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A network approach to a class of multi-facility, multi-product production scheduling problems

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A network approach to a class of multi-facility, multi-product production scheduling problems
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Dorsey, Robert Caleb, 1943-
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English
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vii, 91 leaves. : ill. ; 28 cm.

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Subjects / Keywords:
Algorithms ( jstor )
Capital costs ( jstor )
Integers ( jstor )
Linear programming ( jstor )
Objective functions ( jstor )
Optimal solutions ( jstor )
Overtime ( jstor )
Production costs ( jstor )
Production scheduling ( jstor )
Scheduling ( jstor )
Critical path analysis ( lcsh )
Factory management ( lcsh )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 87-90.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Robert Caleb Dorsey.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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A NETWORK APPROACH TO A CLASS OF MULTI-FACILITY,
MULTI-PRODUCT PRODUCTION SCHEDULING PROBLEMS













By



ROBERT CALEB DORSEY


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTSS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA
1973





































TO LOU
















ACKNOWLEDGMENTS


I wish to acknowledge the considerable contributions to this

work by Dr. H. Donald Ratliff and Dr. Thom J. Hodgson, who acted

respectively as the Chairman and Co-Chairman of my Doctoral Committee.

Their energy, insight and clarity of expression have contributed

greatly to both the form and content of this dissertation.

I thank the remainder of my committee, Dr. Richard L. Francis,

Dr. Kerry E. Kilpatrick and Dr. Richard A. Elnicki, for their efforts

which contributed to the completion of this work. I would also like to

acknowledge Dr. Robert N. Braswell, Dr. Richard S. Leavenworth,

Dr. Eginhard Muth, Dr. B. D. Sivazlian and Dr. M. E. Thomas for their

assistance and support throughout my stay at the University of Florida.

Finally, I thank my wife, Lou, and my parents, Caleb and Dot

Dorsey, for their encouragement throughout the course of this work.

This dissertation was supported in part under grants

DA-ARO-D-31-124-70-G92A and DA-ARO-D-31-124-71-G183 and ONR contract

number N00014-68-A-0173-0013.


iii















TABLE OF CONTENTS

Page

ACKNO WLEDGMENTS ..... ...... iii

LIST OF FIGURES . vi

ABSTRACT . vii

CHAPTER

1 INTRODUCTION . .. .1

An Overview. ... ..... 1
Literature Review . 2
Practical Motivations . 5
A General Problem Description 7
Organization of the Dissertation 9

2 A NETWORK APPROACH TO THE PRODUCTION-INVENTORY
SCHEDULING PROBLEM WITHOUT BACKORDERING .. 10

Introduction . 10
Problem Development and Formulation 11
A Solution Procedure ... .16
An Example .. 20
A Warehouse Space Limitation ... 21
Extensions of the Model .. 22
Conclusions . 25

3 A NETWORK APPROACH TO THE PRODUCTION-INVENTORY
SCHEDULING PROBLEM WITH BACKORDERING .. 28

Introduction . 28
A Mixed Integer Programming Formulation .. 29
Reformulation as an All Integer Program .. 34
Formulation as a Minimal-Cost Flow Problem 37
Conclusions . .. 43

4 A MULTIPLE FACILITY, MULTIPLE PRODUCT PRODUCTION
SCHEDULING PROBLEM WITH OVERTIME .. 45

Introduction .. .. 45
The Overtime Production Scheduling Problem .. 47
A Special Case of the Overtime Scheduling Problem 54
Conclusions . 61












TABLE OF CONTENTS (Continued)


CHAPTER Page

5 AN EFFICIENT INTEGER PROGRAMMIING ALGORITHM FOR A
MULTIPLE BATCH PRODUCTION SCHEDULING PROBLEM 62

Introduction . ... 62
An Integer Program . ... 64
An Algorithm for Solving the Integer Program .. 66
An Example . ... 67
Other Properties of the Integer Program ... 69
The Production Scheduling Problem .. 70
Computational Experience ... 81
Conclusions . ... 81

6 SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH ...... 82

APPENDIX PROOF OF THEOREM 5.1 .. 85

BIBLIOGRAPHY . ... 87

BIOGRAPHICAL SKETCH . ... 91

















LIST OF FIGURES


Figure Page

2.1 Product Inventory Change during a Production Period 13


2.2 Network Example of Problem P2.1 ... 17


2.3 Network Example with Facility and Period
Dependent Costs .... 26


3.1 Network Example of Problem P3.1 ... 33


3.2 Network Example of Problem P3.3 ... 42


4.1 Network Example of Problem P4.2 .. 53


5.1 Geometric Interpretation of the Algorithmic Example 68


5.2 Network Example of Constraint Set P5.6 .. 80










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


A NETWORK APPROACH TO A CLASS OF MULTI-FACILITY,
MULTI-PRODUCT PRODUCTION SCHEDULING PROBLEMS

By

Robert Caleb Dorsey

March, 1973


Chairman: Dr. H. Donald Ratliff
Co-Chairman: Dr. Thom J. Hodgson
Major Department: Industrial and Systems Engineering


A class of multiple facility, multiple product, production-

inventory scheduling problems is considered over a finite planning

horizon. The horizon consists of discrete production periods during

each of which at most one product can be assigned to a facility.

Product demands are constant over a period but not necessarily identi-

cal in all periods. Each product has a minimum batch size which can

be made, and only integer multiples of this basic batch size are allowed.

The problem objective is to determine an assignment of products to facil-

ities which minimizes the sum of any production, inventory, backordering

and facility usage charges which occur over the horizon. Four differ-

ent scenarios for this class of problems are considered.

In a straightforward manner, each of the scenarios which are

considered is formulated as an integer program. It is then shown that

each of the problems can be reformulated as a network flow problem which

can be solved using any of the very efficient algorithms for finding

minimal-cost flows in single commodity networks. In addition, for some











important special cases, single pass procedures are developed which are

more efficient than the standard flow algorithms. In the fourth sce-

nario the horizon length is also considered to be a variable in the

optimization. In this scenario a specially structured integer program-

ming formulation is identified, and an extremely efficient algorithm is

developed for determining its solution.


viii
















CHAPTER 1


INTRODUCTION




An Overview


Inventory management and production control represent two major

components of the manufacturing process. Traditionally, research in

production-inventory systems has been divided into two groups based on

the assumptions regarding the interaction of these components. In one,

the assumption is made that the two components are independent, and

optimization can be achieved by optimizing the control of each component

separately. In the other, inventory management and production control

are not assumed to be independent, but rather the components interact

and optimization must be achieved by considering the two components

jointly.

The research in this dissertation falls in the second group of

thought and deals more specifically with the analysis of a multi-

facility, multi-product, integrated production-inventory system.

Realistic examples of the multiple facility, multiple product system

include textile spinning machines, metal stamping operations, steel

rolling facilities, glass molding lines, container manufacturing machines

and packaging lines.

Contributions within this dissertation include original formu-

lations of the multi-facility, multi-product problem as integer programs











and network flow problems which offer considerable insightsinto the

operation of the production-inventory system. These insights allow us

to enrich the problem generously to include many other realistic aspects

of the production-inventory system heretofore not solvable with present

techniques. In addition, efficient algorithms are developed for solving

different types of associated problems which in many cases are single

pass procedures which can be easily carried out by hand.




Literature Review


Problems of the type considered in this dissertation generally

have been modeled as one-facility, deterministic problems where produc-

tion rates and demand rates for each of the products take on known con-

stant values. Only one product can be produced on the facility at a

time, and a setup is necessary each time a different product is to be

produced. Stockouts are not permitted, and backordering to meet demands

may or may not be allowed. Inventory carrying charges for each of the

products are assessed in direct proportion to the amount of product on

hand. The problem is to determine a schedule for allocating the produc-

tion time of the facility to each of the products so that the costsasso-

ciated with setting up, producing, backordering and carrying inventory

are minimized over a given planning horizon.

One general difference in models dealing with this problem is

in the definition of the planning horizon. In one approach the length

of the planning horizon is assumed to be infinite, and in the other the

length of the horizon is assumed to be fixed and finite. In the finite

horizon case, the optimization problem involves the determination of











a feasible assignment of products to the facilities which minimizes

costs. In the infinite horizon approach, the existence of a finite,

but variable length, production cycle is assumed where this production

cycle repeats itself infinitely many times throughout the horizon.

Hence, the infinite horizon problem involves the joint determination

of the optimal production cycle and the optimal assignment of products

to the facility during this production cycle.

A well-known approach to the infinite horizon form of the problem

is to assume that each product is set up and produced exactly once dur-

ing each production cycle where the amount produced for each product at

each setup is proportional to the length of the production cycle. In

this case the object of the optimization reduces simply to the determi-

nation of the optimal length of the production cycle. This approach is

typically referred to as a model with a pure rotation cycle. An optimal

macro-plan for this model has been determined by Magee [12] and by

Rodgers [14].

The more general approach to the infinite horizon problem is

not to require a pure rotation cycle. Rather each product may be pro-

duced on an individual production cycle where the only restriction is

that feasibility must be maintained with respect to the facility. At

present, there is no general method for determining an optimal macro-

plan for this model. However, upper and lower bounds can be determined

on the cost of the optimal solution. A lower bound is determined by

solving the one-facility, one-product problem with a pure rotation

cycle for each of the products and summing their respective costs.

If, in fact, the optimal production quantities determined by solving












each of these problems can be scheduled on the facility without causing

scheduling conflicts, then this schedule is an optimal macro-plan for

the multiple cycles problem. In like manner, an upper bound on the

optimal cost is determined by solving the one-facility, multiple product

problem with a pure rotation cycle and noting the associated cost.

Researchers who have investigated this problem include Bomberger [1],

Elmaghraby [3,4], Haessler [9], Hodgson [10], Madigan [11], Magee [12],

Maxwell [13], and Stankard and Gupta [15].

Of primary interest in this dissertation is the finite horizon

approach to the problem. In this approach the horizon is assumed to

consist of discrete production periods (shifts, for example) during

which at most one product can be assigned to the facility. During any

period product demands are assumed to be constant but not necessarily

the same in all periods.

The production-inventory scheduling problem was initially

modeled as a finite horizon problem by Wagner and Whitin [16]. They

considered a one-facility problem for the case where backordering is

not allowed and production amounts in any period can vary. They

developed a dynamic programming procedure for solving the problem and

were able to realistically enrich the problem to include period depend-

ent product demands, inventory costs and setup costs.

Zangwill [17] extended the results of Wagner and Whitin to

include backordering and showed that the problem could be formulated

and solved as a minimal-cost flow problem. In addition, he generalized

the cost function to include concave inventory and backordering costs.











In order to bring the finite horizon approach to the problem

more closely in parallel to the real environment, Elmaghraby and Mallik

[2 ] considered the problem in a slightly different context. They

assumed that the production amounts of each product at each assignment

of the product to the facility are fixed and constant. They then

developed a dynamic programming recursion for solving the problem for

the case where demand rates and production costs are assumed to be con-

stant and identical in all periods. For the batch type production sys-

tem where product inventories are updated only at the end of each pro-

duction period, their results apply to a special case of the problem

considered in Chapter 3 of this dissertation.




Practical Motivations


While the assumptions in the previous models provide a basis for

analyzing industrial scheduling problems, it is frequently true that the

characteristics of the real system are much more complex than these

models allow. For example, many production operations consist of numer-

ous facilities (M). When customer demands are received which require

processing in this operation, the system controller must decide, in

addition to what, when and how much of the product to produce, which

facility or facilities should be used. Underlying in the previous

models is the assumption that facility selection is a deterministic

problem which is not part of the decision making, and hence system

optimization can be achieved by solving M one-facility problems.

However, this is not always the case in the real environment as system











controllers frequently interchange products between facilities in order

to meet demand requirements and minimize costs.

Another consideration is initial product inventories. In the

infinite horizon approaches, the assumption is made that the initial

product inventories are "in phase with" the optimal macro-plan of allo-

cating the production time of the facility to each of the products.

In the industrial environment this is not necessarily true, and indus-

trial schedulers in using the infinite horizon approach may still be

left with a subhorizon of unknown time duration during which the facil-

ity must be scheduled "out of phase" with the optimal macro-plan.

In the industrial environment, it is rare that decision making

is based on a single plan of operation as many of the previously

described models assume. For example, an industrial scheduler fre-

quently may vary production run lengths, schedule overtime, schedule

alternate facilities, purchase product from outside sources, etc., in

order to effectively control the system.

Another consideration which is frequently applicable to many

production operations is a fluctuating demand rate. The fluctuations

of the demand rate can arise in numerous ways at least some of which

can be predicted in advance by good forecasting. Many industries have

seasonal demands year-in-and-year-out where the demand might be repre-

sentable by a cyclic function. Some industries have a record of steady,

continued growth which can be represented possibly by a ramp or exponen-

tial growth function. Other industries have consistent sales through-

out the year with large increases around the holiday season (impulse

function). Temporary plant shutdowns for maintenance purposes or












vacations also occur in which case the demand rate for the supplier to

these plants may suffer a sudden drop (negative impulse function). New

plants are being bL. and new uses are being found for different

products either of -nich might correspond to a step function in the

demand rate forecast.

