Multi time period stochastic programming for medium term production planning

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KULTI TIME PERIOD STOCHASTIC PROGRAMMING FOR MEDIUM

TERM PRODUCTION PLANNING

by

Robert W illiam A shford, M.A.

Ph.D. T h e sis

November, 1981

The School o f In d u s tria l and Business Studies

1 4 6 8 9 11 24 29 31 32 35 39 42 44 48 53 63 65 TABLE OF CONTENTS

PART I

INTRODUCTION

1. Medium Term Production Planning

2. S to ch a stic Programming

3. The Eva lu a tio n o f Approximate Solution Techniques

4. Thesis Plan

A REVIEW OF STOCHASTIC PROGRAMMING TECHNIQUES FOR MEDIUM TERM PRODUCTION PLANNING

1. Introduction

2. The Models Studied

3 . Methods of Solution

4 . Conclusions

AN EXPOSITION OF "MULTI-TIME PERIOD STOCHASTIC SCHEDULING" by Beale, F o rre s t and Taylor

1. Introduction

2. The Production/Inventory Model

3. The Formulation o f a Non-Linear Program

4 . Re-Estim ation o f the S to c h a s tic V a r ia b ilit y

4 . Conclusions

PART if

DYNAMIC PROGRAMMING APPROACHES TO PRODUCTION PLANNING

1. Introduction

2. The Dynamic Programming Approach

3. The Case o f a Single Dimensional State Space

4 . The Case of a M ulti-Dim ensional State Space

Pacje

5 . A GENERAL MODEL FOR PRODUCTION PLANNING

1. In tro d u ctio n 67

2. A D e scrip tio n o f the Model 70

3. An A p p lica tio n of the General Model to a Production/Inventory/

Manpower Planning Problem 74

4 . Conclusions 85

6. AN APPROXIMATE SOLUTION TECHNIQUE FOR THE GENERAL MODEL

1. In tro d u ctio n 86

2. Tech nical P re lim in a rie s 89

3. The Reduced Problem 93

4 . The R e stricte d Reduced Problem 109

5. The Reformulation o f the R e stricte d Reduced Problem 119

6. Computational Aspects 137

7. Conclusions 149

PART I I I

7. THEORETICAL APPROACHES TO THE EVALUATION OF SMOOTHING ALGORITHMS

1. In tro d u ctio n 154

2 . Variance Reduction Techniques 159

3. General Functions o f Control V a ria te s 172

4 . Modelling the Expected Future Revenue 184

5 . Conclusions 203

8. THE CONSTRUCTION OF THE MARTINGALE CONTROL STATISTIC FOR THE GENERAL MODEL

1. In tro d u ctio n 206

2 . The General Model 208

3 . The Case ( i ) In te rp re ta tio n o f the M artingale D iffe re n ce

Function 212

4 . The Case (11) In te rp re ta tio n o f the M artingale D ifference

5.

Page

The Case (11 ) In te rp re ta tio n o f the Martingale D ifference

Function: The Quadratic Model 223

6. Some Necessary C a lcu latio n s 229

7. Conclusions 232

9. THE EVALUATION OF FOUR APPROXIMATE ALGORITHMS BY SIMULATION

1. In tro d u ctio n 234

2. The Model on Which the Algorithms Were Tested and the

Control S t a t is t ic s were Used 236

3 . A Simple Numerical Example 243

4 . The R e su lts 245

5 . Conclusions 251

10. CONCLUSIONS

1. Productlon/Inventory Modelling 253

2. The General Model and it s Approximate Solution Technique 254

3. The Eva lu a tio n o f Approximate Solution Techniques 255

4 . Suggestions fo r Further Research 258

Acknowledgements are due to my s u p e rv is o rs , Dr. R.G. Dyson

o f the U n iv e rsity o f Warwick and Pro fe sso r E .M .L. Beale o f Scicon

Computer Services Lim ited . Without th e ir kind and helpful a ssistan ce

the research described in th is th e sis would not have been p o ssib le .

My thanks are also due to T e rri Moss fo r typing the manu­

s c r ip t and fo r her great patience w ith many a lte ra tio n s .

The research described herein was supported by the Science

SUMMARY

Exact so lu tio n s to s to c h a s tic , c ap acitate d , multi-commodity,

m u lti-stag e production/inventory models are in general computationally in t ra c t a b le . The p ra c tic a l a p p lica tio n of such models is therefore

in h ib ite d . In th is th e sis a general s to c h a s tic , ca p a c ita te d , m ulti-

commodity, m u lti-stag e production/inventory model w ith lin e a r cost stru c tu re is proposed. Under convexity conditions i t is a sto chastic

lin e a r program. A good com putationally e f f ic ie n t approximate solution

technique is developed and some numerical re s u lts rep o rted.

I t i s important to assess the m erit of approximate techniques and th is i s done s t a t i s t i c a l l y by r e p lic a t iv e sim u la tio n . But the

accuracy o f th is method improves only as the square root of the number

of sim ulatio n t r i a l s made, so i t is important to e lim in a te any unnecessary

v a r ia b ilit y in each t r i a l . I t is proposed that t h is be done by the

use o f control s t a t i s t i c s . Several novel control s t a t i s t i c s are developed,

the most powerful being a m artingale control s t a t i s t i c constructed

independently fo r each t r i a l from inform ation provided by the approx­

imate technique being te ste d .

R esults are reported o f testin g the approximate solution

technique developed fo r the general model, ordinary lin e a r programming

ignoring a l l the s to c h a stic elements in the problem, and two other approximate techniques, by r e p lic a t iv e sim u la tio n . These suggest that

the penalty incurred by ignoring the sto c h a stic nature o f the problem

is s ig n if ic a n t , but th a t f i r s t order deviation s from optimal decisions

may lead only to second order p e n a ltie s . This is a d e sirab le feature

of the s to c h a stic models, fo r i t in d icates th at approximate solution

techniques to s to c h a stic programs may be more r e lia b le than would be

-1-1. MEDIUM TERM PRODUCTION PLANNING

T h is th e sis develops and stu d ie s a dynamic sto ch a stic model

fo r use in medium term production and inventory planning. In th is

context 'medium term' planning is intended to mean decisions about

such asp ects o f the production system as production le v e ls of in d ivid u al

fin ish e d products and manpower le v e ls fo r d iffe re n t categories o f

employee. Short term scheduling problems which involve a d etailed

a n a ly s is o f the day to day running o f the production system and which

examine, fo r example, which components can be produced in which order

on what machine, are very s p e c ific to the industry and plant being

studied and are excluded. Also excluded are long term s tra te g ic

problems w hich, fo r example, a ris e in decisions to expand or con­

t r a c t production f a c i l i t i e s , to produce a new product lin e or to

enter new markets.

The problems addressed herein are e s s e n t ia lly o f a t a c tic a l

nature and t y p ic a lly concern the s e ttin g of monthly or quarterly

production ta rg e ts, workforce le v e ls and buffer stocks over a planning

horizon o f a y e a r. Some authors r e fe r to th is as production smoothing.

