 S21 = 4.
The optimal sequence is [ Jl 74 72 73 ]. with C] = 6, C4 = 8, C2 = 13, and C3 = 18, which has
one tardy job. Kiscâ€™s algorithm adds 73, which is tardy after 7] and 72, and the subalgorithm
makes 7] a tardy job. When 74 is added to the schedule after J2 and 73, it also is tardy, for a total
of two tardy jobs.
Kise extension. Our algorithm for CSRDD extends Kisc's algorithm by considering two
options when adding a new job to a partial schedule: we can place the job in a position after all of
the ontime jobs or in a position after the last ontime job from the same class (if there is one). In
either case, if an ontime job becomes tardy, we make tardy the job whose removal creates the
shortest partial schedule of ontime jobs. We then choose between the two partial schedules
created, selecting the new partial schedule with the smaller number of late jobs (or smaller
makespan if they tic) as the incumbent before trying to schedule the next job.
Intuitively, it appears that the extended algorithm should outperfomi the Kise algorithm,
since it includes an additional scheduling choice. Due to the complexity of the problem,
however, this is not guaranteed. The following problem is one counterexample:
108
Example 3.4.
2
3
4
0 3 5 2
0 3 13 1
6 3 14 2
14 3 17 2
Ã0i â€” 52i â€” 1  A'o2  ^12 ~ 2.
ITie optimal sequence is  Jx J2J2J4 ). with Cx = 5, C2 = 9, C3 = 14, and C4 = 17, with none
tardy. Kise's algorithm will construct this schedule. In the proposed algorithm, the addition of y3
to [ 7) J2 ] creates a partial schedule with no tardy jobs and a makespan of 14. The scheduling of
73 after 7, and before y2crcates a makespan of 13 (Cj = 5, C3 = 9, C2= 13), so the sequence [ J]J3
J2 ] replaces [ J\J2J2 ]. When we add J4 after y2, y4 is tardy (C4 = 18), and the ontime jobs
complete at time 13; when J4 is scheduled before J2, J2 is tardy (C4= 17, C2= 21). The algorithm
thus yields [ JXJ3 J2J4 ], which is not an optimal schedule.
Tardiness rules. We also tested two heuristics based upon the R & M procedure of
Rachamadugu and Morton (1982) and used for reducing weighted tardiness in Morton and
Ramnath (1992). These are primarily class scheduling extensions.
RM: A dispatching rule where the priority of a job at time t is based upon the weight, the
processing time, and the slack of the job:
RM = Wj / pj * exp(  Sj+ /k* pavg),
where RM is the job priority, wj the job weight, pj the processing time, Sj+ the slack max (0, dj 
t  pj], k a predefined constant, and pavg the average processing time of the jobs in the queue. In
this formulation, jobs with higher weights, shorter processing times, and less slack will be
scheduled first. According to Ramnath and Morton, the constant k is normally set to 2.
XRM: The xdispatch (lookbehind) version of the RM rule includes jobs that will be
arriving soon in an extended queue. That is, they arrive before the completion time of the
shortest job already waiting. The RM priority is discounted by an amount that depends upon the
arrival time:
109
XRM = RM * (1  (1.3 + p)*(rj  t)+/pmin).
where XRM is the job priority, p the utilization factor, rj the arrival time of the job, and pmiAI the
smallest processing time among jobs currently available. Morion and Ramnath (1992) claim that
this procedure reduces weighted tardiness by 40% over the standard RM rule.
These two priorities arc used as dynamic dispatching rules. At a time r, the job with the
highest priority is scheduled next. For our class scheduling problem, we redefine the components
to include the setup times, but otherwise we use the same formulas. This works well for the
objective of weighted tardiness, but for our objective (minimizing the number of tardy jobs), we
would like to postpone the processing of the tardy jobs in order to concentrate on the onlime
jobs. Thus, when we schedule a job that will be tardy, we look for the job whose removal will
result in a shorter schedule with no tardy job and we remove that.
For our class scheduling problem, we include in the processing time the class setup
necessary to process a job and modify the release date by the same amount. That is, if the job
completing at time t is in GÂ¡ and Jj is in GÂ¿, we add to pj and subtract from rj (for the X
RM calculation). We use p = 1 and k 2 and wj = 1 for all Jj.
3.3.5 Analysis of the Heuristic
In this section we discuss the computational effort necessary to perform the extended Kisc
heuristic and the worst ease error of this heuristic. A pseudocode presentation of the heuristic
and its subalgorithm can be found in the Appendix.
It is obvious that the extended version of Rise's heuristic takes more effort than Rise's
algorithm. However, the effort of the algorithm is still O(n2). When adding the next job to a
partial schedule, there are two positions for the new job. The algorithm would take at most O(n)
effort to find the last job from the same class as the new job, to insert the new job, and to
determine which (if any) jobs arc now tardy. If there is a tardy job, a pass of Rise's subalgorithm
must be performed to determine which job to remove. This is also O(n). For the other position,
there is 0(Â«) effort in adding the new job to the end of the partial schedule and performing a pass
no
of the subalgorithm. Thus, the total effort of adding the new job is O(Â«), and since n jobs must be
scheduled, the total etfort of the extended Kise heuristic is OOi2).
Since CSRDD is strongly NPcomplctc, there is no optimal polynomial or pseudoÂ¬
polynomial algorithm. Since our extended Kise heuristic is not guaranteed to find an optimal
solution, we need to look into the worstease error bound. In the following we describe two
families of instances for CSRDD where the extended Kise heuristic cannot find good solutions.
While the first of these examples prove that the heuristic can perform arbitrarily badly, the
examples will also provide problem instances that we can use for testing the performance of the
genetic algorithm. They are especially good for this since we know the optimal solution in
advance.
Example 3.5. In this case, the extended Kise heuristic finds n  1 out of Â«jobs tardy when the
optimal has only two tardy jobs. There are n jobs where y, is in G] and J2 J n are in G2, and
the jobs have the following characteristics:
Jj Ji J2 Jn
rj 0 1
Pj 1 1
dj 1 n
The class setups are as follows:
G; Gi Gi
s0i 0 2
sli  n
s2i 0
The optimal sequence (for n > 2) begins with the n  1 jobs in G2. The last job ends at 1 +
(n  1) = n, so they arc all ontime. Since J\ is scheduled after this, it is tardy. The Kise and Kisc
extension algorithms start by scheduling J{ first. This job completes at time 1. When the first job
from G2 is scheduled to form [ J{ J2 ], the job completes atl+Â«+l=Â« + 2 and is therefore
tardy. Removing J] yields a partial schedule that ends at time 2 and is thus longer than the partial
Ill
schedule consisting of just Jj. Thus72 is made tardy. This continues for all of the jobs from G2,
and they arc all forced to be tardy, for n  1 tardy jobs.
Example 3.6. In this case the optimal solution has no lardy jobs and the extended Kise heuristic
finds n/3 tardy jobs. We construct the problem instance in the following way: choose a nonÂ¬
negative integer k. Let n = 3(k + 1). Let m = 3. For i = 0,...,k, construct three jobs, J31+] in
G\,J3i+2 G2, and J2,1+3, in G3, with the following job characteristics, where 0 < e < 1 and
0 < 8 < 1:
Jj
^3i+l
J3i+2
J3i+3
rj
3i
3i
3i
Pj
1e
1+e
1
dJ
3i+2
3Ã+2+8
3i+3.
Let the class
setups be
: as follows, where s > 8:
Gi
o,
g2
g3
s0i
0
0
0
8li

s
0
s2i
0

s
s3i
1
0
.
The optimal sequence of jobs is [ J2J1 J3J5J4J6 â– â– â– ^3k+2^3k+\ J3k+3 ]â€¢ This schedule
has no inserted idle time and no setup time (see Figure 3.6).
h
J.
J3
J5
h
J3k+2
J3k+1
J3k+3
0 1+e 2 3 4+e 5 6 3k+l+e 3k+2 3k+3
Figure 3.6. Optimal Schedule.
Taking the jobs in ERD order, we schedule ./, first {C\ = 1  e). J2 is scheduled next, but C2
= Cj + i12 + p2 = 2 + s > d2 = 2 + 5. Thus, J2 is tardy, and sincep2>pu Jx remains while J2 is
now tardy. J3 is added next, with C3 = C{ + si3+ p3 = 2  e. 74 is added next, starting at time 3 >
G3 + 531 = 3  e and completing at time C4 = 4  e. (Starting /4 after makes J3 lardy.) By
repeating the above argument, it can be shown that J5 will be made tardy and J6 will follow /4.
112
Iliis process continues for all of the jobs (see Figure 3.7). Thus, the heuristic creates a schedule
where the k + 1 = n/3 jobs in G2 are tardy, while the optimal schedule has no tardy jobs. Note
that we never insert a job into the middle of a partial schedule; thus, Kise's algorithm and the
extended Kise heuristic create the same schedule.
S3I
J3k+1
J3k+3
h
J3k+2
1e 2e 3e 3 4e 3k 3k+le 3k+2e 3k+3
Figure 3.7. Heuristic Schedule.
4k+3+ke
The first of the two above cases will be useful in the proof of the following error bound
theorem. For the purposes of Theorem 3.2, we assume that the triangle equality holds for the
class setups: sag + sÂ¡jC > sac, for all a = 0,..., m, b = 1,..., m, c = 1,..., m.
Theorem 3.2. For an instance of CSRDD, if there exists a schedule in which at least one job
completes ontime, the extended Kise heuristic will create a schedule with at least one ontime
job. Moreover, there exist problem instances where the heuristic will schedule exactly one on
time job, although there exist schedules with more than one ontime job.
Proof. First, we note that the number of scheduled ontime jobs never decreases as the extended
Kise heuristic schedules new jobs. Now, consider a job Jj in class GÂ¡ that completes ontime in
some feasible schedule. Thus, (or Jj, Cj < dj. Now, Cj > rj + pj and Cj > sgÂ¡ + pj, since by the
triangle inequality stated above, if there exist any jobs before Jj, the sum of the setups before Jj is
at least sqÂ¿. Thus, dj > rj + pj, and dj > sqi + pj.
Suppose that the schedule created by the extended Rise heuristic has no ontime jobs.
Then, when the extended Rise heuristic considered Jj, no other ontime jobs were scheduled. The
heuristic started Jj as soon as possible (the maximum of rj and sqÂ¡), but Jj was tardy. Thus,
Cj = max {rj, sqÂ¡} + pj> dj, but this contradicts the result above. Thus, the heuristic creates a
schedule with at least one ontime job.
113
Thus, wc have shown that the extended Kisc heuristic will schedule at least one ontime
job. This bound is tight, as our discussion of Example 3.5 shows that there exist problems for
which the heuristic will schedule exactly one ontime job although the optimal schedule has more
than one ontime job.
3.3.6 The Genetic Algorithm
In this section we present the problem space and discuss the details of the genetic algorithm
we used to find good solutions for CSRDD.
Problem space. In Chapter 2 we described the ideas of alternative search spaces. In this
section we present the problem space that we searched in order to find good solutions for
CSRDD.
We defined a problem space for CSRDD in the following manner: Given a problem p in
problem space, a heuristic h is a function that creates a sequence corresponding to a solution s for
CSRDD, i.e. h(p) = s. Wc defined a problem as a vector of job release dates, using a pass of
Rise's algorithm to create a sequence of jobs by considering the jobs in order of their new release
dates instead of the order imposed by the matching release and due dates. The actual release
dates are used in determining the schedule, however. Note that all solutions for CSRDD
(including the optimal ones) that schedule all tardy jobs last arc in the range of h. Following are
two examples of applying this heuristic to different vectors of deadlines for the problem in
Example 3.3:
Hcuristic(problem) = solution:
Rise (0, 1,5,6 ) = [J2JÂ¡ J i h ]. two jobs tardy.
Rise (0, 8, 5, 4) = [ 7, J4 J3 J2 ], one job tardy.
A genetic algorithm for CSRDD. In this work we developed a genetic algorithm based on
the ideas presented in Davis, 1991, namely, steadystate reproduction without duplicates,
114
fitnesses measured by linear normalization, a uniform crossover operation, operation selection,
and interpolated parameters.
Steadystate reproduction adds new individuals a few at a time, whereas the traditional
method replaces the entire population with a new generation. Steadystate reproduction
(attributed to Whitley, 1988, and Syswcrda, 1989) is used to ensure that good individuals (and
their good characteristics) survive. Steadystate reproduction without duplicates prevents
children that are identical (in chromosome values) to a current member from joining the
population.
Linear normalization is a fitness technique that creates fitness values by ordering the
individuals in a population by their objective function evaluation. The assignment of fitnesses
begins with a constant value and decreases the fitness linearly as it considers each individual in
order. This technique prevents a super individual from dominating the population at the
beginning of a run and yet differentiates between the various very good individuals that exist near
the end of a run. Of course the values of the original constant and the decrement parameter
influence the extent of these two phenomena.
Uniform crossover, an operator first described by Syswcrda (1989), is a way to combine
characteristics in ways that standard one or twopoint crossovers cannot. In a uniform crossover,
two parents are selected and two children produced. Each bit position is considered
independently and the parent that contributes the bit value for that position in the first child is
determined randomly. The second child receives the value for that position from the other parent.
While uniform crossover can destroy a good characteristic by mixing it with a bad string, it can
also combine features that are widely dispersed across the string.
Since onepoint crossover remains a good operator, however, we used both types of
crossover in our genetic algorithm. Before creating a child, we randomly decide on which
operator we wish to perform: uniform crossover, onepoint crossover, or mutation. Each operator
has an operator fitness and the probability of that operator being selected is proportional to that
fitness. If one of the crossovers is selected, two parents are selected and two children are created.
115
If the mutation operator is selected, one parent is selected and a child created by forcing each bit
to undergo a mutation with some small probability. The child or children created arc checked
against the current population for duplication, evaluated, and iaserted into the population,
replacing the worst members of the population. The new population is then reordered and new
fitnesses created using the linear normalization technique.
As we did with the CFTS genetic algorithm, the initial population included one individual
(the dummy, or seed) that was created form the actual problem data. The remaining individuals
in the initial population were constructed by mutating the bits in lite initial (dummy)
chromosome. This initial mutation rate was set at 0.05 per bit.
We interpolate the following parameters over the course of the run: the decrement for linear
normalization is increased, and the operator fitnesses are changed to favor crossovers and
discourage mutations.
Table 3.10. Parameter values for genetic algorithm.
Population size: 100
Linear normalization decrement: 0.2 to 1.2
Probability of selecting mutation operator: 35% to 25%
Probability of selecting uniform crossover operator: 40% to 30%
Probability of selecting onepoint crossover operator: 25% to 45%
Mutation rate: 0.02 per bit
3.3.7 Empirical Tests and Results
In this section we describe the empirical tests conducted to test how well the heuristics and
genetic algorithm perform.
Problem generation. We created problem sets using the due date assignment method of
Hariri and Potts (1989). The release dates can also be created using a similar method. In this
method, setup times and processing times of the jobs arc determined first from random variables.
Then, an estimate of the maximum completion time is made by simply summing the job
116
processing limes. This makespan is used lo define a specific ranges for the due dates and a range
for the release dales. The due dates and release dates are sorted and the matching pairs arc given
to the jobs. By changing the parameters governing the definition of the ranges, problem sets with
different characteristics can be created.
Different sets of 10 problems were created. The number of jobs per problem ranged from
15 lo 100. The processing times ranged from 1 to 20 and the class setup times from 0 to 9. The
jobs were randomly placed into a number of job classes, depending on problem size. The release
dates and due dates were taken from a uniform distribution. The upper and lower bounds of this
distribution were proportional to the sum of the job processing times. The proportions changed
for each problem set. (See Table 3.11.)
Table 3.11. Data on problem sets.
Set
Problems
Jobs
Classes
Release date
Due date
range
range
RH301
10
30
4
00.4
0.4  0.6
RH302
10
30
4
00.4
0.6 1.0
RH151
10
15
4
00.4
0.4  0.6
RH501
10
50
5
00.4
0.4  0.6
RH303
10
30
4
00.4
0.2 1.0
RH304
10
30
4
00.6
0.2  1.0
RHMIXED2 10
30
4
00.4
0.4  0.6
RHMIXED3 10
30
4
0 1.0
0.4  1.2
RHMIXED4 10
30
8
0 1.0
0.4  1.2
Results. After numerical testing on these forty problems, it appears that the Kise and Rise
extension heuristics and the R & M heuristics arc fairly equal. A number of other heuristics were
unable to find as many ontime jobs. Note that the lower bound was derived by using Rise's
algorithm while ignoring all setup times.
117
Table 3.12. Average performance of heuristics.
Set
Jobs
Lower
Bound
Kisc
Extended
Kise
RM
XRM
KH301
30
8.3
12.0
10.7
11.6
11.6
KH302
30
1.1
4.9
2.7
4.2
4.3
KH151
15
4.8
6.8
6.4
6.6
6.7
KH501
50
12.6
19.5
16.0
17.9
17.9
Note: Performance is the average number of tardy jobs found by that heuristic on the ten
problems in each problem set.
In addition to the problems where the release and due dates matched, we created a set of
30job problems where no such correspondence existed. These ten problems had the same
characteristics as the problems in the set KH301. On these problems the R & M heuristic
performed slightly better than the extended Kise heuristic.
Table 3.13. Average performance of heuristics, nonmatching release and due dates.
Set Jobs
Kise
Extended
Kise
RM
XRM
KHMIXED2 30
7.7
5.3
4.5
4.7
KHM1XED3 30
11.1
10.2
8.9
8.7
KHMIXED4 30
10.0
9.2
7.2
7.0
Notes: Performance is the average number of tardy jobs found by that heuristic on the ten
problems in each problem set.
Since the heuristics were finding good solutions, we decided to test the problem space
genetic algorithm on 18 and 30job problems that could not be solved well by the heuristics.
These problems were instances of the problem described above in Example 3.6, where the
extended Kisc heuristic creates a schedule where a third of the jobs are tardy, although an optimal
schedule has no tardy jobs. These results show that the problem space genetic algorithm is able
to find good solutions. (In some of the following graphs we present our results with the number
of ontime jobs, since the objective of the genetic algorithm was to maximize the number of on
time jobs.)
118
Table 3.14. Average performance of heuristics, hard problems.
Set
Jobs
Rise Extended
Rise
Genetic Algorithm*3
1000 3000
Hard A
18
6.0
6.0
0.6
_
Hard B
30
10.0
10.0
4.5
1.5
Notes: Performance is the average number of TARDY jobs found by that heuristic on the five
problems in each problem set.
a: Results reported at 1000 and 3000 individuals.
Figure 3.8. 18job problems, Average number of ONTIME jobs.
119
Table 3.15. 18job problems, number of ONTIME jobs, 10 runs of 1000:
Number of Problem Average
Individuals
Created
1
2
3
4
5
0
13.7
13.9
13.5
13.6
13.4
13.6
100
13.9
14.1
14.1
14.0
13.9
14.0
200
14.7
14.5
14.7
14.8
14.9
14.7
300
15.4
15.3
15.5
15.5
15.3
15.4
400
16.2
16.1
16.1
16.5
15.8
16.1
500
16.5
16.5
16.9
16.6
16.5
16.6
600
16.8
16.8
17.1
17.0
16.8
16.9
700
16.9
17.3
17.3
17.3
17.0
17.2
800
17.0
17.5
17.3
17.3
17.5
17.3
900
17.0
17.5
17.4
17.3
17.5
17.3
1000
17.1
17.5
17.5
17.3
17.5
17.4
Figure 3.9. 30job problems, average number of ONTIME jobs.
120
Table 3.16. 30job problems, number of ONTIME jobs; 3 runs of 3000
Number of
Individuals
Created
Problem
1
2
3
4
5
Avcra
0
20.3
20.3
20.7
21.3
20.7
20.67
100
21.3
21.0
21.0
21.7
21.3
21.27
200
21.7
21.7
21.0
22.0
21.7
21.60
300
22.0
22.0
21.7
22.0
22.0
21.93
400
22.7
22.3
22.3
22.7
22.3
22.47
500
23.3
23.0
22.7
23.0
23.3
23.07
600
24.0
24.0
23.0
23.3
23.7
23.60
700
24.0
24.0
24.0
23.7
23.7
23.87
800
25.0
24.3
24.7
24.7
24.7
24.67
900
25.0
24.7
25.0
24.7
25.0
24.87
1000
25.7
25.0
26.3
25.3
25.3
25.53
1100
26.0
26.0
26.7
26.3
25.7
26.13
1200
26.3
26.0
27.0
26.7
26.0
26.40
1300
26.7
26.7
27.7
26.7
26.3
26.80
1400
27.0
26.7
28.0
26.7
26.7
27.00
1500
27.3
27.0
28.0
27.0
26.7
27.20
1600
27.3
27.3
28.0
27.3
27.0
27.40
1700
27.7
27.3
28.3
27.3
27.0
27.53
1800
28.0
27.7
28.3
27.3
27.3
27.73
1900
28.0
27.7
28.3
28.0
27.3
27.87
2000
28.0
27.7
28.3
28.3
27.7
28.00
2100
28.3
27.7
28.3
28.3
27.7
28.07
2200
28.3
27.7
28.3
28.3
28.0
28.13
2300
28.3
27.7
28.3
28.3
28.3
28.20
2400
28.3
28.3
28.3
28.3
28.3
28.33
2500
28.3
28.3
28.3
28.7
28.3
28.40
2600
28.3
28.3
28.3
28.7
28.3
28.40
2700
28.3
28.3
28.7
28.7
28.3
28.47
2800
28.3
28.3
28.7
28.7
28.3
28.47
2900
28.3
28.3
28.7
28.7
28.3
28.47
3000
28.3
28.7
28.7
28.7
28.3
28.53
3.3.8 Extension to Minimizing Tardiness
In addition to simply minimzing the number of tardy jobs, it is often an objective of
schedulers to minimize the total tardiness of the tardy jobs. To this end, we study the problem of
121
minimizing the total tardiness subject to a constraint on the number of tardy jobs, since the
minimization of total tardiness usually leads to schedules where many jobs are tardy and arc tardy
by a small amount. Since finding the minimum number of tardy jobs is an NPcomplete problem,
we use our heuristic to set the value of the constraint. Then, within that limitation, we minimize
the total tardiness.
We develop some further extensions of Kise's algorithm that create a set of tardy jobs and
then insert the tardy jobs into the schedule of onlime jobs in order to reduce the total tardiness.
We also use our genetic algorithm to search for schedules with low number of tardy jobs and low
total tardiness.
The problem of minimizing tardiness subject to a minimal number of tardy jobs has been
considered for the problem without class setups (or release dates) by Vairaktarakis and Lee
(1993), who develop an algorithm to optimally schedule a given set of tardy jobs and an efficient
branchandbound technique to find the optimal tardy set. Other researchers have studied dual
criteria problems with the same primary objective. Emmons (1975) considered the problem of
minimizing total flowtime subject to minimum number of tardy jobs, using a branchandbound
algorithm to find optimal solutions. Shanthikumar (1983) examined the problem of minimizing
the maximum lateness subject to minimum number of tardy jobs, also using a branchandbound
algorithm.
Our problem, which includes both class scheduling and nonzero release dates, is an NP
complete problem. The tardiness heuristics that we use to find good solutions use the extended
Kise heuristic to determine a set of tardy jobs. The heuristics also use the sequence of ontime
jobs created by the extended Kise heuristic, pushing the jobs to the right, starting them as late as
possible, and attempting to insert the tardy jobs into the gaps in this schedule.
The first heuristic (7j) orders the tardy jobs by their release dales and attempts to
interleave the two sequences of jobs, scheduling tardy jobs to start as soon as possible while
maintaining the feasibility of each ontime job (whose completion time is constrained by the due
date).
122
The second heuristic (T2) considers every tardy job as a candidate for a gap between the
partial schedule and the next ontime job, selecting the tardy job that yields te earliest start time
for the next ontime job. Any remaining tardy jobs arc scheduled by their release dates.
The third heuristic (T3) was a modification of the second that scheduled the remaining
tardy jobs using a version of the Minimum Waste heuristic (see Section 3.2) that didn't consider
deadlines (since all jobs are tardy). This heuristic has been shown to perform well on flowtime
criteria. We use this because minimizing the total tardiness of the set of tardy jobs is identical to
minimizing the total fiowtime of those jobs.
Due to the nonoptimal nature of the extended Kise heuristic, it is possible that the
tardiness heuristics will often be able to schedule a lardy job so that it finishes ontime (reducing
its tardiness to zero). However, the primary objective of these heuristics is to reduce tardiness,
not reduce the number of tardy jobs.
Results. We tested the heuristics on three problem sets, selected because the variance of
the due dates meant that the schedules were more likely to have gaps in which to insert tardy jobs.
Each solution technique was measured by the average total tardiness found by that heuristic on
the ten problems in each problem set and the percent deviation of this average from the average
tardiness found by the extended Kise heuristic. The performance of the genetic algorithm on
each problem is the average of ten trials of one thousand new individuals.
Table 3.17. Data on new problem sets.
Set
Problems
Jobs
Classes
Release date
Due date
range
range
KH303
10
30
4
00.4
0.2 1.0
KH304
10
30
4
00.6
0.2 1.0
123
Table 3.18. Average performance of heuristics.
Set
Extended
Kisc
T,
%
t2
%
t3
%
KH302
236.4
237.3
0.38
234.7
0.72
231.5
2.07
KH303
758.3
756.6
0.22
718.4
5.26
704.1
7.15
KH304
920.0
869.0
5.54
859.4
6.59
844.0
8.26
Note: Performance is the average total tardiness and the percent improvement.
Table 3.19. Average performance of heuristics.
Set
Extended
G.A.
%
G.A.
%
Kisc
3000
KH302
236.4
229.0
3.13
KH303
758.3
884.3
16.62
695.3
8.30
KH304
920.0
771.9
16.10
Note: Performance is the average total tardiness and the percent deviation.
3.3.9 Conclusions
In this section we have introduced a class scheduling problem that we call CSRDD. The
problem is to minimize the number of tardy jobs where some jobs have nonzero release dates,
and we assume that the release and due dates match. We have described a heuristic developed to
find good solutions. We have discussed a problem space and a genetic algorithm to search this
space. We have described our experimental results, from which we make the following
conclusions:
Our extended Kisc heuristic can find good solutions for instances of CSRDD. It can do this
by considering the class setups in both the subalgorithm that decides on which job to make tardy
when a new job is added and the option to insert the new job into the middle of the partial
schedule in order to reduce the number of setups.
124
When the extended Kisc heunstic cannot find good solutions, our problem space genetic
algorithm can. By searching the problem space near the original problem, it can discover
solutions that are improvements on the schedule constructed by Kisc's algorithm.
Also in this section we discussed an extension of this problem to the problem of
minimizing total tardiness in the presence of a constraint on the number of tardy jobs.
3.4 Flowtime with Setups and Release Dates
The third of the class scheduling problems that we consider has jobs with nonzero release
dates, and the objective is to minimize the total flowtimc. We develop some lower bounds and
dominance properties and examine some heuristics for finding good solutions to the problem.
We discuss a problem space genetic algorithm that can improve the performance of a lookbehind
dispatching rule. For this problem we also developed a search technique for comparison
purposes.
3.4,1 Introduction
This problem, like the others we have examined, is motivated by considering the
scheduling of a semiconductor lest area. We have class setups, arriving jobs, and an objective
that mirrors the goal of management to minimize workinprocess inventory.
We will examine the problem of minimizing total flowtimc when the jobs have nonzero
release dates. This strongly NPcomplcte problem is a lookbehind scheduling model, where we
are interested in scheduling a machine by considering the jobs that will be arriving at the machine
soon.
In addition to a lookbehind scheduling rule, we will consider the use of a problem space
genetic algorithm (similar to those developed for CFTS and CSRDD) that can improve the
performance of this rule by adjusting the parameters of the rule. Due to the structure of the
FTSRD problem, we will also use a decomposition heuristic as a means of comparing solution
125
quality. The decomposition heuristic is a search technique that considers a sequence of
subproblcms at each move.
In the next section we will introduce the notation and problem formulation. After that we
will mention some of the previous research on the scheduling problem under consideration (a
review of the literature on class scheduling and genetic algorithms can be found in Chapter 2),
examine some lower bounds and dominance properties that can be used in a branchandbound
technique, discuss our heuristics (including sequencing rules and the genetic algorithm), and
report on the experimental results.
t.4.2 Notation and Pmhlem Formulation
We use the same notation as that for the CFTS and CSRDD problems (Sections 3.2.3 and
3.3.3), except that the jobs do not have due dates. The FTSRD problem is to find a sequence that
minimizes X Cy subject to the constraint that Cj > rj + pj. We will see that FTSRD is NP
complcte and has not been previously considered in the literature on class scheduling.
We make two assumptions in the analysis of the problem. One, a class setup for a job can
begin before the job is available. Two, although all of the release dates arc known, the processing
for a job cannot begin until the release date. These conditions are motivated by our consideration
of the onemachine problem as part of the job shop scheduling problem.
3.4,3 Background
The onemachine problem of minimizing total flowtimc when the jobs have nonzero
release dates has been previously studied in the case where no sequencedependent setups are
present The problem is simple if the jobs are preemptive, that is, if a job that has begun
processing can be interrupted by another job and then resumed later. In this case, the optimal
policy at the next decision point is to schedule the job with the shortest remaining time. The set
of decision points includes all job release times and completion times. If the jobs are non
126
preemptive and have no sequencedependent setup times, the problem (denoted by 1 / rj / Â£ CJ) is
a strongly NPcompletc question, as shown by Lenstra ct al. (1977).
Among class scheduling problems, minimizing the total flowtimc is a strongly NP
complctc problem (Monma and Potts, 1989). Thus, it can be seen that our problem, which is to
minimize total flowtime with class setups and release dates, is also a strongly NPcomplcte task.
Many researchers (see Chapter 2) have considered the problem of minimizing the total
tlowtimc subject to job release dates, 1 / rjt X Cj. Branchandbound algorithms have been the
most popular approaches. None of these researchers included sequencedependent setup times in
their analysis.
3.4,4 Solution Techniques
In this section we will discuss a number of different approaches to solving the FTSRD
problem. These include branchandbound searches, dispatching rules, a decomposition heuristic,
and a genetic algorithm. A job arrives at rj and is available at time t if t > rj. Recall that we
assume that a class setup can be performed before the associated job becomes available.
Branchandbound. A straightforward branchandbound algorithm can be developed for
this scheduling problem. The branchandbound procedure finds the optimal solution by
searching a tree that consists of every possible permutation. Each node of the tree consists of a
partial schedule of jobs processed in that order and as soon as possible. (The root node consists
of an empty schedule.) Unscheduled jobs will be appended to the schedule for a node.
Branching occurs by adding a child node for each unscheduled job. A lower bound can be
calculated for each node, and the search moves to the child with the lowest lower bound.
Local dominance properties. The number of nodes to be examined can be reduced by
applying dominance properties that identify nodes that cannot lead to optimal solutions. Local
dominance properties claim that a node a is dominated by another node P under certain
conditions. Thus, we do not need to branch on node a because we can find a better solution by
branching on p and searching its dcscendents.
127
VVc note here thai matching processing times and release dales do not imply a dominance
property. Even if pj < pt implies rj < rÂ¡ (or even if the release dates arc identical) within each
class, we do not have an optimal order for the jobs in each class. Consider the following
instance:
Jj Pj rj
G, y, 3 0
y2 4 o
y3 4 o
g2 y4 i 6
J5 1 6
J6 1 6
% =521 = 502 = v12 = 5
The optimal solution is [72, J4, J5, J6,JX,J2], with a total flowtimc of 59. The best solution
in which C7, is in SPT order arc [ylt JA, J5, J6, J2, y3], with a total flowtime of 60 (sec Figure
3.10).
X
h
X
h
J6
X
J3
5 7 8 9 13 17
X
J, X
J4
h
h
X
J2 J3
4 7 8 9 14 18
Figure 3.10. Optimal schedule and best SPT schedule.
A number of dominance properties have been suggested for the 1 / rj/X Cj. We have
modified the properties of Dcssouky and Dcogun (1981) for use in our search. We make the
assumption here that the class setups satisfy the triangle inequality: sac < saÂ¡j + s^c for all Ga,
Gb Gc
128
Suppose that we are at a node in the search tree with a partial schedule a that ends at time t
with a job in class Gc and a set K of unscheduled jobs. If Jj is in K and in G^ define the
following earliest start time:
tj = max {/ + sc[j, rj\.
Note that if t + scf} < rj, the necessary class setup can be completed before the job arrives.
At this node we can use any of the following dominance properties:
Property 3.1. Let JÂ¡ be the shortest job in K. If Jj is in GÂ¿ and JÂ¿ is in Ga, the node (a, Jp
is dominated by (o, Jj) if tÂ¡ + s^ < tj.
Property 3.2. The node (a, Jp is dominated by (a, Jj) if + pÂ¡ + s^ < rj.
Property 3.3. If Jj is in GÂ¿ and J, is in Ga, the node (a, Jp is dominated by (a, Jj) if all of
the following statements arc true:
(a) tÂ¡ + pi < tj + pj,
(b) tÂ¡ + pi + sae < tj + pj + Sfre for all job classes Ge : Ge n K * {},
(c) sefr + pj < sea + pi for all Ge : Ge n K * {}, and
(d) seb + pj + sÂ¡jd < sea + pi + Sqj for all Ge :GenK* {},and Gd:GdnK* {}.
Proofs: Property 3.1. Take any schedule (a,Jj, a^,JÂ¿, 02), where there are m jobs in Oj.
If we move JÂ¡ before Jj, the new completion time of is no greater than the old completion time
of Jp, thus the flowtime of is decreased by at least (m + \)pÂ¡ (since JÂ¡ is the shortest job).
Meanwhile, we delay only the start of Jj and the m jobs in 0 j by at most pÂ¡ (since tÂ¿ + + s^ 
tj < pj). The jobs in 02 arc not delayed at all (the triangle inequality of the setups insures this).
Property 3.2. Take any schedule (o, Jj, Oj, JÂ¿, ap) We can move JÂ¡ before Jj without
delaying Jj, decreasing the flowtime of JÂ¿ and possibly that of the jobs in Â©2
Property 3.3. Take any schedule (0, Jj, Oj, JÂ¿, 02). Note that the earliest start times of./,
and Jj are before O]. Interchange JÂ¡ and Jj. The new completion time for 7, is not greater the old
completion time of Jj (by condition a). By condition b (if the first job of Oj is in Ge), the jobs in
Â©1 arc not delayed. Then, by condition c (if the last job of 0] is in Ge), the new completion time
129
for Jj is not greater than the old completion time of JÂ¿. Finally, the last condition (if the First job
of cÂ¡2 is in GÂ¿) implies that no jobs in 02 arc delayed.
In any of these cases, we can take a schedule lliat starts with (a, Jj) and Find a schedule
which starts with (c, JÂ¡Â¡) and which has less total flowtimc. Thus, it is clear that the First node is
dominated and that we do not need to search that part of the tree.
Lower bound. A branchandbound procedure needs a lower bound. A good lower bound
is important to efficiently Finding solutions. We develop two bounds: the First concentrates on the
release dates, the second on the setup times.
The First lower bound for a node completely ignores the class setups. Instead, it solves the
associated 1 / rj, preemption / Â£ Cj problem with the SRPT rule and adds the optimal flowtimc to
that of the partial schedule in the node.
The second lower bound separates the setup times from the job processing times. We
assume that each class will have exactly one remaining setup. If we further assume that this setup
will the shortest possible, then we can easily sequence the job classes to minimize the
contribution of the setup times to the total flowtimc. The unscheduled jobs are scheduled by SPT
without regard to their release dates. The two sums arc added to the flowtimc of the partial
schedule in the node.
The First lower bound should work well when the intcrarrival times are large, and the
second bound should be useful at nodes where all of the unscheduled jobs are available.
Testing. For 30job problems, the branchandbound procedure required excessive
computation time to Find an optimal. Still, improved lower bounds could be found by truncating
the branchandbound search in the following way: We prevent the search from moving below a
certain depth in the tree and take the lower bound at this depth as the objective function value. If
we continue to do this, the truncated search returns the lowest lower bound at this depth. This
will be a lower bound on the optimal solution and will be better than the lower bound computed
at the root node.
130
Dispatching rulos. The SPT rule is known to minimize the total fiowtime for 1 / / X Cj
(Smith, 1956). Since we are studying a total fiowtime problem, we are most interested in SPT
likc heuristics. We will develop a rule that considers the waste associated with a job (the waste is
the sum of the idle time and setup time incurred if the job is scheduled next). We propose the
lookbehind dispatching rule Shortest Waste, "Among the jobs with the minimum waste, schedule
the shortest one." (This rule is similar to the Minimum Waste algorithm for CFTS.)
Let us define a few relevant variables: t is the current time, the completion of the last
scheduled job; Gc is the class of the last scheduled job, and the waste of an unscheduled job Jj in
Gb is
wjâ€” max [rj1, scfj}.
Our dispatching rule can be now stated:
Shortest Waste:
Among all unscheduled Jj, select the job with the minimum wj. Break any tics
by selecting the one with the minimum pj.
Decomposition. For the FTSRD problem, we decided to implement a search heuristic for
comparison purposes. A decomposition heuristic for finding good solutions to sequencing
problems was introduced by Chambers ct al. (1992). The heuristic is a type oflocal search. It
begins with an initial sequence, and forms new sequences until it finds a local minimum. The
critical step is the decomposition of the problem into subproblcms that depend upon the current
solution. A new solution is generated by combining the optimal or nearoptimal solutions to each
subproblcm. The heuristic thus makes very good moves through the search space; only a few
moves are needed before convergence is reached.
Consider, for example, a 12job problem. Start with some initial solution to the problem.
Select the first six jobs of this solution and find a good solution to the 6job subproblem. Take
the first three jobs (in order) of this subproblem solution as the first three jobs of the new
solution. Then combine the remaining three jobs from the subproblcm and with the next three
131
jobs from the initial solution. This forms a new 6job subproblem. Continue solving
subproblems and building the new solution until all of the jobs have been considered.
We use this technique in order to find good solutions against which we could measure the
performance of our other heuristics. (We did not feel that this type of search would be as
effeetive on deadlineoriented CFTS and CSRDD problems.) We use a branchandbound
technique with an approximate dominance property to generate nearoptimal solutions to the
subproblcms. The algorithm has two parameters. We need to select m as the size of the
subproblems which we will solve; the larger the value, the better our subproblem solutions will
be (at the expense of computation time). We also select/as the number of jobs from the
subproblem solution that will be fixed into the new solution. A smaller/requires that more
subproblems be solved per step. The following steps outline the procedure.
Step 1. Set m and/. (We want/to divide n  m.) Let a be the ERD schedule.
Step 2. Take the first m jobs of a. Let it be an empty schedule.
Step 3. Solve the mjob subproblcm by branchandbound.
Step 4. Append the first/jobs of the solution to n.
Step 5. If there are any unconsidcrcd jobs in o, take the next/jobs from o, add to the
mf remaining from the subproblem, and go to Step3; else go to Step 6.
Step 6. Append the m /remaining jobs of the solution to ir. If re is a better schedule
than a, let o = it and return to Step 2; else go to Step 7.
Step 7. The solution found by the heuristic is o.
We experimented with different values of (m,f). The best results (in terms of computation
time and solution quality) were achieved with (9, 3) and (15, 5). We used our branchandbound
algorithm with only the second lower bound and only the following approximate dominance
property:
Property 3.4. Given a partial schedule a, (o, JÂ¿) dominates (a, Jp if/, and Jj are in the same
class, JÂ¿ is the shortest unscheduled job in that class, and wÂ¡ < wj.
This property is a simple extension of a previously considered dominance rule (see, for
instance, Dessouky and Dcogun, 1981). Unlike the dominance rules that we use in the full
132
branehandbound procedure, it has the advantage of being quick to check, since there are fewer
unscheduled jobs from the same class.
In the section on computational results, we will discuss how well the decomposition
heuristic performs.
A problem space genetic algorithm. In this problem, we consider the problem space
defined over the problem release dales. A point in this space is an Â«element vector of nonÂ¬
negative real numbers. When a heuristic is applied to an instance of FTSRD, it uses the actual
release dates to generate a solution. If, however, we adjust the release dates of the problem, we
can change the sequence created by the heuristic. This sequence can be evaluated as a schedule
by using the actual problem data. We can associate, therefore, with the vector of adjusted release
dates a performance value: the total flowtime of the schedule that was ereated. Moreover, we can
search the space of adjusted release dates to find good schedules. This exploration is the
objective of the problem space genetic algorithm. Our purpose is to show that the performance of
a simple heuristic can be improved with a smartandlucky search like a genetic algorithm.
We will use the Shortest Waste heuristic to convert a vector of adjusted release dates into a
sequence of jobs. The optimal solution is within the range of this heuristic: if each adjusted
release dale equals the actual start time of the job in an optimal solution, the Shortest Waste
heuristic will schedule the jobs in the optimal order, since at any time, the job with the shortest
waste will be the one with the next adjusted release dale, which is the job with the next optimal
stan time.
As we did for CFTS and CSRDD, we will use a steadystate genetic algorithm. The initial
population is formed by mutating a source individual that is the digital representation of the
actual release dates. After empirical testing on a number of problem instances, we decided on the
following parameters: The population size is 100 individuals. The four operators are uniform
crossover, onepoint crossover, small mutation, and large mutation: all have the same probability
of being selected. In the small mutation, a bit is flipped with 2% probability; in the large, the
probability increases to 50%. The algorithm uses tournament selection to identify parents. Sec
133
Davis (1991) or Goldberg (1989) for more information about lÃtese aspects of the genetic
algorithm.
Example 3.7. The following problem is used to illustrate some of the issues we have
discussed so far.
j rj pj i
10 5 1
2 0 4 2
3 5 5 2
4 6 2 1
s01=s02=l i12=^ s21=4
The ERD sequence is [J[ J2J2 J4], with a total flowtimc of 55. The EFT sequence is
[J2 Jt, with a total flowtime of 52. The Shortest Waste sequence is identical in this case. If
we adjust the release dates to (1,2, 5, 6), the Shortest Waste sequence is [JÂ¡ J4J2J3], with a
flowtime of 45.
In the branchandbound algorithm, we compute lower bounds at the root node. The first
lower bound for the entire problem is the SRPT schedule, total flowtime of 39, shown below (Jx
is preempted at time 6 by JA)\
h
h
j4
h
J3
0 4 6 8 11 16
Figure 3.11. The SRPT schedule.
The second lower bound (Longest Weighted Batch Size plus SPT) is computed as follows:
Batch sizes: bx = 2. b2 = 2.
Shortest setups: S[ = % = 1. s2 = s02 = 1.
I sÂ¡BÂ¡= 1(4)+ 1(2) = 6.
Processing times in SPT order: 2,4, 5, 5.
Completion times: 2, 6, 11, 16. X Cj = 35.
Lower bound = 35 + 6 = 41.
134
3.4,5 Empirical Testing
Problem generation. In order to test die heuristics, four sets of ten problems were created.
The characteristics of the sets arc shown below (processing, intcrarrival, and setup times
randomly selected from uniform distributions with die given ranges):
Table 3.20. Data on problem sets
Set
Problems
Jobs
Classes
Processing
Times
Intcrarrival
Times
Setup
Times
FT151
10
15
5
1,20
1,10
0,9
FT301
10
30
5
1,20
1,15
0,9
FT302
10
30
10
1,20
1,15
5,9
FT304
10
30
10
1,15
1,20
5,9
Results. In this section we will discuss how well our solution techniques performed. (See
Table 3.21.) The branchandbound could find optimal solutions on only the 15job problems.
On the 30job problems, we used the decomposition heuristic with parameters (9, 3) to quickly
generate solutions and measured the performance of other heuristics against these solutions. The
(15, 5) decomposition was much slower than the (9, 3) decomposition, since the time necessary to
solve each subproblem grew exponentially. Still, it found slightly better solutions, and the
processing time was reasonable (although it varied from problem to problem) if the heuristic
dominance property and second lower bound were used.
Table 3.21. Performance of heuristics.
Problem
Set
Shortest
Waste
(9,3)
(15,5)
Genetic
Algorithm
FT151
1.095
1.005
1.010
FT301
1.067
1.000
0.992
1.006
FT302
1.066
1.000
0.995
1.005
FT304
1.031
1.000
0.994
1.005
Notes: Performance measured against optimal solution for FT151. Against decomposition
(9,3) for 30job problems. All performances are average ratios over 10 problems.
Performance of genetic algorithms averaged over three runs of 3000 individuals.
135
Althuugh the initial lower bounds for the 30job problems were not good, we were able to
improve them using the branchandbound tree to show that the decomposition (9, 3) heuristic
was within ten percent of the optimal flowtime
The Shortest Waste heuristic found good solutions very quickly and generally performed
better than other dispatching rules. On the 15job problems, the genetic algorithm was not an
effective heuristic, since it required more computation lime than the branchandbound and could
not always find optimal solutions.
On the 30job problems, the problem space genetic algorithm found solutions better than
Shortest Waste and as good as the decomposition heuristic. The compulation time was slightly
longer for a 3000individual search than for a (15, 5) decomposition, but a 1000individual search
was much shorter and found solutions with little increase in total flowtime. The exponential
nature of genetic search is exhibited in Figure 3.12. (In other testing, we found that the genetic
algorithm was not as effective when using a simple Earliest Release Date rule to create
sequences).
All programs were run on a 386 PC. Decreases in times were achieved when the programs
were run on a 486 PC, and further decreases could be achieved on a more powerful machine.
Except for the 30job branchandbound (which we could not solve), we do not consider
processing times to be a significant obstacle. The numbers in Table 3.22 arc offered only for
comparison purposes.
Table 3.22. Typical computation times
FT302.1
Decomposition (9,3): 17.8 seconds
Decomposition (15,5): 338.67 seconds
1000individual GA: 134.62 seconds
3000individual GA: 350.37 seconds
Shortest Waste: < 0.1 seconds
FT151.1
Branchandbound: 10.71 seconds
Decomposition (9,3): 0.8 seconds
3000individual GA: 149.24 seconds
Shortest Waste: < 0.1 seconds
136
Figure 3.12. Performance of Shortest Waste Genetic Algorithm
Performance measured against decomposition (9,3).
3.4,6 Special Case
In this section we consider a particular special case of the problem which may be useful in
certain manufacturing situations. We will assume that there exist exactly two job classes and that
the job processing times are equal within each class.
Specifically, we study the following instance: pj = p for all Jj in Gx,pj=q for all Jj in G2.
Since all of the jobs in a class have identical processing times, we may order them by ERD. We
note here that all of the release times are integer.
We will describe a pseudopolynomial dynamic program to solve the problem, a special
case of the strongly NPeomplete FTSRD problem.
137
Dynamic programming. VVc can use a dynamic program lor two reasons: there arc only
two classes, and we have an ordering for the jobs in each class. According to Monma and Potts
(1989), this ordered batch scheduling problem can be solved in pseudopolynomial time. The
following dynamic pargram interleaves the classes. The state variable in the dynamic program
corresponds to a partial schedule that consists of the first ij jobs from G] and the first i2 jobs from
G2 and that ends before or at a specific time with a job from a specific class (if it is cheaper to end
sooner, that schedule should take precedence). At each point in the state space, we will measure
the total fiowtime of the scheduled jobs. The recursion determines the best partial schedule to
which we should add the specified job.
Algorithm 3.3. Lct/(r, a, ij, i2) be the minimum fiowtime of a partial schedule where the
last jobs ends at or before time t, there arc ij jobs from Gl and i2 jobs from G2, and the last
scheduled job is from Ga. t = 0,..., R. a = 1,2. Â¡j = 0,. .. , nx, i2 = 0,. .. , n2. Renumber the
jobs so that j < Â«, if Jj is in Glt rÂ¡ < ... < rnl, and j > nx if Jj is in G2, rnl+1 < ... < rn. R is some
upper bound on the makespan of a schedule. We can find one such R by scheduling all jobs in
ERD order, performing a class setup in front of every job. We also have an upper bound: R <
max {rj} + 'Zpj + ri, s2] + n2 s12.
Initialization:
At, a, ij, id = Â°Â° if t< 0.
At, 1,0, i2) = Â°Â° for all t and for i2 > 0.
At, 2, ij, 0) = Â°Â° for all t and for ij > 0.
At, 1, 1,0) = Â°Â° for t < max {s01, rj) + p, if Jj is the first job in Gt (j = 1).
At, 1, 1,0) = max {s01, rj} + p for t> max {%, rj} + p, if/y is the first job in Gj (j= 1).
At, 2,0, 1) = Â°Â° for t < max {s02, n) + q, ifiy is the first job in G2 (J  nx + 1).
At, 2,0, 1) = max {.v02, ry) + q for / > max [s^, rj) + q, if Jj is the first job in G2 (j = nÂ¡ + 1).
Iteration: (ij + i2> 1)
At, l,rj,i2) = min {Atp, 1, tj1, i2) + t,Atps2i, 2, tj1, i^ + r,/(rl, I,ij,t2)} if t>rj + p,
where j = ij.
At, 1, ij, I2) = oo if Ã < ry + p, where j = ij.
138
At, 2, q, i2) = min {/{tq, 2, q, Ã21) + t,f[tqsn, 1, q, qO + Ã./(Ãl. 2, q, Ã2)} if Ã > ry +
where j = nl + i2.
At, 2, Ã'j, q) = Â«o if t < rj + q, where j = nÂ¡ + i2.
Answer: the optimal total flowtime is min [f{R, 1, rq, 2, rq, n2)}.
Jk
Jj
tp t
h
S21
Jj
tps21 t
Jj
t1 t
Figure 3.13. Schedules for iteration (Jj and J^ in and in GÂ¿).
In the iteration, the first term being considered for/(Ã, 1, /j, is the total flowtime of the
partial schedule formed by adding the Jj (j = iÂ¡) to a schedule ending at or before tp with a job in
G j. The second term is the flowtime if Jj is added to a schedule ending at or before tps2 \ with a
job in G2. The third term corresponds to a schedule that ends at or before r1 with Jj. If t
there is no feasible schedule that ends with Jj, thus,/(Ã, 1, iit iâ– Â£) is set to infinity. The iteration is
similar for the/(r, 2, ix, i2).
The effort for each point in the state space is constant. The effort for the entire program is
thus proportional to the number of points in the state space. This is O(Rn2). Since R is bounded
by a polynomial function of the problem data, the algorithm is pseudopolynomial. The dynamic
program can be implemented to find the optimal objective function value with memory
requirements that are O(Rn): If we are determining the values for points with q + i2 = k, any
points in the space where q + i2
need to keep only ff. and at any time.
139
Test problems. We generated 70 test problems in order to test the dynamic program on a
range of problem sizes. The data for the problem sets are summarized in Table 3.23 (10 problems
in each set).
Table 3.23. Data about problem sets for special case.
Problem
Set
Number
ofjobs
P
q
Class
setup
Interarrival
times
FT201
20
1
i
1
13
FT307
30
1
i
1
13
FT308
30
2
3
1
25
FT309
30
4
2
3
48
FT401
40
1
1
1
13
FT507
50
1
1
1
13
FT601
60
1
1
1
13
Results of special case dynamic program. The dynamic program finds optimal solutions in
time that is nearly proportional to Rri1 (see Table 3.24).
One drawback of the dynamic program is the amount of memory required to perform the
algorithm; we were able to solve only 60job problems. In addition, it requires significant
processing time on a 386 personal computer. These problems can be overcome, however, since a
larger computer could handle more memory (the amount required is O(Rn)), and would run more
quickly.
Table 3.24. Results of dynamic program
Problem
Set
Number
of jobs
Average
R
Average
computation time
FT201
20
41.1
0.808
FT307
30
61.4
2.270
FT308
30
108.1
3.487
FT309
30
185.5
5.899
FT401
40
81.8
4.698
FT507
50
101.6
9.079
FT601
60
121.4
14.836
140
Extensions of special ease. The dynamic program can be modified to solve any FTSRD
problem where there exists a natural order for the jobs within each class. This includes problems
where the jobs in each class have the same processing time (as we have discussed) or problems
where all of the jobs have the same release date (order the jobs in each class by SPT). In any of
these eases we have an ordered batch scheduling problem. If each class has a natural order, the
dynamic program can be used to interleave these sequences since we have an ordered batch
scheduling problem.
Recall from the example presented in Section 3.4.4 that matching processing times and
release dates do not give us a natural order for a class.
3.4,7 Conclusions
In this research we have studied a computationally difficult class scheduling problem. The
objective is to minimize the total flowtime of a set of jobs that have nonzero release dates. We
examined a number of techniques to solve the problem, including a branchandbound search,
lookbehind dispatching rules, a decomposition heuristic, and a problem space genetic algorithm.
We were interested in determining how this type of genetic algorithm can be used to find good
solutions for another class scheduling problem.
Our results arc as follows: While we did develop lower bounds and a number of
dominance properties, our branchandbound approach was unable to solve any 30job problems.
The decomposition heuristic was a successful technique, locating solutions of high quality. The
Shorcst Waste heuristic could sometimes generate good solutions. However, by incorporating
these rules in a genetic algorithm that searched the space of adjusted release dates, we could find
much better solutions.
From these results we can conclude two things: For the onemachine class scheduling
problem we call FTSRD both the problem space genetic algorithm and the decomposition
heuristic can find good solutions in reasonable time. Additionally, lookbehind rules may be
141
useful for job shop scheduling, especially on a bottleneck machine which undergoes class setups
and where jobs continue to arrive while the machine is processing.
3.5 Chapter Summary
In this chapter we have presented the results of research into three onemachine class
scheduling problems. We can make a number of observations about this research. 1. None of
these three problems have been previously considered in the literature. 2. We have presented
analytical results and developed extended heuristics for each of these problems. 3. All three
problems arc motivated by the semiconductor test area job shop environment, and the extended
heuristics developed for these problems may be useful as dispatching rules in the general job shop
scheduling problem. 4. Our problem space genetic algorithm is a robust approach, able to find
good solutions over a variety of onemachine class scheduling problems, and should be
applicable to other difficult combinatorial and scheduling problems.
CHAPTER 4
LOOKAHEAD SCHEDULING PROBLEMS
In this chapter wc discuss the second major area of this research. We study the problem of
scheduling a machine that processes jobs headed for two different secondstage machines. Wc
analyze this threemachine problem with three different objective functions: makespan, total
llowtimc, and number of tardy jobs. We first examine the complexity of die problem and then
identify some lower bounds as well as some special cases that can be solved in polynomial time.
