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Decisions Under Risk Programming with Economic Applications José Hugo Portillo-Campbell Iowa State University

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Decisions Under Risk Programming with Economic Applications José Hugo Portillo-Campbell Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1969 Decisions under risk programming with economic applications José Hugo Portillo-Campbell Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Economics Commons Recommended Citation Portillo-Campbell, José Hugo, "Decisions under risk programming with economic applications " (1969). Retrospective Theses and Dissertations. 4141. https://lib.dr.iastate.edu/rtd/4141 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 70-13,521 PORTILLO-CAMPBELL, José Hugo, 1941- DECISIONS UNDER RISK PROGRAMMING WITH ECONOMIC APPLICATIONS. Iowa State University, Ph.D., 1969 Economics, general University Microfilms, Inc., Ann Arbor, Michigan (£) Copyright by JOSÉ HUGO PORTILLO-CAMPBELL 1970 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED DECISIONS UNDER RISK PROGRAMMING WITH ECONOMIC APPLICATIONS by José Hugo Portillo-Campbell A Dissertation Submitted to the Graduate Faculty in Partial Fulfillmen The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Economics Approved: Signature was redacted for privacy. In Charge ot major worK Signature was redacted for privacy. Head ot Major Department Signature was redacted for privacy. Iowa State University Ames, Iowa 1969 PLEASE NOTE: Some pag--s have small and indistinct type. Filmed as received. University Microfilms ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. REVIEW OF LITERATURE IN RISK PROGRAMMING 3 2.1. Stochastic Linear Programming 5 2.2. Programming under Uncertainty 14 2.3. Chance-Constrained Programming 17 2.4. Reliability Programming 39 3. OPTIMIZATION TECHNIQUES 46 3.1. Sequential Unconstrained Minimization Techniques 47 3.2. Geometric Programming 51 3.3. Generalized Polynomial Programming 55 4. ECONOMIC APPLICATIONS 59 4.1. Production Planning 59 4.1.1. A reliability model 62 4.1.2. Comparison with chance constrained programming 89 4.2. Application of Multi-Period Investment under Uncertainty: Implication of Decision Rules 97 4.2.1. Reliability approach 112 4.2.2. Sample distribution approach 126 5. APPLIED GEOMETRIC PROGRAMMING 136 5.1. A Simple Stochastic Production Model: an Illustration 136 i 5.2. An Aggregative Model of Economic Growth: a Second Illustration 141 iii Page 6. SUMMARY AND GENERALIZATIONS 158 7. BIBLIOGRAPHY 161 8. ACKNOWLEDGMENTS 174 9. APPENDIX A: PRODUCTION PLANNING - A COMPUTER PROGRAM 176 10. APPENDIX B: A NONLINEAR PROGRAMMING SOLUTION 206 11. APPENDIX C: GEOMETRIC PROGRAMMING, COMPUTER OUTPUT 223 1 1. INTRODUCTION Mathematical programming (145, 61) is concerned with the problem of maximizing or minimizing a function of variables that are restricted by a number of constraints. Interest in this problem arose in economics and management sciences, where it was realized that many problems of optimum alloca­ tion of scarce resources could be formulated mathematically as programming problems. The introduction of large light- spaced electronic computers, moreover, made it possible in principle to obtain numerical solutions, provided efficient mathematical methods and computational techniques could be developed. The concept of risk has played a very significant role in the theory of production, resource allocation, and the theory of statistical decisions. This is used here for analyzing the implications of a certain type of probabilistic program­ ming models in a linear and non-linear framework. The plan of the discussion covering this topic will be as follows. In Chapter 2 a general survey of risk programming is presented; the various methods of risk programming discussed in those sections are limited mostly to probabilistic linear programming of static variety, under simplifying assumptions regarding the statistical distributions of the parameters of the problems which are (A,b,c). This is followed by Chapter 2 3 which presents the optimization techniques used in this thesis in order to deal with nonlinear deterministic forms produced by the risk programming problems. Here we are using the most powerful numerical techniques which enable us to solve problems that were impossible to solve in the past. Chapter 4 examined an empirical model of production planning and the emphasis here is on evaluating a reliability level for every period of production planning and comparing with chance constrained programming. We also considered an investment situation, where an investor wished to select a portfolio; this original model first formulated by Naslund (90, 89) is solved here in the general nonlinear fashion, extended to the nonlinear reliability programming case and analyzed in its sample distribution aspects. In Chapter 5 we indicated two applications of geometric programming to economic models, in the first case we use generalized poly­ nomial programming applied to the simple stochastic pro­ duction model, in the second case we applied geometric programming to the posynomials arising in Uzawa's model of economic growth (135). Finally, a broad summary of all principal results and icjeas for future research is presented in Chapter 6. 