There are two p rin cip a l aspects of t h is problem th at require fu rth e r

d is c u s s io n .

F ir s t ly » there is a trade o ff between holding large q u a n titie s

of products in stock and frequent changes of production and manpower

le v e ls . F lu ctu atin g demand might be handled by c o n tin u a lly varying

the production rate and h irin g or la y in g o ff sectio ns of the workforce,

-2-often expensive so i t might be more p ro fita b le to keep the production

rates and manpower le v e ls constant w h ils t meeting flu c tu a tio n s in

demand from high stock le v e ls . In general the best decision w ill lie

between these two extremes. Determination o f p re c is e ly what the best

decisions are in vo lve s q u a n tific a tio n o f the costs involved and study

of the appropriate mathematical model o f the system.

Secondly, the demand requirements themselves are ra re ly known

e x a c tly in the medium term fo r they depend on future decisio ns made

by customers who are outside the control o f the production system.

These demand requirements may only be known p r o b a b ilis t ic a lly . There

is an obvious trade o ff between producing only as much as can d e fin ite ly

be so ld , which keeps stock le v e ls low but takes l i t t l e advantage of

the lik e ly demand, and producing so much th a t demand can always be

s a t is fie d which r is k s carryin g in o rd in ate ly large sto c k s . Determin­

atio n o f the best production t a c tic s in the face o f t h is problem

involves the d e cisio n maker's a ttitu d e toward r i s k , q u a n tific a tio n of

the uncertainty in demand and the study of the appropriate p ro b a b ilis tic

mathematical model.

The problem is u su a lly fu rth e r complicated by co n stra in ts on

perm issible production ra te s and items that can be held in sto ck. These

may require the production o f items to stock in order to take most

advantage o f the peak in c y c lic a l or seasonal demand.

Of the two aspects discussed above the former i s e a s ie r both

from the point o f view of acq uiring s u f fic ie n t cost data and in the

-3-much more d i f f i c u l t from both points of view. The q u a n tific a tio n of

the u n ce rtain ty in future demand is a d i f f i c u l t task and sto ch a stic

models present formidable problems both in th e ir th e o re tic a l and

computational asp ects. H ow ever.it is in a sense more general fo r

models designed to handle the la t t e r problem can e a s ily be extended

to handle the former problem but not v ic e - v e rs a . The model developed

in th is th e s is although motivated by the u ncertainty problem is designed

to handle both. I t is presented in Chapter 5 . In order to exp lain

the stru c tu re of the work some problems associated w ith sto ch a stic

4

-2. STOCHASTIC PROGRAMMING

Models used fo r the a n a ly s is o f the problems o u tlin e d above

f a l l n a tu ra lly into the ambit o f sto ch a stic programming. This is

the study o f c e rta in models (s to c h a s tic programs) which e x p lic it ly

incorporate random v a ria b le s in to th e ir formulation and which reduce

to d e te rm in istic mathematical programmes as the v a r ia b ilit y in the

random v a ria b le s tends to zero . The form ulation of such models

has not only been motivated by production planning problems but also

by the need to control water resources and to tackle problems a ris in g

from economic and fin a n c ia l planning . Each source of " re a l world"

problems has generated d if f e r e n t c la s s e s of sto ch a stic programs.

But there is much common ground between them and th e o re tic a l study

has led to t h e ir being c la s s if ie d on the b asis of th e ir more ab stract

p ro p e rtie s. In consequence most c la sse s of sto ch a stic programmes

have something to o ffe r in the modelling o f production systems. A

b r ie f review of sto c h a stic programming from t h is viewpoint is there­

fore given in Chapter 2. However, fo r the medium term production/

inventory problems described above, one c la s s o f sto c h a stic programs

is more natural to use than any o th e r. This is the c la s s of a ctive

m ultistage programs. Each stage can be id e n tifie d with time periods

in the "re al world" problem, t y p ic a lly months or q u a rte rs , and decisions

which must be made a t each stage are only allowed to depend on the

r e a lis a t io n s o f random v a ria b le s in previous (and p o ssib ly the present)

stages and the d is trib u tio n s o f the random va riab les in la t e r time

periods conditional on these r e a lis a t io n s . Thus production decisions

are only allowed to depend on the demand in previous time periods and

not th at in fu tu re ones.

5

-U nfo rtunately, in g e n e ra l, e x a c tly optimal so lu tio n s to m u lti­

stage sto c h a s tic programs reduce a t best to dynamic programming

methods and these become com putationally in tra c ta b le as the number

of commodities being modelled in c re a s e s . This is shown in Chapter 4

which develops some dynamic programming models. Approximate solution

methods are therefore o f in t e r e s t . T his th e sis contains a general­

is a tio n and development of one o f the most promising approximate

methods due to Beale, Fo rrest and Taylo r [4 ] . Their method is

described in Chapter 3 and the development of i t is presented in

Chapter 6.

I t is important to asse ss the m erit of approximate so lutio n

3 . THE EVALUATION OF APPROXIMATE SOLUTION TECHNIQUES

The optimal u t i l i t y returned by the o b je ctive function of,and

the optimal decision given by,an approximate so lu tio n method to a

s to c h a s tic model may be in e rro r. T h is can be tested on s u f f ic ie n t ly

simple examples by comparisons with those obtained by a method known

to be e x a c t. However, th is comparison may be misleading i f i t is

used to assess the suboptim ality of the d ecisio ns recommended by

the approximate method. F i r s t l y , the u t i l i t y gained by a c t u a lly

using an approximate so lutio n technique may be very d iffe re n t from

that returned by the model's o b je ctive fu n ctio n . Secondly,deviations

from the optimal decisions are not in themselves important. What is

important is the drop in u t i l i t y consequent upon them and t h is may

be hard to gauge.

The method suggested in th is th e s is fo r handling these problems

is that o f s t a t is t ic a l sim ulatio n. The environment w ithin which the

s to c h a s tic program operates is modelled on a computer. The random

v a ria b le s in the problem are simulated by pseudo-random numbers. Under

the in flu e n c e o f these,and the control o f the approximate method being

te ste d , the sto ch a stic process then evolves from the f i r s t time period

in the problem to the time horizon. T h is is known as a sim ulation

t r i a l . I t is repeated a large number o f times in order to assess the

performance of the approximate so lu tio n method s t a t i s t i c a l l y .

U nfortunately s t a t is t ic a l estim ates of a ttrib u te s of in te re s t

in the process made in th is way are unacceptably in accu rate . This

problem i s overcome by the use of control s t a t i s t i c s . These are

described and developed in Chapter 7 , but the ap p licatio n of them

requires the d e riv a tio n o f formulae s p e c ific to the process being

simulated and the algorithm te ste d . These formulae are derived in

Chapter 8 fo r the approximate so lutio n algorithm developed in Chapter

6. However the formulae are not re s t ric te d to th is algorithm . This

is shown in Chapter 9 which reports the re s u lts o f sim ulation experiments

in which four approximate algorithms were tested on two simple

examples. The r e s u lt s of these experiments suggest that f i r s t order

deviations in the decisio ns made by approximate algorithms from th e ir

t r u ly optimal va lu e s produce only second order deviation s in the

u t i l i t y re a lis e d by using algorithm from i t s optimal v a lu e . This

is a very d e sira b le feature o f the process fo r i t in d ica te s that the

suboptim ality of approximate so lutio n methods may be very much sm aller

than the approximations made by i t might suggest. The th e sis ends with

a b rie f summary and conclusions in Chapter 10, in which suggestions

4 . THESIS PLAN

Th is thesis is devoted to the study and development o f sto ch a stic

models fo r medium term production planning. I t divid es into three p a rts .

