A comparison of stochastic linear programming with mean value linear programming for production and profit planning unde...

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A comparison of stochastic linear programming with mean value linear programming for production and profit planning under conditions of uncertainty
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Thesis--University of Florida.
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By Mawsen Liao.
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A COMPARISON OF STOCHASTIC LINEAR PROGRAMMING WITH
MEAN VALUE LINEAR PROGRAMMING FOR PRODUCTION AND
PROFIT PLANNING UNDER CONDITIONS OF UNCERTAINTY











By

Mawsen Liao


A Dissertation Presented to the Graduate Council of
The University of Florida
In Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy






University of Florida


1974



























Dedicated

to

Dr. Williard E. Stone













ACKNOWLEDGEMENTS


Many people deserve credit for assisting in the pre-

paration of this Dissertation. The writer is particularly

grateful to his Supervisory Committee Chairman and Co-

Chairman, Dr. Williard E. Stone and Dr. Gerald L. Salamon,

for their critical reading of the dissertation and for

their helpful support and suggestions. Sincere apprecia-

tion is also expressed to Dr. Warren W. Menke and Dr,

Ronald J. Teichman for their contributions and cooperation

in serving on his Supervisory Committee.

The writer would also like to thank Dr. S.C. Yu for

his teaching and constant encouragement.


iii











TABLE OF CONTENTS


Page

Acknowledgements .................... .. ......... iii

List of Tables ................. ........... ..... vii

List of Figures .................................. ix

ABSTRACT ................................ ........ x

Chapter

I. INTRODUCTION ................ ............. 1

Statement of the Problem .............. 1

Research Methodology and Organization
of the Study .................. 4

II. AN EXAMINATION OF MATHEMATICAL PROGRAMMING
UNDER UNCERTAINTY ............. 6

Introduction ........................ 6

Three Major Stochastic Programming
Techniques .................... 7

Two-Stage Linear Programming ..... 7

Chance-Constrained Programming ... 9

Stochastic Linear Programming .... 11

A Comparison of the Stochastic Program-
ming Techniques ............... 14

III. AN APPLICATION OF ASLP TO THE SHORT-RUN
PRODUCTION PLANNING PROBLEM OF
A SMALL-SIZED FIRM ........... 17

Introduction ......................... 17

Characteristics and Assumptions of the
Selected Production Planning
Problem ....................... 18








Page

Chapter

An Explanation of the ASLP Algorithm .. 24

The Reduction of the Computational
Burden of ASLP ................ 27

The Selection of a Particular Alloca-
tion Plan for the Production
Planning Problem .............. 33

The Determination of Confidence
Intervals for Planning Period
Profits ....................... 36

The Curve Fitting Technique ...... 38

The Integration of a Probability
Function and the Determination
of the Probabilities of Various
Profit Intervals ............. 43

IV. AN APPLICATION OF ASLP TO THE SHORT-RUN
PRODUCTION PLANNING PROBLEM OF
A MEDIUM-SIZED FIRM ........... 46

Introduction .......................... 46

A Hypothetical Production Planning
Problem ....................... 47

The Introduction of a New Dominance
Rule .......................... 51

The Application of the Old 52
Dominance Rule ................

The Introduction of a New
Dominance Rule .............. 53

The ASLP Solution of the Production
Planning Problem .............. 59

The Determination of the Probabilities
of Profit Intervals for the
Planning Period ............... 61

V. A COMPARISON OF ASLP AND MVLP ............ 65

Introduction .......................... 65








Page
Chapter

The Solution Implied by MVLP .......... 67

The MVLP Solutions for Case (1)
and Case (2) ... ..... ........ 67

The MVLP Solutions for Case (3)
and Case (4) .................. 68

The Curve Fitting and Cumulative
Probability Distribution of
Profits for MVLP Solutions .... 73

A Comparison of ASLP and MVLP Solutions 77

The Comparison of ASLP and MVLP
in Case (1) and Case (2) ...... 79

The Comparison of ASLP and MVLP
in Case (3) and Case (4) ...... 81

VI. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 84

Summary and Conclusions ............... 84

Recommendations for Future Research ... 87

APPENDICES ....................................... 88

Appendix (I) ............. ......... ........ 89

Appendix (II) ............................... 108

BIBLIOGRAPHY ..................................... 123

BIOGRAPHICAL SKETCH .............................. 128











LIST OF TABLES


Table Page


1. Values of Problem Constants (for Case
(1) and Case (2)) .................... 19

2. The Correlation Coefficients between a
and a.. (for i = 1,2,3 and j = 1,2,3).. 24

3. An Example of Dominance ................. 28

4. All Possible Ways of Resource Allocation
and the Number of U Matrices should be
Examined .............................. 31

5. Allocation Decisions Resulting in Expected
Profits Greater than $21,000 for Case
(1) and Case (2) ..................... 33

6. Frequency Distribution on Profits and
Calculation of Sums under ASLP, Case
(1) ................................... 39

7. Cumulative Probabilities on Profits (9),
ASLP in Case (1) and Case (2) ......... 44

8. Values of Problem Constants (for Case (3)
and Case (4))............ ........ ..... 48

9. The Correlation Coefficients between a66
and a.. (for i = 1,2,3,4,5,6 and
j = 1,i ,3,4,5,6) ...................... 51

10. The Possible Values of u.. to be considered
in Case (3) and Case '(4) .......... 56

11. The Possible Allocations of Each-Resource
(Based on the Possible Values of uij
in Table 10) ...................... ..... 57

12. Allocation Decisions Resulting in Expected
Profits Greater than K, ASLP in Case
(3) and Case (4) ..................... 60


vii








Table Page

13. Probability Density Functions for ASLP
in Case (3) and Case (4) .......... 62

14. Cumulative Probability Distributions on
Profits (S) under ASLP for Case (3)
and Case (4) ...................... 63

15. The Calculation of M.. for Case (3) ... 70

16. The Calculation of M.. for Case (4) ... 71

17. Minimum Constraints in MVLP for Case (3)
and Case (4) ...................... 72

18. Probability Density Functions for MVLP
in Cases (1), (2), (3), and (4) ... 74

19. Cumulative Probability Distributions of
Profits (9) under MVLP for Cases
(1) and (2) ...................... 75

20. Cumulative Probability Distributions on
Profits (8) under MVLP for Cases
(3) and (4) ...... ...... ......... 76

21. Mean Profits and Standard Deviations of
the Selected ASLP Solution and the
MVLP Solution ..................... 77

22. Cumulative Probability Distribution of
Profits (S) under MVLP and ASLP ... 79

23. The Probabilities of Selected Profit
Intervals under MVLP and ASLP for
Case (1) and Case (2) ............. 81

24. Cumulative Probability Distributions on
Profits (8) under MVLP and ASLP for
Case (3) and Case (4) ............. 82

25. The Probabilities of Selected Profit
Intervals under MVLP and ASLP for
Case (3) and Case (4) ............... 83


viii












LIST OF FIGURES


Page


Figure


1. Efficient Set and Utility Function ........











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 COMPARISON OF STOCHASTIC LINEAR PROGRAMMING WITH
MEAN VALUE LINEAR PROGRAMMING FOR PRODUCTION AND
PROFIT PLANNING UNDER CONDITIONS OF UNCERTAINTY

By

Mawsen Liao

August, 1974
Chairman: Williard E. Stone
Co-Chairman: Gerald L. Salamon
Major Department: Accounting

Mathematical programming has been advocated as a useful

profit and production planning tool. However, many mathemat-

ical programming solution approaches provide optimum solu-

tions only when model parameters are deterministic. In many

production planning problems, this assumption does not hold.

The research which has been done on problems in which inputs

are stochastic has assumed that the stochastic inputs were

independent random variables. However, the planning problems

many managers face contain inputs which are dependent random

variables. Consequently, there exists a need for a

stochastic mathematical programming technique which is ap-

plicable to such cases. One planning tool which shows the

potential of meeting this need is Active Stochastic Linear

Programming (ASLP).








This dissertation is concerned with the application of

the ASLP model to short-run production planning problems of

small and medium sized firms. ASLP is viewed as an approach

which does not ignore the basic uncertainty of the planning

process. The nature of ASLP and implementation issues are

examined in this study. Special attention is paid to the

computational burden of ASLP applications. The ASLP approach

is combined with a statistical analysis to provide useful

probability information on planning period profits.

The absence of an applicable technique has forced man-

agement to solve some planning problems by a deterministic

approach. One such approach that has been used by some

firms is Mean Value Linear Programming (MVLP). Consequently,

ASLP and MVLP are compared in this paper in a variety of

circumstances. Finally, a number of general conclusions are

drawn to provide some basis for choosing one of the planning

approaches over the other.











CHAPTER I


INTRODUCTION

Statement of the Problem

A basic problem in production planning is the efficient

allocation of scarce resources in order to achieve a prede-

termined end. In a business firm, this end may be a maximi-

zation of profit or a minimization of cost. The literature

in management accounting and operations research is filled

with examples of how mathematical programming technique can

be used to solve a firm's production planning problem under

a variety of circumstances. However, many mathematical

programming techniques assume model parameters are deter-

ministic. In most situations, this assumption does not

hold. When model parameters are subject to random variation,

the results obtained from deterministic programming models

may be inadequate. Therefore, there is a need for a

planning model which explicitly considers random variation

in model parameters. All such planning models will be re-

ferred to as stochastic planning models. This research

will be concerned with that subset of stochastic planning


For example, see Robert K. Jaedicke, "Improving Break-
Even Analysis by Linear Programming Technique," N.A.A.
Bulletin, Vol. 42 (March, 1961), pp. 5-12. Thomas H.
Williams and Charles H. Griffin, The Mathematical Dimension
of Accountancy (Cincinnati, Ohio: South-Western Publishing
Co., 1964), pp. 102-120. Charles T. Horngren, Cost Account-
ing: A Managerial Emphasis (Englewood, Cliffs, N.J.:
Prentice-Hall, Inc., 1972), pp. 899-904.








models which are commonly formulated mathematically and are

often referred to as stochastic programming models.

During the past twenty years, much of the research in

stochastic programming has concentrated on problems where

random variations are confined to contribution margins and

to the amount of resources available.2 But, there are many

problems in which the technological coefficients are subject
3,4,5
to random variation.'5 For example, the technological

coefficients in a firm's production planning problem may be

subject to considerable variation because of variance in the

quality of raw materials, variance in working conditions,

unstable machine processing time, and other factors, Further-

more, most previous research on stochastic programming which

examined random variation in technological coefficients as-

sumed that all random variables had independent probability

distributions. It does not seem reasonable to assume that

technological coefficients are independent in a production

planning problem where several products use a set of common

resources. Tintner and Sengupta have pointed out that


For example, see George B. Dantzig, "Linear Programming
under Uncertainty," Management Science, Vol. 2 (1955), pp.
197-206. A. Charnes and W.W. Cooper, "Chance-Constrained
Programming," Management Science, Vol. 6 (1959), pp. 73-76.

3Van DePanne, C. and W. Popp, "Minimum Cost Cattle
Feed under Probabilistic Protein Constraints," Management
Science, Vol. 9 (1963), pp. 405-430.

K.D. Cocks, "Discrete Stochastic Programming," Manage-
ment Science, Vol. 15 (1968), pp. 72-79.

W.H. Evers, "A New Model for Stochastic Linear Pro-
gramming," Management Science, Vol. 13 (1967), pp. 680-693.








variations in the matrix of technological coefficients in-

volve simultaneity and interdependence between activities

(i.e., covariance (aij, a ) 7 0). The production planning

problem many managements face contains dependent random

technological coefficients. The lack of an applicable tool

to solve this kind of problem has forced management to use

other approaches such as mean value linear programming

(MVLP). In MVLP, the numerical value of the technological

coefficients which are used as inputs to the model are the

means of the random technological coefficients. This solu-

tion approach ignores the basic uncertainty of the planning

process by considering only the mean values of random model

parameters. Therefore, when the variation of the model

parameters is considerable, the mean value programming

model may result in misleading information.