As these examples indicate, the real environment is often much

more complex than the assumptions of existing models, and the develop-

ment of new insights is necessary if optimal system control is to be

achieved. This dissertation deals specifically with some of the prob-

lems associated with the operating production-inventory system. The

objective of the research is to provide original insights into the

achievement of optimal control in this complex system as well as to

provide efficient procedures, based on these insights, for solving

related problems in the real environment.




A General Problem Description


An industrial process made up of M facilities in parallel is

considered where there are N different products to be produced over

a finite, but possibly variable length, planning horizon. The horizon

is made up of production periods (shifts, for example) which have equal

time duration. During any period at most one product can be scheduled

on each facility. When any product i is assigned to a facility for one

period, an integer multiple of p. units of product is made. Production
1

completed is added to each product's inventory, and nonnegative demands

(dik units for product i in period k) are satisfied from this inventory.
iK.











Backordering may or may not be allowed (depending on the application),

and setups are assumed to occur between production periods. Initially,

each product i is assumed to have a nonnegative beginning inventory

level denoted by Ii0 (units of product i), and when applicable, a non-

negative initial backorder level denoted by Bi. (units of product i).

During the horizon the system controller must schedule the products to

the facilities in such a manner as to meet all product demands, and at

the same time build up each product's ending inventory level to a

desired nonnegative amount denoted by I.i. The problem objective is to
iH

determine such a production schedule which minimizes the sum of any pro-

duction, inventory, backordering and facility usage charges over the

horizon.

As an example of this problem, consider a simplified version of

a two-product, one-facility problem over a three-period horizon where

at most p. units of each product i can be produced by the facility dur-

ing any period. During any period, then, the system controller has

three options. He can schedule either product one or product two to

the facility, or he can leave the facility idle. Hence in this problem
3
there are 3 or 27 possible solutions to the scheduling problem. Depend-

ing on the imposed constraints, all, some or none of these schedules

may be feasible solutions to the system controller's problem. What the

system controller must attempt to do is determine a feasible schedule

which minimizes costs. In general for a N-product, M-facility, H-period

problem of this type, there will be (N+1) possible schedules to eval-

uate. Hence, it is clear that something significantly better than total

enumeration is needed for the system controller to be able to solve











realistic kinds of problems. One of the goals of this dissertation is

to provide an efficient procedure for solving this problem.




Organization of the Dissertation


In Chapter 2 a fixed horizon problem with production and inven-

tory costs is addressed. A mathematical formulation of this problem is

developed which allows the problem to be solved using any of the algo-

rithms for finding minimal-cost flows in single commodity networks.

However, for an important special case, a single pass algorithm is

given which is even more efficient than the standard flow algorithms.

In Chapter 3 a fixed horizon problem with production, inventory,

and backordering charges is considered. In a straightforward manner,

this problem is formulated as a linear, mixed integer program. It is

shown, however, that the problem can be reformulated as an all integer

program which can be solved as a minimal-cost flow problem.

In Chapter 4 a fixed horizon problem is considered where the

system controller has an overtime option available for use in schedul-

ing the system. This problem is formulated as an integer program, and

a solution procedure is developed for solving the problem which involves

solving a minimal-cost flow problem.

In Chapter 5 the problem of bringing initial product inventories

"in phase with" a desired infinite horizon cycle plan is considered.

The problem is formulated as an integer program which has special struc-

ture, and a very efficient algorithm is developed for solving the problem.

A summary of the results of this research and suggestions for

future research are given in Chapter 6.

















CHAPTER 2


A NETWORK APPROACH TO THE PRODUCTION-INVENTORY
SCHEDULING PROBLEM WITHOUT BACKORDERING




Introduction


The production-inventory scheduling problem is considered over

a fixed planning horizon made up of H production periods. Each of the

M facilities is considered to be identical in that product i can be

produced by any facility at a fixed production rate of p. units per

period and a cost of c. dollars per period. Demands for each product i
1

occur continuously during each period k according to a fixed demand

rate of dik units per period. Inventory carrying charges are assessed

on product inventories. During the horizon the system controller must

schedule the products to the facilities in such a manner as to meet all

demands, without backorders, and at the same time build up each product

i's inventory level to a desired minimum ending amount denoted by I. .

The objective is to determine a production schedule which minimizes the

sum of production and inventory carrying charges over the horizon.

This problem is formulated as a network flow problem for which

efficient solution procedures are known. However, for an important

special case a single pass algorithm is given which is more efficient

than the standard flow algorithms. In addition, the general model can

be enriched generously to include numerous other realistic aspects of











the production system without severely complicating the solution proce-

dure. These generalizations include such things as product and facil-

ity dependent production costs, facility usage costs, period dependent

facility availability, period dependent inventory carrying costs, inven-

tory constraints, facility and product assignment restrictions, and

horizon dependent facility allocation costs. For presentation purposes,

only basic assumptions are used in the development of the problem with

the more detailed aspects of the model being identified in the

"Extensions of the Model" section. Many of these enrichments will no

doubt become apparent in the development.




Problem Development and Formulation


For the problem being considered, backordering is not allowed

and product inventory changes during any period are assumed to occur

linearly (i.e., during any period the production and demand rates are

constant). Thus stockouts on product i will occur during period k if

and only if at least one of the ending period inventories, I or
i,k-l

Iik, is negative. Hence, requiring that ending period inventories be

nonnegative for all products and periods is a necessary and sufficient

condition for preventing stockouts.

Define wik as the minimum number of times (integer) product i

must be produced during the first k periods in order to have a non-

negative inventory level at the end of period k. Then


j=1
ik 1 ( k = 1,2,...,H-1
j=1











where (a) denotes the smallest integer greater than or equal to a.

In addition to being nonnegative, product i's inventory level at the

end of the horizon must be greater than or equal to I. Hence, define

H
H
WiH = max [0, ((iH + E di iO)/Pi> i = 1,2,...,N,
j=1


and let W be an N by H matrix with components [w i].

Define x.k as the number of facilities scheduled to produce

product i during period k where X is defined as an N by H matrix with

component [x ik. The inventory level for product i at the end of per-

iod k can then be written as


k k
I p x. + I d k = 1,2,...,H.
ik 1 10 ij
j=1 j=1


Figure 2.1 gives an example illustration of the change in inventory of

product i during a period in which the amount produced on product i

exceeds the amount demanded. Obviously, the linear change in inventory

of product i during period k is a function of the number of facilities

scheduled to produce product i, and the slope of the line can take on

M+1 possible values (i.e., xik can be 0, 1,2,...,M-1 or M) in period k.

The average amount of inventory on hand for product i during period k

in any of these cases is given by


k-I k-i
(I (-)(p x + 2Z x..+ 21. -d -2 E d.
i,k-1 k 2 Piik j ij 2i0 ik-2 j ij
j = ,2,j=o
k = 1,2,...,H


where xio and di are defined to be zero.
10 i0







































TIME (IN PERIODS)


Figure 2.1


Product Inventory Change during
a Production Period.


I ,k-I







0


ik















0











Denoting by 0. the inventory carrying charge per unit of product
1

i carried in inventory for one period, where 0i > 0, the total inventory

cost over the horizon is given by


N H k-1 N H k-1
( i2)0iPixik+2 E x. i+ Z (-)(2 -d -2E d i
2k 2 1 d i.
i=1 k1 j=0 i=l k=l j=O


1
Letting i = ()0iPi


N H k-1
K= 2 (-)0.2ii -dik -2 E d. .
i=1 k=l j=0


and noting that


H k-1 H
S2 Z x.. = E (2H-2k)x ,
k=l j=0 1 k=l k


the total cost of carrying inventory and producing the N products over

the horizon H can be written as


N H N H
S 2 (2H-2k+l)$.xik + K + E c. E xi (2.1)
ik 1 ik
i=l k=l i=1 k=l


where K is a constant.

The system controller's objective is to minimize (2.1) subject

to the constraints that (a) only M facilities are available for produc-

tion during each period, (b) demands must be met without allowing back-

orders, and (c) at most one product can be scheduled on each facility

each period where a facility, when assigned to product i, is assumed to

produce p. units of product i. A mathematical formulation of this
problem is as follows
problem is as follows:










N H N H
Minimize E E (2H-2k+l) ii + K+ E c. E x
i=1 k=l i ik i=1 I k=1 ik

subject to

N
[P2.1] E xik < M k = 1,2,...,H
i=l

k
k x w i = 1,2,...,N
j= ij ik k = 1,2,...,H


x 0, integer 1,...,N
ik k = 1,2,...,H

It is easily shown that problem P2.1 has a feasible solution if and only

N
if E Wik < AMk for all k = 1,2,....,H.
i=1

Note in problem P2.1 that all costs are positive and that any
H
feasible solution (X) to the problem must have E xik wH for all
k=l
i = 1,2,...,N. Hence, it follows that any optimal solution will have

each product i produced exactly wiH times during the horizon. This

N H
implies that the term, c. Z Xik is a constant in any optimal solu-
i=l1 k=l
tion to problem P2.1. Knowing that constants do not affect the optimi-

zation, problem P2.1 can be reformulated as follows:

N H
Minimize E Z (2H-2k+l) i.x
i=l k=l 1
subject to
N
[P2.2] Z x M k = 1,2,...,H
k2 ik .
i=l

k
E x 2 w i = 1,2,...,N
j=lij ik k = 1,2,..,H


X i. 0, integer 1,2,...,N
ik k = 1,2,...,H .











This problem can be viewed as a flow problem which can be solved

using any of the algorithms for finding minimal-cost flows in single

commodity networks. For specific details, see Ford and Fulkerson [ 5].

An example network for a M-machine, two-product, three-period problem

is shown in Figure 2.2. Note in this network that the number of machines

does not affect the number of nodes and arcs. Rather for an N-product,

H-period problem, there are H(N+1)+2 nodes and H(2N+1) arcs. In this

network the variable xik represents the flow from node k to node ik along

the arc (k,ik). Efficient algorithms are available for solving this

type of problem (e.g., Fulkerson [6 ]). However, because of its special

structure, a more direct solution procedure can be derived.




A Solution Procedure


The problem formulation calls for each product i to be assigned

at least wiH times during the horizon H, and since all costs are posi-

tive, each product will be assigned exactly wiH times in any optimal

solution. For all periods the increase in cost for assigning product

i to a facility in period k' as opposed to a facility in period k where

k < k : H is 2(k-kl)i Hence, from the standpoint of product i, it

is economically more advantageous to assign product i in period k rather

than period k'.

Suppose that the products are numbered so that


N N-1" 2 .21*

If there is an optimal solution to problem P2.2, then there is an

optimal solution (X) to problem P2.2 such that











Costs


Upper
Bounds

M


Lower
Bounds




"w11



W 2 I

W2 1




w12





w22


w23


Figure 2.2 Network Example of Problem P2.1.










if p > q and

H H
if Xpk < (wpH x where E x. 0 (2.2)
j=k+l j=H+1 i

then x = 0.
qk

To see that this result is true, assume that an optimal solution (X*)

to problem P2.2 has

H *
x < (w w ) E x
pk pH p,k-l pj
j=k+l


and x > 0. Let k be the latest period such that k < k and x > 0.
qk pk
Such a period must exist if the solution is feasible. By making a

pairwise swap, a feasible solution (X) can be constructed where


x.. + 1 for (i=p,j=k) and (i=q,j=k')
1J

x.. x. 1 for (i=p,j-k') and (i=q,j=k)


13
x.. otherwise.



This solution has objective function value


z = z + 2(k-k')q 2(k-k')4


Sz since > .
*p q

H
This process can be continued until either xk = (w -w ) E x
pk pH pk-1 j pJ
j=k+l
or x = 0. In either case an optimal solution with respect to p and

q has been constructed. Similarly, by starting with k = H and then

letting k = H-l, k = H-2,...,k = 2, an optimal solution can be con-

structed such that condition (2.2) is satisfied for all products and

periods.










It should also be noted that once the products are numbered

there is at most one feasible solution which satisfies condition (2.2).

Hence, if a solution can be constructed which satisfies condition (2.2)

for all products and periods, the solution is optimal.

An algorithm based on the above result can now be given for

solving problem P2.2. In the algorithm each planning period is con-

sidered once where period H is the first period assigned. During each

period, the priority in which assignments of products to facilities are

made is based on the costs i.. The larger the cost, the greater the

product's priority. Considering each product in the order of its pri-

ority, the number of facility assignments made on product i in period

k is based on (a) the availability of facilities after other products

with a higher cost have been assigned and (b) the maximum desirable num-

ber of assignments of product i in period k. Formally, the algorithm is

as follows where M denotes the availability of facilities at intermedi-

ate stages of the procedure and p and t indicate respectively the cur-

rent product and period under consideration.


Step 0. Order the product indices so that


N *N-1 "'* 2 1

Define wi0 =0 for all i = 1,2,...,N, and set t = H.