Part I reviews estab lished models and so lutio n techniques. Chapter 2

surveys sto ch a stic programming and Chapter 3 presents an exposition

of the methods of B e a le , Fo rrest and Taylo r [ 4 ] . Part I I deals with

novel contributions to modelling production/inventory problems. Chapter

4 describes some dynamic programming techniques, suggests an e f f ic ie n t

algorithm fo r the single-commodity case, and reports some computational

experience with i t . Chapter 5 presents a f a i r l y general production/

inventory model and describes an ap p lica tio n of i t to a production/

manpower/inventory planning problem. In Chapter 6 an approximate

solution technique to i t is developed, and some numerical r e s u lts are

given. The work contained in both Chapters 5 and 6 is a g e n e ra lisa tio n

and extension of that o f Beale et a l . [ 4 ] , I t i s important to assess the

m erit of approximate techniques and th is is done in Part I I I . Chapter

7 describes the techniques of re p lic a t iv e sim ulation and control

s t a t i s t i c s . I t develops some novel ways o f constructing control

s t a t is t ic s . Some o f these are based upon the d e rivatio n o f a martingale

fo r each sim ulation t r i a l from inform ation about the process provided

by the algorithm being te ste d . Detailed formulae fo r the computation

of these are derived in Chapter 8. Chapter 9 describes sim ulation

experiments which te s t both approximate algorithms and the e ffic a c y

o f the control s t a t i s t i c s . The re s u lts are reported and conclusions

drawn from them. This th e sis is concluded with a b rie f summary in

CHAPTER 2

A REVIEW OF STOCHASTIC PROGRAMMING TECHNIQUES

1. INTRODUCTION

The p rin c ip a l d if f ic u lt y o f studying production/inventory

problems is that decisio ns have to be made in the face of u n c e rta in ty ,

not ju s t of u n re lia b le data, but also that in future d ecisio ns made by

o th e rs, fo r example customers, over whom the decision maker has no

c o n tro l. A n a lysis o f the consequent uncertainty in the system is

e sse n tia l in the determination o f the best production stra te g y and

other s a lie n t aspects of the production/inventory s y s te m ,p a rtic u la rly

sa fe ty sto cks. These have t r a d it io n a lly been studied by s t a t i s t i c a l

methods in is o la tio n from the r e s t o f the system. See, fo r example,

Whitin [ 6 1 ] , Nador [ 4 2 ] and Chapter four of Hadley and W hitin [26 ] .

Properly, however, they ought to be studied in the context o f the whole

production process by appropriate modelling.

S to ch a stic programs form the natural choice of models to use

in th is context. Much a tte n tio n has been devoted to them, although

i t has been more directed to a study of th e ir ab stra ct p ro p ertie s

than com putationally e ffe c tiv e methods of so lu tio n . This chapter

presents a b r ie f review of the p rin c ip a l forms o f s to c h a stic programs.

The d iffe re n t forms that have been proposed are surveyed in Section

2 . These d ivid e in to two categories : the passive and the a c t iv e

forms. The form er, in which decisio ns are made a fte r the outcome

of the random v a ria b le s in the problem becomes known,may be o f importance

in s tre g ic planning where the decisio n maker may want to a sse ss the

impact of a new production f a c i l i t y on the p ro b a b ility d is t rib u t io n

of h is to tal revenue. Since the concern of th is th e sis is w ith

passing in t e r e s t here. However, a b r ie f descrip tio n of them has been

included fo r the sake o f completeness.

The a c tiv e sto ch a stic programs re q u ire decisions to be made

before the outcomes of some or a l l of the random va ria b le s in the

problem are known, and themselves d iv id e into two types: the sin g le

o r two-period problems and the more general m ulti-stage problems. The

former a r e , of course, sim pler and the theory behind them b etter

developed than for the la t t e r . However production/inventory problems

are b etter modelled by m ulti-stage a c t iv e programs, each stage

representing a u n it o f tim e, say a month or q u a rter, so i t is these

th at are o f most in te re s t here. A d iscu ssio n o f sin g le and two stage

programs i s given below in order to present a c le a re r p icture of

the com p lexities that a ris e in th e ir m u lti-sta g e g e n e ra lisa tio n s, and

a lso because some o f the techniques used to handle them can be extended

to the m u lti-period case. A review of the approaches that have been

adopted fo r the so lutio n of a c tiv e s to c h a s tic programs is given in

-11-2 .1 . The Basic S tru c tu re of Stochastic Models

Nearly a ll sto c h a s tic models whose formulation has been motivated

by the need to ta c k le production planning problems d iv id e n a tu ra lly

into a f in it e number o f d isc re te time p eriods. Key a ttrib u te s of the

system being modelled are considered to be fixe d during each period,

but may, of co urse, vary between time p erio d s. The models are then

formulated in terms o f these key a t t r ib u t e s , some o f which may be

random v a ria b le s . I t is assumed that the decision maker wishes to

maximise or minimise some function o f these a ttrib u te s subject to the

co n stra in ts imposed upon them by the system.

One o f the most estab lished classe s of models used fo r determ­

i n i s t i c production planning is that o f lin e a r programs. These have

the m erit o f being straightforw ard to formulate and so lu tio n methods

fo r them are w ell-advanced. Developments o f the simplex algorithm

have enabled computer programs to be w ritte n which so lve very large

lin e a r programs indeed. Thus lin e a r programs have formed the natural

s ta rtin g point fo r the development of sto ch a stic models. The concern

of th is chapter w i l l be with these sto c h a stic lin e a r programs

"max" c Tx over x subject to "Ax = b" (1)

where b ,c and x a re column vectors and A i s a m a trix , and (A ,b ,c ) are

random v a ria b le s . There are two d iffe re n t in te rp re ta tio n s to th is

-12-random v a ria b le s (A ,b ,c ) are re a lis e d and the o b je ctive function

and co n stra in ts are w ell defined. In the a ctive approach some or

a l l o f the x 's must be chosen before a ll the random v a ria b le s are

re a lis e d and so both the o b jective function and the co n strain ts have

to be more c a re fu lly s p e c ifie d . The former approach is discussed

f i r s t .

2 .2 . The Passive Approach

In t h is approach otherwise c a lle d the "w ait and see" problem

by Madanasky [ 3 9 ] or " d is trib u tio n " problem by Vajda [54 ] , the

d ecisio ns x are taken a fte r the random va ria b le s (A ,b ,c ) are re a lise d

in the program

max z = c Tx over x su b je ct to Ax = b. (2)

So i t is desired to co nstruct an optimal map or decision ru le from

the outcome space o f the random va ria b le s to the decision space.