Accordingly, the purposes of this study are three in

number. The first objective is to search for a stochastic

programming model which is applicable to solving short-run

production planning problems. The second objective is to

apply that model to a production planning problem which is

characterized by dependent variation in its technological

coefficients. The third objective is to compare MVLP and

the chosen model in a variety of circumstances in order to

provide some basis for choosing one of the approaches over

the other.


Gerhard Tintner and Jati K. Sengupta, Stochastic
Economics (New York: Academic Press, Inc., 1972), p. 209.







Research Methodology and Organization of the Study

This research is principally concerned with solving a

short-run production planning problem where the technolo-

gical parameters are stochastic. A brief description of

the problem was given earlier in this chapter.

A thorough search and review of the literature was

made to identify the characteristics and applicability of

several stochastic programming models to a short-run pro-

duction planning problem. This search and review revealed

that active stochastic linear program (ASLP) is best fitted

to the short-run production planning problem. The results

of this library review are given in Chapter two.

Chapter three focuses on an explanation of the

mechanics of the ASLP approach and an application of ASLP

to the short-run production planning problem of a small

sized (simulated) firm. The chapter includes a description

of the firm and a description of the technique used to gen-

erate the numbers which were used as inputs to the planning

model. A major part of that chapter is an examination of

and a partial resolution of the computational difficulties

encountered in ASLP applications. Finally, the results of

the ASLP application are presented and their potential use-

fulness is discussed.

Chapter four concentrates on the application of the

ASLP approach to a production planning problem of a medium

sized firm. Specifically, this chapter is concerned with

ways in which the ASLP approach may be revised to handle








the great computational burden of problems approaching a

practical size.

Chapters three and four only show the feasibility of

applying ASLP to different sized problems. Chapter five

is devoted to comparing the solutions of ASLP and MVLP in

a variety of circumstances in order to provide some basis

for evaluating the desirability of using ASLP. Specifical-

ly, we will compare the production plans produced by ASLP

to those produced by MVLP when random inputs are relatively

dependent and when random inputs are relatively independ-

ent.

Finally, Chapter six presents the summarizations and

conclusions of this research and offers some suggestions

for future research in the mathematical programming area.












CHAPTER II


AN EXAMINATION OF MATHEMATICAL PROGRAMMING
UNDER UNCERTAINTY

Introduction

A general formulation of a firm's production planning

problem if linear programming is used is:

Max Z = CX

S.T. AX b

Where C is an n-element row vector of contribution
margins

X is an n-element column vector of product
quantities

b is an m-element column vector of resource
availabilities

A is an m by n matrix of technological coef-
ficients

Uncertainty can be introduced into this deterministic

model in a number of ways (e.g., the elements of A, b, and/or

C can be subject to random variations). In this chapter, an

examination will be made of some of the methods by which un-

certainty has been introduced into the deterministic formula-

tion. This examination will include a discussion of how well

the assumptions of each of considered method fit the char-

acteristics of short-run production planning problems. This

chapter will conclude with the observation that the assump-

tions of one of these programming models make it more suit-

able than the other models for short-run production planning

problems.








Three Major Stochastic

Programming Techniques

Previous research on the operational approaches of

solving a mathematical programming problem with random model

parameters can be grouped into three major areas: Two-stage

linear programming, Chance-constrained programming, and

Stochastic linear programming. These areas have some common

characteristics.1 First, they all assume that the probabil-

ity distribution of the random model parameters or their

multivariate probability distribution is known or that past

sample data on all random variables are available. Second,

they employ the known probability distributions (or past

sample values) to convert a stochastic mathematical pro-

gramming model into a deterministic one. However, the

methods of converting a stochastic model into a deterministic

model are different in each of these areas. The nature of

each approach is examined and discussed in this section.


Two-Stage Linear Programming

Two-stage linear programming suggests that the solution

of a stochastic linear programming problem can be broken

down into two stages.2 The general formulation of the firms'

production planning problem under this approach is:


G. Tintner and J.K. Sengupta, Stochastic Economics (New
York: Academic Press, 1972), p. 204.
2G.B. Dantzig, "Linear Programming under Uncertainty,"
Management Science, Vol. 2 (1955), pp. 197-206.








Max Z E(CX) + E (dy)

S.T. AX + By = b

where C, A, b and X are defined as before and

y is a 2m-element column vector of deviations,

d is a 2m-element row vector of penalty costs,

and

B is an m by 2m matrix of the form (I, -I).

This formulation differs from the deterministic linear

programming formulation in the introduction of a new vari-

able (y) with its coefficients (B) in the constraints and

penalty costs (d) in the objective function. The new vari-

able is introduced in order to account for random variation

in A and/or b. The structure of the two-stage problem is

that a decision vector X is selected. Then the random

elements of A and/or b are observed and a vector y is chosen

to compensate for any infeasibility that has occurred. This

subsequent adjustment for infeasibility is made at a penalty

cost given by the vector d.

This approach is particularly applicable to the case

where the column vector b takes only a small number of

possible values. The major limitation of this approach is

that it magnifies the size of the problem.3 The computa-

tional problems are most severe when the elements of A are

subject to dependent random variation which is characteris-

tic of short-run production planning problems. Further and


3H.M. Wagner, Principles of Operations Research,with Ap-
plications to Managerial Decisions (Englewood Cliffs, N.J.:
Prentice-Hall, Inc., 1969), p. 659.








perhaps more important is the fact that the structure of the

two-stage approach assumes that the nature or length of the

problem is such that the firm can make adjustments during

the period to correct for infeasibilities. Very many short-

run production planning problems are such that adjustments

of this kind cannot be made.


Chance Constrained Programming4

A general formulation of a firm's production planning

problem in which Chance-constrained programming may be used

is:

Max Z = CX

S.T. P(AX < b) > a

where P represents the probability, a is a column
vector of specified probabilities and C, A,
X, b are defined as in deterministic linear
programming.

The chance-constrained formulation of the problem deals

with random variations in the constraints. It can under

certain conditions handle random variation in the elements

of A and/or b. This formulation requires that each constraint

i hold with the probability equal to or greater than ai. The

complementary probability, 1 a., represents the allowable
n
risk that > a..X. > b.. This implies that constraint can
j=l 13 3 1
be violated.at most by (1 a.). The difference between the

chance-constrained formulation and the two-stage formulation

is that in the latter case all constraints must hold for


A. Charnes and W.W. Cooper, "Chance-Constrained Pro-
gramming," Management Science, Vol. 6 (1959), pp. 73-76.







all possible combinations of the random variables, whereas

in the former case the constraints are required to hold

with at least a certain level of probability.

As stated by Wagner, the principal advantage of this

approach is that it does not expand the problem size. How-

ever, this approach does not directly consider the cost of

violating a constraint. The cost of violating a constraint

is only indirectly reflected in the probability level of a.

For example, if the cost of violating one constraint was

regarded as higher than the cost of violating another con-

straint, a higher a would be needed on the first constraint.

Therefore, this approach has been critized because it

indirectly evaluates the consequence of violating a con-

straint6 but says nothing about what to do when a constraint

is violated.7 However, Kirby argues that the penalty cost

is extremely difficult to quantify and many firms would be

unable to do so. He concludes that the penalty cost

approach is not meaningful whenever the costs are too dif-

ficult to be determined accurately.8

Chance-constrained programming has been found to be

best suited to problems in which only the elements of the


5Wagner, p. 669.

Ibid., pp. 669-670.

7.D. Cocks, "Discrete Stochastic Programming," Manage-
ment Science, Vol. 15 (1968), p. 73.

M.J.L. Kirby, "The Current State of Chance-Constrained
Programming," Proceedings of the Princeton Symposium on
Mathematical Programminq, edited by Harold W. Kuhn (Princeton
University Press, 1970), p. 94.








vector b are subject to random variation. If the elements

of the matrix A are variable and dependent, the constraints

of chance-constrained programming become non-linear and a

problem of practical size cannot be solved. The argument

advanced in this study is that in many short-run production

planning problems not only will the elements of A be subject

to significant random variation but also it is inconceivable

that all of the elements can be independent. Consequently,

chance-constrained programming is not applicable to many

short-run production planning problems.


Stochastic Linear Programming

Tintner first suggested the use of stochastic linear

programming in the area of agricultural economics in 1955.9

This approach is primarily concerned with analyzing the

statistical distribution of the problem's objective func-

tion. Two approaches are suggested by Tintner: the

"passive approach"l0 and the "active approach." In the

passive approach, the decision maker first observes the

random effects,then he solves a regular linear programming

problem. By repeating this procedure he will solve dif-

ferent linear programming problem each time, due to the

randomness, and thus obtain a probability distribution for

9
G. Tintner, "Stochastic Linear Programming with Appli-
cations to Agricultural Economics," Second Symposium on
Linear Programming (National Bureau of Standards, Washington,
1955), pp. 197-228.
10Ibid., pp. 197-228.

G. Tintner, "A Note on Stochastic Linear Programming,"
Econometrica, Vol. 28 (1960), pp. 490-495.








the objective function. Decisions are then based upon this

probability distribution. Therefore, under the passive ap-

proach, no decision is made until random variables are

observed. But in a short-run production planning problem,

resources allocation decision must be made before the pro-

duction starts (i.e., before random variables are observed).

Consequently, this approach does not fit in production

planning problemsbecause it means production would not be

scheduled until the period is over.

In the active approach, the decision variables are the

amounts of resources to be allocated to the various activi-

ties. The general formulation of a firm's production

planning problem under active approach is:

Max Z = CXS

S.T. AX < bU

where C and A are defined as before,

X is an n by n diagonal matrix of product
quantities

S is an n-element column vector of l's

b is an m by m diagonal matrix of resource
availabilities and

U is an m by n matrix with elements u... The
uij is a proportion of resou ce i de4oted
to product j, uij. 0, and uij = 1 (for
i=1,2,...m). j

Accordingly, one first makes a decision to allocate a

specific amount of resources to each activity. Then, assum-

ing a multivariate probability distribution of the model

parameters, a probability distribution of the objective

function for this particular allocation decision will be








generated. By repeating the same process for different al-

location decisions (i.e., different matrices U), various

probability distributions of the objective function are

obtained. These probability distributions then serve as the

basis for selecting a production plan.

Obviously, this approach takes a different viewpoint of

the production planning problem than the other approaches

discussed. The active approach does not directly focus on

the quantities of products to be produced but instead con-

centrates on the proportion of each resource that should be

allocated to each product (i.e., it focuses on the selection

of a particular matrix U).

The active approach assumes that the planning period is

short enough so that the firm cannot change its allocation

decision during the planning period. In other words, the

firm cannot take any resource from one product which uses

less than was allocated to it and use that resource on an-

other product which needs more than was allocated to it.

This assumption fits well with the realities of many short-

run production planning problems.

The major problem of Tintner's active approach lies in

its implementation. There are two implementation problems.

First, this approach considers a collection of values for

random parameters and solves the problem for each specific

set of random values. Thus, if there are K random para-

meters (i.e., K elements in matrix A) in the problem with

each random parameter taking on one of the L possible








values, then the active approach needs to solve (G-Lk)

linear programming problems, where G is the number of

matrices U to be considered. Solving (G-Lk) linear

programming problems where.K is large can be extremely

burdensome. Second, the active approach expands the size

of the problem through the introduction of the matrices U.

For example, in a production problem of three products and

three constraints, if the permissible values of each uij

are 0, 1/4, 2/4, 3/4, and 1, the number of possible matrices

U would be 3,375. But, when the permissible values are

changed as 0, 1/6, 2/6, 3/6, 4/6, 5/6, and 1, the number of

possible matrices U would be expanded to 21,952. This is

an example of the expansion of problem size that can arise

through the introduction of matrices U. In short, the com-

putational problem of the active approach increases as the

fineness of the partition of the U matrix is increased and

increases as the size of the firm increases.