Step 1. Set M' = M and p = N.

Step 2. Determine:
H
x = min (M', (wH w ) Z x p
t pH ,t- j=t+


M = M x
pt











Step 3. Set p = p-1. If p 1, go to Step 2. Otherwise,

go to Step 4.

Step 4. Set t = t-1. If t 1, go to Step 1. Otherwise,

go to Step 5.

H
Step 5. Terminate. If xi = wiH for all i = 1,2,...,N,
k=l k
then an optimal solution (X) to problem P2 has been

determined. Otherwise, the problem does not have

a feasible solution.


The algorithm is a one pass procedure which terminates after

exactly HN iterations since each product is considered once during each

period. At termination, the constructed sequence is deemed to be

feasible or infeasible depending on whether or not wiH facility assign-

ments have been made for each product. Obviously, an infeasible solu-

tion can be made feasible only if the number of facilities is increased

or if the no stockout restriction is relaxed. Optimality of a con-

structed, feasible solution follows from the result given in (2.2).




An Example


As an example of the solution procedure, consider a two-product,

three-period problem where

M= 102 312
W =
= (10,20) 3 6 8


The product indices are already ordered so that 2 1. Thus the

solution procedure goes as follows:













a. Assign product 2 twice to period 3 since min {10,2} is 2.

b. Assign product 1 eight times to period 3 since min (8,93 is 8.

c. Assign product 2 three times to period 2 since min (10,3] is 3.

d. Assign product 1 twice to period 2 since min {7,23 is 2.

e. Assign product 2 three times to period 1 since min (10,33 is 3.

f. Assign product 1 twice to period 1 since min [7,23 is 2.

2 2 8
g. Stop. An optimal solution is X = with objective
3 3 2

function value z = 760.



A Warehouse Space Limitation


In many production environments, there exists a space limitation

on the maximum amount of material which can be carried in inventory at

any given time. In the problem considered in this chapter, a warehousing

restriction of this type can easily be incorporated into the model for

the special case where one machine-period of production of product i

takes up the same amount of space as one machine-period of production

of any other product. For this special case, let Sk denote the maximum

number of machine-periods of production which can be carried in inven-

tory during period k where S1 is assumed sufficient to store the initial

product inventories. For the inventories to be within limits, then,

it must be the case that the inventory on hand at the end of each period

k is less than or equal to Sk and Sk+.l Equivalently, this implies that


N k N k N
E Z x. min {S Sk + E Z (d. /p.) Z (o /P.) (2.3)
i=1 j= 1 k+l i=l j=1 i=

for all k = 1,2,...,H.











Obviously, (2.3) can never be satisfied if the right-hand side

is strictly negative. Hence, we are only interested in pursuing the

problem for the case where the right-hand side is nonnegative for all

k = 1,2,...,H. Also, xik must be integer valued for all i = 1,2,...,N

and k = 1,2,...,H. Therefore, we can define Uk as the largest integer

less than or equal to the right-hand side of (2.3). The inventory con-

straint can then be included in problem P2.2 by annexing the constraint

N k
E Ex. Uk.
i=1 j= 1


The revised problem is still a network problem which can be

solved using any of the algorithms for finding minimal-cost flows in

single commodity networks. In fact, this warehousing limitation can

easily be incorporated into the previously given solution procedure by

setting X = 0 in step 0 and in step 1 redefining

N r
M = min (M, min (U E x.)}.
r=t,...,H i=l a=t





Extensions of the Model


There are a number of other realistic aspects which can be

easily incorporated into the model. With the exception of (a) below,

the previous algorithm is not satisfactory for determining an optimal

solution when any or all of these additional enrichments are included

in the model. However, an optimal solution can easily be determined

for these cases by a minimum-cost flow algorithm. The enrichments

include:











(a) The number of facilities available in each period may vary.

This assumption is easily included in the previously given solution

procedure by substituting Mt for M in step 1 where Mt denotes the

number of facilities available in period t.

(b) In addition to the cost of producing each product for one

machine-period, there can be a cost (convex) incurred on the number of

facilities in use at any given time. This can be introduced into the

network interpretation of the problem by using the standard trick'of

replacing each arc (s,k) by M different arcs ((s,k) ; I = 1,2,...,M)

each having an upper bound of one on the flow through that arc. Any

flow on arc (s,k)2 would incur a cost A the difference in cost for

using I facilities during period k rather than 1-1 facilities. Any

cost for not using any facilities is simply a constant which does not

affect the optimization. The convex cost assumption implies that


AM M-1 2 1


which insures that any flow in period k will be placed on arc (s,k)2

before arc (s,k)+1 unless equality holds in which case it does not

matter.

(c) There can be a technological restriction on the number of

facilities (m.) which can be used to produce product i simultaneously.

This assumption can be introduced into the network interpretation of

the problem by placing upper bounds of m. on all arcs (k,ik).
1

(d) There can be physical limitations (L.) on the maximum number

of units of product i which can be carried in inventory at any given

time so long as









k
L. r max [PiWik +iO di i = 1,2,...,N .
k=0,1,...,H j=0

This assumption can be introduced into the network interpretation of the

problem by placing upper bounds uik on the arcs (ik;i,k+l) where

r k
Uik = L(Li I1. + Zd )/p k = 1,2,...,-1
j= k = 1,2,...,H-

and [a] is defined as the largest integer less than or equal to a.

(e) Period dependent production costs (c ) can be included in
ik
the network interpretation of the problem by adding cik to the cost

(2H-2k+l)*i already on the arcs (k,ik).

(f) In the formulation of the problem P2.2, it has been assumed

that M facilities have been allocated to the production process for use

throughout the entire horizon H. Another realistic consideration is

a problem where there is competition for these facilities on a long

term basis (i.e., the value of M becomes a variable in the optimization).

This competition might be denoted by a nonnegative charge f(M) where

logically f(M) would be a monotonically nondecreasing function of M.

When this charge is included, a procedure for problem optimization is

to solve problem P2.2 for the maximum possible value of M. From

the optimal solution to this problem, determine the maximum number of

facility assignments (r) in any period. Then an optimal solution to the

allocation problem occurs for that value of M (M = 0,1,...,r) which min-

imizes z(M) + f(M) where z(M) is the corresponding optimal solution to

problem P2.2. This requires that problem P2.2 be solved up to r times.

(g) The cost of producing product i for one machine-period can be

dependent on the facility as well as the period in which the assignment











is made. For this case the problem formulation becomes


N H M N H M
Minimize Z Z (2H-2k+l) i. x. + E E c. CiXik
i=l k=l 1=1 i=1 k=l a=1
subject to

N M
[P2.3] Z E xik < M k = 1,2,...,H
i=1 =1

k M
S x w i = 1,2,...,N
j=l E=1 ij ik k = 1,2,...,H

i = 1,2,...,N
xilk = O or 1 1 = 1,2,...,M
k = 1,2,...,H

where ci k is the cost of producing product i on facility I in period

k and xilk denotes the number of times product i is assigned to facil-

ity I in period k. An example network interpretation of this problem

is given in Figure 2.3. In this example the variable x. k represents

flow from node mek to node nik along the arc (m k,nik).

(h) Realistically, it may be technologically impossible to produce

product i on facility a. This restriction can be included in the model

by omitting arcs (mik,nik) in all periods k = 1,2,...,H.



Conclusions


A multi-facility, production-inventory scheduling problem has

been considered which can be solved as a minimum-cost flow problem.

However, because of the nature of the objective function, an even

simpler solution procedure has been developed which, even for large

problems, can easily be carried out by hand. In addition, the model

can be generalized to allow for the inclusion of other realistic






Costs
m I 5 I + C
V2 +C211
Upper
Bounds

S4V5 22 + c221


m "1 + C 112
1(2 3v 2 + 212 12



\ 22( 3T V I+C122_ "22
3' 2 + C222



\ ('13 +c C113 n
1 32 + c213 13




I 2 I + C 12 n223
Y /2 + C223

Figure 2.3 Network Example with Facility
and Period Dependent Costs.


Lower
Bounds

Iw



w21




"12




W22











assumptions. Efficient minimum-cost flow algorithms are available for

solving the general problem when any or all of these enrichments are

included.

In this model it has been assumed that inventories change

linearly from the end of one period to the end of the following period

as a result of constant production and demand rates. However, the

results (with the exception of the warehouse space limitation) are

equally valid for any system in which inventory changes from the end

of one period to the end of the following period do not drop below the

minimum of the beginning and ending period inventories. This includes

the inventory assumptions of a batch type production system in which

inventories are updated only at the end of each production period.

The only revision in the model would be in the assessment of inventory


carrying charges.
















CHAPTER 3


A NETWORK APPROACH TO THE PRODUCTION-INVENTORY
SCHEDULING PROBLEM WITH BACKORDERING




Introduction


In Chapter 2 the multiple facility, multiple product production-

inventory scheduling problem without backordering was considered. In

this chapter we address the more general problem where backordering to

meet demands is allowed. Specifically, the cost of producing each

product is dependent on the facility on which the product is produced

as well as the period during which the assignment is made. Production

completed during period k is added to each product's inventory at the

end of the period, and outstanding backorders (B. units) and non-

negative demands (di units) for product i are satisfied from this

inventory. Period and product dependent inventory carrying charges

and backordering costs respectively are assessed on material remaining

in inventory and outstanding demands at the end of each period. During

the horizon the system controller must schedule the products to the

facilities in such a manner as to meet all product demands on a first-

come, first-served basis, and at the same time build up each product's

inventory to a desired nonnegative ending amount (iH units). The prob-
iH
lem objective is to determine a production schedule which minimizes the

sum of production, backordering and inventory carrying charges over the

horizon.











In succeeding sections we formulate this multi-product, multi-

facility production scheduling problem as a linear, mixed integer pro-

gram. Once formulated, this problem can be given a network flow inter-

pretation which looks very similar to the network interpretation given

by Zangwill [18] for his one-facility problem. However, in contrast to

Zangwill's problem, this mixed integer programming formulation can be

solved as a minimum-cost flow problem only for a very special case.

We show, however, that this problem can be reformulated so that the

general case can also be solved using any of the algorithms for finding

minimal cost flows in single commodity networks. This reformulation,

in addition to being a model for a very general multi-facility, multi-

product problem can be enriched to include numerous realistic aspects

of the production system without severely complicating the.solution

procedure.


A Mixed Integer Programming Formulation


Initially, we will assume that there is a nonnegative number

of backorders on the books for each product (i.e., B. o 0) and that

initial product inventories for each product i are zero (i.e., I. =0).
iO
This assumption causes no loss in generality since the first-come,

first-served rule implies that the first I units of demand for each

product i will be satisfied from the initial product inventories irre-

spective of any decisions made in the problem optimization. Hence,

we can assume that any initial inventories have been used to adjust

the amount of initial backorders and demands appropriately so that

initial inventories can be assumed to be zero.











The formulation of this production scheduling problem is further

facilitated by redefining demands in terms of cumulative machine-periods

of production. This is done by defining wik as the number of machine-

periods of production of product i required to satisfy all demands in

periods one through k. Hence,



wk = E d.. + B. /p k = 1,2,...,H-1
ik j=1 13 k = 1,2,...,H-

and

H
iiH idik + 0 + iH/Pi
Sin E \d ik + B + IiH3. A 1,2,...,N.
k=


Define xijk as the number of times product i is assigned to

facility j in period k, and denote by c.. the cost of each of these
ijk
assignments where c.ij 0. Define Iik as the number of machine-periods

of production of product i in inventory at the end of period k, and

define Bik as the amount (in machine-periods of production) of unsatis-

fied demand for product i at the end of period k (i.e., the total amount

of product i on backorder at the end of period k). Now let 0ik be the

cost incurred per machine-period of production of product i in inven-

tory at the end of period k, and let bik be the cost incurred per

machine-period of production of unsatisfied demand for product i at the

end of period k where 0i > 0 and bi > 0. The system controller's
ik Ik
problem is to minimize the sum of production, inventory and backorder-

ing costs over the planning horizon subject to the constraints that

(a) all demands must eventually be satisfied, (b) only one product can

be assigned to a facility each period where a facility, when assigned

to produce product i, is assumed to produce exactly p. units of the











product, and (c) the inventory level for each product i at the end of

the horizon must be built up to a minimum ending amount of iH units.
A mathematical formulation of this problem is as follows:
A mathematical formulation of this problem is as follows:


N M H N H-1
Minimize E Z 2 c.. x. + E E 0 I
i=1 j=1 k=l i=l k=l


N H-1
ik + E bik Bik
ik k k=
i=l k=l


subject to


M k
SE x. Iik + B = wi
^ t iji ik ik ik
j=1 1=1

M H
Z E xi. i w
j=1 k=ljk

N
Sx. < 1
1ijk i
i=l k


I B 0
ik Bik


x. i 0, integer
ijk


i = 1,2,...,N
k = 1,2,...,H-1



i = 1,2,...,N



j = 1,2,...,M
k = 1,2,...,H


i = 1,2,...,N
k = 1,2,...,H-1

i = 1,2,...,N
j = 1,2,...,M
k = 1,2,...,H


Problem P3.1 is a linear, mixed integer program which can be

given a network flow interpretation. To construct the network, define

a node m. corresponding to each facility j and period k, and define
jk
a node nik corresponding to each product i and period k. From each node

mjk construct a forward arc (m ,ni ) to each node nk for all
jk jk ik ik
i = 1,2,...,N. The cost per unit of flow on each of these arcs is c.ik.
1Jk
Next connect each node nik to node ni,k+l by two parallel forward arcs

(nik ,n ) and one reverse arc (n ,n. ). The flow through one of
ik ,k+l i,k+l ik
the forward arcs must be exactly Wik. Hence, upper and lower bounds of

Wik are placed on the flow through that arc. Each unit of flow on the


[P3.1]


(3.1)











other forward arc incurs an inventory charge (0ik), and each unit of

flow on-the reverse arc incurs a backordering charge (b i). Finally,
ik
define a source node s and a sink node t where a forward arc (s,m. )
Ik
from node s to node m. is constructed for all j = 1,2,...,M and
Jk
k = 1,2,...,H and a forward arc (n ,t) from node niH to node t is con-
iH iH
structed for all i = 1,2,...,N. Since only one product can be assigned

to a facility each period, a capacity of one is placed on the flow

through each of the arcs (s,m k), and since wiH machine-periods of

product i must be produced during the horizon, a lower bound of wiH is

placed on the flow through the arc (n ,t). Then xik denotes the flow
iH ijk
from node mjk to node nik along the arc (mk ,nik), Iik and wik corre-
jk ik jk ik ik ik
spond to the amount of flow from node nik to node n. along each of
ik 1,k+l
the forward arcs (n ni,k+l ), and Bik denotes the flow from node ni,
ithe forward arcs kk ik 1 i,k+l

to node nik along the reverse arc (ni,k+l,nik). An example network for

a two-product, one-facility, three-period problem is given in Figure 3.1.

In this problem if the w 's are all integer, then problem P3.1

can be solved using any of the algorithms for finding minimal-cost flows

in single commodity networks. For specific details see Ford and

Fulkerson [ 5]. However, because of the way in which the w. 's have

been defined (i.e., in cumulative machine-periods of production), it is

unlikely that these will all be integers. In this case, using a minimal-

cost flow algorithm will not necessarily provide an optimal solution to

the problem which has all xijk integer. In the remainder of this chap-

ter, we will show how problem P3.1 can be reformulated so that it can be

solved using any of the algorithms for finding minimal-cost flows in

single commodity networks when the ik's are not all integer.








Production
Costs Upper and
Backordering Lower Bounds
Costs 1


Figure 3.1 Network Example of Problem P3.1.











Reformulation as an All Integer Program


An optimal solution to problem P3.1 has the property that

inventory carrying charges and backordering costs will not occur simul-

taneously for any product i and period k (i.e., I. B. = 0). This
ik ik
result follows from the fact that I.k and Bik occur in only one con-
ik ik
strain, and the cost coefficients associated with each variable are

strictly positive.

Consider an optimal solution to problem P3.1 having B. > 0
Ik
for some product i and period k. Then, from the above result I = 0
ik
and from constraint (3.1)


M k
2 x. + B = k (3.2)
j=l 1 1J k ik


Now, breaking up each term into its integer part and its fractional

part where [r] denotes the largest integer less than or equal to r and

f(r) denotes the quantity r- [r] (i.e., for nonnegative r the fractional

part of r is denoted by f(r) where 0 f(r) < 1), equation (3.2) becomes


M k
2 E x.j + [Bik] [ ] = f( ik) f(Bik) (3.3)
j=1 2 j1 ik .k


The left-hand side of equation (3.3) is made up of integer terms

only, which implies that f(w ) -f(Bi ) must be an integer. This,

coupled with the fact that 0 : f( ik) < 1 and 0 : f(B i) < 1, implies
ik ik
further that f(wik) -f(B ik) = 0. Hence, an optimal solution to problem
ik Ik
P3.1 having B > 0 has
ik


f(Bk) = f(wi) (
ik ik


(3.4)










In similar fashion consider next an optimal solution to problem

P3.1 having Ik > 0 for some product i and period k. Then B = 0 and
Ik ik
from constraint (3.1)


M k
S E i I =w ik (3.5)
j=1l = i ik ik


Breaking up each term into its integer part and its fractional part,

equation (3.5) becomes


M k
S xij [Iik] ik = f(wik) + f(Iik). (3.6)
j=1 1 k Ik


As in the backordering case, the left-hand side of equation

(3.6) is made up of integer terms only which implies that f(w ) + f(I )

must be an integer. This, coupled with the fact that (a) f(w. ) equals
ik
zero when wik is an integer and is strictly positive when wik is not

an integer, (b) 0 f( ik) < 1, and (c) 0 f(Ii ) < 1, implies further
ik ik
that f(w ik) + f(ik) equals one if wik is not an integer and equals zero

otherwise. Hence, an optimal solution to problem P3.1 having Ik > 0
Ik
has

1i f(w ) if wik integer
Ik ik
f(ik) = (3.7)

0, otherwise.

Finally, by defining I as a 0-1 variable which takes on the
ik
value one if Iik > 0 and wik integer and zero otherwise, relationships

(3.4) and (3.7) become











f(B i) = f(wik) 1i Iik



f(I ) = [I f(wi))i .
ik ik ik


(3.8)


Sufficient relationships have now been developed so that a new

formulation of the production scheduling problem can be given. To

I /
formulate this problem, let I and Bk correspond to [I. ] and [Bk.],
ik ik ik Ik
respectively, and for notational convenience define aik = f(w ik) and

wik = [w ]. Now replacing Ik and Bi in problem P3.1 by their integer
ik ik Ik ik
and fractional parts and then substituting the expressions in (3.8) for

f(B ik) and f(Iik ), we obtain the following formulation:
ik ik


M H N H1-1
E Zc.. x. + 2 E {0 Ii +
jl kCijk ijk+ i=l ik ik
j=1 k=l i=l k=-I


N H-1
+ E bik k + a ik
i=1 k=l


subject to

M k
Sx. I I + B =w
= =1 I ik ik ik ik
j=1 X=1


M H
E E x. w
j=1 k=l jk


x < 1
1 ijk -
i=l

if Ii > 0 and w ik integer
/I ik k ik
Otherwise
O otherwise


X..jk I B k 0, integer
1jk ik' ik


i = 1,2,...,N
k = 1,2,...,H-1



i = 1,2,...,N



j = 1,2,...,M
k = 1,2,...,H


i = 1,2,...,N


k = 1,2,...,H-1


j = 1,2,...,M
k = 1,2,... ,H


N
Min E
i=l


[P3.2]


(3.9)



(3.10)


(3.11)


((1 i-a i k0 a b iklI











Problem P3.2 is an all integer program. It follows from our

previously derived results that solving problem P3.2 is equivalent to

solving problem P3.1. We now proceed to show that problem P3.2 can be

converted into a minimal-cost flow problem.



Formulation as a Minimal-Cost Flow Problem


Note in the objective function of problem P3.2 that the term
N H-1
E E aik b is simply a constant, and since constants do not affect
i=l k=l

the optimization, this term can be omitted. Note also that any feasible

solution to problem P3.2 must have all x integer. Hence w in con-
ijk iH
straint (3.10) can be replaced by the smallest integer greater than or

equal to wH (i.e., (w )) without any effect on the optimization.
In iH
Finally, we will show that constraint (3.11) can be replaced by



I =0 o r e (3.12)
ik ik k =1,2,...,H-1


where eik is a constant defined to be one if wik integer and is zero

otherwise.

To see that (3.11) can be replaced by (3.12), let P3.2' denote

problem P3.2 with (3.12) used in place of (3.11). From equation (3.5),

I.k > 0 if and only if
ik

M k
E x.ij > w.k wi' (3.13)
j=1 1_ ik ik

and from constraint (3.2) and the integrality requirements of

xijk Iik = O if and only if
ik ik











M k
Z Z xi. < w w. < (3.14)
j=l 1= 1 ik ik


Together, (3.13) and (3.14) imply that constraint (3.11) is equivalent

to

M k
1 ,if E xi -wik >0 and w i integer
j=1 2=1
i i = 1,2,... ,N
ik k = 1,2,...,H-1.

S, otherwise

(3.15)

Note next that problem P3.2' is a relaxed version of problem P3.2.

Hence, an optimal solution to problem P3.2 is also an optimal solution

to problem P3.2 if it is feasible to problem P3.2. From (3.15) this

implies that an optimal solution to problem P3.2 is also an optimal

solution to problem P3.2 if in this solution


M k
(a) Ii = 1 implies E x.. -w >0 and w. / integer
ik j 1 iji ik ik
j=1 =1i

and

M k
(b) I = 0 implies E Z x.. -w <0 and/or w. = integer.
ik j= .1 ij ik ik


Case (a):

Obviously, wik cannot be integer valued since this would imply

that I/ would be restricted to the value zero. Hence, to prove this
ik
M k
case we assume that Ii = 1, w i integer, and E E xij -wik<0 for
ik ik l I.J ik
j=1 =1
some product i and period k in an optimal solution to problem P3.2 .

Then in order for constraint (3.9) to be satisfied, it must be the case

that B. 1. This, however, implies that there exists a feasible
ik










solution to problem P3.2' having I" and Bi each reduced in value by
ik ik
one yielding an objective function value improvement given by


Az = (l-a )0ik+ a b b
ikik ik ik ik

< 0 since (1-a k), bik, 0k > 0.


Hence a contradiction since we could not have started with an optimal

solution.


Case (b):

Obviously, (b) holds if Wik= integer. Hence to prove this case

M k
we assume that I integer, and E x.- >0 for some
ik ik j=a- 1 i ik


product i and period k in an optimal solution to problem P3.2 Then in

order for constraint (3.9) to be satisfied, it must be the case that

Iik 1. This, however, implies that there exists a feasible solution

to problem P3.2' having Ik increased in value by one and I' decreased
ik ik
in value by one yielding an objective function value improvement given

by

Az = -ik + (1 -aikik -aikbik


ik ik ik
< 0 since aik, bik' 0ik > 0.


Hence a contradiction since we could not have started with an optimal

solution.

Therefore, cases (a) and (b) do hold at optimality in problem

P3.2 which implies that solving problem P3.2' is equivalent to solving

problem P3.2. Making the specified changes in the objective function











and constraint (3.10) in problem P3.2 and replacing constraint (3.11)

by (3.12), problem P3.2 becomes as follows:


N M H N H-l
Min E E E c x + E E~li- Ik
ijk ijk ik
i=l j=1 k=1 i=l k=1

N H-i
+ (l(-aik) ik-aikb. JI k + E bikBik
i=l k=l

subject to
M k
E x -E' + i = 1,2,. ..,N
j=1 2 1ij ik ik ikik k = 1,2,...,H-1

M H
[P3.3] E E xj (w ) i = 1,2,...,N
j=1 k=l ik

Nj
E x < 1 j = 1,2,...,M
Sijk k = 1,2,...,H


[ < e i = 1,2,... ,N
ik ik k = 1,2,...,H-1

i = 1,2,...,N
ik ik Ik Bi k 0, integer j = 1,2,...,M
ijk' kk = 1,2,...,H .



The multiple facility, multiple product production-inventory

scheduling problem with backordering is now in the format of problems

which can be solved using any of the algorithms for finding minimal-

cost flows in single commodity networks. Efficient algorithms are

available for solving this type of problem (e.g., Fulkerson [ 6]).

The network structure corresponding to problem P3.3 is similar to that

of problem P3.1. The primary difference is in the construction of

a second arc (nik,n i,k+) from each node nik to node ni,k+. Flow

through each of these arcs is limited by the capacity e. and a cost,
Ik











(1-a ik ik -aikbik, is incurred for each unit of flow placed on the

arc (nik ,n k+l). The actual amount of flow placed on this arc is

denoted by the variable I An N-product, M-facility, H-period problem
ik
of this type will have H(M+N)+2 nodes and H(M+MN+4N)-3N arcs. An example

network is shown in Figure 3.2 for a one-facility, two-product, three-

period problem where for notational convenience sik is defined to be

((1-aik)ik aikbik).

In addition to being a model for a very general multi-facility,

multi-product production-scheduling problem, there are a number of other

realistic enrichments to the problem which can be included in the model

without severely complicating the solution procedure. These are dis-

cussed below:

(a) There can be a technological restriction on the number of

facilities (qik) which can be used to produce product i simultaneously

in period k. This generalization can be introduced into problem P3.3

by adding the constraint


x < q i = 1,2,...,N
Mx

j=1 ijk ik k = 1,2,...,H.