We develop a number of heuristics that find good solutions to the problem. We also use a
branchandbound technique to find optimal solutions.
4,1 Introduction
Job shop scheduling includes those scheduling problems in which different jobs may follow
different routes through the shop. These problems arc generally the hardest to solve optimally,
since few properties of optimal schedules arc known and the number of possible solutions
explodes as the problems increase in size.
Because of the complexity of job shop scheduling, algorithms to find the optimal solution
(in a reasonable amount of time) for even the simplest objective functions, e.g. makespan, do not
exist. Recent research has shown that bottleneckbased techniques such as the shifting bottleneck
algorithm (Adams, Balas, and Zawack, 1988) or bottleneck dynamics (see Morton, 1992, for
example) can be successful at finding good schedules. Traditionally, however, researchers have
studied (and schedulers have used) dispatching rules to order the jobs waiting for processing at a
machine.
Normal dispatching rules consider only the jobs currently in the queue for the machine
being scheduled. Wc define lookahead scheduling as the ability of a sequencing procedure to
142
143
include information about the status of machines downstream in the flow, enabling it to make a
better solution. Previous techniques that use this lookahead idea include the workinnextqueue
and numberinnextqueue dispatching rules (Panwalkcr and Iskander, 1977) and the use of
bottleneck starvation avoidance in shop floor control by Glasscy and Petrakian (1989). Robinson
et al. (1993) consider upstream and downstream information in scheduling semiconductor batch
operations. Researchers have also studied lot release policies that look ahead to the status of the
inventory in front of or arriving at a bottleneck; sec for example, Wein (1988), Glasscy and
Resende (1988), and Leachman, Solorzano, and Glassey (1988). Other researchers have studied
procedures that they called lookahead scheduling (Koulamas and Smith, 1988; Zeestraten, 1990)
but the problem setting or interpretation is different.
Consider two examples from the semiconductor test area. In the first, lots of two different
products are processed through the same brand workstation. After brand, the lots require
electrical testing, but the differences between the products indicate that the lots must be tested on
different machines. Or consider the effect of reentrant flows. The various lots waiting for
processing at an electrical test workstation may be at one of two points in their route. At one
point, a lots moves to brand after being tested (and it will return to test at some point in the
future). At another, it moves to visual/mechanical inspection. In the first case, the brand
workstation is sending lots to two different testers; in the second, the tester is sending lots to two
dissimilar workstations. If one of the secondstage machines is a bottleneck, it seems clear that
the sequencing of lots on the firststage machine should try to maximize the efficient use of that
bottleneck.
We can model this scenario with the following threemachine problem; There are three
machines M0, Mi, and M2. Each job follows one of two different flows: M0  M1( or Mq  M2.
Thus, M0 is feeding the other two machines. If one of these secondstage machines is a
bottleneck because the total work required on that machine is the larger than that on the other
machine, the sequencing of jobs on M0 should have as a priority the proper feeding of that
144
machine. This idea of a bottleneck will not affect our analysis of the problem. We will, however,
return to it for the heuristics and the empirical testing.
This problem, which could occur in any number of manufacturing environments, forms an
interesting general flow shop problem and a subproblcm of the job shop scheduling problem. As
a flow shop problem, it is a simple model unlike the multimachine problems previously
discussed in the literature, although work has been done on flexible flow shops with multiple
parallel machines at any given stage.
Moreover, the research into our threemachine problem may improve job shop scheduling
in two ways: one, the solution procedures can be applied directly to the subproblem of scheduling
machines near the bottleneck machine, and two, these techniques may be translated into good
lookahead dispatching rules for scheduling throughout the shop.
In this chapter we investigate three objective functions for this problem: the minimization
of makespan, of total flowtimc, and of the number of tardy jobs. We are concerned with the
analysis of solutions to the problem and the development of heuristics which can be used to find
good solutions.
The major contributions of this work include the proof that minimizing makespan is a
strongly NPcomplctc problem, the identification of optimality properties and special cases that
can be solved in polynomial time, and the development of an approximation algorithm.
The lookahead scheduling problems under investigation arc as follows (using the
numbering given earlier):
4. ThreeMachine LookAhead Scheduling: Makespan (3MLAMS)
5. ThreeMachine LookAhead Scheduling: Flowtimc (3MLAFT)
6. ThreeMachine LookAhead Scheduling: Number of Tardy Jobs (3MNT)
In this chapter we will look at each of these objective functions. In Chapter 2 we discussed
the research relevant to these problems. In the next section we start with the makespan objective.
In Section 4.3 we look at minimizing the total flowtime, and Section 4.4 discusses work on
minimizing the number of tardy jobs.
145
4.2 Minimi/inÃ¼ ihc Makcsnan
As mentioned in the introduction, we arc studying a problem that is a subproblcm of the
general job shop scheduling problem and is also a special ease of the general flow shop problem.
All analysis of the flow shop starts with Johnson (1954), who studied the minimization of
makespan for twomachine flow shop problems and for some special threemachine flow shop
problems. His famous algorithm starts jobs with the smallest firststage tasks as soon as possible
and jobs with the smallest secondstage tasks as late as possible.
Special eases of the flow shop makespan problem have been studied by a number of
researchers, including Mitten (1958), Conway, Maxwell, and Miller (1967), Bums and Rookcr
(1975), and Szwarc (1977). Garcy, Johnson, and Sethi (1976) proved that the general three
machine problem was NPcompletc. Problems with release dates, preemption, precedence
constraints, or more than three machines have also been studied.
In the flexible flow shop, more than one machine may be present at a particular stage.
Heuristics for this type of problem arc discussed by Wittrock (1988), Sriskandarajah and Sethi
(1989), Gupta (1988), and Gupta and Tunc (1991). Lee, Cheng, and Lin (1992) study an
assembly flow shop problem where each job consists of two subassembly tasks that arc
assembled in a third operation.
This threemachine problem is therefore closely related to problems previously studied, but
the preassignment of the jobs to different secondstage machines gives this problem a special
structure and leads to interesting results.
4.2.1 Notation
The following list describes the components of the problem and the notation used.
146
Jj
Job;',y= 1 n.
M0
The firststage machine.
m,,m2
The secondstage machines.
Hx
The set of jobs that visit M0 and then MP
h2
The set of jobs that visit M0 and then M2.
POj
The firststage processing time of Jj on M0.
Pij
The secondstage processing time ofiyon MÂ¡, i =
For a given schedule a, we can calculate the following variables:
Cqj The completion time of Jj on M0.
Cy The completion time of Jj on MÂ¡, i = 1 or 2.
Cj = CÂ¡j, the secondstage completion time of Jj.
Cmax = max {Cj), the makespan of the schedule.
Z Cj the total flowtime of the schedule.
Note that we will call a set of jobs that visit the same secondstage machine a group; thus,
//j and H2 arc each a group of jobs. Each group has a flow. The flow for the jobs in //, is M0 
Mj. The flow for the jobs in H2 is M0  M2. This section is concerned with the problem of
minimizing, over all feasible schedules, the makespan of the jobs. We call this problem the
ThreeMachine LookAhead problem  Makespan (3MLAMS).
4,2.2 Johnson's Algorithm
If we consider just one group and its corresponding flow, the problem of minimizing the
makespan of the set of jobs that visit these two machines is the same as Johnson's twomachine
flow shop problem. Johnson (1954) provided an optimality rule and an algorithm to solve the
problem. If each job to be scheduled has task processing times aj and bj on machines one and
two respectively, then his rule is as follows:
Johnson's Rule: precedes Jj in an optimal sequence if min {a,, bj) < min [aj, }.
This rule is implemented in the following algorithm:
Johnson's Algorithm:
Step 1. Find the smallest unscheduled task processing time. (Break ties arbitrarily.)
147
Step 2. If this minimum is on the first machine, place the job in the first open position in
the schedule. Else, place the job in the last open position in the schedule. Return to Step 1.
4,2.3 Permutation Schedules
In flow shop scheduling, a feasible schedule is called a permutation schedule if the
sequence of jobs on each machine is the same. Thus, the sequence of jobs on the first machine
uniquely identifies a schedule (assuming all tasks arc started as soon as possible). In the three
machine lookahead problem that we arc studying, the two sequences on the secondstage
machines consist form two disjoint sets of jobs. Thus, for our problem, we extend the idea of
permutation schedules to include schedules where the relative order of two jobs in the same flow
is the same at both stages.
It is known that considering permutation schedules is sufficient for finding optimal
solutions for regular twomachine flow shop problems. Thus, for our Lhrecmachinc lookahead
problems, it seems likely that permutation schedules will also be sufficient, since the problem
contains two twomachine flows. We will show that this is indeed true.
Definition: A schedule is a permutation schedule if, for all Jj and JÂ¡ in Hx (HÂ¿), JÂ¡ precedes
Jj on M0 if and only if /,â€¢ precedes Jj on M[ (M;).
Theorem 4.1. For any regular performance measure, there exists a permutation schedule
that is an optimal schedule.
Proof. First we will show that we can interchange two jobs that arc not in the same order at
both stages. Given a schedule o where job Jj directly follows JÂ¡ on machine M] (or but Jj
precedes JÂ¿ on machine M0, move the M0 task of Jj after the M0 task of./,.
If we look at Figure 4.1, we can observe that moving Jj causes all of the tasks (except Jj) on
M0 to start earlier, which docs not delay any (and may expedite some) secondstage tasks. Now,
since Jj will complete on M0 when JÂ¿ did (C'gj = CqÂ¡), the processing of Jj on M, will not be
delayed. Our interchange, therefore, docs not increase the completion time of any job.
148
Thus, for any given schedule, we can create a corresponding permutation schedule by
interchanging the firststage tasks. None of the job completion times arc increased by this
construction. Indeed, some of the completion times may be decreased. Therefore, this
permutation schedule has a better or equal performance on all regular measures (c.g. makespan,
flowtime, number of tardy jobs, maximum lateness). Thus, it is sufficient to consider
permutation schedules when trying to minimize these objective functions. QED.
C0j c0i
Jj
Jb
Jm
h
Jm
JÂ¡
Jj
cli Clj
C0b C'0i C'Qj
Jb
Jjn
Ji
J,
Jb
Jm
Ji
Jj
C'lb c'li C'lj
Figure 4.1. The exchange into a permutation schedule for M0 and M,.
(Machine M2 not shown.)
We would note here that a permutation schedule preserves the relative order of the
sequence for each group. Wc will use the term interleaving for the process of combining two
sequences to form a permutation schedule. We will show in Section 4.2.5 that if wc arc given
sequences for each group, then there is a polynomialtime algorithm to find the interleaving that
minimizes the makespan of the schedule created. However, the optimal secondstage sequences
cannot be determined in polynomial time. In fact, the 3MLAMS problem is strongly NP
completc, as will sec in the next section.
149
4.2.4 NPComnletcncvsx
In this section, we consider the complexity of the 3MLAMS problem. Although other
researchers (including Gonzalez and Salmi, 1978) have shown the NPcomplctcncss of a number
of small shop problems, we cannot determine the complexity of our problem from any of this
previous work.
We will therefore prove that 3MLAMS is strongly NPcomplctc, which will be done by
transforming 3Partition to 3MLAMS. (Recall that it is sufficient to consider permutation
schedules.) The 3Partition problem can be stated as follows: Given a set of aÂ¡, i= 1 3n,
and B, partition these elements into n sets A,,..., An, where each set contains three elements and
the sum of the three elements equals B. We will make the assumption that for all /,
1/4 B < aÂ¿< 1/2 B. (We can transform any problem without this property into one where it is
true.) With this property, there will never be a set of two aÂ¿ or a set of four aÂ¡ where the sum of
the elements equals B.
Theorem 4.2. 3MLAMS is a strongly NPcompletc problem.
Proof. Given an instance of 3Partition, with a set of aÂ¡, i = 1,..., 3n, and B (note all aÂ¡ > 1),
create 4n jobs, 3n of which go from Mâ€ž to M2 and have processing times pgÂ¡ = aÂ¡ and p2Â¡ =
(B+\)aÂ¿, i= 1,..., 3 ir, let these jobs form a set IT. The n remaining jobs that go from M0 to Mj,
J3n+\'J3n+2< â– â– â–
let these jobs form a set X. The desired makespan is M = n(B^ + B) + B.
Part 1. If there exists a partition (AÂ¡ An) such that for all aÂ¿ in AX aÂ¿ = B, then the
sequence on M0 of [A] Jjn+] A2 Jjn+2 â– â– â– An Jjn] will yield a schedule where the completion
time on Iv^ will be M and the time on M2 will be no greater than M. (See Figure 4.2.)
Part 2. Now, suppose there exists no partition. Consider any arbitrary permutation
schedule 0. The sequence is composed of consecutive subsets of jobs from W and of jobs from
X. Let A:,j= 1,.. ., /2+1 be these subsets of W, where Ay directly precedes they'th job from X,
except for Art+j, which is the set of jobs following the last job from X. Note that any of these Ay
150
may be empty. Let Sj equal the sum of the firststage task processing times for the jobs in Aj. Let
AÂ¡. be the first Aj such that SÂ£ is not equal to B. Because there is no partition, there must exist one
such A/, among 4,.... , An.
If Sf. > B, then the makespan on machine M, is delayed by the delay in the Ath job in X:
MS(Mj) > S, + B2 + . .. + Sk + B2 + {n  k)(B2 + B) + B > kB + IcB2 + (n  k)(B2 + B) + B = M.
(This assumes that all of die first k1 jobs have long tasks on Mj, leaving n1 ()k1) long tasks and
one short task on Mj. If one of the first A1 has the short task, then the makespan is even longer.)
If < B, let S = Sj + . .. + S/. < kB (and < kB  l) and suppose Jm is the first job from W
after the k\h job from X. Then, the makespan on M2 is postponed by Jm: MSfM^ > S + kB2 + am
+ (B + l)(/ifi  S) > S + kB2 + am + nB2 + nB  BS  S. Including BS > kB2 + B implies MS(M2)
> am + nB2 + nB + B >M. Thus, there exists no schedule with makespan less than or equal to M.
Part l and Part 2 of the proof show that there exists a partition if and only if there exists a
schedule with the desired makespan. This implies that 3MLAMS is a strongly NPcomplctc
problem. QED.
Mq
Mi
m2
A1 a2 An1
Figure 4.2. A schedule for the 3MLAMS problem.
4.2.5 Makespan Optimality Conditions and PolvnomiallvSolvable Cases
This section discusses the optimal combination of two given sequences, lists a few
properties of optimal schedules, and presents two special eases that have easilyfound optimal
solutions. We will see that we can easily form a permutation schedule by optimally interleaving
151
sequences for each group (Algonthm 4.1). Properties 4.1,4.2, and 4.3 arc dominance properties
between jobs in the same group. Theorems 4.3 and 4.4 describe the special eases.
Because we need to consider only permutation schedules, we can create a single schedule
for all three machines from a sequence of the jobs. That is, given a sequence on the firststage
machine we can create a sequence for each secondstage machine. This is done by considering
the jobs in the order they appear on the first machine. However, there is no comprehensive rule
for determining this sequence. Thus, we will spend some time on how this sequence should be
constructed in certain situations.
Interleaving two sequences. Since the problem is NPcomplctc in the strong sense,
heuristic methods for finding good solutions arc justified. A very natural heuristic is to schedule
the jobs in each group separately. For instance, each group can be scheduled by Johnsonâ€™s rule.
Then we can interleave these two sequences to form a solution to the original problem. The
process of combining two sequences to achieve the minimal makespan is called optimally
interleaving. This is the best combination of these two sequences. Note, however, that the best
combination of the two Johnson sequences may not be an optimal solution; the optimal solution
may be some combination of suboptimal solutions to each subproblcm. In Section 4.2.9 we
present an example where optimally interleaving the Johnson sequences for each group is not
optimal.
So, while interleaving the Johnson sequences is a natural heuristic that works well (see
Section 4.2.8), we may wish to try other heuristics to the subproblcms before interleaving. These
are discussed in Section 4.2.7. In this subsection we will describe how two sequences should be
interleaved.
Now, suppose we arc given two sequences Oj and one sequence for each group (we
have found solutions for each of the subproblems), and we want to find the minimal makespan
that can be achieved by combining these two sequences. The following observations will lead us
to an algorithm fordoing so.
152
Define C as a makespan that we wish to achieve. To minimize the makespan given the two
sequences we need to find the minimal C for which we can find a feasible schedule. We can
schedule the tasks on the secondstage machines in the order given by Oj and 02 and as late as
possible, so that die last task on each machine ends at C. In a feasible schedule, the firststage
task for each job has to complete on Mq before the secondstage task can begin. We need to
determine if there is some ordering of the tasks on Mg so that each task finishes ontime (with
respect the secondstage task).
For each job Jj the start time of the secondstage task can be used as the due date dj for the
corresponding firststage task. We can find a sequence where each firststage task finishes on
time (Cqj < dj) if and only if we can find a sequence where the maximum lateness is less than or
equal to zero. If we wish to minimize the maximum lateness of the firststage tasks, we should
order them by EDD (Earliest Due Date), according to Jackson (1955). The schedule created is an
interleaving of the two given schedules: if JÂ¿ precedes Jj in a^, then dÂ¿ < dj, and JÂ¡ will precede Jy
on Mq.
Each due date dj = C  tj, where tj is the sum of the secondstage processing times of Jj and
the jobs JÂ¡. that follow Jj on the secondstage machine. See Figure 4.3.
0 Cqj dj <â€” tj â€”> C
Figure 4.3. The variables associated with Jj (secondstage started late).
Note: Other secondstage machine not shown.
Since the EDD sequence corresponds to sequencing the tasks in decreasing order of tj, the
sequence of jobs is the same for all values of C, including the optimal one. Therefore, we can
find the optimal makespan from the feasible schedule with the jobs in this order and all tasks
starting as soon as possible. This is the optimal interleaving of two given sequences, described in
Algorithm 4.1.
153
Algorithm 4.1 (Optimal Interleaving): Given a sequence oÂ¡ for the jobs in H] and a
sequence 02 for the jobs in //2, perform the following steps to yield a schedule with die minimal
makespan:
Step 1. For each Jj in Hx, define Aj as the set of jobs (not including Jj) that follow Jj in o .
Then tj = Plj + I.AjpIkr
Step 2. For each Jj in H2, define Aj as tfic set of jobs (not including Jj) that follow Jj in 02
Then tj = P2j+lAjP2t
Step 3. Schedule the jobs on M0 in decreasing order of the tj, starting at time zero, and start
all secondstage tasks as soon as possible.
Note that this algorithm takes O(n) effort, since each group is already in decreasing order of
the tj, and forming the schedule is only combining the two sequences without changing the
relative orderings.
In Example 4.1 we perform Algorithm 4.1 on a problem with five jobs and given sequences
for each group. We arc given sequences f J1J3J2) and [J4J 3] for each group. After
computing the tj, we form the interleaved sequence [ J Â¡ J3J4J5J2 ]â– The corresponding
schedule has a makespan of 11 (see Figure 4.4).
Example 4.1. Given the following five jobs and the two sequences [ JÂ¡ J3 J2 1 and
U4J5Y
Jj HÂ¡ pQj Pjj tj
J! H] 2 4 9
J2 H, 1 1 1
J3 H, 2 4 5
J4 H2 2 1 4
J5 H2 1 3 3
154
Jl
J3
u
J2
J
J3
J2
J4
0
2
4
6
7
8
1U
Figure 4.4. Oplimal schedule for Example 4.1.
Properties of optimal solutions. In general, we still do not know how to construct the
sequences on and M2. These subproblems are what is difficult about this problem. Johnson's
rule, which is optimal for the twomchinc flow shop, is not always applicable for the 3MLAMS
problem. Of some help, however, are a number of dominance properties for jobs in the same
group that limit the number of sequences that need to be considered in searching for the optimal.
The first property establishes a precedence relationship between two jobs that is a restrictive form
of Johnson's rule. The other two relate two jobs that are processed consecutively on some
machine.
Property 4.1. (Absolute Dominance) If both JÂ¿ and Jj are in H^ and pqÂ¿ < pqj and
Pki  Pkp l*ien should precede Jj. (If both equalities hold, then the jobs are obviously
interchangeable.)
Proof. Suppose we have a schedule a such that the inequalities above hold and Jj precedes
JÂ¿. Without loss of generality, suppose both jobs are in //, and that all of the secondstage tasks
are started as late as possible. Then it is easy to sec that exchanging JÂ¡ and Jj on both machines
(see Figure 4.5) docs not increase the makespan. Thus, it is sufficient to consider schedules
where JÂ¿ precedes Jj. QED.
s0j c0j S0i C0i
Jj
Ji
Jj
Ji
Slj Clj Sli Cli
Figure 4.5a. and Jj on M0 and M. (Machine M2 not shown.)
155
C'Ã¼i c'0j
h
Jj
h
Jj
S i i C,i S ij C jj
Figure 4.5b. The exchange of JÂ¡ and Jj on M0 and M,.
(Machine M2 not shown.)
Property 4.2. (Consecutive Dominance) If both JÂ¿ and Jj are in HÂ¿ and arc processed
consecutively on M0, they should be sequenced by Johnson's rule (i.c., if min {pgp p^j} < min
[Pkp PojK ^en // should immediately precede Jj).
Proof. The fact that both jobs arc processed consecutively on both machines implies that
Johnson's Rule applies to them. If the jobs arc not sequenced according to Johnson's Rule, they
can easily be exchanged, and the makespan of the schedule is not increased by the switch. QED.
This property is stated for two jobs from the same group processed consecutively. It can be
extended to a set of three of more consecutive jobs on M0 from the same group:
Corollary 4.1. (Batch Dominance) The jobs in each batch of a schedule should be
ordered by Johnsonâ€™s Rule, where a batch is defined as a set of consecutive jobs on M0 from the
same group.
Property 4.3. (Small Secondstage Dominance) Suppose /,â€¢ and Jj arc in Hf. and are
processed consecutively on Mk. If pgÂ¡ > p^p pgj ^ Pkj> and pf.Â¿ > Pkp then /,â€¢ should precede Jj.
Proof. Without loss of generality, suppose JÂ¿ and Jj arc in Hx. Consider a schedule where
the two groups have been optimally interleaved and the secondstage tasks arc started as late as
possible. Then, if there arc any jobs between JÂ¿ and Jj on M0, they must be from H2, and the start
times of these jobs on M2 arc between the start times of JÂ¡ and Jj on M,. (Sec Figures 4.6a and
4.6b.) Suppose Jj precedes Jp Let us interchange these jobs on both M0 and M, and check that
we still have a feasible schedule without increasing the makespan. Let y be the old start time of
JjOr\ Mj. Let z be the old completion time of/Â¿on M0. For Jj, the new completion time on M0
156
equals z. Since JÂ¡ can be on only one machine at a time, z< y + p jj, the old start lime of JÂ¿ on
M[. Thus, z
POj
Jc
POi
Plj
Pli
Jc
y z
Figure 4.6a. (secondstage tasks started late).
POi
Jc
POj
Pli
Plj
Jc
X
y
7.
Figure 4.6b. (secondstage tasks started late).
Now, let a: be the new start time of J; on M0. Thus y>z  p jj>z  pgj  x. Now, for and
for all jobs Je between Jj and JÂ¿, the new completion time is less than x and thus y and thus the
start time of their secondstage tasks. We can therefore switch Jj and JÂ¡ without increasing the
makespan of the schedule. QED.
Corollary 4.2. If for some //Â¿, pgj > p^j for all Jj, then the jobs in Hf. should be ordered by
LPT on the tasks.
We note that if Corollary 4.2 is true for both Hx and H2, we have the first special case
mentioned below (Theorem 4.3).
Polvnomiallvsolvablc special cases. Like other threemachine flow shop problems, the
3MLAMS problem has special cases that can be solved in polynomial time. Theorems 4.3 and
4.4 present such special cases.
157
Theorem 4.3. 1 f pÂ¡f. < pgÂ¡. for all JÂ¡i '\nHx and P2j 5 Pqj for all Jj in //2, then ihc optimal
solution can be found by ordering the jobs in each group by their secondstage processing limes,
longest processing time first, and optimally interleaving the two sequences.
Proof. This is true because Corollary 4.2 holds for both groups. QED.
Theorem 4.4. If, for some /7Â¿, min {pgj : Jj e //,u//2> max [pÂ¡j : Jj e //Â¿}, then the
optimal schedule can be found by sequencing the jobs in each group by Johnson's Rule and
optimally interleaving these sequences.
Proof. By the given, we know that the conditions of Corollary 4.2 hold for the jobs in HÂ¿.
Thus, the jobs in this group should be ordered by longest secondstage task processing lime first.
For these jobs, this sequence is the same as the sequence given by Johnson's rule. Now, if we can
determine the optimal ordering of the jobs in Hthe other group, we can interleave the groups to
derive an optimal schedule. We will show that these jobs should be ordered by Johnsonâ€™s rule.
Coasidcr an optimallyinterleaved schedule where Jft is processed immediately before J j on
Mk, and where the jobs arc not ordered by Johnsonâ€™s rule. That is, min [pgj, p^} < min [pg^,
Pkj)â€¢
If there arc no jobs from //,â€¢ between J^ and Jj on M0, then Property 4.2 implies that the
jobs should be interchanged.
Else, let y/?1 be the job from HÂ¡ processed just before Jj. IÃ Jj is not the last job on M0, then
we can move Jj to immediately after JÂ¡v Because the jobs that were between J^ and Jj have short
secondstage tasks, delaying these jobs docs not delay the processing of any successive jobs.
Now Jfj and Jj are consecutively processed on Mq, and Property 4.2 implies that the jobs should
be interchanged.
Finally, if Jj is the last job on M0 (and Mk), then the fact that the schedule is optimally
interleaved implies that py < pÂ¡m, since Jm is the last job on MÂ¡. By the given, p^ is less than or
equal to pg^ and pgj. Thus, min [pgj.PuJ < min {pg}vPkj\ implies that p^ < min (Po/j.Pty}
(since Pfy < Pgp and consequently p^ < pp;, and pyx < pÂ¡.j. Property 4.3 implies that Jj should
precede J^
158
Wc have thus shown that the jobs in Hf. should be ordered by Johnsonâ€™s rule. This gives us
an optimal ordering tor H/. which can be interleaved with to solve the problem. QED.
4,2,6 BranehandBound Algorithm
In this section wc will identify three lower bounds on the makespan; we will take the
maximum of these to form our overall lower bound. Wc will also discuss the use of a branch
andbound technique for finding optimal solutions for the 3MLAMS problem. Wc will use the
overall lower bound in the analysis of the worstease performance of an approximation algorithm
(Section 4.2.9). Wc will begin by discussing the three component lower bounds.
For the first component bound, consider only the jobs in Hv Order these jobs using
Johnson's rule and determine for cachiy a completion time CÂ¡j on M,. Then, LBj = max {CÂ¡j}.
Similarly, LB2 = max {C2j}
The third component bound takes into account that all of the jobs use M0. We relax the
problem by dropping the secondstage assignments of the jobs and allowing an infinite number of
secondstage machines. In fact, however, we only need n secondstage machines, one for each
job. If each job has a separate secondstage machine, then the minimal makespan, which will be
a lower bound on the optimal makespan for the original problem, is the maximum sum of first
stage completion time plus secondstage task processing time. By an argument similar to that of
Section 1.4, wc can find the minimal makespan by sequencing the jobs by their secondstage task
processing times, longest first. Wc schedule the jobs in this order on M0. The secondstage
completion time of any job Jj in is Cqj + pÂ¡j. Therefore, LB3 = max {Cqj + pÂ¿j}. Note that
LB3 will be greater than X PqÂ£, since there exists some job Jj that has Cqj = X Pot
Our lower bound for a given problem (hereafter referred to by the variable LB) will be the
maximum of these three lower bounds for the makespan: (i) LB j, the minimum possible
makespan of the jobs in set Hlt (ii) LB2, the makespan of the jobs in set //2, and (iii) LB3, the
minimal makespan if there exist an infinite number of secondstage machines. That is, LB =
max {LB], LB^ LB3}.
159
In a standard branchandbound algorithm, each node will consist of a partial schedule of
jobs. From a node, we exclude certain branches using the above dominance properties and create
a lower bound using straightforward extensions of the above lower bounds.
In order to help prune branches from the search tree, we will use the optimal properties that
we developed in Section 4.2.5. We will also use a dominance property (Property 4.4) that is more
dependent upon the current makespan of each machine (where is the makespan of MÂ¡).
Property 4.4. For a given partial schedule, if Jj is unscheduled and in HÂ¿ (where Hm is the
other group), tf. > X pgÂ¿ (the sum over all jobs), and trn < X pgÂ¿, then Jj should not precede any
job from H/n.
It is easy to show that if, in any schedule constructed from this partial schedule, Jj docs
precede any of the jobs from Hm, we can move Jj to the last position without increasing the
makespan.
We used the branchandbound algorithm to solve a number of problems. On the set of 15
job problems, the running time was usually less than one second (on a 386 PC), although it was
much greater for two problems, one of which had a lower bound that was not tight. See
Table 4.1.
Table 4.1. Statistics on branchandbound procedure for LA 154.
Problem
Lower Bound
Optimal
Time
Nodes
1
2942
2942
0.11
38
2
2080
2080
0.11
39
3
3138
3138
0.16
47
4
3122
3122
0.11
37
5
1956
1956
0.16
56
6
2034
2034
0.22
51
7
2187
2187
0.17
57
8
1919
1919
0.17
71
9
1911
1911
73.92
61916
10
1945
1956
637.30
546654
Note: Time measured in seconds.
160
4.2.7 Heuristics
Since 3MLAMS is a strongly NPcomplcte problem and the branchandbound procedure
may occasionally take too long to find the optimal solution, the use of approximation algorithms
is a preferable alternative. Our discussion so far has led us one very natural heuristic, the
interleaving of the Johnson sequences. We will also introduce another combination that will be
of use later.
Johnson Interleaved. Order the jobs in each group using Johnson's rule. Optimally
interleave them using Algorithm 4.1.
Merged Johnson. Order the jobs in each group using Johnson's rule. Select the group
with the smallest total task processing time on M0 (if the totals are equal, pick one arbitrarily).
Start the schedule with all of the jobs from this group. Follow them with the jobs from the other
group.
4.2.8 Empirical Results
In this section we report on the empirical testing of our heuristics. We discuss the problem
sets generated, our methodology, and our results.
In order to study the scheduling of a bottleneck machine, coasider one of the secondstage
machines, say M,, as a bottleneck operation. As a bottleneck, the workload of M, should be
greater than that of M0 or M2. Therefore, we will construct problems where the total processing
time on M, is likely to be the largest.
Three problem sets (LA154, LA304, LA504) were created using uniform distributions to
generate processing times. The mean fraction of jobs in each group and the mean processing
times were set such that M, would have more work to do than M2. These problems had 15, 30,
and 50 jobs. There were ten problems in each set.
161
Tabic 4.2. Characteristics of 3MLA MS problem instances.
Set
Jobs
Average number
of jobs in H,
M0 times
(range)
M[ limes
M2 limes
LA304
30
15
2060
40120
3090
LA 154
15
5
80160
240480
120200
LA504
50
37
3060
4080
60120
Another problem sel (LA201) was created based upon the problem structure used in the
NPcomplcteness proof. Ten sets of 15 random integers ranging from 1 to 10 was created (with
an adjustment to insure that the total processing time was divisible by 5), and a 20job problem
was created from each set of these aj using the construction in Theorem 4.2. For this problem we
knew a good lower bound, although we determined later that two of the embedded 3Partition
problems had no solution and thus the lower bound could not be achieved.
For each problem, our lower bound on the makespan was the maximum of the three
component bounds (see Section 4.2.6). The best component bound depended upon which
machine had the largest workload for any problem. Since M, generally had the most processing
time, the M0  M] bound was usually greatest.
For the 15job problems, we were able to calculate the value of the optimal solution. For
the other uniform problems we were able to find heuristic solutions that achieved the lower
bound, implying that our lower bounds were tight and that we had found optimal solutions.
On the 20job hard problems, we could solve the imbedded 3Partition problem. For 8 of
the 10 problems, there docs exist a partition and thus we know the optimal makespan. For the
other two, we do not know the optimal, although we came very close with the heuristics.
The results of the Johnson Interleaved and Johnson Merged heuristics on minimizing the
makespan are shown in Table 4.3. Performance is the relative deviation from the optimal (if
known) or the lower bound. We observed that the Johnson Interleaved algorithm found near
optimal or optima] solutions for all of these problems. The performance of the Johnson Merged
algorithm varied from good on the 30 and 50job problems to poor on the 20job problems. The
Johnson Interleaved heuristic outperformed a number of other sequencing rules, and these results
were consistent with testing on a number of other problem sets with different problem structures.
162
Table 4.3. Makcspan summary (able for 3MLAMS problem.
Pcformancc is relative deviation from optimal or lower bound.
Set
Johnson
Interleaved
Johnson
Merged
LA 154
0.50
2.44
LA 31)4
0.00
0.35
LA 504
0.00
0.40
LA 201
0.42
23.14
4,2,9 Heuristic Error Bounds
It is often possible to evaluate a heuristic by analyzing its worstease behavior. Heuristics
that minimize the makcspan of scheduling problems arc especially open for this kind of analysis
due to the simplicity of the objective function. Since the Johnson Interleaved heuristic performed
the best in our empirical study, we arc most interested in determining the worstease performance
of this procedure.
We will begin our analysis with the Merged Johnson procedure. For this bound we will
make use of the lower bounds derived in Section 4.2.6. Recall that our lower bound LB  max
[LBx,LB2, LB3).
Theorem 4.5. For a given instance of 3MLAMS, the schedule created by the Merged
Johnson heuristic has a makcspan less than 3/2 LB.
Proof. Without loss of generality suppose that X//; pqj < X//2 POy Now, LB3 > X Poj1110
sum over all of the jobs, implying X/// Pqj < 1/2 LB2 < 1/2 LB. LBÂ¿ be the minimal makcspan
achievable by the jobs in Hfound by scheduling the jobs using Johnson's rule. The makespan
of the scheduled created using the Merged Johnson heuristic is max {LBh ^HÂ¡P0j+ LB2)
is less than max {LB, 1/2 LB + LB] =3/2 LB. QED.
Now, let us consider the Johnson Interleaved heuristic. Since this algorithm combines the
two Johnson sequences optimally, its performance cannot be worse than that of the Merged
Johnson heuristic. Thus we can make the following statement.
163
Corollary 4.3. For a given instance of 3MLAMS, the schedule created by the Johnson
Interleaved heuristic has a makespan less than 3/2 LB.
Theorem 4.6. The maximum error of the Johnson Interleaved algorithm relative to the
optimal makespan is onehalf.
Proof. If C* is the value of the optimal makespan and CjÂ¡ the makespan of the schedule
created by the Johnson Interleaved algorithm, LB < C* < CjÂ¡ < 3/2 LB < 3/2 C*. Thus,
(Cji  C*) / C* < 1/2. QED.
Let us make a few observations about our algorithm. First, the error bound can be extended
to the ease where there arc m > 2 groups (each group with a different flow to a different second
stage machine). The maximum relative error of the Johnson Interleaved algorithm is 1  l/m in
this case.
Second, the Johnson Interleaved algorithm interleaves the groups by looking ahead to the
future workload of the secondstage machines (the sum of the remaining secondstage task
processing times).
Finally, our error bound of onehalf is tight. In the following problem instance, the bound
is achieved in the limit.
Example 4.2. Given n, let C* be 2n + 1. For Hx, construct jobs Jx,.. ,,Jn with the
following characteristics:
/ = 1,..., n  1: p0i= 1 pÂ¡Â¡= 1
i=n: P0n=] Pln = n
Construct in //2jobs7n+j,... ,J2n with similar characteristics:
/= n+ 1,. .. , 2n  1: P0/=1 Pli= 1
i = 2n: P02n=x Pl2n = n
Now, since ties can be broken arbitrarily, each group is already ordered by Johnsonâ€™s rule
(we could force this ordering by subtracting a small amount from the appropriate firststage
tasks). An optimal schedule can be achieved by interleaving the sequences [J nJ{... J n_\ ] and
164
^2n^n+l â– â– ^2n\ 1 The makespan of such a schedule is C* = 2n + 1. The interleaving of the
Johnson sequences yields a schedule with makespan 3n, as docs merging the Johnson sequences.
As n goes to infinity, the ratio of each of these makespans to C* goes to 1.5. See Figures 4.7, 4.8,
4.9 (Jx = 710) for the optimal, interleaved, and merged schedules with n = 5.
J5
Jx
h
J6
h
J7
CO
j4
J9
J5
h
h
J3
J4
Jx
J6
J7
J8
J9
0
2
7 8 9 10 11
Figure 4.7. Optimal schedule (secondstage tasks started late).
h
J6
J2
J7
J3
J8
J4
J9
J5
^x
Jl
h
J3
J4
J5
J6
J7
J8
J9
^x
0 2 6 10 15
Figure 4.8. Interleaved Johnson schedule (secondstage tasks started late).
Jl
J2
J3
J4
J5
J6
J7
CO
J9
(4
X
Jl
J2
J3
J4
J5
J6
J7
J8
J9
Jx
0 16 15
Figure 4.9. Merged Johnson schedule.
4,3 Minimizing the Total Flowtime
In this section we discuss the ThreeMachine LookAhead problem with the objective of
minimizing the total flowtime (3MLAFT). We know that minimizing total flowtimc in a two
machine flow shop is an strongly NPcomplctc problem (Garcy, Johnson, and Sethi, 1976). Since
165
that problem is a special ease of 3MLAFT, 3MLAFT must also be a strongly NPcomplctc
problem. Hence, we will investigate optimality conditions, lower bounds, and heuristics. Since
total flowtimc is a regular objective function, only permutation schedules need be considered for
optimality (see Section 4.2.3); thus, a sequence of jobs for M0 corresponds to a unique schedule
on all three machines. The problem notation is the same as that for the 3MLAMS problem.
A number of researchers have studied the problem of minimizing the total flowtime in a
flow shop. This work includes lower bounds, branchandbound algorithms, and heuristic
approaches. See Ignall and Schragc (1965) for a branchandbound and Ahmadi and Bagchi
(1990) for improved lower bounds. Szwarc (1983) studies special eases, and Van de Velde
(1990) presents a Lagrangian relaxation. Krone and Stciglitz (1974), Kohler and Steiglitz (1975),
Miyazaki, Nishiyama, and Hashimoto (1978), and Miyazaki and Nishiyama (1980) all present
heuristic approaches. Ahmadi ct al. (1989) includes batch processing.
t.3.1 Total Enumeration
Initially, we wanted the compare the difficulty of finding nearoptimal solutions for 3MLA
FT to the difficulty of finding nearoptimal solutions for 3MLAMS, where we had very good
results. One way in which we could compare the two problems was to compare the range of the
objective functions over the domain of all permutation schedules.
For a ninejob problem, a total enumeration of the permutation schedules includes over
300,000 sequences. We picked an arbitrary instance, created each schedule, and measured the
makespan and total flowtimc of each schedule. This procedure yielded a distribution of
makespans and total flowtimcs over the set of sequences.
The primary result is that 18.5% of all sequences have makespans within 6.7% of the
optimal makespan, but less than one quarter of one percent of all sequences have flowtimes
within 6.7% of the optimal flowtimc. (See Table 4.4 and Figure 4.10.) Because there are much
fewer schedules that are nearoptimal, we can infer that the problem of minimizing the total
flowtimc is a much harder problem than the problem of finding the minimum makespan. Similar
166
results hold for another ninejob problem and a fifteenjob problem where a random sample of the
sequences were examined.
Table 4.4. Distribution of schedule makespans and flowtimcs.
Deviation from
optimal makespan
Percent of
schedules
Deviation from
optimal flowtimc
Percent of
schedules
3.3 %
5.65 %
3.7 %
0.05 %
6.7
18.47
6.7
0.23
10.0
30.13
14.2
2.05
13.3
49.07
21.6
6.85
16.7
62.17
29.1
15.83
20.0
75.41
36.6
29.18
23.3
86.24
44.0
45.37
26.7
92.31
51.5
62.31
30.0
96.43
59.0
77.28
33.3
98.10
66.4
88.65
40.0
100.00
73.9
95.49
Makespan EZ> 
Flowtime
Figure 4.10. Deviation from optimal makespan and flowtime.
167
4.3.2 Lower Bounds
In this section we discuss lower bounds on the total flowtime. These lower bounds provide
us with a way of determining the quality of the solutioas produced by our heuristics and can be
extended into lower bounds for a branchandbound algorithm.
The lower bounds are similar to those proposed in Ignall and Schrage (1965). The initial
lower bound on the total flowtime is calculated by ordering the jobs by their processing times on
M0, shortest processing time first (SPT), and assuming that the secondstage machines have
infinite capacity. That is, LBX = Â£ Cqj + 'Lpjj+Y. P2j
We compute the second bound by considering each of the secondstage machines. Let a =
min {pQj : Jj e //,} and b = min {pQj :Jje H2). Now, a (b) is the earliest time that the second
stage tasks on Mj (M2) could start. Then, form two sequences Oi and o2 of the jobs in H\ and H2
respectively by ordering the jobs in each group by the secondstage task processing times,
shortest processing time first. Schedule on the secondstage tasks of the jobs in //, in the
order given by a,. The first task should begin at a, and each successive task should immediately
follow (we do not schedule the firststage tasks). This forms completion times CÂ¡j for the jobs.
Repeal for the secondstage tasks of the jobs in H2 in order to calculate Ãœ2j
However, only one machine can start at its earliest lime. The other must start at a time no
less than a + b. Let r be the cardinality of Hx and Ã the cardinality of H2. Either the r tasks on Mj
will be delayed by b, or the s tasks on M2 will be delayed by a. After calculating rb and sa, we
can increase our lower bound by adding the smaller quantity. Thus, the second lower bound is
LB2 = X Cjj + Z C2j + min ÃÃJ}
In Example 4.3, we calculate the lower bounds for the problem instance that we introduced
in Example 4.1.
168
Example 4.3. Given ihc following five jobs in two groups:
Jj Hj
POj
Pij
lj
Jj //,
2
4
9
h
1
1
1
J3 //,
2
4
5
J4 h2
2
1
4
J5 H2
1
3
3
Lower Bound 1: Order firststage by SPT and add secondstage:
POj 1 1
2
2
2
C0j 1 2
4
6
8
ZC0J = 21. IPlj=9.
ZP2j
II
r
= 34.
Lower Bound 2: Order each secondstage machine by SPT:
a = 1. r=3. b 1. s = 2.
Plj 1 4
4
P2j 1 3
C1} 2 6
10
C2j 2 5
I Cjj= 18. Z C2j = 7.
min (rb,sa) =
2. LB2 = 27.
4.3.3 Special Case
This section discusses a special case that leads to easilyfound optimal solutions.
Theorem 4.7. IfpyÂ¿ < for all JÂ¡. in //, and P2j ^Pqj for all Jj in H2, then any sequence
o in which the firststage tasks are in SPT order forms an optimal schedule.
Proof. Suppose wc have such a o. Consider//,. If7Â¿ is the first job on M,, then CjÂ¿ = CqÂ¡
+ pjÂ¿. If is the next job from //, on M,, then p]Â¿
completes before (or at the same lime as) the firststage task of7Â¿ and docs not delay the second
stage task of ./Â¿. Thus, C= CqÂ¡.+ pÂ¡Similarly, this equality is true for all JÂ¿ in //,, and C2/
= C0j + P2j for all Jj in H2. The total flowtimc is therefore X Cj = X Cqj +'LpÂ¡j+'L P2jâ€¢ Since
the firststage tasks are in SPT order, this achieves the first lower bound, and this sequence is an
optimal schedule. QED.
169
4.3.4 Empirical Testing
Empirical testing was performed using a number of different heuristics. The three
heuristics that performed the best (and very similarly) arc described below. The 3MLAMS
problem sets were used as a testbed. For each problem, the four lower bounds on the flowtimc
were calculated. The best of these was taken as the lower bound. For the fifteenjob problems,
we were able to find the optimal solutions from the branchandbound algorithm. The
performance of a heuristic on a problem was taken as the relative deviation from the optimal
solution (if known) or the best lower bound.
SPTlookahead. Each group Hx and H2 is ordered by the firststage task processing times
(shortest first), forming two sequences. These sequences arc interleaved by choosing at each step
the first unscheduled job from one sequence or the other. Define at each step the following
variables: t0 is the completion time of the partial schedule on M0 and q is the completion time of
the partial schedule on M,. Let px (pÂ¿> be the processing time on M0 of the next job from Hx
(H2). Note that these are the shortest such task processing times from each set of jobs. If q 10 <
Pi, the workinprocess inventory (W1P) waiting at M, is low; schedule the job from Hx. If q 10
Pi + Pi' then the WIP at Mj is high; schedule the job from H2. Else./q < q 10
the WIP is intermediate; schedule the job with shoncst firststage processing time (px orpÂ¿).
YVIPlookahead. Again, order the jobs in each group by the firststage task processing
times (shortest first). Combine the groups as in SPTLookahead, except for one case; If px < q 
t0
job to schedule.
Johnsonlookahead. Sequence each group by Johnson's rule. Combine the groups as in
SPTLookahead, except for one ease: If px < q  r0 < px + p2, schedule the job from Hx. Thus the
work on Mj is always used to determine which job to schedule.
We tested the heuristics on the problem sets described in Section 1.7. For the fifteenjob
problems, we were able to find the optimal solutions from a branchandbound algorithm. We
170
could not find optimal solutions for the larger problems. Therefore, in order to measure the
heuristics, we computed a lower bound on the flowtimc for each problem. We calculated the
lower bounds described in Section 2.1 and took the largest as the lower bound. The performance
of a heuristic on a problem was taken as the relative deviation from the optimal solution (if
known) or the best lower bound. Due to the special structure of the problem, we could find
improve the second lower bound for the instances in Set 4 by determining the optimal total
flowtimc for the five jobs in Hx.
Minimizing the total flowtimc is harder than minimizing the makespan, and in Table 4.5 we
report the results of the three heuristics described above. These heuristics were selected because
they performed much better than a number of other procedures that combined the groups
differently. The lookahead heuristics found solutions with average total flowtimc within eight
percent of the optimal value. Because of the special structure of the 3MLA instances in Problem
Set LA201, very good solutions were easier to find. The WIPLookahead heuristic was slightly
better on all of the other problem sets. However, the heuristics were very close to each other.
Table 4.5. Heuristic performance for the 3MLAFT Problem.
Performance relative to optimal or lower bound.
Set
SPT
WIP
Johnson
Lookahead
Lookahead
Lookahead
LA 154
7.23
7.13
7.32
LA 304
5.41
5.16
6.15
LA 504
4.95
4.85
6.58
LA 201
1.37
1.37
1.37
4,4 Minimizing the Number of Tardv Jobs
4,4.1 Problem Introduction
The other objective function under consideration in lookahead scheduling is the number of
tardy jobs. This problem models a subproblem of the shop scheduling problem related to the
171
feeding of a bottleneck machine. The objective mirrors the management concern of customer
satisfaction.
Let us call our problem the ThreeMachine Number Tardy (3MNT) problem. Recall that
only permutation schedules need be considered. Thus a sequence for all of the jobs defines a
unique schedule. The problem notation is the same as that for 3MLAMS, except that we have
for each job Jj a due date dy Since 3MLAMS is strongly NPcomplctc, 3MNT is also strongly
NPcomplete. Consider an instance where all of the jobs have the same due date: finding a
schedule with no tardy jobs is equivalent to finding a schedule with makespan less than or equal
to the common due date.
Since the problem is computationally difficult, wc will examine a simple lower bound, a
special case, and some heuristic approaches to finding solutions.
4,4,2 Lower Bound and Special Ca*e
Good lower bounds arc hard to find for the problem of minimizing the number of tardy jobs
in a flow shop (Hariri and Potts, 1989). We will make use of a fairly simple one.
The lower bound makes use of the fact that the MoorcHodgson algorithm will find the
optimal number of tardy jobs for a onemachine problem. Given an instance of 3MNT, the due
date of each job is adjusted by subtracting the processing time of the secondstage task. These
adjusted due dates and the firststage processing times form a onemachine problem for machine
zero. The lower bound is calculated by using the MooreHodgson algorithm to optimally
sequence these tasks. The number of tardy firststage tasks is a lower bound on the minimum
number of tardy jobs for the 3MNT instance.
This lower bound is achievable in the following special case:
Theorem 4.8. If min {pqj : Jj e Hx} > max {p jj : Jj e Hl} and min [pqj : Jj e H2) ^ max
[P2j : Jj e H2], then an optimal sequence can be found be minimizing the number of firststage
tasks that are tardy to the adjusted due dates.
172
Proof. Because of the conditions above, the completion time of a job Jj in HÂ¿ is
Cj = Cqj + pij. Thus, Jj is tardy if and only if Cqj > dj  pÂ¿j. QED.
We also tried lower bounds on each of the secondstage machines, but these were not as
good. This seems reasonable if the interaction of the two flows is significantly contributing to
tardiness.
4,4.3 Heuristics
Simple rules that can be extended for this problem include the MooreHodgson algorithm
and Earliest Due Date (EDD). These can be expanded by including lookahead ideas. We also
developed a simple problem space genetic algorithm to find good solutions.
Lookahead rules. The lookahead extension of the MooreHodgson algorithm includes
both machines. The jobs are ordered by their due dates and added to the schedule until a tardy
job is found. The procedure then determines the critical path of tasks that determines the
completion time of the tardy job. Then, each job in the path is evaluated to determine how much
the completion time would decrease if that job were removed from the partial schedule. This
calculation depends upon whether the job precedes, is, or follows the crossover job (the job
whose first and second tasks arc in the critical path).
We developed two lookahead versions of the EDD rule. The first is similar to the Moore
Hodgson rule. The jobs are sequenced by their due dates. They arc scheduled one at a time.
When a job is tardy, however, we simply remove it to the end of the schedule. (In the Moore
Hodgson rule, we look for the job whose removal helps the most.) This form of the EDD is
called EDDNo Tardy.
The other lookahead version (EDDLookahead) tries to schedule machine one (the
bottleneck) carefully. The jobs in each group arc ordered by their due dates. The primary idea is
to get the jobs from group one to be ontime; group two jobs can be inserted if they donâ€™t
interfere. At any point in constructing the schedule, we consider the next job from each group. If
cither would be tardy if scheduled next, we place it at the end of the schedule. Eventually, we get
173
a job from each group that would be ontime if scheduled next. We then determine if scheduling
the group two job next would cause the group one job to be tardy. If not, we schedule this group
two job and consider the next job from that group. Else wc schedule the group one job next.
Genetic algorithm. After initial testing, wc determined that the lookahead MoorcHodgson
rule found consistently good solutions. Thus, wc used a problem space genetic algorithm to
adjust the problem data used by this rule so that we could find better solutions. The procedure is
similar to those described in Chapter 3. Wc adjust the due dates using a steadystate genetic
algorithm. The population size was 50, and tire algorithm generated 1000 new individuals.
4,4,4 Results
In order to compare the heuristics wc created a number of problem sets. Each set had 10
similar instances. Problems were created with 15, 30, and 50 jobs. The processing times were
chosen randomly, and the due dates were chosen from a range that depended upon the sum of the
processing times.
The lookahead MooreHodgson heuristic performs better than any of the other rules (Sec
Table 4.6). A number of rules that used the firststage due date of a lot were also tested but did
not perform as well. The genetic algorithm was able to find slightly better solutions than those
found by the MoorcHodgson rule.
Table 4.6. Summary table of results for 3MNT.
Set
Jobs
Lower
Bound
EDD
No Tardy
EDD
Moore
Lookahead
Genetic
Algorithm
NT 151
15
5.8
7.8
7.3
7.0
6.5
NT 152
15
5.3
7.6
7.1
7.1
6.3
NT 301
30
9.0
12.7
12.5
12.0
10.6
NT 501
50
17.0
21.6
21.4
20.7
20.4
174
4.5 Application to Job Shop Scheduling
One of the primary motivations for studying these problems was to see if we could develop
useful dispatching rules. After considering our results, it seemed clear that lookahead and look
behind policies similar to those we used on the onemachine and threemachine problems would
be intuitively good ways to dispatch jobs. However, we wanted to determine the tradeoffs of
using such rules on a number of objectives. To this end, we created a job shop scheduling
problem that modeled the semiconductor test area we were studying. This problem had 82 jobs
and 23 machines and included various test operations. Processing time data were gathered from
some historical lots. We scheduled the jobs under a number of dispatching rule combinations,
using standard dispatching rules, lookahead rules, and lookbehind rules. We measured the
schedules on four scales: total flowtime, makespan, number of tardy jobs, and total tardiness.
The bottleneck in this problem was a set of bumin boards. Thus, we developed a lookÂ¬
ahead rule that orders the jobs waiting by their task processing times and uses information about
the downstream bottleneck resource (the bumin board availability) to determine which lot should
be scheduled next. We also developed a lookbehind rule that sequences lots by EDD and
reserves bumin boards for the next late lot that will be arriving soon. The effort of using these
mies is slight for our problem; in a manufacturing environment, the dispatching effort might be
more significant.
We used the following scheme in order to see how lookahead and lookbehind mies would
influence schedule performance. We allocated the mies by dividing the machines into three
areas: electrical test, bumin, and other. In any given policy, all of the machines in the same area
used the same dispatching rule. Our model included one bumin workstation and 15 testers. We
used seven standard mies: SPT, EDD, Slack per Remaining Operation, LPT, Modified Due Dale,
Earliest Finish Time, and FirstInFirstOut in all areas.
For each of the standard mies, we created four policies. In the first, all of the areas used
that rule. In the second, the testers used the lookahead mÃe (since these were the machines
175
feeding the bumin area). In the third, the bumin area used the lookbehind rule. In the fourth,
the testers used the lookahead rule while the bumin area used the lookbehind rule. This
yielded 28 policies.