A Comparison of the Stochastic
Programming Techniques

Three stochastic programming techniques for solving

short-run production planning problems have been examined.

In the previous section we have reasoned that all of these

approaches except Tintner's active approach are suitable to

long-run and not short-run production planning problems. We

have also argued that the active approach fits well with

the characteristics of many short-run production planning

problems. In short, we believe that ASLP will prove to be








a very powerful and practical tool if the computational

burden it poses can be reduced. Therefore, one important

goal of this research is to find a way in which the compu-

tational burden of ASLP can be reduced. Chapter three and

four are concerned with this topic.

The major reasons that Tintner's active approach was

selected for solving the short-run production planning

problem are summarized below. First, the assumptions and

viewpoints taken by the active approach fit very well with

the environment of many short-run production planning

problems. The other two approaches are not as suitable

for short-run production problems. Second, the active ap-

proach permits the solution of problems in which the

elements of A, b, and C are all subject to random varia-

tion. Perhaps more important is the fact that ASLP can

handle dependence among random variables in the matrix A.

Applications of the other techniques previously discussed

have been shown to be deficient in handling dependence

among random variables in matrix A. Third, the active ap-

proach avoids arguments about the consideration of penalty

costs in the objective function with respect to two-stage

programming vis-a-vis chance-constrained programming. Fi-

nally, the .selection of the active approach is motivated

by a recent trend in the literature in decision theory.

This recent trend suggests that techniques which result in

transferring variations in the model parameters to varia-








tions in the objective function provide useful decision-
12,13
making information. 3 Tintner's active approach con-

cerns itself with the derivation of the probability dis-

tribution of a random objective function and therefore is

preferred to those models which hide the variation of

model parameters in a single valued objective function.


































12
Wagner, p. 646.
F.S. Hillier and G.J. Lieberman, Introduction to
Operational Research (San Francisco: Holden-Day, 1967),
p. 531.












CHAPTER III


AN APPLICATION OF ASLP TO THE SHORT-RUN
PRODUCTION PLANNING PROBLEM OF A SMALL-SIZED FIRM


Introduction

The advantages and the disadvantages of applying ASLP

to a short-run production problem were discussed in Chapter

two. It was found that the ASLP model has the potential to

be a very powerful management tool provided its associated

computational burden can be reduced. In this chapter, a

small short-run production planning problem will be solved

by ASLP. An algorithm which significantly reduces the

computational burden of ASLP applications will be presented

and discussed. Subsequently, a probability curve will be

fitted to the profit distribution produced by ASLP in order

to derive confidence intervals on profit outcomes for the

planning period.

This chapter will be divided into five sections:

characteristics and assumptions of the selected production

planning problem, an explanation of the ASLP algorithm, an

algorithm which reduces the computational burden of ASLP,

the selection of a particular U matrix, and the determina-

tion of the confidence intervals for the planning period

profits.








Characteristics and Assumptions of

the Selected Production Planning Problem

In order to illustrate the application of ASLP in a

small short-run production planning, a firm which produces

three products (D,E,F) and uses three limited resources

(1,2,3) will be simulated. In this problem, it is assumed

that the available resources, selling prices of the pro-

ducts, and the unit costs of resources are known constants.

The technological coefficients of the problem are, however,

random variables. Further, the nature of the production

planning problem (i.e., several products using a set of

common resources and one department's output is another de-

partment's input) makes it difficult to foresee a problem

in which the technological coefficients would be independ-

ent random variables. In other words, due to some depend-

ent relationship along the row elements and column elements,

the technological coefficients in each row and in each

column may turn out to be dependent. Therefore, in the

problem chosen for the example, the technological coeffi-

cients are random variables characterized by some amount

of dependence along the row and column of the matrix A. It

is noted that the contribution margins are equal to selling

price minus variable cost. This means that the contribu-

tion margins will be random variables which depend upon the

variation of the technological coefficients.

The above set of assumptions provides a reasonable and

realistic short-run production planning problem which is








amenable to solution by ASLP. However, it should be noted

that the use of ASLP is not restricted to problems of this

sort. For example, when the selling prices of the output,

the unit costs of the input, and the available resources

are random variables, the ASLP model is still applicable.

In summary, the assumptions which have been made allow

us to consider a problem within which selling prices, re-

source availabilities, and variable costs per unit of input

are constants but technological coefficients and contribu-

tion margins are subject to random variation. The values

of the constant factors of the problem are given in Table 1.


Table 1

Values of Problem Constants
(for Case (1) and Case (2))


Variable Cost per Amount
Resource Unit of Resource Available

1 $ 8 4500 units
2 10 6000 units
3 4 3600 units


Products Selling Price/per Unit

D $204
E 193
F 201


Before the characteristics of the stochastic input

data of the modeled firm can be discussed, we must first

examine the two alternative solution approaches of ASLP.

The two basic ASLP solution methods are the sample point

method (indirect method) and the joint distribution method







(direct method). According to the joint distribution

method, one fits empirical probability distribution to

each random technological coefficient, and then determines

the joint probability distribution of the random coef-

ficients. The determination of this joint probability

distribution permits the derivation of a probability dis-

tribution for the objective function. The sample point

method assumes the firm has past "sample" data on the

random technological coefficients. The probability dis-

tribution of the objective function is then derived by as-

signing the different sample values to the technological

coefficients in the constraint equations of a linear pro-

gramming problem. Recently, Tintner and Sastry2 applied

some nonparametric statistics to test differences between

the profit distributions derived by the two methods. Their

study concluded that the objective function probability

distributions derived by these two methods had come from

the same population. Since the sample point and joint dis-

tribution methods give the same results, the basis for

selecting between the two is the availability of the data

required by each of them. The sample point method has been

chosen for examination in this dissertation research

because it is believed a firm will have (or be able to

J.K. Sengupta, G. Tintner, and B. Morrison, "Stochastic
Linear Programming with Application to Economic Models,"
Economic, Vol. 30 (1963), pp. 262-276.
G. Tintner and M.V. Rama Sastry, "A Note on the Use of
Nonparametric Statistics in Stochastic Linear Programming,"
Management Science, Vol. 19 (1972), pp. 205-210.








obtain) information on past values for the random techno-

logical coefficients but will not know the joint probabi-

lity distribution of the random technological coefficients.

In order to provide the necessary inputs for the

sample point method for our example firm, a data bank had

to be created for the elements of the matrix A. For the

purpose of examining a production problem with dependent

technological coefficients, a data generating technique

which would produce dependence among the technological

coefficients had to be discovered. Thirty-six different

matrices A will be generated by using a data generating

technique given in equation (1). The random numbers of

Ri, Cj, and e are generated by the application of the

Monte Carlo method.

(1) a.. = V.. + a R. + 8 C. + E
13 1) 3
where: V.j is the interaction effect unique to
Resource i and product j,

Ri is a random effect of resource i,

C. is a random effect of product j,

e is a random error effect with expected
value of zero, and

a,8 are weights.

The dependence among the technological coefficients in

a particular matrix A will be instituted through the utili-

zation of random values of R. and C. and the selection of

appropriate values for a and 8. An assumption that each Ri

and Cj variable was distributed as the way in the following

list was utilized for the simulation.













Probability
Variable Random Values Occurrence
of Occurrence


2 .50
R
3 .50


3 .33

R 4 .34
2
5 .33


2 .50
R
33 .50


2 .50
C
3 .50


3 .50
C
2 4 .50


2 .50

3 3 .50


This data generation technique allows the amount

of dependence with the A matrix to be controlled.

By using higher values of a or 8 higher amount








of dependence among a (for j = 1,2,...n) or a (for i =
ij ij
1,2,...m) will result than by using lower values of a or 8.

For example, with given values for-V.., 8, and a given

probability distribution on R., Cj, and E, the correlation

between akj and alj will be greater for a = .9 than a= .1.

In order to conduct an examination the effect that

dependence within the A matrix can have on the distribution

of firm profit outcomes, two cases will be considered -

Case (1) where a = 8 = .9 (a case in which the elements of

A are relatively dependent) and Case (2) where a = 8 = .1

(a case in which the elements of A are relatively independ-

ent). In order that the mean values of the a are
ij
approximately the same in the two cases, the following rela-

tion between V2. (the V.. for Case (2)) and V1 (the V.
13 ij 1j
for Case. (1)) has been set:

(2) V2 = + (a a2) E (R.) + (1 ) E (C.)
S) E (Cj)1 3

SV1 + .8 E (R.) + .8 E (C.)


The two sets of thirty-six matrices A and corresponding

C vectors produced by the data generation technique for Case

(1) and Case (2) are given in Table 1 and Table 2 in

Appendix (I). Correlation coefficients between a.. of Case
1)
(1) and Case (2) have been calculated to check that the data

generation technique was producing input data that had the

appropriate characteristics. Table 2 exhibits the sample

correlation coefficients between a23 and a.i in Case (1) and
23Case (2). It can be seen that the sample correlation
Case (2). It can be seen that the sample correlation








coefficients along row two and column three in Case (1)

are greater than those in Case (2).


Table 2

The Correlation Coefficients

between a23 and aij

(for i = 1,2,3 and j = 1,2,3)


Case (1) a = 8 = .9
ail ai2 ai3

a -0.23 -0.29 0.16
lj
a 0.77 0.69 1.00
2j
a3. 0.07 0.18 0.44

Case (2) a = 8 = .1
a a a
il i2 i3

a 0.09 0.24 0.13
Ij
a j 0.13 0.13 1.00

a3j 0.05 0.07 0.01


We have now developed all the data required to solve

the small example firm's production planning problem by

ASLP. We now turn to an examination of the ASLP model.


An Explanation of the ASLP Algorithm

The general formulation of the firm's production

planning problem under ASLP was discussed in Chapter two.

In that chapter, we showed that ASLP adds a matrix U into

the set of constraint equations of the general linear








programming formulation (i.e., the constraint set is

AX < bU instead of AX < b). Any matrix U is a resource al-

location plan and indicates the proportion of each resource

reserved for each activity. The process by which a partic-

ular U matrix is chosen will be discussed in the next sec-

tion of this chapter. Our concern now is to demonstrate

how the ASLP formulation changes the standard linear pro-

gramming formulation. In our hypothetical problem, given a

matrix U, the constraints in the model for a particular

matrix A become the following:



a21 a22 a23 0 x2 0 i 0 b2 0 u21 u22 u23

a31 a32 a33 0 0 x 0 0 b3 u31 U32 u33

or


allxl al2x2 al3x3 blull bll2 blul3
a21xl a22x2 a23x3 b2u21 b2u22 b2u23

a31xl a32x2 a33x3 b3u31 b3u32 b3u33

At this point it is obvious that what ASLP does is

transform a standard linear program with three products

and three constraints into a problem with nine constraints

as follows:


allx1 1 bull; a12x2 5 blul2; al3x3 blul3;
"21xl < b2u21; a22x2 < b2u22; a23x3 < b2u23;

a31x < b3u31; a32x2 < b3u32; a33x3 b3u33-







Therefore, in order to solve a problem by ASLP with a given

U matrix, we simply follow a series of division and minimi-

zation processes. For the general example under considera-

tion these processes are:

x = Min. blu11 b2u21 b3u313
1 all a21 a' 31

S Min.(blul2 b2u22 b3u32
12 a a22 a32

x = Mi. blul3 b2u23 b3u33
x = Min.
3 a13 a23 a33

Consequently, for each A matrix in the sample, the profit is

easily calculated for the given matrix U. Since there are

36 A matrices, each matrix U will produce 36 profit figures

from which the mean profit and the standard deviation of the

profit can be calculated. Presumably the characteristics of

the profit distribution which is produced by a matrix U will

determine whether management selects that plan or another

one. This topic is discussed later in the chapter.