(b) In addition to the cost of producing each product for one

machine-period, there can be a cost incurred on the number of facil-

ities in use at any given time. Let rk denote the cost per facility

in use during period k where rk 2 0. Then this enrichment is included

in the problem formulation by adding the following term to the objec-

tive function,

H N M
Srk Z xjk
k=l i=l j=l









Production
Costs


Upper and
Lower Bounds


Figure 3.2 Network Example of Problemc P3.3.


Upper
Bounds











However, this cost function does not have to be linear. Rather the

costs can be convex. The insertion of a convex cost term is a stan-

dard trick in networks and is discussed in detail in Ford and Fulkerson

[ 5].

(c) The number of facilities available in each period may vary.

This generalization is incorporated in the model by omitting those

nodes mk (and the associated arcs) in period k corresponding to any
jk
facility j which is unavailable.

(d) Upper bounds can be placed on the values of I' and Bk imply-
ik ik
ing physical limitations on the amount of product i which can be carried

in inventory or on backorder during period k+1.

(e) Realistically, it may be technologically impossible to produce

product i on facility j. This restriction can be included in the model

by omitting arcs (mk ,nik) in all periods k = 1,2,...,H.



Conclusions


A multiple facility, multiple product production-inventory

scheduling problem with backordering has been considered. In a

straightforward fashion this problem has been formulated as a linear,

mixed integer program which can be given a network flow interpretation.

However, this formulation can be solved as a minimal-cost flow problem

only for a very special case. In this paper it has been shown that the

general problem can be reformulated so that it can be solved using any

of the algorithms for finding minimal-cost flows in single commodity

networks. This reformulation, in addition to being a model for a very








44


general multi-facility, multi-product problem, can be enriched to include

numerous other realistic aspects of the production system without

severely complicating the solution procedure.
















CHAPTER 4


A MULTIPLE FACILITY, MULTIPLE PRODUCT PRODUCTION
SCHEDULING PROBLEM WITH OVERTIME




Introduction


In the real environment, many production systems face fluctuating

demand patterns which often tax the productive capability of the system

to meet these demands. As a consequence, many schedulers, or system con-

trollers, use overtime to increase the productive capacity of the system.

When this option is available, the cost of the overtime must be taken

into account, and production costs become a function of the amount of

overtime used as well as the straight time production amounts. In this

chapter the problem of scheduling lots to facilities is considered for

the case where an overtime option is available to the system controller.

Specifically, an industrial process made up of M facilities in

parallel is considered where there are N different products to be pro-

duced over a finite planning horizon. During any period k, M facilities

are available for production where Mk < M. All of these facilities are

considered to be identical in that p. units per period of product i can

be produced by any available facility. During any given period at most

one product can be scheduled on each facility. Corresponding to each

of these assignments is a cost, denoted by c.k which includes the cost
of using facility j in period k to produce product i as well as the
of using facility j in period k to produce product i as well as the











associated costs of producing p. units of product i. At the end of

each production period, the system controller has the additional option

of scheduling an overtime shift for producing product i on facility j

when product i is scheduled on straight time on facility j. The incre-

mental cost per period for this overtime production is denoted by gi

where gi > 0. (Alternatively, this problem can be thought of in a batch

type production process where the scheduler must decide whether to pro-

duce zero, one, or two batches of product each time the product is

assigned to a facility for one period. It can also be viewed in the

context of a continuous production process where the scheduler has the

capability of doubling the production rate of each of the facilities.)

Production completed on straight time and overtime during period k is

added to each product's inventory at the end of the period, and nonneg-

ative demands (dik units) are satisfied from this inventory. During

the horizon the system controller must schedule the products to the

facilities in such a manner as to meet all product demands, without

suffering backorders or stockouts, and at the same time build up each

product's ending inventory level to a minimum nonnegative amount denoted

by IiH. The problem objective is to determine such a production sched-

ule which minimizes the sum of straight time and overtime production

costs over the horizon.

In succeeding sections, the overtime production scheduling

problem is formulated as a linear, integer program. This formulation

can be given a network flow interpretation, but in general the problem

cannot be solved using any of the minimal-cost flow algorithms. How-

ever, it is shown that this problem can be solved by solving a relaxed











version of the problem. This relaxed problem has the property that it

can be solved using any of the very efficient algorithms for finding

minimal-cost flows in single commodity networks. Finally, a single pass

algorithm is developed for solving a special case of the overtime

problem.


The Overtime Production Scheduling Problem


In this problem product inventories are updated only at the end

of each production period, and product demands must be satisfied from

these end-of-period inventories without allowing backordering or stock-

outs. Hence, requiring that the net ending period inventories be non-

negative for all products and periods is a necessary and sufficient con-

dition for all demands to be met without backorders or stockouts.

To facilitate the formulation of this problem, define wik as

the minimum number of machine-periods (integer) during which product i

must be produced in the first k periods in order to have a nonnegative

inventory level at the end of period k. Then,


k
ik = max [0, (( Z d. )/Pi k 1,2,...,H-
j=1

where (a) denotes the smallest integer greater than or equal to a.

In addition to being nonnegative, product i's inventory level at the

end of the horizon must be greater than or equal to IiH. Hence, define
1H
H
w =max (O, ((iH + E d.. i )/p i = 1,2,...,N
j=l

and let W be an N by H matrix with components [w ik.
ik











Define yijk as the number of times facility j is scheduled to

produce product i on straight time during period k, and denote by c.i
1jk
the cost of each of these assignments where cijk > 0. Similarly,

define xijk as the number of times facility j is scheduled to produce

product i on overtime during period k, and denote g. as the incremental

cost (product dependent only) of producing product i for any overtime

machine-period where g. > 0. Then the system controller's problem is

to minimize the sum of straight time and overtime production costs sub-

ject to the constraints that (a) only M facilities are available for

production in period k, (b) demands must be satisfied without allowing

backorders or stockouts, (c) product i can be scheduled on overtime on

facility j during period k only if it has already been scheduled on

straight time on facility j during period k, and (d) only one product

can be scheduled on a facility each period where a facility, when

assigned to produce product i, is assumed to produce exactly p. units

of the product each machine-period. A mathematical formulation of this

problem is as follows:

N H k
Minimize E E c yijk + g.x.
i=1 k=l j=l k
subject to
k Mk
Z Z[Y + x ] 2 w i(4.1)
=1 j=1 ij ij ik k = 1,2,... ,H
N < k = 1,2,... ,H
[P4.1] y 1 (4.2)
i= ijk j = 1,2,...,Mk

i = 1,2,... ,N
yjk xjk k 1,2,...,H (4.3)
j = 1,2,... ,Mk

i = 1,2,...,N
Yijk' Xijk O, integer k = 1,2,...,H (4.4)
j = 1,2,...,Mk











Problem P4.1 is a linear, integer program which can be given

a network flow interpretation. However, the problem cannot in general

be solved using any of the minimal-cost flow algorithms. We will show,

though, that a relaxed version of the problem can be formulated and

that this relaxed problem, when solved, will provide an optimal solu-

tion to problem P4.1. This relaxed problem will have the property that

it can be solved using any of the algorithms for finding minimal-cost

flows in single commodity networks. The formulation of this problem is

based, in part, on the following result:


Lemma 4.1

A necessary and sufficient condition for problem P4.1 to have

a feasible solution is that there exists a set of y ijk's satisfying

constraints (4.2) and (4.4) and having


k M

=1 j=l ijI (wik/2) (4.5)


for all i = 1,2,...,N and k = 1,2,...,H.


Proof of Lemma 4.1

A. Necessary Condition:

Assume that there exists a feasible solution (Y*,X*) to

problem P4.1 having

k MI
E Y j (wik /2) (4.6)
1=1 j=1 ij2 ik

for at least one product i and period k. Then (4.6) and constraint

(4.3) together imply that











k MI
E E (y* + x <2 (W /2) -1 < wi
.c=i ij2 + Xij^2 ik-1 -1 ^ik
L=1 j=1 ; ]k k


A contradiction exists since constraint (4.1) cannot be satisfied unless

(4.5) holds.


B. Sufficient Condition:

Set xijk = jk for all i = 1,2,...,N; k = 1,2,...,H; and

j = 1,2,...,Mk. Then, by construction, constraints (4.1), (4.3) and

(4.4) are satisfied. Hence, (Y,X) is a feasible solution to problem

P4.1.

Q.E.D.

Since condition (4.5) must be satisfied by all feasible solu-

tions to problem P4.1, it can be included in the problem as a redundant

constraint. To formulate a relaxed version of problem P4.1, then, we

add constraint (4.5) to the problem, omit constraint (4.3), and make

H Mk
the variable transformation z. = E i x The relaxed problem
S k=l j=l
can be formulated as follows:

N H A N
Minimize E Z E c.ij Yijk + E g.z
i=l k=l j=l i=l g
subject to
k M



M
i 1 j= i ,2,...,
N
[P4.2] yik < 1 k = 1,2,"... (4.8)
i=1 jk j = 1,2,...,M
k M
i1l i = 1,2,...,N (4.9)
= j=1 Yij w Wik/2) k = 1,2,... ,H

i = 1,2,... ,N
yijk zi 0, integer k = 1,2,...,H
j = 1,2,... ,M











Since (a) problem P4.2 is a relaxed version of problem P4.1, and

(b) problem P4.2 includes, as constraints, the necessary and sufficient

conditions specified by Lemma 4.1, it follows that problem P4.1 has a

feasible solution if and only if problem P4.2 has a feasible solution.

Let (Y,Z) be an optimal solution to problem P4.2. Then (Y ,X ) is an

optimal solution to problem P4.1 where, by construction,


o
Yijk = ijk



k j i 1,2,...,N
yj if E E y ir z k = 1,2,...,H (4.10)
1=1 r=l j = 1,2,...,
0
o
xijk


0 otherwise.




To show that (Y ,Xo) is a feasible solution to problem P4.1,

note first that constraints (4.2), (4.3) and (4.4) are satisfied by

construction since (Y,Z) is a feasible solution to problem P4.2. To

show that constraint (4.1) is satisfied, we note from (4.10) that

Y = Y and that either


k aM k
(a) E E x = E y ij (4.11)
a=1 j=1 1 =1 j=l1

or

k MI
o
(b) E E x = z (4.12)
=1l j=l j1 i


for all i = 1,2,...,N and k = 1,2,...,H.










Case (a):

After substituting Y for Y, constraint (4.9) and equation (4.11)

together imply that


k M
SZ yo( + xo. 2(wik/2) > wi.
11 j=1 i k

Hence constraint (4.1) is satisfied for this case.


Case (b):

Constraint (4.7) and equation (4.12), together with the fact

that Yo = Y implies that

k M
Z Z ( +y + x .
1=1 j=1 ij ij ik

Hence, constraint (4.1) is also satisfied for this case, and (Y ,X ) is

a feasible solution to problem P4.1.

In minimization problems, it is well known that the optimal

objective function value for a relaxed version of a problem represents

a lower bound on the value of the objective function of any feasible

solution to the unrelaxed problem. Therefore, since (a) (Y ,X ) is

a feasible solution to problem P4.1, and (b) the objective function

value of the solution (Y ,X ) is equal to the objective function value

of the optimal solution (Y,Z), it follows that (Y ,X ) is an optimal

solution to problem P4.1.

Problem P4.2 can be viewed as a flow problem which can be solved

using any of the algorithms for finding minimal-cost flows in single

commodity networks. An example network for a one-product, three-period

problem with two facilities available in each period is shown in

Figure 4.1. In this network the variable y.ij represents the flow from
ijk











































c
0



0(0
tn-0
0-0











node mj to node n along the arc (mjk,ni ), and the variable zi rep-
jk ik jk ik 1
resents the flow from node s to node oil along the arc (s,o ).

Efficient algorithms are available for solving this type of problem

(e.g., Fulkerson [6]).




A Special Case of the Overtime
Scheduling Problem


For the special case where the production costs are not facility

and period dependent (i.e., c.. = c. for all k = 1,2,...,H and
ijk i

j = 1,2,...,Mk), a single pass procedure can be developed for solving

the overtime production scheduling problem. This procedure, even for

large problems, can easily be carried out by hand. Before describing

the procedure, however, note first that facility discrimination is no

"k
longer necessary, and we can let Yik = 2y Problem P4.1, then,
j=1
reduces to the following formulation:


N H N
Minimize E E c.y + Z g.z.
ik i.i
i=l k=l i=l

subject to
k
z + y w = 1,2,... (4.13)
i ai ik k = 1,2,...H

[P4.3] N
ik y Mk k = 1,2,...,H (4.14)
i=l

k

I iik k = 1,2,...,H


i = 1,2,...,N
ik' zi 0, integer k = 1,2,...,H











The procedure to be given for solving problem P4.3 involves

two phases. Phase I determines a set of yik' s (if they exist) which

satisfy the necessary and sufficient conditions for a feasible solution

to problem P4.3 to exist as stated in Lemma 4.1. Phase II, then, builds

onto this Phase I solution in order to determine an optimal solution

to problem P4.3.