The average results over the seven standard rules arc summarized in Table 4.7. We observe
that the use of a lookbehind rule, which is concerned with expediting late jobs, has a drastic
effect on due daterelated measures. It is able to reduce total tardiness while increasing the
number of tardy jobs. This is a common tradeoff in scheduling problems. The lookahead rule,
which is concerned with avoiding unnecessary delays, reduces the total flowtime and makespan
objectives. Our results are the consequence of the specific definitions of these lookahead and
lookbehind rules. For other problems, alternative definitions may yield different results. While
our results are not proof that lookahead and lookbehind rules are the answer to solving the job
shop scheduling problem, the decreases in total fiowtime (when the lookahead rule was used)
and total tardiness (with the lookbehind rule) did encourage us to use them in our procedures to
find good solutions (discussed in Chapter 5).
Table 4.7. Performance of 28 dispatching rule combinations.
Fiowtime
Makespan
Tardy
Tardiness
All policies0
2,179,344
88,677
17.1
192,394
Single rules6
2,168,014
89,324
16.6
249,968
With Lookahead6
2,135,800
88,850
16.6
243,253
With Lookbehind6
2,222,639
88,534
17.6
138,130
With both6
2,190,921
88,001
17.6
138,226
Notes: a: Average over all 28 policies.
b: Average over seven policies, one for each standard rule.
4.6 Chapter Summary
In this chapter we have examined a special case of the general threemachine flow shop. In
this problem, the jobs to be scheduled form two classes with different groups, and we wish to
minimize the makespan, the total flowtime, or the number of tardy jobs.
176
We proved that minimizing the makespan is a strongly NPcompIete problem, and we also
identified some properties of optimal solutions and some speeial cases that can be solved in
polynomial time. We developed an approximation algorithm, Johnson Interleaved, that can find
nearoptimal solutions by looking ahead to the future workload of the secondstage machines.
We showed that the worstcase error bound for this procedure is onehalf and that this bound is
tight in the limit. We also developed a branchandbound algorithm that can find exact solutions
to the problem.
Then we described the problem of minimizing the total flowlime. We presented lower
bounds, optimality conditions, and the results of testing on selected problem instances a number
of heuristics that look ahead to current workload at the secondstage machines.
Finally we examined the problem of minimizing the number of tardy jobs. We discussed a
simple lower bound, a special case that achieves this bound, and a number of simple and lookÂ¬
ahead heuristics. We also showed that a problem space genetic algorithm can find better
solutions.
These results have two contributions. First is the analysis of these threemachine problems,
problems previously unstudied in the literature. We conclude from the results of our empirical
testing that lookahead heuristics can find good solutions for the problems of minimizing total
flowlime and minimizing the number of tardy jobs, and the interleaving procedure minimizes
makespan.
Second, while these problems are important questions in their own right, they are also
significant as subproblems in a job shop. It is possible to apply our results to the problem of job
shop scheduling, either as part of a general scheduling procedure or as an attempt to schedule the
bottleneck of a job shop more efficiently.
CHAPTER 5
GLOBAL JOB SHOP SCHEDULING
In this chapter we describe a global job shop scheduling procedure that uses a genetic
algorithm to find a good schedule. We have implemented the scheduling system in a
semiconductor test area. The test area is a job shop and has sequencedependent setup times at
some operations. The concern of management is to meet their customer due dales and to increase
throughput. This requires the coordination of many resources, a task beyond the ability of simple
dispatching rules. We discuss a centralized procedure that can find a good schedule through the
use of a detailed scheduling model and a genetic algorithm that searches over combinations of
dispatching rules. We discuss our effort in developing a system that models the shop, creates
schedules for the test area personnel, and contributes to test area management.
5.1 Introduction
In many areas of manufacturing, the ability of a facility to meet its objectives depends
upon the close coordination of resources. This coordination must occur on different levels:
capacity planning, release planning, and lot dispatching. Effective approaches exist (and new
techniques are being developed) for the first two levels. The third level, meanwhile, continues to
pose very difficult scheduling problems. These are the problems that we address. Like other
researchers, we arc interested in creating systems to better schedule resources in a manufacturing
process, since effective scheduling can lead to improvements in throughput, customer satisfaction
(measured by meeting due dates), and other performance measures.
We arc concerned with the effective scheduling of a semiconductor test area, a job shop
environment. In this facility, a lot is a number of identical semiconductor devices. Each lot must
undergo a number of tests (electrical and physical) and other operations before the product can be
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shipped lo ihe customer. Associated with die lot is the due date of the customer order it will be
used to fill. Because each product has a unique route, different lots take different paths through
the lest area.
A number of characteristics make the semiconductor test area difficult to control. These
include the conflicting goals of management (balancing increased throughput against meeting due
dates), work centers with sequencedependent setup limes, operations where multiple lots can be
processed simultaneously, and the relationships between operations.
The purpose of this chapter is to describe the global job shop scheduling system developed
for semiconductor testing. We will examine the needs of the semiconductor test area, their
requirements for a scheduling system, and our approach to this problem.
The primary goal of the global job shop scheduling system is to use information about the
current status of the shop, the jobs to be manufactured, and the production process in order to
create a schedule of activities for each work center in the shop over a fixed time period. By using
a centralized procedure with global information, the system can search for a schedule better than
that computed by making nearsighted, local decisions.
The system finds good solutions with a genetic algorithm, a type of heuristic search. One
interesting characteristic of this search is that it looks for a good combination of dispatching rules
to find an efficient schedule. The use of dispatching rules is an effective local procedure to create
a job shop schedule; such techniques do not, however, use any global information. Our
procedure addresses this shortcoming. It goes beyond simply finding good local schedules; it
executes a search that looks for a better global schedule. While global scheduling techniques
have been proposed before, the important contributions of our work are the development of a
genetic algorithm for job shop scheduling and the implementation of a scheduling system that
uses such an advanced solution procedure.
The next section of this chapter will review some background and related research on job
shop scheduling. The genetic algorithm for global scheduling is introduced in Section 5.3, where
we also discuss a small example of our search space. In Section 5.4 we describe the scheduling
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problem in ihc semiconductor test area under consideration and the global job shop scheduling
system that wc developed. We summarize our report in Section 5.5 and describe how a similar
system may be useful in other manufacturing environments.
5.2 Job Shop Schcdulimi
As mentioned in the introduction, the problem of coordinating resources to ensure efficient
manufacturing is a difficult problem. When the production process is fairly straightforward,
techniques such as JustInTime (JIT) manufacturing result in efficient scheduling. In many
situations, however, the process is much more complicated, and finding an optimal or near
optimal schedule is an impossible task. In this section we will review some of the approaches to
job shop scheduling. More details on these papers can be found in Chapter 2.
The traditional method of controlling job shops is the use of dispatching rules. A
dispatching rule is a sequencing policy that orders the jobs waiting for processing at a machine;
the ordering depends upon the particular dispatching rule used. Common rules include the
Shortest Processing Time (SPT) and Earliest Due Date (EDD) rules.
While such rules have been the subject of much research, standard dispatching rules have a
narrow perspective on the scheduling problem, since they ignore information about other jobs and
other resources in the shop. More advanced lookahead and lookbehind rules attempt to include
more information, but their reach is still limited. Effective job shop scheduling depends upon the
interaction of a number of factors. The complexity of these interactions makes the development
of a global scheduling system a difficult task.
A number of researchers have looked at semiconductor manufacturing at all levels, from
production planning to scheduling. Approaches to shop floor control include lot release policies,
dispatching mies, deterministic scheduling, controltheoretic approaches, knowledgebased
approaches, and simulation. Uzsoy et al. (1992a, 1993) review a substantial number of papers
that consider these approaches to semiconductor scheduling.
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This work in semiconductor manufacturing includes corporatewide production planning
that uses a ratebased model of production and linear programming (Lcachman, 1993, and
Hackman and Lcachman, 1989). Such a global planning system has been developed under the
name IMPReSS at Harris Semiconductor. Hung and Lcachman (1992) have developed an
iterative method that uses a linear program and a discreteevent simulation to develop longterm
production plans for a wafer fab.
While most of the research in semiconductor scheduling has concentrated on the fabrication
of semiconductor wafers, a number of other papers have addressed the problems of
semiconductor lest. These include Lee, Uzsoy, and MartinVega (1992), and Uzsoy etal. (1991a,
1991b, and 1992b). Lee et al. (1993) report on the implementation of a decision support system
for the dispatching of lots in a semiconductor test area. Our work is the natural extension of this
system.
Previous approaches to the production of detailed shop schedules for planning and shop
floor control include expert systems like ISIS (Fox and Smith, 1984), OP1S (Smith, Fox, and Ow,
1986, and Ow and Smith, 1988), MICROBOSS (Sadch, 1991), and OPAL (Bensana, Bel, and
Dubois, 1988); costbased procedures such as OPT (Optimized Production Technology, reviewed
by a number of authors, including Jacobs, 1984), Faaland and Schmitt (1993), and the bottleneck
dynamics of SCHEDSTAR (Monon et al., 1988); simulation (Lcachman and Sohoni, Najmi and
Lozinski); and leitstands (Adclsbcrgcr and Kanct, 1991). Adler et al. (1993) describe the
implementation of a bottleneckbased scheduling support system for a paper bag production
flexible flow shop. While these approaches have been developed for a number of different
manufacturing processes, the complications of the semiconductor test area forced us to consider a
new design.
Job shop scheduling, as one of the most difficult scheduling problems, has attracted a lot of
attention from researchers. Techniques such as the shifting bottleneck algorithm (Adams, Balas,
and Zawack, 1988) or bottleneck dynamics (see Morton, 1992, for example) concentrate on
solving the problem at one machine at a time. More work has gone into the development and
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evaluation of various dispatching mies. Panwalkar and Iskander (1977) present a list ol over 100
mies. Recent studies include Fry, Philipoom, and Blackstonc (1988), Vepsalaincn and Morton
(1988), and Bhasakaran and Pinedo (1991).
More sophisticated lookahead and lookbchind mies have also been discussed. Look
behind mies (called xdispatch by Morton, 1992) consider the jobs that will be arriving soon from
upstream machines. Lookahead mies consider information about the downstream machines.
This includes the workinnextqueue and the numberinnextqueue mies of Panwalkar and
Iskander (1977), bottleneck starvation avoidance (Glasscy and Petrakian, 1989), and lot release
policies that lookahead to the bottleneck (Wein, 1988; Glasscy and Resende, 1988; and
Leachman, Solorzano, and Glasscy, 1988). Lookahead and lookbchind scheduling problems
have been studied in this dissertation and by Lee and Herrmann (1993).
Finally, heuristic searches have also been developed for job shop scheduling, and a number
of these are discussed in Section 2.8 of this dissertation.
5.3 A Genetic Algorithm for Job Shon Scheduling
In this section we will describe how a genetic algorithm can be used to find good schedules
for the job shop scheduling problem.
One approach to difficult scheduling problems such as job shop scheduling is local search,
an iterative procedure that moves from solution to solution until it finds a local optimum. Smart
andlucky searches (or heuristic searches, or probabilistic search heuristics) attempt to overcome
the primary problem of these simple searches; convergence to local optima. These more complex
searches arc smart enough to escape from most local optima; they still must be lucky, however, in
order to find the global optimum.
In previous sections we reviewed the basic concepts of genetic algorithms and mention
some application of smartandlucky techniques for job shop scheduling. In this section we
describe our application of these ideas.
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5.3.1 The Heuristic Space
Many heuristic searches for the job shop scheduling problem have been considered. One
recentlyintroduced idea is to search the problem and heuristic spaces, since a solution is the
result of applying a heuristic to a problem. A change to the parameters of a heuristic or a
problem yields a slightly different solution. This section will describe how a heuristic space can
be used for job shop scheduling. We will show why a genetic algorithm is especially suited for
searching this space.
The idea of searching heuristic and problem spaces was reported by Storer, Wu, and
Vaeeari (1992). In their paper, the authors examine the general job shop scheduling problem.
They define a heuristic space composed of vectors of dispatching rules. Each vector in the space
can be used to determine a schedule. Each rule in the vector is used for a fixed number of
dispatching decisions, regardless of the machine being scheduled. That is, all of the machines use
the same dispatching rule at the same time until the next rule replaces it.
Our approach uses a different perspective. Since we will be working in a dynamic job shop
scheduling environment, we may not know how many operations will be scheduled. Thus, it is
impossible to divide the scheduling horizon by allocating each rules to a fixed number of
operations. Instead, it seems more fair to assign a dispatching rule to each machine. The system
can evaluate this combination of dispatching rules (which we call a policy) by measuring the
performance of the schedule that is created by using this policy. Changing the policy by
modifying the dispatching rule on one or more machines changes the schedule created.
In order to demonstrate this idea, consider the following fourjob, threemachine problem
and two policies used to dispatch the jobs wailing for processing (see Figure 5.1 for the task
processing times).
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Job Due Date Sequence of Machines to Visit:
J, 8
J2 12
J3 17
J4 15
Machine 1, Machine 2, Machine 3
Machine 1, Machine 2
Machine 2, Machine 1, Machine 3
Machine 2, Machine 3
Machine 1 Machine 2 Machine 3
Job 1
Job 2
Job 3
Job 4
Figure 5.1. Task processing times for each job.
Tasks grouped by a dotted line arc on the same machine.
Policy 1 = [EDD, EDD, EDD], Under this policy, the tasks waiting for processing at Mi
are sequenced by earliest due date First. Thus, when both 7, and/2 arc at M,7, is sequenced first
since its due date is 8.
The dispatching rule For M2 is EDD, so the tasks arc sequenced by the job due date. When
73 and 74 are at M2,74 is selected. After this task is completed, Jl has arrived and its due date is
also smaller than 73. The same thing occurs when 7, Finishes and 72 simultaneously arrives.
184
Policy 1 : [EDD, EDD, EDD]
Schedule: Makespan = 21.
Jl
h
â€¢*3
h
J,
h
J 3
u
h
0 4 7 9 10 11 12 17 21
In Policy 2, ihc altered dispatching rules, [SPT, SPT, SPT], change the selection of jobs.
On M, the job /2 has the shortest task processing time and is thus preferred. On M2 the job J3 has
the shorter task processing time and is scheduled first.
Policy 2 : [SPT, SPT, SPT]
Schedule: Makespan = 18.
h
J.
h
h
h
h
h
h
0 1 3 5 7 8 10 11 14 18
Obviously, the application of a different policy creates a different schedule with a different
evaluation on any objective we could wish to measure (makespan, defined as the maximum job
completion time, was chosen here only as an example).
These policies can be easily manipulated by a genetic algorithm. Traditional genetic
algorithms must be modified for machine scheduling problems since the components of the
strings (the jobs in the sequence) arc not independent of each other. The heuristic space
described above, however, consists of vectors that have independent elements. That is, the
dispatching rule for the first machine docs not affect what values the other elements can have.
Thus, a crossover operation that breaks two strings (vectors) and joins the separate pieces yields
offspring that arc valid points in the search space.
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5.3.2 A Genetic Algorithm for Global Scheduling
We will describe in this section the Genetic Algorithm for Global Scheduling (GAGS).
This search procedure is the engine that finds a good schedule. After discussing it in this section,
we will turn away and address the scheduling system that it drives.
GAGS consists of two primary components: the genetic search and the model of the shop
floor. The interaction between these two functions is outlined in Figure 5.2. The genetic
algorithm starts with a number of policies. (Each policy is a combination of dispatching rules,
one rule for each machine. See Table 5.1 for a list of the dispatching rules that were used in the
global scheduling procedure.) Each policy is evaluated by the schedule that is created if the jobs
in the shop are dispatched according to this policy. The model of the shop floor is employed to
build this schedule from the set of shop, job, and process information.
GAGS
Genetic
Operator
Parents
Population of Policies
Input Data
Offspring
Policy Scheduling Model
Offspring
Fitness
Performance
Schedule
Best Policy
Schedule
Figure 5.2. GAGS  Scheduling Model Interface
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Table 5.1. List of dispatching rules considered.
SPT
Minimum Setup
EDD
Shop: LateSetup MatchEDD
Slack per remaining operation
Modified Due Date
Longest Processing Time
Shortest Remaining Processing Time
Earliest Finish Time
First In First Out
Least work in next queue
EDD Lookahead to bumin
SPT Lookahead to bumin
Lookbchind from bumin
The genetic algorithm creates new offspring policies by combining two policies in the
population (crossover) or changing a solitary parent (mutation). For instance, consider again the
above threemachine problem and the two policies [EDD, EDD, EDD] and (SPT, SPT, SPT],
These policies were evaluated by creating schedules for all three machines. Their evaluations (on
the makespan objective function) were 21 and 18. If we split each policy after the first rule and
link the beginning of the first policy with the end of the second policy, we create the offspring
policy [EDD, SPT, SPT], which the reader can confirm creates a schedule with a makespan of 17.
Policy 3 : [EDD, SPT, SPT]
Schedule: Makespan = 17.
Jl
J 2
J3
h
h
Jl
J 2
J4
J.
J 3
01 4 5 7 8 10 11 13 17
These offspring policies are evaluated using the scheduling model. The genetic algorithm
converges to a number of good policies, and die best is used to create a final schedule.
The model of the shop floor is the problem to be solved, and the genetic algorithm provides
different heuristics. The system depends upon this model, therefore: only with a valid model can
187
a scheduling system create plans that match reality. For our project, the model was a
deterministic simulation of the shop. It uses as input a set of resources that correspond to the
equipment and staffing of the test area, a set of jobs that correspond to the lots that need
processing in the test area, and expert knowledge about the production processes. The policy
provided by the genetic algorithm sequences the jobs at each resource. Events in the simulation
correspond to resources beginning work on a task and resources finishing tasks. Other events arc
added as necessary to control the simulation of the production process.
Note that the search is not affected by the complexity of the scheduling problem being
solved. The genetic algorithm can Find solutions to classical problems and to problems with
prcviouslyunconsidcrcd characteristics. The genetic algorithm can take advantage of complexity
by including dispatching rules designed for use in that environment.
5.4 Global Job Shop Scheduling
In this section we will discuss the manufacturing process and scheduling needs of the
semiconductor test area for which we are trying to create good schedules. As we will sec, the
problem is quite difficult. This will lead us into the description of the design, implementation,
and contributions of our global job shop scheduling system.
5.4,1 The Semiconductor Test Process
The manufacturing of semiconductors consists of many complex steps. This includes four
primary activities: wafer fabrication, probe, assembly, and test. This research is concerned with
the last facility. Although the routes of lots through the test process vary significantly over the
many different types of products, general trends can be described (see Section 2.1 of this
dissertation).
The area under study tests commercialuse semiconductor devices and consists of two
domains: one dedicated to Digital products and another to Analog products. We directed our
work toward the Digital side, where there arc over 1400 product lines. The test area has three
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shifts per day, five days a week. The resources in the Digital side include nearly sixteen electrical
test heads and a dozen branders. The staffing on a normal shift consisted of nearly fifteen
personnel in eight areas. The product mix changes continuously, and staffing and machine
resources often change to reflect this.
Although a typical product may have a route that consists of over 30 numbered operations,
only a fraction of those are steps where significant time and resources are required. In our job
shop model of the test area, we concentrated on the following operations: electrical test (room,
low, and high temperature), brand, bumin, bumin load and unload, visualmechanical test, and
document review. Due to the reentrant nature of the process, an average lot will have twelve
steps that need processing.
Most of the operations in the semiconductor test area are electrical or physical tests,
performed in an automated fashion or by hand. During these tests, each device in a lot must be
examined. Thus, the processing times depend directly upon the size of the lot. Also, the products
in the test area have different package types. A package is the shell in which the semiconductor
chip resides. These packages can be plastic or ceramic and come in many different sizes and
styles. The package of a product also influences how the devices are tested.
In the job shop model, a work center for each of the resources in the shop (except the bum
in ovens) is a tester, machine, or operator that needs to be scheduled. Each type of work center
has a list of the operations that can be performed at the station. Distinct resources that perform
the same operations are modelled as separate stations, each with a queue of jobs that next need
processing at that station. These queues arc sequenced by the dispatching rule for that station.
Two unique operations in semiconductor testing are electrical test and bumin. Electrical
test operations have sequencedependent setups, and bumin is a bulkservice process.
The sequencedependent setups at electrical test are a result of the test requirements. First,
a handler must be attached to the test head to automatically feed the devices in the lot. However,
due to the physical attributes of the packages, different handlers arc required for different
packages. Testing also occurs at various temperatures, which requires additional equipment. The
189
amount of setup is thus affected by the setup of the previous test, since it will be necessary to put
into place new equipment only when the previous setup was different.
The other unique operation is bumin. At this operation, the devices in the lot are placed
into one of a number of ovens for a fixed period of time. These limes range from 24 hours to
over 4 days. Many lots can be bumedin at once, and the capacity constraint is more often on the
availability of bumin boards than on the space in the oveas. Additionally, after the boards are
removed from the oven, the devices arc unloaded from the boards, and the lot must undergo an
electrical test within 96 hours to locate any faulty devices. The combination of bulk service,
secondary capacity constraints, and operation deadlines requires special modeling.
Access to bumin is controlled by the bumin load work center, since each lot must be
loaded onto bumin boards before being placed in the oven. If sufficient boards are not available,
the lot cannot be processed. After bumin load, the lot moves directly to the bumin oven and
begins the bumin period. The factory control system provides information on which day the lot
should be removed from bumin. On that date, the lot is scheduled for unload, and the bumin
boards become available for another lot.
As Lee et al. (1993) mention, the test area has five features that distinguish it from classical
job shop scheduling:
1. Sequencedependent setup times and reentrant product flows,
2. Machines with different scheduling characteristics,
3. Complex interactions between machines,
4. Dynamic production environment, and
5. Multiple, conflicting objectives.
These characteristics make effective scheduling of the semiconductor test area a difficult
task. They also encourage us to design a global scheduling system that can search for good
solutions without being obstructed by complexity.
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5.4,2 The Previous Scheduling System
In this subsection we briefly discuss how scheduling was done previously in the test area
using dispatching rules and some drawbacks to this system. (The development of this system is
described in Lee ct al. 1993.)
The test area was using a set of dispatching stations to sequence the lots awaiting
processing at each work center at the beginning of the shift. During the shift, this ordered list of
lots was updated by using the dispatch station to resequence the lots in the queue, since more lots
may have arrived. The dispatching stations and rules arc part of a software module called Short
Interval Scheduling (SIS).
SIS is a component of WORKSTREAM, a product of Consilium, Inc. (Consilium, 1988).
WORKSTREAM is the test areaâ€™s computerintegrated manufacturing (CIM) system and is
implemented on the corporate VAX mainframe. Transactions such as processing a lot or
beginning a setup are logged into WORKSTREAM, which maintains information about the
current status of each lot and each machine. This information is used by SIS to sequence the lots
waiting for processing.
In addition to WORKSTREAM, the company uses higherlevel systems such as Activities
Planning & Dispatching (AP/D) to match the production lots to customer orders and IMPReSS to
determine the amount of product that each area of the company (including the test area) should be
produced each week. All of these systems provide critical data about the lots to be processed and
the characteristics of the test area. We will need this information to model the test area.
While SIS has led to better scheduling in the test area, it has a number of disadvantages
related to the structure and capabilities of the CIM system. The primary obstacle is that
dispatching rules are making local decisions (even when they attempt to lookahead or look
behind). Thus they are unable to know how their decisions affect the work at other areas in the
test area. And they are unable to advantage of information that may lead to better dispatching
191
decisions. In addition, the dispatching rules are unable to use information about processing and
setup times.
Since the implemented dispatching rules cannot use global information, we began to
consider job shop scheduling procedures. While a number of optimization procedures have been
suggested (see Section 2), these have concentrated on the classical job shop problem of
minimizing makespan. In particular, the more wellknown procedures use the disjunctive graph
to represent the problem. Unfortunately, manufacturing environments like semiconductor test
area often have more complicated problems that require more complicated scheduling models.
We wanted to make use of global information in a complex production process and to
search for a better schedule. Therefore, we decided to implement the genetic algorithm for global
scheduling. We described the primary characteristics of this procedure in Section 3. In the
remainder of Section 4 we will describe the development of our global job shop scheduling
system.
5.4.3 Scheduling Needs
Over the course of a number of years, the scheduling system in the semiconductor test area
progressed from manual dispatching to an integrated rulebased decision support system. As we
mentioned above, this system had significant limitations, since it was based on dispatching rules.
However, the planners in the area need to know in what order the lots waiting at a station should
be processed, and this system gave them this information. They would also like to know when
these lots will be completed.
In addition to these tactical decisions, the planners need to have some idea of how their
limited resources (in both personnel and equipment) on the shop floor affect the performance of
the shop in relation to meeting the shop goals. Finally, they need to measure the performance of
the personnel on the floor.
The managers of this test area have two objectives. The first is to meet customer due dates.
The company stresses customer satisfaction, and the responsibility of the test area is to ship the
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required number of parts to the customer by the requested time. The test area is measured on the
number of delinquent line items, the number of customer orders that arc not shipped ontime.
Thus, the most important of the two primary objectives is to minimize the number of tardy jobs.
The second objective is to test as much product as possible. This goal is both positively
and negatively correlated with the first goal. If the area can test product more quickly, it is more
likely to meet customer due dates. However, the push to increase the quantity tested can interfere
with the need to process lots ontime (since the lots with the earliest due dates may be those with
the largest processing times). At the time we were working on this project, the test area had extra
capacity, so we considered the objective of minimizing flowtimc to be a subordinate one.
5.4,4 Scheduling System Design
We have looked at some of the complexity that occurs in scheduling a semiconductor test
area. These characteristics make the problem in the semiconductor test area much more difficult
than the job shop problems previously considered. At the same time, however, the CIM systems
in place arc able to provide the type of areawide information needed to create a schedule.
Therefore, we arc motivated to try a new approach: a global scheduling system that uses a genetic
algorithm to find good schedules in the presence of shop floor complexity. In this section we will
describe the basic design features.
This system includes a simulation model of the test floor and an optimization procedure
that can search for schedules using a genetic algorithm. The search finds a schedule that is better
than any schedule that a predefined set of dispatching rules could create. This schedule specifies
the order in which the lots should be processed and the times at which the operations should be
done.
The system can be used to react to unforeseen events that might occur during the course of
a shift by including new information and producing a new schedule; for example, a machine may
fail, or an important lot may arrive. The system also serves in a decision support manner,
allowing the planners to determine the effects that changing resources has upon the schedule.
193
Through the use of fair processing times in an accurate model, the schedule can be used as a
target for the shift; the work completed by the shift personnel can be compared to the work on the
schedule.
The foundation for the system is the test area's extensive C1M system, which will provide
the information necessary to model the test area. Finally, the global scheduling system is under
the control of production planners, who use the computing power of the system to evaluate
alternative plans and to create a detailed schedule for the shop floor.
The most important component of a global job shop scheduling system is the model of the
shop floor. Only with a valid model can a scheduling system create plans that match reality. For
our project, the model was a deterministic simulation of the test area. It uses as input a set of
resources that correspond to the equipment and staffing of the test area, a set of jobs that
correspond to the lots that need processing in the test area, and a set of dispatching rules that
sequence the jobs at the resources. Thus, the model creates a schedule for the current scenario.
This is the central relationship in the system. Because the schedule depends upon the
dispatching rules, the optimization procedure in this global scheduling system is a search of the
combinations of dispatching rules in order to find a good schedule (one that meets management
goals efficiently). A genetic algorithm is employed to search over the rules, evaluating each set
by running the simulation with those rules and measuring how well the schedule performed at
keeping jobs ontime. This pan of tlte system is called the Genetic Algorithm for Global
Scheduling, and it is discussed in Section 5.3 of this chapter.
5.4,5 Information Requirements
The scheduling system makes use of data from a number sources inside and outside the test
area's CIM system. In this subsection we will mention the types of information that are used in
the system. See Figure 4 for a diagram of how the primary data structures interact.
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Figure 4. Data Structures
Three general types of files can be described: data extracted from the CIM, data collected
by the project team, and data entered by the user. The first category includes data that is fairly
stable and data that constantly changes. Stable data includes information about the product
routes. This ProductOperation (PROP) file matches each product to the operations that need to
be performed. This file is updated weekly as new products are added to the product mix. Other
files of extracted information list package types, bumin board requirements, and tester
requirements.
The dynamic data is primarily the lot status information. The status of a lot changes as it
undergoes different operations. Before each shift the CIM system takes a snapshot of the status
for each lot in the test area. Our system extracts lot status information for each digital lot from
this report. Although the status of a lot continues to change during the day, our system cannot sec
the changes until the next report.
195
The project team also collected data on processing times and test area resources. The
collection of processing times will be discussed in Section 4.7. The resources files matches each
type of resource (tester, brandcr) to the operations that it can process.
Finally, the user controls other information needed for the scheduling system, including the
current test area resources (how many of each type), the dispatching rules to use, the breaks
schedule, the status of any down machines, and the arrival of any new hot lots.
5.4,6 Implementation of Global Scheduling
The implementation of the global scheduling system followed the design of the system and
the collection of data sources. In this subsection we will describe how we implemented the
system and how the test area creates schedules.
After creating the scheduling model and the genetic algorithm offline, we installed the
basic programs on the corporate mainframe. We began testing the system and developing the
utilities that collect the necessary job, shop, and process information. Working closely with the
test area personnel, we began to run schedules each day. Feedback from these initial schedules
led to improvements in all parts of the system. A technology transfer session formally introduced
the system capabilities to the test area personnel, and soon they began to run the system
themselves. The project team created user interface programs and documentation so that the test
area would be able to use, understand, and maintain the system.
The scheduler performs shift scheduling for the Digital side of the test area. Output from
the system includes the shift schedule, which lists each work center, the lots to be scheduled, and
the approximate start and finish times for each operation. Also available are postshift
performance reports that compare the production of the test area to the schedule.
User interface programs make it easy for the users to collect and modify data, create a
schedule, view the output, and create performance reports.
196
Creating a schedule consists of a number of steps. First, the current status of all oÃ the lots
in the lest area is determined from the preshift lot status report. Nondigital lots and lots in a
stores operation are ignored.
Second, the lot status information is combined widi the PROP file, which converts lots into
jobs. The job consists of relevant lot information and a number of operations. The lot
information includes product class, due date, and lot quantity. For each operation a processing
time is computed along with a remaining cycle time.
Third, the user must cheek the current shop settings on resources, dispatching rules, and
breaks. The user is shown the default settings and has the opportunity to make any changes.
Fourth, the schedule is created using the job, shop, and process information collected to this
point. This schedule can be created using the genetic algorithm, which searches over
combinations of dispatching rules, or the dispatching rules which the user chose. In the latter
case, no search is performed; the scheduling model uses the userâ€™s rules and outputs the schedule
that these rules yield.
The genetic algorithm begins with a population of randomlycreated policies. The genetic
operators shift the population towards more successful policies (measured on their ability to
create a schedule with more ontime jobs). The algorithm stops after a Fixed period of time; this
time was selected after experimentation into the tradeoff between schedule quality and search
effort.
In addition to creating a shift schedule, the system creates a number of other files that arc
used to drive performance reports. There arc three performance reports: shift summary, daily
summary, and daily detailed. The summary reports compare what was done with what was
scheduled. The scheduled operations for each machine are compared to the operations that have
been those lots. The detailed report compares processing times. The scheduled processing times
are compared to die actual times that are derived from daily lot history files.
If the user makes no changes to the default shop settings, die entire process can take as litdc
as ten minutes. This compares to the 15 minutes necessary in SIS to sequence just the lots
197
waiting at one work center. Therefore, the process can be done as part ol the preshift planning,
and the user has ample opportunity to update any data and to experiment with different shop
settings.
The system is currently being used by the test area personnel, who will soon be expanding
the system to include the Analog side of the floor. After discussing some of the issues that our
implementation raised, we will describe the contributions of our system.
5.4,7 Implementation Issues
Researchers engaged in projects like this one often encounter diflicultics that are
considered in no classroom and occasionally learn things that arc in no textbook.
Processing times. One primary problem was a result of our attempt to model an uncertain
process with a deterministic procedure. We wanted to use processing times that were realistic but
that also set fair targets for the test floor. Thus, we needed good estimates of what the processing
times should be. The collection of these estimates was an important concern. We had historical
data from the factory control system, which monitors when each lot begins and ends each
operation. We began with an average for each operation, but this average ignored the variability
in processing time due to lot quantity. We then took the historical data for the subset of products
and fitted a linear regression to the data in an effort to predict processing time from lot size. The
regression was run for different package types in order to remove another source of variability.
Perfection. We learned that perfection is an unattainable goal in a setting as complex as
semiconductor test. However, we also learned that falling short of perfection is sufficient if our
system improves the ability of the facility to satisfy customer demand efficiently. From the
beginning, management knew (better than the project team did) that we could not hope to capture
all of the activities that occur. This attitude allowed us to build a significant model instead of
being overwhelmed by the complexity of finding an optimal solution. With modeling and
controlling the test area perfectly beyond our reach, we concentrated on developing a system
198
which meets the needs of the test area planners and provides them with a tool which they can use
to intelligently manage their facility.
Simplicity. A significant feature of the implemented system is the ability of production
planners to modify data related to resources and other factors. This called for a number of
modules that could manipulate the data without requiring undue effort from the planners. These
modules were userfriendly (easy to understand and failsafe), flexible routines that the planners
felt comfortable using. We feel that this has contributed to the success of the implementation.
Timeliness. Next to the estimates of processing times, accurate information about lot status
was the hardest data to gather. Since there are hundreds of lots in the facility's inventory,
determining which lots arc waiting at which stations is not a trivial task. The system depends
upon WIP extracts that were run from the factory control system before each shift. Although the
planners have the ability through the factory control system to check on the status of individual
lots, it was not possible to develop procedures to gather the status of the lots during the middle of
the shift.
Because the system could not gather the current lot status during the shift, it could only
react to unforeseen events by rebuilding the original schedule from the beginning of the shift and
incorporating the new information about machine breakdowns and lot arrival. Searches to find
good schedules were not possible.
Optimization procedure. A number of other researchers continue to work on ways to solve
the job shop scheduling problem. Although we cannot guarantee that our heuristic space genetic
algorithm is the best procedure for finding a good shop schedule, a search over dispatching rules
seemed to be an ideal approach for the needs and requirements that we faced, however.
The system makes use of the scheduling model and a separate genetic algorithm program.
The two procedures arc distinct modules developed and modified independently. This gave us a
great deal of flexibility as we began to test and implement the scheduling system. We chose the
genetic algorithm to gain the power of its parallelism and its simplicity and to avoid the problems
199
ol local searches. The test area personnel could understand how policies combined to form new
policies and how these policies could be used to create schedules.
Due to the impossibility of finding an optimal schedule to the dynamic job shop scheduling
problem, it is sufficient for our heuristic space search (an approximation algorithm) to determine
a quality schedule quickly.
5.4,8 Contributions of Global Scheduling
In this section we will discuss the benefits of our implementation of the global scheduling
system. While primarily unquantifiable, the gains arc real.
The test area now has a tool that can substantially improve scheduling with a combination
of global information, detailed scheduling model, genetic search for good schedules, reaction to
unforeseen events, and shortterm performance reports.
The global system has a number of strengths compared to SIS, the previous scheduling
system. As a centralized procedure, it can make use of information from around the test area
including processing times, queue lengths, current setups, resource availability, and job arrivals.
The genetic algorithm searches for a schedule that has more ontime lots than a schedule created
by any fixed set of dispatching rules.
As a shift scheduler, the system gives the test area a plan that shows how work at one area
depends upon work at another and that can be used to measure the performance of that shift. In
fact, the ability to accurately model the test area and compute a shift schedule was just as
important to the test area as the ability to find better schedules. The simulation component of the
system allows the lest area planners to forecast how modifying the level of available resources
will affect the output of the test area. The system can also react to certain unforeseen events that
may necessitate a change in the schedule.
As mentioned earlier, the product mix in the semiconductor lest area is constantly
changing. This variety is a significant factor on the test area performance. Thus, we are unable
to measure any longterm quantitative benefits. The test area managers have expressed their
200
confidence that the system will lead to improvements in the performance of the test area through
improved schedules and planning tools.
5.5 Chapter Summar\
In tliis chapter we have described the development of a global scheduling system that uses
a genetic algorithm to find good schedules. This system has been successfully implemented in a
semiconductor test area. Controlling the test area is a complex, dynamic job shop scheduling
problem where it is difficult to meet the management objectives of satisfying customer demand
ontime and increasing throughput. Our scheduling system uses the extensive data available in
the CIM databases in order to simulate the operation of the test area. This system uses a genetic
algorithm to search for combinations of dispatching rules that yield schedules that arc better than
the sequencing that could be done with fixed dispatching rules. The primary practical
accomplishment of this research is the implementation of an advanced job shop scheduling
system in a manufacturing environment. Moreover, this implementation makes use of a new
heuristic procedure that searches the combinations of dispatching rules to find a good schedule.
Thus it is able to adapt each shift to changing conditions in the jobs to be scheduled and the shop
resources.
This approach could be extended to any manufacturing area that has a complicated shop
scheduling problem and a computerintegrated factory control system that can supply the
necessary data for such a global scheduling system. In fact, the semiconductor manufacturing
firm where we have implemented the system is considering exporting this system to other areas.
Additionally, other optimization techniques may be useful in finding good schedules.
CHAPTER 6
SUMMARY AND CONCLUSIONS
This dissertation has reported on a number of production scheduling problems that were
motivated by considering the testing of semiconductors. The research into these topics,
summarized below, adds to the body of knowledge about scheduling. This work is especially
relevant to the study of the harder and broader problem of job shop scheduling. Benefits of the
work include results on specific onemachine class scheduling problems, results on threemachine
lookahead problems, and the use of genetic algorithms and new search spaces on different types
of scheduling problems.
6.1 Onemachine Class Scheduling Problems
The research on the three onemachine class scheduling problems has yielded a number of
results. Most notably, the problem space genetic algorithm is a robust tool for finding high
quality solutions to difficult scheduling problems.
For the problem of minimizing total flowtime subject to deadline constraints (CFTS), we
developed an multiplepass heuristic that makes use of an optimal property for jobs in the same
class. By considering the effect of wasted setup time, it is able to find reasonable solutions. We
can improve upon these solutions with a problem space genetic algorithm that adjusts the job
deadlines in order to create better schedules.
For the problem of minimizing the number of tardy jobs where the jobs have nonzero
release dates (CSRDD), we extend a nonsetup procedure to create a heuristic for the class
scheduling problem. The average performance of the heuristic is good compared to a number of
other dispatching rules. A problem space genetic algorithm is able to find better solutions on
some especially difficult problems where the heuristic performs poorly.
201
202
Our extended heuristics for die problem of minimizing the total flowtimc of jobs with nonÂ¬
zero release dates (FrSRD), were outclassed by a decomposition procedure and a problem space
genetic algorithm. We developed a number of dominance properties for use in a branchand
bound technique.
6.2 Lookahead Scheduling
The threemachine problems that have been studied show that lookahead rules can perform
better than standard dispatching rules.
For the problem of minimizing makespan, the interleaving of the Johnson sequences is able
to provide nearoptimal solutions. The worst case relative error of this heuristic is fifty percent.
There are, however, special cases of the problem that can be solved in polynomial time.
The problem of minimizing the total fiowtime is more difficult. There do exist special
eases where the lower bounds can be achieved. Lookahead rules that consider the queue at the
secondstage machines are able to find good solutions.
The last problem was that of minimizing the number of tardy jobs. Again, special eases
exist where optimal solutions can be easily found. Lookahead rules were able to find better
schedules than other sequencing rules. A problem space genetic algorithm found improved
solutions.
6.3 Searching for Job Shoo Schedules
This research has investigated die development of a procedure for the job shop scheduling
problem. The genetic algorithm makes use of known heuristics (dispatching rules) but increases
their effectiveness by searching over combinations of rules to find a good schedule. This
procedure provides a way to find good schedules under any objective function and in any
scheduling environment, since it makes use of a detailed shop floor scheduler. These
characteristics make this procedure unique.
203
In addition, this procedure has been implemented as pan of a global scheduling system for
a semiconductor lest area. In this environment, it creates realtime schedules for the next shift
using information about the current status of the lots, the current resources in the shop, and the
manufacturing process. The system includes not only the simulation model and the genetic
algorithm but also utility functions for the collection of data from multiple sources and the
generation of performance reports, in addition, the system can respond to unforeseen events, and
the test area planners can use the system to determine the effect of changing the shop resources.
6.4 Conclusions
The scheduling of a manufacturing process is a complicated problem. The sialic job shop
scheduling problem is incredibly difficult to solve, and no system has been able to optimize the
scheduling of an entire dynamic job shop, which is the environment present in many
manufacturing facilities. Thus, continued research into procedures that can find good schedules
is necessary.
Many researchers have studied this problem, introducing systems which range in scope
from companywide planning to machine scheduling. This research is concerned with efforts at
the level of the shop floor and machine.
This research investigated production scheduling problems that arc motivated by
semiconductor test operations and are expected to hold widely in other production environments.
Of particular concern are those problems that occur in the testing of semiconductor devices. Let
us now take a moment to provide some perspective.
This dissertation is concerned with two types of operations research: management science
and management engineering (the terms of Corbett and Van Wassenhove, 1993). Management
science is the discovery of new results that add to the body of knowledge about a subject.
Management engineering (a less active area of the field) is the solution of a practical problem by
modifying existing tools or by using existing tools in original ways.
204
The scientific contributions arc clear. While much research in the areas of scheduling and
semiconductor manufacturing has been performed, this research investigated a number of
previously unstudied problems and methods. Our research into these problems has yielded a
number of useful properties and effective heuristics.
This research has shown that smartandlucky searches and the new problem and heuristic
spaces can be used for the problems under consideration. Problem space genetic algorithms can
find good solutions to machine scheduling problems. For the job shop scheduling problem, a
genetic algorithm can search combinations of dispatching rules and use a shop floor simulation to
determine a good schedule.
Additionally, this research ties together separate problems in an effort to improve the
scheduling of the manufacturing process being studied. This is an engineering question. The
cooperation of the semiconductor test facility motivated research into the problems of an actual
system and provided an opportunity to implement our solution procedures. (This docs not
preclude the potential of our approach to solve problems in other manufacturing environments.)
Our global job shop scheduling system (with its detailed simulation model and heuristic
space genetic algorithm) is a new technique for the problem of creating good shift schedules for a
semiconductor test area in realtime. Since lookahead dispatching rules can be more effective
than standard rules, the results of the work into the threemachine subproblcms have been used as
dispatching procedures for the shop floor and as part of the job shop scheduling procedure.
This research opens some chapters of scheduling that need to further pursued. While the
problem space genetic algorithm is a robust procedure that finds good solutions, better solutions
to particular problems may be achievable through the use of solutionspecific heuristics that can
improve the schedules that the genetic algorithm constructs. Also, it may be possible to create
searches that combine different spaces for other types of combinatorial problems.
The set of class scheduling problems includes a number of other interesting problems; so
does the set oflookahead problems. Even more interesting is the combination of these problems.
205
Lookahead class scheduling problems may yield more insights into the job shop scheduling
problem.
There also exist issues in global job shop scheduling that still need to be addressed: job
release, resource planning, and uncertainty in die manufacturing process. It is difficult to obtain
satisfactory solutions in these circumstances using current scheduling techniques.
The general approach of this research (problem and heuristic space searches) can be applied
to almost any difficult problem. It may be profitable to consider this approach for problems
where good solutions arc hard to find.
Why investigate production scheduling problems motivated by semiconductor
manufacturing? Because organizations and individuals often have difficulty meeting their goals
efficiently, and this dissertation, which has applications beyond the problems contained herein,
offers some ways to solve people's problems.
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BIOGRAPHICAL SKETCH
Jeffrey W. Herrmann has a degree in applied mathematics from Georgia Tech and is a
Ph.D. candidate in the Department of Industrial and Systems Engineering at the University of
Florida. He held an NSF Graduate Research Fellowship from 1990 until 1993 and has been
working on an applied research project with Harris Semiconductor. His research interests include
operations research, production scheduling, and heuristic search.
219
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
ChungYce Lee, Chairman
Associate Professor of Industrial and Systems
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
1 certify that I have read this study and that in my opinion it conlorms to acceptable
standards of scholarly presentation and is fully adequatCTin spopc and quality, as a dissertation
for the degree of Doctor of Philosophy.
"ftt
Sherman Bai
Assistant Professor of Industrial and Systems
Engineering
I certify that 1 have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
Gary Koehler
ProfessOT of Decision and Information Sciences
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
â€ž/SclcUk Eren
Professor of
â€¢ccision and Information Sciences
This dissertation was submitted to the Graduate Faculty of the College of Engineering and
to the Graduate School and was accepted as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
December, 1993
fa
Winfred M. Phillips
Dean, College of Engineering
Karen Holbrook
Dean, Graduate School
â€™s
MARSTON SCIENCE UBRARY
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JULY 94
PAGE 7
AN INVESTIGATION OF PRODUCTION SCHEDULING PROBLEMS MOTIVATED BY SEMICONDUCTOR MANUFACTURING By JEFFREY WILLIAM HERRMANN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993
PAGE 9
ACKNOWLEDGMENTS Years of learning are not sufficient for one who wishes to prepare a dissertation; also necessary is an environment which encourages one to develop the talents which God has given and to become an intelligent person, a responsible person, a kind person, a successful person. For providing such a setting I thank my family. To many others am I indebted: to the chairman of my supervisory committee, Dr. ChungYee Lee, who has been an exceptional influence on my development as a researcher, teacher, and scholar, to department chairman Dr. Jack Elzinga, Dr. Boghos Sivazlian, and the members of my committee; to all who were involved in the Harris project and whose effort was critical to this work; and to the quality friends, classmates, and teachers whom I have known and who have all added something valuable to my life. DISCLAIMER This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. Any opinions, findings, conclusions or recommendations expressed in this dissertation are those of the author and do not necessarily reflect the views of the National Science Foundation. u
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TABLE OF CONTENTS ACKNOWLEDGMENTS page ABSTRACT . CHAPTERS vn L 2. INIRODUCTION ....
PAGE 12
i
PAGE 13
2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.4.5 General Topics 2.4.6 Summary Lookahead and Lookbehind SchcduUng Class Scheduling Some Onemachine Problems 2.7.1 Constrained Rowtime 2.7.2 Release and Due Dales 2.7.3 Flowtime and Release Dates Smartandiucky Searches . 2.8.1 Introduction . 2.8.2 Simulated Annealing . 2.8.3 Tabu Search . 2.8.4 Genetic Algorithms . 2.8.5 Summary Problem and Heuristic Space NPComplcteness . Chapter Summary . 52 55 55 57 60 61 61 62 64 64 64 67 70 75 75 77 78 3. ONEMACHINE CLASS SCHEDULING PROBLEMS 3.1 Introduction .... 3.2 Constrained Flowtime with Setups 3.2.1 Introduction . 3.2.2 Literature Review 3.2.3 Notation and an Optimal Property. 3.2.4 The Heuristic 3.2.5 The Genetic Algorithm 3.2.6 Empirical Testing . 3.2.7 Conclusions . 3.3 Cla.ss Scheduling with Release and Due Dales 3.3.1 Introduction . 3.3.2 Literature Review 3.3.3 Notation and Problem Formulation 3.3.4 Heuristics 3.3.5 Analysis of the Heuristic 3.3.6 The Genetic Algorithm 3.3.7 Empirical Tests and Results 3.3.8 Extension to Minimizing Tardiness 3.3.9 Conclusions . 3.4 Flowtime with Setups and Release Dates 3.4.1 Introduction . 3.4.2 Notation and Problem Formulation 3.4.3 Background . 3.4.4 Solution Techniques 80 80 81 81 84 85 87 91 100 104 104 105 105 106 106 109 113 115 120 123 124 124 125 125 126 IV
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3.4.5 Empirical Testing . 3.4.6 Special Case . 3.4.7 Conclusions . 3.5 Chapter Summary . LOOKAHEAD SCHEDULING PROBLEMS 4.1 Introduction . 4.2 Minimizing the Makespan . 4.2.1 Notation 4.2.2 Johnson's Algorillim 4.2.3 Permutation Schedules 4.2.4 NPComplctencss . 4.2.5 Makespan Optimality Conditions and PolynomiallySolvable Cases 4.2.6 BranchandBound Algorithm 4.2.7 Heuristics .... 4.2.8 Empirical Results . 4.2.9 Heuristic Error Bounds 4.3 Minimizing the Total Flowtime . 4.3.1 Total Enumeration . 4.3.2 Lower Bounds 4.3.3 SpeciiU Case .... 4.3.4 Empirical Testing . 4.4 Minimizing the Number of Tardy Jobs . 4.4.1 Problem Introduction 4.4.2 Lower Bound and Special Case . 4.4.3 Heuristics .... 4.4.4 Results .... 4.5 Application to Job Shop Scheduling 4.6 Chapter Summary .... GLOBAL JOB SHOP SCHEDULING . 5.1 Introduction ..... Job Shop Scheduling ... A Genetic Algorithm for Job Shop Scheduling 5.3.1 The Heuristic Space. 5.3.2 A Genetic Algorithm for Global Scheduhng Global Job Shop Scheduling 5.4.1 The Semiconductor Test Process . 5.4.2 The Previous Scheduling System . 5.4.3 Scheduling Needs 5.4.4 Scheduling System Design. 5.4.5 Information Requirements . 5.2 5.3 5.4 134 136 140 141 142 142 145 145 146 147 149 150 158 160 160 162 164 165 167 168 169 170 170 171 172 173 174 175 177 177 179 181 182 185 187 187 190 191 192 193
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5.4.6 Implcmcniation of Global Scheduling . . . 195 5.4.7 Implementation I.s.sucs ..... 197 5.4.8 Conlribution.s of Global Scheduling ... 199 5.5 Chapter Summary ....... 200 6. SUMMARY AND CONCLUSIONS 201 6.1 Onemachine Class Scheduling Problems .... 201 6.2 Lookahead Scheduling ...... 202 6.3 Searching for Job Shop Schedules ..... 202 6.4 Conclusions 203 REFERENCES 206 BIOGRAPHICAL SKETCH 219 VI
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillmcni of the Requirements for the Degree of Doctor of Philosophy AN INVESTIGATION OF PRODUCTION SCHEDULING PROBLEMS MOTIVATED BY SEMICONDUCTOR MANUFACTURING By Jeffrey William Herrmann December. 1993 Chairman: Dr. ChungYee Lee Major Department: Industrial and Systems Engineering Manufacturing and service organizations frequently face the challenges of making highquality products quickly and of delivering those products to their customers ontime. Improvements in the scheduling of their operations can often contribute to their success in meeting these goals. However, as manufacturing processes become more complex, the difficulty of finding good production schedules increases. This dissertation addresses dynamic deterministic job shop scheduling, a problem that occurs in many manufacturing environments. The problem is among the most difficult scheduling problems, and few solution procedures have been implemented. The approach in this research is to consider specific subproblems that are motivated by semiconductor test operations and to develop genetic algorithms that exploit alternative search spaces. The research includes new analytical and empirical results for previously unstudied onemachine class scheduling problems and threemachine lookahead problems. The onemachine problems include sequencedependent setup times. In the threemachine problems, two groups of jobs are processed on separate secondstage machines. Testing shows that a new type of genetic algorithm can find good schedules for the onemachine problems by adjusting the problem data while using an appropriate heuristic. An approximation algorithm for the threemachine problem is able to find nearoptimal schedules. vu
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Moreover, this dissertation describes the development and application of a global job shop scheduling system for the semiconductor test area. This system uses a detailed deterministic simulation model of the shop iloor, data about the current status of the shop, and a genetic algorithm to search over combinations of dispatching rules in order to create a good shift schedule. These rules include those motivated by this research into the onemachine and threemachine problems. The scheduling system is able to adapt to changing conditions each shift. The benefits of this work consist of the identification of dominance properties for the onemachine class scheduling and threemachine lookahead problems, the development of a problem space genetic algorithm, the definition of new lookahead heuristics, the creation of a new genetic algorithm for global scheduling, and the implementation of these results to the actual semiconductor test floor that is the motivation for this work. viu
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CHAPTER 1 INTRODUCTION Why investigate production scheduling problems motivated by semiconductor manufacturing? Because scheduling works. Scheduling, the term, refers to the process of assigning tasks to resources and determining when each task will be done. Scheduling is not, however, limited to manufacturing, since this type of problem occurs in other activities. Scheduling, the science, embodies knowledge about models, techniques, and insights related to actual systems (Baker, 1992). This field is an important part of operations research. In order to compete effectively in the marketplace, firms have used operations research as a tool for understanding and improving their manufacturing systems. This is especially true in the manufacturing of hightechnology products, where the new, complicated processes lead to production scheduling problems that are not wellsolved. The postassembly testing of semiconductors, for instance, is a complicated job shop scheduling problem; among the difficulties is the presence of sequencedependent setups at certain operations. Moreover, scheduling, which has been studied from the eariiest days of operations research, has received new attention due to the successful implementation of justintime systems that emphasize the close coordination of resources and the maintenance of low woricinprocess inventory levels. Although the research in production scheduling has yielded a large body of knowledge which has been successfully applied to a number of areas, there still remain problems to be solved. The problems previously studied have always simplified reality in some way. Adding realism creates problems that resist simple solution techniques. Further research into these types of problems is necessary in order to increase our ability to control manufacturing processes effectively. 1
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The research described in ihis dissertation falls into this category. It includes factors that model reality more closely and studies a number of different problems in an effort to gain iasighl into how improved job shop scheduling may be achieved. This insight will be applied to the semiconductor test area being studied. The problems that this research investigates are motivated by a hightechnology field where much time has been spent investigating manufacturing systems, namely the semiconductor industry. The research is particularly motivated by the operations of semiconductor testing, although the problems can also be found in the processes of other manufacturing industries. This introduction includes a number of sections about topics that will arise in the scheduling of semiconductor test operations. These topics are introduced here in order to describe how they are related to this research. The first of these is an overview of semiconductor manufacturing. 1.1 Semiconductor Manufacturing In order to give some context to the problems being studied, this section describes the general flow of semiconductor manufacturing. The manufacturing of semiconductors consists of many complex steps. The first process is that of wafer fabrication, done in a superclean environment in order to protect the delicate structures on the wafers from contaminants. The wafers of silicon undergo repeated applications of the cycle of photolithography, etching, and diffusion in order to build, layer by layer, the different electronic devices. A wafer will contain a number of identical circuits laid out in a grid pattern on the wafer. In the next process, probe, these individual circuits are inspected with a wafer probe and marked if defective. In the assembly process, another clean area, the wafers are cut into the separate circuits. Good circuits are placed into the packages that protect them from the environment and that provide electrical connections to allow them to interact with the world. The identical devices from a lot of identical wafers become a lot that then moves into the test
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facility, an area where cnvironmenial safeguards protect the devices from static electricity that could still damage the circuits. The lest process consists of initial testing, bumin, and Tmal testing. The testing includes electrical testing of each device and other testing of the device packages. Bumin consists of subjecting the semiconductor chips to extreme thermal stresses. Other nontest operations, such as serialization and brand, take place during the process. Not all of the lots follow the same route through the lest floor. Different products are tested on different machines and undergo a different set of tests. Thus, the test area is a job shop, and the scheduling problem is a job shop scheduling problem. This job shop of semiconductor test operations has sequencedependent setup times, reentrant flows, batch processing, routes that temporarily leave the test area, and other complications that lead to many interesting scheduling questions. The primary management objectives for this area are customer satisfaction and making profit. Because testing is the last process in the manufacturing flow, the output of this area affects how well the company can get orders to its customers on time and how much product the company can ship to generate revenue. The particular semiconductor test area being studied has implemented a decision support system to assist in the scheduling of lots through the area. The programs in the system can order the lots waiting for an woricstation with priority rules that include information about lateness, priority, and setups. As part of the facilitywide computerintegrated manufacturing system, it is able to access realtime information about the lots. The system is used to create schedules for a short time period (two to four hours), since dispatching more often would require excessive computer resources. Lots that are tardy and lots that have some extcmallyimposed priority are expedited whenever possible.