A FORTRAN IV computer program has been written to

calculate the quantities of each product produced and the

objective function value for each of the thirty-six matrices

A given a matrix U. The program prints out for each matrix

U the quantities of each product produced and the value of

the objective function for each matrix A. In addition, the

print out contains the expected value and standard devia-

tion of the objective function.

The practical problem with ASLP is the potential number

of matrices U that can be considered. In other words, the








potential number can be as large as people want to make it.

In the next section, the computational burden of ASLP and

the ways in which this burden can be significantly reduced

will be discussed.


The Reduction of the Computational Burden of ASLP

The firm is more likely to find an allocation plan

best suited to it when a large number of possible values

for each u.. is considered. However, as the potential num-

ber of values for the individual u.. is increased, the num-

ber of potential matrices U to be considered is increased

geometrically. For example, 0, 1/6, 2/6, 3/6, 4/6, 5/6,

and 1 have been considered for the values of u.. in the pro-
13
duction planning problem considered in this chapter. For

this particular division of the values of u.., the potential

number of U matrices to be considered is 283 = 21,952. The

use of 0, 1/4, 2/4, 3/4, and 1 for the values of uij reduces

the number of possible matrices U to only 153 = 3,375. By

reducing the number of possible u.. values, the firm does

reduce the computational burden of ASLP but the firm is like-

ly to miss a good allocation plan whenever it does so.

This study will show that for a particular division of

the values of u.., not all possible matrices U need to be
13
considered. In other words, some allocation plans can be

dominated by others. The dominance rule that has been con-

structed for use in this problem is: A matrix U' will be

dominated by another matrix U" whenever the matrix U" pro-








duces a greater profit than the matrix U' for each set of

technological coefficients in the sample. An example of

the dominance rule for our production problem is given in

Table 3:


Table 3

An Example of Dominance

-~~-Products
Allocation Decision D E F

Resource 1 1 0 0
U" 2 1 0 0
3 1 0 0


Resource 1 1 0 0
U' 2 1/3 2/3 0
3 1/3 1/3 1/3



If in Table 3 we assume that each product requires

some of each resource then matrix U' is dominated by matrix

U" because both matrices result in only units of D being

produced but matrix U" will produce a larger number of D

(and a higher profit) than matrix U' for every matrix A of

technological coefficients.

By applying this dominance rule to the sample problem,

those U matrices which allocate some of resource i to the

production of one product but which allocate no other re-

source (or resources) to that same product are not consider-

ed. In other words, the U matrices remaining for considera-

tion are those U matrices which allocate some of each

resource to a product (or products). Based upon this








dominance rule, the number of U matrices that should be

examined in the sample problem is reduced. Furthermore,

the number of U matrices that need to be considered after

the applications of this dominance rule can be determined

by the following step-wise procedure: First step is to

list the possible product combinations. Since there are

three products (D, E, F) in the sample problem, the pos-

sible product combinations are seven in total. They are:

(1) To produce product D only;

(2) To produce product E only;

(3) To produce product F only;

(4) To produce products D and E;

(5) To produce products D and F;

(6) To produce products E and F;

(7) To produce products D, E, and F.

The next step is to determine the number of ways that

each resource can be allocated to the product (or products).

For instance, in order to produce products D and E in our

sample problem, we can allocate each resource in five dif-

ferent ways as stated in Table 4.

Then the number of U matrices that should be examined

for each possible product combination can be determined by

the possible ways each resource can be allocated to the

products in the given combination. For example, the number

of U matrices that should be examined for the production of

D and E is 5x5x5 = 125.

Finally, the number of U matrices that should be exam-







ined in the sample problem is the sum of the number of U

matrices that should be examined for each possible product

combination. In our sample problem, the number of U

matrices to be examined is 1,378 as stated in Table 4.

Table 4 shows all possible resource allocations for

each possible product combination. As a result of the ap-

plication of this single dominance rule, the number of U

matrices to be considered is decreased from 283 = 21,952 to

1,378. While this is a remarkable decrease in problem size,

the remaining problem of having to examine 1,378 profit

frequency distributions is by no means trivial. Conse-

quently, it has been assumed that management can rule out

any allocation plan producing an expected profit below some

specified figure. Consequently, in our problem, any matrix

U resulting in an expected profit of less than $20,000 was

ruled out from further consideration. This cut-off figure

is somewhat arbitrary but has been based upon the expected

profit that results from a mean-value linear programming so-

lution to the problem (see Chapter five). The utilization

of this cut-off figure results in reducing the number .of

profit frequency distributions that management has to examine

from 1,378 to 32 or less. This reduces the problem of

choosing the desired profit distribution to manageable size.

Table 5 presents the mean and standard deviation of the

firm's profits during the planning period for those U matrices

resulting in mean profits greater than $20,000. In the fol-

lowing section, our attention will be focused on management's

selection of a particular allocation plan as the solution for

the problem.














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Table 5

Allocation Decisions Resulting in Expected
Profits Greater than $21,000


Case (1)
a = 8 = .9
U Expected Profit Standard Deviation
1/3 0 2/3
1/2 0 1/2 $21,348 $7,350
1/2 0 1/2
2/3 0 1/3
5/6 0 1/6 $21,378 $8,007
5/6 0 1/6
1/2 0 1/2
2/3 0 1/3 $21,697 $7,714
2/3 0 1/3
Case (2)
_______________ a = 8= .1_______________
U Expected Profit Standard Deviation
1/2 0 1/2
2/3 0 1/3 $21,103 $8,031
1/2 0 1/2
1/3 1/6 1/2
1/2 1/6 1/3 $21,184 $7,491
1/2 1/6 1/3
1 0 0
1 0 0 $21,313 $11,544
100
2/3 0 2/3
5/6 0 1/6 $21,600 $ 9,565
5/6 0 1/6
1/3 0 2/3
1/2 0 1/2 $21,768 $ 7,976
1/2 0 1/2
1/2 0 1/2
2/3 0 1/3 $21,870 $ 8,283
2/3 0 1/3

The Selection of a Particular Allocation Plan

for the Production Planning Problem

It has-been previously noted that each allocation plan

(i.e., a U matrix) under ASLP produces a profit distribu-

tion. Management must select one of these profit frequen-

cy distributions as being the most preferred distribution

for the problem. The traditional method suggests the








solution of the plan which results in the highest expected

profit as the solution. However, it has been pointed out

that expected value may not be the only relevant figure of
3,4
merit for decision-making purposes.34 Hertz has suggest-

ed discriminating between distributions on the basis of

(1) expected values, (2) variability of distribution, and

(3) risks involved.5 His method is an improvement of the

traditional method under which the selection is based only

upon the highest expected value. The matrix U which

results in the highest expected profit and the lowest va-

riance is the best solution. In the problem examined here

the matrix U which produces the highest expected profit is

not the one which produces the lowest variance. There-

fore, the decision-maker has many choices between the

various combinations of expected profit and variance.

Markowitz6 suggests the consideration of a set of efficient

expected profit and variance (E-V) combinations from which

a particular E-V combination can be selected. The efficient

set is described as the locus of points representing those

combinations with the lowest variance for a given expected

profit and the highest expected profit for a given


3R.M. Barefield, "Comments on a Measure of Forecasting
Performance," Journal of Accounting Research, Vol. 7
(Autumn, 1969), pp. 324-327.

4David B. Hertz, "Risk Analysis in Capital Investment,"
Harvard Business Review, Vol. 42 (January-February, 1964),
pp. 95-106.

5Ibid, pp. 95-106.

Harry Markowitz, "Portfolio Selection," Journal of
Finance, Vol. 7 (1952), pp. 71-91.








variance. Given a decision maker's utility function,

U.= f(E,V), a preferred combination can be selected from

the efficient set. Figure 1 illustrates the efficient set

and a utility function:


Variance (V)

U
u1
S2

B 3




S



A Expected Profit
(E)
Figure 1

Efficient Set and Utility Function


Point A is the minimum variance.

Point B is the maximum expected value.

Curve AB represents efficient set.

U. = f(E,V) is utility function.


The preferred combination is the point of tangency of

the utility.function and the efficient set. In Figure 1,

this is at point S which determines a particular production

plan for the production problem. It should be noted that

the assumption of the linear utility function is made

simply as a matter of convenience. The utility function in

Figure 1 can be a curve.








In summary, the ASLP approach considers the un-

certainty of the planning process by transforming the

random variation of the model parameters into random

variation in the objective function. It does not hide

the stochastic characteristics of the objective func-

tion in one single figure. Therefore, the utilization

of ASLP approach allows a selection of a preferred

matrix U based upon the stochastic characteristics of

the objective function.


The Determination of Confidence

Intervals for Planning Period Profits

The fact that several characteristics of a probabi-

lity distribution may be useful for decision-making

purposes has been pointed out in the previous section.

The application of ASLP to production planning problems

allows management to consider these stochastic character-

istics of a probability distribution. In order to

demonstrate the determination of confidence intervals on

planning period profits in the problem at hand, we need

to pick a preferred production plan. We have arbitrari-

ly assumed that in our problem management has selected

the matrix U which resulted in the highest expected

profit. This arbitrary assumption was made simply as a

matter of convenience. It should be noted that the

method of this selection is in no way limited to this

particular selection. The method will work for any U








matrix selected. The 36 profit figures produced by ASLP

.for that U matrix which produces the highest expected

profit are given in Table 5 in Appendix (I).

Two methods may be considered for deriving probabi-

lities associated with specified profit intervals given

the profit distribution of an allocation plan. One

method utilizes the cumulative observed frequency distri-

bution on profits and the other method utilizes the

estimated theoretical probability distribution profits.

The cumulative observed frequency distribution may be

used to determine the probabilities of a number of profit

intervals. Unfortunately, these estimates may be too

rough because the particular set of profit figures which

leads to an observed probability for a specified profit

interval is only one set of all possible sets of profit

outcomes. Theoretically, it is possible to avoid this

problem by determining probabilities of profit intervals

from the theoretical probability distribution of profit.

The practical problem is that the theoretical probability

distribution can never be known with certainty. However,

there is a curve-fitting method the Pearson system -

which estimates the theoretical probability distribution

underlying a given frequency distribution. The Pearson

curve-fitting system has been used in actuarial science

to smooth out the roughness in underlying insurance

statistics and is appropriate to our purpose of smoothing

out the roughness in the firm's planned profit data.








In the next section, the curve-fitting technique is

applied to the profit distribution associated with the

choose plan. Then, all subsequent confidence intervals

are based directly upon the smooth curve rather than the

actual data.


The Curve-Fitting Technique

The purpose of this section is to demonstrate the ap-

plication of the Pearson curve-fitting system.7 Basically,

the curve-fitting technique involves the calculation of

the first four moments of the given data and subsequently

calculates some functions of these moments to determine a

curve type. The parameters of the curve type are then

calculated from other functions of the moments. The

demonstration of the fitting of a Pearson curve to the

profit frequency distribution ASLP Case (1) is given as

follows:

First, the data are arranged in a frequency distribu-

tion and various sums are calculated as in Table 6.

The first through fourth sums are called factorial
8
moments. These sums form the basis for calculating the

criterion values to determine a curve type. From the

totals of the columns in Table 4 we define:


7William P. Elderton and Norman L. Johnson, Systems
of Frequency Curves (New York: Cambridge University Press,
1969), pp. 35-110.

Ibid., p. 21.



















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S2 = 264/36 = 7.3333

S3 = 1355/36 = 37.6389

S4 = 5544/36 = 154.0000

S5 = 19342/36 = 537.2778


The next step is to find the moments about the centroid

vertical as follows:


d = S = 7.333
2
V2 = 2S3 d(l+d) = 14.1672

V3 = 6S4 3V2(l+d) d(l+d)(2+d) = -0.5421

V4 = 2485 2V3{2(l+d)+l) V2{ 6(l+d)(2+d) 1}

d(l+d)(2+d)3+d)
= 422.9419


Since the data do not display high contact at either

end of the distribution, Sheppard's adjustments will not be

needed.9 Therefore, we can calculate the values of 1 and

02 directly from V2, V3, and V4 as follows:


S= V3/V2 = 0.0001

2
2 = V4/V2 = 2.1072

From the values of 1 and 82 the criterion (K) can be

calculated.