I t can be shown th e o re tic a lly (See Dempster [ 1 7 ]) that the

outcome space can be p artitio ned into a f in it e set o f decision regions

such th at the optimal d e cisio n , x ° , is constant in each decision

regio n. Furthermore, each decision region can be id e n tifie d with

a b asis of (2) and the decision regions form a c e llu la r stru ctu re

whose faces have Lebesgue measure ze ro .

Having found the se t of decisio n regions and the optimal

-13-x ° a -13-x ° (A ,b ,c ) (3)

the problem i s then to compute the d is trib u tio n functio n of

z = CT x ° ( A ,b ,c ) .

The c h a ra c te ris a tio n o f t h is d is trib u tio n functio n in terms

o f general random (A ,b ,c ) has not y e t been obtained, but sp ecial

cases in which A and c (d u a lly A and b) are fix e d have been studied.

For example, Bereanu [ 6 ] has treated the case where there is only

a sin g le random v a ria b le in the problem and la t e r extended h is work

[ 7 ] to the case where A is s to c h a stic but imposing r e s t r ic t io n s

on the random v a ria b le s .

In g e n e ra l, the a lte rn a tiv e a c tiv e approach is a more natural

one fo r the modelling o f production planning problems and i t is th is

to which a tte n tio n is now d ire c te d .

2 .3 . The A ctive Approach

In t h is approach, also known as the "here and now" approach, some

or a l l of the decisio n va ria b le s must be chosen before the outcome of

a l l the random v a ria b le s in the problem is known. When the process is

e x p lic it ly p e rio d ic and the decisio n v a ria b le s and random v a ria b le s

p e rtain to in d ivid u a l time p e rio d s, i t is common to make the decision

v a ria b le s in each period a function o f the random v a ria b le s re a lise d

up to (and perhaps includ ing ) that period.

-14-Care must be exercised over the d e fin itio n s of the o b je ctive

function and c o n s tra in ts . The o b je c tiv e function is designed to

model the decisio n maker's preference about how the system should

behave. There are three such models commonly treated in the l i t e r ­

a tu re . These are

(a ) E-models, in which i t is assumed that the decision maker has

a neutral a ttitu d e towards r is k and so wishes to maximise (m inim ise)

h is expected p r o fit (c o s t) ,

(b) P-models, in which i t is assumed that the decision maker wishes

to maximise (m inim ise) the p ro b a b ility of h is p ro fit (co st) being

greater (le s s than) some target v a lu e , and

(c ) V-models, in which i t is assumed that the decision maker wishes

to minimise the to tal v a r ia b ilit y o f h is p r o fit or c o st.

See, fo r example Charnes and Cooper [ 9 ] fo r a fu rth e r d iscussio n

of such models with reference to chance constrained programming. The

m a jo rity of work published in t h is area deals with E-models. In

what follow s reference w ill only be made to these. However, P and V

model analogues should be re a d ily apparent.

There are two a lte rn a tiv e in te rp re ta tio n s of the co n stra in ts

Ax = b. They can be regarded as holding almost su re ly ( a . s . ) i . e .

with p ro b a b ility one, or with some prescribed high p ro b a b ility . The

la t t e r approach is known as chance constrained programming.

The remainder of th is sub-section w ill be devoted to one or two

stage models. T h e ir g e n e ra lisa tio n to many stages leads to an even

greater v a rie ty of in te rp re ta tio n s and is discussed in Section 2.4

-15-The two-stage model in which the c o n s tra in ts hold almost su re ly

is now addressed. E x p lic it ly stated i t is

A,B and R are m atrices and b .c .d .r .x and y are v e c to rs . Formally

t h is is c a lle d the two stage sto ch astic lin e a r program with recourse.

The i n i t i a l d e c is io n , x , must be made before the random v a ria b le s

(A ,B ,b ,c ,d ) a re re a lis e d ; the re a lis e d c o n stra in t discrepancy b - Ax

y ie ld s a lo ss by the second stage which is to choose a recourse decision

y to

Minimise dTy subject to By = b-Ax, y 2 0 . (7)

The problem i s considerably sim p lifie d i f the recourse m a trix, B,

is fix e d and equal to (1 ,- 1 ) where I is the id e n tity m a trix. Dempster

[ 1 7 ] re fe rs to the problem thus obtained as that with simple recourse.

Beale [ 3 ] and Wets [58 ] re fe r to i t as the complete problem.

The s in g le stage chance constrained problem may be w ritten

Max E { c Tx - min d^y} ₍₅₎

x

_y

su b je ct to Rx

Ax + By = b a .s .

and x 20, y 20, a .s .

(6)

(8) over x and su b je c t to

P{Ax s b}

2

0

2

(9)

-16-where A,b and c are random v a ria b le s ,

a

and the decision x must be chosen before the random va ria b le s are

r e a lis e d . There are a v a r ie t y of ways in which the co n stra in ts may

be regarded. The two p rin c ip a l ones are

(a ) Total chance c o n s tra in ts , where (9) may be w ritte n

P {(Ax)i sb., V i } i a

(11)

and

(b) J o in t chance c o n s tra in ts where (9) may be w ritte n

P {(A x )1 s b j} * c y V i. (12)

U sually 6 i s taken to be 1, so (10) holds almost s u re ly , but Charnes

and Kirby [12 ] allo w 3 to be le s s than one.

The study o f such models fo r general random A ,b , and c is very

complex. U sually authors r e s t r i c t th e ir attention to the case where

only b is random. See, fo r example, M ille r and Wagner [4 1 ] and

Charnes, Kirby and Raike [13 ] . However, I s h i i , Shiode, Nitshida

and Iguchi [34 ] study a model in which one row of the technology

m atrix is random.

Under c e rta in conditions the two-stage sto ch a stic lin e a r program

with recourse is equivalen t to the sin g le stage chance constrained

problem. Gartska [23 ] reviews re s u lts dealing with t h is equivalence.

Ju s t as the a c tiv e approach provides a more natural se ttin g fo r

the modelling o f production planning problems, so m ulti-stage versions

of i t are more appropriate than single or two stage programs. These

-17-2 .4 . M ulti-stage Versions of the Active Problem.

In m ulti-stage problems, the process being modelled divid es

n a tu ra lly into time p erio d s. Each decision va ria b le and random

va ria b le can be associated with a p a rtic u la r time period. I t w ill

be expedient to review those models in which the constraints hold

almost su re ly f i r s t . Four p rin c ip a l va ria n ts have received a tte n tio n .

The most general is discussed f i r s t .

(a ) The General Lower T rian g u lar Model

E x p lic it ly stated i t is

su b je ct to

l

u=0 fo r t = 0 ...T (14)

where the A ^ 's are m a trice s, the bt 's , ct 's and x t ‘ s are v e cto rs.

re a lis e d a t period t . Let a ll the random v a ria b le s re alise d in period

t be a fun ctio n of a more general random va ria b le Then the

d ecisio ns x t are re s tric te d to be functions o f

When a l l the A ^ 's are id e n t ic a lly zero fo r u s t-2 the

follow ing problem is obtained

r T

Max E

l

t=0 t

and x t * 0 a . s .

