Configuring cell systems to handle variable demand

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Configuring cell systems to handle variable demand
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Thesis (Ph. D.)--University of Florida, 1996.
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Includes bibliographical references (leaves 178-199).
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by Jeffrey Schaller.
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CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND


By

Jeffrey Schaller



















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


1996















ACKNOWLEDGMENTS


I would like to thank Dr. Selcuk Erenguc, my

supervisory committee chairman, for his invaluable guidance

and patient encouragement throughout my years in the

program. My supervisory committee cochairman Dr. Asoo J.

Vakharia has also provided invaluable guidance during the

past year and I am thankful that he brought his expertise in

cellular manufacturing to our department. I also would like

to thank Dr. Harold Benson, Dr. Patrick Thomson, and Dr.

Chung-Yee Lee for serving on my supervisory committee and

for the fine instruction they provided in the classes I

attended. I would also like to thank Dr. Antal Majthay for

the direction he provided while supervising my assistantship

assignments.

Finally, and most importantly, I want to thank my

mother and father for the love and support they have

provided my entire life. Anything good that I have ever

accomplished is a result of having the best parents.










TABLE OF CONTENTS





ACKNOWLEDGMENTS. . ii

LIST OF TABLES vii

LIST OF FIGURES. ix

Abstract x

Chapters

1 INTRODUCTION 1

A Brief Description of Cellular Layouts 1
Product Line Layout 1
Process Layout. 3
Cellular Layouts 4
Cell Formation 7
Early Cell Formation Techniques 8
Later Cell Formation Techniques 9
The Impact of Dynamic Demand on Cell Formation
Decisions 11
Dynamic Demand's Impact on Cellular Layouts 11
Strategies to Handle Dynamic Demand while
using a Cellular layout 16
Current Practices 18
Automotive Parts Manufacturer. 18
Enclosure Manufacturer 20
The Problem and its Terminology. 23
The Organization of this Proposal 24

2 REVIEW OF CELL FORMATION LITERATURE 26

Introduction 26
Procedures Based solely on a Part-Machine Incidence
Matrix 30
Objectives and Constraints. 31
Frequently used objectives 31


iii









Frequently used constraints. 34
Other objectives . 35
Methods 37
Manual Identification. .. 38
Matrix Manipulation 39
Clustering using Similarity Coefficients 40
Graph Theory. 42
Mathematical Models 43
Other Methods 44
Evaluation of Procedures 44
Procedures Which use a Part-Machine Incidence Matrix
and Incorporate Data to Provide Weights for Parts
or Machines. 46
Objectives and Constraints. 47
Methods 48
Routing Based Procedures Without Machine Capacities 49
Operations Sequences. . 50
Alternative Routings. . 51
Procedures Which Incorporate Setup Information. 52
Procedures Which Incorporate Machine Capacities 55
Model 1 55
Model 2 57
Methods 59

Procedures That Consider Multiple Scenarios for Part
Volumes. 60
Other Procedures 63
Summary 64

3 MODELS 66

Introduction 66
Model I. 67
Description of Model I . 67
Model I Formulation . 69
Description of Model I Equations. 71
An Example of Model I . 74
Model II 77
Description of Model II. 77
Model II Formulation. . 79
Description of Model II Equations 81
An Example of Model II . 83









4 SOLUTION PROCEDURES.


Introduction .
Model I Heuristic .
Lower Bounds for Model II. .
Linear Programming Relaxation.
Lagrangean Relaxation .
Linear Programming Relaxation Tightened
Adding Valid Inequalities
Valid Inequality Set 1 .
Valid Inequality Set 2 .
Valid Inequality Set 3 .
Heuristics for Model II .
Heuristic 1. .
Description of Heuristic 1 .
Formal Procedure for Heuristic 1
Discussion of Heuristic 1 .
Heuristic 2. .
Heuristic 3A .
Description of Heuristic 3A.
Formal Procedure .
Discussion of Heuristic 3A .
Heuristic 3B .

5 DESCRIPTION OF EXPERIMENTS AND DATA


S 87
88
91
91
92
by
94
94
94
96
S 97
98
98
S 98
100
101
S 101
S 105
.105
.107
S 110
111


114


Introduction 114
Experiments 115
Experiment 1 Data . 116
Part-Machine Incidence Matrices 117
Processing Times . 119
Part Demand. 120
Inventory and Backorder Costs. 120
Machine Capacities and Number of Machines of Each
Type 121
List of Potential Cells. 122
Cell Production and Intercell Production Costs 123
Summary 124
Experiment 2 Data . 125

6 RESULTS. 127


Results of Experiment 1


127









Lower Bounds for Model II
Heuristic Results for Model
Discussion of Results
Lower Bounds. .
Heuristics .
Results of Experiment 2 .

7 CONCLUSIONS AND FUTURE RESEARCH.

Conclusions .
Future Research .

APPENDICES

A FORMULATION OF MODEL I EXAMPLE

B FORMULATION OF MODEL II EXAMPLE

C PROBLEM 1 DATA AND SOLUTION
Problem 1 Data .
Problem 1 Solution .

LIST OF REFERENCES. .

BIOGRAPHICAL SKETCH .


S 127
I 130
. 131
131
S 136
. 138

S 142


S 142
. 144




147

S 149

S 151
. 151
S 171

S 178

S 200












LIST OF TABLES


TABLE pace


1. Part Routing. 5

2. Cellular Layout. 5

3. An Example of a Part-Machine Incidence Matrix. 8

4. Capacity Consideration . 10

5. Part Family Machine Requirements.. 12

6. Load on Machine Type by Period. 13

7. Average Load. 13

8. Part-Machine Incidence Matrix for Model I Example. 75

9. By Part Demand.. 75

10. Machine Data. 75

11. Potential Cells. 84

12. Production Costs 85

13. Part-Machine Data Sets . 118

14. Summary of Experimental Problems 124

15. L.P. and Lagrangean Relaxations 128

16. Valid Inequality Relaxations 129

17. Heuristics 1 and 2. 132

18. Heuristics 3A and 3B 133









19. Problem 1 Results . 140

20. Problem 2 Results 140

21. Problem 1 Selected Cells . 172

22. Part/Cell Processing . 173









LIST OF FIGURES

FIGURE page




1. Product Line Layout . 2

2. Process Layout 3

3. A Process Layout. 6

4. A Cellular Layout.. . 6

5. Family 1 Load by Period.. .. 14

6. Family 2 Load By Period.. 14

7. Total Load by Period.. 15















Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND

By

Jeffrey Schaller

May 1996


Chairperson: Dr. S. Selcuk Erenguc
Major Department: Decision and Information Sciences

This research addresses the problem of configuring cell

systems for environments that experience variable demand.

Past cell formation models consider only average product

demand to determine configurations of cells. Short term

fluctuations in demand may cause a cell configuration that

would appear to be good, when considering only average

demand, to perform poorly because during some periods

machines of a certain type may be under utilized in one cell

while machines in another cell may not be able to produce

the required products. For these reasons, a methodology for

configuring cell systems in environments that experience

variable demand is needed.









In this dissertation, two mathematical models are

formulated. The first model is used to develop a list of

potential cells, and the second model is used to select

cells to include in a shop. Since both models are

computationally complex, this research develops heuristic

procedures to solve the models. Given that other approaches

can also be used to develop a list of potential cells, the

primary focus of this research is on the second cell

selection model. Lower bounding procedures are proposed for

this model.

In order to test the effectiveness of the lower

bounding and heuristic procedures for the second model,

fifteen problems from the literature were modified and

solved optimally. The lower bounding and heuristic

procedures were then used on these problems, and the lower

bounds and objective function values are compared to the

optimal objective function values. A second experiment was

conducted, in order to test the effectiveness of the lower

bounding and heuristic procedures on larger problems. Using

two larger published data sets the results are compared to

the results obtained using the lower bounding and heuristic

procedures.















CHAPTER 1
INTRODUCTION



1.1 A Brief Description of Cellular Layouts



Cellular Manufacturing is a manufacturing application

of the group technology philosophy where groups of processes

are dedicated to sets of products or parts. The associated

layout (referred to as cellular layout) is a form of

production layout that combines features of two traditional

forms of plant layout: product line layout and process

layout.

1.1.1 Product Line Layout


In a plant with a product line layout, machinery and

equipment are dedicated to a specific product, and thus,

machines needed to manufacture each product are laid out in

a way that facilitates efficient material flow. The

advantages of product line layout are fast throughput times,

efficient material handling, simplified scheduling, and











control and low work-in-process inventories. The

disadvantages of product line layout are that it requires a

high level of investment in specialized machinery and

equipment, and it lacks flexibility because only one type of

product can be manufactured without incurring a high

changeover cost. This type of layout is typically used for

high volume, standardized products. Examples are

automobiles, television sets and appliances. An example of a

product line layout is shown in Figure 1.





,- Product Flow

[] Machines


S-B6 > Product 1



)I.> Product 2



J A- N .-> Product3


Figure 1. Product Line Layout.











1.1.2 Process Lavout


A process layout groups machines which perform similar

functions into departments. Many different types of products

may use a specific machine type and each product may visit

different machine types in various sequences. An example of

a process layout is shown in Figure 2.





--+ Product Flow

l Machines


Product 3

SProduct 2









Product 2


Figure 2. Process Layout.


The products in this example have the same routings as

the products in Figure 1 but notice that the material flows

are much more jumbled than in Figure 1. The advantage of a

process layout is that many different types of products can











be manufactured on the same set of machines which leads to

flexibility in the utilization of machinery. The

disadvantages of a process layout are that material handling

costs are high, material flows are complex which causes

planning and control to be difficult, flow time is long,

setup costs are high and work-in-process inventory is high.

Typically low volume or nonstandardized products are

produced using a process layout. Examples include (Hyer

1984): aerospace, agricultural machinery, business machines,

control devices, diesel engine assemblies, machine tools,

and machined parts.

1.1.3 Cellular Layouts


Cellular layouts are hybrid layouts which combine

features of both product and process layout. The objective

of implementing a cellular layout is to enable a

manufacturer of low volume specialized or custom products to

obtain some of the benefits of product line layout while

maintaining some of the flexibility of a process layout.