Ibid, p. 51.








1( 2+3)2
K = 4(42-381) (282-381-6) = -0.00004


Since 1 = 0, K and 82 < 3, we are dealing with a Type


II Pearson Curve.10 The general form of the Type II curve

is:

Y = Y (1-x2/a2) origin at mode = mean


where -a < x < a and

5 8 -9
m 0. 2 = 0.8602
2(3-8) -


a -382 8.1777


N r(m+1.5) 3.1692
o a/-' r(m+l)

This means that we have the following probability density

function (Y) on a function of x:

Y = 3.1692 (1-x2/66.8753)0.8602 where -8.1777 < x < 8.1777.

This curve has its origin at the mode of the profit (21,667)

and can be used to find the probability distribution on

profits since profits and x are related by the following

formula:


Profit = 21,667 + 32,711 x
16.355

The curves for ASLP in Case (1) and Case (2) are:


10id p. 45.
Ibid., p. 45.









Case (1) Y = 3.1692 (1-x2/66.8753)0.8602, where
(a=B=.9) -8.1777 < x < 8.1777,
ASLP origin at the mode = mean,
profits = 31,667 +'(32,711/16.355) x

0 9405
Case (2) Y = 3.143 (1+x/6.5582) 9 where
(a=B=.l) -6.5582 S x < 13.6916,
ASLP origin at the mode,
profits = 18,889 + (40,496/20.2498) x


The calculations necessary to derive the curve's parameters

for ASLP in Case (2) are given in Appendix (II).

Before the smooth curves can be used to determine the

probability of profit intervals, the goodness of fit should

be tested. Unfortunately, a good measure of how well a

smooth probability density function fits a given frequency

distribution does not exist. The available tests only

examine whether the given frequency distribution is likely

to have come from the theoretical distribution specified

by the smooth curve.1

The Kolmogorov-Smirnov one-sample test is a powerful
12
test of goodness of fit when samples are small. It

treats individual observations separately and thus, unlike

the Chi-square test, need not lose information through the

combining of categories. In the problems at hand, the

frequencies falling into each category are small. There-

fore, the Kolmogorov-Smirnov test is believed to be more

applicable than the Chi-square test. The Kolmogorov-

11
1Ibid., pp. 165-173.
12Sidney Siegel, Nonparametric Statistics (New York:
McGraw-Hill Book Company, Inc., 1956), p. 51.








Smirnov test has been conducted on the ASLP data for both

cases (1) and (2). The results of the test are that (in

each case) the null hypothesis that the known frequency

distribution came from the theoretical probability distri-

bution specified by the smooth curve cannot be rejected.13


The Integration of a Probability Density Function and the
Determination of the Probabilities of Various Profit
Intervals

In the previous section an estimate of the theoretic-

al probability distribution on planning period profits was

derived. The estimate of the theoretical probability dis-

tribution then provides a basis to determine the cumulative

probability distribution on planning period profits.

Simpson's Rulel4 was used to numerically integrate the

estimated theoretical probability distribution function in

order to determine the cumulative probability distributions

on profits. Table 7 exhibits the cumulative probability

distribution for selected profit numbers in Case (1) and

Case (2).


1At a 5 percent level of significance, the critical
value for the Kolmogorov-Smirnov test for N = 36 is 0.227.
But the maximum absolute deviation between the cumulative
frequency distribution and the theoretical cumulative
distribution is only 0.101 in Case (1) and 0.042 in Case
(2).

14
1Merlin L. James, Gerald M. Smith, and James C.
Wolford, Applied Numerical Methods for Digital Computation
with FORTRAN (Scranton: International Textbook Co., 1967),
pp. 284-290.








Table 7

Cumulative Probabilities on Profits (8)

ASLP


Case (1) Case (2)
K (a = 8 = .9) (a = 8 = .1)
P (9 K) P (6 < K)

12,000 .12 .12
14,000 .18 .19
16,000 .26 .28
18,000 .34 .36
20,000 .43 .45
22,000 .51 .54
24,000 .60 .62
26,000 .69 .69
28,000 .77 .76
30,000 .84 .82
32,000 .90 .87
34,000 .95 .91
36,000 .99 .95


The information in Table 7 provides a basis for

calculating probabilities of various profit intervals. For

example, in Case (1) there is an 81% probability that

profit falls in the interval of $14,000 to $36,000 and a

61% probability that the profit will fall in the $18,000

to $34,000 interval.

We have assumed for the sake of brevity that managers

have been able to select a preferred production plan before

the smoothed confidence interval data are generated. How-

ever, the confidence interval data for several production

plans could be made available to managers whenever they

were unable to choose a single production plan on the basis

of the rough profit distribution produced by the sample

data. Thus, the ASLP algorithm coupled with the Pearson








curve-fitting technique can be used to give managers confi-

dence interval data for several production plans. Conse-

quently, the data produced by ASLP accentuate the uncertain

nature of the planning problem and they also give managers

a rational basis for choosing among several production

plans.

In this chapter, we have examined an ASLP algorithm

for considering uncertainty in a firm's short-run production

plans. We have discussed and investigated the computational

burden and the reduction of the computational burden in the

application of ASLP to a small short-run production planning

problem. We have also shown how ASLP coupled with a curve-

fitting technique can be used to develop a probability den-

sity function on planning period profits. Further, we have

demonstrated how this probability density function can be

used to determine the probabilities associated with various

profit intervals for the planning period. In next chapter,

an application of ASLP to a production planning problem

with a larger matrix A will be conducted.











CHAPTER IV


AN APPLICATION OF ASLP TO THE SHORT-RUN PRODUCTION
PLANNING PROBLEM OF A MEDIUM-SIZED FIRM


Introduction

In Chapters two and three, it was pointed out that

problem size is a critical factor in ASLP applications.

The potential expansion in problem size in ASLP is greater

for a large matrix A than for a small matrix A. An ap-

plication of ASLP to a production planning problem of

three products and three resources (i.e., a 3x3 matrix A)

has been discussed in Chapter three. The purpose of this

chapter is to present an application of ASLP to a produc-

tion planning problem with a larger matrix A than was

considered in Chapter three. Specifically, in this chap-

ter ASLP will be applied to a production planning problem

which has six products and six resource constraints (i.e.,

a 6.x 6 matrix A). This production planning problem will

have thirty-six elements in its matrix A which is four

times the size of a 3 x 3 production planning problem.

The potential expansion in problem size under these new

conditions is much greater than the one discussed in Chap-

ter three. The application of the dominance rule of Chap-

ter three to this new problem results in a remarkable

decrease in computational burden, but it still leaves too

many U matrices to be considered. However, if there are








minimum product constraints in problem, then ASLP can be

used to solve the larger problem of this chapter.

The presentation in this chapter is divided into

four parts: a hypothetical production planning problem,

the introduction of a new dominance rule, the ASLP solu-

tion of the production planning problem, the determination

of the probabilities associated with specified profit in-

tervals for the planning period.


A Hypothetical Production Planning Problem

In order to illustrate the application of ASLP to a

larger production planning problem, a problem which has

six products (D,E,F,G,H,I) and six resources (1,2,3,4,5,6)

has been constructed. This problem is an expansion of the

3 x 3 production problem discussed in Chapter III. The

3 x 3 production problem has been expanded by adding three

more products and three more resources. The basic assump-

tions of the problem are the same way as in Chapter three.

Recall that in the original production planning problem,

selling prices were constant, resource availabilities were

constant, and unit input costs were constant. The techno-

logical coefficients and the product contribution margins

were assumed to be random variables. The values of the

constant factors of the expanded problem are given in

Table 8.








Table 8

Values of Problem Constants

(for Case (3) and Case (4))


Resource Input Cost/per Unit Amount Available

1 $10 9,030
2 8 11,180
3 5 8,610
4 4 9,800
5 3 9,330
6 3 7,820

Products Selling Price/per Unit

D $ 265
E 270
F 273
G 275
H 271
I 280



The formula aij = V.. + aRi + 8C + e has been

used to generate the values of a.. in the expanded pro-
13
duction planning problem. The values of Ri, Cj, and e

in the formula are random numbers and are generated by

the utilization of the Monte Carlo method. The distribu-

tion of each random variable R. and Cj utilized for the

simulation is given as the following.










Probability
Variable Random Values of Occurrence

R1 2 .50
3 .50


3 .33
R 4 .34
2 5 .33


2 .50
R3 3 .50


4 .50
R4 5 .50


R 3 .50
4 .50


R 2 .50
6 3 .50


C 2 .50
1 3 .50


C 3 .50
2 4 .50


C 2 .50
3 3 .50


C 3 .50
4 4 .50


C 2 .50
5 3 .50


C 4 .50
6 5 .50








Different levels of dependence among the technological coef-

ficients has been built into the problem through the use of

different values for a and 8 in Case (3) and Case (4). Case

(3) has a = 8 = .9 (the case in which the elements of the A

matrix are relatively dependent). Case (4) has a = 8 = .1

(the case in which the elements of the A matrix are rela-

tively independent). Also, the mean values of the aij are

set to be approximately the same in the two cases through

the use of the equation -



V.. = Vij + .8 E(R.) + .8 E(C.) which was discussed in


Chapter three.

The two sets of thirty-six matrices A and correspond-

ing contribution margins produced by this data generation

technique are given in Table 3 (Case (3)) and Table 4 (Case

(4)) in Appendix (I). As in Chapter three, sample correla-

tion coefficients between a of Case (3) and Case (4) have
1j
been calculated to make sure the data generation technique

was successful. Table 9 exhibits the sample correlation

coefficients between a66 and a.. in Case (3) and Case (4).

It is observed that the sample correlation coefficients

along row six and column six in Case (3) are greater than

those in Case (4) so that the cases do display the desired

characteristics.








Table 9

The Correlation Coefficients
between a66 and aij
(for i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6)
(for i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6)


The Introduction


of a New Dominance Rule


In chapter three, a


dominance rule was


discussed and


applied to the production problem described in that chapter.

That dominance rule reduced the computational burden of

ASLP in that problem to a manageable size. However, the


Case (3) a = 8 = .9
ail ai2 ai3 ai4 ai5 ai6
_____ i2 13 i4 i5 i6
alj -0.12 0.03 0.19 -0.11 0.00 0.27

a2j
a2j 0.18 0.12 0.14 0.13 0.10 0.43

aj -0.17 0.14 -0.05 -0.11 -0.03 0.43

a4ji
a4j -0.11 0.02 -0.08 -0.06 0.00 0.28

a5j
0.17 0.27 0.11 0.19 0.13 0.60

a6j 0.47 0.62 0.48 0.59 0.54 1.00

Case (4) a = 8 = .1
ail ai2 ai3 ai4 ai5 ai6

j -0.15 0.06 0.03 0.00 0.15 0.11

_aj 0.05 0.28 0.12 0.06 -0.05 0.01

a3j 0.16 -0.02 -0.21 0.37 -0.07 -0.15

a4j -0.26 -0.12 0.15 -0.15 0.18 -0.11

a5j 0.09 -0.03 -0.16 -0.08 -0.25 0.06
a6j -0.0 13 -0.02 -0.04 0.20 1.00
-0.10 -0.13 -0.02 -0.04 0.20 1.00








potential computational burden of ASLP is greater for a

large matrix A than for a small matrix A. Therefore, in

a problem having a larger matrix A than three by three

there may still be a large number of matrices U to con-

sider even after the dominance rule of Chapter three has

been utilized. This section will be divided into two

parts. In the first part, the dominance rule suggested

in chapter three will be applied to the new problem. In

the second part, a new dominance rule will be introduced.