A* , b.. and c f are random v a ria b le s fo r a ll t i l , and are supposed

tu t t

(b) The S ta irc a s e Model

Max E (1 6)

t=0

sub ject to A x

o o = bo

-B x + Ai x i _{o o 1 1} = b, a . s . _{1 (17)}

and xt a 0.

Again, the t th^ stage decision is r e s t ric te d to be a function

of the random v a ria b le s re a lis e d up to and including that p erio d.

The special case when only the b 's are random has received atte n tio n

from Dantzig [15 ] , Wets [57 ] and Birge [ 8 ] .

(c ) The Control Theoretic Formulation

This i s a special case of the s ta irc a s e problem in which only

the b 's are random. I f the decision v a ria b le s x t are p a rtitio n e d

into ( y l , u J ) T where y . is a sta te v a ria b le and u. a control v a ria b le ,

and i f the system m atrices At and Bt can be correspondingly p artitio ned

and i f the random vector bt is p a rtitio n e d correspondingly in to A,

t

and Bt

1 9

-Ft * t + Gt Ut = * t+ I

and V t + Dt ut s et

Gaalman [20] studies a sp ecial case o f th is where (18) 1s

* t+ l = V t + Bt ut +Ct 5 t <19)

and the process evolves over the in f i n i t e horizon. Using modern

control theoretic techniques and making assumptions about the

d is trib u tio n of the random v a ria b le s and s t a b ilit y of the process,

Gaalman derives the optimal d ecisio n ru le s ut as a function of the state

v a r ia b le y ^ _ j. However, the no n-negativity co n strain ts on ut and

y t have been dropped and h is model cannot handle capacitated production/

inventory systems th at are o f in t e r e s t here.

(d) The M ulti-Stage Recourse Problem

This is a natural m ulti-stage g en eralisatio n of the two-stage

s to c h a stic program with recourse. Dempster [ 17] w rite s i t as

su b je ct to Max {E

X

i ‘ A

t=i

-m in d^y }

y

(20)

A01 X1 = b0

A11 X1 - » I = bj a .s . (21)

k u

\

-20-In general Atu> Bt> t>t and c t are random fo r t i l . I f these

t th stage random v a ria b le s are regarded as fun ctio ns of a more general

random v a r ia b le , ? t , then the decisions x t are re s tric te d to be

functions of ^ and the recourse d ecisio n s y t are re s tric te d

to be functions of

I t can be shown that the m ulti-stage general tria n g u la r problem

(a ) is e q u ivale n t to the m ulti-stage recourse problem ( d ) , fo r each

can be regarded as a special case of the o th e r.

To see t h is le t the su p e rscrip t a or d denote an a ttrib u te

pertaining to problem (a) or (d) re s p e c tiv e ly . Then to show th at

problem (a ) is a special case of problem (d) s e t y d = xd+1 and

x t “ x t+l* Then x t is a 'function and furthermore ^

Aat is defined to be -Bd and Aau to be Adu-1 then

k - V ad yd Rd vd

bt " * Atu u ‘ Bt xt u=l

Xa

u*

Also defining to be cd+1 - dd fo r 1 s t ¡s T - l ,

cd and ca to be -dd i t is seen that

ca to be c? and

o 1

Max E

l

xd

t-i

yd

t=0

To show th a t problem (d) is a special case of problem ( a ) ,

-21-and corresponding p a r tit io n c“ :

" » )

a

c t = fo r 1 s t < T - l

a

co = and c .

-d:

i t is seen that x “ is a function of ^ ...1 and

T J . T ,T . J .

Max E J c l x? = Max E { 7 c? x :. - min d* y^}

**

t-0

*d

fl

,d

Also i f A^u is correspondingly p artitio n e d into ( A^u+1»0) for

0 s u s t - l and (0 ,- B b) fo r u = t , then

bt '

l

t

p . d d

n

l

U=1

so problem (d) is a sp e cial case o f problem (a) and therefore the

Hence the s ta irc a s e problem (b) and the control th e o re tic

problem (c ) can be regarded as sp ecial cases o f the m ulti-stage

recourse problem ( d ) .

(e ) M ulti-stage Chance Constrained Problems

The sin g le stage chance-constrained problem g eneralises e a s ily

to the m ulti-stage case , although there is a greater v a rie ty of

p o ssib le in te rp re ta tio n s of the c o n s tra in ts . As in the m ulti-stage

recourse problem the technology m atrix has a lower tria n g u la r block

s tru c tu re . In general terms the model may be stated

T T

Max E

l

t= l

over x and subject to

P{^llxl

5 bl* * al

P{A21x1+A22x2< b2) * a2

t

Pi

l

U=1

and (P x t 2 0} 2 (23)

the Atu 's are m a trice s, the bt ' s , c t 's and x t 's are vectors and

A^ , bx and c . are in general random v a ria b le s fo r t i l . I f

tu t t 3

At u ’ bt and Ct are re9arclecl as bein9 functions of a more general

-23-to be functions of Most authors r e s t r ic t t h e ir attention

to the case o f fixe d technology m atrices Atu when in c e rta in cases

the problem is equivalent to the m u lti-stag e recourse problem (d ),

Gartska [23 ] . The fu rth e r r e s t r ic t io n that the matrices At should

a l l be of dimension (1 x l ) is also made in most of the lit e r a t u r e .

The d iffe re n t ways in which the p ro b a b ility of the c o n s tra in ts (22)

and (23) can be interp reted needs fu rth e r d iscu ssio n .

I f the p ro b a b ilitie s in (22) and (23) are computed using the

jo in t d is trib u tio n o f a l l the random v a ria b le s in the problem then

Charnes and Kirb y [ 1 1 ] term the problem one of "to ta l chance

c o n s tra in ts ". I t was t h is approach th a t Charnes, Cooper and Symonds

[10 ] used in th e ir o rig in a l form ulation o f a chance-constrained

problem to model the production o f heating o i l .

I f the p ro b a b ilitie s in (22) and (23) are computed using the

d is trib u tio n o f conditional on the re a lis e d values of ^ , . . . , 5 1

then the problem is termed one o f "co nditio nal chance co n stra in ts"

by Charnes and Kihby [1 1 ] .

E is n e r, Kaplan and Soden [ 1 8 ] a lso considered another in t e r ­

p retation which they c a lle d the "conditional-go" approach. At stage

t the u th stage c o n stra in t (where u > t) is regarded as not a c tu a lly

being revealed u n til the u th stage. So the p ro b a b ilitie s in (22)

are computed w ith the marginal d is trib u tio n of given j

i . e . the u th stage c o n stra in t becomes u

PI I Auvx#(5l... 5v.ll s b ulEl... (24>

V=1

where t < u.

Other v a ria n ts o f the problem a r is e out of d iffe re n t choices of

adm issible decisio n r u le s . This is discussed in Section 3 on solution

3 . METHODS OF SOLUTION

In th is se ctio n some so lutio n techniques that have been proposed

fo r the sto ch a stic programs presented above w ill be b r ie f ly discussed.

