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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 TwoStage Linear Programming ..... 7 ChanceConstrained Programming ... 9 Stochastic Linear Programming .... 11 A Comparison of the Stochastic Program ming Techniques ............... 14 III. AN APPLICATION OF ASLP TO THE SHORTRUN PRODUCTION PLANNING PROBLEM OF A SMALLSIZED 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 SHORTRUN PRODUCTION PLANNING PROBLEM OF A MEDIUMSIZED 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 EachResource (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 CoChairman: 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 shortrun 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. 512. Thomas H. Williams and Charles H. Griffin, The Mathematical Dimension of Accountancy (Cincinnati, Ohio: SouthWestern Publishing Co., 1964), pp. 102120. Charles T. Horngren, Cost Account ing: A Managerial Emphasis (Englewood, Cliffs, N.J.: PrenticeHall, Inc., 1972), pp. 899904. 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. 197206. A. Charnes and W.W. Cooper, "ChanceConstrained Programming," Management Science, Vol. 6 (1959), pp. 7376. 3Van DePanne, C. and W. Popp, "Minimum Cost Cattle Feed under Probabilistic Protein Constraints," Management Science, Vol. 9 (1963), pp. 405430. K.D. Cocks, "Discrete Stochastic Programming," Manage ment Science, Vol. 15 (1968), pp. 7279. W.H. Evers, "A New Model for Stochastic Linear Pro gramming," Management Science, Vol. 13 (1967), pp. 680693. 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 shortrun 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 shortrun 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 shortrun pro duction planning problem. This search and review revealed that active stochastic linear program (ASLP) is best fitted to the shortrun 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 shortrun 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 nelement row vector of contribution margins X is an nelement column vector of product quantities b is an melement 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 shortrun 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 shortrun 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: Twostage linear programming, Chanceconstrained 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. TwoStage Linear Programming Twostage 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. 197206. 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 2melement column vector of deviations, d is a 2melement 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 twostage 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 shortrun production planning problems. Further and 3H.M. Wagner, Principles of Operations Research,with Ap plications to Managerial Decisions (Englewood Cliffs, N.J.: PrenticeHall, Inc., 1969), p. 659. perhaps more important is the fact that the structure of the twostage 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 Chanceconstrained 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 chanceconstrained 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 chanceconstrained formulation and the twostage formulation is that in the latter case all constraints must hold for A. Charnes and W.W. Cooper, "ChanceConstrained Pro gramming," Management Science, Vol. 6 (1959), pp. 7376. 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 Chanceconstrained programming has been found to be best suited to problems in which only the elements of the 5Wagner, p. 669. Ibid., pp. 669670. 7.D. Cocks, "Discrete Stochastic Programming," Manage ment Science, Vol. 15 (1968), p. 73. M.J.L. Kirby, "The Current State of ChanceConstrained 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 chanceconstrained programming become nonlinear and a problem of practical size cannot be solved. The argument advanced in this study is that in many shortrun 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, chanceconstrained programming is not applicable to many shortrun 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. 197228. 10Ibid., pp. 197228. G. Tintner, "A Note on Stochastic Linear Programming," Econometrica, Vol. 28 (1960), pp. 490495. 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 shortrun 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 nelement 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 (GLk) linear programming problems, where G is the number of matrices U to be considered. Solving (GLk) 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 shortrun 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 longrun and not shortrun production planning problems. We have also argued that the active approach fits well with the characteristics of many shortrun 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 shortrun production planning problem are summarized below. First, the assumptions and viewpoints taken by the active approach fit very well with the environment of many shortrun production planning problems. The other two approaches are not as suitable for shortrun 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 twostage programming visavis chanceconstrained 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: HoldenDay, 1967), p. 531. CHAPTER III AN APPLICATION OF ASLP TO THE SHORTRUN PRODUCTION PLANNING PROBLEM OF A SMALLSIZED FIRM Introduction The advantages and the disadvantages of applying ASLP to a shortrun 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 shortrun 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 shortrun 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 shortrun 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. 262276. 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. 205210. 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. Thirtysix 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 forV.., 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 thirtysix 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 thirtysix 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 stepwise 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 cutoff figure is somewhat arbitrary but has been based upon the expected profit that results from a meanvalue linear programming so lution to the problem (see Chapter five). The utilization of this cutoff 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. S It) LI n in N cV N ra ij r rP 4 rI 4JLA tLn IA H H M 114 __ _ CCDH 00000 ,DD. ,'. LAH\JV ) (D "S 0 x o.r V4J) to A rdQ U HH I40 4 0) a) 44dV 0 'r $4 >if1 rq 44 '1 V)( 0 A4 rI 4J r00 %.0 w %D t.0ko 0 C'CO %% i V) N IV o Ln Cq~v 00000 O4 0 4 0 a .0 *( (D 4J 8 r8 C r 4 \\\\ \\ 0 0 C 0 0 0 Ln r4 4:r c, m rl i t c 4 v m OHQ IAHDCOIDIID'4mOCD ~ 00 NN.NN1 00000 \ \\ 4) 4t 8 I kD kD w %D t %D kD %D tD =1 HOO0 \ N 000 Hr ~ ~ ~ \ \ OO 0 00 0''.C 0 a H a 0 0 0 0 'N N,0 0 0N N N%\ .