This is done by forming product families which consist of

products that require the same types of machines (preferably

in the same sequence). A machine of each machine type (or











more than one if required) required to produce the products

in a family is put into a cell and is dedicated to producing

a family of products. As an example consider the products

shown in Table 1.


Table 1. Part Routing.

Product

1

2

3

4

5

6


Table 2.


Machine Sequence

1 4 5

5 4 1

2 3 5

2 3 4 5

1 2 3 4

1 3 2 4


Cellular Layout.


Machines

1,4,5

2,3,4,5

1,2,3,4


Products

1,2

3,4

5,6


Figure 3 (on the following page) shows the process

layout for producing these products. Figure 4 shows the


Cell

1

2

3












resulting product flows if the three cell layout in table 2

is used.



Process Layout
--p Product Flow
1 Machines


Product 2


- Product 3
- Product 4



> Product 5

Product 6


Figure 3. A Process Layout.





Cellular Layout
*-0 Product Flow
] Machines


Figure 4. A Cellular Layout.









7

Notice the simpler product flows in the cellular layout

vs the process layout. Creating cells in this manner enables

material handling costs to be reduced and materials handling

flows to be simplified. This in turn can lead to reduced

flow time, reduced work-in-process inventories and

simplified planning and control. Setup time and cost for

parts manufactured in a cell can typically be reduced by

developing specialized procedures for setups required in the

cell. Quality improvement is also usually a benefit because

cell operators have more experience manufacturing a specific

set of products.


1.2 Cell Formation



In order to implement cellular manufacturing cells must

be formed. The cell formation problem includes the following

decisions. How many cells should there be? Which machine

types should be included in a cell? How many machines of

each type should be included in a cell? Which parts should a

cell produce?











1.2.1 Early Cell Formation Techniques


Most of the early cell formation algorithms used some

form of part-machine incidence matrix. In a part-machine

incidence matrix machines are represented as rows and parts

are represented as columns in the matrix (or vice versa).

Each element in the matrix is '1' if part j requires machine

i for processing and '0' otherwise.


Table 3. An example

Machine
Type

A

B

C

D


a part-machine incidence matrix


Table 3 shows an example of a part-machine incidence

matrix. In this example part 1 requires machine types A and

C but not B and D, and part 5 requires machine types A, B,

and D but not C.

The algorithms use the part-machine incidence matrix as

input and then use various techniques to break up the parts











and machines into cells. These techniques include manual

identification (Burbidge 1963); clustering using measures of

similarity (McAuley 1972, Carrie 1973); graph theory using

measures of similarity (Rajagopalan and Batra 1975,

Chandrasekharan and Rajagopalan 1986, Vannelli and Kumar

1986, and Kumar, Kusiak, and Vannelli 1986); and matrix

manipulation (King 1980, King and Nakornchai 1982, Chan and

Milner 1981, and Chandrasekharan and Rajagopalan 1986).

1.2.2 Later Cell Formation Techniques


Many of the algorithms that were developed later tend

to be more sophisticated and are based on mathematical

models that include operational considerations in the

objectives and constraints. These models also usually

include some sort of integrality constraint. Operational

considerations that have been incorporated into algorithms

include operations sequences, machine setup similarities

between parts, part volumes, and machine capacities. A

typical integrality constraint is that the number of

machines of a machine type that are assigned to a cell is

required to be integer.











Table 4. Capacity Consideration

Part Hours required
Family on machine type 1

1 2700

2 2500

Total 5200

# of type 1 machines available 3

Capacity per type 1 machine 2000 hrs

Total capacity 6000 hrs




For an example of why capacity, volume, and machine

integrality are important considerations when making cell

formation decisions, consider Table 4. In this example two

part families have been formed and both families have parts

which require processing on machine type 1. The table shows

the hours required on machine type 1 for each family, the

number of type 1 machines available, and the capacity per

type 1 machine. If we could assign 1.5 type 1 machines to

each family, we would have ample capacity to produce both

families and everything would be fine. However, we cannot

dedicate fractions of machines to families. If we assign two

machines to family 1 and one machine to family 2, then there









11

will be idle capacity dedicated to family 1 while we do not

have enough capacity to produce family 2. If we assign two

machines to family 2 and one machine to family 1, a similar

situation arises. One alternative to this problem is to

combine both families and create a larger cell by assigning

all the machines to this single cell. Using this alternative

there is adequate capacity to produce both part families

but, because the cell is larger and the parts that are

produced in the cell are more diverse, some of the benefits

of cellular manufacturing are likely to be lost.

1.3 The Impact of Dynamic Demand on Cell Formation Decisions


1.3.1 Dynamic Demand's Impact on Cellular Layouts


Demand for a product is said to be dynamic if it varies

from period to period. Past cell formation models consider

only average product demand to determine the configurations

of cells and which products to produce in a cell. Short term

fluctuations in demand may cause a cell configuration that

would appear to be good, when considering only average

demand, to perform poorly. During some periods machines of a

certain type may be underutilized in one cell while machines

in another cell may not be able to produce the required











Table 5. Part Family Machine Requirements.


Part Family Required Machines

1 2 6 7

2 1 5 6



quantities of products. For an example consider the part

families in Table 5. Suppose there are 2 machines of type 6

available and the capacity of a type 6 machine is 2500 units

per period. The load on machine type 6 by period for each

family is shown in Table 6. Table 7 (on the following page)

shows the average load on machine type 6 for each period. If

one machine of type 6 was dedicated to producing family 1

and the other was dedicated to producing family 2 there

would be enough capacity in each cell to meet the average

load. If you look at the by-period requirements in Table 6,

however, you can see that insufficient capacity would exist

to produce family 2 while capacity would be underutilized in

family l's cell during the first three periods. During the

last three periods the situation is reversed. When looking

at the load of both families in aggregate there is enough











Table 6. Load on Machine Type by Period.


Load on machine type 6 by period

Period 1 2 3 4 5 6

Family 1 1480 2068 1256 2676 2831 3360

Family 2 2856 2914 3144 2154 738 674

Total 4336 4982 4400 4830 3569 4034


Table 7. Average Load

Average

Family 1 2278.5

Family 2 2080.5

Total 4358.5


Capacity

2500

2500

5000


capacity to produce both families during every period.

Figures 5, 6, and 7 show the load variations graphically.

Figure 5 (on the following page) shows the load caused by

family 1, Figure 6 the load caused by family 2, and Figure 7

shows the load in aggregate. The aggregate load is much more

stable than the individual family loads. This illustrates an

advantage of a process layout when there is dynamic demand.

By combining all the machines of a given type into a





















3500


Family 1
Load by period


...ii I -.


500

1 2 3 4 5 e
PklWN


Figure 5. Family 1 Load by Period.







Family 2
Load by period


Figure 6. Family 2 Load by Period.


Y


PloM













Total
Load by period




WM-




1000/


1 2 3 4 5 6



Figure 7. Total Load by Period.

department and allowing any product that requires that

machine type to use any machine of that type flexibility is

provided so that the peaks and valleys of a product's demand

can be offset by other products.

Studies have shown the adverse effects of variable

demand on a cellular layout. Flynn and Jacobs (1986, 1987)

conducted simulation studies which show that when demand

variability is high a process layout is favored over a

cellular layout in terms of mean flowtime. Garza and Smunt

(1991) showed that demand variability can cause intercell

flows in a cellular layout and this results in less stable

flow time performance as compared to a process layout.











1.3.2 Strategies to Handle Dynamic Demand While Using a
Cellular Layout


There are strategies which can be used to offset the

effects of demand variability in a cellular manufacturing

system. These include combining cells to smooth out load

variations, routing parts through alternative cells, let

parts move between cells (intercell movement), hold

inventories of parts, and incur backorders for parts.

Combining cells to form larger cells allows the large

cell to produce a greater variety of parts than the

individual small cells. Since there are more parts it is

likely that machine loads will be more stable in the large

cell. The large number of parts produced by the cell will

also cause more complex materials flows and higher material

handling cost. Therefore, there is a tradeoff that must be

considered when choosing a cell's size; stable loads in

large cells vs lower material handling cost in small cells.

Cells can also be formed so that there is more than one

cell that can produce a part. Hence, if there is

insufficient capacity to produce the part in its preferred

cell, it may be possible to produce the part in a different

cell. By having alternative cells, load variations can be









17

smoothed out while meeting product demand. The disadvantage

of this approach is that it may be more costly to produce

parts in their alternative cells and/or there may be a need

to duplicate more machines to develop these types of

configurations.

If there is insufficient capacity for a machine type to

produce all the parts required, then some parts can be moved

to another cell which has that machine type for processing.

This is called an intercell transfer. Ang and Willey (1984)

showed that low levels of intercell transfer can greatly

improve the flow time performance of a cellular layout. The

disadvantages of intercell transfer are that this requires

materials handling capability between cells, there is

increased paper work required to track part movement, and

quality could be adversely affected because of the loss of

responsibility by a single cell for the part's quality.

A final strategy that could be used is that during

periods of low demand, inventories of parts can be produced

which can be used to meet demands for parts during periods

when insufficient capacity exists in the cells to produce

all the parts required. Alternatively parts could be

backordered when capacity is not available in a cell and











then produced during a later period when there is adequate

capacity in the cell. Using inventories and backorders will

level out the load requirements in the cells, but the cost

of holding inventories or backordering parts must be

incurred.


1.4 Current Practices



In order to find out how cellular layout is used in

industry, we visited two plants that utilize a cellular

layout. The first plant produces parts for automotive

manufacturers. The second plant produces steel enclosures

that are used for industrial applications to hold electrical

circuitry. A brief description of how each plant uses a

cellular layout and how they deal with demand variability

presented next.

1.4.1 Automotive Parts Manufacturer


This plant produces parts that are major components in

the assembly of automobiles and some service parts. It uses

product line, process, and cellular layout. Major factors in

the choice of layout are maintaining machine utilization and

efficient use of space. The plant has over 20 cells that are









19

used for production. In some cases the cells are production

lines that produce one part. One cell is a small job shop

that produces approximately 150 parts. The other cells are

of varying sizes in terms of the number of machines they

contain and the number of products they produce. There are

certain processes in the plant which are strictly process

departments. An example of this is the paint department. All

parts that require painting go to the paint department

(there are no paint facilities in the cells). The reason why

this department was chosen to be a process layout is that

one oven was sufficient to handle the entire plant's

requirements.