Since production planning problems are more n a tu ra lly modelled by the

a c tiv e or "here and now" sto ch a stic programs than the "passive" or

"w ait and see" v a r ie t y i t is the so lu tio n of the former which is of

in te re s t here; lin e s of attack on the la t t e r have already been b r ie f ly

mentioned.

The two or sin g le -stag e a c tiv e s to c h a stic programs are e asie r

to so lve than t h e ir m ulti-stage co u n te rp arts. Exact com putationally

e ffe c tiv e methods o f so lutio n have been devised fo r the former, but

not fo r the la t t e r when there are many c o n stra in ts per stage, and

these are necessary fo r the e x p lic it modelling of many commodities.

The only e ffe c tiv e approximate method t h a t has been proposed is that

of B e ale , F o rre st and Taylor [ 4 ] which solves a simple production/

inventory model.

3 .1 . Two Stage Sto ch a stic Programs w ith Recourse

As with the other sto ch a stic programs, solution of the general

case of (5) and (6) in which A ,B ,b ,c and d are a l l random is very

d i f f i c u l t both th e o re t ic a lly and com putationally. Attention has u su ally

been re s tric te d to the special case in which only b is random, although

Evers[19 ] tackle s a random A m atrix by Monte Carlo methods and

El-A g izy [ 1 ] has shown that i f c is random then i t can without lo ss of

g e n e ra lity be replaced by i t s expected value and the co rre la tio n between

-25-Dantzig [15 ] tackled the problem with simple recourse 1n which

only b i s random. He showed that the sto ch a stic program i s equivalent

to the d e te rm in istic program

Here the recourse m a trix B has been p artitio ned into (1 ,- 1 ) and the

vectors d and y have been correspondingly p artitio ned into

F is the d is trib u tio n function o f b. This program can be shown to be

convex i f (d+ + d’ ) z 0.

Solutions to the above program can be approximated by assuming

a d isc re te d is trib u tio n fo r the random vector b, in which case a d iffe re n t

recourse d e cisio n , y , must be associated with each point o f the d iscre te

d is t rib u t io n . Dantzig and Madanasky [16 ] adopt th is approach and use

decomposition methods to e x p lo it the program's stru c tu re . S tra zick y

[52 ] takes th is approach fu rth e r by using b asis decomposition and reports

some numerical r e s u lt s .

Wets [60 ] has investigated the d e rivatio n o f d e te rm in istic

equivalents fo r problems with general fixe d recourse.

Other approaches have been proposed by, fo r example, Van Slyke

and Wets [ 51] who use gradient methods and Garkska and Rutenberg [24 ]

who use la t t ic e p o in ts .

I (x-b)dF(b) + d 'T [ (b - x )d F (b )}

'bsx 'bzx

subject to

Rx

Ax + y + - y" = b

x,y+,y"

2

0 .

-26-Of most in te re s t in modelling sto ch a stic production planning

problems are m u lti-period models. I t 1s to the so lutio n of these that

a tte n tio n is now d ire c te d .

3 .2 . M ulti-Stage S to ch a stic Programs with Almost Sure Constraints

These are s to c h a s tic programs in which the co n stra in ts hold

almost su re ly ( i . e . with p ro b a b ility one). As w ith the two-stage case,

the so lutio n o f the general m u lti-stag e recourse problem i s very

d i f f i c u l t , although Wets [59 ] shows th e o re tic a lly that any solution

algorithm fo r the two stage case can be extended to the m ulti-stage

case.

Dantzig [15 ] was the f i r s t to study the special case of programs

w ith s ta irc a s e stru ctu re in which o nly the rig h t hand sid e vecto r, b,

is random. Again he suggested d is c re tiz in g the d is trib u tio n of b to

d erive an equivalent d e te rm in istic program. B ir ge [ 8 ] does the same

and extends D antzig's methods of e x p lo itin g the stru ctu re to the problem

thus generated by using la rg e scale decomposition, p a rtitio n in g and

basis fa c to r iz a tio n . He presents a number of ways of doing th is one

of which uses the in te re s tin g re s u lt due to Wets [ 5 7 ] th at the s t a ir ­

case problem thus d is c re tiz e d is equivalen t to a d e te rm in istic convex

program of the form

Max c Tx + Q(x) x

subject to Ax 3 b

x e D

-27-Birge [ 8 ] goes on to show that his techniques are re a lly

methods of dynamic programming, d iffe rin g only 1n the way in which

d e cisio n s are approximated as functions of the state v a ria b le s .

3 .3 . Chance Constrained Problems

In d iscussing solution methods fo r these, the general m ulti-stage

chance constrained problem w ill be addressed. Apart from Is h ii et a l .

[34 ] who only deal w ith a sin g le stage model, solution methods have

o n ly been proposed fo r the case in which the technology matrices are

f ix e d , i . e . j u s t b and c are random. As in the models in which the

c o n s tra in ts hold almost s u re ly , so lu tio n techniques proceed by the

d e riv a tio n of an equivalent d e te rm in istic program. The ease with which

these d e te rm in istic programs can be solved depends upon th e ir convexity

p ro p e rtie s. The work of Prekopa [43 ] , [44 ] and [45 ] on lo g arith m ically

concave measures has shed much lig h t on th is .

U sually authors r e s t r ic t t h e ir attention to searching fo r optimal

f i r s t order d ecisio n ru le s in which the decisio n x t made a t stage t

i s re s tric te d to be a lin e a r functio n of the random va ria b le s already

r e a lis e d , ...bf l ’ c t - l* Charnes* Cooper and Symonds [ 10] adopted

t h is technique in modelling the production of heating o i l . The model

which they studied had total chance-constraints and only the b 's were

random. Moreover, there was only one c o n stra in t per stage in th e ir

model. They were able to c a lc u la te the optimal lin e a r decision ru le s by

dynamic programming s ta rtin g a t the time horizon and working backwards.

as was shown by Charnes and K irby [11 ] , even when there 1s more than

one c o n stra in t per stage. Kortonek and Soden [36 ] give another proof

o f th is r e s u lt and also consider the case where the cost vector c 1s

random. L a te r, Charnes and Kirby [12 ] proved th at piecewise lin e a r

ru le s are optimal under conditional chance c o n s tra in ts , although there

they re s t ric te d th e ir a tte n tio n to only one c o n stra in t per stag e. This

enabled them to derive com putationally e f f ic ie n t so lutio n techniques,

invo lving in some instances a s e rie s of simple one va ria b le non-linear

optim isation problems.

This completes the d iscussion of so lu tio n techniques fo r the

-29-4 . CONCLUSIONS

In th is chapter 1t has been proposed that production/inventory

problems be modelled by sto c h a stic programs. A review has been provided

o f those most freq u en tly studied in the lit e r a t u r e and the lin e s of

approach that have been suggested fo r t h e ir so lu tio n . The sto chastic

programs divide into two c la s s e s , the p assive and the a c t iv e . The

a c tiv e ones then d ivid e into separate c la s s e s according to whether the

co n stra in ts hold only w ith some prescribed high p ro b a b ility (chance

constrained programs) or almost su re ly (1.e . with p ro b ab ility one),

and according to whether one, two or many time periods or stages are

modelled. Further d e ta ils may be found in Sengupta and Tintner [48 ]

who review s to c h a stic lin e a r programming and Kirby [ 35] who surveys

chance constrained programming.