% U4 C> ri % C) a o a CD 's, , *,,\ LnrI NmrIL y P4 C)oa kD to o %D t ", NINNI'. l 0 0 II J4 0  0 o 3 4 0 Un S:rl r^a a4 : 5 :: Q m f a03 a are$ ( O l, l rl I O 1 m 1 INl vr4 lHMN mC HNri N 4l , H UI N N m 10 ~~ mO HO N f NOL D O I IV H NM CMrIC o w wI wL w % %iC to SrI RWH rfi N N rn 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 hasbeen 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 decisionmaking 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 decisionmaker 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 (EV) combinations from which a particular EV 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. 324327. 4David B. Hertz, "Risk Analysis in Capital Investment," Harvard Business Review, Vol. 42 (JanuaryFebruary, 1964), pp. 95106. 5Ibid, pp. 95106. Harry Markowitz, "Portfolio Selection," Journal of Finance, Vol. 7 (1952), pp. 7191. 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 decisionmaking 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 curvefitting method the Pearson system  which estimates the theoretical probability distribution underlying a given frequency distribution. The Pearson curvefitting 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 curvefitting 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 CurveFitting Technique The purpose of this section is to demonstrate the ap plication of the Pearson curvefitting system.7 Basically, the curvefitting 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. 35110. Ibid., p. 21. 44 3 TrMO N o4rin i ON o0 0i00 %DMn C'4rIH n 0 s .. 0i tt4 4HH . wO iN N L L(PA co 1D w mw mf m m m %D E0 r l r 'l In C 0 aa 00000000000000 .0 m rl m m r l a% ..q m tn r m o H SHMMr4HrCNNNMmm 00000000000000 O 0 0000000000000 0C4 *r 00000000000000 0 m 0 w M CO W4, .H 0 i .4 W0 00 0 U14 4r 1 0 *rl CO >1 0 Sal 3 U O1 P Cd fo rt 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(42381) (2823816) = 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 (1x2/a2) origin at mode = mean where a < x < a and 5 8 9 m 0. 2 = 0.8602 2(38)  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 (1x2/66.8753)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 (1x2/66.8753)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 KolmogorovSmirnov onesample test is a powerful 12 test of goodness of fit when samples are small. It treats individual observations separately and thus, unlike the Chisquare 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 KolmogorovSmirnov test is believed to be more applicable than the Chisquare test. The Kolmogorov 11 1Ibid., pp. 165173. 12Sidney Siegel, Nonparametric Statistics (New York: McGrawHill 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 KolmogorovSmirnov 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. 284290. 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 curvefitting 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 shortrun 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 shortrun 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 SHORTRUN PRODUCTION PLANNING PROBLEM OF A MEDIUMSIZED 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 thirtysix 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 thirtysix 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 r. .l r. . on on Ln on in o Va lblbrbOb . .O tn Ln 0n c) 0 > w W t %DDO HHHHHH O D D . .l Ln n n or 0in iL 4 H1 H rH Hi H1 o OH 414 O 0 0n 0 H 1W I) > OH C) OH 4 0 *rl 40 02 Or HA E4m O 0 a 4) V 0oooomoaooo00aoo0momomo0tooo Um mmio m 0 m In 0 wiN r ruo NrNi in ewin Nr m n mom 0C i() ia c )rlo iC> C i)C ia 4= in c) c) rl ca a McS ra l uLn io it l it l 1 0  I  04 AP H M 4 mo o 000o a in b t oo aIn nooi i nu in n in Ln Ln Linin L 4 S 0 mommimonMoooomomocomocooo 3co o ococo00000 co0 0000000000000000 Do Doorrrwtr0 s H rl r H r 4 H rA r ci H r4 rI ri rI H H rq H v r4 Cl 0 Hl r4V l rL O n a a o o a n : : aao c: o3 c) c) ( c) 0 a o a c 0 0 o 6 o . . 0 0 1I H I N I 0 Ulnoo0 3oi 30 ooooo0 30 3i on o Ln Ln Lo H r r rH rH * S S00oC00 0o HHHHHlr 00 00 r ***C*a** HHHHHHllr 0000)00 HHHHHHrlr H O0 r0 ll rl rH l riH aU ) ouLmn 4.) oo oo UW0 %Dm m 0 00 OD 00 00 O * r *c 0 4) 0 . o . 0 mrmm r % rl r r, r oC ooc oo M11 9Tv: The ASLP Solution of the Production Planning Problem A computer program was written that calculates the thirtysix 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 thirtysix 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 curvefitting 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 (1x2/86.26967)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 x36.5764e 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 onesample 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) andthe 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 rI rO H H 00 N 0 Hr HrIO 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 E4 C N N (i SD N O r (N N 0 U HOD Hq 0 o ri 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 rl 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 %0 11 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 . 1ito rl H r rur rHco %0 r1 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 rr 0 vo co co r' 0 o n C0 0 . cr~ vco Vn 431 l r1 (HI ON O H N l rl rl H N rH CN C4 N CM CM 04 0 C4 in om cOm on 0r Ln t I' 4r C4 . Hr rl il 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 r4 1, n r4t I H 0 v4 O r) 0 rq I 0 4O tu rA (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 CurveFitting and Cumulative Probability Distribution of Profits for MVLP Solutions Probability density functions have been fitted to the thirtysix 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 onesample 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) (1x/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 e0.66x where (2) x > 7.415, origin at the mode, profits = 18,137 + (41,693/20.8465) x. Y = 2.8523(1x2/85.57942)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 KolmogorovSmirnov 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 smallsized 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 shortrun production planning problem. As a result, ASLP was found to be the model best fitted to a shortrun production planning prob lem. Next an investigation of the problems involved in the applying ASLP to a firm's shortrun 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 shortrun 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 curvefitting 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 shortrun 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. 