The plant also has 20 high speed roll mills. Fourteen

of the mills are grouped together in a department and six

have been moved to cells. The roll mills which have not been

moved are large mills which perform complicated processing.

It was decided that it would be too expensive and require

too much space to duplicate these machines in multiple

cells. The mills that were moved to cells are smaller and

the capacity of the mills matches the requirements demanded

by their cells. Some of the cells were created by combining

production lines. This enabled the plant to better utilize

equipment and save space.











The plant experiences demand for products that is

variable. This is particularly true with service demand

which is very lumpy. A variety of strategies are used to

handle this variability. Many of the low volume parts,

including service parts, are produced in the large job shop

cell. The varying demands of the individual parts tend to

offset each other and help stabilize the load in the cell.

Sometimes a large order is produced partially in advance of

its due date and parts are inventoried. There are instances

when orders are rescheduled to a later date. There are also

times when production capacities in other cells are used to

perform some operations and then intercell transfers of

parts occur. When this happens, it is a disruption in the

plant because all the presses for the job must be setup at

the same time to maintain quality.

The primary benefits obtained by using a cellular

layout in this plant are reduced equipment and space,

reduced material handling cost, and reduced setups.

1.4.2 Enclosure Manufacturer


Approximately 2800 standard end items are produced in

this plant. The plant also produces some custom jobs. The

standard end items are produced and then sent to regional











warehouses. Roughly 200 items account for 90% of the units

produced by the plant.

This plant originally used a process layout. The

current layout slowly evolved to a hybrid cellular process

layout. Currently there are three cells and plans for more

cells are being formulated. The cells are designed to

produce high volume products very efficiently and are setup

as flow lines. Currently 75% of the unit volume and 40% of

the dollar volume is produced using cells. High volume

production in cells is undertaken to justify the investment

in dedicated equipment and tooling.

The demand for standard end items is variable and

seasonal while demand for custom items tends to be lumpy.

The plant responds to demand variability using a variety of

methods. A portion of the plant is still using a process

layout. This portion of the plant is generally used to

produce low volume standard items and custom items. The

demand for these items tend to be more erratic, but by

producing many items on the same equipment, the load

generated is fairly stable.

High volume items are produced in the cells and the

demand for these items tends to be fairly stable. There are









22

times, however, when some facilities in the plant are idle

while other facilities become bottlenecks. The plant also

had to invest in additional equipment to build cells that

could meet the seasonality of demand. Sometimes custom jobs

will be processed in cells if the equipment in the cell is

underutilized. At other times, a product normally produced

in a cell will be produced in the process portion of the

plant if the cell capacity is exceeded. Inventory is also

held and customer orders are backordered to help deal with

demand variability.

The following benefits have been attributed to the use

of cellular layout in the plant. 1) Material handling cost

has been significantly reduced because conveyors are used to

move materials in the cells. 2) Setups have been

significantly reduced because tools have been dedicated to

products and designed for quick setup. 3) Productivity has

increased. Labor costs have been cut in half. 4) Leadtimes

have been reduced from 4 weeks to 2 weeks. 5) WIP inventory

has been dramatically reduced. 6) Quality has been improved

due to the fast feedback provided by the cellular layout. 7)

The plant has become a safer place to work because there is









23

less material handling and simpler setups. 8) The fill rate

on customer orders has increased from 50% to 80%.


1.5 The Problem and its Terminology



Based on our experience and industry visits, the

problem we address is as follows. Given a set of parts, a

finite planning horizon with T periods, and a set of machine

types, the problem is to specify a set of cells to be

included in the shop and a production and inventory plan

that will minimize total cost. For each cell the number of

machines of each type must be specified. The set of cells to

be included in the shop can contain cells of varying sizes

in terms of the total number of machines included in each

cell. For example the shop could include several small cells

and a large "remainder cell." A remainder cell is a portion

of the shop that typically is organized as a process layout.

The production and inventory plan must specify the quantity

produced of each part in each cell during each period, and

the quantities of each part produced using intercell

movement during each period.











Each part has associated with it the following: (a) a

set of machine types that are required to produce the part

and the processing time required by the part on each machine

type; (b) a demand deterministicc) for each period during

the planning horizon; (c) a cost to hold one unit of

inventory for one period (unit inventory holding cost); (d)

a cost to hold one unit on backorder for one period (unit

backorder cost); (e) costs to produce a unit of the part in

various size cells (in terms of the number of machines

contained in the cell). (These costs are referred to as cell

production costs. It is assumed that larger cells have a

higher cost); (f) a cost to produce a unit of the part in

two or more cells (intercell movement cost). Each machine

type has associated with it the following: (a) the number of

machines of that type that are available and (b) the

capacity of a single machine of that type.


1.6 The Organization of this Dissertation



The remainder of this dissertation is organized as

follows. Chapter 2 contains a survey of cell formation

literature. Chapter 3 introduces the two models that can be









25

used to develop cellular configurations, and Chapter 4

describes solution procedures for these models. Chapter 5

describes the experimental data used to test the solution

procedures described in Chapter 4, and Chapter 6 presents

computational results of these tests. Finally Chapter 7

proposes future research directions.















CHAPTER 2
REVIEW OF CELL FORMATION LITERATURE




2.1 Introduction



The cell formation problem has attracted more academic

interest than any other aspect of cellular manufacturing.

The reason research has focused so heavily on cell formation

problem is due to the complex nature of the cell problem.

In order to design a cellular system decisions must be made

as to how many cells there should be, which machine types

should be included in a cell, and which parts should a cell

produce. Arriving at these decisions constitutes a solution

to the cell formation problem. Many aspects of the

effectiveness and efficiency of an operation will be

affected by the cell formation decisions. Examples are labor

costs, investment, leadtimes for products, supervision of

personnel, quality, and amount of paperwork needed to

control the operation. This leads to the consideration of

many possible objectives in arriving at a solution. The

26











problem is also complex because an entire machine usually

must be assigned to a cell. Because of this most

mathematical models of the problem are integer programming

models which are difficult to solve.

The procedures that have been developed to solve the

cell formation problem are diverse with respect to 1) the

objectives and constraints considered by the procedure, 2)

the data required as input for the procedure, and 3) the

techniques used by the procedure to solve the problem. The

first two items are somewhat related. Usually, to

incorporate an objective or a constraint into a problem,

data are required to support that objective or constraint.

For example, if a constraint of the problem is to use only

existing equipment to configure cells, then data by machine

type of the number of machines available are required. Many

procedures use objectives that are surrogates for other

objectives which if incorporated would require additional

data and cause the problem to become more complex. For

example, a possible objective when designing a cellular

system is to maintain a minimum level of machine utilization

for each machine. This objective would require data on part

volumes, part processing times, and machine capacities. Many











cell formation procedures use a surrogate measure for

machine utilization based on the percentage of parts to be

produced by the cell that uses a machine. This measure is

used in the hope that insight into machine utilization will

be gained without explicitly considering the data required

to determine machine utilization. Of course, surrogate

measures can sometimes turn out to be a poor representation

for the desired objective or constraint.

The diversity of the cell formation procedures with

respect to the items mentioned above causes the development

of a framework for reviewing the procedures to be difficult.

The most comprehensive review of cell formation procedures

to date was done by Wemmerl6v and Hyer (1986). Over 70

procedures are categorized in Wemmerl6v and Hyer's paper.

The authors classify cell formation procedures based on

which of four general approaches was used in the procedure.

In all four approaches parts and machines must be selected

for possible inclusion in the cellular manufacturing system.

An evaluation step of the proposed cells is also required.

The four approaches are 1) identify part families without

the help of machine routings (including procedures that use

group technology codes and those that do not); 2) identify









29
machining groups; 3) identify part families using routings;

4) identify part families and machine groups simultaneously.

The advantages of this method of classification are that it

is inclusive and provides classifications for procedures

which perform similar functions. Within each of the last

three approaches, however, the procedures can vary greatly

with respect to the objectives and constraints considered

explicitly by the procedure as well as the data required for

the procedure.

In this literature review we propose six categories

which are based on the data used, by a procedure, to

classify procedures. There are two reasons for classifying

procedures this way. 1) Identifying data used by a procedure

provides insight into factors that are explicitly considered

by the procedure and helps determine the strengths and

limitations of the procedure, and 2) procedures that use

similar data can be compared with one another. The six

categories that were chosen are 1) procedures that are based

solely on data that can be represented by a part-machine

incidence matrix, 2) procedures that use a part-machine

incidence matrix and incorporate data which can be used to

provide weights for parts or machines, 3) procedures that











incorporate operations sequences, 4) procedures that

consider part setup times or costs, 5) procedures that

include part volumes, part processing times, and machine

capacities, and 6) procedures that consider multiple

scenarios for part volumes. Some procedures may fall into

two or more categories. There, also, are a few procedures

which use data that do not fall into any of the categories,

therefore, a seventh category called "other" is included.

For each category discussions of the objectives and

constraints considered by procedures, and methods used by

procedures to develop a solution are included.


2.2 Procedures Based solely on a Part-Machine
Incidence Matrix



These are the most common of the cell formation

procedures. The concept of a part-machine incidence matrix

was described in chapter 1. As a brief refresher, in a part-

machine incidence matrix machines are represented as rows

and parts are represented as columns in the matrix (or vice

versa). An element in the matrix is '1' if part j requires

machine i for processing and '0' otherwise. The advantages

of these procedures are that they have low data requirements











(all the data can be obtained from part routings) and they

tend to be relatively simple and efficient. The disadvantage

of these procedures is they do not consider factors such as

operations sequences, part volumes, machine capacities, and

demand variability. When a solution is developed using one

of these procedures the cell designer must somehow factor in

other important considerations to arrive at a final

solution.

There are several objectives and constraints considered

by procedures in this category and many techniques have been

used to develop solutions.

2.2.1 Obiectives and Constraints


The most common objectives used in these procedures are

to minimize exceptional elements, minimize duplicate

machines, and maximize a measure of similarity between parts

or machines assigned to a cell. The two most common

constraints are maximum cell size and measure of similarity

threshold.