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1.2 Job Shop Scheduling The semiconductor lest area is a job shop, and optimizing the scheduling of the area is a job shop scheduling problem. This section discusses job shop scheduling. Job shop scheduling consists of those scheduling problems in which different jobs may follow different routes tlirough the shop. A job consists of a number of operations or tasks, which must be processed in the given order. There exists a specified machine (or workstation) that can perform each operation. These problems are generally the hardest to solve optimally, since few properties of optimal schedules are known and the number of possible solutions explodes as the problems increase in size. However, in many cases, the job shop is a better model of reality than onemachine, parallelmachine, or flow shop problems. Because of the complexity of job shop scheduling, algorithms to find the optimal solution (in a reasonable amount of time) for any arbitrary objective function do not exist. Recent research has shown that the shifting bottleneck heuristic is successful at finding good solutions for a simple job shop scheduling problem. Traditionally, however, researchers have studied and schedulers have used dispatching rules to order the jobs waiting for processing at a machine. When a machine becomes available, it chooses from among the jobs in its queue by using a dispatching rule. Common dispatching rules employ processing times and due dates in both simple rules and complex combinations. These dispatching rules are often extensions of the algorithms used to solve onemachine problems. In any shop, there may exist a bottleneck machine whose throughput is a limiting factor on the capacity of the shop. If so, the improved scheduling of this one machine becomes an important objective. 1.3 Lookahead and Lookbehind Scheduling This section addresses a weakness of the traditional dispatching rule approach to scheduling job shops and defines two types of rules that overcome this weakness.
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Traditional dispatching rules arc myopic; they arc concerned only with the machine to be scheduled and the jobs waiting for that machine. Improved scheduling may be realized by using rules that can consider morc information, in order to see that critical lots arc arriving soon or that a certain machine has an excessive queue. Lookahead and lookbehind scheduling includes procedures that look around the shop for more information to use in making a scheduling decision. With this additional information, they can hopefully produce better decisions. Lookahead models consider the machines where the jobs will be headed after this stage. Lookbehind models consider the jobs that will be arriving at this machine soon. The terms lookahead and lookbehind are used to designate scheduling procedures that do more than consider just the state of one machine. Both types of models look into the future (where jobs will go and what jobs will be arriving). The difference is what part of the future they consider. 1.4 Setups One of the complications of scheduling the operations in semiconductor testing is the presence of different product types that require different configurations on the same machine. The task of configuring a machine in order to process a job is a setup. If the setup for a job is independent of the job that was scheduled before it on the same machine, the setup time can be included in the processing requirements of that job. If the setup for a job is sequencedependent, that is, the time of the setup depends upon the immediate predecessor job, the scheduling problem becomes quite difficult. Because the problems with sequencedependent setups are quite difficult, researchers have examined special ca.scs of the general problem. The most common problem is the class scheduling problem, where the jobs to be processed are grouped into a number of job classes. There exists no sequencedependent setup between jobs from the same class, although there does exist a special setup, the class setup, when a job from one class is processed after one from another class. There may also exist a class setup for the first job processed. These class setups
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may be sequencedependent in iliai tiiey depend upon the class that was last processed. Class scheduling is still dirficult. though: the class scheduling extensions of onemachine problems that are easy to solve are usually NPcomplete problems. An example of class setups occurs in the electrical testing of assembled semiconductor devices on a machine which can test a number of different types of semiconductors. If a machine is scheduled to test a lot consisting of devices that arc different from the devices tested in the previous lot, various setup tasks are required. These tasks may include changing a handler and load board that can process only certain types of semiconductor packages or loading a new test program on the tester. However, if the new lot consists of devices that are the same as the previous type, none of this setup is required. This is a class scheduling problem. 1.5 Smartandluckv Searches As mentioned before, algorithms to solve the job shop scheduling problem optimally in reasonable time do not exist. In addition to the shifting bottleneck heuristic and the use of dispatching rules, one way to find good solutions is to search for them. This section will discuss standard local searches and new, more sophisticated heuristic searches. One method of finding good solutions to hard optimization problems has been local search. A local search begins with some initial solution and moves from an incumbent solution to a better neighboring solution, ending when no improvement can be found. At this point, the search has reached a local optimum. Two common local searches are hillclimbing and steepest descent. In a hillclimbing search, the neighbors are chosen at random, and the first better neighbor found is chosen as the new solution. In a steepest descent search, the entire neighborhood of the incumbent solution is searched and the neighbor that has the best performance improvement is selected. In order to overcome the fact that these searches converge only to local optima, new heuristic searches have been developed. These include simulated annealing, tabu search, and genetic algorithms.
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These heuristic searches arc complex searches that are smart enough to escape from most local optima and are lucky enough to generally find good solutions. Simulated annealing (SA) is a variant of the hillclimbing algorithm. Simulated annealing is so called because the algorithm views optimization as a process analogous to physical annealing, the cooling of a system until it reaches a lowenergy slate. In the same way that a system may pass through higherenergy states during the cooling process, the simulated annealing search occasionally moves to worse neighbors before settling into a good solution. Tabu search (TS) is a variant of steepest descent. Given an incumbent solution, a TS canvasses the neighborhood of this solution in order to find the best allowable move. If the search is at a local optimum, it is forced to move away, and the move that would return to the previous solution is temporarily prohibited (tabu) by adding it to the tabu list for a number of moves. In this way, TS may continue to move away from a local optimum and find an area of the space that leads to another local optimum. An aspiration level insures that the search will make a move that leads to a solution better than any yet found, even if tabu. Thus, a tabu search works because the tabu list forces the search to explore new areas of the solution space. The shortterm aspect of the memory and the aspiration level allow, however, the search to get to a global optimum. A genetic algorithm (GA) is a procedure that mimics the adaptation that nature uses to find an optimal state. In genetic algorithms, solutions are represented as strings (chromosomes) of alleles. An allele is a bit of information about the solution. The search performs operations on the population of solutions. TTiese operations are 1) the evaluation of individual fitness, 2) the formation of a gene pool, and 3) the recombination and mutaUon of genes to form a new population. After a period of time, good strings dominate the population, providing a good solution. When applied to scheduling problems, simulated annealing and tabu search easily search the complex solution spaces with simple types of moves, generally find good solutions fast, and are smart and lucky variations of standard searches such as hillclimbing and steepest descent.
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g Gcnclic algorithms arc something completely diffcrcnt, work well by using good strings and implicit parallelism, and arc harder to implement. 1.6 Problem Space and Heuristic Space Typical problemsolving searches like those described in the above section have examined the solution space. A solution can be a vector of values or a sequence of objects. Applying a heuristic to a problem yields a point in the solution space. This implies that a search over all of the possible heuristics would fmd a solution that gives the optimal objective function value. Moreover, applying a heuristic to another, similar problem also generates a point in the solution space which can be evaluated for the original problem using the objective function. Thus, a search over the new problem space provides a way to solve the problem. An example of a heuristic space is the space of vectors of m dispatching rules for a wjob and mmachine job shop scheduling problem, where each dispatching rule is applied to a different machine. Searching over the vectors of dispatching rules is similar to machine learning. For a complicated onemachine problem, a sample problem space would be the space of ^element vectors, where each element corresponds to an alternative due date for a job. Sorting the jobs by these alternative due dates creates a new solution. 1.7 Objective Functions The management of a manufacturing system is concerned with many different performance measures, including customer service, inventory holding costs, throughput, and machine utilization. This section describe how the objectives used in scheduling mirror these concerns. Scheduling problems include many different objective functions that attempt to model the concerns of those who are running the system. Most objectives are functions of the completion times of the jobs to be scheduled. The makespan is the maximum completion time, which is when all of the jobs are finished, and is some measure of the throughput of the system. The total flowtime is the total time that the jobs spend in the system and is a measure of the workin
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process inventory held while processing ihe jobs. Other objective functions include due dales for each job. where a job is lardy if il compleies after iis due dale. These objective functions are concerned with customer satisfaction, and include maximum lateness, which is the maximum difference between a completion time and a due date, the number of lardy jobs, and the total tardiness. Another interesting objective function is earliness, which is a measure of the holding cost incurred by storing finished product until it is delivered at its due date. While all of these objective functions (and many more) have been studied for onemachine problems, most analysis of flow shop and job shop scheduling has focused on the minimization of makespan. Since this objective is not appropriate in semiconductor manufacturing, this research is especially interested in solving job shop scheduling problems with other objectives. 1.8 Overview of Research There exist different approaches to attacking the job shop scheduling problem. One can use the shifting bottleneck algorithm to minimize the makespan, use good dispatching rules in a dynamic environment, or search for good global schedules. This research is concerned with the last two methods (see Figure 1.1). This research investigates the use of smartandlucky searches over the new search spaces to find good solutions to the job shop scheduling problem. Also investigated are a number of interesting onemachine class scheduling problems and lookahead models in order to devise techniques that can be used as good dispatching rules. Finally, this research studies how these techniques may be applied to the actual semiconductor test floor that is the motivation for this work. The benefits of this work include three areas: 1 . The addition of results to the body of knowledge about specific scheduling problems, including the difficult class scheduling problems and little studied lookahead models. 2. The construction of a robust genetic algorithm for onemachine class scheduling problems. 3. The development and implementation of a genetic algorithm for global job shop scheduling. Moreover, this work continues the investigation of scheduling semiconductor test operations that has only recently begun.
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10 SCHEDULING Semiconductor Manufacturing Job Shop Scheduling Shifting Bottleneck Dispatching Rules x T Onemachine Problems 1 Searches LookAhead Problems Class Scheduling Flow Shop Smart &c Lucky Searches Problem and Heuristic Space Figure 1.1. Relationship of dissertation topics. 1.9 Plan of Dissertation This research investigates both onemachine and multimachine problems. The onemachine problems studied are class scheduling problems that model the complicating factor of machine setups in the manufacturing process. This work also considers some threemachine general flow shop problems that are attempts to model how scheduling can be improved by considering the states of other machines. The research also investigates a general job shop scheduling problem in an attempt to see how new search spaces may be used for global job shop scheduling. For these problems, the research has focused on finding good lower bounds, developing heuristics to find good solutions for the problems, and using genetic algorithms to find better solutions. Empirical testing of these bounds and heuristics serves as a way of evaluating the
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11 heuristics and the searches. Also, good heuristics for the subproblems can be implemented as advanced dispatching rules for solving the job shop scheduling problem. In addition, analytical results such as NPcompleteness, lower bounds, error bounds, and oplimality conditions arc presented for the problems being studied. The machine scheduling problems under investigation arc as follows: 1. Constrained Flowtime with Setups (CFTS) 2. Class Scheduling with Release and Due Dates (CSRDD) 3. Flowtime with Setups and Release Dates (FTSRD) 4. ThreeMachine LookAhead Scheduling: Makespan (3MLAMS) 5. ThreeMachine LookAhead Scheduling: Flowtime (3MLAFT) 6. ThreeMachine LookAhead Scheduling: Number of Tardy Jobs (3MNT) 7. Job shop scheduling The first problem is the class scheduling extension of the onemachine problem of minimizing the total flowtime of a set of jobs that have deadlines. The second problem is the class scheduling problem where each job has a release date and a due date. The objective is to minimize the number of tardy jobs. The objective in the third problem is to minimize the total flowtime where each job has a release date. The research next considers the threemachine lookahead problems with the objectives of makespan, total flowtime, and number of tardy jobs. Finally, this research investigates the general job shop scheduling problem and a heuristic space approach to finding good solutions with a genetic algorithm. The dissertation consists of the following five chapters: the background; the research on onemachine class scheduling problems, on threemachine lookahead problems, and on job shop scheduling; and the conclusions. The background material contains detailed information about the problems and methods under investigation and summaries of a number of papers that are related to this work. The topics include semiconductor test operations and semiconductor scheduling, job shop scheduling
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12 techniques, class scheduling, some onemachine problems, smartandlucky searches, problem and heuristic space, lookahead and flow shop scheduling, bumin scheduling, and NPcompleteness. The discussion of the research contains reports on the problems that have been investigated: three onemachine class scheduling problems, three threemachine general flow shop problems, and a heuristic search for use on the general job shop scheduling problem. In the conclusions we summarize this research and identify some directions for future research.
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CHAPTER 2 BACKGROUND This chapter of the dissertation perfonns two functions: it elaborates on the topics in the introduction and provides a review of the relevant literature. The discussion in this chapter is both wideranging and descriptive. In Chapters 3, 4, and 5, discussions of previous research will be problemspecific and less explicit. The first section will describe in more detail the particular operations associated with the testing of semiconductor devices. Remaining sections will discuss papers on semiconductor manufacturing, job shop scheduling techniques, and flow shop scheduling problems. Also included in this chapter are sections on lookahead and lookbehind scheduling, the scheduling of the bumin operation, class scheduling, some onemachine problems, smartandlucky searches, problem and heuristic space, and NPcompleteness. 2. 1 Semiconductor Test Operations The manufacturing of semiconductors consists of many complex steps. As described in the introduction, this includes wafer fabrication, probe, assembly, and test. This research is concerned with this last facility. Although the routes of lots through the test process vary significantly over the many different types of products, general trends can be described. Figure 2. 1 shows what these routes look like for seventeen representative products of a test area. The numbers on each arc are the number of products that follow that path. Presented here is a description of the operations in a typical product route. The terms semiconductor and device describe the same thing. 13
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14 LOT FORM INSPECTION SERIALIZATION XRAY INITIAL TEST BURN IN1 INTERIM TEST BURN IN2
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15 The loi arrives at the lolform area with an assembly traveler that describes the history of the lot up to that point. A lot is started through the test floor when it is needed to meet backordered demand or when the area planner includes it on the weekly release plan. The test traveler for this product is printed, and engineering information is extracted from the database and attached to the traveler. The tubes of devices arc placed into a box, and the box is moved to a waiting area. The lot is then serialized. Done in the brand area, serialization is the process of giving each device a unique serial number. This process is laborintensive as someone must stamp each device with a branding machine. The lot then undergoes Xray testing. This operation includes six tasks; loading the devices into racks, shooting the Xrays, unloading the devices, developing the Xrays, reading the Xrays for internal defects in the package assembly, and culling the rejects. The lot may experience significant waiting before and after the reading of the Xrays. Three people handle the lot: one who loads and shoots, one who reads, and one who culls. Electrical testing, the next step, is an important, machineintensive operation. During an electrical test, the lot is processed by a handler machine that positions each device over a contact with the tester, a computer that performs a number of tests by subjecting the device to a number of different electrical inputs. After the test of that semiconductor is done, the device is fed to an output bin. The test floor contains a number of different types of testers and different handlers. Since each product requires a different combination of handler and tester, significant setup time can be incurred if the operator must find and install a handler to process the lot. After the initial test, the lot must be bumedin. The devices in the lot are loaded onto boards that hold from 10 to 100 devices each. These boards are placed into ovens, where the devices are subjected to thermal stresses and electrical inputs in order to cause infant mortality. The bumin period lasts a minimum of 24 hours, although no penalty is associated with keeping the devices in the oven for longer than this time. After the boards are removed from the oven, the devices are unloaded from the boards.
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16 After the bumin period, Uie lol returns to a tester and undergoes interim test. Interim lest consists of testing at room, low, and high temperatures. For the low temperature test, a botUc of liquid nitrogen must be attached to the handler and the handler must chill the testing environment to the desired temperature. In a test at high temperature, the handler warms the test chamber to the desired temperature. After the last bumin and interim test, a final test is conducted. The lot then goes to the assembly facility, where fine and gross leak testing of the package seal is performed. During this time, the lot is out of the test facility's area of control. The lot retums and moves to brand, where each device is stamped with the company logo. The devices are placed into pans and then sit through a threehour bake in a kiln. After this step, testing is performed to verify that no damage was done to the devices during brand. After another trip to assembly, the lot undergoes a visual test, where each device is inspected under magnification for obvious defects in the packaging. The lot enters document review next. At this time, all of the paperwork associated with the lot must be reviewed for completeness. The lot then undergoes customer source inspection, where a representative of the customer personally supervises the testing of a sample of the lot. In the shipping operation, the devices are placed in boxes and the boxes are addressed, ready for delivery to the customer. 2.2 Semiconductor Scheduling The manufacturing of semiconductors has received much attention form production planning and scheduling researchers. This section reviews a number of papers that address the issue of semiconductor production planning and scheduling directly or indirectly. As in Uzsoy, Lee, and MartinVega (1992a, 1993), the papers are classified into the following topics: production planning; shop fioor scheduling: dispatching rules and work release, deterministic scheduling, batch processing, controltheoretic approaches, knowledgebased approaches, and simulation; and performance evaluation: queueing models, and simulation. This review is
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17 intended to display the numerous methods that have been used in the research of production and scheduling issues of semiconductor manufacturing. 2.2.1 Production Planning A number of authors have addressed the largescale problem of production planning for the semiconductor industry. The most common approach has been a hierarchical decomposition of the problem. Leachman (1986) describes a corpraratelevel production planning system. This system divides the manufacturing process into the stages of wafer fab, probe, assembly and test, linked by inventories. Its model may include multiple facilities. The production processes in a plant are treated as a single entity, and problem complexity is reduced by the aggregation of products that are in the same process at each stage. Information from corpxjrate databases is used in a linear program with dynamic production functions that capture the relationships between the processes. This yields a production plan. Through a set of linear programs, the aggregate plan is then decomposed into a capacityfeasible weekly start schedule for various facilities. In a later work, this system was implemented by Harris Semiconductor. Hadavi and Voigt (1987) describe the planning system for a Siemens development wafer wafer fab. In their approach, they create different levels of abstraction for different levels of planning and localize rescheduling while making minimal resource constraints. The system architecture is a hierarchy of constraint sets that represent different time windows. Quarterly requirements are divided into months, weeks, and days. Arriving orders start feasibility analysis (can it be done?) and then a scheduling heuristic (when will it be done?) that tries to satisfies the constraint sets (based on the hierarchy). Scheduled orders are sequenced by an algorithm that maximizes throughput. A rulebased expert system recovers from disturbances by local rescheduling.
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18 Harrison, Holloway, and Palcll (1989) discuss the production planning question and present a case study of National Semiconductor Corporation. They list the following programs that can improve customer service: management information systems; logic and algorithms used to generate delivery quotations and to schedule production; and performance evaluations and incentive systems based on measures of delivery performance. They place the emphasis on operational decisionmaking, from booking orders to scheduling lots on machines. The authors also consider performance measurement, which should give the upper management information about the different groups, all of whom are acting to improve their performance on these scales. These measures should encourage mangers to act optimally for the company. The authors make the following comments about scheduling. Marketing wants commitments to be inviolate; production planners thus want control over factory loading. Managers view scheduling as a MIS problem that provides reliable routine execution. The company must get the right information to the right people and then coerce them to do the right thing. Golovin (1986) describes various attempts to solve the problem of production planning and factory scheduling. Mathematically, an integer programming problem of the factory scheduling problem considers all costs. The solution philosophies are diverse and involve certain tradeoffs. Mathematical optimization gives the best solution but is computationally costly in data and time and ignores local conditions. Dispatching rules are concerned only with the present. They may make poor scheduling decisions by ignoring global conditions, but they work quickly and cheaply. JustInTime maintains a balance of work in the factory and attempts to maintain a high level of quality. The authors find most promising hierarchical systems that decompose the problem into sets of decisions made at different times: capacity planning, release planning, and lot dispatching. This approach may yield suboptimal policies but it gives control of decisions to the users who are responsible for the results. Scheduling is done under the assumption that sufficient capacity
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19 exists. Dealing with uncertainly in yields and equipment calls for keeping safety slock of standard product and buffers in front of each machine to prevent starvation. 2.2.2 Shop Floor Control Methods used to control the shop floor vary widely in their approaches to solving the problems encountered there. The basic areas include lot release policies, dispatching rules, deterministic scheduling, controltheoretic approaches, knowledgebased approaches, and simulation. The production of detailed shop schedules for planning and shop floor control has also been considered by researchers looking for some way to go beyond materials requirements planning, which docs not consider capacity constraints, and to search for schedules better than those found by dispatching rules. Vollmann, Berry, and Whybark (Chapter 5, 1988) discuss three simple ways in which shop floor control can be done: Gantt charts (often created backwards from the job due dates), priority sequencing (dispatching) rules, and finite loading schemes, which create a schedule for the time horizon by simulating the operation of the shop. Bai and Gershwin (1989) initiate a discussion of all important phenomena in semiconductor fabrication and characterization of all events and scheduling objectives and factory types. They start with the observation that two types of events exist: controllable and uncontrollable. Controllable events are those that the person in charge of scheduling the shop can start. The state of the system is a complete description of the production variables, such as the status of each machine and each worker. The variables in the state are chosen by the scheduler for his purposes, and he may ignore certain inconsequential quantities. Events change the state of the system, and the current state limits the options of the scheduler, who must make a decision based upon the state. The authors note that schedules in the real worid are subject to disruption by uncontrollable events and that schedulers have three weapons to reduce the effects of disruption: realtime
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20 scheduling systems, prediction, and inherentlyrobust schedules. The first is the most powerful, but the computational effort makes combinatorial optimization impractical; thus, scheduling heuristics are necessary. The paper includes a summary of the semiconductor manufacturing processes and a detailed description of wafer fabrication. This description includes an explanation of the different machines, workers, and events in the fabrication process. The uncontrollable events are classified into those that are predictable and unpredictable. The latter include machine failure and defective wafers and are the ones that make life difficult for schedulers. The authors also include the constraints on scheduling and the objectives of scheduling, which are the minimization of WIP, throughput, variability in throughput, and costs and the meeting of demand. Lot release policies and dispatching rules . A number of researchers have realized that the release of work into the shop has large ramifications on the performance of that shop. Dispatching rules are in use in many shops, and some work has been devoted to developing better rules. Wein (1988) considers the problem of minimizing cycle time in a semiconductor wafer fabrication. His approach is to study the input regulation policy. He studies four alternatives: no control (Poisson arrivals); uniform start policy (constant release rate); constant WIP (closed loop); and workload regulating. This last rule focuses on the heavilyutilized workstations and uses a Brownian network model to approximate a multiclass queueing model with dynamic control capability. A lot is released when the work in the system for a bottleneck machine falls below a certain level. Bottleneck machines use various dispatching rules while other machines use FIFO. The workload regulation rule was found through simulation studies to reduce the mean and variance of cycle time, and the effects of dispatching rules were less significant and varied by system and input type. The study uses data gathered at HewlettPackard Technology Research Center in Palo Alto. Glassey and Resende (1988 a,b) examine a wafer wafer fab with a single bottleneck workstation, single product, and constant demand. Their approach is to control input regulation
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21 by considering global informalion. The measure of performance is the cyclelime versus throughput curve. Their answer is to release jobs so that they arrive at the bottleneck just in time to avoid starving that machine (starvation avoidance). This requires the determination of the virtual inventory, the amount of work for the bottleneck machine that is there now or will be soon (within the lead time it takes a released job to reach the machine). If this inventory is less than the lead time, release a job. This is similar to a safetystock inventory policy. Simulation studies reveal that starvation avoidance results in lower cycletime vs. throughput curves than other release rules: uniform, fixedWIP, and workload regulation. Solorzano (1989) describes the implementation at Harris Semiconductor. Glassey and Pctrakian (1989) consider the problem of minimizing the queue of a bottleneck machine in a shop using starvation avoidance input regulation. They do so by using queue predictions and dispatching rules that give higher priority to lots that should encounter a shorter queue on their next visit to the bottleneck. This extra queue prediction computation is intensive, although certain simplifying assumptions are made. Objectoriented simulation studies show good behavior (better than dispatching rules such as SPT and FIFO) in both a oneproduct wafer fab and a twoproduct wafer fab. Wein and Chevalier (1992) take a broader view of the jobshop scheduling problem by considering three dynamic decisions: assigning duedates to arriving jobs, releasing jobs from backlog to shop floor, sequencing jobs at each workstation. Their objective is to minimize WIP and duedate lead time, subject to an upper bound constraint on fraction tardy. They take a twostep approach: (1) release and sequence jobs to minimize WIP subject to throughput rate; and (2) set due dates that minimize duedate lead time. The authors propose three principles: (1) while maintaining fraction tardy, average due date lead times can be reduced by dynamically basing duedates on the status of the backlog and shop floor, the type of arriving job, and the job release and sequencing policies used; (2) without affecting the throughput, WIP can be reduced by regulating the amount of work on the shop floor
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22 for bottleneck stations; and (3) belter duedate performance can be achieved by focusing on efficient system performance and ignoring due dates when sequencing. The proposed job release policy is to inject a customer into the shop whenever the workload at the bottlenecks is at a certain level, determining the customer by a workload balancing input heuristic. Priority .sequencing uses dynamic reduced costs from an LP. In step two, due dates are set using rough approximations that follow the spirit of principle 1 . Simulation experiments considered a twomachine, twoproduct shop. The proposed policies beat standard policies. Lee et al. (1993) describe a decision support system for shopfloor control in a test facility. The system uses the ShortInterval Scheduler (SIS) module of COMETS to perform dispatching. The main contribution of the work is the development and implementation of a mechanism that considers sequencedependent setups while making despatching decisions. This is done by classifying the setups and assigning each operation a setup code representing the setup configuration (determined by handler, temperature, and package type). This allows the operator to select operations with desirable setup characteristics in addition to the due date or operation type allowed by COMETS. This system has been implemented in a test facility and has been running successfully for over two years. Detenninistic scheduling . Under certain conditions, or using simplifying assumptions, controlling the shop floor can be represented by a scheduling problem that can be solved deterministicaUy. The solution to this problem gives a schedule that can be used on the shop floor. The papers by Uzsoy, Lee, and MartinVega (1992b) and Uzsoy etal. (1991a, 1991b) consider the scheduling of a semiconductor test facility and the associated singlemachine problems. This work addresses backend and duedate issues. Their first approach is the use of the shifting bottleneck algorithm, which iteratively schedules workcenters based on some measure of criticality.
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23 In another result, they use dynamic programming algorithms and heuristics to minimize maximum lateness and number tardy on a singlemachine problem with sequencedependent setup times. Finally. Ihcy minimize maximum lateness on a single tester with a branch and bound algorithm and local search improvements to heuristics. They also describe the prototype implementation of an approximation scheme for an entire test facility (Harris Semiconductor), which may be a practical alternative to dispatching rules. Graves et al. (1983) study the problem of scheduling a production facility with reentrant product flows for identical products. Their objective is to minimize average cycle time while meeting a target production rate. They develop cyclic schedules that process each operation of a job once every cycle, where the length of the cycle is the reciprocal of the production rate. Thus, in each cycle, one job is started and one job is finished. The scheduling of tasks on machines in the cycle is done with a greedy heuristic that maintains feasibility. The machines in the problem may be multichannel or batch facilities. The authors compare their rule to the simple FIFO dispatching rule. Kubiak, Lou, and Wang (1990) consider a reentrant job shop with a hub machine, the machine to which jobs return repeatedly. They wish to minimize total completion time under the following assumptions: (1) the shortest operation on the bottleneck is longer than any other operation (allowing the singlemachine simplification); and (2) the jobs possess a hereditary order, where a smaller total processing time implies a shorter processing time for any operation on hub machine. An optimal wellordered schedule sequences jobs at the same operation for the hub machine by SPT. They develop a dynamic program and present a heuristic that develops clustered schedules, where groups of jobs scheduled together, finishing one cluster before moving to the next. Lee, Uzsoy, and MartinVega (1992) study bumin as a singlemachine problem. They assume that each lot of devices has been loaded onto a number of boards, and each of these boards forms a job for a bumin oven. If a batch of jobs is placed into the oven, the entire batch must remain in the oven until all of the jobs have been processed long enough. Thus, the
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24 processing time of the batch equals the maximum processing time of the jobs in the batch. The authors examine the performance measures of maximum tardiness, number of tardy jobs, maximum lateness, and makespan. In this mathematical paper, the authors present dynamic programming algorithms to solve batch scheduling problems with release dates and parallel batch scheduling problems. Bitran and Tirupali (1988 a,b) consider the problem of epitaxial wafer manufacturing. They identify the bottleneck as the epitaxial growth operation, which takes place in a number of different reactors. Their model is a singlestage parallel unrelated machine problem, with the objectives of makespan and total tardiness. They develop static scheduling heuristics that create schedules in two phases: product group priorities (by workload and due date measures) and then job priorities (by due date) within each group. They also address the planning problem of assigning reactors to product groups in order to decomfX)se the problem into independent shops. This approach reduces the complexity of the problem and results in the following observation: the choice of heuristic to solve the problem should be guided by the homogeneity of product set and the objective function. An implemented scheduling system provides shop floor schedule for each reactor, job status, leadtime estimates, and reactor load profile. Periodic resolving of model after planning preprocessing creates uniform reactor loads and homogeneous product mixes. Control theoretic approaches . In an attempt to find policies that perform better than standard rules, some researchers have created controltheoretic models. The solutions to these models are then used to manage the shop floor. Bai, Srivatsan, and Gershwin (1990) consider a hierarchical production planning and scheduling system for a semiconductor wafer fabrication facility. They attempt to meet throughput goals while treating random disruptions explicitly. They integrate the scheduler with the system data base. Events in the wafer wafer fab are classified by their frequency and whether they are controllable (starling a lot vs. machine breakdown). Planning hierarchy is organized by these frequencies. Each level has events with the same magnitude of frequency, capacity
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25 constraints, and objectives passed to lower levels. When something with a low frequency happens, the target frequencies of higherfrequency events arc rccalculalcd. The realtime scheduling is derived from control theory and trcats random events as part of the system. This decomposition allows small but very detailed models. The approach is implemented at the MIT Integrated Circuit Lab. Recent results on this work are covered in Bai and Gershwin (1992). Bai and Gershwin (1990) cover previous work on scheduling singlepart, multiplepart, and reentrant flow systems using a realtime feedback control algorithm. For singlepart systems, the linear programming problem for the feedback control law divides the surplus space into regions where the optimal production rate is constant. There also exists a spot called the hedging point where the surplus is enough to compensate for any disruptions. The feedback controller attempts to drive the system to this point. The authors measure the system's performance in keeping the amount of surplus close to zero with low buffer inventories. They also attempt to improve behavior by separating the machines from each other to reduce effect of machine failure. Solving a nonlinear program yields an optimal hedging point (minimizes buffer inventory and buffer sizes). The current buffer inventories and the hedging point are then used to calculate the desired production rate for the system. Loading times for machines are calculated heuristically to be close to the optimal production rate. For multiplepart systems, machines are divided into singlepart subsystems with the same failure and repair rates and capacities proportional to the demand for that part. In reentrant systems, machines are again divided by part and also by operation, with appropriate capacities calculated. In Lou and Kager (1989), the authors consider VLSI wafer fabrication with the goal of reducing WIP while following target production and observing machine capacity. Their approach is to use lot release and lot dispatching control rules based on flow rate control, a stochastic optimal control problem. Assuming a continuous flow, the authors divide the shop into workstations and determine a production rate for each according to control rules that consider the
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26 inventories across the floor and predetermined hedging points. The advantages of this model include reducing tlie dimension (by ignoring machines) and providing dynamic feedback control (by responding to surpluses, down machines). The authors compare their policy to an eventdriven simulation using uniformloading by cosus of inventory at all stages, and claim that the flowrate control reduces costs by 50%. Gong and Matsuo (1990a) examine a multiperiod production system with random yield, with the objective of minimizing fluctuation of WIP inventory, leading to more predictable system performance. Their approach is to consider control rules for starting product into each stage and raw material. They report that intuitive control rules can be destabilizing. Their new control rule Minimum Weighted Variance is the optimal control policy for a stochastic dynamic program with a quadratic objective function that penalizes WIP deviation from targets and inrinite time horizon. Rule performs well in systems close to capacity. Gong and Matsuo (1990b) also consider a problem with multiple products, Umited workccnter capacity, rework, and reentry. Again, they wish to minimize weighted WIP variance, where the weights in the objective are determined by a nonlinear program that minimizes total expected WIP. With sufficient conditions on the variance of yield and rework fractions, the authors find an optimal control policy for the associated dynamic program. An important conclusion is that the development of stable control policies in uncertain environments is challenging and depends upon the yield and rework distributions. In Ou and Wein (1991), the wafer fabrication system has a single bottleneck, multiple processes with reentry, and byproducts. The authors use for their model a singleserver queue with job classes that correspond to different operations. The model's objective is to minimize total cost: the sum of holding costs for WIP (jobs waiting for reentry) and finished goods and backorder costs. Their approach is to develop a control problem approximation involving Brownian motion. The optimal control policy can be interpreted as the optimal schedule. The authors compare their policy to two statedependent heuristic policies with a simulation and find that their policy reduces costs.
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27 Knowlcdgcbascd approaches . Many researchers have attempted to give managers better control of the shop floor with decision support and expert systems, which incorporate knowledge that can be used to set control policies or schedule machines. Some expert systems can even perform scheduling events themselves. Adachi, Moodie, and Talavage (1988) consider a production system with reentrant product flows. They use a simulation model to examine the effect of management decision variables on system performance. The variables are lot size, dispatching rules, start rate, frequency of urgent jobs, and frequency of machine breakdowns. The measures are product mix, throughput, cycle time, and WIP. The authors report that the start rate was a more important factor than the dispatching rules. The lot sizes also had significant effects. The dispatching rules were important policies only in overcapacity shops. The authors then develop a pattern recognition DSS that, through an iterative procedure, helps the user find a system that matches the userdefined goals. A prototype is implemented on the printed circuit board fabrication line at NEC Corporation. In Adachi, Moodie, and Talavage (1989), the authors extend their work on production systems with reentrant product flows. This time, the decision support system includes a rulebased component and simulation model. The simulation model comes from the previous work and evaluates the effect of control variables on measures of inventory, product mix, cycle time, throughput. The results are used to obtain regression coefficients, which are stored in a database for the rulebased component, which organizes the control variables into the hierarchy of start rate, lot size, and priority rule. This DSS was implemented for a printed circuit board fabrication line, resulting in superior control policies to the previous pattern recognition DSS. Adelsbergcr and Kanet (1991) report on a new tool in computeraided manufacturing scheduling: the leitstand, a decision support system with a computeraided graphical interface. Leitstands include the following components: the graphics component is a electronic Gantt chart; a schedule editor allows a user to easily change an existing production schedule; the data base manager incorporates data from production planning system, engineering, and shop floor and
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28 uses specific knowledge; an evaluation component measures the schedules on objective functions and creates reports; and an advanced automatic schedule generator produces feasible schedules. Savell, Perez, and Koh (1989) describe scheduling semiconductor wafer production with an expert system that takes advantage of the modularity of workcenters and sophisticated data analysis. The wafer fab is divided into into thirty cells. The expert system, which is an offtheshelf PC package with some external routines for data manipulation, creates daily schedules. It includes two modules: (1) the priority assignment module uses information from CAM about the operational slack of each lot and knowledge about special lots; and (2) the equipment scheduling module for each cell uses specific knowledge about the cell and the lot priorities to create a schedule for the lots in the cell and those arriving at the cell within a certain time frame. The expert system is implemented in two cells (PDiffusion and Aluminum Deposition) at Harris Semiconductor. Sullivan and Fordyce (1990) report on IBM Burlington's Logistics Management System for wafer fabrication. The main function is the shop fioor dispatching of lots. It replaces slack lead times with information to handle the coupling of strategic and operational decisions. The LMS includes reallime lot and machine status and proactive intervention, with an expert system that has knowledge about generating alerts in certain conditions and responding to some of these by making dispatch decisions that consider five conflicting objectives: lot priority, ontime delivery, fiow requirements, increasing throughput, meeting engineering specifications. The LMS is implemented in various areas and realizes the importance of accurate data and continual updating to reflect changing environment. Fordyce et al. (1992) discuss the current version of the Logistics Management System (LMS) in place at the IBM Buriington Semiconductor manufacturing site. The emphasis of this paper is on the last of the four tiers of the scheduling decision hierarchy. This is dispatch or shortinterval scheduling for periods from one hour lo two weeks. This tier contains the decisions concerning the actual manufacturing flow, including the scheduling of an operation. The LMS contains a dispatcher/shortintcrvalschedulcr, that creates zones of control around the botUcneck
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29 points, which may a sequence of operations that lots must visit repeatedly. Kanbans within these zones monitor the WIP level for the different passes through the zone. The dispatching is done by something called a Judge. The Judge receives information from goal advocates in order to make its decision, and the different advocates have different goals, including meeting due dates, meeting the daily plan, maintaining low WIP (this is the goal of the kanbans), and increasing machine utilization (this includes minimizing setups). Hadavi etal. (1991) present a distributed architecture for realtime scheduling and describe its implementation in a wafer fabrication factory. The system (ReDS) works to meet management objectives of meeting due dates, reducing WIP and finished goods inventory, reducing cycle times, and maximizing machine utilization. The system abstracts constraints and time into a tree with nodes that correspond to an interval of time; it also abstracts an order into an "essence function" that describes its critical resources. The release policy uses "continuity indices" to reduce cycle times. The modules in the system include a preprocessor to abstract the constraints and orders and a feasibility analysis to release orders. Also in the system are a detailed scheduling module that uses least commitment planning in determining a daily or shift schedule. A sequencer then dispatches the operations scheduled for a time period by using a dynamic sequencing rule that responds in real time to a changing floor. Rao and Lingaraj (1988) review a number of expert systems for production and operations management decision making. They classify the systems along two dimensions: strategic decisions versus tactical decisions and operations orientation versus resources orientation. Scheduling systems are classified as tactical, operationsoriented applicaUons. They review a number of systems in scheduling, capacity planning, facility layout, process & product design, quality control, aggregate planning, inventory control, and maintenance & reliability. They conclude that expert systems should combine technological knowledge and logistical data in order to be effective.
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30 ISIS (Fox and Smith. 1984) is an expert system for scheduling that uses a hierarchical approach to scheduling: job selection, capacity analysis, resource analysis, and reservation selection. It uses decisions made at each level as constraints for the lower levels. Opportunistic scheduling is a knowledge based approach introduced in the OPIS system (Smith, Fox, and Ow, 1986, and Ow and Smith, 1988). In this system, the authors extend the joboriented scheduler of ISIS with a machineoriented scheduling procedure. After identifying an initial bottleneck, the search for a good schedule proceeds by directing activity towards the bottleneck subproblems of the job shop scheduling problem. The characteristics of this system include alternative problem decompositions, multiple scheduling heuristics, and multiple problem abstraction. Sadeh (1991) discusses a system called MICROBOSS, another opportunistic scheduler that identifies an initial bottleneck and revises its strategy as new bottlenecks emerge during the construction of the schedule. This system is a more flexible procedure, however, for it can revise its scheduling strategy after each operation. Thus, it avoids having to scheduling large subproblems for a machine or a job. Bensana, Bel, and Dubois (1988) describe a system called OPAL, a job shop scheduling software that combines three types of knowledge: theoretical knowledge about scheduling problems, empirical knowledge about scheduling heuristics, and practical knowledge of the scheduling environment. They claim that combining artificial intelligence techniques with operations research should be an effective approach to scheduling problems. OPT and other approaches . OPT (Optimized Production Technology) is a proprietary scheduling system that focuses on the scheduling of the bottleneck resource. The system has been reviewed by a number of authors, including Jacobs (1984), Meleton (1986), Lundrigan (1986), and Vollman (1986). OPT uses a forward finiteloading scheduling procedure to schedule the identified bottleneck. The remaining, noncritical operations are scheduled using an backward infiniteloading procedure. According to Morton (1992), the advantages of OPT are its good solutions to large scheduling problems and its focus on the bottlenecks. Disadvantages
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31 include the inability to do reactive scheduling without resolving the whole problem. Details about OFT can be found in the above articles and in VoUmann, Berry and Whybark (Chapter 20, 1988). Faaland and Schmitt (1993) use a costbased system to enhance the planning function of MRP in an assembly shop. They developed a system that can create a detailed schedule of jobs, workers, and work centers for a maker of aircraft audio, power, and light systems. They include inventory holding costs associated with finishing early, opportunity costs associated with late delivery, and payroll costs, since the model includes the crosstraining of workers. Morton (Chapter 16, 1992) describes an early version of bottleneck dynamics called SCHEDSTAR (initially reported in Morton et al., 1988). This system iterated lead times and prices over the bottleneck and considered an objective function that included revenue, tardiness, direct completion costs, and holding costs. A number of release and dispatch heuristics were studied on a variety of shop situations, and the authors conclude that bottleneck dynamics (and iterated pricing and lead times) leads to better schedules. Simulation scheduling . Primarily, simulation models are used to predict the performance of certain policies. However, once the policies are set, the simulation model can be used to create a feasible schedule for the shop floor. Atherton (1988) states that simulation can be used on all levels of planning. For shortterm scheduling, models may perform shortinterval and shopnAoor scheduling. This form of scheduling considers factory capacity (dispatching rules don't). A validated simulation model, provided with current information on the system status, can use rules to determine what will happen in the short term, providing a shop floor schedule. If the projected lot completions do not meet production requirements, further simulations may be necessary to find a schedule that is close to the goal. Leachman and Sohoni attack the problem of semiconductor manufacturing using an automated scheduling system and teamworkfostering management. For every shift, a target is set that considers the real Ooor. The entire staff meets to identify problems and propose
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32 solutions. The targets arc generated by the scheduling system: a simulation model of the wafer fab in BLOCS that is linked to the WIP tracking system, providing it with realtime data. The simulation considers availability but assumes perfect execution. Thus, it is fair and accurate. To make the schedule, it uses logic accepted by the staff, including least slack, FIFO, and starvation avoidance rules and furnace waiting cost analysis. The authors discuss how a shift proceeds: meetings before and after the shift with explicit incentives for meeting the target. They also review the organizational effectiveness of their approach in terms of motivation, satisfaction, communication and coordination, problem solving capability, acceptance of change. Najmi and Lozinski (1989) discuss the implementation of the system in two wafer fabs at NCR, Inc. The simulation model is written in BLOCS, an objectoriented language where each physical object or collection of data is a object (composed of data and procedures), and the objects communicate to each other with messages that represent the interaction of objects. 2.2.3 Performance Evaluation The complexity of semiconductor fabrication facilities makes direct analytical evaluation of them difficult at best. In many cases, researchers use queueing models or simulation models in order to gain some insight on how the system performs in the current or some proposed environment. Burman et al. (1986) discuss the ways that OR tools and techniques are used to analyze IC manufacturing hnes: simulation, queueing analysis, and deterministic capacity models. The authors describe a simulation study for direct step on wafer printers in photolithography. The study determined the effect of lot size, number of products, and setup time on capacity and WIP. It also used largescale simulations to obtain arrival distributions to this area. A deterministic capacity model of a clean room uses the mean number of steps and the mean process time. The analysis took much less computing time than a simulation. The model was useful for estimating a room's capacity and determining the minimal number of machines required for a proposed
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33 program. They conclude that simulation models have the greatest flexibility and best success but take considerable time to develop and run. The other techniques arc quicker and, if not quite accurate, do provide estimates that may be verified in simulation. Oucucinp models . Queueing models attempt to model the shop floor as a set of servers with service time distributions and possible stochastic routings. The system yields a set of equations that describe its behavior. Because the success of solving these equations is usually undermined by the complexity of the shop being modelled, researchers are forced to make simplifying assumptions. Chen et al. (1988) develop a simple queueing network model to predict performance measures in an research and development wafer fab. The HewlettPackard Technology Research Center Silicon Wafer fab serves as the model. The model is a classic job shop, and the performance measures are throughput and cycle time. Machine breakdown rates added to actual service time to create an effective service time. The naive queueing network model includes different types of customers that have different routing distributions. Straightforward formulas from previous researchers yield performance measures. The authors report that the model yielded performance measures that were within 10% of the actual numbers. Wein (1991) investigates a wafer wafer fab with timedependent stochastic defects, using a simple queueing theoretic model to determine the relationship of yield and cycle time to throughput. The model is a singleserver queueing system with exponential arrival and service times. The author derives a closed form relationship between the mean cycle time and the throughput of good die. In standard models, as the start rate is increased, the throughput increases and asymptotically approaches some upper bound as cycle times increase to infinity. In this model, the cycle times increase without bound but throughput reaches some maximum and then decreases as the increased cycle time leads to a higher defect rate (and thus less yield). Simulation . Simulation models remain as a popular approach to measuring the performance of manufacturing system, and many packages have been developed. The state of the
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34 art is probably the BLOCS package developed at CaliforniaBerkeley and partially described in Najmi and Lozinski (1989). The following papers report on some uses of simulation. In two papers, Dayhoff and Atherion (1986 a.b) develop simulation models of wafer fabs. They model wafer fab as a qucucing network with components for each lot and process flow. Models are used to analyze system performance, measured by its signature: the plots of cycle time, inventory level, and throughput versus start rate. The signature provides a way to gain information from the large quantity of data provided by a simulation by displaying the relationships among different variables. As a specific example, they model a bipolar wafer fab. They compare the signatures for a number of dispatch schemes for a bipolar wafer fab where products visit the masking station seven times. In the wafer fab, there exist three levels of dispatching: (1) among lots of one product waiting for one process, (2) among process steps of the same product, (3) among products. This paper works at level two; the first level is done by FIFO, third by product priority. The study compares the following dispatch schemes for masking: earlier steps first (a form of longest remaining processing time), later steps first (a form of shortest remaining processing fime), roundrobin (changing priority). Qualitative analysis shows that earlier steps first failed at higher start rates by building inventory and that the later steps first was better by getting higher throughput with no increase in inventory. Dayhoff and Atherton (1987) provide definitions of the elements of wafer fabrication and their interactions. They idenfify the important parameters that govern wafer wafer fab and result from wafer fab operations. The elements defined include work station, process flow, products, wafer lot, process step, batch, batching down, service, dispatch system, rework, and wafer fab graph. They discuss the following types of simulation results: average wait times, queue lengths, inventory, cycle time, equipment utilization, yield, throughput. Atherton and Pool (1989) discuss the ACHILLES simulation model in a wafer wafer fab of Silicon Systems, Inc. The describe the model and its initialization, calibration, and validation by comparison to actual measures of factory performance. The discuss the use of the model to predict the effect of reducing inventory levels on cycleumes.
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35 Spcncc and Welter (1987) discuss a project by Stanford University and Motorola Inc. that allempLs to improve the performance of a semiconductor wafer fabrication line. The goal is to increase the tha>ughpul of the line without sacrificing cycle lime. The analysis was done with a Monte Carlo simulation to determine capacity, defined as throughput rate, cycle time, and WIP. The performance was measured on the steadystate cycletime vs throughput tradeoff curve, instead of using transient signatures (c.f. Dayhoff and Atherton). The photolithography cell was studied for its analytical difficulty, the build up of WIP in this cell, and the proposal of system changes. The authors report that adding resources (operators, aligner equipment) reduced cycle lime at higher throughput. Reducing reworks, setup times, and the time to wait for repair reduced cycle times at all levels. Larger lot sizes were preferable at higher throughputs. 2.2.4 Summary From this review, it can be seen that many different approaches have been applied to the problems of production planning and scheduling. However, most of this work addresses the questions raised in wafer fabrication. The primary exceptions are Uzsoy, Lee, and MartinVega (1992b), Uzsoy et al. (1991a, 1991b), Lee, Uzsoy, and MartinVega (1992), and Lee etal. (1993) which do explicitly consider the problems of semiconductor test. However, this work does not make use of the new, sophisticated heuristic searches and is not concerned with the ideas of class scheduling and lookahead and lookbehind scheduling. Although Savell, Perez, and Koh (1989) do develop a system that is lookbehind, it is implemented in an offline system in a wafer fabrication cell. The research in this dissertation, however, investigates the application of both lookbehind and lookahead rules into the current realtime dispatching system of a semiconductor test area.
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36 2.3 Job Shop Scheduling Before beginning the analysis of scheduling problems, it is useful to review the notation to be used and the basics of scheduling problems. Because even the simplest job shop scheduling problems arc NPcomplclc, the literature consists of different heuristics. Job shop scheduling, as one of the most difficult scheduling problems, has attracted a lot of attention from researchers. Techniques such as the shifting bottleneck algorithm (Adams, Balas, and Zawack, 1988) or bottleneck dynamics (described in Morton, 1992) concentrate on solving the problem at one machine at a time. Other researchers have studied how well different dispatching rules perform in minimizing makespan and other objective functions. Panwalkar and Iskander (1977) present a list of over 1(X) rules. Recent studies include Fry, Philipoom, and Blackstone (1988) and Vepsalainen and Morton (1988). Various scheduling systems for shop floor control are reviewed in Section 2.2. More sophisticated lookahead and lookbehind rules have also been introduced; see Section 2.5 for a discussion of these ideas. In this section we review scheduling notation, the shifting bottleneck algorithm, and dispatching rules. 2.3.1 Scheduling Notation In the general job shop scheduling problem, there exists a set of jobs Jj,j = 1 , ..., n, and set of machine Mj, / = 1, ..., m. Eachjob/.has /i; operations (or tasks) Oy, i = 1, ..., nj, where 0,.= k if the ith operation of 7.is to be processed on machine Mj^. In a flow shop, all nj = m and Oij = / for aU /. In general, each operation has a processing time /?,.> 0. A job can have release dates r: and due dates d: and deadlines D .. A feasible schedule o is a plan that determines when each operation is processed on each machine subject to the following constraints: the operations of each job must be performed in order, and no machine can process more than one operation at at time. For a schedule, certain performance measures can be associated with a job/.: the completion time C., the lateness L: = C: d., the tardiness T: = max {L., 0), and whether or not
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37 the job is lardy, U; = 1 if T.> and otherwise. Objective functions that measure the performance of the schedule include the makcspan, Cf^^^j^^ = max [Cj], the total flowtime X C., and the number of lardy jobs X U:. The scheduling problem is to find a feasible schedule that minimizes the objective function. For the regular performance measures mentioned above, the set of schedules that need lo be considered is the set of active schedules, where no operation can be started earlier without causing another operation to be delayed. In this set, there exists a onetoone correspondence between a schedule and its representation by a sequence of operations for each machine. Thus, a sequence that orders ihe jobs on each machine can be considered a solution to the problem. Standard scheduling problems can be classified in a concise way by using the following threeelement description: x/y / z. The field x describes the machine environment as onemachine, parallelmachine, or shop. The second field (y) describes any constraints or special characteristics of the problem. The third field (z) describes the objective function. Consider the following examples: 1 / '/ / S Uj is a onemachine problem where the jobs have release dates and the objective is to minimize the maximum lateness. I / Dj/'Z Cj is the onemachine problem where the jobs have deadlines and the objective is to minimize the total nowtimc. F2 / / Cf^i^jj^ is the twomachine flowshop problem of minimizing the makespan. J / / C^^j^j^ is the general job shop scheduling problem of minimizing makespan. 2.3.2 Shifting Bottleneck Adams, Balas, and Zawack (1988) introduce the Shifting Bottleneck procedure to minimize the makespan of job shop scheduling. This algorithm sequences the machines in a job shop successively by identifying the machine that is a bottleneck among the machines not yet sequenced. After the scheduling of the new bottleneck, all of the previouslyconsidered machines
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38 have their schedules readjusted. The individual steps involve finding the solution to onemachine problems. The authors lean heavily on the disjunctive graph associated with the job shop scheduling problem. In this graph, there exists a node for each operation to be performed as well as dummy start and termination nodes. For each job there exist directed arcs from each operation to its successor and an arc from the start to the first operation and an arc from the last operation to the termination. For each machine, there exist a pair of disjunctive arcs between each pair of operations on this machine. All of the arcs have a length equal to the processing time of the head operation. If a selection is made for the disjunctive arcs of each machine and the corresponding graph is noncyclic, a schedule for the entire shop is made by scheduling each job as soon as possible under the constraints imposed by the selections and the job operation sequence. The makespan of the schedule is equal to the longest path from the start node to the termination node. The Shifting Bottleneck procedure iteratively selects a bottleneck machine, schedules this machine, and readjusts all previouslyscheduled machines. Given that a set of machines has been scheduled, procedure determines the bottleneck by considering each unscheduled machine individually. This yields a onemachine problem where the minimization of the makespan for the shop is equivalent to minimizing the maximum lateness of the operations on this machine. The minimization problem is NPcomplete, but the authors use the algorithm of Carlier, a branchandbound procedure that gives excellent results. The machine that has the largest such maximum lateness is the bottleneck, and that machine is added to the set of scheduled machines. The local reoptimization procedure considers the set of scheduled machines and performs a number of cycles. Each cycle consists of solving the one machine problem for each machine in turn, using the selections found for all of the other machines. The authors report that the Shifting Bottleneck procedure performs exceptionally well, including finding the optimal makespan for the notorious 10job and 10machine problem of Giffier and Thompson.