Problem P4.3 has a feasible solution if and only if there exists

a set of ik 's which satisfy constraints (4.14) and (4.15) for all

i = 1,2,...,N and k = 1,2,...,H. The set of yik's is determined by

omitting constraint (4.13) and solving the relaxed problem. In terms

of the network given in Figure 4.1, this is equivalent to omitting those

nodes oik for all i = 1,2,...,N and k = 1,2,...,H and all arcs leading

into and out of these nodes. To solve this relaxed problem, a backward

procedure is used which is essentially equivalent to a flow algorithm.

The procedure is as follows where R denotes the availability of facil-

ities at intermediate stages of the procedure and p and t indicate

respectively the current product and period under consideration.


Phase I

Step 0. Define (w i/2) = 0 for all i = 1,2,...,N and set t = H.

Step 1. Set R = Mt and p = N.

Step 2. Determine:

H
pt = min {R, (w pH2) -(Wp t_/2) Z y p
j=t+l

H
where Z y 0
j=H+1


R = R pt












Step 3. Set p = p-1. If p 1,.go to Step 2. Otherwise,

go to Step 4.

Step 4. Set t = t-1. If t 1, go to Step 1. Otherwise,

go to Step 5.
H
Step 5. Terminate. If yik = (w iH/2) for all i = 1,2,...,N,
k=l
then an optimal solution has been determined for the

relaxed problem. Otherwise, problem P4.3 does not

have a feasible solution.


If, in Phase I, an optimal solution to the relaxed problem is

determined, then we proceed to Phase II, since from Lemma 4.1 a feasible

solution to problem P4.3 is known to exist. Otherwise, we stop, since

problem P4.3 does not have a feasible solution. Before proceeding,

however, note that any feasible solution to the relaxed problem having

each product i assigned exactly (w i/2) times during the horizon is an

optimal solution to the relaxed problem. Hence, the relaxed problem

may have many alternate optimal solutions. Using the backward proce-

dure approach to determine one of these optimal solutions has the fol-

lowing important advantage, however, in determining an optimal solution.

The Phase I procedure at termination for each period k has the property

that either (a) all MA facilities are assigned, or (b) the only product

assignments which can be reassigned to period k (without violating

feasibility or assigning a product more than (wiH 2) times) are those

which are already assigned to a period r where r k.











Example 1

As an example of the Phase I procedure for solving the relaxed

version of problem P4.3, consider a two-product, three-period problem

where


c = (6,3)
W
M = (8,8,10)


An optimal solution is det

a. Assign product

b. Assign product

min (9,93 = 9.

c. Assign product

min (3,8) = 3.

d. Assign product

e. Assign product

min (3,8] = 3.

f. Assign product


4 6 24 (w/2)
S =
6 12 14]


ermined as follows:


12

7 .


two once to period 3 since min (1,10) = 1.

one nine times to period three since



two three times to period two since


one once to period

two three times to


two since min (1,53 = 1.

period one since


one twice to period one since min (2,5] = 2.


2 1 9
g. Stop. An optimal solution is Y =
3 3 1


Assuming Phase I yields a feasible solution to the relaxed

problem, we now proceed to Phase II. In this phase an optimal solution

to problem P4.3 is determined by building on to the Phase I solution.

This is equivalent to assuming that the part of an optimal solution

which has been determined in Phase I is fixed (i.e., let Y' denote the

solution obtained in Phase I) and now all that remains is the determin-

ation of the remaining portion of the solution (i.e., denote the remain-

ing portion of the solution by (Y',Z)). The Phase II problem to be

solved is formulated as follows:











N H N
Minimize E E [cik + ciYk + E g.z
i=l k=l i=l

subject to
k k
S// i i = 1,2,...,N
zi + Y i ik Yi k = 1,2,...,H
2=1 i =I

[P4.4] N N
yik Mk ik k = 1,2,...,H
i=1 1=1

a i = 1,2,...,N
Yik' z. 0, integer i 1,2,...,H
ik 1 k = 1,2,...,H


A network interpretation of problem P4.4 is obtained from

Figure 4.1 by omitting all arcs (nik,ni,k+l) for all i = 1,2,...,N and

k = 1,2,...,H-1, and all arcs (n. ,t) for all i = 1,2,...,N. Also

N
redefine MA = h E yik for all k = 1,2,...,H. Now note from this
i=l
network interpretation that for all products having ci g gi that it is

at least as advantageous to assign overtime as it is to assign straight

time. Since this can be done without causing facility conflicts, all

demands for these products are satisfied by overtime assignments. For

those products having ci < gi, it is more advantageous to assign straight

time than it is overtime. However, there are a limited number of avail-

able facilities left. Hence, it is more advantageous to assign a product

i having (c.-g.) < (c.-g.) to a vacant facility than it is a product j.

To determine an optimal solution to problem P4.4, we again use a back-

ward procedure because of the useful property which it allowed in the

Phase I procedure. In the procedure, sets S and S are defined where S

contains those products for which an optimal form is known (i.e., ini-

tially those products having c. > g.), and S contains those products
1 1











for which an optimal solution form is not known. The procedure then

considers one product in S at a time(i.e., the first product considered

is that product in S having the minimum value of c.-g.). Once the form

of an optimal solution is determined for this product, then it is trans-

ferred from set S to S. This process continues until S is an empty set.

In the procedure we build on to the yik's directly rather than using the

Y" and Y notation.


Phase II

Step 0. (a) Order the product indices so that


C1-1 c2-g2 ... CN -gN.

(b) Define the sets

S = [ilc-gi < 03 and S = ilc.-gi a 03.
I i 1 i

H
(c) Set z. = W Z yik for all i = 1,2,...,N.
k=1
Step 1. If S Z 0, go to Step 2. Otherwise, stop. An optimal

solution is (Y,Z).

Step 2. Set p = max i and t = H.
ViES

Step 3. (a) Determine

N t-1
a = min z p, Mt Ey i z p + E y w 3
i=1 &=0

where y and w are defined to be zero for all

i = 1,2,...,N.

(b) Set


pt = pt + a

and

z. = z. a.
1 1











Step 4. If t = 1 or zi = 0, go to Step 5. Otherwise, set

t = t-1 and return to Step 3.

Step 5. Set S = (S-p3 and S = [S+p], and go to Step 1.


Since overtime periods are assigned without facility conflict,

it is clear that the only way there could exist a better optimal solu-

tion to problem P4.3 than that obtained by Phases I and II is if there

exists an optimal solution (Y*,Z) having

H H H H
E k= E y + 1 and E = k y 1
ik ik jk ik
k=l k=l k=l kk=

for at least two products i and j where c.-g. < c.-g.. However such

a solution cannot exist since (a) in Phase II product i was assigned

to all periods before product j, and (b) in Phase I the backward proce-

dure has the property that there does not exist a feasible interchange

of these two product assignments.


Example 2

As an example of the Phase II procedure, reconsider the problem

given in Example 1 where, in addition, g = (5,9). An optimal solution

is determined as follows:

a. The products are already indexed so that cl-g 1 c2-g2.

Hence, since c -g1 = 1 and c2-g2 = -6, we define

S = '13 and S = (2].

b. Set y23 = 1 + 0 = 1 since min (7,0,1} = 0.

c. Set y22 = 3 + 4 = 7 since min [7,4,4] = 4.

d. Set y21 = 3 + 3 = 6 since min [4,3,3j = 3.

e. Stop. An optimal solution is Y = and Z = (12,0).
16 7 1]











Conclusions


A multiple facility, multiple product overtime scheduling

problem has been considered. A linear, integer programming formulation

was given for the problem which in this case can be solved by solving

a relaxed version of the problem. This relaxed problem was shown to

have the property that it can be solved using any of the algorithms for

finding minimal-cost flows in single commodity networks. However, for

an important special case a single pass algorithm was developed which

is more efficient than the standard flow algorithms.

















CHAPTER 5


AN EFFICIENT INTEGER PROGRAMMING ALGORITHM
FOR A MULTIPLE BATCH PRODUCTION SCHEDULING PROBLEM




Introduction


In this chapter a multiple batch production process is considered

where the horizon length is a variable in the optimization. During any

production period, each of the M facilities has the capability of pro-

ducing up to m. batches of product. A learning effect is experienced
1

in the production of these batches and is exhibited in the production

costs. The cost (c ) of the Ith batch produced on a facility during

any period is no less than the cost (ci, +) of the 4+1st batch produced

for all I = 1,2,...,m.-1. A batch of product i contains p. units, and
1

only whole batches are produced. Production completed during period k

is added to each product's inventory at the end of the period, and non-

negative demands (dik units) for product i are satisfied from this

inventory.

It is assumed here that under normal operating conditions the

scheduler, or system controller, for this process has a production plan

for scheduling the N products to the M facilities. This production plan

may be a schedule based on the well-known infinite horizon economic manu-

facturing quantity which was discussed in Chapter 1. In these formula-

tions of the problem all demands must be met without allowing backorder-

ing, and the problem objective is to determine a repetitive production











cycle for allocating the production time of a facility to each of the

N products so that costs are minimized. In most treatments the addi-

tional assumption is made that initial product inventory levels are

"in phase with" the infinite horizon cycle plan meaning that the product

inventory levels are all sufficiently large for an infinite horizon

cycle plan to begin immediately without suffering stockouts. When ini-

tial product inventories are considered in the problem, there may exist

a subhorizon of unknown time duration (h) during which the facilities

must be scheduled "out of phase with" the infinite horizon cycle plan

in order to avoid stockouts or backorders. In this chapter an extremely

efficient procedure is developed for bringing a system into phase with

a given production pattern such as an infinite horizon cycle plan.

Let iH denote the minimal inventory level in units for each
iH
product i which, when taken collectively for all products, represents

an acceptable condition or "state of the system" for the system con-

troller's desired production pattern to begin. Realistically, there is

usually more than one such state which can be used since the entry point

into a repetitive production cycle is ordinarily not unique. (However,

the number of possible entry states is finite since time is discrete.)

In this chapter we develop a solution procedure for a single entry

state and leave to the reader the straightforward reapplication of the

solution procedure for other entry states. The scheduler's problem

then is to determine an assignment of products to facilities and the

corresponding production amounts at each of these assignments over a

variable horizon h subject to the constraints that (a) all demands are

satisfied without allowing backordering, and (b) the nonnegative,











initial product inventory level I (units of product i) for all

products is converted to a desired nonnegative, ending inventory level

liH. The objective is to minimize the sum of production costs and a

general cost function f(h) which is a monotonically nondecreasing func-

tion of the horizon length h.

In succeeding sections, we identify a specially structured

integer program and present an extremely efficient algorithm for its

solution. After identifying special properties of this general problem,

the multiple facility, multiple product production scheduling problem

is formulated as an integer program and shown to have the same special

structure.


An Integer Program


Noteworthy contributions have been made to the present state-of-

the-art of solving discrete optimization problems (i.e., for a survey

of many of these, see Geoffrion and Marsten [7 ]), but to date the

research has not provided an efficient procedure for solving the general

problem. Rather, researchers have been most successful in (a) identify-

ing problem structures which insure integer termination of the corre-

sponding relaxed (non-integer) optimization problem, and (b) determin-

ing solution techniques which are applicable to specific classes of

discrete problems.

In this section a specially structured integer programming

formulation is identified for which a very efficient algorithm will be

presented for determining its solution. The problem formulation is

given below where (a) J is the index set of integers, (b) all functions











-1
are real valued, (c) gk (*) exists for all k = 1,2,...,, and (d) for

any vectors e = (e ,e ,... ,e) and x = (x ,x 2...,x ) having e. > 0
1 n 2' n i

and x. 2 L. for all i = 1,2,...,n, f(x) < f(x + e) and
-1 -1
gk (hk (x) gk hk(x + e)3.


Minimize z = f(x)

subject to


[P5.1] k(x ) hk(x) W O k = 1,2,...,m
3k k k EJ5


xi > L., integer i = 1,2,...,n (5.2)
1 1

A result which provides the basis for an algorithm for solving

problem P5.1 is the following:


Lemma 5.1

If lower bounds, Li, are known for all variables x i = 1,2,...,n,

in problem P5.1, then for the variable x. in any constraint k of (5.1),


1
3k

L. = max {L., ( Jk k k I k

is also a lower bound on the variable x in problem P5.1. (Note that
Jk
(a) implies the smallest integer greater than or equal to a.)


Proof of Lemma 5.1

Assume that there exists an x. < L. in a feasible solution to
3k Jk
problem P5.1. By assumption, L. is a known lower bound on the value
3k
of x in any feasible solution to the problem. Hence it must be the

case that x. 2 L. which implies from (5.3) that
3k 3k











-1
L. = (g hk(L))> (5.4)
k

But (5.4) is just constraint k of (5.1) rewritten as an equality with

-1
each variable x. set at its known lower bound. Since gk h k(L)- <
-1 *
g [h (L + e)) for any e 2 0, the only way a variable x. < L. can be
3k 3k
feasible to problem P5.1 is if some x. = L. is decreased in value. Hence,
1 1
a contradiction exists since L is a lower bound on the value of x. in
1 1

any feasible solution to problem P5.1.