2.2.1.1 Frequently used objectives

Exceptional elements. Many procedures (Burbidge 1963,

Vannelli and Kumar 1986, King 1980, Chan and Milner 1982,









32
King and Nakornchai 1982, Waghodekar and Sahu 1984, Boctor

1989, Chow and Hawaleshka 1993, Khator and Irani 1987, and

Boctor 1991) have as an objective the minimization of

exceptional elements. An exceptional element occurs when a

part requires two or more cells for processing. This

objective is important because exceptional elements would

require increased material handling cost and paperwork cost,

or subcontracting cost to remove the exceptional element

from the plant. If many parts are exceptional elements when

a cellular system is designed the benefits of the system

will be greatly diminished.

Machine duplication. Minimization of machine

duplication is also an objective used by many procedures

(Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and

Milner 1982, King and Nakornchai 1982, Khator and Irani

1987, and Al-Qattan 1990). Machine duplication (also

sometimes referred to as a bottleneck machine) occurs when a

machine type is required in two or more cells. When a

machine type is required in two or more cells additional

machines may be required which increases machine investment

and may cause machine utilization to be lower vs a process

layout. This objective is usually used in conjunction with











the exceptional element constraint. Most procedures trade

off exceptional elements against machine duplication.

Vannelli and Kumar's (1986) procedure attempts to minimize

the number of machine types that must be duplicated while

allowing no exceptional elements.

Measures of similarity. Several procedures attempt to

maximize measures of similarity between pairs of parts or

pairs of machines (Rajagopalan and Batra 1975, Carrie 1973,

Mcauley 1972, Kusiak 1987, De Witte 1980, Chandrasekharan

and Rajagopalan 1986a, Luong 1993, Askin et al. 1991,

Srinivasan et al. 1990, Gunasingh and Lashkari 1989b, and

Wei and Kern 1991). Measures of similarity between pairs of

parts are based on how similar the processing requirements

are for the parts. Measures of similarity between pairs of

machines are based on which parts require the machines.

Creating cells that produce parts that have similar

processing requirements and machines that process similar

parts is one of the main objectives of cellular

manufacturing, therefore, this is an important objective.

Most of the measures of similarity that are used in cell

formation are based on some sort of similarity coefficient

(the concept of a similarity coefficient will be described









34
in the next section) which can take on a value between 0 and

1.

2.2.1.2 Frequently used constraints

Cell size. The most popular constraint used by these

procedures is a cell size constraint (Chandrasekharan and

Rajagopalan 1986, Boctor 1989, Luong 1993, Stanfel 1985,

Boctor 1991, Wei and Kern 1989, Al-Qattan 1990, and Vannelli

and Kumar 1986). The cell size constraint limits the number

of machine types allowed in a cell. The constraint is

imposed to limit the physical size of cells. It is generally

assumed that intracell material handling cost will be

insignificant but this assumption loses validity as cell

size grows. This constraint is used with each of the

objectives described above. Without this constraint, a cell

consisting of all machine types could be created which would

have no exceptional elements and would not require any

duplicate machines.

Similarity threshold. A second constraint which is

sometimes imposed is a measure of similarity threshold

(Carrie 1973, McAuley 1972, and Chandrasekharan and

Rajagopalan). This constraint is usually used in conjunction

with the objective that maximizes some similarity measure.









35
The similarity threshold constraint does not allow a machine

or a part into a cell unless the cell's similarity measure

is above the threshold. The threshold can be applied to a

single pair of items (single linkage) in a cell, any pair of

items (complete linkage) in the cell, or the average

similarity between items (average linkage) in the cell. If

the threshold constraint is not applied then a single large

cell would result and the shop would be a process layout. If

a high threshold is applied to a measure of similarity

between machines the resulting layout would have small cells

and possibly many exceptional elements. If a high threshold

is applied to a measure of similarity between parts then the

result would be many part families and to create cells for

each family could require many machine types to be

duplicated. Picking a similarity threshold requires

judgement by the cell designer. Usually the procedures would

be run with several similarity thresholds and then a part-

machine structure would be selected.

2.2.1.3 Other objectives

Chandrasekharan and Rajagopalan (1986) and Kumar and

Chandrasekharan (1990), and Stanfel (1985) have developed

three measures which are slightly more sophisticated and use

these measures as objectives.











Grouping efficiency. The first measure is called

grouping efficiency (Chandrasekharan and Rajagopalan 1986).

This measure is a convex combination of within-group

utilization and inter-cell efficiency. Within-group

utilization is a measure of how heavily parts visit machines

in a cell (summation of the number of machine types used by

each part in each cell divided by the summation of the

number of machine types times the number of parts in each

cell). Inter-cell efficiency equals the fraction of

exceptional elements divided by the number of non-

exceptional elements. This measure serves as a surrogate for

a tradeoff between machine investment and materials handling

cost. It is speculated that cell layouts with a high within-

group utilization will result in a shop in which machine

types have high utilization because most of the parts in the

cell use each machine type in the cell. This should result

in a lower required machine investment than a shop that has

a low within-group utilization. A cell layout with a high

inter-cell efficiency should have a low material handling

cost because there are relatively few exceptional elements.

Grouping efficacy. The second measure is called

grouping efficacy (Kumar and Chandrasekharan 1990). Grouping











efficacy is similar to grouping efficiency but is adjusted

to take into consideration the density of the part-machine

incidence matrix (the density of the part-machine incidence

matrix is equal to the 1 entries in the matrix divided by

total entries).

Weighted transfers. Stanfel (1985) uses an objective

that considers both inter-cell and intra-cell transfers.

Intra-cell transfers for a part are based on the number of

machine types that are in a cell that are not required to

process the part. The objective weights both inter-cell and

intra-cell transfers (intra-cell transfers usually have a

lower weight). This objective serves as a surrogate for

material handling cost and also considers cell size. Cells

with more machine types are likely to be larger and require

a higher intra-cell movement cost.

2.2.2 Methods


A wide variety of methods have been used to solve the

cell formation problem with only a part-machine incidence

matrix as input. The most popular methods have been manual

identification, matrix manipulation, clustering using

similarity coefficients, graph theory, and solving

mathematical models.











2.2.2.1 Manual identification

Probably the earliest procedure to be used for cell

formation is a manual procedure called production flow

analysis developed by Burbidge (1963). This procedure uses

part routings to analyze inter and intra departmental flows.

There are two stages in production flow analysis: 1) factory

flow analysis and 2) departmental flow analysis. Factory

Flow analysis studies the basic routes in the factory and

then eliminates exceptions. Departmental analysis is broken

into group analysis and line analysis. In group analysis

parts are divided into families so groups of machines can be

put together so that the parts in a family can be completely

manufactured by the group. The main criterion is that the

division into groups should not necessitate an increase in

the number of machines required.

El-Essawy and Torrance (1972) also develop a manual

procedure called component flow analysis. This procedure is

similar to production flow analysis.

Since production flow analysis and component flow

analysis are manual procedures they become cumbersome to

apply as the problem size becomes large.











2.2.2.2 Matrix manipulation

This method attempts to create a block diagonal

structure, in which almost all l's occupy the diagonal

submatices, and almost all zeros occupy the off-diagonal

submatrices, by manipulating the data in the part-machine

incidence matrix. King (1980) was the first to use this

approach. His procedure is called rank order clustering

(ROC). Each row and column are considered as binary words,

and rows and columns are alternately sorted until a block

diagonal structure is obtained. King and Nakornchai (1982)

revised the procedure so it would be more efficient to run

on a computer (if there are many rows or columns in the

part-machine incidence matrix the binary word approach

cannot be used) and used a relaxation if there are

bottleneck machines (bottleneck machines are machines which

process many parts). Chan and Milner (1981) developed a

procedure called the direct clustering algorithm (DCA). The

advantage of this procedure is that it can start with any

matrix and arrive at the same result (if the order of parts

or machines is changed the ROC procedure may obtain a

different result). The authors also show that DCA will

converge to a solution in a limited number of iterations.









40

Chandrasekharan and Rajagopalan (1986) developed a procedure

called modified rank ordering clustering algorithm (MODROC).

This procedure builds on the ROC procedure by taking ROC

output and using clustering to create cells. Khator and

Irani (1987) developed a procedure called the occupancy

value method (OV). The OV method eliminates limitations of

other methods (ROC, DCA) by building up clusters along the

diagonal using small selected sections of the larger matrix.

2.2.2.3 Clustering using similarity coefficients

Clustering methods using similarity coefficients have

been used by many researchers to develop procedures to solve

the cell formation problem (Carrie 1973, McAuley 1972,

Chandrasekharan and Rajagopalan 1987, Waghodekar and Sahu

1984, DeWitte 1980, Chow and Hawaleshka 1993, Luong 1993,

Shafer and Rogers 1993, Askin et al. 1991, Srinivasan and

Narendran 1991, and Wei and Kern 1989). These procedures use

some measure of similarity between pairs or groups of parts

(or pairs or groups of machines) based on the part-machine

incidence matrix. A clustering procedure is then used to

group similar items into clusters which become the basis for

forming cells.











The earliest clustering procedure for cell formation

was developed by McAuley (1972). He used the Jaccard

similarity coefficient. The Jaccard similarity coefficient

for a pair of machines is defined as the number of parts

processed by both machines divided by the number of parts

processed by either machine (the similarity coefficient

ranges between 0 and 1). McAuley then used single linkage

cluster analysis to form groups of machines. Single linkage

cluster analysis uses the most similar pairs of items

between two groups to define the similarity between two

groups. Other clustering methods have also been used. Two

common methods are complete linkage cluster analysis and

average linkage cluster analysis. Complete linkage cluster

analysis defines the similarity between two groups by the

least similar pair of items between the groups and average

linkage cluster analysis uses the average similarity between

the items in the groups. Seifoddini (1988, 1989a) compares

single linkage clustering and average linkage clustering for

cell formation and outlines the advantages and disadvantages

of each method.

Many similarity coefficients and distance measures have

been used in clustering procedures for cell formation.











Shafer and Rogers (1993a) did a comprehensive survey of

these measures.