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39 2.3.3 Dispatching Rules Because of the complexity of job shop scheduhng, algorithms to find the optimal solution for any arbitrary objective function do not exist. Thus, researchers have studied and schedulers have used dispatching rules to order the jobs wailing for processing at a machine. Most of the research in this area has examined the performance of various despatching rules, which sequence the jobs that arc waiting for a machine according to some statistic that is a function of the jobs' characteri.stics. When a machine becomes available, it chooses from among the jobs in its queue by using a rule that sorts the jobs by some criteria. Common dispatching rules employ processing times and due dates in simple rules and complex combinations. These dispatching rules are sometimes extensions from simple onemachine problems. For instance, the Shortest Processing Time (SPT) algorithm is known to minimize the total flowtime of jobs processed on one machine. The SPT dispatching rule sorts jobs waiting for a machine by the amount of processing time they require on the machine. Also, the Earliest Due Date (EDD) algorithm is known to minimize the maximum lateness of a set of jobs being processed on one machine. The EDD dispatching rule is used in job shop scheduling in an attempt to reduce maximum lateness. While the list of known dispatching rules includes over 100 items, only a handful are commonly used. And most of the rest are combinations of the most common rules. Day and Hottenstein (1970) present a review of sequencing research, in which they discuss Jackson's Decomposition Principle (1967), which assumes that the arrival times for each job arriving from outside the system are exponentially distributed, the processing times at each machine are exponentially distributed, the jobs are routed to a machine by a fixed probability transition matrix, and the priority rule at each machine is FirstComeFirstServed (FCFS). They also discuss several due dale assignment schemes presented by Conway (1965). These are CON (constant from the order to the due date), RAN (random: due date chosen by customer and accepted by salesman), TWK (Total Work Content: the allowable shop time is
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40 proportional to the sum of the processing times of the operations of the job), and NOP (Number of Operations: the allowable shop time is proportional to the number of operations). The authors also review the previous research on dispatching rules by Conway (1965), who compared SI to 3 1 other rules and found that it dominated the set of rules tested. It is simpler and easier to implement. However, the biggest objection to SI is the following fact: if the mean time which jobs spend in the system is small, individual jobs (those with long operations) may be intolerably delayed. Thus the variance of the lateness distribution is the basic disadvantage of the SI mle. Conway tried three methods to reduce the variance: (i) Alternating the SI rule with a low variance rule (with respect to flowtime) to periodically clean out the shop, (ii) forcibly truncating the SI rule by imposing a limit on the delay that individual jobs will tolerate; and (iii) dividing jobs into two classes, preferred and regular, as the primary criteria, leaving SI as a secondary criteria. The authors then move to the COVERT dispatching rule. Buffa (1968) retained the performance of the SI rule and tended to minimize the extreme completion delays of a few orders. Trilling (1966) used a ratio of delay cost rate of a job to the processing time of that job on the machine in question. The job with the highest ratio is dispatched first. The authors also report on the priority rules employed in industry, where job lateness is a primary concern. Hence, EDD and least slack rules are most used. However, LPT becomes a popular rule because schedulers rank jobs by the index of importance and it is reasonable to expect some positive correlation between importance and processing time. HoUoway and Nelson (1974) study the problem of job shop scheduling with stochastic processing times. Their performance measures are the mean, variance, and maximum of tardiness. For dispatching, the authors use HSP, a multipass heuristic that produces delay schedules, HSPNDT (a nondelay version), DDATE, SPT, SLACK, and a SPTSLACK combination. The authors study three problems of different size, tightness, and utilization, where the processing times were constant or from one of three distributions. They report that HSP did well
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41 on lowvariance problems and on maximum tardiness. HSPNDT beat HSP in some situations. Of the nondelay ailcs, the deterministic problems were a good predictor of relative performance on problems with variability. DDATE improved its relative performance as variability of processing times increased. Panwalkar and Iskander (1977) list 100 dispatching rules. They categorize these rules into five classes: simple dispatching rules, combinations of simple rules, weighted priority indexes, heuristic scheduling rules, and other rules. The simple dispatching rules are organized into processing time, due date, number of operations, cost, setup time, arrival time, and machine rules. Combination rules sequence jobs by considering first one characteristic and then another. Weighted priority indexes .sequence jobs by combining values from different job characteristics (by adding or dividing). Blackstone, Phillips, and Hogg (1982) state that the best measure of performance is cost effectiveness. Their work covers this objective function and others: tardiness, lateness, flowtime, inventory. They note that analytic measures depend upon Jackson's decomposition, which assumes a FIFO dispatching rule. Other rules lead to interrelated queues and thus researchers use simulation. For the singleserver model, it is known that SI (shortest imminent processing time) minimizes mean flowtime and mean lateness and EDD (earliest due date) minimizes maximum lateness and maximum tardiness. Mean tardiness cannot be optimized by any dispatching rule; thus, the authors conclude that "no single dispatching rule yet developed will optimize delay costs in the job shop environment." The authors consider a number of dispatching rules. Their first is SI. They report that SI is not affected much by imperfect data, it performed best on mean flowtime, and it minimized the number of tardy jobs and mean lateness for exogenous (constant and random) due date assignment. It also perfonned better when internal due dates are less than seven times processing time and utilization is high. Modifications to SI to clear jobs that have been waiting include
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42 alternating rules and truncation. Truncation versions seem useful for shops having control over due dales and concerned about very late jobs. The authors also consider a number of due date rules: EDD, Slack, and Slackpcropcralion. Slackperopcrdtion is the best and, compared to SI or FIFO, produces a smaller variance of job lateness independent of due date assignment. Compared to EDD and slack, it performs best on lateness variance, costperorder, job tardiness, number of lardy jobs. Slack may be defined in static or dynamic terms, although rules ihal use ihe latter are better. The authors also report that critical ratio rules are also in use. Among rules that used neither processing time nor due date information are FIFO and NOP (number of operations). The authors report that both perform worse than other good rules. They also conclude that the look aheadrules NINQ (number in next queue) and WINQ (work in next queue) are not as good as SI in flowtime and inventory criteria and that a valuebased rule usually becomes longest processing time, which generally behaves poorly. The authors report that weighted combinatorial rules are not belter than any single rule, although COVERT (delay cost over time remaining) may be useful in shops willing to estimate delay costs. The authors also discuss dispatching heuristics. The lookahead heuristic LAH allows insertion of idle time in order to process a critical job. Heuristics improve the performance of dispatching rules but their implementation may not be cost effective; a complete study has not been done. Green and Appel (1981) examine the problem of job shop scheduling by asking the following questions: What traditional dispatching rules do experienced schedulers select? Would dispatch rule selection be influenced by urgency? Would schedulers select a dispatch order based on organizational influence and/or peer pressure? The authors asked schedulers in a number of plants to denote which of the following rules they used: Due Date, Slack, Operations Due Date, Slack per Operations, SPT, FCFS, COVERT, Program in Greatest Trouble (PGT), or Friend Needs a Favor (FNF). The authors report that influence systems affect scheduling. PGT (a
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43 coalition rule) was highly valued, but FNF (an individual nilc) was rejected. Traditional and theoretical rules were not highly valued. Kanet and Hayya (1982) also consider the problem of job shop scheduling. Their paper tries to determine if operation due dates are better. Their comparison is done by running controlled simulation experiments using the following dispatching rules: Earliest Due Date: DDATE and OPNDD; Smallest Job Slack: SLACK and OPSLK; Critical ratio: CR and OPCR. The slack is static; the critical ratio is dynamic: time until duedate over remaining allowance or operation allowance. For critical ratio, each operation must be assigned an allowance. This allowance is proportional to the processing time. The allowance multiple controls the difficulty of meeting due dates. The authors consider the performance measures of lateness (mean and standard deviation), fraction tardy, conditional mean tardiness, flowtime (mean and standard deviation), and maximum tardiness. The authors report that all rules were outperformed by their operation counterparts on all measures. Introducing operation due dates shifts distribution of job lateness to the left (less) and compresses it. One unexpected result was that OPNDD (due date) outperformed OPSLK (slack) and always minimized maximum tardiness. The CR rule gives lower lateness variances, but the OPCR rule was even better. The authors note that, as the allowance multiple is increased, due dates become larger and DDATE and SLACK rules act more like SPT, minimizing flowtime. For the critical ratio rules, a larger allowance means longer jobs get lower ratios, causing CR to act like LPT. Baker and Bertrand (1982) take up the problem of singlemachine dynamic scheduling and study different due date assignment methods and dispatching rules. They are concerned with average tardiness. The authors introduce a new dispatching rule: the modified due date rule (MDD), where the modified due date is defined as the maximum of the due date and earliest finish date. This rule is a combination of EDD and SPT that implicitly responds to changes in the amount of slack. The authors report that MDD dominates both SPT and EDD on average tardiness.
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44 Baker and Kanct (1983) extend the MDD rule to attack the problem of job shop scheduling. They consider the performance measures of mean job tardiness and proportion of tardy jobs. They introduce the modified operation due dale rule (MOD), where the modified operation due date is the maximum of operation due date and earliest operation finish time. The authors also consider other dynamic rules: Slack per operation, Critical ratio, and COVERT under different allowances and utilization levels. The authors report the following results: MOD performs better than MDD and SPT and is sometimes better than operation due date, critical ratio, and slack per operation rules. The authors thus conclude that MOD is an important dispatching mechanism. Baker (1984) considers the job shop scheduling problem and attempts to clarify some conflicting results between different rules of dispatching and duedate assignments. He points out that two factors are of primary interest: flowtime and duedate performance. He reports that SPT is the best tactic for reducing mean flowtime. However, no single priority rule dominates performance comparisons. He reports that the critical ratio is best for minimizing conditional mean tardiness (that is, the average tardiness of tardy jobs); SPT is best for number of tardy jobs; however, for mean tardiness, the results are mixed. Fry, Philipoom, and Blackstone (1988) consider the problem of job shop scheduling with 90% utilization. For the due date assignment, they use the total work method with two different allowances. They consider the following performance measures: mean flowtime, mean tardiness, and root mean square tardiness (designed to punish large tardiness). They study truncated and alternating versions of the SPT dispatching rule. The authors define the critical percentage (%C) as the percentage of jobs that enter a higher priority queue because they have been waiting a long time. They report that the best rule depends upon the this critical value. Vepsalaincn and Morton (1987, 1988) consider the problem of minimizing weighted tardiness in job shop scheduling. They use the following dispatching rules: FCFS, EDD, Slack per remaining processing time (S/RPT), WSPT, weighted COVERT, and Apparent Tardiness
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45 Cost (ATC). A lead lime eslimalion is necessary for last two rules. Lead time estimates arc computed as a fixed multiple of processing times, derived from rough waitingline analysis and job priority indices, or found from iterating simulations. The authors first studied the flow shop problem, where ATC and COVERT dominated using any leadtime estimate. Using the iteration estimate resulted in better weighted tardiness. ATC did better on fraction of jobs tardy and inventory measures. In the job shop, under different loads and due date tightnesses, the authors report that the ATC rule did better, followed closely by COVERT. The use of the priority estimates yielded a smaller fraction of jobs tardy. 2.3.4 Summary In addition to introducing some notation, this section covered two important methods of job shop scheduling: the shifting bottleneck procedure and dispatching rules. Both methods have limitations, however. The shifting bottleneck procedure searches for a schedule and continuously improves upon it. Thus, it is only a local search technique. The dispatching rules that have been studied are mainly shortsighted techniques that do not consider other machines. The research in this dissertation extends this work by considering smartandlucky searches and investigating lookahead and lookbehind dispatching rules. 2.4 Flow Shop Scheduling This dissertation considers a threemachine problem where all jobs are processed on a certain machine and then go to one of two secondstage machines. This problem is similar to a flow shop, and it was useful to review the flow shop scheduling problems that have been previously studied. The flow shop problem is actually a collection of problems that deal with the minimization of some regular objective function for a set of jobs in a flow shop, where each job consists of a number of different operations that must be processed on a set of machines. The primary feature
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46 of a flow shop is thai the sequence of operations is the same for each job. That is, each follows the same path through the shop. All now shop analysis starts with Johnson (1954) who studied the minimization of makespan for twomachine flow shop problems and for some special threemachine flow shop problems. His algorithm starts jobs with the smallest tasks on machine 1 as soon as possible and jobs with the smallest tasks on machine 2 as late as possible. For the twoand threemachine cases, it can be shown that only permutation schedules need be considered. Permutation schedules are schedules for the machines in which the sequence of jobs is the same for each machine. Johnson showed that for four machines, passing may be necessary for optimality. Although most analytical research in shop scheduling has dealt with the makespan objective function, this research is interested in the minimization of total flowtime, also known as the mean completion time, the sum of completion times, the mean time in system, or the total time in system. Therefore, this section concentrates on a number of papers on the minimization of total flowtime, a lesscommonly studied objective, and then moves on to other objectives, including maximum lateness, and the number of tardy jobs. Also included in this section are reports on some NPcompleteness results and problems with separated setup times. 2.4.1 Makespan Special cases of the flow shop makespan problem have been studied by a number of researchers, including Mitten (1958), Conway, Maxwell, and Miller (1967), Bums and Rooker (1975), and Szwarc (1977). Carey, Johnson, and Sethi (1976) proved that the general threemachine problem was NfPcomplete. Problems with release dates, preemption, precedence constraints, or more than three machines have also been studied. 2.4.2 Total Flowtime Ignall and Schrage (1965) describe a branchandbound algorithm for F2 // X C: and for F3 // C^ax^^^ ^^'^ '"^'^' nowiimc problem, the authors consider two values: the sum of
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47 completion times if the first machine is the only capacity constraint and the sum if the second machine is the capacity constraint. The larger of these two values is the bound at each node. Krone and Slciglitz (1974) consider the static flow shop scheduling problem with the objective of minimizing the mean flowtime of n jobs that must visit m machines. They present a heuristic method. The authors note that two previous researchers performed separate sampling experiments of the flow shop scheduling problem. Heller concluded that permutation schedules should be investigated while Nugent found that good schedules tended to be permutation schedules that had allowed some jobs to pass others (a local reversal). For the flow shop problem, the authors consider semiactive schedules (where no operation can be shifted forward in time) and define a schedule as an array S that gives a sequence for each machine. S,. is the job that is 7th on the /th machine. The authors define a twophase heuristic. In the first phase, the search considers only permutation schedules. In the second phase, the search takes an initial permutation schedule and allows deviations from the uniform ordering. Each phase is a local search, although the neighborhood structures are different for each phase. Kohler and Steiglitz (1975) study the twomachine flow shop problem with the objective of minimizing the mean fiowtime. They present algorithms that they combine with lower bounds that guarantee the accuracy of the heuristics. The authors use the lower bound of Ignall and Schragc. They compute an initial solution by moving down a branchandbound tree without backtracking, selecting the node with the lowest lower bound at each stage. They consider different local searches, with either a random start or the above initialization, hillclimbing (first improvement), and one of four neighborhood structures: backward insertion, forward insertion, a double adjacent pair interchange (switches two pairs), and finally the union of the first and third. The authors report that the good iniualization helped the local search perform better than the random starting search. The authors then present three different branchandbound algorithms that differ in the way the nodes are eliminated. Since some of the algorithms exceed computational limits, they determine a bracket that is the relative difference between the final upper bound and the greatest
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48 lower bound. If the goal is to find a solution that is within a certain amount of the optimal, this bracketing idea allows great speedup since a good (sometimes optimal) initial upper bound was used and the lower bound is very close. For a given r< 1 , the algorithm stops when the lower bound L is greater than or equal to r times the best upper bound i/, i.e., ril
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49 holds for the iwomachinc case where the r/^ > t2r for all jobs J^ and the case where the columns and rows form a perfect order under dominance. Ahmadi et al. (1989) study the twomachine flow shop subproblcm with one or two batch processors as a segment of a larger job shop. They study the objective functions of makespan and total flowiimc. They consider two types of processors. The first is a batch processor that has a fixed capacity and the time necessary to process the jobs in a batch is a constant. The other type of processor is called a discrete processor, which refers to a standard singlemachine processor. The authors consider the six types twomachine problems that can be formed. To minimize the makespan of a batchdiscrete shop, full batches are optimal, permutation schedules dominate nonpcrmutalion schedules, and the optimal solution is to sequence jobs by LPT and fill each batch. For the discretebatch shop, the analysis is similar, but in this case the jobs should be ordered by SPT. For twobatchmachineshop, all of the jobs are equivalent. The optimal policy is to completely fill the batches on machine one and to use a dynamic program to fill the batches on machine 2, using the completion times on machine 1 as release dates. To minimize the total flowtime of a discretebatch shop, SPT should sequence the jobs on the discrete machine 1. A dynamic program can be used to perform the batch dispatching on machine 2. For the twobatchmachineshop, the batches on machine 1 should be completely filled and the completion times from this can be used as release dates to batch machine 2, which is dispatched using the same dynamic program. For the problem where a discrete processor is fed by a batch machine, the batches should be completely filled, but the authors prove that this case is strongly NPcomplete. They do consider special cases relating the processing times on one machine to another and derive two optimal algorithms. For the NPcomplete problem, the authors propose a heurisUc determines the number of shortest batches that are shorter than the batch processing time. To these batches the shortest jobs are dealt. The remaining jobs are sequenced by SPT to form batches. The authors show that the error tends to 1/2 as the number of batches tends to infinity.
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50 Ahmadi and Bagchi (1990) present a lower bound on the total nowiimc. Given a partial schedule J^. of r jobs and fixing a machine 7, the authors bound the final completion times based on the completion times on machine;. Then, they search over sequences on machine 7, considering the sequence constraints and the machine) release dates for each job. This is the NPhard problem of solving l/rVX C.. The authors solve the problem by allowing preemption. The lower bound is the greatest found by performing this procedure over each of the machines. Van de Velde (1990) studies the twomachine flow shop problem and minimizes the sum of completion times by using a Lagrangcan relaxation. The author decides to relax the precedence constraints between the two operations of a job (instead of the capacity constraints on each machine). Let the vector of multipliers be X = (^i, X2 A.Â„). The relaxed problem is one of minimizing the weighted sum of operation completion times, solved by SWPT on each machine separately. The author, however, wants to restrict the solution to permutation schedules. Thus, the author reformulates the problem into a linear ordering problem, which is polynomially solvable in certain cases. The author lets all X,j = c, where < c < 1, and if c = or c = 1 , the problem yields the bounds of Ignall and Schrage. Thus, for c, the relaxation can be solved optimally with a permutation schedule. The author claims that the lower bound as a function of c is continuous, concave, piecewiseIinear function, and the maximum can be found in polynomial time. One partial sequence dominates another if the sum of completion times is smaller and the makespan is also smaller. The author develops a number of sufficient conditions for dominance. Also, if P2/ P2i ^^^ Pli Plh '^'^" ^^ author proves that there exists an optimal sequence where job i precedes job j. 2.4.3 Maximum Lateness (Lmax) Masuda, Ishii, and Nishida (1983) find a solvable case oiVll I L^^^^ and present an algorithm for the general problem with a worst case bound. The authors first prove that the HDD sequence is an optimal schedule if for i and j : rf^
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51 This says ihat the EDD and Johnson sequence arc the same. However, this property docs not provide any precedence for the general case, but the authors schedule the jobs according to EDD and give an error bound that is tight asymptotically. Grabowski, Skubalska, and Smutnicki (1983) the flow shop problem of minimizing maximum lateness with release dates. They analyze two lower bounds, one with one bottleneck machine and a nonbottleneck preceding it and the other with two bottleneck machines and a nonbotdcneck between them. 2.4.4 Numbcrof Tardv Jobs Hariri and Potts (1989) consider a permutation flow shop and develop a lower bound and a branchandbound procedure to minimize the number of tardy jobs. This problem is NPhard. For a given partial sequence and a machine j, we can consider the time this machine is available, disregard the processing before machine j and suppose that the machines after machine j have infinite capacity. The additional processing can be subtracted from the due date and The MooreHodgson algorithm applied to find a minimum number of late jobs. Applying this procedure to each of the machines yields a lower bound. The authors derive another bound using the consistent early set, a set of jobs that is feasibly early for each of the machine subproblems. The size of the smallest consistent late set (a set whose complement is a consistent early set) is the lower bound. The authors extend this to the idea of a consistent feasible early sequence for singlemachine and rmachine problems. In generating problems to test their bounds, the authors set due dates that were uniformly distributed in a range proportional to an estimate of the total processing time. The fractions ranged from 0.2 to 1.0. The branchandbound algorithm with the simple bound was good enough for the small problems, but the use of consistent sequences yielded the most efficient procedure on the larger problems. The due date range of 0.4 to 0.6 yielded the hardest problems since the range was not great enough to guide sequencing and the due dates were tight.
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52 2.4.5 General Tonics Garcy, Johnson, and Sethi (1976) prove a number of NPcomplclcness proofs for scheduling problems. They do allow zero processing times. The authors begin by discussing some of the terms of NPthcory. All of the proofs will be by transformation from 3panition, which is NPcomplete in the strong sense. The first proof is that of the ^^ 1 1 Cmax problem, where dummy jobs are created to give slots on the second machine that jobs corresponding the elements of A must fit. The next proof is for the F2 // E C: problem. Dummy jobs create slots on the second machine that dummy jobs of intermediate length and the jobs corresponding to the partition problem must fill. Long tasks are added to the end to insure that the spacers are completed as soon as possible. Lastly, they prove that ill I C^^ is NPcomplete, although this depends upon a job that is reentrant n times to machines 1 and 2. Gonzalez and Sahni (1978) consider a number of flow shop and job shop problems, with and without preemption, minimizing makespan and flowtime. The authors extend the makespan results to the preemptive problems. F3/pmtn/C^^j^ is NPcomplete even if no job has more than 2 operations. They have jobs with 12, 13, and 23 flows. They transform Partition into this, and the result and proof are the same for nonpreemptive jobs. F3/pmtn/C,Â„^jj^ is strongly NPcomplete, including jobs with operations on all three machines. J2/pmtn/C^^ is NPcomplete if all but two jobs have only two operations. The two other jobs go 121 and 212. The result also holds for nonpreemptive jobs. J2/pmtn/C,^^j;j is strongly NPcomplete with one job that visits each of the two machines n times. The authors move to some approximation algorithms and bounds. For the total flowtime job shop problem, the bound on the ratio of the flowtimes for any busy schedule to the optimal is m. And this holds if the busy schedule is SPT. The ratio for makespan is also bounded by m. A variation of Johnson's algorithm that considers pairs of machines has an error bound of mil.
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53 Lagcweg. Lcnslra, and Rinnooy Kan (1978) report on a number of differcnl lower bounds and a classification of them as well as showing that a new lower bound for the makespan problem is better than the rest. They present two dominance criteria and then move to the lower bounds, which relax the problem by allowing some machines to have infinite capacity. The unitcapacity machines are called bottlenecks and the infinitecapacity one are nonbottlenecks. The authors limit consideration to at most two bottleneck machines M^ and My and combine consecutive nonbottlenecks into one by summing the processing times for each job. Additionally, a nonbottleneck can be eliminated by adding the minimum processing requirement on this machine to the lower bound. This leads to nine different schemes (excluding symmetric ones). The authors describe each of the nine and analyze the effort required for each. They describe some upper bound calculations. They test all of the lower bounds on problems with six jobs and three, five, or eight machines. The better ones were tested on larger problems with 10 to 50 jobs and 3 to 5 machines. The best results were obtained using elimination criteria and the two bottleneck machine bound with a nonbotileneck between them and the head and tail nonbottleneck machines removed. Some researchers have looked at the flexible flowshop problem, where there may exist two or more parallel machines at a stage of the flow. Wittrock (1988) attempts to minimize the makespan and the queueing time. He creates three subproblems: machine allocation, sequencing, and timing. In order to simultaneously minimize makespan and queueing, he develops a heuristic procedure to greedily sequence the jobs given an allocation of the jobs to machines. He applies the LPT heuristic to do this allocation at each stage. Timing consists of loading the parts into the system as late as possible while not delaying any subsequent operation. He also considers buffer limits. Sriskandarajah and Sethi (1989) consider a number of simple heuristics for the flexible flowshop problem. They derive worstcase error bounds for list scheduling and a Johnsonlike sequence on the twostage case where the first stage has only one machine. Their best relative
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54 error bound is 2. They also examine the problem where each of the iwo stages has m machines. Their heuristic creates m twomachine flowshops and uses list scheduling and LPT to allocate the jobs to the different shops. Each shop is sequenced by Johnson's algorithm. Gupta (1988) and Gupta and Tunc (1991) study the special twostage cases where one of the stages has only one machine. Both papers propose heuristics that sequence the jobs (by Johnson's or some algorithm rule) and then assign the jobs to the parallel machines. Lower bounds on the makespan and the results of empirical testing are also discussed. Lcc, Cheng, and Lin (1993) study a threemachine problem where each job consists of two tasks that are assembled in a third operation. Each of the two firststage tasks is processed on a different machine and the tasks can be processed in parallel. The assembly task is performed on the third machine and cannot begin until both firststage tasks are finished. They show that the problem is NPcomplete. The authors provide some restricted versions that are polynomially complete. These include cases where one of the tasks dominates the other, where the assembly tasks is dominated by both firststage tasks, and where the assembly tasks dominates the firststage tasks. The authors present a branchandbound approach that uses the cases presented to trim the search tree. The authors present heuristics (with error bounds) that create new firststage tasks from the job data and scheduling according to Johnson's algorithm to minimize the makespan of a twomachine flowshop. The best relative error bound is 1/3. This threemachine problem is therefore closely related to problems previously studied, but the special structure of the problem (the different flows) leads to interesting twists on these results. Lee and Herrmann (1993) look at the threemachine lookbehind problem with the objective of minimizing the makespan. They derive results similar to the ones we derive for the threemachine lookahead case.
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55 2.4.6 Summary The papers covered in this section provide a number of ideas that may be useful in the thrccmachinc problems to be .studied. The threemachine problem docs not fit into the flow shop model, and thus work must be done on how to modify the above approaches. Also, the NPcomplcteness proofs do not apply, and the NfPcompleteness of the threemachine problem must be proved. The flexible flowshop models and the a.ssembly flowshop model of Lee, Cheng, and Lin (1993) may be provide some results on minimizing makespan that can be applied, although the problem is fundamentally different for the flowtimc objective. 2.5 Lookahead and Lookbehind Scheduling This research includes in its investigation of the complicated job shop scheduling problem the use of more sophisticated dispatching techniques that consider more information in their decisionmaking. This research tries to be precise in its use of the terms lookahead and lookbehind. The idea of looking at the other machines in the shop when dispatching is not wellresearched, although the workinnextqucue and numberinnextqueue dispatching rules mentioned by Panwalker and Iskandcr (1977) are simple lookahead rules. Lookbehind rules consider the jobs that will be arriving soon (called xdispatch by Morton and Ramnath, 1992). Lookahead rules consider information about the machines downstream in the flow. This includes the workinnextqueue and the numberinnextqueue rules of Panwalkar and Iskander (1977), bottleneck starvation avoidance (Glassey and Pctrakian, 1989), bottleneck dynamics (described in Morton, 1992), and lot release policies that lookahead to the bottleneck (Wein, 1988; Glassey and Resende, 1988a; and Leachman, Solorzano, and Glassey, 1988). A threemachine lookbehind scheduling problems similar to problems studied in this dissertation is considered in Lee and Herrmann (1993).
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56 Also, Blacksionc, Phillips, and Hogg (1982) report on a heuristic that is called lookahead but would be classified here as lookbehind, and Savell, Perez, and Koh (1989) did develop a lookbehind system. More specifically, lookahead models consider the states of the machines where the jobs will be processed after being processed on the machine being scheduled. Information about the workload of other machines may be useful in balancing the flow of product through the shop. Lookbehind models consider the machines the precede the machine being scheduled in the flow and the jobs being processed on these machines. Some of these machines will be processing jobs that will next need processing at the machine being scheduled. If the times that these jobs will complete is known, these times form release dates to the machine being scheduled, and the scheduling decision can explicitly consider these imminent arrivals. Other researchers have studied procedures that they called lookahead scheduling, but the problem setting or interpretation is slightly different. Vepsalainen and Morton (1987) called their weighted COVERT and apparent tardiness cost rules "lookahead" since they are concerned with the remaining waiting time of a job. They use, however, average waiting times without looking at the current queues. Koulamas and Smith (1988) are concerned with the scheduling of jobs on machines that are attended by a server that must unload a job from a machine and load another job onto the machine. Interference results from one machine finishing while the server is busy at another machine. This interference degrades the system performance by preventing the machine from being maximally utilized. The authors study a twomachine system where each machine has a distinct set of job types arriving to it. The authors propose a lookahead rule for scheduling a machine that considers the state of the other machine when deciding what job to sequence next. The rule attempts to schedule a job whose completion will not interfere with the completion of the job on the other machine.
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57 This work is similar lo the definition of lookahead in this paper, although its objective (to minimize interference) is not related to the completiontime objectives being studied. Zeestratcn (1990) is concerned with minimizing makespan in a job shop with routing flexibility. Thai is, some operations may be able lo choose from more than one machine for their processing. The author defines a lookahead rule as one that considers the entire state of the system and all of the unscheduled operations and creates a partial schedule that specifies a few operations for each machine. This schedule is followed until another scheduling decision is made. At this point, the procedure is repeated, using the new information about state of the system. The lookahead rule is so called because it searches through the states that the system could reach in a period that is approximately twice the average cycle time. That is, it looks ahead in time without attempting to schedule all of the remaining operations. Thus, it falls somewhere between fixed schedules and realtime dispatching. This type of scheduling is actually more global in nature, since it considers the entire shop when scheduling. It is not looking at specific machines. In summary, although scattered lookahead and lookbehind techniques have been considered, this research attempts to categorize these ideas, conduct analysis of lookahead and lookbehind scheduling problems in order to derive good rules, and integrate the results into a job shop scheduling environment. 2.6 Class Scheduling Manufacturing often involves machines that process different product types, and this phenomenon can be modelled as a class scheduling problem, a topic mentioned in the introduction (Section 1 .4). A number of researchers have studied the class scheduling case of sequencedependent setup times. Examples reported by Monma and Potts (1989) include paint production machines that are cleaned between the production of different colors, a computer
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58 system that must load the appropriate compiler for a type of task, and a limited labor force with workers switching between two or more machines. Bruno and IDowncy (1978) study class scheduling problems with deadlines. Their problem is the singlemachine scheduling of classes of tasks with deadlines and a setup time or changeover cost between classes. They prove that, with setup times, the question of fmding a feasible schedule is NPcomplelc. With changeover costs, the problem of finding a minimal cost feasible schedule is also NPcomplcte. Monma and Potts (1989) present complexity results for class scheduling problems without deadlines. The assume that the class setups satisfy the triangle property. They show that minimizing makespan, maximum lateness, the number of tardy jobs, and the unweighted and weighted flowtime is NPcomplete, although they discuss a number of optimal properties and dynamic programming algorithms that are polynomial in number of jobs but exponential in the number of batches. They also show that the corresponding parallel machine problems are NPcomplete. The authors conclude that the design and analysis of heuristics for these problems is important Ahn and Hyun (1990) study the problem of minimizing flowtime and present a dynamic programming algorithm that is similar to those of Monma and Potts. This algorithm is exponential in the number of classes, and the authors develop an iterative improvement heuristic that finds nearoptimal schedules. Sahney (1972) considers the problem of scheduling one worker to operate two machines in order to minimize the flowtime of jobs that need processing on one of the two machines. Sahney derives a number of optimal properties and uses these to derive an intuitive branchandbound algorithm for the problem. Gupta (1984) also studies the twoclass scheduling problem and derives an 0(rt log n) algorithm to minimize flowtime. Potts (1991) presents an example that Gupta's algorithm does not solve and goes on to describe an O(n^) dynamic program to minimize flowtime and an Oiii^) algorithm to minimize weighted flowtime.
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59 Coffman, Nozari, and Yannakakis (1989) consider a class scheduling problem wilh two subassembly part types. A product consists of two parts, one of each type, both made on the same machine, and the product cannot be delivered until both parts are finished. Their objective is to minimize the flowtime of delivered products, and using properties of an optimal schedules, they authors create an O(Vrt) algorithm. The authors describe extensions to multiple copies, limits on the number of changeovers, and limits on batch size as solvable cases without details. Mason and Anderson (1991) study the onemachine class scheduling problem with the objective of minimizing weighted completion times under sequenceindependent setups that fulfill the triangle inequality. By using properties of an optimal sequence and aggregating jobs into composite jobs, the authors develop a brandandbound algorithm using their dominance criteria and lower bound. Dobson, Karmarkar, and Rummel (1987) consider a class scheduling problem with the objective of minimizing flowtime under both the itemHow and batchflow delivery schedules. The batch (or class) setups are sequenceindependent and the processing times are equal for all parts (jobs) in one part type (class). They formulate integer programs for both problems, solving the itemflow problem and singleproduct batchHow problems optimally. For the multipleproduct batch flow, the authors can only provide some heuristics based on their other results. Dobson, Karmarkar, and Rummel (1989) consider the above singleitem problems on uniform parallel machines. They assume that the work is continuously divisible and determine the amount of work to allocate to each machine by solving a convex programming problem. The authors also apply their results to solve the itemflow problem. Woodruff and Spearman (1992) consider an interesting class scheduling problem where the objective is to maximize the profit of feasible schedules. That is, jobs must be completed by their deadlines. The profit calculation includes the value of jobs that are not required to be processed but add some revenue and two different costs: holding and setup. The authors search for good solutions with a tabu search. Discussion of this search can be found in Section 2.8.3.
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60 A more typical problem is studied by Ho (1992), who examines the problem of minimizing the number of tardy jobs where the jobs fall into exactly two classes. He develops and efficient branchandbound algorithm. Gupta (1988) studies the class scheduling problem of minimizing the total flowtimc. He develops a heuristic that is based upon the standard shortest processing time rule. Empirical testing shows that the heuristic produces good results on small problems. A more sophisticated class scheduling problem is the minimization of the maximum lateness of a set of jobs with nonzero release dates. This problem is considered by Schutten and Zijm (1993), who develop a branchandbound algorithm and a tabu search over the sequences of jobs. They report good results on problems with up to 50 jobs. In summary, this body of research does not yet include much work on problems with additional criteria, such as deadlines or release dates, or on shop problems. Moreover, this research has concentrated on branchandbound techniques and optimal algorithms for problems with just two classes. Little work has been done on heuristics for problems with an arbitrary number of job classes or on the use of smartandlucky searches to solve class scheduling problems. 2.7 Some Onemachine Problems In order to gain insight into new dispatching rules that may be useful in the job shop scheduling question, some onemachine problems will be investigated. The three onemachine problems reviewed in this section are the problem of minimizing total flowtime with deadline constraints, the problem of minimizing the number of tardy jobs subject to matching release dates, and the problem of minimizing the total flowtime of jobs with release dates. These questions will be the first to be studied as classscheduling problems.
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61 2.7.1 Constrained Flowtimc The first one machine problem studied in the preliminary work is a class scheduling extension of 1 / D . / X C., the constrained flowtime problem first studied by Smith (1956). Smith, in this paper, proves the oplimality of ordering the jobs by Shortest Processing Time to minimize the total fiowtime problem 1 / / X Cj. He extends this to the problem where the jobs all have deadlines that must be met. The makespan of the jobs is known, and following the SPT property of minimizing total fiowtime, the algorithm tries to schedule the longest job to end at this time. However, the longest job's deadline may be less than the makespan, in which case the job is not available to end at this time. So. the algorithm searches for the longest job that can end at this time and schedules it. The algorithm then moves to the start time of this scheduled job. This time is now the completion time of the remaining unscheduled jobs, and the algorithm again searches for the longest available job. This algorithm is hereafter referred to as Smith's algorithm. The weighted problem is strongly NPcomplete (Lenstra, Rinnooy Kan, and Brucker, 1977), and different elimination criteria and branchandbound techniques have appeared. Potts and van Wassenhove (1983) study the Lagrangean relaxation of the deadline constraints, leading to the discovery of optimal solutions. Work by Posner (1985) and Bagchi and Ahmadi (1987) present improved varieties of this bound. Many other problems have been studied; see Herrmann, Lee, and Snowdon (1993) for a survey of dual criteria problems. 2.7.2 Release and Due Dates The other onemachine problem in the preliminary results is the lookbehind problem 1 / r./ S Uj, which is a strongly NPcomplete problem in general (Lawler, 1982). The problem has been solved optimally if all release dates are zero by the MooreHodgson algorithm (Moore, 1968). The problem has been solved optimally by Kise, Ibaraki, and Mine (1978) if the release
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62 and due dates match (r. < r^ implies d: ^ d0. They present an 0(/i2) algorithm (Kise's algorithm, described below) in this case. Kise's algorithm orders the jobs by their release and due dates (a nonambiguous ordering since the dates must match). The algorithm is an extension of MooreHodgson algorithm for minimizing the number of late jobs. Each job is scheduled after the partial schedule of ontime jobs while maintaining release availability. If the new job is tardy, the algorithm searches the ontime jobs for the job whose removal leaves the shortest schedule of ontime jobs. The removed job is made tardy and will be processed with the other tardy jobs after the ontime jobs. In this manner, the algorithm finds the largest subset of the jobs that can be delivered ontime. These jobs are scheduled in order of their release and due dates. Unlike the MooreHodgson algorithm, the subalgorithm that removes a job cannot just choose the longest job, since the presence of release dates limits how the removal a single job affects the completion times of later tasks. They present an efficient way to determine which job should be removed. 2.7.3 Flowtimc and Release Dates A last onemachine problem that may yield some good results when studied as a class scheduling problem is closely related to the constrained fiowtime problem. This is the problem of minimizing total fiowtime subject to job release dates. Written as 1 / r./ X C:, this problem is a strongly NPcompletc question, as shown by Lenstra, Rinnooy Kan, and Brucker (1977). Dessouky and Deogun (1981) present a branchandbound algorithm with a lower bound and dominance properties used to prune the search tree. They list a number of dominance properties that hold for a given partial sequence of the jobs. Their lower bound is derived from the EFT schedule but starts the selected job at the first release time. They can solve problems with up to 50 jobs. They also report that the EFT sequence generally finds good solutions (within 3% of optimal).
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63 Bianco and Ricciardelli (1982) consider the weighted version of ihc problem. Let the weighted processing time be the processing time divided by the weight of the job. They present an improved branchandbound algorithm with further dominance properties for nodes with a partial schedule. They compute a lower bound by allowing preemption. The hardest 10job problems to solve were those with a larger range of weights and a maximum release date near the expected sum of processing times (that is, the average intcrarrival and processing times were nearly equal). Hariri and Potts (1983) also attack the problem of minimizing weighted flowtime with release dates. They use a Lagrangean relaxation of the constraints Cj > r.+ pj to find a lower bound. They make use of previous dominance properties and add another that measures the effect of interchanging two consecutive jobs. They solve problems with 10 to 50 jobs, and the hardest problems (some of which remain unsolved) were those with a range of release dates approximately the same as the expected sum of processing times. Gazmuri (1985) studies the question probabilistically. He develops two cases. In the undersaturated case, where the expected processing time is stricUy less than the expected time between release dates, the author develops an algorithm that partitions the jobs into smaller sets and schedules each of the sets optimally. The algorithm for the oversaturated case starts with the optimal preemptive schedule and patches preempted jobs by shifting the delayed segments to the left until the job is whole. For both cases, the heuristic is asymptotically optimal as the number of jobs goes to infinity. Liu and MacCarthy (1991) use both mixedinteger linear programming and branchandbound techniques to solve the problem. They present a number of heuristics that are different priority rules and report that the heuristics generally find nearoptimal solutions (within 1%) with little computational effort until the problem sizes exceed 100 jobs. The MILP can solve problems with only 10 jobs, the branchandbound procedure problems with 25 jobs. Finally, Rinaldi and Sassano (1977) also report on a branchandbound technique for the weighted problem.
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64 2.8 Smartandluckv Searches In this section we will discuss a class of searches that have been receiving much attention from researchers recently. This class include tabu search, simulated annealing, and genetic algorithms. We will use the last of these in the research into different scheduling problems. 2.8.1 Introduction One approach to difficult scheduling problems such as job shop scheduling is local search, an iterative procedure that moves from solution to solution until it finds a local optimum. Two examples are hillclimbing and steepest descent. The hillclimbing algorithm chooses a nearby point at random and moves there if the point improves the value of the objective function. The steepest descent procedure examines the entire neighborhood of an incumbent point and selects the point (if any exist) that is most improving. Heuristic searches (or probabilistic search heuristics) attempt to improve upon the primary problem of these simple searches: convergence to local optima. We can use the term smartandlucky to describe these more complex searches: they are smart enough to escape from most local optima; they still must be lucky, however, in order to find the global optimum. Simulated annealing, the most popular of the methods described in this work, was developed independently by Kirkpatrick, Gelatt, and Vecchi (1983) and Cemy (1985). Glover claims that tabu search, which is also widely used, goes back to a paper of his from 1977. Holland developed the ideas of genetic algorithms, which have had the least success so far, in his 1975 book. This section reviews the basic concepts of these searches and a number of scheduling applications. 2.8.2 Simulated Annealing Simulated annealing (SA) is a variant of the hill climbing algorithm. As mentioned before, both Kirkpatrick, Gelatt, and Vecchi (1983) and Cemy (1985) produced the initial papers on the
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65 simulated annealing algorithm, so called ixicausc the algorithm views opUmizalion as a process analogous to physical annealing, the cooling of a system until it reaches a lowenergy stale. In annealing, as a system cools, configurations occur with weight dependent upon their energy and the system's temperature. In 1953 Metropolis et al. developed a statistical simulation of an annealing system. This algorithm randomly shifted the system from one configuration to another, using the physical laws that governed behavior to guide the procedure. Kirkpatrick, Gelatt, and Vecchi (and Cemy) modified his procedure to solve optimization problems. The crucial element was the acceptance of a nonimproving move with a probability that was a function of the difference D in objective function and some cooling constant T. The standard formula was e^'^'. This acceptance of a nonimproving move is what differentiates a simulated annealing from hillclimbing. Kiricpatrick, Gelatt, and Vecchi (1983) list four ingredients for a simulated annealing algorithm: a precise description of system (that is, a solution space), the random generation of moves, a quantitative objective function, and an annealing schedule. Simulated annealing makes a move by picking from a set of possible operations that can be applied to the incumbent solution. Usually, the choice is made randomly, although the set may be ordered somehow. At a given temperature, the algorithm may stop when some equilibrium or maximum number of moves is reached. Then the algorithm reduces the temperature geometrically or linearly. The initialization of the algorithm with a starting point may be random or may use a heuristic. It can be proved that simulated annealing will converge to a global optimum. Aarts and van Laarhoven (1985) use a homogeneous Markov chain that reaches equilibrium at each temperature to prove this. They use a general form of acceptance probabilities, of which the standard exponential is a special case. However, reaching equilibrium at a temperature may require an exponential number of moves. They avoid the exponential chain length by defining quasicquilibrium conditions and conclude with a polynomial algorithm. Kirkpatrick, Gelatt, and Vecchi (1983) studies a number of chip design problems and the traveling salesman problem. In his example, he places the cities in nine clumps and examines the
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66 effect of temperature on the solutions found. At high temperature, the algorithm minimizes the number of long jumps; at a medium temperature, the coarse structure changes but stays minimal; and at low temperatures, the procedure improves the solutions at the small scale. Cemy also studies the traveling salesman problem. Van Laarhoven, Aarts, and Lenstra (1988) study the job shop scheduling problem by making use of the disjunctive graph, where the makespan of a solution is the length of the longest path. The disjunctive graph has an arc for each precedence among operations for a job and an (imdirected) edge connecting those operations on different jobs that use the same machine. A schedule can be found by directing the edges to form an acyclic graph. Given a feasible solution, any swap of a critical disjunctive edge will form another feasible solution. This becomes their move, and they claim that their simulated annealing is as good as shifting bottleneck and simpler. A different type of simulated annealing is included in the Matsuo, Suh, and Sullivan papers (1988, 1989) on controlled search simulated annealing (CSSA). In this approach, the algorithm uses a good initialization, independent acceptance probabilities, a low initial acceptance probability, and a sequential search of smaller neighborhoods. The motivation for acceptance probabilities independent of the change in objective is its use in empirical testing of different cooling schedules. Their first paper addresses the singlemachine weighted tardiness problem. The CSSA uses a fixed number of iterations and its move swaps an adjacent pair. The authors compare different cooling schedules and claim that the CSSA is a better procedure than standard SA. In the paper on job shop scheduling, the authors use the shifting bottleneck algorithm for iniualization. The CSSA identifies the critical path and then switches two operations on the critical path with !ookback and lookahead options that swap other operations to improve makespan. If a nonimproving move is not accepted, they perform a local search to find a possible improving solution. They claim better makespans with the same effort as shifting bottleneck.
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67 For now shops. Ogbu and Smith (1990) present a simulated annealing that they call "probabilisticexhaustive" because it uses independent acceptance probabilities with a random search of the entire neighborhood, moving to the last point accepted. They minimize makespan, and their moves are pair swaps and insertions. They get better processing lime than the standard SA and belter results than repetitive local searches. Vakharia and Chang (1990) look at a flow shop with jobs in families and family setup limes. They also use independent acceptance probabilities. Their moves swap jobs within a family or swap families. They use different initializations and compare their search to other heuristics. 2.8.3 Tabu Search Tabu search (TS) is a variant of steepest descent. Glover (1989, 1990) presents a good discussion of tabu searches. Given an incumbent solution, a TS searches the neighborhood of this solution, finding the best allowable move. A TS allows bad move away from local optima, prohibits moves which lead backwards through shortterm memory (the tabu list) and has an aspiration level to override the tabu list under certain conditions (usually best found). A tabu search works because the tabu list forces the search to explore new areas of the solution space. The shortterm aspect of the memory and the aspiration level allow the search to get to a global optimum however. Table 2.1. Outline of a Simple Tabu Search. 1. Pick initial x. let x'= x. T, the tabu list, is empty. 2. Let S(x) be the neighborhood of x. Take s' as best member of S(x) \ T. 3. Letx = 5Y^)Uc(x)
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68 Updating the tabu list T is the crucial pari of a tabu search. Usually this means adding moves that would reverse the step just taken and removing old tabu moves. Laguna, Barnes, and Glover (1989) investigate a tabu search for a singlemachine scheduling problem with linear delay penalties and setup cost dependencies. They use a heuristic initialization and iascriion and swap moves. They claim results much belter than a branchandbound. Barnes and Chambers (1991) investigate the use of a tabu search to solve the job shop scheduling problem of minimizing makespan. The authors describe an improved approach that uses an actual makespan change when evaluating new schedules instead of a projected change. The authors compare their job shop scheduling approach to Applegate and Cook's shuffle algorithm and the Shifting Bottleneck algorithm of Adams, Balas, and Zawack. They observe that their tabu search achieves better makespans in most cases over a range of problem sizes. Barnes and Laguna (1992) examine the multiplemachine weighted flowtime problem with a similar tabu search. This problem reduces to a partition problem, and their search does prevent nonimproving swaps: they claim that swaps are too small to move through the solution space effectively. Again, they claim results that beat a branch and boimd approach. Widmer and Hertz (1989) take up the flow shop and use permutation schedules. They define a distance between two jobs as the approximate increase in makespan and perform a TS upon the open TSP. A move in the solution space is the swap of a pair. They get slightly better makespans than six heuristics at the cost of terrible processing times. An ambitious paper by Malek et al. (1989) examines parallel and serial tabu searches and simulated annealing on the traveling salesman problem. For each search, a move was a 2opt (subsequence reversal). The parallel runs communicate periodically, sharing good solutions. Parallel SA performed a quick annealing on each processor and achieved superlinear speedup. According to the authors, parallel TS performed best, but the simulated annealing was more robust (less sensitive to parameter changes).
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69 In an inlcresling article, Woodoilf and Spearman (1992) study the problem of maximizing the profit gained from scheduling a set of jobs on one machine. The authors claim that the onemachine problem has significance for shops with a bottleneck operation, where the sequencing of the bottleneck determines how the shop performs. These jobs include jobs that arc required in some sense, say to meet specific orders, and jobs that are filler, in that they arc not required now, but some profit is realized by producing them. The jobs have due dates, and the schedule should be feasible with respect to these. The costs include the holding costs of finishing jobs too early and setup costs incurred when the machine must be switched from processing jobs of one family to those of another. Thus, this problem is a class scheduling problem, and it is an NPcomplete question just to ask if there exists a feasible schedule. The authors show that if the holding costs are zero, it is optimal to order the jobs in each family by EDD. If the holding costs are positive, this EDD within a family is still used as a heuristic in order to simplify the problem of finding a good schedule. The authors use a tabu search to search the solution space for good sequences. The moves of the tabu search are insertions of jobs while maintaining EDD within a family. Complications are added by the presence of filler jobs that are not required to be in the schedule. The authors use a diversification parameter for two reasons: first, the parameter diversifies the search by being included in a modified cost function that provides a penalty for infeasible solutions while allowing the search to use these as passes into new areas. Secondly, this parameter allows the search to optimize its performance. This is done by initially performing tabu searches over a number of different values of the diversification parameter and then continuing the search with the best values. Empirical testing of the algorithm on data motivated by an actual manufacturing environment showed that the search is a useful method of finding good solutions for this problem. Glover, Taillard, and de Werra (1991) describe the main aspects of tabu search and discuss various refinements that may lead to a new generation of search. These refinements occur on the tactical, technical, and computational levels. Tactical improvements are concerned with
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70 improving the neighborhood structure and defined moves. This may include the use of learning approaches, shifting penally tactics, and strategic oscillation. Technical improvements focus on the issues of neighborhood size, tabu list structures, and aspiration conditions. Computational improvements include reducing the time required to compute the objective function over the neighborhood and the parallclization of the search. Laguna and Glover (1991) use target analysis to promote diversification in a tabu search used to find good solutions to a single machine problem with linear delay and setup costs. The tabu search uses both swap and insert moves. The goal of using target analysis is to teach the tabu search to use good rules. The authors describe five phases of target analysis. The first phase is to take some specific problems and find very good solutions to those problems. In the second phase, with these very good solutions as targets, the problems are solved again and information is gathered on how well the current decision rules lead to the target. In the third phase, this information is integrated into a master decision rule. The fourth phase finds good parameter values for this master. Finally, the master is applied to the original problems to verify its merit. Reeves (1993) examines how the definition of the neighborhood in a tabu search affects the balance between exploration and exploitation (or diversification and intensification). Given the proper balance, he finds that tabu search is a more efficient procedure than simulated annealing for the permutation fiowshop problem. 2.8.4 Genetic Algorithms A genetic algorithm is a smartandlucky search that manipulates a population of points in the effort to find the optimal solution. Each individual in the population is a string of genes, where each gene describes some feature of the solution. Genetic algorithms mimic the processes of natural evolution, including reproduction and mutation. The most powerful operator of a genetic algorithm is the crossover operator: the recombination of the genes of two parents to
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71 create two offspring. This crossover allows the offspring to acquire the good characteristics of different points in the search space. Most genetic algorithms perform the following steps, stopping when a Fixed number of offspring has been created: Step 0: Form an initial population. Step 1; Evaluate the individuals in the population. Step 2: Select individuals to become parents with probability based on their fitness. Step 3: For each pair of parents, perform a crossover to form two offspring. Mutate each offspring with some small probability. Step 4: Place the offspring into the population. Return to Step 1. Holland (1975), Davis (1987, 1991), and Goldberg (1989) pix)vide good descriptions of genetic algorithms. Holland's 1975 book introduced the genetic algorithm (GA), a procedure that mimics the adaptation that nature uses to find an optimal state. In genetic algorithms, solutions are represented as strings (chromosomes) of alleles, and the search performs operations on the population of solutions. Liepins and Hilliard (1989) identify these operations as 1) the evaluation of individual fitness, 2) the formation of a gene pool, and 3) the recombination and mutation of genes to form a new population. After a period of time, good strings dominate the population, providing an optimal (or nearoptimal) solution. The solution strings may be a sequence of binary bits or a permutation. The fitness of a solution is its objective function evaluation or ranking. In forming the gene pool, the algorithm takes, through some random process, those solutions that are more fit. The recombination is some type of crossover, onepoint or multipoint. It is this powerful crossover mechanism that makes GAs special. The mutation operation, so important in SA, is a secondary operation here and serves only to maintain diversity. Why do GAs work? A schemata is a pattern possessed by individuals in population. The fitness of a schema is the average fitness of those solutions that possess it. Genetic algorithms work because good (short) schema survive exponentially, and the population of solutions provides implicit parallelism.
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72 When do ihcy fail? A gcnclic algorilhm may fail if ihc strings arc inappropriate representations of solutions, the strings exclude important problem information (such as constraints), or the algorithm converges to a local optimum. The first issue is the largest for scheduling problems: representation is difficult for combinatorial problems because standard crossovers may lead to infeasibility. In an effort to address this problem for the traveling salesman problem, where a solution is a permutation of cities, Oliver, Smith, and Holland (1987) propose a number of unusual crossovers. The first is the Order crossover, which substitutes a subsequence from the first parent into the second and then visits the remaining cities in the same relative order as before. The crossover maintains short schema. Their PMX crossover compares subsequences from both parents and then performs a swapping in the second parent. The Cycle crossover matches cities to form independent tours. Then, for each tour, the crossover picks a parent to supply the cities. After comparing the tree crossovers, they claim that the Order is better on TSP by preserving the short schema, which are important for this problem. The job shop scheduling problem is much more complicated problem. Davis (1985) solves a simple job shop problem with a GA. Another method is an idea by Storer, Wu, and Vaccari (1990, 1992): searching problem spaces and heuristic spaces, which are discussed in detail in Section 2.9. Storer, Wu, and Vaccari (1990) note that a solution is simply the application of a heuristic to a problem. Changing the problem or the heuristic generates a new schedule and thus a new solution. A problem is a vector of processing times, and a heuristic is a vector of dispatching rules that can be used to create a nondelay or active schedule. These data structures create simple strings and perform well under standard crossover as well as simulated annealing and tabu search. The authors perform all three searches on both types of spaces. For the tabu search, the tabu list was a set of tabu makespans. Otherwise, the SA and TS were standard. The authors also tried shifting bottleneck, probabilistic dispatching, and random search over their new spaces. They test their heuristics on a number of problems ranging in size from 10 to 50 jobs. The
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73 problems were also classified as easy or hard. The easy problems had completely random routings, and the hard problems tended have greater competition for the machine resources at any point in time. The genetic algorithms on problem space generally found higherquality solutions. Fox and McMahon (1990) are interested in the job shop scheduling problem but study the traveling salesman problem in order to gain insight into using genetic algorithms for sequencing problems. The authors consider the binary precedence matrix M where m,.= 1 if city / precedes cityy. m^i = for all cities /. This matrix incorporates microand macroinformation. The authors consider row and column swap operators that exchange the successors or predecessors of two cities. The first new operator is the intersection operator, in which an offspring inherits the precedences that exist in both parents. That is c^ = a^ and ft^,' o^^r all ij. This will yield a matrix that is consistent (no cycles) but incomplete; in other words, a partial ordering. The matrix is completed through an analysis of the row and column sums. The second new operator is the union operator. First, the cities are partitioned into two sets. This forms for each parent two precedence submatrices corresponding to the two subsets. The operator then takes one submatrix from one parent and the submatrix for the other subset from the other parent. Again, this leads to a partial ordering that must be completed. The authors claim that this operator contains little microinformation from the parents, unless a Markov process is used to partition the cities. The authors compare different operators for the TSP, including a random operator used as a benchmark. If the proposed operators are no better than this, they should not be pursued. Their city topologies included random distances, clustering, concentric circles, and a 30city problem from Oliver, Smith, and Holland (1987). The population size was 900 with no mutation. The also tried searches with and without elitism (the keeping of the best solution). Among nonelite searches, the two new operators were great, even when considering processing time. Their advantage disappeared when the searches got to use elitism.