3 2. REVIEW OF LITERATURE IN RISK PROGRAMMING Since the origin of the species, men have been making decisions, and experts have been telling them how they make, or should make decisions. Von Neumann and Morgenstern (91) developed a theory of maximizing the expected utility. In order for their results to be valid, however, their assump­ tion that rational individuals are choosing the right utility function must hold true. The fundamental problem of production is the optimum allocation of scarce resources between alternative ways of achieving an objective. It can be seen that the objective may be the maximization of the firm's profit or the minimi­ zation of costs. Cases exist, however, in which besides profit maximization or cost minimization, the objectives include risk minimization. If the decision-maker is willing to sacrifice profit in exchange for security, the result depends on his behavior. The difference between uncertainty and risk must be ^ ^ t pointed out here. Each term has had distinct meaning in different parts of economic literature. The term "risk" is characterized in a model in which the entire probability distribution of the outcomes has formally been taken into account, whether the character of that distribution is c^m- sidered subjective or objective. The term "uncertainty" is 4 applied to models in which the above stated conditions are not the case. It is very important to have a realistic theory explaining how individuals choose among alternate courses of action when the consequences of their actions are not fully known to them. A survey of the literature of approaches to the theory of choices in risk taking situations has been given by Arrow (3). The probability theory represents the sustained efforts of mathematicians and philosophers to provide a rational basis on which expectations may be derived from past events. Roy (102) stated that there are major objections when one attempts to maximize expected gain or profit. The ordinary man has to consider the possible outcomes of a given course of action on one occasion only, and the average or expected outcome, if this conduct were repeated a large number of times under similar conditions, is irrelevant. Also, the well-known phenomenon of the diversification of resources among a wide range of project or investment situations is not explained. Mathematical programming under risk has been developed during the last ten years and there exist many published articles in the field. The research, to date, can be divided in three major areas (88) namely: a) stochastic linear programming b) linear programming under uncertainty 5 c) chance constrained programming All methods start from the linear programming formulation namely minimizing or maximizing a linear function subject to linear constraints. Stochastic linear programming is mainly concerned with studying the statistical distribution of max z. This work has followed two lines namely the so called "passive" and "active" approaches. Programming under uncertainty partitions the problem into two or more stages. First the decision maker selects an initial decision, then the random effects occur. Chance-constrained programming allows for constraint violations a certain proportion of the time. Reliability programming is an extension of chance constrained programming where the probability levels are not preassigned, but optimized in some sense. 2.1. Stochastic Linear Programming The ordinary linear programming problem can be formulated as follows (35, 57, 60) Max c'x [2.1.1] subject to Ax < b [2.1.2] x > 0 6 where c is a row vector with n elements X is a column vector with n elements A is a m by n matrix b is a column vector with m elements. The problem is to find values for the elements of x which maximize [2.1.1] and satisfy [2.1.2]. Risk is intro­ duced to the problem when either some elements of c, A or b or any combination of them include random elements. Stochastic linear programming which was first suggested by Tintner (130) is primarily concerned with the statistical distribution of max z = c'x in [2.1.1]. It is assumed that a multivariate probability distribution for the ele­ ments of A, b, c in [2.1.1] and [2.1.2] is known. We may be represented, for instance, as Prob (A,b,c) [2.1.3] where Prob refers to the probability of simultaneous occurrence of specified values of the matrix A and the vectors b and c. 1 Thus, given a Prob function as in [2.1.3] one may ask how max c'x will be distributed. There are three types of distribution problems; the passive approach, active approach, terme# distribution problems and expected values problems by 7 Vajda (136), and other distributional approaches. The passive approach assumes that for each possible configuration of the random variables the optimal activities, are selected.
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