2.2.2.4 Graph theory

Several procedures use graph theory to solve the cell

formation problem. Rajagopalan and Batra (1975) were the

first to use a graph-theoretic approach. In their procedure

machines are considered vertices, and edges are

relationships between the machines. The Jaccard similarity

coefficient is computed for each pair of machines and an

edge is included if the similarity coefficient is above a

threshold value. The weight of an edge is equal to the

similarity coefficient. A graph partitioning approach is

then used to create machine groups and parts are allocated

to machine groups to create manufacturing cells.

Chandrasekharan and Rajagopalan (1986a) show that the

problem can be represented as a bipartite graph in which one

set of vertices represents machines and the other set of

vertices represents parts. An edge exists between a machine

vertex and a part vertex if the part uses the machine.

Vannelli and Kumar (1986), and Kumar, Kusiak and Vannelli

(1986) also use this approach and propose heuristics to find

the minimal cut nodes needed to create the required number

of cells. A cut represents intercell movement.











2.2.2.5 Mathematical models

Several researchers have formulated models that require

only the information provided by a part-machine incidence

matrix. These models can then be solved optimally; or if the

problem is too large a lower bound can be obtained, and

heuristic procedures can be evaluated against the lower

bound.

Kusiak (1987) formulated the problem using the p-median

model. The objective is to maximize the similarities between

parts that are assigned to the same part family

(similarities refer to a similarity coefficient). The

constraints in this model assign each part to exactly one

family, specifies the number of part families, ensure that

part i belongs to family j only if family j is formed, and

ensure integrality. Shtub (1989) shows that Kusiak's model

can be formulated as a generalized assignment problem.

Srinivasan, Narendran, and Mahaderan (1990) formulate the

problem as an assignment model which differs from Kusiak's

model in that the number of groups to be created is not

required as an input for the model.

Boctor (1989, 1991) formulated a 0-1 integer program

that has as its objective the minimization of intercell











transfers (a machine type is assigned to a cell that is

different from that of a part that requires processing on

it). Constraint sets that assign each part and each machine

type to only one cell are included and a constraint set for

cell size is included. The author proves that the

integrality constraints for part assignment can be dropped

and an integer solution will still be obtained.

2.2.2.6 Other methods

Many other methods have been used to develop procedures

for cell formation. Our review of the literature found the

following set theoretic-Purcheck (1975), Sundaram and Fu

(1987), and Purcheck (1985); string manipulation-Wu,

Venugopal, and Barash (1986); polyhedral dynamics-Robinson

and Duckstein (1986); neural network-Chu (1993), and

Karparthi and Surresh (1993); Hamiltonian path-Askin et al.

(1991); branch and bound-Al-Qattan (1990); and simulated

annealing-Boctor (1991).

2.2.3 Evaluation of Procedures


Considering the amount of research that has been done

to develop cell formation procedures, based on part-machine

incidence matrices, there has been comparatively little











research that compares the effectiveness of various

procedures.

The majority of research that has been done in

comparing cell formation procedures is a comparison of

various similarity measures and clustering procedures.

Harrigan and Mosier (1988) tested four similarity

coefficients used in clustering procedures. The objective of

their study was to asses the appropriateness of cellular

manufacturing within a particular manufacturing environment.

Mosier (1989) tested similarity coefficients and clustering

procedures using 30 generated problems which varied by

cluster definition and block diagonal density. Shafer and

Rogers (1993b) tested 16 similarity measures and four

clustering procedures and compared the procedures based on

various performance measures.

Miltenburg and Zhang (1991) tested 9 procedures that

used a variety of methods on 8 well known problems from the

literature and 60 generated problems. Chu and Tsai (1990)

tested three procedures that used matrix manipulation on 11

problems from the literature.

Shafer and Meredith (1990) conducted a study that used

actual data from three companies. Cell layouts were











developed using 7 procedures. Two of the procedures used

matrix manipulation, four procedures used clustering and one

procedure was based on operations sequences. A simulation

model was then built and the performance of each of the cell

layouts generated by the procedures was tested. The

performance measures used were average distance traveled,

number of extra-cellular operations, average WIP, maximum

WIP, and longest average queue.



2.3 Procedures Which Use a Part-Machine Incidence Matrix and
Incorporate Data to Provide Weights for Parts or Machines




These procedures are similar to the procedures of

section 2.2 with the exception that these procedures

recognize that some parts are more important than others,

and some machine types are more important than others. Data

is incorporated into these procedures which allows parts and

machine types to have various weights of importance. Part

weights are usually based on the volume of units required

for a part times a cost per unit. The cost per unit is

usually based on the cost of intercell movement or an

incremental subcontracting cost to remove the part from the











system. Machine type weights are usually based on the cost

of a single machine of a given type.

2.3.1 Objectives and Constraints


The objectives used by these procedures are 1) minimize

the cost of exceptional elements (Kusiak and Chow 1987,

Seifoddini 1989c, Seifoddini and Wolfe 1987, and Kumar and

Vannelli 1987); 2) minimize the cost to duplicate machines

(Seifoddini 1989c, and Sundaran and Fu 1987); 3) maximize a

measure of similarity (Steudel and Ballakur 1987,

Balasubramanian and Panneerselvam 1993, Okogbaa et al. 1992,

and Gupta and Seifoddini 1990).

The two most popular constraints are cell size and

measure of similarity threshold.

The first objective is sometimes used by itself subject

to the constraint that a machine type can only appear in one

cell and a cell size constraint (Kusiak and Chow 1987, Kumar

and Vannelli 1987); or in conjunction with the second

objective (Seifoddini 1989).

Some of the similarity measures used by these

procedures incorporate weights. Steudel and Ballakur (1987)

use a similarity measure called cell bond strength (CBS)

which is based on the processing times of parts.











Balasubramanian and Panneerselvam's (1993) measure of

similarity is based on the number of excess moves a part

requires in each cell. Okogbaa et al. (1992) base their

measure on flows between machine types. Gupta and Seifoddini

(1990) incorporate part volumes and processing times into a

similarity coefficient.

2.3.2 Methods


Several methods are used by these procedures.

Seifoddini and Wolfe (1987) cluster using several similarity

thresholds and then check the intercell movement cost for

each threshold and choose the best to use as a layout. Gupta

and Seifoddini (1990) cluster using complete linkage with a

threshold constraint. Steudel and Ballakur (1987) use a

dynamic programming algorithm to maximize their similarity

coefficient. Balasubramanian and Panneerselvam (1993) first

create a set of potential cells by creating a cell for each

part and then creating additional potential cells using the

rank ordering clustering procedure. They then select cells

based on a warehouse covering algorithm. Kumar and Vannelli

(1987) use a procedure which starts with part seeds or

machine seeds. Part seeds are parts that do not use any











common machine types and machine seeds are machine types

which do not process any common parts. The procedure then

uses objective function criteria to add parts and machines

to cells.





2.4 Routing Based Procedures Without Machine Capacities



These procedures require part routings to be used as

opposed to a part-machine incidence matrix. The part

routings provide additional information about a part's

sequence of operations and possible alternative processing

plans. The procedures in this category also can incorporate

cost and volume information, but do not incorporate any

information about machine capacities. The procedures are

further categorized into two groups: 1) procedures which use

operations sequences, and 2) procedures which use

alternative routings. Procedures which consider operations

sequences focus on efficient materials flows when forming

cells. Procedures which consider alternate routings focus on

equipment utilization and flexibility when forming cells.











2.4.1 Operations Sequences


Vakharia and Wemmerl6v (1990), Selvam and

Balasubramanian (1985), Tam (1990), and Logendran (1991)

have developed procedures which incorporate operations

sequences. The first three procedures use similarity

coefficients. Vakharia and Wemmerl6v (1990) attempt to

create cells that do not require parts to backtrack (An

attempt is being made to create flow-line cells in which

there is a unidirectional flow thru the machines in the

cell. If a part must move in the opposite direction of the

flow then a backtrack has occurred). A similarity

coefficient, that compares the number of machines in two

groups that are used in the same sequence to the total

number of machines in the groups, is used to cluster groups.

Selvam and Balasubramanian (1985) develop a set of potential

cells using a similarity coefficient based on operations

sequences, and then select the desired number of cells by

using a covering algorithm. The objective is to minimize

material handling cost. Tam (1990) also develops a

similarity coefficient based on operations sequences, and

then uses a clustering method called k-nearest-neighbor











(KNN) to form cells. Logendran aggregates machine types in

an attempt to minimize weighted intra-cell and inter-cell

moves based on operations sequences.



2.4.2 Alternative Routings


Kusiak (1987), Shtub (1989), and Sankaran and

Kasilingam (1990) have developed models which incorporate

the possibility of alternate routings. Kusiak (1987)

formulates the problem as a p-median model. The objective is

to maximize the similarities between part routings assigned

to the same family (similarities are based on a comparison

of process plans for pairs of parts). The constraints in

this model are only one routing for a part is selected, each

part is assigned to one family, only N part families are

formed where N is an input parameter, part i belongs to

family j only if family j is formed, and integrality. Shtub

(1989) shows that Kusiak's (1987) formulation is equivalent

to the generalized assignment problem (GAP). Sankaran and

Kasilingam (1990) formulate a model that maximizes the

number of routings that can be completed in a cell. The

purpose of this model is to design a cellular system that is











flexible (parts will have alternate cells that can be used

for processing). The constraints of the model are at least

one routing is chosen for each part, cell size limit, a

budget limit that can be spent on machines, and if a routing

is assigned to a cell then all the machine types required by

the routing are assigned to the cell.





2.5 Procedures Which Incorporate Setup Information



Several procedures incorporate setup objectives into

the cell formation problem. Setup is an important

consideration in cell formation because by combining items

with similar setups and dedicating equipment to producing

them setup time can be substantially reduced which reduces

labor costs and increases plant capacity.

Shafer and Rogers (1991) formulated models which

include minimizing setup time as an objective (minimizing

intercellular movements and investment in machines are also

objectives). Shafer and Roger's models also consider

sequence dependent setups in the cell formation problem. The

models combine a p-median formulation with the travelling











salesman problem (TSP). A two stage heuristic is used to

solve the problem. In stage I a p-median model is solved to

form part families and cells. Stage II then solves a

travelling salesman problem to determine sequences for each

family.