The Application of the Old Dominance Rule

According to the old dominance rule, only those U

matrices which allocate some of each resource to the same

product (or products) are remaining for consideration. By

applying this dominance rule to the new problem, if the

possible values of uij are 0, 1/10, 2/10, 3/10, 4/10, 5/10,

6/10, 7/10, 8/10, 9/10, and 1, the number of matrices U to

be considered is decreased from (6+135+720+1260+750+120)6

to (6+1356+7206+12606+7506+1206). While this is a remark-

able decrease in computational burden, the calculation of

(6+1356+7206+12606+7506+1206) is very expensive. This is

an example that the old dominance rule is not very power-

ful in the reduction of computational burden for a problem

with a large matrix A. We, therefore, turn to the problem

of finding a new mechanism to reduce the computational

burden of ASLP for these problems.








The Introduction of a New Dominance Rule

The structure of ASLP is such that additional matrices

U may be eliminated from consideration if the firm has some

minimum output constraints. During a short period, it seems

likely that a firm may have some minimum output constraints

because of the characteristics of the market situation. Con-

sequently, the assumption of minimum output constraints

provides a reasonable and probably a more realistic short-

run production planning problem than if these constraints

are omitted. The assumption of minimum output constraints

simplifies the ASLP procedure because the firm does not

consider those matrices U which produce less than the

minimum desired output on any product for any sample matrix

A. Consequently, the firm only considers those matrices U

which satisfy the minimum output constraints. For purposes

of illustration, an assumption of 180 units of minimum

desired output for each product is made in the production

planning problem of this chapter.

The minimum desired output is arbitrarily selected for

the purpose of further illustration. One should note that

a higher minimum desired output will result in a greater

number of matrices U being eliminated than a lower minimum
6
desired output. Given the condition of u.. = 1, the
j=l 13
higher the minimum desired output is set, the fewer the

number of possible values of u.. that will satisfy the

minimum requirement.

Given the minimum output constraint and the maximum








value of each aij, the minimum quantity of each resource

which has to be allocated to each product can be easily

calculated. Specifically, the quantity is calculated by

the following formula:

The minimum quantity of resource i allocated to

product j = (the minimum desired output of product j) x

(the maximum value of aij).

Dividing the amount of each resource available into

the minimum quantity of each resource needed for each

product will result in the required minimum uij. The

required minimum uij is the minimum proportion of

resource i which will be allocated to produce product j

at a level of 180 units. The matrix U containing the

required minimum u.. for case (3) and for case (4) is given

as follows:

Case (3)

.123 .175 .197 .151 .141 .169
.196 .148 .149 .149 .144 .159
.134 .163 .121 .182 .205 .154
.154 .162 .154 .156 .143 .174
.169 .151 .135 .160 .174 .170
.133 .161 .138 .172 .181 .172

Case (4)

.123 .175 .197 .151 .141 .169
.196 .148 .149 .149 .144 .159
.134 .163 .121 .182 .205 .161
.154 .162 .154 .156 .143 .174
.169 .151 .135 .160 .174 .170
.133 .161 .138 .184 .181 .172
1
For example, in case (3), the required minimum u =
180 x the maximum a /the quantity of resource 1 available =
180 x 6.2/9030 = .1.1








It should be noted that many of the minimum u.. in case
1]
(3) are equal to those in case (4). In these cases, the

identity is due to the fact that the maximum values of aij

are the same for both cases.

At this point, it is obvious that the firm needs only

to consider those matrices U which contain uij equal to or

greater than those minimum u..'s derived above. Here a
6 13
constraint u.. < 1 must be satisfied. When the constraint
j=l 3
is less than one, the resources available are not completely

consumed. This requires an adjustment so that a full employ-

ment of available resources is assured. Given the tableau

of required minimum u..'s and the understanding that
6 13
>u. = 1, an alternate tableau as in Table 10 may be
j=l 13
constructed of possible values for each uij based upon

management's experience or preference.

Since the required minimum u..'s in case (3) are ap-

proximately equivalent to those of case (4), one set of

possible values for each u.. is derived as an illustration

for both cases (3) and (4) in Table 10.








Table 10

The Possible Values of uij to Be


Considered in Case (3)


This particular set of

the possible allocations of


and Case (4)


possible values of uij suggests

each resource to different


products in Table 11. The suggested resource allocations

in Table 11 provide 38,880 (6x6x6x6x5x6 = 38,880) U matrices

for consideration in the problem. This is a remarkable

decrease in problem size for ASLP in a problem with a 6 x 6

matrix A.


Products
D E F G H I
Resources ______

1 .125 .180 .200 .155 .145 .170
.130 .185 .205 .160 .150 .175


2 .205 .155 .155 .155 .145 .160
.210 .166 .160 .160 .150 .165


3 .140 .165 .125 .185 .215 .160
.170 .190 .220 .165


.165 .165 .160 .160 .145 .180
.170 .170 .165 .165 .150 .185


5 .175 155 .140 .160 .180 .170
.180 .145 .165 .185 .175


.135 .165 .140 .175 .185 .175
6 .140 .170 .145 .180 .190 .180

















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The ASLP Solution of the Production Planning
Problem

A computer program was written that calculates the

thirty-six optimal product quantity vectors and the cor-

responding profit figures for each of 38,880 U matrices

which were derived for consideration in the preceding

section. The computer program also calculates the aver-

age profit for each U matrix and then drops from further

consideration any U matrix resulting in average profits

less than $47,780 in case (3) and $45,000 in case (4).

The program finally prints out the 36 product quantity

vectors, the corresponding optimal profits, and the mean

value and standard deviation of profit for each remain-

ing matrix U. Table 12 presents the mean and standard

deviation of the 36 profit values for those U matrices

which result in expected profits greater than $47,789

in case (3) and $45,010 in case (4).








Table 12

Allocation Decisions Resulting in
Expected Profits Greater than K
(ASLP)


Case (3)
K Z $ 47,780.00
Expected Standard
Matrix U Profit- Deviation
.130 .180 .205 .160 .150 .175
.210 .160 .160 .160 .150 .160
.140 .165 .125 .190 .215 .165
.165 .170 .165 .165 .150 .185 $47,792.92 $12,789.17
.180 .155 .140 .165 .185 .175
.140 .170 .140 .180 .190 .185
.130 .185 .200 .160 .150 .175
.210 .160 .160 .160 .150 .160
.140 .165 .125 .190 .215 .165
.165 .170 .165 .165 .150 .185 $47,789.28 $12,795.84
.180 .155 .140 .165 .185 .175
.140 .170 .140 .180 .190 .180
.130 .185 .205 .155 .150 .175
.210 .160 .160 .160 .150 .160
.140 .165 .125 .190 .215 .165
.165 .170 .165 .165 .150 .185 $47,790.33 $12,799.22
.180 .155 .140 .165 .185 .175
.140 .170 .140 .180 .190 .180
Case (4)
K $ 45,010.00
Matrix U Expected Standard
Profit Deviation
.130 .185 .205 .155 .150 .175
.210 .155 .160 .160 .150 .165
.140 .170 .125 .185 .215 .165
.165 .170 .165 .165 .150 .185 $45,011.16 $ 7,833.16
.180 .155 .145 .165 .185 .170
.140 .170 .145 .175 .190 .180
.130 .185 .205 .155 .150 .175
.210 .155 .160 .160 .150 .165
.140 .170 .125 .185 .215 .165
.165 .170 ..165 .165 .150 .185 $45,018.05 $ 7,861.05
.180 .155 .145 .165 .185 .170
.140 .170 .145 .180 .185 .180
.130 .185 .205 .155 .150 .175
.210 .155 .160 .160 .150 .165
.140 .170 .125 .185 .215 .165
.165 .170 .165 .165 .150 .185 $45,013.19 $ 7,812.95
.180 .155 .145 .165 .185 .175
.140 .170 .145 .180 .185 .180






61


Chapter three discussed the possibility that manage-

ment may consider not only the expected value of profits

but also the range, median, mode, and variance of these

profit values. Accordingly, the ASLP solution for the

sample (or any) problem is determined by the management's

selection of a preferred matrix U based on the appropriate

characteristics of the corresponding profit distribution.

For the purpose of further discussion, it is arbitrarily as-

sumed that management has selected the matrix U which as-

sociates the lowest standard deviation of .profit in Table 12.

The thirty-six values of the objective function for that

selected U matrix in case (3) and case (4) are given in

Table 6 in Appendix (I). In the next section, the discus-

sion will turn to the derivation of the probabilities of

various profit intervals for the planning period given that

an U matrix has been selected.


The Determination of the Probabilities

of Profit Intervals for the Planning Period

The Pearson system of curve-fitting was used to fit a

probability distribution curve to the profit distributions

of the selected ASLP solution in case (3) and case (4).

Table 13 presents the fitted probability density functions

for ASLP in case (3) and case (4). The necessary calcula-

tions to derive the parameters of these curves are given in

Appendix (II).








Table 13

Probability Density Functions for

ASLP in Case (3) and Case (4)


Case Probability Density Function

Case (3) Y = 2.8693 (1-x2/86.26967)09542, where
-9.2881 < x < 9.2881, origin at the
mode = mean, profits = 47,788.11 + 3,000 (x).


Case (4) Y = 214.379(1)56 x-36.5764e -516.09/x
where x > 0, origin at the mode, profits
= 30000 + 3,000 (x).



Before these smooth curves were used to determine

cumulative probabilities of selected profit numbers, the

goodness of curve fit had to be examined. The Kolmogorov-

Smirnov one-sample test was conducted to determine how

well the smooth curves fit the given profit distribution.

In each case, the results of the test are that the null

hypothesis that the given frequency distribution came from

the specified theoretical distribution cannot be rejected

at a 5 percent significance level.

The fitted curves were then integrated over selected

profit intervals to derive the cumulative probabilities

for selected profit numbers. The resulting cumulative

profit distributions on the selected ASLP solution for

Case (3) and Case (4) are given in Table 14.







Table 14

Cumulative Probability Distributions

on Profits (B) under ASLP for

Case (3) and Case (4)


_P (2 < K)
SCase (3) Case (4)
K a = 8 = .9 a = 8 = .

$ 24,000 0.0162
27,000 0.0473
30,000 0.0912
33,000 0.1458 0.0271
36,000 0.2094 0.0973
39,000 0.2799 0.2198
42,000 0.3556 0.3789
45,000 0.4346 0.5445
48,000 0.5153 0.6902
51,000 0.5957 0.8030
54,000 0.6741 0.8821
57,000 0.7487 0.9336
60,000 0.8176 0.9652
63,000 0.8792 0.9837
66,000 0.9314 0.9942
69,000 0.9723 1.0000
72,000 1.0000


The information in Table 14 provides the basis for

calculating probabilities associated with various profit

intervals for the planning period in each case. For

example, in case (3), these is a 42% probability that the

profit will fall in the interval of $48,000 to $66,000

and an 88% probability that the profit will fall in the

$30,000 to $69,000 interval.

The application of ASLP to a sample problem of six

products and six constraints has been discussed in this

chapter. A new dominance rule has been developed and ap-

plied in order to reduce the number of matrices U that








need to be considered. It has been shown that, without this

new dominance rule, the application of ASLP to a problem

with a six by six A matrix would be very expensive to solve.

With this dominance rule the solution of this problem by

ASLP is feasible.

Chapters three and four have concentrated on the feasi-

bility of applying ASLP to certain production planning

problems. Chapter five deals with the question of whether

it is desirable to apply ASLP to production planning

problems. This question is examined by comparing ASLP solu-

tions to the solutions which would be derived by mean value

linear programming (MVLP).












CHAPTER V


A COMPARISON OF ASLP AND MVLP


Introduction

The purpose of this chapter is to conduct a compara-

tive study of ASLP and MVLP in the circumstances depicted

in Chapter three and four. It was pointed out in Chapter

one that MVLP ignores the uncertainty of production plan-

ning problems by considering only the mean values of the

model's random parameters. However, a reinterpretation of

the MVLP solution is required before ASLP and MVLP results

can be properly contrasted. This study will assume that

the optimal vector of product quantities derived by MVLP

are to be interpreted as a decision to allocate the appro-

priate fraction of resources to the appropriate products.