The most useful c la s s o f sto ch a stic programs from the point of

view of medium term production planning is th at of m ulti-stage a c tiv e

programs in which the co n stra in ts hold almost s u re ly . Unfortunately

in general th is 1s the hardest c la s s to s o lv e . Approaches to the solution

g en erally involve the d is c re tiz a tio n of the random va ria b le s involved

and the use of advanced large sc a le programming techniques to take

advantage of the stru c tu re of the problem thus generated. These can be

shown to be equivalen t to dynamic programming techniques, the other

candidate for handling m ulti-stage a c tive sto c h a stic programs. These

methods are unsuitable fo r ta c k lin g multi-commodity problems because

of t h e ir computational com plexity. See Chapter 4 fo r a discussion o f

In view of the d if f ic u lt y o f solving m ulti-stage a c tiv e problems

e x a c tly even i f the random v a ria b le s are assumed to be d is c re te ,

approximate techniques deserve serious co n sid e ratio n . A promising

method is th a t of B e a le , Fo rre st and Taylor [ 4 ] who study a simple

multi-commodity production/inventory model which has an upper bound

on the to ta l production in any period. Their approach has provided

one of the foundations o f the research described in th is th e s is ,

notably the development of a more general production/inventory model

which is described in Chapter 5 and an approximate solution technique

described in Chapter 6. Accordingly an exposition of th e ir work is

CHAPTER 3

AN EXPOSITION OF "MULTI-TIME PERIOD STOCHASTIC SCHEDULING"

-31-1 . INTRODUCTION

Beale, F o rre st and T a ylo r [4 ] aim to provide a su ite o f computer

programs that would enable production planners to obtain good re lia b le

medium term production s tra te g ie s in the face o f uncertainty in the

demand fo r th e ir products. The authors do t h is by studying a simple

sto ch a stic multi-product production/inventory model and proposing a

com putationally tra cta b le approximate so lutio n technique. This technique

i s num erically fe a s ib le in the sense that the s iz e of problem that can

be reasonably tackled (measured by the number o f product lin e s that

can be treated in d iv id u a lly ) is of the same order as the s iz e of

problem that could be handled i f the demand requirements were known

w ith c e r ta in t y .

T h e ir paper has provided much of the Impetus fo r the research

described in Part I I and so an exposition of t h e ir work together w ith

a d iscussion of it s m erits and lim ita tio n s i s appropriate here. In

an e ffo r t to overcome the lim ita tio n s inherent in th e ir technique, a

much more general production/inventory model was formulated in Chapter

5 and studied in Chapter 6.

The production/inventory model which they study is given in Section

2 . They approximate i t by a non-linear program and the method by which

they do th is is described in Section 3 . Some c o e ffic ie n ts in i t a re ,

however, s t i l l unknown. They estimate these it e r a t iv e ly by a process

described in Section 4 . The chapter ends w ith a b rie f summary and

-32-2 . THE PRODUCTION/INVENTORY MODEL

Production, sales and inventory le v e ls are to be planned fo r each

of T time periods. Demand requirements fo r each time period are

ch ara cte rise d by p ro b a b ility d is t rib u t io n s . Production ra te s are con­

sidered to be fixe d during each time period but may vary between time

p e rio d s. At the s t a r t o f any time period production le vels are decided.

During that period the demand is re a lis e d and a t the end of i t sa le s

are made and stock le v e ls become apparent. A ll the costs are considered

to be lin e a r in the production decisions made a t the s t a r t o f, sa le s 1n,

and stock le v e ls re a lise d a t the end o f.each time period. There is an

upper bound on the total production in each period. I t is assumed that

the d e cisio n maker has a neutral a ttitu d e towards r is k and so de sire s

to maximise h is to tal expected p r o fit .

L e t the column vecto rs pt , at> s t and dt denote the production in ,

sa le s in , stock a t the end o f , and demand in time period t . Id e n tify

the i th component of each vector with the i th_ product.

L e t Cp t, Pt and CSt be column vectors of u n it production c o s ts ,

sale p ric e s and stockholding c o s ts . Let TCApt be the maximum to tal

production permitted 1n each period and le t 1 be a vector of l ' s .

Then the model which B e ale , Fo rre st and Taylo r study may be

e x p lic it ly stated :

Maximise E j pt a t " CPt pt " t-1

(1)

over a t , pt and s t sub ject to

(2)

-33-t - i + pr a -33-t _ s -33-t " 0 and (4)

a t ,p t and S j i O

( 5)

fo r t ■ 1,2, t * 0 Is the I n i t ia l o r sta rtin g state so sQ, fo r

example, are the I n i t i a l stocks and are thus part o f the model's

Input data.

Notice that a l l that is a c tu a lly required from a so lutio n to the

above model 1s the f i r s t time period production d e cisio n s. In sub­

sequent periods the model would be re-run w ith new s ta rtin g stocks

and more accurate d a ta .

The authors propose that the standard deviation of each component

o f the demand in each time period should be d ir e c tly proportional to

i t s mean, which in turn is a lin e a r fu n ctio n o f the sales in the

previous time period.

I f vt 1s a vector pertaining to the t th time period, le t v i t be

i t s i th component. Then e x p lic it ly stated t h e ir demand model is

d1t * dMi t^1 + C1t nt + R1t e1t) (6)

dM1t* B1ot + E Bi j t aj t - l (7 )

j

where nt and are independent real Gaussian random v a ria b le s , and

Bi o t ’ 8i j t * Ci t Ri t are known ^ xed c o n sta n ts. The term nt is intended

to model the global v a r ia b ilit y o f a ll products in each time period

and is intended to model the v a r ia b ilit y in demand between individual

-34-Notice that a p a rt from the in i t ia l production le v e ls p^ a ll

the va ria b le s in the model are a c tu a lly random v a ria b le s . T his is

because decisions made 1n future time periods are allowed to be

functions of the demand re alise d up to that period.