Chakravarty and Shtub (1984) developed a procedure to

form cells that considers inventory and setup costs

associated with batch sizes as well as materials handling

costs. The authors develop a cost function for the total

setup and inventory holding cost for all components at a

machine and then derive an optimal production cycle time for

a machine (lotsize). The authors then show how to assign a

machine, that could be assigned to more than one group

(based on processing sequences), to one of the groups based

on the cost function.

Askin and Subramanian (1987) constructed an economic

model of manufacturing costs as a function of cell

configuration. The model includes the following costs:

setup, variable production, production cycle inventory, WIP,

material handling, and fixed machine. Setup costs include

both product and family setup costs. A heuristic, that is an

extension of the approach of Boucher (1984) is used to









54
develop product and family cycle times and considers family

setup costs, WIP costs and finished goods inventory costs.

The procedure used to create cells consists of three

sequential stages. ROC is used in the first stage to cluster

the part-machine incidence matrix. Stage two attempts to

combine adjacent groups based on an economic evaluation.

Stage three aggregates groups based on machine capacities

and further economic evaluation. Rajamani, Singh, and Aneja

(1992) formulate a mathematical model for cell formation in

environments where sequence dependent setups exist. The

model considers the tradeoff between setup costs and machine

investment (by dedicating machines to produce certain parts

setup cost is reduced but machine investment increases). The

model's objective function is to minimize the sum of total

discounted cost of machines assigned to all the cells, and

setup costs incurred due to sequence dependence of parts in

each cell. The constraints in the model ensure each part is

produced in a cell, each part has a place in a sequence, a

sequence is defined, relationships between pairs of parts,

and capacity constraints (these include processing time and

setup time).











2.6 Procedures Which Incorporate Machine Capacities



These procedures recognize that it may not be possible

to process all parts in a desired cell due to limited

machine capacity. These procedures require for each part a

demand requirement and the processing time on each machine

type, and for each machine type the time available for

processing. The procedures in this category are based on

mathematical models which require an integer number of

machines of each type assigned to a cell. Two common models

can be identified, although, there are several variations to

each of these models.

2.6.1 Model 1


Shafer, Kern and Wei (1992), Shafer and Rogers (1991),

Askin and Chin (1990), Choobineh (1988), Askin and

Subramanian (1987), Vakharia, Chang and Selim (1993),

Rajamani, Singh and Aneja (1990), Sankaran (1990), Sule

(1991), and Kamrani and Pansaei (1993) use models which are

variations of this basic model.

These models include at least two items in the

objective function. The first item deals with the cost of











parts which cannot be produced in a single cell. Shafer,

Kern and Wei (1992) consider a subcontracting cost to remove

the part, the other models consider the cost to move parts

between cells. The second item is the cost to use additional

machines. If there is an infinite number of machines

available and there is no cost to use machines a cell could

be created for each part and no intercell transfers or

subcontracting would be required. The models recognize that

there is a tradeoff between the two objectives that were

described. The cost of machines can be decreased at the

expense of intercell material handling cost or

subcontracting, and the reverse is true.

These models contain constraints which do not allow the

processing time of a machine type in a cell exceed the time

available of a machine type in a cell. The models, with one

exception, also contain constraints which restrict the size

of a cell to k machine types where k is a required input

parameter, or require cells defined by machine types as

input (Shafer, Kern and Wei 1992, Askin and Subramanian

1987, Sule 1991, and Choobineh 1988). The exception is

Sankaran (1990) who uses a constraint which requires a

minimum level of similarity (based on parts which are











processed) between machines in a cell. These models, with

the exception of Sule (1991), require that a part is

assigned to a single cell (a part's production requirement

cannot be split among cells).

There are several variations to this model. Askin and

Chin (1990), Choobineh (1988), Vakharia, Chang and Selim

(1993), and Sule (1991) incorporate operations sequences

into the model. Choobineh (1988) and Rajamani, Singh and

Aneja (1990) consider multiple process plans or alternate

routings in their formulation. Shafer and Rogers (1991)

include setup time in the objective function; and Askin and

Subramanian (1987) include setup costs and inventory costs

in the objective function. Kamrani and Pansaei (1993) do not

include a cost for parts that cannot be produced in a cell

in the objective function but include constraints which do

not allow intercell transfers.

2.6.2 Model 2


Wei and Gaither (1990a, 1990b), Dahel and Smith (1991,

1993), Nagi, Harhalakis, and Proth (1990), and Logendran

(1990, 1993) use models which are variations of this basic

model.









58

These models include the cost of parts which cannot be

produced in a single cell. Wei and Gaither (1990a, 1990b)

consider a subcontracting cost to remove the part, Dahel and

Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and

Logendran (1990, 1993) consider the cost to move parts

between cells.

These models also contain constraints which do not

allow the processing time on a machine type in a cell exceed

the time available of the machine type in the cell. These

models contain two other sets of constraints: one that

limits the number of machines that can be assigned to a cell

to a prespecified number; and one that limits the number of

machines that are available of each machine type. The second

constraint is important because if an unlimited number of

machines are available for each type then a cell could be

created for each part and there would be no need for

subcontracting or intercell transfers.

Dahel and Smith (1991, 1993) incorporate part

operations detail into their model. Nagi, Harhalakis, and

Proth (1990) consider alternate routings in their

formulation and allow a part's production requirement to be

spread over several routings so machines can be better











utilized. Wei and Gaither (1990b) include, along with the

cost of bottleneck parts, average cell utilization,

intracell load imbalances, and intercell load imbalances in

the objective function. Each of the items is given a weight

and the objective is to minimize the weighted sum of the

items. Logendran (1990, 1993) includes both intracell and

intercell moves in the objective. Weights are given to both

types of moves (moves are based on operations sequences and

a lower weight is usually given to intracell moves) and the

objective is to minimize weighted moves. Logendran (1990)

includes constraints that do not allow a machine's

utilization to fall below a specified level; and Logendran

(1993) includes machine utilization in the objective

function.

2.6.3 Methods


Since the models contain integer variables, reasonable

size problems can be computationally difficult to solve,

therefore, many of the procedures used are heuristics which

may not solve the problem optimally. As in other categories

a variety of methods are used in the procedures.











Several procedures (Shafer and Rogers 1991, Askin and

Chin 1990, Askin and Subramanian 1987, Logendran 1990, Nagi,

Harhalakis, and Proth 1990, and Sule 1991) break the problem

into stages and solve each stage sequentially. Some of the

procedures solve the subproblems in each stage optimally.

Shafer and Rogers (1991) break their formulation into two

subproblems, a p-median problem and a travelling salesman

problem, and solve the subproblems optimally. Nagi,

Harhalakis, and Proth (1990) create two subproblems and then

iterate between the two subproblems until convergence to a

local minimum is achieved.

Other methods are also used. Wei and Gaither (1990) fix

seed machines and then form cells by solving the integer

program. Vakharia, Chang and Selim (1993) use simulated

annealing and tabu search. Dahel and Smith (1991, 1993) use

the constraint method to solve their multi objective model.





2.7 Procedures that Consider Multiple Scenarios
for Part Volumes






Procedures in this category recognize that demand for

parts may not remain the same over time, or that demand for











parts may be stochastic. Previous procedures, which

considered part volumes, were based on annual or average

demand and were considered to be deterministic. If demand

varies from period to period or estimates for demand turn

out to be inaccurate a cell layout may not perform as well

as expected. Cell layouts created by procedures in this

category should have the flexibility to perform well even if

there are expected volume swings or volume differs from what

was estimated.

Little work has been done in this area. Vakharia and

Kaku (1988, 1993) developed a procedure to handle long-term

demand changes. Seifoddini (1990) developed a procedure for

probabilistic demand estimates.

Vakharia and Kaku (1988, 1993) focus on a strategy of

part reallocation to respond to demand changes as opposed to

partial cell system redesign or complete cell system

redesign. This strategy is chosen because it has the lowest

cost of change (if cell design is robust changes can be

handled by this strategy). Vakharia and Kaku proposed a

model. The objective is to minimize material handling cost

and additional machine investment costs. The constraints of

the model are each operation for each part is assigned to a











cell, determination of the number of new machines required

based on machine capacity, a cell is visited if an operation

is performed in a cell, integrality, and nonnegativity.

Additional constraints can be added to limit cell size to a

specified number of machines. A heuristic is used to solve

the model.

Seifoddini's (1990) procedure incorporates the

uncertainty of product mix into the final solution. The

procedure has three major steps: 1) the product mix and the

associated part-machine incidence matrix are expressed in

the form of a probability function, 2) a set of alternate

solutions are developed based on all possible product mixes,

3) a measure of effectiveness is developed and used as a

criterion for evaluating different solutions for the purpose

of choosing the best solution. This procedure assumes there

is a discrete distribution of product mixes and part-machine

incidence matrices. The procedure uses similarity

coefficients and a clustering algorithm to develop a cell

layout for each product mix that minimizes intercell

material handling cost. The intercell material handling cost

is then calculated for each product mix and each layout, and

an expected intercell material handling cost is calculated









63

for each layout. The layout with the lowest expected cost is

chosen.





2.8 Other Procedures



Procedures in this category incorporate data that is

not included in the previous categories.

Gunasingh and Lashkari (1989b) formulate an integer

programming model that groups machines based on machine

tooling availability and parts tooling requirements. The

authors define a compatibility index between a machine and a

part. The index is based on the number of common tools

between part i and machine j divided by the minimum of the

number of tools part i requires or machine j has available.

The objective of the model is to maximize the compatibility

index. Constraints of the model include a cell size

restriction, allocation of each machine to at least one

cell, and integrality.

Irani, Cavalier and Cohen (1993) consider both the

layout of cells and machines in order to minimize the effect

of intercell movement. The authors use a from-to travel









64

chart to layout machines within a cell and layout cells with

respect to each other. The procedure also considers options

such as placing a machine that is common to two cells, that

are adjacent to each other, between the two cells so that

greater flexibility of machine utilization can be achieved.

Min and Shin (1991) formulated a model that

incorporates human factors into cell formation. The authors

believe that, in addition to the normal functions, cell

formation should also allocate operators to cells, assign

machines to each operator, and determine the job scope for

each operator. The model includes many sets of constraints

and is a multiobjective goal program. Some of the

constraints and objectives that stem from incorporating

human factors are maximum number of operators allowed in a

cell, maximum number of parts that operator p can handle,

assignment of parts to operators, wage rates of operators,

and skill matching of an operator with a part.