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74 The authors claim thai this work will be useful in scheduling problems, although execution time remains a big problem. Biegcl and Davem (1990) begin with a general discussion of genetic algorithms. They then describe how a genetic algorithm could play a part in the scheduling of a shop. They discuss what data are essential and how the GA could form a schedule for a known group of jobs that follow the same route. The authors move to the singlemachine problem and consider how the analogous GA would work. Moving to the 2and //imachine flow shop, the authors present a simple discussion of what the GA would do. For the dynamic problem, the authors consider performing a rescheduling in realtime to deal with arrivals and use a FIFO dispatching rules on all other machines. The authors then list a number of areas for future research. Cleveland and Smith (1989) use a genetic algorithm to schedule the release of jobs into a manufacturing cell to minimize weighted tardiness. They consider both the sequencing problem and the problem of determining the release times. Nakano and Yamada (1991) introduce a binary representation to solve the job shop scheduling problem. This representation denotes the relative ordering of each pair of jobs on each machine. If an illegal chromosome is formed by a genetic operator, a nearby legal one replaces it. The authors report that their algorithm finds good solutions on some standard problems. Whitley, Starkweather, and Shancr (1991) develop an edge recombination operator for the traveling salesman problem and report that a genetic algorithm with this operator is able to find very good schedules. Starkweather et al. (1991) compare six sequencing operators for the traveling salesman problem. They discover that the different operators stress different types of information; since the TSP depends upon adjacency, the edge recombination operator is the best.
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75 In Syswercla(1991), the author discusses the scheduling of a lighter simulation lab. Finding a feasible scheduling is made difficult by a number of constraints. A genetic algorithm that manipulates sequences of jobs and employs a schedule builder to translate the sequence into a legal schedule was able to find good schedules. 2.8.5 Summary Simulated annealing and tabu search easily search complex spaces by lending themselves to usually simple types of moves, generally find good solutions fast, and are smart variations of standard searches such as hillclimbing and steepest descent. Genetic algorithms work in a different manner. They work well due to the survival of good schema and their implicit parallelism. They have been harder to implement on scheduling problems, however. This research in this dissertation builds upon this work into smartandlucky searches by looking into some alternative search spaces. 2.9 Problem and Heuristic Space This dissertation includes the development of global job shop scheduling models, whose solution will yield a schedule that can be followed for time period like an eighthour shift. As mentioned in the last section, one way to find good schedules is to use a search. One recentlyprofX)sed idea is to search problem and heuristic spaces. This section will define mathematically how these spaces can be used and will review the ideas of a few papers in this area. Define a problem p as a set of data about which an optimization question can be asked. A solution 5 is a point that is consistent with the problem structure, where the solution has some performance z measured by applying the objective function/to the solution with the problem data, z =f(p,s). If the problem is fixed, there is a performance function CÂ„ over the solution space such that z = Cp(s) =f(p,s). Solving a problem translates as finding the solution that gives the minimum function value.
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76 Traditionally, problemsolving searches have occurred in the solution space. In the solution space live points (for standard optimization) or sequences (for combinatorial optimization). Heuristics arc also applied to problems. A heuristic h, applied to a problem p, yields a solution s, s = G(hjj) or .V = h(p). Thus, for a given problem p, there exists a performance function Dp(h) over the heuristic space such that z = Dp(h) = Cp(h(p)). This implies that a search over different heuristics could fmd a solution that gives the optimal objective function value. Moreover, a solution can be generated in ways besides applying a heuristic to the original problem. Applying a heuristic /z to a problem y in another space can generate a solution s = h(y), which can be evaluated using the objective function. Thus, given a problem p and the heuristic h, there exists a performance function Â£Â„ ^fy) over the other problem space, where z = Ep ^(y) = Cp(h(y)). And as before, a search over this new problem space provides a way to solve the problem. The idea of searching heuristic and problems spaces was investigated by Storer, Wu, and Vaccari (1990, 1992). In these papers, the authors are investigating the general job shop scheduling problem. They define a heuristic space composed of vectors of dispatching rules. The heuristic uses each rule in turn for a fixed number of dispatching decisions. For example, if the problem is a tenjob by tenmachine problem, the vector might have five elements, where the dispatching rule in the first element is used for the first window of twenty decisions, the next rule for the next window, and so on, until all one hundred operations have been processed. For a problem space, they use a space of vectors of processing Umes, where each position corresponds to a different operation. At each scheduling decision in the formation of the schedule, the dispatching rule selects a job by considering these alternative processing times. The operation of that job is scheduled using the actual job data in order to calculate the finish time of the operation, thus ensuring that the schedule produced will be feasible. Thus, a vector of processing times yields a sequence of operations for each machine.
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77 The authors include a number of concepts related to searching these new spaces, including the definitions of neighborhoods, the use of these spaces in genetic algorithms, the fact that the spaces contain optimal solutions, and details of local search implementations. The application to genetic algorithms merits special comment. Traditional genetic algorithms have difficulty with scheduling problems since the components of the strings (the jobs in the sequence) are not independent of each other. The problem space and heuristic space described above, however, consist of vectors that have independent elements. That is, the dispatching rule (or processing time) in the first element does not affect what values the other elements can have. Thus, a crossover operation that breaks two strings (vectors) and joins the separate pieces yields offspring that are valid points in the search space. The authors extend this work in Wu, Storer, and Chang (1993) to the onemachine rescheduling problem. They use heuristics that adjust "artificial tails" to find schedules that are efficient and deviate little from the original schedule. Bean (1992) uses the idea of random keys to provide an alternative search space for a number of problems: multiplemachine scheduling, resource allocation, and quadratic assignment: the keys are used to sequence the jobs (or other variables). Then some simple rule is used to generate a solution from this sequence. Good empirical results are reported. Our research extends these ideas to onemachine class scheduling problems and investigates a similar heuristic space for the job shop scheduling problem. 2.10 NPComplctcncss Job shop scheduling problems are usually quite difficult to solve, and they are among the hardest optimization problems known. Combinatorial problems, as well as other types of problems, can be classified by their complexity, or how difficult they are to solve. The hardest problems are called NPcomplete. The primary guide to the theory and use of NPcompleteness is the book by Garcy and Johnson (1979).
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78 Most ot the problems studied in operations research have been classiTicd by their complexity, and as researchers study new problems, their complexity is idcntiried also. There are three primary classifications: polynomial, NPcomplelc, and strongly NPcomplete. Problems in the first set can be solved with algorithms that have a running time proportional to some polynomial function of the problem size. Algorithms to solve problems in the second set require effort that is a polynomial function of the problem size and the problem data. Such an algorithm is called a pseudopolynomial algorithm. Algorithms to solve problems that are strongly NPcomplete require an exponential amount of effort. The complexity of a problem can be determined by identifying the problem as a more difficult case of a hard problem, by developing a polynomial algorithm to solve the problem, or by transforming a previously classified problem into the new problem. By being able to determine the complexity of a problem, researchers are able to know what types of approaches may be possible in finding optimal solutions to the problem. A problem that is known to be NPcomplete will not be solved optimally by any polynomialtime algorithm. Pseudopolynomial dynamic programs will not solve strongly NPcomplete problems. This does leave the possibility, however, that certain special cases may be more easily solved, that algorithms that search for an optimal solution may have good average performance (e.g. the simplex method of solving linear programs), and that polynomialtime heuristic algorithms may be able to find generally highquality solutions. This research is concerned with a number of different scheduling problems, which will classified by their complexity as necessary. We will directly prove NPcompleteness for one of the problems that we study. Each of the other problems is a more difficult case of a previously considered NPcomplete problem. 2.11 Chapter Summary This chapter has attempted to cover a broad spectrum of topics and literature. Obviously, much research has been done in the fields of job and fiow shop scheduling and the field of
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79 managing semiconductor manufacturing. However, no system to optimize the scheduling of such a process has emerged, despite the use of many different and sophisticated procedures. Still, new tools are needed that may have good performance when applied to specific settings. There do exist new ideas, such as class scheduling, lookahead and lookbehind approaches, smartandlucky searches, and problem and heuristic spaces. The literature in these areas reports on some initial research. However, the full capabilities of the these methods, especially when joined to solve a scheduling problem, have not been fully explored. I I
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CHAPTER 3 ONEMACHINE CLASS SCHEDULING PROBLEMS In Chapters 3, 4, and 5. we discuss the results of our research into a number of different scheduling problems and the general job shop scheduling problem. The primary motivation is to examine subproblems of the job shop scheduling problems in order to gain insight into the larger problem. The subproblems that we will examine are interesting scheduling problems that have not been previously considered. We will start with the onemachine class scheduling problems. Our approach in this chapter is to develop analytical results, to test extended heuristics, and to show that a problem space genetic algorithm can be a good procedure for a variety of scheduling problems. 3.1 Introduction The three onemachine problems studied in this work are class scheduling problems that model the complicating factor of machine setups in the manufacturing process. Most problems with sequencedependent setup times are NPcomplcte. Class scheduling problems have a special structure that makes them good candidates for further research: the jobs to be scheduled form a number of disjoint job classes and setups occur whenever the machine processes consecutive jobs from different classes. And although we will study a number of different problems, we will sec that a problem space genetic algorithm will be a useful procedure for all of them. The onemachine class scheduling problems under investigation are as follows: 1. Constrained Flowtime with Setups (CFTS) 2. Class Scheduling with Release and Due Dates (CSRDD) 3. Flowtime with Setups and Release Dates (FTSRD) 80
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81 The first problem studied is the class scheduling extension of the onemachine problem of minimizing the total flowtime of a set of jobs that have deadlines. The research includes an optimalily property for the jobs in the same class, a good heuristic, and the development of a problem space genetic algorithm that can find better solutions. The second problem is the class scheduling problem where each job has a release dale and a due date. The objective is to minimize the number of tardy jobs. We investigate a number of heuristics and the use of a problem space genetic algorithm. We also look at a secondary criteria, minimizing the total tardiness. The objective in the third problem is to minimize the total Howtime where each job has a release date. Wc consider a number of approaches to finding good solutions, including a problem space genetic algorithm, and present a special case that can be solved with a pseudopolynomial dynamic programming algorithm. This chapter considers the research on each of these problems in turn. Research relevant to these problems is also discussed in Chapter 2. See especially Sections 2.6 and 2.7. 3.2 Constrained Flowtime with Setups We will first consider the onemachine class scheduling problem of minimizing the total flowtime subject to the constraint that each job must finish before its deadline. A new heuristic is proposed for the problem. We investigate the use of a genetic algorithm to improve solution quality by adjusting the inputs of the heuristic. We present experimental results that show that the use of such a search can be a successful technique. 3.2.1 Introduction This research is motivated by the scheduling of semiconductor test operations. Assembled semiconductor devices must undergo electrical testing on machines that can test a number of different types of semiconductors. If a machine is scheduled to test a lot consisUng of devices that are different from the devices tested in the previous lot, various setup tasks are required.
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82 These tasks may include changing a handler and load board thai can process only certain types of semiconductor packages or loading a new test program for the new part. However, if the new lot consists of circuits that are the same as the previous type, none of this setup is required. This type of change is a sequencedependent setup that can be modelled by class scheduling. Since postassembly testing is the last stage in semiconductor manufacturing, meeting a job's due date is a very important objective for the manager of a test facility. A secondary criterion is the minimization of total flowtime (the sum of the job completion times), which reflects the manager's desire to increase throughput and decrease inventory holding costs. This is a dual criteria problem, in which the primary criterion is used as a constraint and the secondary criterion is optimized under this restriction. The problem of minimizing the total flowtime subject to deadlines is an old problem. Smith (1956) provides an optimal solution technique that repeatedly schedules the longest eligible job last. The class scheduling version of the problem, however, is more difficult. For our problem, finding a feasible schedule is an NPcomplete problem. (A schedule is feasible if every job finishes before or at its deadline.) Thus, there exist no exact algorithms to minimize in polynomial time the total flowtime subject to the deadline constraints. (For a discussion of the theory of NPcompleteness, see Section 2. 10 and Garey and Johnson, 1979.) Thus, we are motivated to try different heuristics. In this work we develop a multiplepass heuristic that finds good solutions quickly. The first contribution of our investigafion of this problem is the extension of Smith's algorithm into a heuristic which considers the setup times while sequencing the jobs by their deadlines and processing times. We are also interested in using a genetic algorithm to improve the quality of our solutions. A genetic algorithm is a heuristic search that has been used to find good solufions to a number of different optimization problems, but genetic algorithms searching for good schedules must overcome the difficulty of manipulating the sequences of jobs. We investigate the use of a genetic algorithm to search a new type of space, the problem space. This type of approach was introduced in Slorer, Wu, and Vaccari (1992), who consider alternative search spaces for the
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83 general job shop scheduling problem. In this worii, wc extend the idea to the problem of onemachine class scheduling. Our search attempts to adjust the deadlines of the given problem so that our heuristic will find even better solutions. The space of adjusted deadlines that we search forms n problem space (we will return to this pwinl in Section 3.2.5). The second contribution of this work is our use of this method to improve the scheduling of a singlemachine problem. If we use the principles of Davis (1991), then we can classify our genetic algorithm as a type of hybrid genetic algorithm. However, the only unusual characteristic of our algorithm is the decoding (the bit string does not describe a point in the solution space; instead it must be mapped to a solution via the heuristic). Moreover, the range of hybrid genetic algorithms is so large (for instance, Goldberg, 1989, describes hybrids differently) that our use of the term problem space genetic algorithm is a more precise description of the search. Finally, this problem space exists independently of the genetic algorithm, and the use of this new search space is not limited to our search. Other searches (including steepest descent, simulated annealing, and tabu search) could be used to explore the space. Therefore, we will continue to refer to our search space as a problem space and to our search as a problem space genetic algorithm. The next subsection summarizes some of the relevant literature on class scheduling problems and the dual criteria objective under consideration. In Section 3.2.3, we discuss our notation, an example instance of the problem, and a number of basic results. We discuss in Section 3.2.4 the heuristic developed for the problem. Our genetic algorithm will employ this heuristic. In Section 3.2.5, we present a problem space, introduce genetic algorithms, and discuss the details of the genetic algorithm we developed to search the problem space. Section 3.2.6 describes the generation of the sample problems, the computational experiments, and the results. Finally, in Section 3.2.7, we present our conclusions.
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I 84 3.2.2 Literature Review In this section we will briefly mention some of the relevant research on class scheduling and on the dual criteria problem of minimizing total flowtime subject to job deadlines. This owrk and the literature on genetic algorithms are discussed in more detail in Chapter 2. One of the first papers on problems with class scheduling characteristics is Sahney (1972), who considers the problem of scheduling one worker to operate two machines in order to minimize the flowtime of jobs that need processing on one of the two machines. Sahney derives a number of optimal properties and uses these to derive an branchandbound algorithm for the problem. Gupta (1984) defines the class scheduling problem, and Potts (1991), Coffman, Nozari, and Yannakakis (1989), and Ho (1992) also study twoclass scheduling problems. Bruno and Downey (1978) prove that, for more general class scheduling problems, the question of finding a schedule with no tardy jobs is NPcomplete. Monma and Potts (1989) prove that many class scheduling problems are NPcompleie, including minimizing makespan, maximum lateness, the number of tardy jobs, total flowtime, and weighted flowtime. Dobson, Karmarkar, and Rummel (1987, 1989), Gupta (1988), Ahn and Hyun (1990), and Mason and Anderson (1991) all study the class scheduling problem under different objective functions. We will modify the procedure of Ahn and Hyun in order to use it as a comparative heuristic. The only other dual criteria problem in this area is studied by Woodruff and Spearman (1992); they consider a class scheduling problem with profit maximization and deadlines. In the dual criteria literature, the problem of minimizing total flowtime subject to job deadlines (a deadline is a constraint on the completion time) is among the oldest questions, being first studied by Smith (1956). The problem of minimizing the weighted flowtime subject to job deadlines is a strongly NPcomplete problem (Lenstra, Rinnooy Kan, and Brucker, 1977), and a number of researchers have examined branchandbound techniques.
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85 3.2.3 Notation and an Opiimal Proocnv In this section we introduce our problem and notation, give an example instance of the class scheduling paiblem under consideration, and present some basic results. Our class scheduling problem is the minimization of the total flowtime of a set of jobs where the jobs have deadlines on their completion. The problem can be formulated with the following notation: Jj }ohjJ=l,. . . ,n Pj processing time of /.Dj deadline of 7. G,job class i,i= I, . . . ,m rtjnumber of jobs in Gj Sq^ time of initial setup if first job is in G,Sf^ time of setup between jobs in G^ and G,Cj completion lime of 7,5J C; total flowtime. The problem is to find a sequence that minimizes the total flowtime (Z Cj) subject to the deadline constraints (C; < D . for all Jj). We name this problem the Constrained Flowtime with Setups problem (CFTS). Since job preemption or inserted idle time leads to a nonoptimal solution, we will assume that schedules being considered have neither. Any schedule that is a solution for CFTS will have a number of batches or runs that are sets of jobs from one class processed consecutively. Before each batch will be a class setup. The problem involves determining the composition and order of batches from different classes. CFTS is an extension of a onemachine problem studied by Smith (1956). In his problem, which we name the Constrained Flowtime problem (CFT), there exist no sequencedependent setup umes. An instance that we will use to illustrate our work is described in Example 3. 1 . Example 3.1. The data in Table 3.1 form an instance of a class scheduling problem with five jobs in two job classes. The first three jobs form one class, with the remaining two jobs in the
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86 second class. Recall thai no setup is required between jobs in the same class. However, a class setup is necessary between jobs of different classes. Table 3.1. Job and Class Data for Example 3.1. j
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87 The following property extends Smith's mle to each class in CFTS. This property will then be extended to consider all classes in order to generate approximate solutions to CFTS. Lemma 3.1 (Smith's property for CFTS). For each class in an optimal schedule for CFTS. the only job that could be scheduled to complete at a time t is the longest eligible one. Proof. It suffices to show that if two jobs 7, and J: in the same class are both eligible at time / and J I is longer than J: (p,> p.), then scheduling 7. to complete at time / leads to a nonoptimal solution. Suppose we do. Then C: = t, and 7,precedes J: in the schedule formed. Create a new schedule by interchanging the two jobs. Since J: is moved to the left, it is still feasible, and the new completion time is less than Cj, the old completion time of 7, (p,>p/). The completion times of any jobs between 7.and 7,are decreased. Meanwhile, 7/ completes when Jj did, but this is feasible since t < D,. We have therefore created a feasible schedule with less total flowtime, and the original schedule cannot be optimal. QED. 3.2.4 The Heuristic Quick methods of finding good solutions are sometimes effective ways to attack difficult problems. In this section we describe a multiplepass heuristic that extends the idea of Smith's rule. We illustrate how this heuristic works using Example 3.1. Our heuristic finds solutions for CFTS by scheduling jobs in the spirit of Smith's rule, working backwards from the end of the schedule. Since the makespan (the maximum completion time) of the optimal solution is not known, the heuristic starts with a trial makespan. After scheduling all of the jobs, we compute the actual makespan (by removing any idle time) and use this makespan as the starting point for another iteration. We continue this process until some limiting makespan is reached. At this point, another pass of the heuristic yields a schedule with the same makespan or a schedule that is infeasible (because some job or setup starts before time zero). This heuristic consirucLs schedules that satisfy Smith's property for CFTS (Lemma 3.1). While that lemma applies only to jobs in one class, our algorithm extends the idea of longest
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88 eligible job by coasidering all of ihe job classes. Wc schedule the longest job with the minimum wasted time. Wasted time is time spent in a setup and idle time. This (singlepass) Minimum Waste algorithm schedules an eligible job from the same class as the previously scheduled job if one exists. Else, it selects a job from the class with the smallest setup or selects the job with the latest deadline. It does this by measuring each job's gap: the wasted time incurred by selecting that job. Note that if no class setups exist, this algorithm is the same as Smith's rule for CFT (Algorithm 3. 1). Algorithm 3.2 (Singlepass) Minimum Waste. Step 0: Given a completion lime t, select for the last job the longest job eligible at this time Jj, i.e. r < Dj. Schedule this job to end at t, and reduce t by pj. Step 1 (a modification to Smith's rule): Suppose that at time t, a job from class G,starts. Then, for each unscheduled job J:, define q: as the gap between the last possible completion time of Ty and t. liJj is in class G^, q: = max {/ Dj, 5^,} (see Remarks below for an explanation of this definition). Let ^ = min {q: over unscheduled 7, } . Select the longest job Jj with qj = ^ and schedule this job to end altq. Any necessary setup sj^i can begin altq. Reduce f by
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89 MultiplePass Minimum Waste Heuristic. To find a good solution for CFTS. we can use the following procedure that makes use of the singlepass Minimum Waste algorithm. Step A: Lclt'=mdi\ [Dj.j = \, . . . ,n]. StepB: Letr = t'. Step C: Perform one pass of the Minimum Waste algorithm (Algorithm 3.2) with completion lime t. This creates a trial schedule. Step D: Let t' be the sum of processing times and setup times of this schedule. \f t' < t and the trial schedule is feasible, go to Step B. (The smaller makespan may yield another schedule.) Step E: If the trial schedule was infeasible or t' = t, take the last feasible schedule created and remove the inserted idle time, starling all jobs as soon as possible. This schedule is the result of the heuristic. If an infeasible schedule was created on the first pass, then lake the sequence of jobs from the schedule and process the jobs in this order, starting at time zero. This will yield a schedule with some violated deadline constraints. Because the problem of finding a feasible schedule is NPcomplele, a single pass of the Minimum Waste algorithm is not guaranteed to find one. Still, as we shall see, it is usually able to find a feasible schedule if one exists. If a feasible schedule exists, it must finish by the maximum deadline, which is the first trial makespan. Initially, the heuristic is concemed with reducing the makespan. Eventually, as the makespan reaches a lower limit, the algorithm concentrates on the flowtime objective through its use of Smith's property to select a job. Example 3.2. Let us apply the MultiplePass Minimum Waste heuristic to Example 3. 1 . In the first iteration, f = 1 8, the maximum deadline. The first pass of the Minimum Waste algorithm performs the following calculations (see Table 3.2 for complete algorithm): at time 18, no jobs have been scheduled, and the only eligible job is J^. After choosing 75, t is reduced by pj = 2 to 16. Foryy,yj, and y^, the waste is the gap until the deadline. Thus.t^y = 16Dy = 13, and similariy for the other two jobs. For 72however, D2 = 16. and the deadline gap is zero, but because J2 is in a different class than J^,q2 = sj2=\. Thus, J 2 has the smallest waste and is scheduled to end at lime 15. After five steps, all of the jobs arc scheduled (see Figure 3.1). There
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90 are two units of inserted idle time, however, so the actual makcspan of the schedule can be reduced to 16. Table 3.2. Calculations of the first pass of the Minimum Waste algorithm. Initial makespan ; 18. Time:
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91 Table 3.3. The second pass of the Minimum Waste algorithm. Initial makcspan 16.
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92 one adjusts the problem data that are used by the heuristic, one generates a different solution. This set of adjusted problem data is tlic problem space. The idea is applied to the job shop scheduling problem, and different heuristic searches over the spaces are pcrfomied, including hill climbing, genetic algorithms, simulated annealing, and tabu search. Our research extends this idea by defining a problem space for the onemachine class scheduling problem. If we adjust the deadlines that are inputs (along with the other problem data) to a pass of the Minimum Waste algorithm, we will create a possibly different schedule. We will use as a problem space for CFTS these adjusted deadlines, and we will use one pass of the Minimum Waste algorithm to create a sequence of jobs using the adjusted deadlines. The feasibility (against the actual deadlines) and total flowtime of the sequence can be evaluated by scheduling the jobs to start at time zero with no inserted idle time. The idea is to force jobs to be done earlier or later by decreasing or increasing the deadlines. We will prove that every solution for CFTS (including the optimal one) is in the range of /i. Theorem 3.1. For each solution to an instance of CFTS, there exists a vector of adjusted deadlines that can be mapped to that solution using one pass of the Minimum Waste algorithm. Proof. Suppose that o is a solution (a feasible schedule with no preemption or inserted idle time) for an instance of CFTS. For each job, consider adjusting the deadline so that it equals the job completion time. Then, if we use one pass of the Minimum Waste algorithm with the adjusted deadlines, the job selected for the last position will be the job with the maximum adjusted deadline. This job is the one with the maximum completion time C,and thus was the last job in a. It will be scheduled to complete at its adjusted deadline, which is C,. Now we are at the start time t of a job J^ and the job with the smallest gap is the unscheduled job J: that immediately precedes 7,in o, since the adjusted deadline is Cj, and Cj < t. Any setup necessary between 7 . and J^ is already included in the difference between Cj and t. Thus the gap cannot be larger for this job, and the gap for any other job 7^ is larger since Q < Cj. This job will be scheduled to complete at its adjusted deadline, which is Cj. If we continue in this
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93 manner, all of the jobs will be sequenced in ihc same order as Ihey were in a, and we create the same schedule. QED. The more we adjust the deadlines, the more change we create in the schedule. For instance, consider the following examples of applying one pass of the Minimum Waste algorithm to vectors of adjusted deadlines where we have changed only the second and fifth deadlines: (Note that the fourth schedule created is infeasible since J 2 completes at time 18, which is greater than the actual deadline: D2 = 16. The adjusted deadline of 19 was used only to sequence the jobs.) Heuristic(problem) = solution: Minimum Waste (3, 6, 14, 10, 20) = [ 1 2 4 3 5 ], flowtime 46. Minimum Waste (3, 16, 14, 10, 18) = [ 1 4 3 2 5 ], flowtime 50 (original deadlines). Minimum Waste (3, 17, 14, 10, 16) = ( 1 5 4 3 2 ], Howiimc 46. Minimum Waste (3, 19, 14, 10, 17) = [ 1 4 3 5 2 ], infeasible (C2 = 18). In Figure 3.4 we show a graph that illustrates how adjusting just two of the five deadlines of Example 3.1 can create a number of different schedules. The first, third, and fourth deadlines were not adjusted. Each point in the plane (only nonnegative deadlines were considered) corresponds to a pair of values for the second and fifth adjusted deadlines. The points in each region of the plane are mapped by a pass of the Minimum Waste algorithm to the job sequence denoted by the fivedigit sequence shown in that region. The dot marks the point that corresponds to the unadjusted deadlines {D2 = 16, D^ = 18). The best sequences achievable by adjusting these deadlines are 12435 and 15432 (total fiowtimc = 46), and the only other feasible sequences are 14235 and 14325 (total fiowUme = 50). The optimal solution (which cannot be found by adjusting only the second and fifth deadlines) is 13452, with total flowtime = 43. Since the actual problem space consists of all of the problem data and there are numerous heuristics that can be used, we can investigate other spaces and heuristics that might be usefiil. Our first search was to adjust the job processing times and to use the Shortest Processing Time (SPT) rule. However, it is difficult to find feasible solutions since SFT ignores the deadline constraints entirely. We also tried the using the Earliest Due Date (EDD) rule while adjusting the
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94 deadlines, but EDD docs not give enough allenlion to the flowtime objective. Sequencing by either SPT or EDD is a fairly naive heuristic, since neither makes use ol the other available information. The Minimum Waste algorithm, however, considers due dates, processing times, and setups, and using it improves our searches. In addition, while it would be possible to use the Minimum Waste algorithm while adjusting the processing or setup times, the effect of these variables on the sequencing of jobs is more indirect than that of the deadlines. The use of a heuristic space seems to be hard for this problem. Feasibility is a large concern, and there are very few heuristics we can use to find feasible schedules. Also, the Minimum Waste algorithm has no parameters to adjust. Job 5 Deadline 18 14
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9S A genetic al^oriihm lor CHTS . In this section we discuss ihc details of the genetic algorithm we developed lo find better solutions for CFTS. As mentioned before, genetic algorithms arc heuristic searches that use a population of points in the effort to find the optimal solution. The stronger members of the population survive, mate and produce offspring that may undergo a mutation. These offspring form a new generation. Genetic algorithms have been used on sequencing problems before, although they cannot use natural crossover techniques in searching the solution space. The advantage of the problem space is that the genetic algorithm can use standard techniques to create offspring. Our genetic algorithm searches the space of adjusted deadlines. We use a binary coding for the adjusted deadlines and a single pass of the Minimum Waste algorithm as the heuristic. For this genetic algorithm, we use many of the ideas presented in Davis (1991), to which we refer readers who wish to learn more about the issues discussed here. In the problem space, each point is a vector of integers that are deadlines used as input for one pass of the Minimum Waste algorithm. We will use a binary representation of the points in problem space. In the population of the genetic algorithm, each individual is a string of bits. Each successive sixbit substring represents a deadline for a specific job. The integer decoded from tills binary number ranges from zero to 63 and linearly maps to a real number in the range from zero to the maximum deadline in the given problem data. This discretization reduces the problem space but still allows the deadlines to vary significantly with respect to each other. The adjusted deadlines are used as input to a single pass of tiie Minimum Waste algorithm, which outputs a sequence of jobs. The algorithm uses the largest of the adjusted deadlines as the initial makespan and schedules Uie jobs accordingly, using tiie actual job processing and class setup times where necessary but using the adjusted deadlines to determine when a job is eligible. If necessary, the algorithm can start jobs before time zero. Using the actual problem data and the sequence of jobs output from one application of tiie Minimum Waste Algoritiim, we can create a schedule of jobs Uiat starts at time zero and has no
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96 inserted idle time. Wc evaluate the original string of bits by computing the feasibility and the total flowtimc of this schedule. Since we cannot guarantee that the algorithm will produce a schedule with no tardy jobs, we use a penalty function to make undesirable those individuals in the population that yielded infeasible schedules (with respect to the actual deadlines). This penally function is C> = X 7".^, where T: = max (0, C. D.} . Our objective function/is defined as/= 'LCj + r . In order to encourage solutions with good total flowtime (regardless of feasibility) at the beginning of the search and to encourage feasibility near the end of the search, we start the search with the constant r small and increase it periodically. The initial population includes one individual (the dummy, or seed) that we create by dividing each actual deadline by the maximum deadline, multiplying by 63, rounding down to the nearest integer, and converting this integer (which is in the range to 63) to its binary representation. The remaining individuals in the initial population arc constructed by mutating the bits in the initial (dummy) chromosome. The mutation rate is set at fifty percent (0.5). Note that using an initial mutation rate of 0.5 is equivalent to choosing a random chromosome from the entire search space. Let us illustrate this procedure using Example 3.1, the problem we introduced earlier (see Table 3.1 for problem data). Also, let us define LxJ as the greatest integer less than or equal to x. Mapping the deadlines to the bit strings yields the dummy, shown in Table 3.4. Let us create another member on the initial population. If two bits, the fourth of the fourth substring and the first of the last substring, are flipped in the mutation, we have the point in problem space shown in Table 3.5. Performing one pass of the Minimum Waste algorithm on the new deadlines (see Table 3.6) yields the sequence [15 4 3 2], from which we create the feasible schedule shown in Figure 3.5, with a makespan of 15 and a total flowtimc of 46.
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97 Table 3.4. Bit representation of the dummy. L63Dj/18j
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98 Table 3.7. Flowlimc performance while tuning 2(XX)individual genetic algorithm on 30job problems (Ncwprot>4). Population Size Operator Probabilities (in percent) Increase in r Frequency of Increase Mutation Average Rate Ratio Tuning Increase in r and Frequency of Increase, Population Size = 10 10 10 10 10 10 10 10 10 10 25. 25. 25. 25. 25. 25. 25, 25. 25. 25. 25. 25. 25. 25. 25.25. 25.25. 25.25 25.25 25.25 25.25 25.25 25.25 25.25 25.25 25.25 +2 + 10 +50 +2 + 10 +50 +2 +10 +50 10 10 10 50 50 50 100 100 100 Tuning Increase in r and Frequency of Increase. Population Size 50 50 50 50 50 50 50 50 50 50 25. 25. 25. 25. 25. 25. 25, 25, 25, 25. 25, 25. 25.25. 25, 25. 25, 25, 25.25 25.25 25.25 25,25 25,25 25,25 25.25 25.25 25,25 +2 + 10 +50 +2 +10 +50 +2 +10 +50 10 10 10 50 50 50 100 100 100 Tuning Increase in r and Frequency of Increase, Population Size 100 100 100 100 100 100 100 Tuning Mutation Rale 10 10 10 10 10 10 50 50 50 25.25.25.25 25.25.25.25 25.25.25.25 25. 25. 25. 25 25, 25, 25, 25 25.25.25.25 25. 25. 25. 25, 25, 25. 25. 25. 25. 25. 25. 25. 25. 25. 25.25. 25.25. 25.25 25,25 25,25 25.25 25.25 25.25 25,25 25,25 25,25 +2 +50 +2 +50 +2 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 10 10 50 50 100 100 10 10 10 50 50 50 10 10 10 2%
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99 Table 3.7. (Continued) Population Operator
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100 solution. Thus, a search that incorporates some kind of local search at the end of the genetic algorithm may be useful. 3.2,0 Empirical Tv\stins In this section we describe a set of experiments performed in order to test the genetic algorithm and the heuristics. Wc discuss the generation of sample problems and the computational results. Problem generation . In order to test the heuristics and the genetic algorithm described above, it is necessary to create a set of test problems. We describe in this section how we can create problems that have at least one feasible solution and problems where finding a feasible solution is more difficult. The problems in the first problem set have 30 jobs in four classes, with random processing times in the range [1, 20] and sequencedependent setup times in the range [0, 5]. For this set, we want to determine random deadlines in order to insure that some feasible schedule did exist. We use the following procedure: after computing the random class setup limes, each job is given a random processing time, and an initial completion time is computed by scheduling it after all previously constructed jobs. This firstgenerated, firstserved schedule yields a makespan that becomes an upper bound for the deadlines, and each job is given a deadline determined by sampling a random variable uniformly distributed between the job completion time (in this schedule) and the makespan, i.e. the interval [ C., C^^ ]. Thus, the initial sequence is a feasible solution. In order to determine the performance of the genetic algorithm on minimizing the flowiime when feasible solutions are harder to locate, a number of additional problem sets are created. In addition to the problem set described eariicr, which includes problems that were known to have a feasible solution, we generate 30job and 50job problems with tighter deadlines. The 30job problems have four job classes and the 50job problems ten job classes. Tighter deadlines are achieved by extending the range of values that a random deadline could lake. Let us define a value k thai can range from zero to one. The deadline for 7. is taken from the interval
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101 [k Cj, Cf^^, where the C.and C^^^. arc from the original generated schedule. U k= 1 , this plan is the same as the original one, and the generated problem is guaranteed to have a feasible schedule. U k. = 0, all of the deadlines vary equally, and there may exist no feasible schedule. As k decreases from one to zero, the problems we generate have a higher probability of having fewer feasible schedules. We generated problems with k = 0, 0.2, and 1. Since the MultiplePass Minimum Waste Heuristic cannot find feasible solutions for some of these problems, we will see if the genetic algorithm can find solutions that are feasible. Results . In this section we discuss the results of our experiments with the solution procedures on the generated problem sets. We summarize the findings and present tables of the collected data. Since the optimal solutions are not known (a branchandbound algorithm to find optima requires excessive computational time) and no good lower bound can be determined, we measure the performance of the solution procedures relative to each other. Each procedure was run once on each of the problems in the problem sets. The procedures include the MultiplePass Minimum Waste Heuristic and the problem space genetic algorithm. For comparison purposes, we also implemented a version of the heuristic that Ahn and Hyun (1990) use to reduce the total fiowtime in class scheduling problems. They proposed an iterative heuristic that starts with an initial feasible sequence where the jobs in each class are in SPT order (since they are not concerned with deadlines) and applies both a forward and backward procedure to it, repealing the steps until no strict improvement is found. Each of the forward and backward procedures interchanges different subschedules where the second subschedule consists of jobs from one class and the first subschedule has no jobs from this class. If the interchange reduces the total fiowtime, the subschedules are switched; this maintains the class SPT propeny. Our version of this algorithm, called the Modified Ahn & Hyun heuristic, uses one pass of the Minimum Waste algorithm to form the initial schedule. In addition, a potential swap of two subschedules is performed only if the swap reduces the total fiowtime and maintains deadline feasibility.
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102 The results (see Table 3.8) show that the genetic algorithm can find solutions that arc much better than those found by the MultiplePass Minimum Waste Heuristic and arc slightly better than those that Modified Ahn & Hyun heuristic produces. The genetic algorithm needs more time to find good solutions on the larger problems, although additional tuning may help improve the performance of the search. The searches for the 30job problems were for 2000 iterations using the following parameter settings: population size of 10, all operator fitnesses equal, increase of 50 in r every 10 individuals. The searches for the 50job problems were for 3000 iterations using the same population size, no large mutations, and an increase of 50 in r every 50 individuals. Table 3.8. Total flowtime performance of heuristics on problems where a feasible was found. Problem Problems
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103 compared to that of running the MulliplcPass Minimum Waste and Modified Ahn & Hyun heuristics. The heuristics need less than one second to find a good solution to a problem. A faster computer, however, would be able to reduce the computation time necessary for the genetic algorithm. Still, these results show that genetic algorithms that search the problem space can find very good solutions to scheduling problems. The improvement in total flowtime of the solutions that the genetic algorithm can find is a result of two things: the multiple sampling of the search space and the evolutionary process. This leads to the following question: Arc the genetic characteristics of the search a significant factor? We answer this question by changing the genetic algorithm so that each individual is a completely new one. Instead of choosing parents and creating offspring, we create new individuals by again mutating the dummy individual. This is a random sampling approach. (See Table 3.9.) These results imply that even though the same number of individuals are evaluated, the random sampling performs well but does not generate the same quality of solutions that the genetic algorithm does. Thus, we feel that the evolutionary process does contribute significantly to the improvement in total flowtime. Table 3.9. Total flowtime performance of random sampling on problems where a feasible was found. Problem Problems
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104 solutions from adjusted deadlines; and the ability of the genetic algorithm to combine the best characteristics of the points in the initial population. 3.2.7 Conclusions This portion of the work has two contributioas: it introduces an extended heuristic for the dual criteria class scheduling problem that we call CFTS, and it describes a problem space genetic algorithm used to find good solutions. The problem is to minimize the total fiowtime subject to deadline constraints. In this section we present a multiplepass heuristic for finding good solutions and discuss problem space and the genetic algorithm. Finally, we describe our experimental results, in which we compared the genetic algorithm to some heuristic approaches. From these results we make the following conclusions: The MultiplePass Minimum Waste heuristic performs well at minimizing the total flowtime of CFTS. Though not an exact procedure, it is usually able to find feasible, highquality solutions. A genetic algorithm that searches a problem space of the Minimum Waste algorithm for CFTS can find solutions with lower total flowtime. This genetic algorithm includes a penalty function for infeasible points that increases the cost of tardiness as the search progresses. In addition, it produces slightly better solutions than another procedure modified for this problem. 3.3 Class Scheduling with Release and Due Dates In this section, we study the onemachine class scheduling problem of minimizing the number of tardy jobs. Moreover, some of the jobs have nonzero release dates. We describe an extended heuristic developed for this problem and a genetic algorithm used to find good solutions. We also discuss an extension of this problem to the question of minimizing tardiness with minimum number of tardy jobs.
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lOS 3.3.1 Iniroduction The class scheduling problem studied in this section is to schedule a set of jobs, where some jobs have nonzero release dates, in order to minimize the number of lardy jobs. This problem is motivated by the semiconductor lest area. Since postassembly testing is the last stage in semiconductor manufacturing, meeting a job's due date is a very important objective for the manager of a lest facility. The consideration of release dates is an attempt to model the lookbehind situation thai exists in the job shop, where the scheduling of a machine (the bottleneck, for instance) may be improved by including infonmation about the jobs that are arriving soon. This problem, like most class scheduling problems, is a difficult case. Since even finding a schedule with no tardy jobs is an NPcomplete problem, exact algorithms to solve our problem in polynomial time do not exist. Thus, wc are motivated to try different heuristics and searches. Our approach was to modify an existing algorithm to include class setups and see how such an algorithm performs on this problem. Since the heuristic was not guaranteed to find good solutions, we also investigated a genetic algorithm. Thus, this research presents contributions in the extension of class scheduling problems to include a problem that has not been previously investigated and the use of both genetic algorithms and problem spaces to include the search for good solutions to class scheduling problems. 3.3.2 Literature Review In this section we will mention some of the most relevant research on class scheduling and on the problem of minimizing the number of tardy jobs in the presence of release dales. A full discussion can be found in Chapter 2. Bruno and Downey (1978) prove that, for general class scheduling problems, the problem of finding a schedule with no tardy jobs is NPcomplete. Monma and Potts (1989) prove that class scheduling to minimize the number of tardy jobs is an NPcomplete problem.
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106 As discussed in Chapter 2, the onemachine problem of minimizing ihe number of lardy jobs when some have nonzero release dales (1 / r.7 Z Uj) is a slrongly NPcompleie problem (Lawler, 1982). A rcsiricied version of the problem has been considered by Kise, Ibaraki, and Mine (1978), who solve the problem optimally if the release and due dales match (r.< r^ implies dj < df.). They present an 0(n~) algorithm (Kise's algorithm, described in Section 3.3.4) for this case. 3.3.3 Notation and Problem Formulation We will use the basic notation introduced in Section 3.2.3. For CSRDD, each job Jj has a release date r: and a due date d:. For a given schedule, (J: = 1 if C.> d: and otherwise. The problem is to find a sequence that minimizes X (J; subject to the constraint that C: > n + pj. An NPcomplcie problem, CSRDD is unstudied in the literature on class scheduling. In order to simplify the problem, it is assumed that the release and due dates match; that is, there exists an ordering where the jobs are simultaneously in Earliest Release Dale (ERD) order and in Earliest Due Date (EDD) order. Our primary heuristic for CSRDD extends Kise's algorithm for the problem without setups to form a heuristic for finding good solutions. 3.3.4 Heuristics In this section we will describe a number of heuristics: Kise's algorithm for the problem without class setups, our extension of this algorithm, and other heuristics used for testing purposes. Kise's algorithm. Kise's algorithm orders the jobs by their release and due dates (a nonambiguous ordering since the dates must match). The algorithm is an extension of the MooreHodgson algorithm (Moore, 1968) for minimizing the number of late jobs. Eachjob is scheduled after the partial schedule of ontime jobs while maintaining release date availability. If the new job is tardy, the algorithm searches the ontime jobs for the job whose removal leaves the shortest schedule of ontime jobs. The removed job is made tardy and will be processed with the other
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107 tardy jobs after the feasible jobs. In this manner, the algoritlmi finds the largest subset of the jobs that can be delivered ontime. These jobs arc scheduled in order of their release and due dates. The search subalgorithm has effort that is linear in the number of jobs in the partial schedule. Since the subalgoriihm may be performed up to n times, the total effort of Kise's algorithm is 0(Â«2). Rise's heuristic is not optimal for CSRDD, although it can be modified to include setup times. Take the following example: Example 3 J. J rj pj dj i 10 5 6 1 2 4 13 2 3 5 5 14 2 4 6 2 15 1 ^01 =^02 = 1Â•^la^ 1Â•^21 ='*Â• The optimal sequence is [7, 74/2 ^ ], with Cj = 6, C4 = 8, C2 = 13, and C^^ 18, which has one tardy job. Kise's algorithm adds J^, which is tardy after 7] and 72. and the subalgorithm makes 7) a tardy job. When 74 is added to the schedule after 72 and 73, it also is tardy, for a total of two tardy jobs. Kise extension. Our algorithm for CSRDD extends Kise's algorithm by considering two options when adding a new job to a partial schedule: we can place the job in a position after all of the ontime jobs or in a position after the last ontime job from the same class (if there is one). In either case, if an ontime job becomes tardy, we make tardy the job whose removal creates the shortest partial schedule of ontime jobs. We then choose between the two partial schedules created, selecting the new partial schedule with the smaller number of late jobs (or smaller makespan if they tie) as the incumbent before trying to schedule the next job. Intuitively, it appears that the extended algorithm should outperform the Kise algorithm, since it includes an additional scheduling choice. Due to the complexity of the problem, however, this is not guaranteed. The following problem is one counterexample:
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108 Example 3.4. j rj pj dj i 10 3 5 2 2 3 13 1 3 6 3 14 2 4 14 3 17 2 ^01 =Â•^21 = 'Â• ^02 = Â•^la^ 2. The optimal sequence is 1 J^ J2J3 J4 ], with C, = 5, C2 = 9, C3 = 14, and C4 = 17, with none tardy. Kise's algorithm will construct this schedule. In the proposed algorithm, the addition of 73 to [JiJj] creates a partial schedule with no tardy jobs and a makespan of 14. The scheduling of Jj after/, and before ^2 ^^reates a makespan of 13 (C, = 5, Cj = 9, Â€2= 13), so the sequence [ J^J^ J2 ] replaces [ 7, J2J3 ]. When we add J^ after 72 . J4 is tardy (C4 = 18), and the ontime jobs complete at time 13; when J^ is scheduled before 72, 72 is tardy (C4 = 17, Â€2= 21). The algorithm thus yields [ 7, 73 72/4 ), which is not an optimal schedule. Tardiness rules. We also tested two heuristics based upon the R & M procedure of Rachamadugu and Morton (1982) and used for reducing weighted tardiness in Morton and Ramnath (1992). These are primarily class scheduling extensions. RM: A dispatching rule where the priority of a job at time t is based upon the weight, the processing time, and the slack of the job: RM = Wj/pj * exp( Sj'^ I k * p^^g), where RM is the job priority, w.the job weight, pj the processing time, Sj^ the slack max (0, dj r p.} , ^ a predefined constant, and p^^^ the average processing time of the jobs in the queue. In this formulation, jobs with higher weights, shorter processing times, and less slack will be scheduled first. According to Ramnath and Morton, the constant k is normally set to 2. XRM: The xdispatch (lookbehind) version of the RM rule includes jobs that will be arriving soon in an extended queue. That is, they arrive before the completion time of the shortest job already waiting. The RM priority is discounted by an amount that depends upon the arrival time:
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109 XRM = RM * (1 (1.3 + p)*(rj t)^IPmin>' where XRM is the job priority, p the utilization factor, r: the arrival time of the job, and /7^,Â„ the smallest processing time among jobs currently available. Morton and Ramnath (1992) claim that this procedure reduces weighted tardiness by 40% over the standard RM rule. These two priorities are used as dynamic dispatching rules. At a time r, the job with the highest priority is scheduled next. For our class scheduling problem, we redefine the components to include the setup times, but otherwise we use the same formulas. This works well for the objective of weighted tardiness, but for our objective (minimizing the number of tardy jobs), we would like to postpone the processing of the tardy jobs in order to concentrate on the ontime jobs. Thus, when we schedule a job that will be tardy, we look for the job whose removal will result in a shorter schedule with no tardy job and we remove that. For our class scheduling problem, we include in the processing time the class setup necessary to process a job and modify the release date by the same amount. That is, if the job completing at time t is in G^ and J : is in G^, we add s^j^ to p.and subtract s^j^ from r.(for the XRM calculation). We use p = 1 and k = 2 and w. = 1 for all J:. 3.3.5 Analysis of the Heuristic In this section we discuss the computational effort necessary to perform the extended Kise heuristic and the worst case error of this heuristic. A pseudocode presentation of the heuristic and its subalgorilhm can be found in the Appendix. It is obvious that the extended version of Kise's heuristic takes more effort than Kise's algorithm. However, the effort of the algorithm is still 0{rfi). When adding the next job to a partial schedule, there arc two positions for the new job. The algorithm would take at most 0(n) effort to find the last job from the same class as the new job, to insert the new job, and to determine which (if any) jobs arc now tardy. If there is a tardy job, a pass of Kise's subalgorithm must be performed to determine which job to remove. This is also 0(/i). For the other position, there is 0(Â«) effort in adding the new job to the end of the partial schedule and pcrfomiing a pass
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110 of the subalgoriihm. Thus, the total effort of adding the new job is 0(n). and since n jobs must be scheduled, the total effort of the extended Kise heuristic is 0(/2). Since CSRDD is strongly NPcomplcte, there is no optimal polynomial or pseudopolynomial algorithm. Since our extended Kise heuristic is not guaranteed to find an optimal solution, we need to look into the worstcase error bound. In the following we describe two families of instances for CSRDD where the extended Kise heuristic cannot find good solutions. While the first of these examples prove that the heuristic can perform arbitrarily badly, the examples will also provide problem instances that we can use for testing the performance of the genetic algorithm. They are especially good for this since we know the optimal solution in advance. Example 3S. In this case, the extended Kise heuristic finds n 1 out of n jobs tardy when the optimal has only two tardy jobs. There are n jobs where 7) is in G, and J2, . . ,Jn are in Gj, and the jobs have the following characteristics: Jj Jj h' < ^n Fj 1 Pj 1 1 dj 1 n The class setups are as follows: Gi
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Ill schedule consisting of jusl 7. Thus J 2 is made lardy. This continues for all of the jobs from Gi, and they are all forced to be tardy, for n 1 tardy jobs. Example 3.6. In this case the optimal solution has no tardy jobs and the extended Kise heuristic finds n/3 tardy jobs. We construct the problem instance in the following way: choose a nonnegative integer it. Letrt = 3(/k+ 1). Letm = 3. For / = 0, ..., /t, construct three jobs, 73,^.1 in Gi,J2i+2 Â•" ^2' 2nd J2i+3 in G3, with the following job characteristics, where < e < 1 and 0<6<1: 'J Pj Jsi+l 3i 1e 3i+2 Â•'31+2 h\+3 3i 3i 1+e 1 3i+2+6 3i+3. Let the class setups be as follows, where s>5: Gi
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112 This process conlinucs for all of the jobs (see Figure 3.7). Thus, the heuristic creates a schedule where the )t + 1 = /i/3 jobs in G2 are tardy, while the optimal schedule has no tardy jobs. Note that we never insert a job into the middle of a partial schedule; thus, Kise's algorithm and the extended Kisc heuristic create the same schedule. Jl
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113 Thus, wc have shown ihat Uic extended Kise hcurislic will schedule at least one ontime job. This bound is tight, as our discussion of Example 3.5 shows that there exist problems for which the heuristic will schedule exactly one ontime job although the optimal schedule has more than one ontime job. 3.3.6 The Genetic Algorithm In this section we present the problem space and discuss the details of the genetic algorithm we used to fmd good solutions for CSRDD. Problem space. In Chapter 2 we described the ideas of alternative search spaces. In this section we present the problem space that we searched in order to find good solutions for CSRDD. We defined a problem space for CSRDD in the following manner: Given a problem p in problem space, a heuristic /i is a function that creates a sequence corresponding to a solution s for CSRDD, i.e. h(p) = s. We defined a problem as a vector of job release dates, using a pass of Rise's algorithm to create a sequence of jobs by considering the jobs in order of their new release dates instead of the order imposed by the matching release and due dates. The actual release dates are used in determining the schedule, however. Note that all solutions for CSRDD (including the optimal ones) that schedule all tardy jobs last are in the range of h. Following are two examples of applying this heuristic to different vectors of deadlines for the problem in Example 3.3: Heuristic(problcm) = solution: Kise (0, 1 , 5, 6) = [ 72 ^s ^i ^4 ]Â• two jobs tardy. Kise (0, 8, 5, 4) = [ 7] J^ Jj Jj ], one job urdy. A genetic algorithm for CSRDD. In this work we developed a genetic algorithm based on the ideas presented in Davis, 1991, namely, steadystate reproduction without duplicates.
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114 fitnesses measured by linear normali/ation, a unilomi crossover operation, operation selection, and interpolated parameters. Steadystate reproduction adds new individuals a few at a time, whereas the traditional method replaces the entire population with a new generation. Steadysute reproduction (attributed to Whitley, 1988, and Syswerda, 1989) is used to ensure that good individuals (and their good characteristics) survive. Steadystate reproduction without duplicates prevents children that are identical (in chromosome values) to a current member from joining the population. Linear normalization is a fitness technique that creates fitness values by ordering the individuals in a population by their objective function evaluation. The assignment of fimesses begins with a constant value and decreases the fitness linearly as it considers each individual in order. This technique prevents a super individual from dominating the population at the beginning of a run and yet differentiates between the various very good individuals that exist near the end of a run. Of course the values of the original constant and the decrement parameter influence the extent of these two phenomena. Uniform crossover, an operator first described by Syswerda (1989), is a way to combine characteristics in ways that standard oneor twopoint crossovers cannot. In a uniform crossover, two parents are selected and two children produced. Each bit position is considered independently and the parent that contributes the bit value for that position in the first child is determined randomly. The second child receives the value for that position from the other parent. While uniform crossover can destroy a good characteristic by mixing it with a bad string, it can also combine features that are widely dispersed across the string. Since onepoint crossover remains a good operator, however, we used both types of crossover in our genetic algorithm. Before creating a child, we randomly decide on which operator we wish to perform: uniform crossover, onepoint crossover, or mutation. Each operator has an operator fitness and the probability of that operator being selected is proportional to that fitness. If one of the crossovers is selected, two parents are selected and two children are created.