The authors cla im that although th e ir model i s simple, i t can

e a s ily be extended by the addition of extra c o n stra in ts to cover

the more complicated problems that are lik e ly to be met in p ra c tic e ,

without a lte rin g i t s fundamental structure and approximate so lutio n

algorithm . This is o nly p a rtly tru e . The production c o n stra in t (2)

can be replaced by a more r e a lis t ic set of technological co n stra in ts

without a lte rin g t h e ir solution procedure. But t h e ir model cannot

accommodate bounds on storage cap acitie s or the c o st of changes in

production le v e l. N either can a more comprehensive demand model, fo r

example one in which the mean demand is modelled as a lin e a r function

of a moving average o f past s a le s , be used with t h e ir solution

-35-3 . THE FORMULATION OF A NON-LINEAR PROGRAM

The authors id e n tify the c ru c ia l quantity of In te re s t in th e ir

model to be the excess of supply over demand, and they are p a r tic u la rly

interested in i t s v a r i a b i l i t y . They c a ll the excess of supply over

demand in the t th_ time period e^, where i t is defined by

can then be replaced by a c o n s tra in t which r e s t r ic t s sa le s to be le ss

than both the stock a v a ila b le fo r sa le and demand

i- e - a t s min ( s ^ + Pt .d t )

which can be e q u ivalen tly w ritte n

Su b stitu tio n fo r dt given by (6) and (7 ) in (8) shows th a t the i th

component of et is

j

The authors now take expected values in the problem defined by

rows ( 1 ) , ( 2 ) , ( 4 ) , (9) and (10) to y ie ld the problem

(

8

2

and le t the variance of i t s 1 th component be o ^ . The c o n s tra in t (3)

a t * s t - l + pt ‘ max ( et ’ ° ) ' (9)

Maximise

l

t=l

over a t , pt , and ¡ t sub ject to

-36-S TCAPt (13)

5t - 5t - l - Pt* _{“ bt} (14)

at - - pt+E{max(et ,0 )) VI O ₍₁₅₎

s t+ l + Pt " a t ’ s t ■ 0 and (16)

a t ’ s t* Pt

fo r t ■ 1 , 2 , . . . , T .

* 0 (17)

The i n i t i a l production d e cisio n , pl t has been treated fo r con­

venience as a random va ria b le equal to i t s expected value with prob­

a b i lit y one. at ,p t , i t and et denote the expected values o f a t , pt> s t

and et re s p e c tiv e ly . bt is a vector whose 1 th_ component is B.-ot and

Bt i s a m atrix whose (1, j ) t h component is

Thus the o rig in a l problem has been approximated by one which

would be a d e te rm in istic lin e a r program except fo r the term

E{max(et , 0 ) } (18)

They ta c k le th is by supposing that ei t can be treated as though i t

has a normal d is trib u tio n N(ei t ,c^t ) , whence the i th component of

(18) i s

°1t f l ( i1t /01t ) {19)

where f1 is a fu n ctio n : F -*• F defined by

f ^ x ) ■ £ (?+ x)d *(0 .

$ being the Gaussian d is trib u tio n fu n c tio n . Thus f j ( x ) i s the mean o f a

-37-with mode x but truncated a t zero such that the p ro b a b ility o f it s

being non-positive is concentrated 1n a point mass a t zero.

I f the o 's were known, (19) could be substituted into (15) to

y ie ld a d e te rm in istic non-linear program. However, the a 's are not

known and have to be estim ated. They derive a re cu rsiv e procedure

fo r t h is which is described below.

Moreover, they assume that is d ire c tly proportional to the

i th component o f the mean demand so they could se t

a i t = T1t a i t

fo r some constant Ti t . But th is would lead to paradoxical consequences.

I f i t i s not desired to s e ll a p a rtic u la r product, say the k th , then

sk t 1 + pkt must be P0 S lt ^ve order to s a t is f y (15) and (1 9 ).

The cause of th is paradox i s the assumption that the demand i s normally

d is trib u te d so there is always a p o s itiv e p ro b a b ility that i t w ill be

negative. The authors avoid th is by instead se ttin g

°1t E T1t a'i t and {2 0 )

x i t " ° 1 t /dM1t (21)

where oi t denotes an estimate of a 1 t . Thus they enable s a fe ty stocks

to be reduced considerab ly i f i t i s not desired to meet demand in

f u l l . However, t h is changes the stru c tu re of the problem. For i f

i t i s not desired to meet demand in f u ll and a ^ d ^ ^ is s m a ll, then

the v a r ia b ilit y in the problem represented by oi t is treated as being

-38-S u b stltu tln g fo r E{max(et ,0 ) } by (19) and (20) the 1 th

component of c o n stra in t (15) can be w ritte n

a i t " s1t - l " P i t + T1t ai t f l (e i t / (T i t ai t ^ * 0

^

The problem defined by (1 2 ), (1 3 ), ( 1 4 ) , (2 2 ), (16) and (17) is

the non-linear program which they solve by the introduction of

separable va ria b le s to d erive good f i r s t time period production

d e c isio n s. T h e ir procedure fo r estim ating and hence T n i S an

it e r a t iv e one. t is i n i t i a l l y se t to i t s minimum value ( i . e . the

value obtained by ignoring the sto ch a stic v a r ia b ilit y in everything

except demand), which is

/ ( C ^ + R ^ ) (23)

and then re-estim ated. The procedure by which they do th is deserves

4 . RE-ESTIMATION OF THE STOCHASTIC VARIABILITY

In th is section the method whereby the authors re-estim ate

T1t b r ie f ly d escrib ed . They im p lic itly assume, but do not

e x p lic it ly state th a t, a t any stage the state of the production/

inventory process which they model can be characterised by a

s ta te vector £t , which they define by

« Î - (« I» *?> (24)

So a t the end of time period t , given the input d a ta , the process is

completely described by the stock le v e ls , s t and s a le s ju s t made, a^.

Therefore, the production decision made a t the s t a r t of time

period t w ill be a functio n of the previous time period state vector

Beale e t a l . assume that th is function can be approximated

by a lin e a r one:

(25)

where p° is a constant vector and A ^ and a|,^ are constant m atrices.

Only A ^ and A ^ need be estimated and the way 1n which they do

th is i s described below.

i s , of c o u rse , simply the i th diagonal e n try o f the dispersion

m atrix o f the excess o f supply over demand, et , and is so

the authors d esire to estimate the dispersion m atrix o f e t . This they

do by using the above approximation to derive an expression fo r i t in

terms o f the d isp ersio n matrices of the previous time period state ve cto r,

They then seek to d e riv e an expression fo r the dispersion matrix

o f In terms o f those of ^t _1 and dfc. But to f a c il it a t e t lie lr

a n a ly s is they make one fu rth e r approximation. They approximate

S1t " (e1f w1 1) (26)

where wi t are the sla ck s associated with (15) by

s 1t " SC it + SV it e1t i 27)

Hence Sc i t and Sv i t are co n sta n ts, the la t t e r being defined by

SV1t “ f 2 ^ ei t " wit ^ ° 1 t^

f2 being a fun ctio n: K -*■ F + such that

[ f 2(x)]2 = *(x) + [l-i(x )]*(x )x 2-[l-2*(x)]*(x)-[<Mx)]2

where $ , * are the Gaussian p ro b a b ility density function and d i s t r i ­

bution function re s p e c tiv e ly . I t 1s not necessary to estimate S ^ .

The m erit o f th is value of Syi t is that I f et were normally

d is trib u te d then s i t given by (26) has the same variance as i f I t were

given by (2 7 ). U nfortunately th is does not preserve the covariances

between the si t , c fo r given t . So the variance o f the total number of

items in stock 1s not preserved e ith e r.

So, having made the two approximations above, an expression for

x .j£ in terms o f the d isp ersio n matrices of the previous time period

s ta te vecto r, ^t l , and demand,dt>Is derived as Is an expression for