Q.E.D.


An Algorithm for Solving the Integer Program


As is, the problem formulation has an interesting structure.

The objective function is a nondecreasing function for all increasing

values of x.. Each of the constraints in (5.1) has one vari-able which

can be expressed as a nondecreasing function for all increasing values

of x.. A feasible solution to the problem requires that all variables

be integer valued and bounded from below. Using these properties and

assuming that a feasible solution to the problem is known to exist, an

algorithm for solving the problem can be described as follows:

Step 0. Set x = L.

Step 1. If any constraint in (5.1) is not satisfied, go to

Step 2. Otherwise, stop. An optimal solution is (x).

Step 2. Pick any unsatisfied constraint in (5.1), redefine

-1
x. = (g kh (x)3)

and return to Step 1.
and return to Step 1.











At each iteration of the procedure, the algorithm either

terminates or the value of a variable in an unsatisfied constraint is

increased by at least one. Therefore, since (a) a feasible solution

is known to exist, and (b) from Lemma 5.1 each variable is always reset

at a value equivalent to a known lower bound on that variable, the

algorithm must terminate in a finite number of iterations. Furthermore,

since the value of the objective function in problem P5.1 is nondecreas-

ing for an increasing value of any x., optimality of the solution at

termination also follows from Lemma 5.1.


An Example


Consider the following example which is in the format of

problem P5.1.

Minimize z= 0.5x + 2x


subject to


2x1 0.5x2 2 10

- x + 4x 2 8

xI 0.1x2 2 1

x x2 2 0, integer


(5.5)

(5.6)


The problem is illustrated in Figure 5.1 where an optimal solution is

obtained by the iterative scheme in the following manner.

a. Set x = 0.

b. Constraint (5.5) is not satisfied so redefine

x1 = (<10 + 0.5(0)5/2) = 5.

c. Constraint (5.4) is not satisfied so redefine

x2 = ((8 + 1(5)/4) = 4.

















I 0.1x2= I


2x- 0.Ix2 =10














Feasible Integer Region


20




15




10




5


20


Geometric Interpretation of the Algorithmic Example.


-xi +4x2= 8


Figure 5.1












d. Constraint (5.5) is not satisfied so redefine

x1 = ([10 + 0.5(4))/2) = 6.

e. Stop. All constraints are satisfied. An optimal solution

is x = (6,4) with objective function value

z = 0.5(6) + 2(4) = 11.



Other Properties of the Integer Program


In addition to the ease of solution, problem P5.1 has a number

of additional properties which are of interest.


Property 1

Consider any two problems having identical constraint sets of

the form required in problem P5.1, but different objective functions.

The same optimal algorithmic solution solves both problems irrespective

of the objective function f(x) so long as each satisfies the condition

f(x) : f(x + e) for all x and any e 0.


Property 2

Any feasible solution (x) to a problem P where problem P

includes the constraint set of problem P5.1, has x L x* where x* is

the optimal algorithmic solution to problem P5.1.


Property 3

The algorithmic solution to problem P5.1 is the unique optimal

solution to the problem if f(x) < f(x + e) for all x and all e 2 0

where at least one component of e is strictly positive.












Property 4

The corresponding continuous portion of problem P5.1 (omitting

the integer requirements on x in constraint (5.2)3 has an optimal solu-

O O
tion (x ) which has all x. integer when a feasible solution exists if
-1
1
gk [(hk(x)] is integer valued for all x = integer and k = 1,2,...,m.


Note finally that in the development i the algorithmic proce-

dure, termination was insured since a feasil ,. solution to problem P5.1

is known to exist. If a feasible solution is not known to exist, then

an additional stopping rule must be added to the procedL.e. This can

be done in a straightforward manner when necessary (e.g., place upper

bounds on each of the decision variables and terminate the solution pro-

cedure when any one of these bounds is exceeded).




The Production Scheduling Problem


The system controller's problem is to determine an assignment

of N products to M facilities and the corresponding production amounts

at each of these assignments over a variable horizon h. This is to be

done in such a manner as to [a] satisfy all demands (di units for
ik
product i in period k) without allowing backordering, [b] convert the

nonnegative, initial inventory level (I ) for all products i into
10
a desired nonnegative, minimum ending inventory level (IiH), and [c]

minimize costs. The costs include production charges (i.e., c.i denotes
*th
the incremental cost of producing an th batch of product i during any

machine-period) and a general cost function f(h) which is a monotonically

nondecreasing function of the horizon length h. A batch of product i












contains exactly pi units of the product and only whole batches can be

produced. Only one product can be assigned to a facility each period,

and up to m. batches of that product i can be produced during any
1

machine-period.

By assumption, the initial and required ending period inventory

levels are nonnegative which implies that backordering, if it occurs,

must occur at the end of at least one of the intermediate periods

(k = 1,2,...,h-l). Define wik as the minimum number (integer) of

batches of product i which must be produced during the first k periods

in order to prevent backorders from occurring at the end of period k.

Then,
k
wik = max [0, ( d _- i )/Pi k = 1,2,...,h-1.
kkl 0- k = 1,2,...,h-1.

j=1

Likewise, define


h
w = max [0, ((Ih + d.. I. )/Pi i = 1,2,...,N.
j=l 1

Now by defining xilk as the number of facilities scheduled to

produce exactly I batches of product i during period k, a mathematical

formulation of this problem can be given as follows:
N h mi
Minimize f(h) + Z E E E c. x.i
i=l k=l 2=1 r=l

subject to
k mi
Z E2 x. > i wN (5.7)
j=l =1 j ik k = 1,2,...,h
[P5.2] N m.
S 1 x. i=1 R=1
i = 1,2,... ,N
h, x. i 0, integer I = 1,2,... ,mi
k 1,2,...,h
k = 1,2,...,h .












Problem P5.2 is a nonlinear, integer program since the index h

is a variable in the optimization. Rather than solve problem P5.2

directly, a relaxed version of the problem will be formulated and

solved. This relaxed problem can be put into the format of problem

P5.1 and solved very efficiently by the previously given algorithm.

Once this relaxed problem has been solved, a simple procedure for solv-

ing problem P5.2 will be given.

The relaxed problem is formulated by (a) omitting those con-

straints in (5.7) corresponding to k = 1,2,...,h-1, (b) relaxing the

k constraints given in (5.8) to the single constraint given in (5.10)
h
below, and (c) making the variable transformation yil = Z xi. .
k=l
The problem, then, is as follows:


N i
Minimize f(h) + E E c. y.
i=l &=1 r=l

subject to

m.
1
[P5.3] E ly Wih i = 1,2,... ,N (5.9)


N mi
h (1/M) E E yi (5.10)
i=1 1=1


h, y 0, integer 1,2,...,N
ia = 1,2,...,mi.


Theorem 5.1

A sufficient condition for problem P5.3 to have a feasible

solution is that
N
2 (d ik/m.p.) < M (5.11)
i=l ik k 1
i=l

for all k = 1,2,...,h.











Relation (5.11) is a facility capacity condition which is

common in continuous infinite horizon production-inventory problems.

For example, see Hadley and Whitin [8 ]. A proof for this multiple

facility, multiple batch discrete problem is given in the Appendix

where the proof given is along the lines of that given by Elmaghraby

and Mallik [2 ] for the case m. = 1 and M = 1. In this paper it is
1
assumed that condition (5.11) is satisfied.

Note that problem P5.3 is in the format of problem P5.1 for

the special case where m. equals one for all i = 1,2,...,N. However,

in general, it is not in the correct format. Define the variable v.

as the number of machine-periods having at least I batches of product

i produced. Then, for all i = 1,2,...,N,


vi vi,V +l 2 = 1,2,...,m.-1
1
Yi=
v. A = mi .
1 M1

Making this variable transformation in problem P5.3, we obtain the

following equivalent program.

N mi
Minimize f(h) + S ci vi.
i=l i=1
subject to

mi



[P5.4] v v = 12,...,N
1 i,^l u = 1,2,...,m.-1
1
N
h & (1/M) Z vi
i=l


h, v 0, integer = 1,2,...,
S= 1,2,...,m. .
1











Now making the additional transformation,


t. = v.
1i ir
r=l


2 = 1,2,..,m,


problem P5.4 can be reformulated equivalently as follows:

m.-l
N 1
Minimize f(h) + [c.i t. Z (c. i ci )ti 2
u i,m. i,m 1,+1

subject to


t.
1,m.
1


Wih


2til t2 0


2t ti,-2 ti 0



ti ti,1 0
t21i,2-1


i = 1,2,...,N


i = 1,2,...,N


i = 1,2,...,N
L = 3,4,...,m.


i = 1,2,...,N
S= 2,3,...,m.
i


N
h >- (1/M) 2 t.
i=l


h, t. 0, integer
12


= 1,2,...,N
= 1,2,...,m..


Note that for the case m. = 2, constraints (5.12) should be omitted.

The constraint set of problem P5.5 is now in the format of

problem P5.1 (i.e., each constraint has one variable which can be

written as a nondecreasing function of other nondecreasing variables

and a feasible solution is known to exist). Hence, an optimal solution

to problem P5.5 can be obtained by the algorithmic procedure since


c c c ... > c. 0
il i2 i,m.
1
implies that the objective function is a nondecreasing function of any

increasing variable.


[P5.5]


(5.12)











Upon solving problem P5.5 by the algorithmic procedure, the

optimal solution (h,T) can be transformed back into an optimal solution

(h,Y) to problem P5.3. This solution has the property that m. batches
1
of each product i will be produced each time product i is assigned for

one machine-period with the possible exception of one assignment.

m.-1
i
Hence, Z yi, equals 0 or 1 for all products i = 1,2,...,N.
a=l


Lemma 5.2

The optimal algorithmic solution (h,T) to problem P5.3 has

m.-1

E y i= 0 or 1 (5.13)
=1

for all products i = 1,2,...,N.


Proof of Lemma 5.2

From Property 1, the optimal algorithmic solution to problem

P5.3 is the same irrespective of whether the objective function value

is strictly increasing or just nondecreasing in each variable. Hence,

in problem P5.3 it can be assumed that

c. >c > ..> c
il i2 m.
1

for all products i = 1,2,...,N. Assume that the optimal solution to

problem P5.3 does not satisfy (5.13). Then an optimal solution (h*,Y*)
mi-1 *
must have E yi. > 2 which implies that either
2=1
m.-1
(a) Z lYi = < mi

or
m.-l

(b) 'E y = 9 > m.
acilyi











Case (a):

There exists a feasible solution (h,Y) having h = h* and


1 if = e

if M.
YiI = if I= m

0 otherwise


with objective function value


z: z (Ci ci )


< z since (ci c ) > 0.


Hence, there is a contradiction since (h*,Y*) cannot be an optimal

solution.


Case (b):

There exists a feasible solution (h,Y) having h = h* and


(1 if = [6/m]m. and i > 0


Yi i + [/m if = m.

S, otherwise


with objective function value

z z (c c. )
il i,m.

< z since (c.i ) > 0 .
il ,m.

Hence, there is a contradiction since (h*,Y*) cannot be an optimal

solution.

Q.E.D.

The optimal algorithmic solution (h ,Y ) to problem P5.3 implies

that the shortest horizon over which all product inventories can be











0
built up simultaneously to an amount IiH is h production periods.

mi

During this horizon each product should be assigned Z y times where


all but at most one of these assignments should include the production

m.-l
i
o
of m. batches of product i. If Z yi. equals one, then the other

m.-i
1
assignment of product i should include the production of E 0yi
i iA

batches. Clearly, since problem P5.3 is a relaxed version of problem

P5.2, (h ,Yo) is also an optimal solution to problem P5.2 if it is

feasible to problem P5.2.

Without loss of generality, it can be assumed that the first

0
imi assignments of product i during the horizon h have m. batches
m.
of product produced where E y assignments are made. Let z denote
21i as ik
i1
the number of these machine-period assignments which are made in period k.

Then, obviously, (h ,Y ) can be converted into a feasible solution to

problem P5.2 if and only if there exists a feasible solution to the

following constraint set:

ho mi
h 1i
Z -k = i = 1,2,...,N
k=l k =1

i
Z z (w / m /M i = 1,2,... ,N
l ij ik k = 1,2,...,ho-1
j=1
[P5.6]
N
Sz < M k = 1,2,...,h
ik
i=l

i = 1,2,...,N
z 0, integer
ik k = 1,2,...,ho.











m
i
Since 2 yi (w iho/mi) for all i = 1,2,...,N, it follows
L=1
directly that there exists a feasible solution to constraint set P5.6

if and only if

N
2 (w /m) 5 Mk
i=1

for all k = 1,2,...,h This, coupled with the following result and

the fact that any feasible solution to problem P5.2 must have h ho

(Property 2), implies that problem P5.2 has a feasible solution if and

only if (h ,Y ) can be converted into a feasible solution to problem P5.2.


Lemma 5.3

Problem P5.2 has a feasible solution if and only if

N
S(w i/m.) Mk (5.14)
i=l
for all k = 1,2,...,h.