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115 If the mutation operator is selected, one parent is selected and a child created by forcing each bit to undergo a mutation with some small probability. The child or children created are checked against the current population for duplication, evaluated, and iascricd into the population, replacing the worst members of the population. The new population is then reordered and new fitnesses created using the linear normalization teclmique. As we did with the CFTS genetic algorithm, the initial population included one individual (the dummy, or seed) that was created form the actual problem data. The remaining individuals in the initial population were constructed by mutating the bits in the initial (dummy) chromosome. This initial mutation rate was set at 0.05 per bit. We interpolate the following parameters over the course of the run: the decrement for linear normalization is increased, and the operator fitnesses are changed to favor crossovers and discourage mutations. Table 3.10. Parameter values for genetic algorithm. Population size: 100
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116 processing times. This makcspan is used to define a specific ranges for the due dales and a range for the release dales. The due dales and release dates arc sorted and the matching pairs are given to the jobs. By changing the paramclcrs governing the definition of the ranges, problem sets with different characteristics can be created. Different sets of 10 problems were created. The number of jobs fxr problem ranged from 15 to 100. The processing times ranged from 1 to 20 and the class setup times from to 9. The jobs were randomly placed into a number of job classes, depending on problem size. The release dales and due dates were taken from a uniform distribution. The upper and lower bounds of this distribution were proportional to the sum of the job processing limes. The proportions changed for each problem set. (See Table 3.11.) Table 3.11. Data on problem sets. Set Problems Jobs Classes Release date Due date range range KH301 10 30 4 00.4 0.40.6 KH302 10 30 4 00.4 0.61.0 KH151 10 15 4 00.4 0.40.6 KH501 10 50 5 00.4 0.40.6 KH303 10 30 4 00.4 0.21.0 KH304 10 30 4 00.6 0.21.0 KHM1XED2 10 30 4 0.4 0.4 0.6 KHM1XED3 10 30 4 01.0 0.41.2 KHMIXED4 10 30 8 01.0 0.41.2 Results. After numerical testing on these forty problems, it appears that the Kise and Kise extension heuristics and the R & M heuristics arc fairiy equal. A number of other heuristics werc unable to find as many ontime jobs. Note thai the lower bound was derived by using Rise's algorithm while ignoring all setup limes.
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117 Table 3.12. Average jjerformance of heuristics. Set
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118 Set Table 3.14. Average performance of heurislics, hard problems. Jobs Kise Extended Genetic Algorithm" Kise 1 ()()() 3(XX) Hard A
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119 Table 3.15. 18job problems, number of ONTIME jobs. lOrunsoflOOO: Number of
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120 Table 3.16. 30job problems, number of ONTIME jobs; 3 runs of 3000 Number of Problem Average Individuals Created 12 3 4 5
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121 minimizing the total tardiness subject to a constraint on the number of tardy jobs, since the minimization of total tardiness usually leads to schedules where many jobs are tardy and arc tardy by a small amount. Since finding the minimum number of lardy jobs is an NPcompletc problem, we use our heuristic to set the value of the constraint. Then, within that limitation, we minimize the total tardiness. We develop some further extensions of Kise's algorithm that create a set of tardy jobs and then insert the tardy jobs into the schedule of ontime jobs in order to reduce the total tardiness. We also use our genetic algorithm to search for schedules with low number of tardy jobs and low total tardiness. The problem of minimizing tardiness subject to a minimal number of tardy jobs has been considered for the problem without class setups (or release dates) by Vairaktarakis and Lee (1993), who develop an algorithm to optimally schedule a given set of tardy jobs and an efficient branchandbound technique to find the optimal tardy set. Other researchers have studied dual criteria problems with the same primary objective. Emmons (1975) considered the problem of minimizing total fiowtime subject to minimum number of tardy jobs, using a branchandboimd algorithm to find optimal solutions. Shanthikumar (1983) examined the problem of minimizing the maximum lateness subject to minimum number of tardy jobs, also using a branchandbound algorithm. Our problem, which includes both class scheduling and nonzero release dates, is an NPcomplete problem. The tardiness heuristics that we use to find good solutions use the extended Kise heuristic to determine a set of tardy jobs. The heuristics also use the sequence of ontime jobs created by the extended Kisc heuristic, pushing the jobs to the right, staning them as late as possible, and attempting to insert the tardy jobs into the gaps in this schedule. The first heuristic (7,) orders the tardy jobs by their release dates and attempts to interieave the two sequences of jobs, scheduling tardy jobs to start as soon as possible while maintaining the feasibility of each ontime job (whose completion time is constrained by the due date).
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122 The second heuristic (T2) considers every tardy job as a candidate for a gap between the partial schedule and the next ontime job, selecting the tardy job that yields te earliest start time for the next ontime job. Any remaining tardy jobs arc scheduled by their release dates. The third heuristic (Tj) was a modification of the second that scheduled the remaining tardy jobs using a version of the Minimum Waste heuristic (see Section 3.2) that didn't consider deadlines (since all jobs arc tardy). This heuristic has been shown to perform well on fiowtime criteria. We use this because minimizing the total tardiness of the set of tardy jobs is identical lo minimizing the total fiowtime of those jobs. Due to the nonoptimal nature of the extended Kise heuristic, it is possible that the tardiness heuristics will often be able to schedule a tardy job so that it finishes ontime (reducing its tardiness to zero). However, the primary objective of these heuristics is to reduce tardiness, not reduce the number of tardy jobs. Results . We tested the heuristics on three problem sets, selected because the variance of the due dates meant that the schedules were more likely to have gaps in which to insert tardy jobs. Each solution technique was measured by the average total tardiness found by that heuristic on the ten problems in each problem set and the percent deviation of this average from the average tardiness found by the extended Kise heuristic. The performance of the genetic algorithm on each problem is the average of ten trials of one thousand new individuals. Table 3.17. Data on new problem sets. Problems
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Set KH302 KH303 KH304 Extended Kise 236.4 758.3 920.0 123 Table 3.18. Average performance of heuristics. T, % T2 % 237.3 756.6 869.0 0.38 0.22 5.54 234.7 718.4 859.4 0.72 5.26 6.59 231.5 704.1 844.0 Note: Performance is the average total tardiness and the percent improvement. Table 3.19. Average performance of heuristics. % 8.30 Note: Performance is the average total tardiness and the percent deviation. Set
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124 When the extended Kise heuristic cannot find good solutions, our problem space genetic algorithm can. By searching tlic problem space near the original problem, it can discover solutions that arc improvements on the schedule constructed by Kisc's algorithm. Also in this section wc discussed an extension of this problem to the problem of minimizing total tardiness in the presence of a constraint on the number of tardy jobs. 3.4 Flowtime with Setups and Release Dates The third of the class scheduling problems that we consider has jobs with nonzero release dates, and the objective is to minimize the total flowtime. We develop some lower bounds and dominance properties and examine some heuristics for finding good solutions to the problem. We discuss a problem space genetic algorithm that can improve the performance of a lookbehind dispatching rule. For this problem we also developed a search technique for comparison purposes. 3.4.1 Introduction This problem, like the others we have examined, is motivated by considering the scheduling of a semiconductor test area. We have class setups, arriving jobs, and an objective that mirrors the goal of management to minimize workinprocess inventory. We will examine the problem of minimizing total fiowtime when the jobs have nonzero release dates. This strongly NPcomplcte problem is a lookbehind scheduling model, where we are interested in scheduling a machine by considering the jobs that will be arriving at the machine soon. In addition to a lookbehind scheduling rule, we will consider the use of a problem space genetic algorithm (similar to those developed for CFTS and CSRDD) that can improve the performance of this rule by adjusting the parameters of the rule. Due to the structure of the FTSRD problem, we will also use a decomposition heuristic as a means of comparing solution
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125 quality. The decomposition heuristic is a search technique that considers a sequence of subproblcms at each move. In the next section we will introduce the notation and problem formulation. After that we will mention some of the previous research on the scheduling problem under consideration (a review of the literature on class scheduling and genetic algorithms can be found in Chapter 2), examine some lower bounds and dominance properties that can be used in a branchandbound technique, discuss our heuristics (including sequencing rules and the genetic algorithm), and report on the experimental results. 3.4.2 Notation and Problem Formulation We use the same notation as that for the CFTS and CSRDD problems (Sections 3.2.3 and 3.3.3), except that the jobs do not have due dates. The FTSRD problem is to find a sequence that minimizes X C.subject to the constraint that C.> r.+ p .. We will see that FTSRD is NPcomplete and has not been previously considered in the literature on class scheduling. We make two assumptions in the analysis of the problem. One, a class setup for a job can begin before the job is available. Two, although all of the release dates are known, the processing for a job cannot begin until the release date. These conditions are motivated by our consideration of the onemachine problem as part of the job shop scheduling problem. 3.4.3 Background The onemachine problem of minimizing total flowtime when the jobs have nonzero release dates has been previously studied in the case where no sequencedependent setups are present The problem is simple if the jobs are preemptive, that is, if a job that has begun processing can be interrupted by another job and then resumed later. In this case, the optimal policy at the next decision point is to schedule the job with the shortest remaining time. The set of decision points includes all job release times and completion times. If the jobs are non
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126 preemptive and have no sequence dependent setup times, the problem (denoted by 1 / r./ Z Cp is a strongly NPcomplete question, as shown by Lenstra et al. (1977). Among class scheduling problems, minimizing the total flowtime is a strongly NPcomplete problem (Monma and Potts. 1989). Thus, it can be seen that our problem, which is to minimize total flowtime with class setups and release dates, is also a strongly NPcomplete task. Many researchers (see Chapter 2) have considered the problem of minimizing the total flowtime subject to job release dates, 1 / r./ X C:. Branchandbound algorithms have been the most popular approaches. None of these researchers included sequencedependent setup times in their analysis. 3.4.4 Solution Techniques In this section we will discuss a number of different approaches to solving the FTSRD problem. These include branchandbound searches, dispatching rules, a decomposition heuristic, and a genetic algorithm. A job arrives at r .and is available at time r if f > r.. Recall that we assume that a class setup can be performed before the associated job becomes available. Branchandbound . A straightforward branchandbound algorithm can be developed for this scheduling problem. The branchandbound procedure finds the optimal solution by searching a tree that consists of every possible permutation. Each node of the tree consists of a partial schedule of jobs processed in that order and as soon as possible. (The root node consists of an empty schedule.) Unscheduled jobs will be appended to the schedule for a node. Branching occurs by adding a child node for each unscheduled job. A lower bound can be calculated for each node, and the search moves to the child with the lowest lower bound. Local dominance properties . The number of nodes to be examined can be reduced by applying dominance properties that identify nodes that cannot lead to optimal solutions. Local dominance properties claim that a node a is dominated by another node (3 under certain conditions. Thus, we do not need to branch on node a because we can find a better solution by branching on p and searching its dcsccndcnts.
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127 We note here that matching processing limes and release dates do not imply a dominance property. Even if p.< pi implies r.< r,(or even if the release dales are identical) within each class, we do not have an optimal order for the jobs in each class. Consider the following instance: A
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128 Suppose Uial we are al a node in ihc search tree wiih a partial schedule o that ends al lime t with a job in class G^ and a set K of unscheduled jobs. If 7. is in K and in Gfj, define the following earliest start time: f. = max {t + s^ij.rj}. Note that if r + j^.^ < r., the necessary class setup can be completed before the job arrives. At this node we can use any of the following dominance properties: Property 3.1. Let J,be the shortest job in K. UJj is in G^ and 7,is in G^, the node (a, Jj) is dominated by (a, J^) iff,*s^ < t:. Property 3.2. The node (o, 7.) is dominated by (o, J^) if /,Â• + p,+ s^ < rj. Property 3.3. If 7.is in G^ and 7, is in G^, the node (o, JJ) is dominated by (a, 7,) if all of the following statements arc true: (a) ti + Pi < tj + pj, (b) f J + Pi + s^g < tj + Pj + Sfjg for all job classes Gg.G^nK^ [}, (c) s^Ij + Pj < SgQ + pi for all Gg : G^ n /c: ?i { ) , and (d) Sgij + pj + Sfyfi < Sgg + p^+ 5^ for all Gg : Gg n ^ ?t ( } , and G^ : G^ n a: ?t { } . Proofs: Property 3.1. Take any schedule (o, 7., ai,7j, 02), where there are m jobs in Oj. If we move Jj before 7., the new completion time of 7^is no greater than the old completion time of 7;; thus the flowiime of 7,is decreased by at least {m + l)p,(since Jj is the shortest job). Meanwhile, we delay only the start of 7; and the m jobs in Oj by at most/?,(since ?/ + p, + s^ tj < Pi). The jobs in aj are not delayed at all (the triangle inequality of the setups insures this). Property 3.2. Take any schedule (a, Jj, O] , 7,, 02). We can move 7,before Jj without delaying 7,, decreasing the flowtime of 7^ and possibly that of the jobs in 02Property 3.3. Take any schedule (o, Jj, O] , 7,, 02). Note that the earliest start times of 7, and Jj are before Oj . Interchange 7,and Jj. The new completion time for 7,is not greater the old completion time of 7.(by condition a). By condition b (if the first job of O] is in Gg), the jobs in 0 are not delayed. Then, by condition c (if the last job of O] is in G^), the new completion time
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129 fori, is not greater than the old completion time ofy^. Finally, tlic last condition (if the first job of 02 is in Gj) implies that no jobs in 02 are delayed. In any of these cases, we can take a schedule that starts with (o, 7.) and find a schedule which starts with (.a, 7,) and which has less total fiowtime. Thus, it is clear that the first node is dominated and that we do not need to search that part of the tree. Lower bound . A branchandbound procedure needs a lower bound. A good lower bound is important to efficiently finding solutions. We develop two bounds: the first concentrates on the release dates, the second on the setup times. The first lower bound for a node completely ignores the class setups. Instead, it solves the associated 1 / r., preemption / X C. problem with the SRPT rule and adds the optimal fiowtime to that of the partial schedule in the node. The second lower bound separates the setup times from the job processing times. We assume that each class will have exactly one remaining setup. If we further assume that this setup wiU the shortest possible, then we can easily sequence the job classes to minimize the contribution of the setup times to the total fiowtime. The unscheduled jobs are scheduled by SPT without regard to their release dates. The two sums are added to the fiowtime of the partial schedule in the node. The first lower bound should work well when the interarrival times are large, and the second bound should be useful at nodes where all of the unscheduled jobs are available. Testing . For 30job problems, the branchandbound procedure required excessive computation time to find an optimal. Still, improved lower bounds could be found by truncating the branchandbound search in the following way: We prevent the search from moving below a certain depth in the tree and take the lower bound at this depth as the objective function value. If we continue to do this, the truncated search returns the lowest lower bound at this depth. This will be a lower bound on the optimal solution and will be better than the lower bound computed at the root node.
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130 Dispatching rules . The SPT rule is known lo minimize the total flowiime for 1 / / Z C: (Smith. 1956). Since we are studying a total llowtime problem, we are most interested in SPTlike heuristics. We will develop a rule that considers the waste associated with a job (the waste is the sum of the idle time and setup time incurred if the job is scheduled next). We propose the lookbehind dispatching rule Shortest Waste. "Among the jobs with the minimum waste, schedule the shortest one." (This rule is similar to the Minimum Waste algorithm for CFTS.) Let us define a few relevant variables: t is the current time, the completion of the last scheduled job; G^, is the class of the last scheduled job, and the waste of an unscheduled job J: in G^is Wj = max {r.t,s^fy]. Our dispatching rule can be now stated: Shortest Waste: Among all unscheduled J;, select the job with the minimum wj. Break any ties by selecting the one with the minimum pj. Decomposition . For the FTSRD problem, we decided lo implement a search heuristic for comparison purposes. A decomposition heuristic for finding good solutions to sequencing problems was introduced by Chambers et al. (1992). The heuristic is a type of local search. It begins with an initial sequence, and forms new sequences until it finds a local minimum. The critical step is the decomposition of the problem into subproblems that depend upon the current solution. A new solution is generated by combining the optimal or nearoptimal solutions to each subproblem. The heuristic thus makes very good moves through the search space; only a few moves are needed before convergence is reached. Consider, for example, a 12job problem. Start with some initial solution to the problem. Select the first six jobs of this solution and find a good solution to the 6job subproblem. Take the first three jobs (in order) of this subproblem solution as the first three jobs of the new solution. Then combine the remaining three jobs from the subproblem and with the next three
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131 jobs from the initial solution. This forms a new 6job subproblem. Continue solving subproblems and building the new solution until all of the jobs have been considered. We use this technique in order to fmd good solutions against which we could measure the performance of our other heuristics. (We did not feel that this type of search would be as effective on deadlineoriented CFTS and CSRDD problems.) We use a branchandbound technique with an approximate dominance property to generate nearoptimal solutions to the subproblems. The algorithm has two parameters. We need to select m as the size of the subproblems which we will solve; the larger the value, the better our subproblem solutions will be (at the expense of computation time). We also select/as the number of jobs from the subproblem solution that will be fixed into the new solution. A smaller/ requires that more subproblems be solved per step. The following steps outline the procedure. Stepl. Set m and/. (We want/ to divide Â«m.) Let o be the ERD schedule. Step 2. Take the first m jobs of a. Let rt be an empty schedule. Step 3. Solve the mjob subproblem by branchandbound. Step 4. Append the first /jobs of the solution to k. Step 5. If there are any unconsidered jobs in o, take the next /jobs from o, add to the m /remaining from the subproblem, and go to Step 3; else go to Step 6. Step 6. Append the m /remaining jobs of the solution to n. If n is a better schedule than o, let o = 7C and return to Step 2; else go to Step 7. Step 7. The solution found by the heuristic is o. We experimented with different values of (m.f). The best results (in terms of computation time and solution quality) were achieved with (9, 3) and (15, 5). We used our branchandbound algorithm with only the second lower bound and only the following approximate dominance property: Property 3.4. Given a partial schedule o, (o, J[) dominates (o. Jj) if 7/ and Jj are in the same class, /j is the shortest unscheduled job in that class, and w,< Wj. This property is a simple extension of a previously considered dominance rule (see, for instance, Dessouky and Deogun, 1981). Unlike the dominance rules that we use in the full
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132 branchandbound procedure, it has the advantage of being quick to check, since there arc fewer unscheduled jobs from the same class. In the section on computational results, we will discuss how well the decomposition heuristic performs. A problem space genetic algorithm . In this problem, we consider the problem space defined over the problem release dates. A point in this space is an nelement vector of nonnegative real numbers. When a heuristic is applied to an instance of FTSRD, it uses the actual release dates to generate a solution. If, however, we adjust the release dates of the problem, we can change the sequence created by the heuristic. This sequence can be evaluated as a schedule by using the actual problem data. We can associate, therefore, with the vector of adjusted release dates a performance value: the total fiowtime of the schedule that was created. Moreover, we can search the space of adjusted release dates to find good schedules. This exploration is the objective of the problem space genetic algorithm. Our purpose is to show that the performance of a simple heuristic can be improved with a smartandlucky search like a genetic algorithm. We will use the Shortest Waste heuristic to convert a vector of adjusted release dates into a sequence of jobs. The optimal solution is within the range of this heuristic: if each adjusted release date equals the actual start time of the job in an optimal solution, the Shortest Waste heuristic will schedule the jobs in the optimal order, since at any time, the job with the shortest waste will be the one with the next adjusted release date, which is the job with the next optimal start time. As we did for CFTS and CSRDD, we will use a steadystate genetic algorithm. The initial population is formed by mutating a source individual that is the digital representation of the actual release dates. After empirical testing on a number of problem instances, we decided on the following parameters: The population size is 100 individuals. The four operators arc uniform crossover, onepoint crossover, small mutation, and large mutation; all have the same probability of being selected. In the small mutation, a bit is Hipped with 2% probability; in the large, the probability increases to 50%. The algorithm uses tournament selection to identify parents. See
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133 Davis (1991) or Goldberg (1989) for more infomialion aboul these aspects of the genetic algorithm. Example 3.7. The following problem is used to illustrate some of the issues we have discussed so far. 'y 1
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134 3.4.5 Empirical Testing Problem generation . In order to test the heuri.slics, four sets of ten problems were created. The characteristics of the sets arc shown below (processing, interarrival, and setup times randomly selected from uniform distributions with the given ranges): Table 3.20. Data on problem sets Set Problems Jobs Qasscs Processing Interarrival Setup Times Times Times FTISI 10 15 5 1,20 1,10 0,9 FT301 10 30 5 1,20 1,15 0,9 FT302 10 30 10 1,20 1,15 5,9 FT304 10 30 10 1,15 1,20 5,9 Results . In this section we will discuss how well our solution techniques performed. (See Table 3.21 .) The branchandbound could find optimal solutions on only the 15job problems. On the 30job problems, we used the decomposition heuristic with parameters (9, 3) to quickly generate solutions and measured the performance of other heuristics against these solutions. The (15, 5) decomposition was much slower than the (9, 3) decomposition, since the time necessary to solve each subproblem grew exponentially. Still, it found slightly better solutions, and the processing time was reasonable (although it varied from problem to problem) if the heuristic dominance property and second lower bound were used. Table 3.21. Performance of heuristics. Problem Shortest (9,3) (15,5) Genetic Set Waste Algorithm FT151 1.095 1.005 1.010 FT301 1.067 1.000 0.992 1.006 Fr302 1.066 1.000 0.995 1.005 FT3(M 1.031 1.000 0.994 1.005 Notes: Performance measured against optimal solution for FT151. Against decomposition (9,3) for 30job problems. All performances are average ratios over 10 problems. Performance of genetic algorithms averaged over three runs of 30(X) individuals.
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135 Although the initial lower bounds for the 30job problems were not good, we were able lo improve them using the branchandbound tree lo show that the decomposition (9, 3) heuristic was within ten percent of the optimal flowtimc. The Shortest Waste heuristic found good solutions very quickly and generally performed better than other dispatching rules. On the 15job problems, the genetic algorithm was not an effective heuristic, since it required more compulation time than the branchandbound and could not always find optimal solutions. On the 30job problems, the problem space genetic algorithm found solutions better than Shortest Waste and as good as the decomposition heuristic. The computation time was slightly longer for a 3000individual search than for a (15, 5) decomposition, but a 1000individual search was much shorter and found solutions with little increase in total flowtime. The exponential nature of genetic search is exhibited in Figure 3. 12. (In other testing, we found that the genetic algorithm was not as effective when using a simple Earliest Release Date rule to create sequences). All programs were run on a 386 PC. Decreases in times were achieved when the programs were run on a 486 PC, and further decreases could be achieved on a more powerful machine. Except for the 30job branchandbound (which we could not solve), we do not consider processing times to be a significant obstacle. The numbers in Table 3.22 are offered only for comparison puiposes. Table 3.22. Typical computation times FT302.1 Decomposition (9,3): 17.8 seconds Decomposition (15.5): 338.67 seconds 1000individual G A: 134.62 seconds 3000individual GA: 350.37 seconds Shortest Waste: < 0.1 seconds FT151.1 Branchandbound: 10.71 seconds Decomposition (9,3): 0.8 seconds 3000individual GA: 149.24 seconds Shortest Waste: < 0.1 seconds
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136 1.07 500 1000 1500 2000 Number of New Individuals 3000 Figure 3.12. Performance of Shortest Waste Genetic Algoritlim Performance measured against decomposition (9,3). 3.4.6 Special Case In this section we consider a particular special case of the problem which may be useful in certain manufacturing situations. We will assume that there exist exactly two job classes and that the job processing times are equal within each class. Specifically, we study the following instance: p.= p for all J: in Gj, p: = q for all J: in GiSince all of the jobs in a class have identical processing times, we may order them by ERD. We note here that all of the release times are integer. We will describe a pseudopolynomial dynamic program to solve the problem, a spjccial case of the strongly NfPcomplete FTSRD problem.
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137 Dynamic programming . Wc can use a dynamic program for iwo reasons: there arc only two classes, and we have an ordering for the jobs in each class. According to Monma and Potts (1989), this ordered batch scheduling problem can be solved in pseudopolynomial time. The following dynamic program interleaves the classes. The state variable in the dynamic program corresponds to a partial schedule that consists of the first j'l jobs from G] and the first iiSoh?, from Gi and that ends before or at a specific time with a job from a specific class (if it is cheaper to end sooner, that schedule should take precedence). At each point in the state space, we will measure the total fiowtime of the scheduled jobs. The recursion determines the best partial schedule to which we should add the specified job. Algorithm 3.3. Lety(f, a, /], i^) be the minimum fiowtime of a partial schedule where the last jobs ends at or before time r, there are tj jobs from G) and J2 jobs from G2, and the last scheduled job is from G^. f = R. a = 1, 2. /, = Â«,, /j = 0, . . . , ^2Renumber the jobs so that y<Â«, ify.isinGi, r, < . . . < r^,, andy'>/ii if/.is in G2, rn,+, < . . . < r^. /? is some upper bound on the makespan of a schedule. We can find one such R by scheduling all jobs in ERD order, performing a class setup in front of every job. We also have an upper bound: R < max [rj] + 'Lpj + Â«! S21 + Â«2 ^12Initialization: /(r,a,/Â„/2) = Â°Â°iff<0. fit, 1, 0, 12) = Â°Â° for all t and for ij > 0. jit, 2, /j, 0) = Â°Â° for all t and for /j > 0. y(r, 1, 1,0) = Â°Â° for f< max {sq^, rA + p, if/.isthe first job in Gj (j=l). fit, 1, 1,0) = max {5oi,r } +pforf >max {%, ry} ip, ifjy is the first job inG, 0= !)Â• fit, 2, 0, 1) = Â°Â° for t < max {502. ''/} + (}< >f Â•{/ is the first job in G2 (/ = "i + !)Â• fit, 2, 0, 1) = max {^02. rj) + max { jqj' '}} + ?Â• ^^Jj 's the first job in G2 0' = Â«i + !) Iteration: (j, + /2> fit, l.i,,j2) = min Uitp, I, iyl, 12) + t,AtpS2i,2, ir\, k) + t,Ai^, 1. J,, J2)) iff>')+p, where) = /i. fit, 1, /,, ^2) = Â°Â°\i t
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138 fit. 2. J,, /j) = min [/{iq. 2. <,. /jl) + t,AtqS\i, 1. /i. /20 + './('L 2. ii. h)) iit^rj + q, where 7 n^ + 12. fit, 2, /,, 12) = Â°Â° if r < r. + ver: the optimal total flowtime is min [/(/?, \, n^, n^,/iR, 2, /i], n2)].
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139 Test problems . Wc generated 70 test problems in order to test the dynamic program on a range of problem sizes. The data for the problem sets are summarized in Table 3.23 (10 problems in each set). Table 3.23. Data about problem sets for special case. Problem Set FT201 FT307 FT308 FT309 FT401 FT507 FT601 Number of jobs 20 30 30 30 40 50 60 1 1 2 4 1 1 1 Class setup 1 1 1 3 1 1 1 Interarrival times 13 13 25 48 13 13 13 Results of special case dynamic program . The dynamic program finds optimal solutions in time that is nearly proportional to Rrfi (see Table 3.24). One drawback of the dynamic program is the amount of memory required to perform the algorithm; we were able to solve only 60job problems. In addition, it requires significant processing time on a 386 personal computer. These problems can be overcome, however, since a larger computer could handle more memory (the amount required is 0(y?n)), and would run more quickly. Table 3.24. Results of dynamic program Problem
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140 Extensions of special case . The dynamic program can be modi Tied lo solve any FTSRD problem where there exists* a natural order for the jobs within each class. This includes problems where the jobs in each class have the same processing time (as we have discussed) or problems where all of the jobs have the same release date (order the jobs in each class by SPT). In any of these cases we have an ordered batch scheduling problem. If each class has a natural order, the dynamic program can be used to interleave these sequences since we have an ordered batch scheduling problem. Recall from the example presented in Section 3.4.4 that matching processing times and release dates do not give us a natural order for a class. 3.4.7 Conclusions In this research we have studied a computationally difficult class scheduling problem. The objective is to minimize the total flowtime of a set of jobs that have nonzero release dates. We examined a number of techniques to solve the problem, including a branchandbound search, lookbehind dispatching rules, a decomposition heuristic, and a problem space genetic algorithm. We were interested in determining how this type of genetic algorithm can be used to find good solutions for another class scheduling problem. Our results are as follows: While we did develop lower bounds and a number of dominance properties, our branchandbound approach was unable to solve any 30job problems. The decomposition heuristic was a successful technique, locating solutions of high quality. The Shorest Waste heuristic could sometimes generate good solutions. However, by incorporating these rules in a genetic algorithm that searched the space of adjusted release dates, we could find much better solutions. From these results we can conclude two things: For the onemachine class scheduling problem we call FTSRD both the problem space genetic algorithm and the decomposition heuristic can find good solutions in reasonable time. Additionally, lookbehind rules may be
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141 useful for job shop scheduling, especially on a bottleneck machine which undergoes class setups and where jobs continue to arrive while the machine is processing. 3.5 Chapter Summary In this chapter we have presented the results of research into three onemachine class scheduling problems. We can make a number of observations about this research. 1. None of these three problems have been previously considered in the literature. 2. We have presented analytical results and developed extended heuristics for each of these problems. 3. All three problems are motivated by the semiconductor test area job shop environment, and the extended heuristics developed for these problems may be useful as dispatching rules in the general job shop scheduling problem. 4. Our problem space genetic algorithm is a robust approach, able to find good solutions over a variety of onemachine class scheduling problems, and should be applicable to other difficult combinatorial and scheduling problems.
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CHAPTER 4 LOOKAHEAD SCHEDULING PROBLEMS In this chapter we discuss the second major area of this research. We study the problem of scheduling a machine that processes jobs headed for two different secondstage machines. We analyze this threemachine problem with three different objective functions: makespan, total flowtime, and number of tardy jobs. We first examine the complexity of the problem and then identify some lower bounds as well as some special cases that can be solved in polynomial time. We develop a number of heuristics that find good solutions to the problem. We also use a branchandbound technique to find optimal solutions. 4.1 Introduction Job shop scheduling includes those scheduling problems in which different jobs may follow different routes through the shop. These problems are generally the hardest to solve optimally, since few properties of optimal schedules are known and the number of possible solutions explodes as the problems increase in size. Because of the complexity of job shop scheduling, algorithms to find the optimal solution (in a reasonable amount of lime) for even the simplest objective functions, e.g. makespan, do not exist. Recent research has shown that bottleneckbased techniques such as the shifting bottleneck algorithm (Adams, Balas, and Zawack, 1988) or bottleneck dynamics (see Morton, 1992, for example) can be successful at finding good schedules. Traditionally, however, researchers have studied (and schedulers have used) dispatching rules to order the jobs waiting for processing at a machine. Normal dispatching rules consider only the jobs currently in the queue for the machine being scheduled. Wc define lookahead scheduling as the ability of a sequencing procedure to 142
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143 include information about the status of machines downstream in the flow, enabling it to make a better solution. Previous techniques that use this lookahead idea include the workinnextqucue and numbcrinnextqueue dispatching rules (Panwalker and Iskandcr, 1977) and the use of bottleneck starvation avoidance in shop floor control by Glasscy and Pctrakian (1989). Robinson et al. (1993) consider upstream and downstream information in scheduling semiconductor batch operations. Researchers have also studied lot release policies that look ahead to the status of the inventory in front of or arriving at a bottleneck; see for example, Wein (1988), Glasscy and Resende (1988), and Leachman, Solorzano, and Glassey (1988). Other researchers have studied procedures that they called lookahead scheduling (Koulamas and Smith, 1988; Zeestraten, 1990) but the problem setting or interpretation is different. Consider two examples from the semiconductor test area. In the first, lots of two different products are processed through the same brand workstation. After brand, the lots require electrical testing, but the differences between the products indicate that the lots must be tested on different machines. Or consider the effect of reentrant flows. The various lots waiting for processing at an electrical test workstation may be at one of two points in their route. At one point, a lots moves to brand after being tested (and it will return to test at some point in the future). At another, it moves to visual/mechanical inspection. In the first case, the brand workstation is sending lots to two different testers; in the second, the tester is sending lots to two dissimilar workstations. If one of the secondstage machines is a bottleneck, it seems clear that the sequencing of lots on the firststage machine should try to maximize the efficient use of that bottleneck. We can model this scenario with the following threemachine problem: There are three machines Mq, M,, and Mj. Each job follows one of two different flows: Mq Mj, or Mq M2. Thus, Mq is feeding the other two machines. If one of these secondstage machines is a bottleneck because the total work required on that machine is the larger than that on the other machine, the sequencing of jobs on Mq should have as a priority the proper feeding of that
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144 machine. This idea of a txMlleneck will nol affect our analysis of the problem. Wc will, however, return to it for the heuristics and the empirical testing. This problem, which could occur in any number of manufacturing environments, forms an interesting general tlow shop problem and a subproblem of the job shop scheduling problem. As a flow shop problem, it is a simple model unlike the multimachine problems previously discussed in the literature, although work has been done on flexible flow shops with multiple parallel machines at any given stage. Moreover, the research into our threemachine problem may impix)ve job shop scheduling in two ways: one, the solution procedures can be applied directly to the subproblem of scheduling machines near the bottleneck machine, and two, these techniques may be translated into good lookahead dispatching rules for scheduling throughout the shop. In this chapter we investigate three objective functions for this problem: the minimization of makespan, of total flowtime, and of the number of tardy jobs. We are concerned with the analysis of solutions to the problem and the development of heuristics which can be used to find good solutions. The major contributions of this work include the proof that minimizing makespan is a strongly NPcomplete problem, the identification of optimality properties and special cases that can be solved in polynomial time, and the development of an approximation algorithm. The lookahead scheduling problems under investigation are as follows (using the numbering given earlier): 4. ThreeMachine LookAhead Scheduling: Makespan (3MLAMS) 5. ThreeMachine LookAhead Scheduling: Flowtime (3MLAFT) 6. ThreeMachine LookAhead Scheduling: Number of Tardy Jobs (3MNT) In this chapter we will look at each of these objective functions. In Chapter 2 we discussed the research relevant to these problems. In the next section we start with the makespan objective. In Section 4.3 we look at minimizing the total flowtime, and Section 4.4 discusses work on minimizing the number of tardy jobs.
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145 4.2 Minimi/in^ ihc Makcspan As mcniioncd in ihc iniroduclion, wc arc studying a problem that is a subproblcm of the general job shop scheduling problem and is also a special case of the general flow shop problem. All analysis of the flow shop starts with Johnson (1954), who studied the minimization of makespan for twomachine flow shop problems and for some special threemachine flow shop problems. His famous algorithm starts jobs with the smallest firststage tasks as soon as possible and jobs with the smallest secondstage tasks as late as possible. Special cases of the flow shop makespan problem have been studied by a number of researchers, including Mitten (1958), Conway, Maxwell, and Miller (1967), Bums and Rooker (1975), and Szwarc (1977). Garey, Johnson, and Sethi (1976) proved that the general threemachine problem was NPcomplcte. Problems with release dates, preemption, precedence constraints, or more than three machines have also been studied. In the flexible flow shop, more than one machine may be present at a particular stage. Heuristics for this type of problem are discussed by Wittrock (1988), Sriskandarajah and Sethi (1989), Gupta (1988), and Gupta and Tunc (1991). Lee, Cheng, and Lin (1992) study an assembly flow shop problem where each job consists of two subas.sembly tasks that are assembled in a third operation. This threemachine problem is therefore closely related to problems previously studied, but the preassignment of the jobs to different secondstage machines gives this problem a special structure and leads to interesting results. 4.2.1 Notation The following list describes the components of the problem and the notation used.
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146 /; iobj,j= 1, .... /I. Mo The firststage machine. M], M2 The secondstage machines. W] The set of jobs tliat visit Mq and then Mp H2 The set of jobs that visit Mq and then M2. Pqj The firststage processing time of 7. on Mq. Plj The secondstage processing time of 7. on Mj, / = 1 or 2. For a given schedule o, we can calculate the following variables: Cqj The completion time of 7. on Mq. Cy The completion time of 7.on Mj, / = 1 or 2. Cj = Cy, the secondstage completion time of 7.. Cmax = "13" {Cj}' the makespan of the schedule. Z Cj the total flowtimc of the schedule. Note that we will call a set of jobs that visit the same secondstage machine a group; thus. Hi and H2 are each a group of jobs. Each group has a flow. The flow for the jobs in Hy is Mq M]. The flow for the jobs in H2 is Mq M2. This section is concerned with the problem of minimizing, over all feasible schedules, the makespan of the jobs. We call this problem the ThreeMachine LookAhead problem Makespan (3MLAMS). 4.2.2 Johnson's Algorithm If we consider just one group and its corresponding flow, the problem of minimizing the makespan of the set of jobs that visit these two machines is the same as Johnson's twomachine flow shop problem. Johnson (1954) provided an oplimality rule and an algorithm to solve the problem. If each job to be scheduled has task processing times a . and b: on machines one and two respectively, then his rule is as follows: Johnson's Rule: 7, precedes 7; in an optimal sequence if min {a,, b:] < min {a., ft,}. This rule is implemented in the following algorithm: Johnson's Algorithm: Step 1. Find the smallest unscheduled task processing time. (Break ties arbitrarily.)
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147 Step 2. If this minimum is on the llrsl machine, place ihe job in the first open position in the schedule. Else, place the job in the last open position in the schedule. Return to Step 1 . 4.2.3 Permutation Schedules In flow shop scheduling, a feasible schedule is called a permutation schedule if the sequence of jobs on each machine is the same. Thus, the sequence of jobs on the first machine uniquely identifies a schedule (assuming all tasks arc started as soon as possible). In the threemachine lookahead problem that we are studying, the two sequences on the secondstage machines consist form two disjoint sets of jobs. Thus, for our problem, we extend the idea of permutation schedules to include schedules where the relative order of two jobs in the same flow is the same at both stages. It is known that considering permutation schedules is sufficient for finding optimal solutions for regular twomachine flow shop problems. Thus, for our threemachine lookahead problems, it seems likely that permutation schedules will also be sufficient, since the problem contains two twomachine flows. We will show that this is indeed true. Definition: A schedule is a permutation schedule if, for all J: and 7,in //, (H^), Ji precedes Jj on Mq if and only if 7,precedes J: on M] (M^). Theorem 4.1. For any regular performance measure, there exists a permutation schedule that is an optimal schedule. Proof. First we will show that we can interchange two jobs that are not in the same order at both stages. Given a schedule a where job J: directly follows 7,on machine M, (or M2) but Jj precedes 7,on machine Mq, move the Mq task of 7y after the Mq task of 7,. If we look at Figure 4. 1 , we can observe that moving ./.causes all of the tasks (except Jj) on Mq to start earlier, which docs not delay any (and may expedite some) secondstage tasks. Now, since 7,will complete on Mq when 7, did {C'qj = Cqi), the processing of 7y on M, will not be delayed. Our interchange, therefore, does not increase the completion time of any job.
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148 Thus, for any given schedule, wc can create a corresponding pcrmulalion schedule by interchanging the firststage tasks. None of the job completion times are increased by this construction. Indeed, some of the completion times may be decreased. Therefore, this permutation schedule has a better or equal performance on all regular measures (e.g. makespan, flowiime, number of tardy jobs, maximum lateness). Thus, it is sufficient to consider permutation schedules when trying to minimize these objective functions. QED.
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149 4.2.4 NPComplctcncss In this section, wc consider the complexity of the 3MLAMS problem. Although other researchers (including Gonzalez and Sahni, 1978) have shown the NPcomplclcness of a number of small shop problems, we cannot determine the complexity of our problem from any of this previous work. We will therefore prove that 3MLAMS is strongly NPcompletc, which will be done by transforming 3Parlition to 3MLAMS. (Recall that it is sufficient to consider permutation schedules.) The 3Parlition problem can be stated as follows: Given a set of a,, j = 1, . . . , 3Â«, and B, partition these elements into n sets y4,, . . . , 4Â„, where each set contains three elements and the sum of the three elements equals B. We will make the assumption that for all /, 1/4 B 1), create 4Â« jobs, 3n of which go from Mq to Mj and have processing times pQi = a,and P2j = (S+l)a,, J = 1 , . . . , 3rt; let these jobs form a set W. The n remaining jobs that go from Mq to M,, Â•^3/1+1' Â•^3n+2' Â• Â• ' UnPOk = ^^ fo"" ^^ * ^nd pj/, {B+\)B, ^ = 3ai, .... 4nl, and p; 4Â„ = B; let these jobs form a set X. The desired makespan is M = n(B^ + B) + B. Part 1. If there exists a partition (/I, AÂ„) such that for all a,in A/^, I a,= B, then the sequence on Mq of [4 iJjn+X ^2 ^Sn+l '^n Un^ ^''^ y'^^'^^ ^ schedule where the completion time on M, will be M and the time on Mj will be no greater than M. (See Figure 4.2.) Part 2. Now, suppose there exists no partition. Consider any arbitrary permutation schedule a. The sequence is composed of consecutive subsets of jobs from W and of jobs from X. LetAy.y = 1. n+1 be these subsets of W, where Aj directly precedes theylh job from X, except for/\Â„+i, which is the set of jobs following the last job from X. Note that any of these Aj
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150 may be cmply. Let S: equal the sum of the firslslagc task processing times for the jobs in A:. Let Aj^ be the first A: such that 5^ is not equal to B. Because there is no partition, there must exist one such i4^ among y4i /\Â„. If S^ > B, then the makespan on machine M] is delayed by the delay in the /ih job inX: MS(M,) > S, + B+ . . . + S^ + fi"2 + (n k){B^ + B) + B > kB + kB^ + (n k.){B^ + B) + B = M. (This assumes that all of the first k\ jobs have long tasks on M,, leaving n\(k\) long tasks and one shon task on M , . if one of the first k1 has the short task, then the makespan is even longer.) If S^ < B, let S = Si + . . . + S^ < A:B (and < kB 1) and suppose 7^ is the first job from W after the /tth job from X. Then, the makespan on Mj is postponed by 7^: MS(M2) ^ 5 + kB'^ + a^^ + (B + \){nB S)>S + kB^ + a^ + nB^ + nB BS S. Including BS > kS^ + B implies MS(M2) > a^ + nB^ + nB + B>M. Thus, there exists no schedule with makespan less than or equal to A/. Part 1 and Part 2 of the proof show that there exists a partition if and only if there exists a schedule with the desired makespan. This implies that 3MLAMS is a strongly NPcomplete problem. QED. Mr Ml M^ Al
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151 sequences for each group (Algorithm 4.1). Properties 4.1, 4.2. and 4.3 arc dominance properties between jobs in the same group. Theorems 4.3 and 4.4 describe the special cases. Because we need to consider only permutation schedules, we can create a single schedule for all three machines from a sequence of the jobs. That is, given a sequence on the firststage machine we can create a sequence for each secondstage machine. This is done by considering the jobs in the order they appear on the first machine. However, there is no comprehensive rule for determining this sequence. Thus, we will spend some time on how this sequence should be constructed in certain situations. Interleaving two sequences . Since the problem is NPcomplete in the strong sense, heuristic methods for finding good solutions are justified. A very natural heuristic is to schedule the jobs in each group separately. For instance, each group can be scheduled by Johnson's rule. Then we can interleave these two sequences to form a solution to the original problem. The process of combining two sequences to achieve the minimal makespan is called optimally interleaving. This is the best combination of these two sequences. Note, however, that the best combination of the two Johnson sequences may not be an optimal solution; the optimal solution may be some combination of suboptimal solutions to each subproblem. In Section 4.2.9 we present an example where optimally interleaving the Johnson sequences for each group is not optimal. So, while interleaving the Johnson sequences is a natural heuristic that worics well (see Section 4.2.8), we may wish to try other heuristics to the subproblcms before interieaving. These are discussed in Section 4.2.7. In this subsection we will describe how two sequences should be interleaved. Now, suppose we are given two sequences c j and 02one sequence for each group (we have found solutions for each of the subproblcms), and we want to find the minimal makespan that can be achieved by combining these two sequences. The following observations will lead us to an algorithm for doing so.
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1S2 DcTinc C as a makcspan that wc wish lo achieve. To minimi/.c the makespan given the Iwo sequences we need to find the minimal C for which we can find a feasible schedule. Wc can schedule the tasks on the secondstage machines in the order given by O] and aj and as late as possible, so thai the last task on each machine ends at C. In a feasible schedule, the firststage task for each job has to complete on Mq before the secondstage task can begin. We need to delcrmine if there is some ordering of the tasks on Mq so that each task finishes ontime (with respect the secondstage task). For each job J: the start time of the secondstage task can be used as the due date d.for the corresponding firststage task. We can find a sequence where each firststage task finishes ontime (Cq: < dj) if and only if we can find a sequence where the maximum lateness is less than or equal to zero. If we wish to minimize the maximum lateness of the firststage tasks, we should order them by HDD (Earliest Due Date), according to Jackson (1955). The schedule created is an interleaving of the two given schedules: if /^ precedes J: in a^, then d^ < dj, and /, will precede Jj on Mq. Each due date dj = C r., where t: is the sum of the secondstage processing times of iy and the jobs Jfr that follow Jj on the secondstage machine. See Figure 4.3.
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153 Algorithm 4.1 (Optimal Interleaving): Given a sequence Oj for the jobs in //, and a sequence 02 for the jobs in H2, pcrfomi the following steps to yield a schedule with the minimal makespan: Step 1 . For each 7. in //,, define A: as the set of jobs (not including J J) that follow Jj in Oj. Then tj = pjj+'L^jPjl^. Step 2. For each 7. in Wj. define A: as the set of jobs (not including Jj) that follow Jj in 02Thenry=/72y+lAyP2)tStep 3. Schedule the jobs on Mq in decreasing order of the tj, starting at time zero, and start all secondstage tasks as soon as possible. Note that this algorithm takes 0(Â«) effort, since each group is already in decreasing order of the tj, and forming the schedule is only combining the two sequences without changing the relative orderings. In Example 4.1 we perform Algorithm 4.1 on a problem with five jobs and given sequences for each group. We are given sequences [JjJ^J2]2ind[ J 4 J^ ] for each group. After computing the f ., we form the interleaved sequence [ Jj Jj J 4 J$J2^^*^ corresponding schedule has a makespan of 1 1 (see Figure 4.4). Example 4.1. Given the following five jobs and the two sequences [ 7/ JjJ2 1 and Jj Hi POj Pij \ '} Jl Hj 2 4 9 h Hi 1 * ' J3 Hi 2 4 5 J4 H2 2 1 4 J5 H2 1 3 3
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154 h
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155
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156 equals z. Since 7, can be on only one machine at a time, z
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157 Theorem 4.3. If p/^ ^ Po^t ^^^ ^'* ^k '" ^i ^'^ P2j ^ POj f"*" ^^* Â•'/ '" ''z^^^ l^c optimal solution can be found by ordering the jobs in each group by their secondstage processing times, longest processing time first, and optimally interleaving the two sequences. Proof. This is true because Corollary 4.2 holds for both groups. QED. Theorem 4.4. If, for some H^, min [pg: .Jje Hi^jHi) ^ max (/?,. : J: e //,), then the optimal schedule can be found by sequencing the jobs in each group by Johnson's Rule and optimally interleaving these sequences. Proof. By the given, we know that the conditions of Corollary 4.2 hold for the jobs in //,. Thus, the jobs in this group should be ordered by longest secondstage task processing time first. For these jobs, this sequence is the same as the sequence given by Johnson's rule. Now, if we can determine the optimal ordering of the jobs in //^, the other group, we can interieave the groups to derive an optimal schedule. We will show that these jobs should be ordered by Johnson's rule. Consider an optimallyinterleaved schedule where J^ is processed immediately before Jj on Mj^, and where the jobs are not ordered by Johnson's rule. That is, min {pQj, pj^f^} < min {p^;,, Pkj^ If there are no jobs from //, between Jf^ and 7. on Mq, then Property 4.2 implies that the jobs should be interchanged. Else, let y^ be the job from //, processed just before Jj. \Uj is not the last job on Mq, then we can move Jj to immediately after Jf^. Because the jobs that were between 7^ and Jj have short secondstage tasks, delaying these jobs does not delay the processing of any successive jobs. Now Jf^ and Jj are consecutively processed on Mq. and Property 4.2 implies that the jobs should be interchanged. Finally, if 7.is the last job on Mq (and M^, then the fact that the schedule is optimally interieaved implies that p^y < Pi^, since 7^ is the last job on Mj. By the given, PyÂ„ is less than or equal to pQf^ and pqj. Thus, min [pQj, pu^ ] < min [pgh^ Pkj) implies that p^/, < min {pQf^, pj^j) (since p^y < pqj) and consequently p^ < pq^ and p^/j < p^y. Property 4.3 implies that Jj should precede 7^.
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158 Wc have thus shown that the jobs in //^ should be ordered by Johason's rule. This gives us an optimal ordering for //^ which can be interleaved with //,Â• to solve the problem. QED. 4.2.6 BranchanjBound Alj^oriihm In this section we will identify three lower bounds on the makcspan; we will take the maximum of these to form our overall lower bound. We will also discuss the use of a branchandbound technique for finding optimal solutions for the 3MLAMS problem. We will use the overall lower bound in the analysis of the worstcase performance of an approximation algorithm (Section 4.2.9). We will begin by discussing the three component lower bounds. For the first component bound, consider only the jobs in //]. Order these jobs using Johnson's rule and determine for each 7.a complefion time Cj: on M,. Then, LBj = max {Cjj]. Similarly, LB2 = max {Â€21]The third component bound takes into account that all of the jobs use Mq. We relax the problem by dropping the secondstage assignments of the jobs and allowing an infinite number of secondstage machines. In fact, however, we only need n secondstage machines, one for each job. If each job has a separate secondstage machine, then the minimal makespan, which will be a lower bound on the optimal makcspan for the original problem, is the maximum sum of firststage completion time plus secondstage task processing lime. By an argument similar to that of Section 1.4, we can find the minimal makespan by sequencing the jobs by their secondstage task processing times, longest first. We schedule the jobs in this order on Mq. The secondstage completion time of any job Jj in //, is Cqj + pij. Therefore, LB3 = max {Cqj + Pij] . Note that LS3 will be greater than S pqj^, since there exists some job Jj that has Cqj = Z pgic Our lower bound for a given problem (hereafter referred to by the variable LB) will be the maximum of these three lower bounds for the makespan: (i) LBj, the minimum possible makespan of the jobs in set //,, (ii) LB2, the makespan of the jobs in set Hj, and (iii) LB^, the minimal makespan if there exist an infinite number of sccondslagc machines. That is, LB = max [LBi,LB2,LB2].
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159 In a standard branchandbound algorithm, each node will consist of a partial schedule of jobs. From a node, wc exclude certain branches using the above dominance properties and create a lower bound using straightforward extensions of the above lower bounds. In order to help prune branches from the search tree, we will use the optimal properties that we developed in Section 4.2.5. We will also use a dominance property (Properly 4.4) that is more dependent upon the current makespan of each machine (where f, is the makespan of Mj). Property 4.4. For a given partial schedule, if 7. is unscheduled and in //^ (where H^ is the other group), ';t ^ POi ^^^ ^^^ Â°^^'' ^' Jot's), and f^ < S Poi< ihen J: should not precede any job from Z/^. It is easy to show that if, in any schedule constructed from this partial schedule, 7. does precede any of the jobs from //^, we can move Jj to the last position without increasing the makespan. We used the branchandbound algorithm to solve a number of problems. On the set of 15job problems, the running time was usually less than one second (on a 386 PC), although it was much greater for two problems, one of which had a lower bound that was not tight. See Table 4.1. Table 4.1. Statistics on branchandbound procedure for LA154. Problem Lower Bound Optimal Note: Time measured in seconds. Time Nodes 1
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160 4.3.7 Heuristics Since 3MLAMS is a strongly NPcomplctc problem and the branchandbound procedure may occasionally take loo long to find the optimal solution, the use of approximation algorithms is a preferable alternative. Our discussion so far has led us one very natural heuristic, the interieaving of the Johnson sequences. We will also introduce another combination that will be of use later. Johnson Interleaved. Order the jobs in each group using Johnson's rule. Optimally interleave them using Algorithm 4. 1. Merged Johnson. Order the jobs in each group using Johnson's rule. Select the group with the smallest total task processing time on Mq (if the totals are equal, pick one arbitrarily). Start the schedule with all of the jobs from this group. Follow them with the jobs from the other group. 4.2.8 Empirical Results In this section we report on the empirical testing of our heuristics. We discuss the problem sets generated, our methodology, and our results. In order to study the scheduling of a bottleneck machine, consider one of the secondstage machines, say Mj, as a bottleneck operation. As a bottleneck, the workload of M, should be greater than that of Mo or M2. Therefore, we will construct problems where the total processing time on M, is likely to be the largest. Three problem sets (LA 154, LA304, LA504) were created using uniform distributions to generate processing times. The mean fraction of jobs in each group and the mean processing times were set such that M, would have more work to do than M2. These problems had 15, 30, and 50 jobs. There were ten problems in each set.