This interpretation implies that when the random parameters

deviate from their expected values the ex post vector of

product quantities may differ from the a priori "optimal"

solution vector of product quantities. This interpreta-

tion of the MVLP solution allows us to compare the MVLP

and ASLP solutions for the sample cases described in

Chapter three and Chapter four.

Two variables which appear to have a bearing on

whether MVLP or ASLP should be used in a particular context

will be studied in this chapter. These two variables are








the amount of dependence among the problem's technological

coefficients and the size of the problem. The dependence

variable will be considered in two cases: low (a = 8 = .1)

and high (a = 8 = .9) The problem size variable will be

considered in two sizes: small (three products with three

constraints) and large (six products with six constraints).

The two variables under study provide four different

cases for observation. They are:

Case (1): A problem of three products and three
constraints with a "high" amount of dependence
among the technological coefficients.

Case (2): A problem of three products and three
constraints with a "small" amount of dependence
among the technological coefficients.

Case (3): A problem of six products and six constraints
with a "high" amount of dependence among the
technological coefficients.

Case (4): A problem of six products and six constraints
with a "small" amount of dependence among the
technological coefficients.

The ASLP approach has been applied to each of these

cases in chapters three and four. The MVLP approach will

be applied to solve each of the cases in this chapter and

the results will then be compared to the ASLP results. The

purposes of these comparisons are to provide some basis

for choosing one planning approach over the other.

For the purpose of convenience, this chapter is par-

titioned into three sections as follows: The solutions

implied by MVLP, the curve fitting and cumulative probabi-

lity distribution of profits for MVLP solutions, a

comparison of ASLP and MVLP solutions.







The Solutions Implied by MVLP

The ASLP solutions have been discussed in chapter

three and chapter four. In order to compare the solutions

resulting from ASLP and MVLP, the MVLP solutions need to

be discussed first. The first section presents the MVLP

solutions for cases (1) and (2) and-the second section

presents the MVLP solutions for cases (3) and (4).


The MVLP Solutions for Case (1) and Case (2)

The mean values of each of the technological coeffi-

cients and contribution margins in cases (1) and (2) have

been calculated and displayed in Table 1 and Table 2 of

Appendix (I). These mean values have been treated as de-

terministic model parameters to solve the production plan-

ring problems. The a priori vectors of optimal product

quantities have been computed to be as follows:

Case (1) xl = 272.7; x2 = 229.5; x3 = 163.7.

Case (2) xI = 288.3; x2 = 190.5; x = 190.5.

By combining these solution vectors and the mean values of

each of the technological coefficients, the following two

U matrices are derived as the MVLP solution to the firm's

production planning problem in Case (1) and Case (2):

Case (1) Case (2)

.312 .377 .311 .329 .309 .362
U = .485 .295 .220 U = .510 .240 .250
383 .405 .212 .412 .338 .250

A u.. in the above U matrix was calculated by a formula as

the following:
aijxj
uij =
.2 ---(-.x-
j ..xj)
j=l 1)








For example ull in the U matrix of Case (1) is calculated

as:

(5.15)(272.70)
U11 (5.15)(272.7)+(7.39)(229.5))+(8.55) (163.7)

S1404.4 312
4500

These two allocation plans derived from the MVLP solution

generate 36 profit figures in each case. These profit values

for cases (1) and (2) are exhibited in Table 5 of Appendix

(I).


The MVLP Solutions for Case (3) and Case (4)

In Case (3) and Case (4), a consideration of minimum

output constraints has been included in the ASLP solution

processes. For the purpose of presenting a comparison of

ASLP and MVLP results in cases (3) and (4), the minimum out-

put constraints must also be considered in the MVLP approach

in these two cases. Specifically, the allocation decision

resulting from MVLP should also satisfy the minimum output

constraint of 180 units on each product for each sample

matrix A. In order to satisfy this constraint, the MVLP

solution needs to consider a higher minimum constraint than

180 units for each product. This requirement of a higher

minimum constraint in MVLP is due to the fact that some a
ij
may be larger than their mean values. The steps used to

determine the minimum constraint for each product in the

MVLP constraint equations are as follows:

First, calculate 180)(maximum value of ai.)
First, calculate Mij = Mean of a..
SMean of a..
13






69

Second, select the largest value of M.., for i = 1,2,3,4,
13
5,6, as the required minimum constraint for product

j in the MVLP constraint equations.

Table 15 and Table 16 show these calculation of M.. for

Case (3) and Case (4) respectively.















H H- 0 0 H 0
N N N CN N N
H II II II II ii

oC N 0, 0H oCa




r -4 --I o --I o r--I r-O
H H 00 N 0 Hr




Hr-IO H0 H HI o1
SH C H oCM O t --









f v
N N N N N N




4. Ii % II gI II ||













S0
O



E-4 C N N (i




SD N O r-





(N N 0
U- HOD H-q 0 o r-i H H5I J


I C O M C CN






N N N N N N
S11 II II II II II

"n r wh m n w o4
(Y 0N ( CD 00 OJ 'C


L n l
(-I CO N C r-l H










N N N N N N


0 DN MY NH VN
H O 11 II II H | O| IN









( N (M (M Cl C'0



*- C 0il -












S00 0 0
.
*N 0 0 N 0
SN 0 C HM 0
N N N N N
H II I II I Ii II II

c o c1 0 co o',I ojen ft IO o
rl L rH n r L



0 O O O 0 0
*

r H O0 0 Hl
N N N MN N N

m II II II II II II

NO rto .oio or on N oL0
HID HI Hc N D
%0 1-1 o rii rrX rl tlo
cm N


00 N 0 O 00 0v4
.
N l k 9y % 9l vo
H H o 0 H o
N N N N N N

11 II II II II II

%0 oon o v co
0 0O n .
1-ito r-l H r r-ur rHco %0 r-1


o M r i
.
o o (' 4
0 nH o( 0 C0 N
N C4 CN N NM C4

r II I II II II II

oDm r--r 0 vo co co
r'- 0 o n C0 0



.
cr~ vco Vn 431 l r1 (HI


ON O H N

-l r-l r-l H N rH
CN C4 N CM CM 04



0 C4 in om cOm on 0r
|Ln t I' 4r C4 .
Hr rl- i-l w H r- rHi rHin


,

H 0 CM 0C H NC
N< C N C4 0 CN

% II II II II II II

mID to o NC oo L L n
H- q n n 0 o *
N C>n r-4 1, |n r-4t I- H


0




v4-
O






r-) 0


rq
I 0


4O-

tu


r-A
(d


C-)
0)





72

Table 17 presents the minimum constraint for each

product which must be included in the MVLP constraint e-

quations.



Table 17

Minimum Constraints in MVLP for Cases (3) and (4)


Psodu Product Product Product Product Product Product
Case D E F G H I

Case (3) 228 222 224 219 221 220

Case (4) 228 222 222 218 221 223



Cases (3) and (4) are solved by MVLP utilizing the

above minimum constraints.. Their optimal product quantities

were found to be as follows:


Case (3)

X1 = 228.05;


X4 = 219.00;

Case (4)

X1 = 228.00;


X4 = 218.00;


X2 = 220.00;


X5 = 221.96;




X2 = 222.00;


X5 = 221.25;


X = 224.00;


X = 220.00.


X3 = 222.00;


X6 = 223.00.


By combining these solution vectors and the mean

values of each of the technological coefficients, two U

matrices are derived. These two U matrices are the MVLP

solutions to the firm's production planning problem in








Case (3) and Case (4) and are as follows:


Case (3)

.1303 .1817 .2121 .1550 .1465 .1744
.2177 .1531 .1617 .1555 .1497 .1623
U = .1359 .1661 .1230 .1897 .2199 .1654
.1706 .1677 .1674 .1659 .1447 .1837
.1838 .1537 .1430 .1616 .1868 .1711
.1392 .1684 .1450 .1804 .1855 .1815


Case (4)

.1296 .1794 .2108 .1561 .1476 .1765
.2195 .1523 .1587 .1560 .1521 .1614
U = .1362 .1649 .1221 .1890 .2191 .1687
.1680 .1670 .1660 .1688 .1429 .1873
.1848 .1516 .1440 .1614 .1850 .1732
.1338 .1650 .1385 .1903 .1866 .1858


These two MVLP matrices U generated 36 profit figures

in each Case. These profit values are exhibited in Table

6 of Appendix (I).


The Curve-Fitting and Cumulative Probability

Distribution of Profits for MVLP Solutions

Probability density functions have been fitted to the

thirty-six profit figures produced by MVLP for each case

in the same manners as in Chapters three and four. The

necessary calculations in order to derive the curve's

parameters are given in Appendix (II). The Kolmogorov-

Smirnov one-sample test has been conducted to determine

how well the smooth curves fit the given frequency distri-

bution under MVLP. Again, the results of the tests do not

reject the null hypothesis (at a 5 percent level of








significance) that the given frequency distributions came

from the specified theoretical distributions.1 Table 18

presents the fitted probability density functions for MVLP

in each case.

Table 18

Probability Density Functions

for MVLP in Cases (1), (2), (3), and (4)


Case Probability Density Function

1.21849 2.80152
Y = 3.981(1+x/5.5214) (1-x/12.6946) ,80152
(1) where -5.5214 < x < 12.6946, origin at the mode,
profits = 18,617 + (36,432/18,216) x.
Y = 5.8444(l+x/7.415)4.8954 e-0.66x where
(2) x > -7.415, origin at the mode,
profits = 18,137 + (41,693/20.8465) x.

Y = 2.8523(1-x2/85.57942)0.92, where
(3) -9.2509 < x < 9.2509, origin at the mode = mean,
profits = 47,686.91 + 3,000 (x).
Y = 2.653(+x/5.65117) (0.96692) (5.65117) -(0.96692)/x,
(4) where x > -5.65117, origin at the mode,
profits = 41,631.70 + 3,000 (x).

These fitted curves were then integrated over selected

profit intervals to derive the cumulative probabilities on

profits for the planning period. The resulting cumulative

probability distributions on selected profits for MVLP in

Cases (1) and (2) are given in Table 19, for MVLP in Cases

(3) and (4) are given in Table 20.

At a 5 percent level of significance, the critical value
for the Kolmogorov-Smirnov test for N = 36 is 0.227. But the
maximum absolute deviation between the cumulative frequency
distribution and the theoretical cumulative distribution is
0.069 in Case (1), 0.045 in Case (2), 0.0764 in Case (3), and
0.0841 in Case (4).









Table 19

Cumulative Probability Distributions

of Profits (9) under MVLP for Cases (1) and (2)

P (9 < K)



K Case (3) Case (2)
a = 8 = .9 a = 8 = .1

$12,000 .08 .08

14,000 .16 .16

16,000 .26 .26

18,000 .37 .38

20,000 .48 .49

22,000 .58 .60

24,000 .68 .70

26,000 .77 .77

28,000 .84 .84

30,000 .89 .89

32,000 .94 .89

34,000 .97 .92

36,000 .98 .97








Table 20

Cumulative Probability Distributions

on Profits (8) under MVLP for Cases (3) and (4)

P (9 < K)


Case (3) Case (4)
K = = .9 = = .


$24,000 .0170_

27,000 .0491_

30,000 .0937 -

33,000 .1490 .0334

36,000 .2128 .1194

39,000 .2833 .2558

42,000 .3588 .4174

45,000 .4375 .5752

48,000 .5176 .7093

51,000 .5975 .8123

54,000 .6753 .8853

57,000 .7494 .9338

60,000 .8181 .9645

63,000 .8793 .9831

66,000 .9314 .9939

69,000 .9724 1.0000

72,000 1.0000_








A Comparison of ASLP and MVLP Solutions


Table 21 presents the mean profits and the standard

deviation of the selected ASLP solution and the MVLP so-

lution in each case.