2.9 Summary



The review of procedures in the previous sections

illustrates how diverse cell formation procedures are with









65
respect to factors considered and data required, as well as

the methods used. A single procedure probably cannot be

developed which considers all the possible factors that

could impact on the performance of a cell layout. When

choosing a cell formation procedure to apply to a particular

situation selection should depend on the factors that appear

to be most important. In many instances it may be

appropriate to use several procedures and then compare the

layouts that are generated.

The review of the procedures in section seven indicates

that more work could be done to develop procedures that

create good cell layouts in situations that experience

demand variability or demand uncertainty. Cellular layouts

usually tradeoff strengths and weaknesses of product line

layouts and process layouts. A key strength of a process

layout is its flexibility to handle volume changes or

uncertainty, product line layouts tend to be less flexible

in these environments. Procedures which can evaluate whether

a shop should be organized with small efficient cells

dedicated to a few parts or larger less efficient cells

which can be used to produce many parts, or a mixture of

large and small cells should be of aid to the shops which

exist in the environment described above.















CHAPTER 3
MODELS



3.1 Introduction



This chapter presents two models which can be used for

identifying cellular configurations in the presence of

dynamic demand. Model I attempts to form cells that do not

exceed a predetermined cell size in terms of the number of

machines contained in each cell. This model considers

materials handling costs as well as inventory and

backordering costs to select a solution. Model II develops a

cell configuration for a shop by selecting cells from a set

of potential cells, which could be of varying sizes.

To develop a cell configuration, first several runs of

model I could be made with varying cell size parameters as

input. The cells that were formed in each of the runs can be

included in the set of potential cells. Model II would then

be run using this set of potential cells to create a cell

configuration.












3.2 Model I



3.2.1 Description of Model I


This model attempts to form cells that do not exceed a

cell size constraint which is specified in terms of the

number of machines a cell can contain. The model can also

form a remainder cell which has no cell size constraint.

The model uses a planning horizon which consists of T

periods. The parts to be produced during the planning

horizon are known. The demand for each part is known and can

vary from period to period. Each part requires a set of

machine types for production, and the processing time per

unit for each part on each machine type, that is required by

the part, is a known parameter.

There are a limited number of machines of each type

that can be used to produce parts. Each machine has a

limited capacity available during each period in the

planning horizon. The capacity of the machine types within

each cell is an important constraint in this model. The

total processing time required of a machine type within a











cell during any period cannot exceed the total capacity of

that machine type within that cell.

In order to effectively utilize machine capacity the

model allows for inventory planning decisions. The number of

units produced of a part during a period can exceed the

demand requirement for the period. The excess production can

be stored in inventory and used to satisfy demand

requirements in later periods. A part's demand requirement,

for a period, may also be partially or completely

backordered and excess production of the part during a later

period may be used to fill backorders. If an inventory of a

part is held then there is a cost associated with the

inventory. If a backorder is incurred there is a cost

associated with the backorder. Each part has a unit

inventory holding cost and a unit backorder cost associated

with it. These costs are on a per period basis.

The unit cost of producing a part depends on where in

the shop the unit is produced. If the unit is produced

entirely in a cell that meets the cell size constraint then

the production cost is zero. If the unit is produced

entirely in the remainder cell there is a cost associated

(by part) with producing the unit. If the unit is produced










69

using machines that reside in two or more cells there is an

intercellular movement cost associated (by part) with

producing the unit. Each part has a unit remainder cell

production cost and a unit intercellular movement production

cost associated with it.

3.2.2 Model I Formulation



The formulation of model I is shown below.


I T
Min Z= E (MMRiXitr+MMIiXiti+hilit+biBit)
i-I t-I


Subject to


K
Iit-l-Bit-1+1: Xitk+Xiti-Iit+Bit=dit
k-I




PijXitk CjYjk
i-1


for i= .,I; (2)
t=l,...,T.


for
for
for


I K 1
jE PijXitk+ Pij (Xiti+Xitr) < CjAj for
- k-i -1 for


k=l, ...,K & r;
j=l, ... ,J;
t=l,...,T.



j=l, .. .,J;
t=l, .,T


for j=l,...,J.


K
E YiA3
Yk-


J

Yjk j-1


for k=l,...,K











Yjk is integer for all j & k. (7)


All variables are a 0 (8)




where

I=The total number of parts.

T=The total number of periods.

J=The total number of machine types.

K=The total number of cells not including a remainder

cell.

i=part index (i=1,...,I).

t=period index (t=l,...,T).

j=machine type index (j=l,...,J).

k=cell index (k=l,...,K & r).

MMRi=The cost to produce 1 unit of part i in the

remainder cell.

MMIi=The cost to produce 1 unit of part i using two or

more cells (uses intercell movement).

hi=The cost to hold 1 unit of part i for 1 period.

bi=The cost to backorder 1 unit of part i for 1 period.

dit=The demand for part i in period t.

Pij=The processing time for part i on machine type j.











Cj=The capacity in hours of 1 machine of type j.

Aj=The number of machines available of type j.

CS=The maximum number of machines allowed in a cell

excluding the remainder cell.

Xik=The number of units of part i produced in cell k

during period t.

Xi,,=The number of units of part i produced using

intercell movement during period t.

Iit=The inventory of part i at the end of period t.

Bit=The number of units on backorder of part i at the

end of period t.

Yjk=The number of machines of type j assigned to cell

k.

3.2.3 Description of Model I Equations


(1) Objective function: The objective function is the sum

over all parts and all periods of four terms for each part

during each period. The first term is the cost of producing

a unit of part i in the remainder cell times the number of

units produced of part i in the remainder cell during period

t. The second term is the cost of producing a unit of part i

using intercell movement times the number of units produced











of part i using intercell movement during period t. The

third term is the cost to hold one unit of part i in

inventory for one period times the number of units of part i

that are in inventory at the end of period t. The last term

is the cost to hold one unit of part i on backorder for one

period times the number of units on backorder at the end of

period t.

(2) This set of equations represents material balance

constraints. The first two terms on the left side of the

equality (Iit_--Bit-,) represent the net inventory position of

part i at the beginning of period t (or the end of period

t-l) and the last two terms on the left side of the equality

(-Iit+Bi,) represent the negative of the net inventory

position of part i at the end of period t. The middle two
K
terms on the left side of the equality ( Xltk+Xiti) equals

production in units of part i during period t. Therefore,

the net inventory position of part i at the beginning of

period t plus production of part i during period t minus the

net inventory position of part i at the end of period t must

equal the demand for part i during period t (the right side

of the equality). There is a constraint for each part during

each period.









73
(3) These constraints ensure that the total processing time

required by parts produced in cell k during period t does

not violate the capacity of machine type j in cell k. There

is a constraint for each machine type, in each cell during

each period.

(4) These constraints ensure that the total processing time

required by parts produced during period t does not violate

the capacity of machine type j. There is a constraint for

each machine type during each period.

(5) These constraints ensure that the total number of

machines of type j that are assigned does not exceed the

number of machines of type j that are available. There is a

constraint for each machine type.

(6) These constraints require that cell k not contain more

than a total of CS (the cell size parameter) machines. There

is a constraint for every cell with the exception of the

remainder cell.

(7) These constraints require that the number of type j

machines assigned to cell k be integer. There is a

constraint for each machine type in each cell.

(8) These constraints require all the production, inventory,

and backorder variables to be nonnegative.















3.2.4 An Example of Model I


The following example shows how model I is applied to a

problem with 2 parts, 3 machine types, and two periods of

demand. The part-machine incidence matrix for this example

is shown in table 8. Part 1 requires machine types 2 and 7,

and part 2 requires machine types 1 and 2. The cell size

limit used in this example is two machines. A remainder cell

can also be formed. Table 9 shows the demand by period, the

unit inventory cost, and the unit backorder cost for each

part. Table 10 shows the number of machines of each type

that are available and the capacity in units per machine of

each type. The cost to manufacture either part in a cell

that meets the cell size restriction is 0, the cost to

manufacture either part in the remainder cell is 1, and the

cost to manufacture either part using intercell movement is

3.

In the formulation of this problem we allow for two

cells that meet the cell size constraint and a remainder

cell. The objective for this problem is

Min Z=Xllr+Xi2r+X21r+X22r+3Xii+3X12i+3X21i+3X22I+4B11+4B12+4B21

+4B22+I +112+I21 +I22











Table 8 Part-Machine Incidence Matrix for Model I Example.

Machine Type
Part 1 2 7
1 1 1
2 1 1


Table 9 By Part Demand.

Period Unit Unit
Part 1 2 Inv. Cost B/o Cost
1 450 0 1 4

2 0 450 1 4


Table 10 Machine Data.

Machine Type # Available Capacity/Machine

1 1 450

2 2 225

7 1 450


The full formulation for this problem is shown in appendix

A. Examples of each of the different types of constraints

will be shown below. The demand constraint for part 1,

period 1 is

Xiii+Xi2+Xinr+XII-I,+B,,=450

The capacity constraint for machine type 2 in cell 1 during

period 1 is

X,,,+X211225Y21











The shop capacity constraint for machine type 2 during

period 2 is

X121+Xi22+X12r+X121+X221+X222+X22r+X22,i450

The machine availability constraint for machine type 7 is

Y,7+Y72+Y7r:1

The cell size constraint for cell 1 is

Y11+Y21+Y,712

The solution to this model is to form one cell with all

the machines in it (a process layout), produce 450 units of

part 1 during period 1, and produce 450 units of part 2

during period 2. With this solution no inventory or

backorders would be required, and the objective function

value equals 900. It is interesting to see how the solution

would change if we had assumed a constant demand of 225

units for each part during each period. With this

assumption, the model forms two cells. The first cell would

have a type 2 machine and a type 7 machine. This cell would

produce 225 units of part 1 in each period. The second cell

would have a type 1 machine and a type 2 machine. This cell

would produce 225 units of part 2 in each period. The

objective function value would equal 0. If this plan were

implemented and the demand turned out to be the same as in

table 9 the cost would be 1125.