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162 Table 4.3. Makcspan summary tabic for 3MLAMS problem. Pcformancc is relative deviation from optimal or lower bound. Set Johnson Johason Interleaved Merged LA 154
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163 Corollary 4.3. For a given instance of 3MLAMS, the schedule created by the Johnson Interleaved heuristic has a makespan less than 3/2 LB. Theorem 4.6. The maximum error of the Johnson Interleaved algorithm relative to the optimal makespan is onehalf. Proof. If C is the value of the optimal makespan and Cj/ the makespan of the schedule created by the Johnson Interieaved algorithm, LB < C* < Cjj < 3/2 LB < 3/2 C*. Thus, (CjjC*)/C*< 1/2. QED. Let us make a few observations about our algorithm. First, the error bound can be extended to the case where there arc m > 2 groups (each group with a different flow to a different secondstage machine). The maximum relative error of the Johnson Interieaved algorithm is 1 1/m in this case. Second, the Johnson Interleaved algorithm interleaves the groups by looking ahead to the future workload of the secondstage machines (the sum of the remaining secondstage task processing times). Finally, our error bound of onehalf is tight. In the following problem instance, the bound is achieved in the limit. Example 4.2. Given n, let C be 2Â« i1 . For A/] , construct jobs /j , . . . , 7Â„ with the following characteristics: J =!,...,Â«1: Poi=^ P]i='^ Construct in Hj jobs J^^ ],..., J2n with similar characteristics; i = n+ \, . . . ,2n\: pgi = 1 pji = I ' = 2i: P02n='^ Pl2n = '^Now, since ties can be broken arbitrarily, each group is already ordered by Johnson's rule (we could force this ordering by subtracting a small amount from the appropriate firststage tasks). An optimal schedule can be achieved by interleaving the sequences [ JfiJ\ J nl 1 ^^^
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164 ^Â•^2n^n+\ ^In} 1The makespan of such a schedule is C =2n+ 1. The interleaving of the Johnson sequences yields a schedule with malccspan 3n, as docs merging the Johnson sequences. As n goes to infinity, the ratio of each of these maicespans to C goes to 1 .5. See Figures 4.7, 4.8, 4.9 (J^ = y,o) for the optimal, interleaved, and merged schedules with Â« = 5. h
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165 that problem is a special case of 3MLAFT, 3MLAFT must also be a strongly NPcomplcte problem. Hence, we will investigate opiimality conditions, lower bounds, and heuristics. Since total flowtime is a regular objective function, only permutation schedules need be considered for optimality (see Section 4.2.3); thus, a sequence of jobs for Mq corresponds to a unique schedule on all three machines. The problem notation is the same as that for the 3MLAMS problem. A number of researchers have studied the problem of minimizing the total flowlime in a flow shop. This work includes lower bounds, branchandbound algorithms, and heuristic approaches. See Ignall and Schrage (1965) for a branchandbound and Ahmadi and Bagchi (1990) for improved lower bounds. Szwarc (1983) studies special cases, and Van de Velde (1990) presents a Lagrangian relaxation. Krone and Steiglitz (1974), Kohlcr and Steiglitz (1975), Miyazaki, Nishiyama, and Hashimoto (1978), and Miyazaki and Nishiyama (1980) all present heuristic approaches. Ahmadi et al. (1989) includes batch processing. 4.3.1 Total Enumeration Initially, we wanted the compare Lhe difficulty of finding nearoptimal solutions for 3MLAFT to the difficulty of finding nearoptimal solutions for 3MLAMS, where we had very good results. One way in which we could compare the two problems was to compare the range of the objective functions over the domain of all permutation schedules. For a nine job problem, a total enumeration of the permutation schedules includes over 300,000 sequences. We picked an arbitrary instance, created each schedule, and measured the makespan and total Howtime of each schedule. This procedure yielded a distribuUon of makespans and total tlowtimcs over the set of sequences. The primary result is that 18.5% of all sequences have makespans within 6.7% of the optimal makespan, but less than one quarter of one percent of all sequences have flowtimes within 6.7% of the optimal fiowlimc. (See Table 4.4 and Figure 4.10.) Because there are much fewer schedules that are nearoptimal, we can infer that the problem of minimizing the total flowtime is a much harder problem than the problem of finding the minimum makespan. Similar
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166 results hold for another ninejob problem and a fifteenjob problem where a random sample of the sequences were examined. Table 4.4. Distribution of schedule makespans and flowiimes. Deviation from
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167 4.3.2 Lower Bounds In this section we discuss lower bounds on the total flowtimc. These lower bounds provide us with a way of determining the quality of the solutions produced by our heuristics and can be extended into lower bounds for a branchandbound algorithm. The lower bounds are similar to those proposed in Ignall and Schrage (1965). The initial lower bound on the total flowtime is calculated by ordering the jobs by their processing times on Mq, shortest processing time first (SPT), and assuming that the secondstage machines have infinite capacity. That is, LB^ X Cq: + Z Pjj + S P2jWe compute the second bound by considering each of the secondstage machines. Let a = min [pQj .Jje //, ) and ft = min {pg: :Jje Hj}. Now, a (b) is the earliest lime that the secondstage tasks on Mj (Mi) could start. Then, form two sequences Oj and O2of the jobs in//, and H2 respectively by ordering the jobs in each group by the secondstage task processing times, shortest processing time first. Schedule on M, the secondstage tasks of the jobs in //, in the order given by O]. The first task should begin at a, and each successive task should immediately follow (we do not schedule the firststage tasks). This forms completion times Cjj for the jobs. Repeat for the secondstage tasks of the jobs in H2 in order to calculate C2/. However, only one machine can start at its cariiest time. The other must start at a time no less than a + b. Let r be the cardinality of//; and s the cardinality oi Hj. Either the r tasks on M, will be delayed by b, or the s tasks on M2 will be delayed by a. After calculating rb and sa, we can increase our lower bound by adding the smaller quantity. Thus, the second lower bound is LBj = X C]j + Z C2j + min {rb, sa}. In Example 4.3, we calculate the lower bounds for the problem instance that we introduced in Example 4.1.
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168 Example 4 J. Given the following five jobs in iwo groups: Jj
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169 4.3.4 Empirical Testing Empirical testing was performed using a number of different heuristics. The three heuristics that performed the best (and very similarly) are described below. The 3MLAMS problem sets were used as a testbcd. For each problem, the four lower bounds on the flowtime were calculated. The best of these was taken as the lower bound. For the fifteenjob problems, we were able to find the optimal solutions from the branchandbound algorithm. The performance of a heuristic on a problem was taken as the relative deviation from the optimal solution (if known) or the best lower bound. SPTlookahead. Each group //, and H2 is ordered by the firststage task processing times (shortest first), forming two sequences. These sequences are interieavcd by choosing at each step the first unscheduled job from one sequence or the other. Define at each step the following variables: fg is the completion lime of the partial schedule on Mq and f 1 is the completion time of the partial schedule on Mj. Letpi (P2) be the processing time on Mq of the next job from H^ (H2). Note that these are the shortest such task processing times from each set of jobs. If f 1 ^o Px, the workinprocess inventory (WIP) waiting at Mj is low; schedule the job from //j. If ^i 1^ >Pi+ P2, then the WIP at Mj is high; schedule the job from H2. Else, Pi < ^1 fo < Pi + ^2' ^d the WIP is intermediate; schedule the job with shortest firststage processing time (pi or/72)WIPlookahead. Again, order the jobs in each group by the firststage task processing times (shortest first). Combine the groups as in SPTLookahcad, except for one case: If /?i < t^ h schedule the job from //j. Thus the work on M] is always used to determine which job to schedule. We tested the heuristics on the problem sets described in Section 1.7. For the fifteenjob problems, we were able to find the optimal solutions from a branchandbound algorithm. We
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170 could not find optimal solutions for the larger problems. Therefore, in order to measure the heuristics, we computed a lower bound on the flowtime for each problem. We calculated the lower bounds described in Section 2. 1 and took the largest as the lower bound. The performance of a heuristic on a problem was taken as the relative deviation from the optimal solution (if known) or the best lower bound. Due to the special structure of the problem, we could find improve the second lower bound for the instances in Set 4 by determining the optimal total flowtime for the five jobs in H^. Minimizing the total flowtime is harder than minimizing the makespan, and in Table 4.5 we report the results of the three heuristics described above. These heuristics were selected because they performed much better than a number of other procedures that combined the groups differently. The lookahead heuristics found solutions with average total flowtime within eight percent of the optimal value. Because of the special structure of the 3MLA instances in Problem Set LA201, very good solutions were easier to find. The WIPLookahead heuristic was slightly better on all of the other problem sets. However, the heuristics were very close to each other. Table 4.5. Heuristic performance for the 3MLAFT Problem. Performance relative to optimal or lower bound. Set SPT WIP Johnson Lookahead Lookahead Lookahead LA154
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171 feeding of a bottleneck machine. The objective mirrors the management concern of customer satisfaction. Let us call our problem the ThreeMachine Number Tardy (3MNT) problem. Recall that only permutation schedules need be considered. Thus a sequence for all of the jobs defines a unique schedule. The problem notation is the same as that for 3MLAMS, except that we have for each job J: a due date d:. Since 3MLAMS is strongly NPcomplctc, 3MNT is also strongly NPcomplete. Consider an instance where all of the jobs have the same due date: finding a schedule with no tardy jobs is equivalent to finding a schedule with makespan less than or equal to the common due date. Since the problem is computationally difficult, we will examine a simple lower bound, a special case, and some heuristic approaches to finding solutions. 4.4.2 Lower Bound and Special Case Good lower bounds are hard to find for the problem of minimizing the number of tardy jobs in a flow shop (Hariri and Potts, 1989). We will make use of a fairly simple one. The lower bound makes use of the fact that the MooreHodgson algorithm will find the optimal number of tardy jobs for a onemachine problem. Given an instance of 3MNT, the due date of each job is adjusted by subtracting the processing time of the secondstage task. These adjusted due dates and the firststage processing times form a onemachine problem for machine zero. The lower bound is calculated by using the MooreHodgson algorithm to optimally sequence these tasks. The number of tardy firststage tasks is a lower bound on the minimum number of tardy jobs for the 3MNT instance. This lower bound is achievable in the following special case: Theorem 4.8. If min [pq: :Jje //, } > max [pj: : J.Â€ //, } and min {pQj :Jje H2} ^ max (P2j '.Jie H2], then an optimal sequence can be found be minimizing the number of firststage tasks that are tardy to the adjusted due dates.
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172 Proof. Because of ihe condilions above, the completion time of a job J: in //, is C: = Cq; + Pij. Thus. Jj is tardy if and only if Cq: > d; Piy QED. We also tried lower bounds on each of the secondstage machines, but these were not as good. This seems reasonable if the interaction of the two flows is significantly contributing to tardiness. 4.4.3 Heuristics Simple rules that can be extended for this problem include the MooreHodgson algorithm and Eariiest Due Date (EDD). These can be expanded by including lookahead ideas. We also developed a simple problem space genetic algorithm to find good solutions. Lookahead rules . The lookahead extension of the MooreHodgson algorithm includes both machines. The jobs are ordered by their due dates and added to the schedule until a tardy job is found. The procedure then determines the critical path of tasks that determines the completion time of the tardy job. Then, each job in the path is evaluated to determine how much the completion time would decrease if that job were removed from the partial schedule. This calculation depends upon whether the job precedes, is, or follows the crossover job (the job whose first and second tasks are in the critical path). We developed two lookahead versions of the EDD rule. The first is similar to the MooreHodgson rule. The jobs are sequenced by their due dates. They are scheduled one at a time. When a job is tardy, however, we simply remove it to the end of the schedule. (In the MooreHodgson rule, we look for the job whose removal helps the most.) This form of the EDD is called EDDNo Tardy. The other lookahead version (EDDLookahead) tries to schedule machine one (the bottleneck) carefully. The jobs in each group arc ordered by their due dates. The primary idea is to get the jobs from group one to be ontime; group two jobs can be inserted if they don't interfere. At any point in constructing the schedule, we consider the next job from each group. If either would be lardy if scheduled next, we place it at the end of the schedule. Eventually, we get
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173 a job from each group that would be ontime if scheduled next. We then determine if scheduling the group two job next would cause the group one job to be tardy. If not, we schedule this group two job and consider the next job from that group. Else we schedule the group one job next. Genetic algorithm . After initial testing, we determined that the lookahead MooreHodgson rule found consistently good solutions. Thus, we used a problem space genetic algorithm to adjust the problem data used by this rule so that we could find belter solutions. The procedure is similar to those described in Chapter 3. We adjust the due dates using a steadyslate genetic algorithm. The population size was 50, and the algorithm generated 1000 new individuals. 4.4.4 Results In order to compare the heuristics we created a number of problem sets. Each set had 10 similar instances. Problems were created with 15, 30, and 50 jobs. The processing times were chosen randomly, and the due dales were chosen from a range that depended upon the sum of the processing times. The lookahead MooreHodgson heuristic performs better than any of the other rules (See Table 4.6). A number of rules that used the firststage due date of a lot were also tested but did not perform as well. The genetic algorithm was able to find slightly better solutions than those found by the MooreHodgson rule. Table 4.6. Summary table of results for 3MNT. Set Jobs Lower EDD EDD Moore Genetic Bound No Tardy Lookahead Algorithm NT 151
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174 4.5 Application lo Job Stiop Scheduling One of the primary motivations for studying these problems was to see if we could develop useful dispatching rules. After considering our results, it seemed clear that lookahead and lookbehind policies similar to those we used on the onemachine and threemachine problems would be intuitively good ways to dispatch jobs. However, we wanted to determine the tradeoffs of using such rules on a number of objectives. To this end, we created a job shop scheduling problem that modeled the semiconductor test area we were studying. This problem had 82 jobs and 23 machines and included various test operations. Processing time data were gathered from some historical lots. We scheduled the jobs under a number of dispatching rule combinations, using standard dispatching rules, lookahead rules, and lookbehind rules. We measured the schedules on four scales: total flowiime, makespan, number of tardy jobs, and total tardiness. The bottleneck in this problem was a set of bumin boards. Thus, we developed a lookahead rule that orders the jobs wailing by their task processing times and uses information about the downstream bottleneck resource (the bumin board availability) to determine which lot should be scheduled next. We also developed a lookbehind rule that sequences lots by EDD and reserves bumin boards for the next late lot that will be arriving soon. The effort of using these rules is slight for our problem; in a manufacturing environment, the dispatching effort might be more significant. We used the following scheme in order to see how lookahead and lookbehind rules would influence schedule performance. We allocated the rules by dividing the machines into three areas: electrical test, bumin, and other. In any given policy, all of the machines in the same area used the same dispatching mle. Our model included one bumin workstation and 15 testers. We used seven standard rules: SPT, EDD, Slack per Remaining Operation, LPT, Modified Due Date, Earliest Finish Time, and FirstInFirstOut in all areas. For each of the standard rules, we created four policies. In the first, all of the areas used that rule. In the second, the testers used the lookahead rule (since these were the machines
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175 feeding the bumin area). In the third, the bumin area used the lookbehind rule. In the fourth, the testers used the lookahead rule while the bumin area used the lookbehind rule. This yielded 28 policies. The average results over the seven standard rules arc summarized in Table 4.7. We observe that the use of a lookbehind rule, which is concemed with expediting late jobs, has a drastic effect on due daterelated measures. It is able to reduce total tardiness while increasing the number of tardy jobs. This is a common tradeoff in scheduling problems. The lookahead rule, which is concemed with avoiding unnecessary delays, reduces the total flowtime and makespan objectives. Our results are the consequence of the specific definitions of these lookahead and lookbehind rules. For other problems, altemative definitions may yield different results. While our results are not proof that lookahead and lookbehind rules are the answer to solving the job shop scheduling problem, the decreases in total flowtime (when the lookahead rule was used) and total tardiness (with the lookbehind rule) did encourage us to use them in our procedures to find good solutions (discussed in Chapter 5). Table 4.7, Performance of 28 dispatching rule combinations. Flowtime Makespan Tardy Tardiness All policies^ 2,179,344 88,677 17.1 192,394 Single mles^ 2,168,014 With Lookahead'' 2,135,800 With Lookbehind* 2,222,639 With both* 2,190,921 Notes: a: Average over all 28 policies. b: Average over seven policies, one for each standard rule. 4.6 Chapter Summarv In this chapter we have examined a special case of the general threemachine flow shop. In this problem, the jobs to be scheduled form two classes with different groups, and we wish to minimize the makespan, the total fiowiime, or the number of tardy jobs. 89,324
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176 We proved that minimizing the makcspan is a strongly NPcomplctc problem, and we also identified some properties of optimal solutions and some special cases that can be solved in polynomial time. We developed an approximation algorithm, Johnson Interleaved, that can find nearoptimal solutions by looking ahead to the future workload of the secondstage machines. We showed that the worstcase error bound for this procedure is onehalf and that this bound is tight in the limit. We also developed a branchandbound algorithm that can find exact solutions to the problem. Then we described the problem of minimizing the total flowtime. We presented lower bounds, optimality conditions, and the results of testing on selected problem instances a number of heuristics that look ahead to current workload at the secondstage machines. Finally we examined the problem of minimizing the number of tardy jobs. We discussed a simple lower bound, a special case that achieves this bound, and a number of simple and lookahead heuristics. We also showed that a problem space genetic algorithm can find better solutions. These results have two contributions. First is the analysis of these threemachine problems, problems previously unstudied in the literature. We conclude from the results of our empirical testing that lookahead heuristics can find good solutions for the problems of minimizing total flowtime and minimizing the number of tardy jobs, and the interieaving procedure minimizes makespan. Second, while these problems are important questions in their own right, they are also significant as subproblems in a job shop. It is possible to apply our results to the problem of job shop scheduling, either as part of a general scheduling procedure or as an attempt to schedule the bottleneck of a job shop more efficiently.
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CHAPTER 5 GLOBAL JOB SHOP SCHEDULING In this chapter wc describe a global job shop scheduling procedure that uses a genetic algorithm to find a good schedule. We have implemented the scheduling system in a semiconductor test area. The test area is a job shop and has sequencedependent setup times at some operations. The concern of management is to meet their customer due dates and to increase throughput. This requires the coordination of many resources, a task beyond the ability of simple dispatching rules. We discuss a centralized procedure that can find a good schedule through the use of a detailed scheduling model and a genetic algorithm that searches over combinations of dispatching rules. We discuss our effort in developing a system that models the shop, creates schedules for the test area personnel, and contributes to test area management. 5.1 Introduction In many areas of manufacturing, the ability of a facility to meet its objectives depends upon the close coordination of resources. This coordination must occur on different levels: capacity planning, release planning, and lot dispatching. Effective approaches exist (and new techniques are being developed) for the first two levels. The third level, meanwhile, continues to pose very difficult scheduling problems. These are the problems that we address. Like other researchers, we are interested in creating systems to better schedule resources in a manufacturing process, since effective scheduling can lead to improvements in throughput, customer satisfaction (measured by meeting due dates), and other pxjrformance measures. We are concerned with the effective scheduling of a semiconductor test area, a job shop environment. In this facility, a lot is a number of identical semiconductor devices. Each lot must undergo a number of tests (electrical and physical) and other operations before the product can be 177
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178 shipped to the customer. Associated with the lot is the due date of the customer order it will be used to fill. Because each product has a unique route, different lots take different paths through the test area. A number of characteristics make the semiconductor test area difficult to control. These include the conflicting goals of management (balancing increased throughput against meeting due dates), work centers with sequence de pendent setup limes, operations where multiple lots can be processed simultaneously, and the relationships between operations. The purpose of this chapter is to describe the global job shop scheduling system developed for semiconductor testing. We will examine the needs of the semiconductor test area, their requiremenLs for a scheduling system, and our approach to this problem. The primary goal of the global job shop scheduling system is to use information about the current status of the shop, the jobs to be manufactured, and the production process in order to create a schedule of activities for each work center in the shop over a fixed time period. By using a centralized procedure with global information, the system can search for a schedule better than that computed by making nearsighted, local decisions. The system finds good solutions with a genetic algorithm, a type of heuristic search. One interesting characteristic of this search is that it looks for a good combination of dispatching rules to find an efficient schedule. The use of dispatching rules is an effective local procedure to create a job shop schedule; such techniques do not, however, use any global information. Our procedure addresses this shortcoming. It goes beyond simply finding good local schedules; it executes a search that looks for a bener global schedule. While global scheduling techniques have been proposed before, the important contributions of our work are the development of a genetic algorithm for job shop scheduling and the implementation of a scheduling system that uses such an advanced solution procedure. The next section of this chapter will review some background and related research on job shop scheduling. The genetic algorithm for global scheduling is introduced in Section 5.3, where we also discuss a small example of our search space. In Section 5.4 we describe the scheduling
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179 problem in the semiconductor test area under consideration and the global job shop scheduling system that we developed. We summarize our report in Section 5.5 and describe how a similar system may be useful in other manufacturing environments. 5.2 Job Shop Schcduiint! As mentioned in the introduction, the problem of coordinating resources to ensure efficient manufacturing is a difficult problem. When the production process is fairly straightforward, techniques such as JustInTime (JIT) manufacturing result in efficient scheduling. In many situations, however, the process is much more complicated, and finding an optimal or nearoptimal schedule is an impossible task. In this section we will review some of the approaches to job shop scheduling. More details on these papers can be found in Chapter 2. The traditional method of controlling job shops is the use of dispatching rules. A dispatching rule is a sequencing policy that orders the jobs waiting for processing at a machine; the ordering depends upon the particular dispatching rule used. Common rules include the Shortest Processing Time (SPT) and Eariiest Due Date (EDD) rules. While such rules have been the subject of much research, standard dispatching rules have a narrow perspective on the scheduling problem, since they ignore information about other jobs and other resources in the shop. More advanced lookahead and lookbehind rules attempt to include more information, but their reach is still limited. Effective job shop scheduling depends upon the interaction of a number of factors. The complexity of these interactions makes the development of a global scheduling system a difficult task. A number of researchers have looked at semiconductor manufacturing at all levels, from production planning to scheduling. Approaches to shop floor control include lot release policies, dispatching rules, deterministic scheduling, controltheoretic approaches, knowledgebased approaches, and simulation. Uzsoy et al. (1992a, 1993) review a substantial number of papers that consider these approaches to semiconductor scheduling.
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180 This work in semiconductor manufacturing includes corporatewide production planning that uses a ralebased model of production and linear programming (Leachman, 1993, and Hackman and Leachman, 1989). Such a global planning system has been developed under the name IMPReSS at Harris Semiconductor. Hung and Leachman (1992) have developed an iterative method that uses a linear program and a discreteevent simulation to develop longterm production plans for a wafer fab. While most of the research in semiconductor scheduling has concentrated on the fabrication of semiconductor wafers, a number of other papers have addressed the problems of semiconductor test. These include Lee, Uzsoy, and MartinVega (1992), and Uzsoy etal. (1991a, 1991b, and 1992b). Lee et al. (1993) report on the implementation of a decision support system for the dispatching of lots in a semiconductor test area. Our work is the natural extension of this system. Previous approaches to the production of detailed shop schedules for planning and shop floor control include expert systems like ISIS (Fox and Smith, 1984), OPIS (Smith, Fox, and Ow, 1986, and Ow and Smith, 1988), MICROBOSS (Sadeh, 1991), and OPAL (Bensana, Bel, and Dubois, 1988); costbased procedures such as OPT (Optimized Production Technology, reviewed by a number of authors, including Jacobs, 1984), Faaland and Schmitt (1993), and the bottleneck dynamics of SCHEDSTAR (Morton et ai, 1988); simulation (Leachman and Sohoni, Najmi and Lozinski); and leitstands (Adelsberger and Kanet, 1991). Adlcr et al. (1993) describe the implementation of a bottleneckbased scheduling support system for a paper bag production flexible flow shop. While these approaches have been developed for a number of different manufacturing processes, the complications of the semiconductor test area forced us to consider a new design. Job shop scheduling, as one of the most difficult scheduling problems, has attracted a lot of attention from researchers. Techniques such as the shifting bottleneck algorithm (Adams, Balas, and Zawack, 1988) or bottleneck dynamics (see Morton, 1992, for example) concentrate on solving the problem at one machine al a time. More work has gone into the development and
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181 evaluation of various dispatching nilcs. Panwalkar and Iskandcr (1977) present a list of over 100 rules. Recent studies include Fry, Philipoom, and Blackslone (1988), Vcpsalainen and Morton (1988), and Bhasakaran and Pinedo (1991). More sophisticated lookahead and lookbehind rules have also been discussed. Lookbehind rules (called xdispatch by Morton, 1992) consider the jobs that will be arriving soon from upstream machines. Lookahead rules consider information about the downstream machines. This includes the workinncxtqucuc and the numbcrinnextqucue rules of Panwalkar and Iskander (1977), bottleneck starvation avoidance (Glassey and Petrakian, 1989), and lot release policies that lookahead to the bottleneck (Wein, 1988; Glassey and Resende, 1988; and Leachman, Solor/ano, and Glassey, 1988). Lookahead and lookbehind scheduling problems have been studied in this dissertation and by Lee and Herrmann (1993). Finally, heuristic searches have also been developed for job shop scheduling, and a number of these are discussed in Section 2.8 of this dissertation. 5.3 A Genetic Algorithm for Job Shop Scheduling In this section we will describe how a genetic algorithm can be used to find good schedules for the job shop scheduling problem. One approach to difficult scheduling problems such as job shop scheduling is local search, an iterative procedure that moves from solution to solution until it finds a local optimum. Smartandlucky searches (or heuristic searches, or probabilistic search heuristics) attempt to overcome the primary problem of these simple searches: convergence to local optima. These more complex searches are smart enough to escape from most local optima; they still must be lucky, however, in order to find the global optimum. In previous sections we reviewed the basic concepts of genetic algorithms and mention some application of smartandlucky techniques for job shop scheduling. In this section we describe our application of these ideas.
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182 5.3.1 The Heuristic Space Many heuristic searches for the job shop scheduling problem have been considered. One reccnilyiniroduced idea is to search the problem and heuristic spaces, since a solution is the result of applying a heuristic to a problem. A change to the parameters of a heuristic or a problem yields a slightly dilTcrcnt solution. This section will describe how a heuristic space can be used for job shop scheduling. We will show why a genetic algorithm is especially suited for searching this space. The idea of searching heuri.stic and problem spaces was reported by Storer, Wu, and Vaccari (1992). In their paper, the authors examine the general job shop scheduling problem. They define a heuristic space composed of vectors of dispatching rules. Each vector in the space can be used to determine a schedule. Each rule in the vector is used for a fixed number of dispatching decisions, regardless of the machine being scheduled. That is, all of the machines use the same dispatching rule at the same time until the next rule replaces it. Our approach uses a different perspective. Since we will be working in a dynamic job shop scheduling environment, we may not know how many operations will be scheduled. Thus, it is impossible to divide the scheduling horizon by allocating each rules to a fixed number of operations. Instead, it seems more fair to assign a dispatching rule to each machine. The system can evaluate this combination of dispatching rules (which we call a policy) by measuring the performance of the schedule that is created by using this policy. Changing the policy by modifying the dispatching rule on one or more machines changes the schedule created. In order to demonstrate this idea, consider the following fourjob, threemachine problem and two policies used to dispatch the jobs wailing for processing (see Figure 5.1 for the task processing times).
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183 Job
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184 Policy 1 : lEDD, EDD, EDD] Schedule: Makcspan = 21. Ji J^ 9 10 11 12 17 21 In Policy 2, the altered dispatching rules, [SPT, SPT. SPT], change the selection of jobs. On M, the job Ji has the shortest task processing time and is thus preferred. On M2 the job J^ has the shorter task processing time and is scheduled first. Policy 2 : [SPT. SPT, SPT] Schedule: Makespan = 18. h
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185 5.3.2 A Genetic Algorithm for Global Scheduling Wc will describe in Ihi.s section the Genetic Algorithm for Global Scheduling (GAGS). This search procedure is the engine that finds a good schedule. After discussing it in this section, we will turn away and address the scheduling system that it drives. GAGS consists of two primary components: the genetic search and the model of the shop floor. The interaction between these two functions is outlined in Figure 5.2. The genetic algorithm starts with a number of policies. (Each policy is a combination of dispatching rules, one rule for each machine. See Table 5. 1 for a list of the dispatching rules that were used in the global scheduling procedure.) Each policy is evaluated by the schedule that is created if the jobs in the shop are dispatched according to this piolicy. The model of the shop floor is employed to build this schedule from the set of shop, job, and process information. GAGS Genetic Operator Parents Population of Policies Best Policy Schedule Offspring Policy Offspring Fitness Input Data Scheduling Model Performance Schedule Figure 5.2. GAGS Scheduling Model Interface
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186 Table 5.1. List of dispatching rules considered. SPT Minimum Setup EDD Shop: LateSetup MatchEDD Slack per remaining operation Modified Due Date Longest Processing Time Shortest Remaining Processing Time Earliest Finish Time First In First Out Least work in next queue EDD Lookahead to bumin SPT Lookahead to bumin Lookbehind from bumin The genetic algorithm creates new offspring policies by combining two policies in the population (crossover) or changing a solitary parent (mutation). For instance, consider again the above threemachine problem and the two policies [EDD, EDD, EDD] and [SPT, SPT, SPT). These policies were evaluated by creating schedules for all three machines. Their evaluations (on the makespan objective function) were 21 and 18. If we split each policy after the first rule and link the beginning of the first policy with the end of the second policy, we create the offspring policy [EDD, SPT, SPT], which the reader can confirm creates a schedule with a makespan of 17. Policy 3 : [EDD, SPT, SPT] Schedule: Makespan = 17. Jl
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187 a scheduling system create plans that match reality. For our project, the model was a deterministic simulation of the shop. It uses as input a set of resources that correspond to the equipment and staffing of the test area, a set of jobs that correspond to the lots that need processing in the test area, and expert knowledge about the production processes. The policy provided by the genetic algorithm sequences the jobs at each resource. Events in the simulation correspond to resources beginning work on a task and resources finishing tasks. Other events are added as necessary to control the simulation of the producUon process. Note that the search is not affected by the complexity of the scheduling problem being solved. The genetic algorithm can find solutions to classical problems and to problems with prcviouslyunconsidered characteristics. The genetic algorithm can take advantage of complexity by including dispatching rules designed for use in that environment. 5.4 Global Job Shop Scheduling In this section we will discuss the manufacturing process and scheduling needs of the semiconductor test area for which we are trying to create good schedules. As we will see, the problem is quite difficult. This will lead us into the description of the design, implementation, and contributions of our global job shop scheduling system. 5.4. 1 The Semiconductor Test Process The manufacturing of semiconductors consists of many complex steps. This includes four primary activities: wafer fabrication, probe, assembly, and test. This research is concerned with the last facility. Although the routes of lots through the test process vary significantly over the many different types of products, general trends can be described (see Section 2.1 of this dissertation). The area under study tests commercialuse semiconductor devices and consists of two domains: one dedicated to Digital products and another to Analog products. We directed our work toward the Digital side, where there are over 1400 product lines. The test area has three
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188 shifts per day. five days a week. The resources in the Digital side include nearly sixteen electrical test heads and a dozen brandcrs. The staffing on a normal shift consisted of nearly fifteen personnel in eight areas. The product mix changes continuously, and staffing and machine resources often change to rcfiect this. Although a typical product may have a route that consists of over 30 numbered operations, only a fraction of those are steps where significant time and resources are required. In our job shop model of the lest area, we concentrated on the following operations: electrical test (room, low, and high temperature), brand, bumin, bumin load and unload, visualmechanical test, and document review. Due to the reentrant nature of the process, an average lot will have twelve steps that need processing. Most of the operations in the semiconductor test area are electrical or physical tests, performed in an automated fashion or by hand. During these tests, each device in a lot must be examined. Thus, the processing times depend directly upon the size of the lot. Also, the products in the lest area have different package types. A package is the shell in which the semiconductor chip resides. These packages can be plastic or ceramic and come in many different sizes and styles. The package of a product also influences how the devices are tested. In the job shop model, a work center for each of the resources in the shop (except the bumin ovens) is a tester, machine, or operator that needs to be scheduled. Each type of work center has a list of the operations that can be performed at the station. Distinct resources that perform the same operations are modelled as separate stations, each with a queue of jobs that next need processing at that station. These queues are sequenced by the dispatching rule for that station. Two unique operations in semiconductor testing are electrical test and bumin. Electrical test operations have sequencedependent setups, and bumin is a bulkservice process. The sequencedependent setups at electrical test are a result of the test requirements. First, a handler must be attached to the test head to automatically feed the devices in the lot. However, due to the physical attributes of the packages, different handlers arc required for different packages. Testing also occurs at various temperatures, which requires additional equipment. The
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189 amount of setup is thus affected by the setup of the previous test, since It will be necessary to put into place new equipment only when the previous setup was different. The other unique operation is bumin. At this operation, the devices in the lot are placed into one of a number of ovens for a fixed period of time. These times range from 24 hours to over 4 days. Many lots can be bumedin at once, and the capacity constraint is more often on the availability of bumin boards than on the space in the ovens. Additionally, after the boards are removed from the oven, the devices are unloaded from the boards, and the lot must undergo an electrical test within 96 hours to locate any faulty devices. The combination of bulk service, secondary capacity constraints, and operation deadlines requires special modeling. Access to bumin is controlled by the bumin load work center, since each lot must be loaded onto bumin boards before being placed in the oven. If sufficient boards are not available, the lot cannot be processed. After bumin load, the lot moves directly to the bumin oven and begins the bumin period. The factory control system provides information on which day the lot should be removed from bumin. On that date, the lot is scheduled for unload, and the bumin boards become available for another lot. As Lee et al. (1993) mention, the test area has five features that distinguish it from classical job shop scheduling: 1. Sequencedependent setup times and reentrant product flows, 2. Machines with different scheduling characteristics, 3. Complex interactions between machines, 4. Dynamic production environment, and 5. Multiple, conflicting objectives. These characteristics make effective scheduling of the semiconductor test area a difficixlt task. They also encourage us to design a global scheduling system that can search for good solutions without being obstructed by complexity.
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190 5.4.2 The Pn?viou.s Sclicdulint; System In this subsection we briefly discuss how scheduling was done previously in the test area using dispatching rules and some drawbacks to this system. (The development of this system is described in Lee ci al. 1993.) The test area was using a set of dispatching stations to sequence the lots awaiting processing at each work center at the beginning of the .shift. During the shift, this ordered list of lots was updated by using the dispatch station to resequence the loLs in the queue, since more lots may have arrived. The dispatching stations and rules are part of a software module called ShortInterval Scheduling (SIS). SIS is a component of WORKSTREAM, a product of Consilium, Inc. (Consilium, 1988). WORKSTREAM is the test area's computerintegrated manufacturing (CIM) system and is implemented on the corporate VAX mainframe. Transactions such as processing a lot or beginning a setup are logged into WORKSTREAM, which maintains information about the current status of each lot and each machine. This information is used by SIS to sequence the lots waiting for processing. In addition to WORKSTREAM, the company uses higherlevel systems such as Activities Planning & Dispatching (AP/D) to match the production lots to customer orders and IMPReSS to determine the amount of product that each area of the company (including the test area) should be produced each week. All of these systems provide critical data about the lots to be processed and the characteristics of the test area. We will need this information to model the test area. While SIS has led to better scheduling in the test area, it has a number of disadvantages related to the structure and capabilities of the CIM system. The primary obstacle is that dispatching rules are making local decisions (even when they attempt to lookahead or lookbehind). Thus they are unable to know how their decisions affect the work at other areas in the test area. And they are unable to advantage of information that may lead to better dispatching
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191 decisions. In addition, the dispatching rules arc unable to use information about processing and setup times. Since the implemented dispatching rules cannot use global information, we began to consider job shop scheduling procedures. While a number of optimization procedures have been suggested (see Section 2), these have concentrated on the classical job shop problem of minimizing makespan. In particular, the more wellknown procedures use the disjunctive graph to represent the problem. Unfortunately, manufacturing environments like semiconductor lest area often have more complicated problems that require more complicated scheduling models. We wanted to make use of global information in a complex production process and to search for a better schedule. Therefore, we decided to implement the genetic algorithm for global scheduling. We described the primary characteristics of this procedure in Section 3. In the remainder of Section 4 we will describe the development of our global job shop scheduling system. 5.4.3 Scheduling Needs Over the course of a number of years, the scheduling system in the semiconductor test area progressed from manual dispatching to an integrated rulebased decision support system. As we mentioned above, this system had significant limitations, since it was based on dispatching rules. However, the planners in the area need to know in what order the lots waiting at a station should be processed, and this system gave them this information. They would also like to know when these lots will be completed. In addition to these tactical decisions, the planners need to have some idea of how their limited resources (in both personnel and equipment) on the shop floor affect the performance of the shop in relation to meeting the shop goals. Finally, they need to measure the performance of the personnel on the floor. The managers of this test area have two objectives. The first is to meet customer due dates. The company stresses customer satisfaction, and the responsibility of the test area is to ship the
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192 required number of parts to the customer by the requested time. The test area is measured on the number of delinquent line items, the number of customer orders that are not shipped ontime. Thus, the most important of the two primary objectives is to minimi/.c the numberof tardy jobs. The second objective is to test as much product as possible. This goal is both positively and negatively correlated with the first goal. If the area can test product more quickly, it is more likely to meet customer due dates. However, the push to increase the quantity tested can interfere with the need to process lots ontime (since the lots with the earliest due dates may be those with the largest processing times). At the time we were working on this project, the test area had extra capacity, so we considered the objective of minimizing llowtime to be a subordinate one. 5.4.4 Scheduling System Design We have looked at some of the complexity that occurs in scheduling a semiconductor test area. These characteristics make the problem in the semiconductor test area much more difficult than the job shop problems previously considered. At the same time, however, the CIM systems in place are able to provide the type of areawide information needed to create a schedule. Therefore, we are motivated to try a new approach: a global scheduling system that uses a genetic algorithm to find good schedules in the presence of shop floor complexity. In this section we will describe the basic design features. This system includes a simulation model of the test floor and an optimization procedure that can search for schedules using a genetic algorithm. The search finds a schedule that is better than any schedule that a predefined set of dispatching rules could create. This schedule specifies the order in which the lots should be processed and the times at which the operations should be done. The system can be used to react to unforeseen events that might occur during the course of a shift by including new information and producing a new schedule; for example, a machine may fail, or an important lot may arrive. The system also serves in a decision support manner, allowing the planners to determine the effects that changing resources has upon the schedule.
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193 Through the use of fair processing limes in an accurate model, the schedule can be used as a target for the shift; the work completed by the shift personnel can be compared to the work on the schedule. The foundation for the system is the test area's extensive CIM system, which will provide the information necessary to model the test area. Finally, the global scheduling system is under the control of production planners, who use the computing power of the system to evaluate alternative plans and to create a detailed schedule for the shop floor. The most important component of a global job shop scheduling system is the model of the shop floor. Only with a valid model can a scheduling system create plans that match reality. For our project, the model was a deterministic simulation of the test area. It uses as input a set of resources that correspond to the equipment and staffrng of the test area, a set of jobs that correspond to the lots that need processing in the test area, and a set of dispatching rules that sequence the jobs at the resources. Thus, the model creates a schedule for the current scenario. This is the central relationship in the system. Because the schedule depends upon the dispatching rules, the optimization procedure in this global scheduling system is a search of the combinations of dispatching rules in order to find a good schedule (one that meets management goals efficiently). A genetic algorithm is employed to search over the rules, evaluating each set by running the simulation with those rules and measuring how well the schedule performed at keeping jobs ontime. This part of the system is called the Genetic Algorithm for Global Scheduling, and it is discussed in Section 5.3 of this chapter. 5.4.5 Information Requirements The scheduling system makes use of data from a number sources inside and outside the test area's CIM system. In this subsection we will mention the types of information that are used in the system. See Figure 4 for a diagram of how the primary data structures interact.
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195 The project leum also collcclcd data on processing times and lest area resources. The collection of processing times will be discussed in Section 4.7. The resources files matches each type of resource (tester, brandcr) to the operations that it can process. Finally, the user controls other information needed for the scheduling system, including the current test area resources (how many of each type), the dispatching rules to use, the breaks schedule, the status of any down machines, and the arrival of any new hot lots. 5.4.6 Implementation of Global Scheduling The implementation of the global scheduling system followed the design of the system and the collection of data sources. In this subsection we will describe how we implemented the system and how the test area creates schedules. After creating the scheduling model and the genetic algorithm offline, we installed the basic programs on the corporate mainframe. We began testing the system and developing the utilities that collect the necessary job, shop, and process information. Working closely with the test area personnel, we began to run schedules each day. Feedback from these initial schedules led to improvements in all parts of the system. A technology transfer session formally introduced the system capabilities to the test area personnel, and soon they began to run the system themselves. The project team created user interface programs and documentation so that the test area would be able to use, understand, and maintain the system. The scheduler perfomis shift scheduling for the Digital side of the test area. Output from the system includes the shift schedule, which lists each work center, the lots to be scheduled, and the approximate start and finish times for each operation. Also available are postshift performance reports that compare the production of the test area to the schedule. User interface programs make it easy for the users to collect and modify data, create a schedule, view the output, and create performance reports.
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196 Creating a schedule consists of a number of steps. First, the current status of all of the lots in the lest area is determined from the prcshift lot status report. Nondigital lots and lots in a stores operation arc ignored. Second, the lot status information is combined with the PROP file, which converts lots into jobs. The job consists of relevant lot information and a number of operations. The lot information includes product class, due date, and lot quantity. For each operation a processing time is computed along with a remaining cycle lime. Third, the user must check the current shop settings on resources, dispatching rules, and breaks. The user is shown the default settings and has the opportunity to make any changes. Fourth, the schedule is created using the job, shop, and process information collected to this point. This schedule can be created using the genetic algorithm, which searches over combinations of dispatching rules, or the dispatching rules which the user chose. In the latter case, no search is performed; the scheduling model uses the user's rules and outputs the schedule that these rules yield. The genetic algorithm begins with a population of randomlycreated policies. The genetic operators shift the population towards more successful policies (measured on their ability to create a schedule with more ontime jobs). The algorithm stops after a fixed period of time; this time was selected after experimentation into the tradeoff between schedule quality and search effort. In addition to creating a shift schedule, the system creates a number of other files that are used to drive pcrfonmance reports. There are three performance reports: shift summary, daily summary, and daily detailed. The summary reports compare what was done with what was scheduled. The scheduled operations for each machine are compared to the operations that have been those lots. The detailed report compares processing times. The scheduled processing times are compared to the actual times that are derived from daily lot history files. If the user makes no changes to the default shop settings, the entire process can take as little as ten minutes. This compares to the 15 minutes necessary in SIS to sequence just the lots
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197 waiting at one work center. Therefore, the process can be done as part of the prcshift planning, and the user has ample opportunity to update any data and to experiment with different shop settings. The system is currently being used by the test area personnel, who will soon be expanding the system to include the Analog side of the floor. After discussing some of the issues that our implementation raised, we will describe the contributions of our system. 5.4.7 Implementation Issues Researchers engaged in projects like this one often encounter difficulties that are considered in no classroom and occasionally learn things that are in no textbook. Processing times . One primary problem was a result of our attempt to model an uncertain process with a deterministic procedure. We wanted to use processing times that were realistic but that also set fair targets for the test floor. Thus, we needed good estimates of what the processing times should be. The collection of these estimates was an important concern. We had historical data from the factory control system, which monitors when each lot begins and ends each operation. We began with an average for each operation, but this average ignored the variability in processing time due to lot quantity. We then took the historical data for the subset of products and fitted a linear regression to the data in an effort to predict processing time from lot size. The regression was run for different package types in order to remove another source of variability. Perfection . We learned that perfection is an unattainable goal in a setting as complex as semiconductor test. However, we also learned that falling short of perfection is sufficient if our system improves the ability of the facility to satisfy customer demand efficiently. From the beginning, management knew (better than the project team did) that we could not hope to capture all of the activities that occur. This attitude allowed us to build a significant model instead of being overwhelmed by the complexity of finding an optimal solution. With modeling and controlling the test area perfectly beyond our reach, we concentrated on developing a system
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198 which meets ihc needs of the test area planners and provides them with a tool which they can use to intelligently manage their facility. Simplicity . A significant feature of the implemented system is the ability of production planners to modify data related to resources and other factors. This called for a number of modules that could manipulate the data without requiring undue effort from the planners. These modules were userfriendly (easy to understand and failsafe), flexible routines that the planners felt comfortable using. We feel that this has contributed to the success of the implementation. Timeliness . Next to the estimates of processing times, accurate information about lot status was the hardest data to gather. Since there are hundreds of lots in the facility's inventory, determining which lots are wailing at which stations is not a trivial task. The system depends upon WIP extracts that were run from the factory control system before each shift. Although the planners have the ability through the factory control system to check on the status of individual lots, it was not possible to develop procedures to gather the status of the lots during the middle of the shift. Because the system could not gather the current lot status during the shift, it could only react to unforeseen events by rebuilding the original schedule from the beginning of the shift and incorporating the new information about machine breakdowns and lot arrival. Searches to find good schedules were not possible. Optimiy.ation procedure . A number of other researchers continue to work on ways to solve the job shop scheduling problem. Although we cannot guarantee that our heuristic space genetic algorithm is the best procedure for finding a good shop schedule, a search over dispatching rules seemed to be an ideal approach for the needs and requirements that we faced, however. The system makes use of the scheduling model and a separate genetic algorithm program. The two procedures are distinct modules developed and modified independently. This gave us a great deal of ficxibility as we began to test and implement the scheduling system. We chose the genetic algorithm to gain the power of its parallelism and its simplicity and to avoid the problems
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199 of local searches. The test area personnel could understand how policies combined to form new policies and how these policies could be used to create schedules. Due to the impossibility of fmding an optimal .schedule to the dynamic job shop scheduling problem, it is sufficient for our heuristic space search (an approximation algorithm) to determine a quality schedule quickly. 5.4.8 Contributions of Global Scheduling In this section we will discuss the benefits of our implementation of the global scheduling system. While primarily unquantifiable, the gains are real. The test area now has a tool that can substantially improve scheduling with a combination of global information, detailed scheduling model, genetic search for good schedules, reaction to unforeseen events, and shortterm performance reports. The global system has a number of strengths compared to SIS, the previous scheduling system. As a centralized procedure, it can make use of information from around the test area including processing times, queue lengths, current setups, resource availability, and job arrivals. The genetic algorithm searches for a schedule that has more ontime lots than a schedule created by any fixed set of dispatching rules. As a shift scheduler, the system gives the test area a plan that shows how work at one area depends upon work at another and that can be used to measure the performance of that shift. In fact, the ability to accurately model the test area and compute a shift schedule was just as important to the test area as the ability to find better schedules. The simulation component of the system allows the test area plaimers to forecast how modifying the level of available resources will affect the output of the test area. The system can also react to certain unforeseen events that may necessitate a change in the schedule. As mentioned earlier, the product mix in the semiconductor test area is constantly changing. This variety is a significant factor on the lest area performance. Thus, we are unable to measure any longtcnm quantitative benefits. The test area managers have expressed their
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200 confidence that the system will lead to improvements in the performance of the test area through improved schedules and planning tools. 5.5 Chapter Summary In this chapter we have described the development of a global scheduling system that uses a genetic algorithm to find good schedules. This system has been successfully implemented in a semiconductor test area. Controlling the test area is a complex, dynamic job shop scheduling problem where it is difficult to meet the management objectives of satisfying customer demand ontime and increasing throughput. Our scheduling system uses the extensive data available in the CIM databases in order to simulate the operation of the test area. This system uses a genetic algorithm to search for combinations of dispatching rules that yield schedules that are better than the sequencing that could be done with fixed dispatching rules. The primary practical accomplishment of this research is the implementation of an advanced job shop scheduling system in a manufacturing environment. Moreover, this implementation makes use of a new heuristic procedure that searches the combinations of dispatching rules to find a good schedule. Thus it is able to adapt each shift to changing conditions in the jobs to be scheduled and the shop resources. This approach could be extended to any manufacturing area that has a complicated shop scheduling problem and a computerintegrated factory control system that can supply the necessary data for such a global scheduling system. In fact, the semiconductor manufacturing firm where we have implemented the system is considering exporting this system to other areas. Additionally, other optimization techniques may be u.scful in finding good schedules.
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CHAPTER 6 SUMMARY AND CONCLUSIONS This dissertation has reported on a number of production scheduling problems that were motivated by considering the testing of semiconductors. The research into these topics, summarized below, adds to the body of knowledge about scheduling. This work is especially relevant to the study of the harder and broader problem of job shop scheduling. Benefits of the work include results on specific onemachine class scheduling problems, results on threemachine lookahead problems, and the use of genetic algorithms and new search spaces on different types of scheduling problems. 6.1 Onemachine Class Scheduling Problems The research on the three onemachine class scheduling problems has yielded a number of results. Most notably, the problem space genetic algorithm is a robust tool for finding highquality solutions to difficult scheduling problems. For the problem of minimizing total flowtime subject to deadline constraints (CFTS), we developed an multiplepass heuristic that makes use of an optimal property for jobs in the same class. By considering the effect of wasted setup time, it is able to find reasonable solutions. We can improve upon these solutions with a problem space genetic algorithm that adjusts the job deadlines in order to create better schedules. For the problem of minimizing the number of tardy jobs where the jobs have nonzero rclca.se dates (CSRDD), we extend a nonsetup procedure to create a heuristic for the class scheduling problem. The average performance of the heuristic is good compared to a number of other dispatching rules. A problem space genetic algorithm is able to find better solutions on some especially difficult problems where the heuristic performs poorly. 201
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202 Our extended heuristics for the problem of minimizing the total flowtime of jobs with nonzero release dales (FTSRD), were outclassed by a decomposition procedure and a problem space genetic algoriilim. \Vc developed a number of dominance properties for use in a branchandbound technique. 6.2 Lookahead Scheduling The threemachine problems that have been studied show that lookahead rules can perform better than standard dispatching rules. For the problem of minimizing makespan, the interleaving of the Johnson sequences is able to provide nearoptimal solutions. The worst case relative error of this heuristic is fifty percent. There are, however, special cases of the problem that can be solved in polynomial time. The problem of minimizing the total flowtime is more difficult. There do exist special cases where the lower bounds can be achieved. Lookahead rules that consider the queue at the secondstage machines are able to find good solutions. The last problem was that of minimizing the number of tardy jobs. Again, special cases exist where optimal solutions can be easily found. Lookahead rules were able to find better schedules than other sequencing rules. A problem space genetic algorithm found improved solutions. 6.3 Searching for Job Shop Schedules This research has investigated the development of a procedure for the job shop scheduling problem. The genetic algorithm makes u.se of known heuristics (dispatching rules) but increases their effectiveness by searching over combinations of rules to find a good schedule. This procedure provides a way to find good schedules under any objective function and in any scheduling environment, since it makes use of a detailed shop floor scheduler. These characteristics make this procedure unique.
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203 In addition, this procedure has been implemented as part of a global scheduling system for a semiconductor test area. In this environment, it creates realtime schedules for the next shift using information about the current status of the lots, the current resources in the shop, and the manufacturing process. The system includes not only the simulation model and the genetic algorithm but also utility functions for the collection of data from multiple sources and the generation of performance reports. In addition, the system can respond to unforeseen events, and the test area planners can use the system to determine the effect of changing the shop resources. 6.4 Conclusions The scheduling of a manufacturing process is a complicated problem. The static job shop scheduling problem is incredibly difficult to solve, and no system has been able to optimize the scheduling of an entire dynamic job shop, which is the environment present in many manufacturing facilities. Thus, continued research into procedures that can find good schedules is necessary. Many researchers have studied thiis problem, introducing systems which range in scope from companywide planning to machine scheduling. This research is concerned with efforts at the level of the shop floor and machine. This research investigated production scheduling problems that are motivated by semiconductor test operations and are expected to hold widely in other production environments. Of particular concern are those problems that occur in the testing of semiconductor devices. Let us now take a moment to provide some pcrsj^cctive. This dissertation is concerned with two types of operations research: management science and management engineering (the terms of Corbett and Van Wassenhove, 1993). Management science is the discovery of new results that add to the body of knowledge about a subject. Management engineering (a less active area of the field) is the solution of a practical problem by modifying existing tools or by using existing tools in original ways.
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204 The scicntinc coniribulions arc clear. While much research in the areas of scheduling and semiconductor manufacluring has been pcrlormed. this research investigated a number of previously unstudied problems and methods. Our research into these problems has yielded a number of useful properties and effective heuristics. This research has shown that smarlandlucky searches and the new problem and heuristic spaces can be used for the problems under consideration. Problem space genetic algorithms can find good solutions to machine scheduling problems. For the job shop scheduling problem, a genetic algorithm can search combinations of dispatching rules and u.se a shop floor simulation to determine a good schedule. Additionally, this research lies together separate problems in an effort to improve the scheduling of the manufacturing process being studied. This is an engineering question. The cooperation of the semiconductor test facility motivated research into the problems of an actual system and provided an opportunity to implement our solution procedures. (This docs not preclude the potential of our approach to solve problems in other manufacturing environments.) Our global job shop scheduling system (with its detailed simulation model and heuristic space genetic algorithm) is a new technique for the problem of creating good shift schedules for a semiconductor test area in realtime. Since lookahead dispatching rules can be more effective than standard rules, the results of the work into the threemachine subproblems have been used as dispatching procedures for the shop floor and as pan of the job shop scheduling procedure. This research opens some chapters of scheduling that need to further pursued. While the problem space genetic algorithm is a robust procedure that finds good solutions, better solutions to particular problems may be achievable through the use of solutionspecific heuristics that can improve the schedules that the genetic algorithm constructs. Also, it may be possible to create searches that combine different spaces for other types of combinatorial problems. The set of class scheduling problems includes a number of other interesting problems; so does the set of lookahead problems. Even more interesting is the combination of these problems.
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205 Lookahead class scheduling problems may yield more insights into the job shop scheduling problem. There also exist issues in global job shop scheduling that still need to be addressed: job release, resource planning, and uncertainty in the manufacturing process. It is difficult to obtain satisfactory solutions in these circumstances using current scheduling techniques. The general approach of this research (problem and heuristic space searches) can be applied to almost any difficult problem. It may be profitable to consider this approach for problems where good solutions are hard to find. Why investigate production scheduling problems motivated by semiconductor manufacturing? Because organizations and individuals often have difficulty meeting their goals efficiently, and this dissertation, which has applications beyond the problems contained herein, offers some ways to solve people's problems.
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BIOGRAPHICAL SKETCH Jeffrey W. Herrmann has a degree in applied malhcmalics from Georgia Tech and is a Ph.D. candidate in the Dcparimeni of Industrial and Systems Engineering at the University of Florida. He held an NSF Graduate Research Fellowship from 1990 until 1993 and has been working on an applied research project with Harris Semiconductor. His research interests include operations research, production scheduling, and heuristic search. 219
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I certify thai I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. pY~^^^ L <2J^ ChungYee Lee, Chairman Associate Professor of Industrial and Systems Engineering I certify that I have ixrad this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. izmga ^y ^ fessor of Industriaf;arid Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequatp.jin sppe and quality, ^as a dissertation for the degree of Doctor of Philosophy. Sherman Bai Assistant Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gary K Profcssi of Decision and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /^elclik Ercn Professor of cision and Information Sciences This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1993 fn Winfrcd M. Phillips Dean, College of Engineering Karen Holbrook Dean, Graduate School
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J 1262 05234 5005 MARSTON SCIENCE LIBRARY LD 17S0 SCIENCE LIBRARY Ir,/ HECKMAN BINDERY INC. JULY 94