Table 21

Mean Profits and Standard Deviations

of the Selected ASLP Solution and the MVLP Solution


Case (1) Case (2)

ASLP MVLP ASLP MVLP

Mean
Profit $21,697 $21,091 $21,870 $21,409

Standard
Deviation $ 7,714 $ 6,529 $ 8,283 $ 7,557
Case (3) Case (4)

ASLP MVLP ASLP MVLP

Mean
Profit $47,793 $47,687 $45,013 $44,734

Standard
Deviation $12,789 $12,789 $ 7,813 $ 7,860



An analysis of the information in Table 21 gives us

an idea that the ASLP approach resulted in higher mean

profits than the MVLP approach in each case. However,

the standard deviations associated with ASLP in Case (1)

and Case (2) are higher than those associated with the MVLP

approach. Consequently, it is very difficult, without a









utility function, to compare the results of ASLP and MVLP

based only on the information in Table 21. One way to

extend the comparisons of ASLP and MVLP is simply to

compare the probabilities of various profit intervals pro-

duced by ASLP and MVLP.

The probability of a specified profit interval can

be derived from an observed profit frequency distribution

on an estimated profit probability distribution. Chapter

three discussed and compared these two methods in some

detail. The conclusion of that chapter is that the

estimates of the probabilities of profit intervals

directly through the use of observed profit frequency dis-

tributions are too rough. Conversely, fitting a profit

frequency distribution into a probability distribution

curve enables us to determine the probability of profit

intervals at any desired level. Therefore, in this

study, the comparison of the probabilities associated

with various profit intervals has been based upon the

profit's estimated probability density function rather

than the observed profit frequency.

For the sake of convenience, we have divided the

comparison between ASLP and MVLP into two parts. In








part one, we compare ASLP and MVLP for cases (1) and (2).

In part two, we compare ASLP and MVLP for cases (3) and (4).


The Comparison of ASLP ahd MVLP in Case (1) and Case (2)

The cumulative probability distributions on ASLP and

MVLP planning period profits for cases (1) and (2) were

shown in Tables 7 and 19. Table 22 combines the information

contained in those two tables.


Table 22

Cumulative Probability Distribution
of Profits (9) under MVLP and ASLP

P (98 K)


Case (1) a = 8 = .9 Case (2) a = 8 = .1
K
MVLP ASLP MVLP ASLP

$12,000 .08 .12 .08 .12
14,000 .16 .18 .16 .19
16,000 .26 .26 .26 .28
18,000 .37 .34 .38 .36
20,000 .48 .43 .49 .45
22,000 .58 .51 .60 .54
24,000 .68 .60 .70 .62
26,000 .77 .69 .77 .69
28,000 .84 .77 .84 .76
30,000 .89 .84 .89 .82
32,000 .94 .90 .92 .87
34,000 .97 .95 .95 .91
36,000 .98 .99 .97 .95



Table 22 provides the basis for calculating probabi-

lities of various profit intervals for the planning period.

For instance, in the ASLP results of Case (1), there is an

81 percent probability that profit falls with the interval

of $14,000 to $36,000 and a 61 percent probability is






80

associated with the profit interval $18,000 to $34,000. This

type of information provides the basis for the comparisons

between ASLP and MVLP. Management usually is more interest-

ed in the probabilities of profit intervals at upper profit

levels than the probabilities of profit intervals at lower

profit levels. Therefore, Table 23 exhibits the

probabilities of several profit intervals at upper profit

level for ASLP and MVLP in Case (1) and Case (2). A

comparison of the probabilities in Table 23 indicates that

the probability associated with each selected profit interval

is higher under ASLP than MVLP for Case (1). However, in

Case (2) (the case of low dependence) this consistent

result is not present for the selected profit intervals.

These results suggest that high or low dependence in the

variation of technological coefficients may be an important

factor in the decision of whether to use ASLP or MVLP in a

production planning problem of a small-sized firm.








Table 23

The Probabilities of Selected Profit Intervals
under MVLP and ASLP for Case (1) and Case (2)


The Comparison of ASLP and MVLP in Case (3) and Case (4)

The cumulative probability distributions of ASLP and

MVLP in cases (3) and (4) have been shown in Tables 14 and

20. Those two tables are combined andexhibited in Table 24.


Probability of Selected Profit Intervals
Profit Case (1) Case (2)
Interval a = 8 = .9 a = 8= .1
MVLP ASLP MVLP ASLP

$18,000-$34,000 .60 .61 .57 .55
$18,000-$36,000 .61 .65 .59 .59
$20,000-$36,000 .50 .56 .48 .50
$22,000-$34,000 .39 .44 .35 .37
$22,000-$36,000 .40 .48 .37 .41








Table 24

Cumulative Probability Distributions
on Profits (M) Under MVLP and ASLP
for Case (3) and Case (4)

P (9 < K)


Case (3) a = = .9 Case (4) a = = .1
K
MVLP ASLP MVLP ASLP

24,000 .0170 .0162
27,000 .0491 .0473
30,000 .0937 .0912 -
33,000 .1490 .1458 0.0334 0.0271
36,000 .2128 .2094 0.1194 0.0973
39,000 .2833 .2799 0.2558 0.2198
42,000 .3588 .3556 0.4174 0.3789
45,000 .4375 .4346 0.5752 0.5445
48,000 .5176 .5153 0.7093 0.6902
51,000 .5975 .5957 0.8123 0.8030
54,000 .6753 .6741 0.8853 0.8821
57,000 .7494 .7487 0.9338 0.9336
60,000 .8181 .8176 0.9645 0.9652
63,000 .8793 .8792 0.9831 0.9837
66,000 .9314 .9314 0.9939 0.9942
69,000 .9724 .9723 1.0000 1.0000
72,000 1.0000 1.0000 -



The information in Table 24 provides the basis for

deriving the probabilities of selected profit intervals.

In Table 25 the probabilities associated with selected

upper profit levels for ASLP and MVLP in Case (3) and

Case (4) are displayed. We can see from Table 25 that

the probabilities produced by ASLP are greater than those

produced by MVLP for all selected intervals for both cases.

This means that the ASLP rather than the MVLP provides

more appropriate solutions to the given production plan-

ring problems. However, it should be noted that the dif-








ference between the ASLP and MVLP, profit interval probabi-

lities is not great. Table 23 also shows that the differ-

ence of the probabilities between ASLP and MVLP in Case

(4) are more significant than those in Case (3). This

result suggests that ASLP does better in Case (4) (the case

of low dependence among technological coefficients). This

finding is different from the findings for the small sized

firm where it was found that ASLP was comparatively better

for the case of high dependence. This suggests that the

effect of dependence may not be the same in large and

small firms. Furthermore, perhaps a characteristic of the


Table 25

The Probabilities of Selected Profit Intervals
under MVLP and ASLP for Case (3) and Case (4)


Probability of Selected Profit Intervals
Profit Case (3) Case (4)
Interval a = 8 = .9 a = = .1
MVLP ASLP MVLP ASLP

$30,000- $60,000 0.7244 0.7264 0.9645 0.9652
$30,000- $66,000 0.8377 0.8402 0.9939 0.9942
$33,000-$66,000 0.7824 0.7856 0.9605 0.9671
$36,000- $60,000 0.6053 0.6082 0.8451 0.8679
$42,000- $66,000 0.5726 0.5758 0.5765 0.6153
$42,000- $69,000 .0.6136 0.6168 0.5826 0.6211



random variables besides their covariation (such as their

individual variance) may be important in the decision to use

MVLP or ASLP. This issue needs further research.












CHAPTER VI


SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS


Summary and Conclusions

A literature search and review was conducted to iden-

tify the characteristics and applicability of several

stochastic programming models to a short-run production

planning problem. As a result, ASLP was found to be the

model best fitted to a short-run production planning prob-

lem. Next an investigation of the problems involved in

the applying ASLP to a firm's short-run production planning

problem was conducted in order to determine whether the

technique is feasible. Finally, a comparison of ASLP and

MVLP was made in order to determine whether and under what

circumstances ASLP provides production plans which are

preferable to MVLP production plans.

In Chapter three, it was found that the computational

burden of ASLP increases geometrically due to the introduc-

tion of U matrices into the solution process. However, in

that chapter it was shown that this computational difficulty

can be overcome through the concept of dominance. Two do-

minance rules were suggested and applied in Chapter three.

First, a matrix U will be dominated by another matrix U

whenever the latter produce a greater profit than the former









for every set of technological coefficients in the sample.

Second, by assuming a minimum desired expected profit, the

number of profit frequency distributions necessary for con-

sideration is decreased to a manageable size. These two

dominance rules reduced the computational burden of ASLP

for the planning problems considered in Chapter three.

In Chapter four, ASLP was applied to a larger short-run

production planning problem. Although the application of

the first dominance rule suggested in Chapter three resulted

in a great reduction of the computational burden for the

larger problem, the number of U matrices remaining for con-

sideration was still very large. Therefore, another domi-

nance rule was developed to eliminate additional U matrices

from consideration. By introducing minimum output

constraints into the problem, the number of possible u..

values in a matrix U was decreased and the number of

matrices U that needed to be considered was significantly

reduced. The application of this latter dominance rule

reduced the computational burden of ASLP in the larger prob-

lem to a reasonable level. We have noted that without con-

sideration of this latter dominance rule, the application

of ASLP to the larger production planning problems would

have been very expensive.

The comparison of the ASLP and MVLP solutions was made

in Chapter five. The two variables which were employed in

the comparison were the amount of dependence among the

technological coefficients and the size of the matrix A.









The comparison between ASLP and MVLP was made by comparing

the probabilities of various profit intervals produced by

the two models. The results of these comparisons are sum-

marized as follows:

1. In a production planning problem with 3x3 matrix A,
ASLP does better in the case where the amount of
dependence among the technological coefficients is
"high."

2. In a production planning problem with 6x6 matrix A,
ASLP does better in a case where the amount of
dependence among the technological coefficients is
"low."

In conclusion, the ASLP approach considers the un-

certainty of the planning process by transforming the random

variation of the model parameters into random variation of

the objective function. The assumptions of and the view-

point taken by ASLP were found to fit very well with the

environment of many production planning problems. By

combining ASLP with a curve-fitting method, a probability

density function on planning period profit was derived.

This probability density function on profits was used to

determine the probabilities of various profit intervals for

the planning period. The conclusion of this study is that

ASLP is a powerful and effective planning aid to management.

But the choice between ASLP and MVLP to a short-run produc-

tion planning problem is dependent upon the amount of

dependence among the technological coefficients, size of

matrix A, and other factors to be determined in future

research work.








Recommendations for Future Research

This research provides evidence that ASLP has the

potential for being a powerful and effective planning aid

to management. Future research could be done in three di-

rections. In the first direction, further investigation of

the dominance concept would aid the application of ASLP

problems of a large size. The major problem in the applica-

tion of ASLP is caused by the introduction of matrices U in

the solution process. This study utilizes three dominance

rules and there may be more which would improve the results.

The second direction of the future research could be

a further investigation of the effect that dependence of

technological coefficients and the size of matrix A has on

the solutions derived by ASLP and MVLP. The conclusions

derived from this research have been based upon two dif-

ferent levels of dependence with matrix A and two differ-

ent sizes of matrix A. More empirical research is needed

before a general conclusion on how to choose one approach

over the other can be drawn.

The third direction of the future research could be

extended to develop a stochastic control model based upon

the ASLP approach. Planning is only one of the responsi-

bilities of .management. Management's function is also to

control. Therefore, a model for the latter purposes is

needed to aid management in the control of the production

plans. It is conceivable that a control model based upon

ASLP could be developed to increase the efficiency of man-

agement in production scheduling problems.































APPENDICES



























APPENDIX I


This appendix contains tables referred to in the body

of the dissertation.