Proof of Lemma 5.3

The "if" part follows since the algorithmic solution (h ,Y )

has already been shown to be a feasible solution when (5.14) is satis-

fied. The "only if" part follows by summing over all i in constraint

(5.7), assuming m. batches are produced at each facility assignment,
1
and summing over all k in constraint (5.8). From (5.7), then, any

feasible solution must satisfy

m. m.
N k i i
Z E E x. > i(w /m.) (5.15)
1 ik I
i=1 j=l =1 i=l

for all k = 1,2,...,h and from (5.8) any feasible solution must satisfy











N k mi
E E E x.. Mk (5.16)
i=l j=1 Y1


for all k = 1,2,...,h. Together, relations (5.15) and (5.16) along

with the integrality requirement on xilk imply that (5.14) must be

satisfied by any feasible solution to problem P5.2.

Q.E.D.

A procedure for solving problem P5.2 is to solve the relaxed

version of the problem, and then convert this solution to a feasible

solution to problem P5.2 by finding a feasible solution to constraint

set P5.6. Any feasible solution determined by the procedure represents

an optimal solution to problem P5.2 since the optimal objective function

value of a relaxed version of a problem is a lower bound on the optimal

objective function value of the general problem. In addition, if a

feasible solution cannot be determined to constraint set P5.6, then it

is known that problem P5.2 does not have a feasible solution.

Constraint set P5.6 describes a network for which a two-product,

M-facility, three-period example is shown in Figure 5.2. From the net-

work it can be seen that a simple procedure for determining a solution

to this constraint set is to assign each product i (w i/mi) times to

vacant facilities in period one. Then assign each product i (wi2/mi) -

(w i/m.) times to vacant facilities in periods one and two, and in gen-

eral assign each product i (w ik/m) -'w.i,kl/mi) additional times to
ik i i,k-1 i
vacant facilities in periods one through k for all remaining periods

k = 3,4,...,h The solution obtained is optimal unless at some stage

in the assignment procedure there are insufficient facilities available

to complete all assignments in which case problem P5.2 does not have

a feasible solution.







Lower
Bounds


Figure 5.1 Network Example of Constraint Set P5.6.












Computational Experience


The algorithmic procedure was programmed in Fortran H and used

to solve the production scheduling problem. An IBM 370/165 computer

was used, and 200 different problems were considered where parameters

were generated randomly. Problems having as many as 20,000 variables

and 20,000 constraints were generated. All problems were solved, and

the computation times which were obtained were very nearly a linear

function of the number of constraints. The average solution time (as

-4
measured in terms of constraints in the problem) was 5.8 X 10-4

-4
seconds per constraint with a range of approximately 3 10-4

seconds per constraint about the mean. The computer program completed

approximately 100,000 iterations of the algorithm each second.




Conclusions


A specially structured integer program has been identified,

and an extremely efficient algorithm for determining its solution has

been developed. The specially structured program is a generalization

of a multi-facility, multi-product production scheduling problem which

has been presented. The scheduling problem arises, realistically, in

industrial situations where schedulers use repetitive cycle plans such

as the well-known economic manufacturing quantity models for controlling

production systems.
















CHAPTER 6


SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH




In this dissertation a class of multiple facility, multiple

product, production inventory scheduling problems has been considered.

Specifically, four different scenarios for this class have been addressed.

In a straightforward manner, each of these was formulated as an integer

program. It was then shown that each of the problems could be reformu-

lated as a network flow problem which can be solved using any of the

very efficient algorithms for minimal-cost flows in single commodity

networks. In addition, for some important special cases, single pass

procedures were developed which are more efficient than the standard

flow algorithms. In the fourth scenario, a specially structured integer

programming formulation was identified, and an extremely efficient algo-

rithm was developed for determining its solution.

The scenarios which have been considered have included very

general and realistic assumptions. The originality of the formulations

for this class of problems, in addition to allowing us to determine

optimal solutions efficiently, has offered considerable insight into

the operation of the production inventory system. These insights have

allowed us to enrich the problems generously to include many other real-

istic aspects of the production system heretofore not solvable with

present techniques.











As should be the case, however, the research for this disserta-

tion has been open-ended in character. Each question answered has

allowed new questions to be posed. Some of the logical and important

extensions which it would seem desirable to pursue in future research in

this area include:

1. It has been assumed throughout the dissertation that setups

occur between production periods, whereas in many production systems,

setups actually occur during production periods. In these situations

the productive capacity of the system is depleted when setups occur,

and an important extension of the results in this research would be to

obtain equivalent results for the case where setups are assumed to occur

during the production period.

2. The facility usage costs which have been used do not take

into account the product assigned to the facility during the previous

period. Realistically, however, the usage costs often should only be

assessed when a different product is produced on the facility in each

of the two periods. Solving the production-inventory scheduling prob-

lems for this more general case would be a considerable contribution,

but the problem appears to be quite difficult.

3. Each time a product is assigned to a facility for one period,

it has been assumed that only an integer multiple of the basic batch

size for each product can be produced. An important generalization of

the results in this research could be obtained by determining equiva-

lent results for the case where partial batches are allowed.







84


4. Product demand rates, though not necessarily identical in

all periods, have been assumed to be deterministic. Considerable addi-

tional insight into the multi-facility, multi-product production-

inventory system could be achieved by studying similar problems having

stochastic demand rates.

















APPENDIX


PROOF OF THEOREM 5.1


Define


N
h = TT (m.p.)
i=l 1 1


Z (dik/m.p.)
=1
Yi =
0


if I = m.,

otherwise

otherwise


i = 1,2,... ,N.


Then, from (Al),


Iy =


h
S(di /p )
k=l


= ih (ill


- i )/Pi
i0O


for all i = 1,2,...,N, and

N mi h N
E E yi = Z (d ik/mii)
i=1 i k=1 i=1

h
< Z M, since from (5.11)
k=l

N
Z (d ikmpPi) < M.
i=l k 1 1


Relationship (A2) implies that

N mi
Mh E i y, = s > 1.
i=l A=1


(Al)


(A2)












Thus, during the horizon h, all product demands are satisfied

exactly, and there are s additional unassigned periods available.

Hence, by assigning each product i, pi additional times during the

horizon h where


0 5 .i max (0 (i io )/Pi)


and
N
Z = s,
i=l

at least one product must become discretely closer to satisfying con-

straint (5.9). If all constraints are now satisfied, then a feasible

solution has been determined. Otherwise, redefine



(ill i = (iH i im

and

h+1, h+2, ... = 1,2,...,


and repeat the procedure. A feasible solution must be obtained in

a finite number of iterations since at each iteration, all demands

are satisfied and at least one product is assigned at least one

additional time.


Q.E.D.
















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Model of a Dynamic Economic Lot Size Production System A Net-
work Approach," Management Science, 15, pp. 507-527, 1969.




Additional References


Baker, Kenneth R., Control Policies for an Integrated Production
and Inventory Systems, Ph.D. Thesis, Cornell University, 1969.

Bowman, E. H., "Production Scheduling by the Transportation Method
of Linear Programming," Operations Research, 4, pp. 100-103, 1956.

Brown, Robert G., Decision Rules for Inventory Management, Holt,
Rinehart and Winston, Inc., New York, New York, 1967.

Buffa, Elwood S., Production-Inventory Systems, Richard D. Irwin,
Inc., Homewood, Illinois, 1968.

Carroll, Donald C., Heuristic Sequencing of Single and Multiple
Jobs, Ph.D. Thesis, Massachusetts Institute of Technology, 1965.

Conway, R. W., W. L. Maxwell and L. W. Miller, Theory of Schedul-
ing, Addison-Wesley Publishing Company, Reading, Massachusetts,
1967.

Dooley, J. E. and Ben Bernholtz, "A Many-Product Single-Machine
Recursive Scheduling Procedure," Canadian Operational Research
Society Journal, 7, pp. 165-176, 1969.












Eilon, Samuel, Elements of Production Planning and Control,
John Wiley and Sons, Inc., New York, New York, 1962.

Elmaghraby, Salah E., Arup K. Mallik and Henry L. W. Nuttle,
"The Scheduling of Lots on a Single Facility," AIIE Transactions,
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Hanssman, Fred, Operations Research in Production and Inventory
Control, John Wiley and Sons, Inc., New, New York, 1962.

Hodgson, Thom J. "An Analytic Model of a Two-Product, One-
Machine, Production-Inventory System," Management Science, 19,
pp. 391-405, 1972.

Hodgson, Thom J., Sequence-Lot Size Policies in a Production-
Inventory System, Ph.D. Thesis, University of Michigan, 1970.

Holt, C., F. Modigliani, J. Muth and H. Simon, Planning Production
Inventories and Work Force, Prentice-Hall, Inc., Englewood Cliffs,
New Jersey, 1960.

Kanodia, Lalit, Heuristic Sequencing of Jobs and Batch Size Deter-
mination in the Dynamic Closed Job Shop, Ph.D. Thesis, Massachusetts
Institute of Technology, 1967.

Keachie, E. C. and Robert J. Fontana, "Effects of Learning on
Optimal Lot Size," Management Science, 13, pp. B102-B108, 1966.

Kortanek, K. O. and A. L. Soyster, "On the Status of Some Multi-
Product Multi-Period Production Scheduling Models," Management
Science, 17, pp. B560-B561, 1971.

Kunreuther, Howard, "Production-Planning Algorithms for the
Inventory-Overtime Tradeoff," Operations Research, 19, pp. 1717-
1719, 1971.

Love, Stephen F., "A Facilities in Series Inventory Model with
Nested Schedules," Management Science, 18, pp. 327-338, 1972.

Manne, Alan S., "Programming of Economic Lot Sizes," Management
Science, 4, pp. 115-135, 1958.

Parsons, J A., "Multiproduct Lotsize Determination when Certain
Restrictions Are Active," Journal of Industrial Engineering, 17,
pp. 360-365, 1966.

Salveson, M. E., "A Mathematical Theory of Production and Schedul-
ing," Journal of Industrial Engineering, 4, pp. 3-6, 1953.

Taha, Hamby A. and Ronald W. Skeith, "The Economic Lot Sizes in
Multistage Production Systems," AIIE Transactions, II, pp. 157-
165, 1970.













Veinott, Jr., Arthur F., "The Status of Mathematical Inventory
Theory," Management Science, 12, pp. 745-777, 1966.

Veinott, Jr., Arthur F., "Optimal Policy for a Multi-Product,
Dynamic, Nonstationary Inventory Problem," Management Y-.ionce,
12, pp. 206-222, 1965.

Veinott, Jr., Arthur F., "Minimum Concave-Cost Solution of
Leontief Substitution Models of Multi-Facility Inventory Systems,"
Operations Research, 17, pp. 262-291, 1969.

Zangwill, Willard I., "Minimum Concave Cost Flows in Certain
Networks," Management Science, 14, pp. 429-450, 1968.

Zangwill, Willard I., "A Deterministic Multiproduct, Multifacility
Production and Inventory Model," Operations Research, 14, pp. 486-
507, 1966.
















BIOGRAPHICAL SKETCH


Robert Caleb Dorsey was born May 5, 1943, in Memphis, Tennessee.

He received his elementary and secondary education in the Memphis school

system, graduating from White Station High School in June, 1961. He

attended Georgia Institute of Technology from 1961 to 1964 and received

a Bachelor's degree with a major in Industrial Engineering in August,

1964. After graduating from Georgia Tech, he worked for Tennessee

Eastman Company in Kingsport, Tennessee, from September, 1964, through

March, 1970. While working with Tennessee Eastman Company, Robert

Dorsey pursued his studies through a night program at the University of

Tennessee and was awarded a Master of Science degree with a major in

Industrial Engineering in December, 1967.

In March, 1970, Robert Dorsey took an educational leave of

absence from Tennessee Eastman in order to enter the graduate program

in the Industrial and Systems Engineering Department at the University

of Florida. He received a Master of Engineering degree from the Uni-

versity of Florida in June, 1971.

Robert Dorsey is a registered Professional Engineer and is a

member of Alpha Tau Omega fraternity, Alpha Pi Mu honorary engineering

society, and the American Institute of Industrial Engineers, The Insti-

tute of Management Science, and Operations Research Society of America

professional societies. He is married to the former Lou Gilliam of

Kingsport, Tennessee, and is the father of one son, Trent.












I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.





11. Donald Ratliff ,lairnian
Assistant Professor of Industrial and
Systems Engineering



I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the decree of
Doctor of Philosophy.





Thom J. Hod son, Co-Chairman
Assistant Pr fessor of Industrial and
Systems Engin ering


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.





Richard L. Francis
Professor of Industrial and
Systems Engineering



I certify that I have read this study and that in my opinion it
co.nior',Ls to acceptable standards of scholarly presentation and is fully
ad'q!u-.te, in .-c:ope and quality, as a dissertation for the degree of
)DoCr tor of Pil 'o'hy.





KerF. E. Ki trick
As. istant i ofcssor of Industrial and
Systen's engineering g




Full Text

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