3.3 Model II



3.3.1 Description of Model


This model selects cells, which will be formed to

produce parts, from a set of potential cells. The objective

of the model is to minimize production, inventory, and

backordering costs.

The model uses a planning horizon which consists of T

periods. The parts to be produced during the planning

horizon are known. The demand for each part can vary from

period to period. Each part requires a set of machine types

for production, and the processing time per unit for each

part on each machine type, that is required by the part, is

a known parameter.

There are several potential cells that are being

considered for inclusion in the shop. All the cells cannot

be included because there are not enough machines available.

There are several different types of machines in the shop

and the number of machines available of each type must be

specified as an input parameter for the model. The

configuration of each potential cell in terms of the number

of machines of each type required by the cell is known.











Each machine has a limited capacity available during

each period in the planning horizon. The capacity of the

machine types within each cell is an important constraint in

this model. The total processing time required of a machine

type within a cell during any period cannot exceed the total

capacity of that machine type within that cell.

In order to effectively utilize machine capacity the

model allows for inventory planning decisions. The number of

units produced of a part during a period can exceed the

demand requirement for the period. The excess production can

be stored in inventory and used to satisfy demand

requirements in later periods. A part's demand requirement,

for a period, may also be partially or completely

backordered and excess production of the part during a later

period may be used to fill backorders. If an inventory of a

part is held then there is a cost associated with the

inventory. If a backorder is incurred there is a cost

associated with the backorder. Each part has a unit

inventory holding cost and a unit backorder cost associated

with it. These costs are on a per period basis.

The unit cost of producing a part varies depending on

which cell the unit is produced in. Each part has a cost










79

assigned for each cell. These costs can reflect the material

handling and other costs associated with producing a unit of

the part in a particular cell. A unit of a part can also be

produced by using machines that reside in two or more cells.

If this occurs the unit will incur costs associated with

intercell material handling and other costs, such as the

paperwork required for intercell movement and tracking.

3.3.2 Model II Formulation



The formulation of model II is shown below.


1 T K I T
Min Z= (CiiXit,+hiit+biBiJt + (CikXik)
i-I t-1 k-1 i-I t-I


Subject to


K
Iit-1-Bit-i+ E Xitk+XitI- Ii+Bit=dit




I

PijXitk< CjZkMjk
i-1





SPijXitk+ pij (Xitz) CA
i-I k-I i-I

K
E MjZk Aj
k-I


for i=l, ..,I; (10)
t=1,...,T.





for k=l,...,K & r; (11)
for j=1, ...,J;
for t=l,...,T.


for j=l, ...,J;
for t=l, ...,T


(12)


for j=1,...,J. (13)









80

Zk, {0,1} for k=l,...,K. (14)


All other variables a 0 (15)



where

I=The total number of parts.

T=The total number of periods.

J=The total number of machine types.

K=The total number of cells.

i=part index (i=l,...,I).

t=period index (t=l,...,T).

j=machine type index (j=1,...,J).

k=cell index (k-l,...,K).

Cik=The cost to produce 1 unit of part i in cell k.

Cii=The cost to produce 1 unit of part i using two or

more cells (uses intercell movement).

hi=The cost to hold 1 unit of part i for 1 period.

bi=The cost to backorder 1 unit of part i for 1 period.

dit=The demand for part i in period t.

Pij=The processing time for part i on machine type j.

Cj=The capacity in hours of 1 machine of type j.

Aj=The number of machines available of type j.











Mjk=The number of machines of type j required to form

cell k.

Xilk=The number of units of part i produced in cell k

during period t.

Xi,,=The number of units of part i produced using

intercell movement during period t.

Iit=The inventory of part i at the end of period t.

Bit=The number of units on backorder of part i at the

end of period t.

Zk=1 if cell k is formed; 0 otherwise.

3.3.3 Description of Model II Equations


(9) Objective function: The objective function sums

production, inventory, and backorder costs for all parts

over all periods. There are four terms in the objective

function. The first term sums across all parts, cells, and

periods the cost to produce a unit of part i in cell k times

the number of units produced of part i produced in cell k

during period t. The second term sums the cost to produce a

unit of part i using intercell movement times the number of

units produced of part i using intercell movement during

period t across all parts and all periods. The third term is











the cost to hold one unit of part i in inventory for one

period times the number of units of part i that are in

inventory at the end of period t summed across all parts and

all periods. The last term is the cost to hold one unit of

part i on backorder for one period times the number of units

on backorder at the end of period t summed across all parts

and all periods.

(10) This set of equations represents material balance

constraints. The first two terms on the left side of the

equality (Iit-_-Bit_1) represent the net inventory position of

part i at the beginning of period t (or the end of period

t-l) and the last two terms on the left side of the equality

(-Iit+Bit) represent the negative of the net inventory

position of part i at the end of period t. The middle two

terms on the left side of the equality (F Xit+Xiti) equals
k-i
production in units of part i during period t. Therefore,

the net inventory position of part i at the beginning of

period t plus production of part i during period t minus the

net inventory position of part i at the end of period t must

equal the demand for part i during period t (the right side

of the equality). There is a constraint for each part during

each period.









83

(11) These constraints ensure that the total processing time

required by parts produced in cell k during period t does

not violate the capacity of machine type j in cell k. There

is a constraint for each machine type, in each cell during

each period.

(12) These constraints ensure that the total processing time

required by parts produced during period t does not violate

the capacity of machine type j. There is a constraint for

each machine type during each period.

(13) These constraints ensure that the cells selected for

inclusion in the shop do not require more type j machines

than are available. There is a constraint for each machine

type.

(14) These constraints require that cell k is either

selected (Zk=l) or not selected (Zk=0) for inclusion in the

shop.

(15) These constraints require all the production,

inventory, and backorder variables to be nonnegative.

3.3.4 An Example of Model II


This example uses the data presented in tables 8 thru

10 of section 3.2.4. Table 8 shows the part-machine











incidence matrix; table 9 shows the demand by period, the

unit inventory cost, and the unit backorder cost for each

part; and table 10 shows the number of machines of each type

that are available and the capacity in units per machine of

each type. Three potential cells are considered for

inclusion in the shop. Table 11 shows the configuration of

each cell in terms of the number of machines of each type

that would be included in the cell if the cell is selected.


Table 11 Potential Cells

Machine Type
Cell 1 2 7

1 0 1 1

2 1 1 0

3 1 2 1


Table 12 shows unit production costs for each part in each

cell that the part can be completely manufactured in. Table

12 also shows unit production costs using intercell

movement.

The objective for this problem is

Min Z=X13+X+X12 +X213+X223+3X1I+3X12I+3X21I+3X22I+4B11+4B12+4B21

+4B22+I11+I1+I1+I22











Table 12: Production Costs

Part cell 1 cell 2 cell 3 Intercell

1 0 n/a 1 3

2 n/a 0 1 3


The full formulation for this problem is shown in appendix

B. Examples of each of the different types of constraints

will be shown below. The demand constraint for part 1,

period 1 is

Xiii+X13,+XI -In+B11=450

The capacity constraint for machine type 2 in cell 1 during

period 1 is

X,111225Z,

The shop capacity constraint for machine type 2 during

period 2 is

X121+X222+X123 +X12+X223+X22I 450

The machine availability constraint for machine type 2 is:

Zi+Z2+2Z32

The solution to this model is to select only cell 3.

Cell 3 contains all the machines in the shop so this

solution is a Process layout. 450 units of part 1 should be

produced in period 1, and 450 units of part 2 should be









86
produced in period 2. This solution requires no inventory or

backorders, and the objective function value equals 900. It

is interesting to see how the solution would change if we

had assumed a constant demand of 225 units for each part

during each period. With this assumption, the model selects

cells 1 and 2 for inclusion in the shop. 225 units of part 1

would be produced in cell 1 during each period, and 225

units of part 2 would be produced in cell 2 during each

period. The objective function value would equal 0. If this

plan were implemented and the demand turned out to be the

same as table 9 the cost would be 1125.















CHAPTER 4
SOLUTION PROCEDURES



4.1 Introduction



In this chapter, solution procedures for the models are

proposed. Since model II develops a shop configuration from

a list of potential cells, that may have been created by

solving model I several times with different cell size

parameters, the focus of the procedures proposed is on model

II. A heuristic for solving model I is proposed in section

4.2. This heuristic can then be run, using several cell size

parameters, to create a list of potential cells. Section 4.3

proposes lower bounding schemes for model II. Heuristics for

solving model II are proposed in section 4.4 The lower

bounding schemes in section 4.3 can be used, for large

problems, to evaluate the heuristics of section 4.4.












4.2 Model I Heuristic



This heuristic requires several iterations to develop a

solution for model I. During each iteration an attempt is

made to form one cell that meets the cell size constraint.

If a cell is formed (that meets the cell size constraint)

then the integer variables for that cell are fixed, and then

the next iteration of the heuristic attempts to form another

cell. If a cell is not formed (that meets the cell size

constraint) the heuristic stops and a solution has been

developed. The heuristic also stops if all the machines have

been used to form cells that meet the cell size constraint.

In addition to the notation of section 3.2.2, the

following notation is required.

Let n=The iteration index for the heuristic.

Let Mjk=The number of type j machines required by the cell

formed in iteration n of the heuristic.

The following steps are required:

Step 0. Initialization.

set n=l;

Step 1. Solve problem P1 (described below).












Step 2.

a. set a=1;

for j=l to J do

if Yjk>0 then a=0;

b. set b=l;

for j=l to J do
K
if Aj- Mjk>0 then b=O;
k-i
c. if a+b>0 then stop

else n=n+l and repeat step 1.

Problem P1 is as follows:



I T
Min Z=- E (MMRiXitr+MMIiXiti+hilit+biBit)
i-1 t-1


Subject to



Iit-l-Bit-1+ Xitk +Xitr+Xit- Iit+Bit=dit
k-1


I
PijXitk CjMjk






Pi XinC
7 i~tn j~
Ep=!.Y


(16)


for i=l, ...,I; (17)
t=1,...,T.


for k=l, ...,n-l;
for j=l, ...,J;
for t=l,...,T.



for j=l, ...,J;
for t=l, ...,T.


(18)





(19)




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