Multiple Criteria Decision Making and its Applications to Economic Problems PDF
Preview Multiple Criteria Decision Making and its Applications to Economic Problems
MULTIPLE CRITERIA DECISION MAKING AND ITS APPLICA TIONS TO ECONOMIC PROBLEMS MUL TIPLE CRITERIA DECISION MAKING AND ITS APPLICATIONS TO ECONOMIC PROBLEMS by ENRIQUE BALLESTERO and CARLOS ROMERO Technical University of Madrid Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data ISBN 978-1-4419-5053-6 ISBN 978-1-4757-2827-9 (eBook) DOI 10.1007/978-1-4757-2827-9 Printed on acid-free paper All Rights Reserved © 1998 Springer Science+Business Media New York Origina11y published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1s t edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, e1ectronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. PREFACE The importance of connecting Operational ResearchIManagement Sciences (ORIMS) methods to economic analysis has been upheld for many years. A pioneer work in this direction is Dorfinan, Samuelson & Solows' book Linear Programming & Economic Analysis which connects linear programming models to economic analysis in general and to Leontiefs input-output model, in particular. Another remarkable foregoing is the introduction of multi-objective optimisation into portfolio selection theory and capital asset pricing models through Markowitz's mean-variance frontier. Prior to these mathematical operational efforts, fmancial analysis could hardly be considered a quantitative science. Despite this endeavour most recent and powerful ORIMS approaches, such as Multiple Criteria Decision Making (MCDM), are seldom present in today's economic literature. Nevertheless, it is a common fact that any real world decision in economics has to take into account multiple and eonflieting eriteria, as has been extensively demonstrated in fields such as investment analysis, produetion planning, finanee, manpower planning, natural resourees management, and so on. In short, we fail to understand why the searce presenee of MCDM in economics. Standard approaches in eeonomics· are based on the optimisation of a decision makers' objeetives as a one-criterion function. At times, it is argued that eeonomie methodology stays away from the multi-criteria teehniques in order to measure preferences and optimise objeetives. However, one eould conclude that the standard paradigm in economics should remain impervious to new multi-criteria approaches. Such an attitude runs completely counter to all scientifie advances and praetieal economic applieations that the analysis intends to foster. Indeed, our intention is not aimed at ehanging the sound traditional paradigm in economies but seeks to demonstrate that MCDM ean reinforce it. If MCDM methodology is worthy for economists, we also think that the traditional economic thought is fruitful for the MCDM analysts themselves. Part One of the book introduces the MCDM approaches, which are most interesting to economists. This first part, comprising Chapters 1-5, is basically an overview ofMCDM methods that can most likely be used to address a wide range of economic problems. Our intention is that readers looking for an in depth discussion of multi-eriteria analysis ean grasp and beeome aequainted with the initial MCDM tools, language and definitions. vi Preface Part two, which comprises Chapters 6-8, focuses on the theoretical core of the book. Thus in Chapter 6, an economic meaning is given to several key concepts on MCDM, such as ideal point, distance function, etc. It illustrates how Compromise Programming (CP) can support the standard premise of utility optimisation in economics as well as how it is capable of approximating the standard utility optimum when the decision-makers' preferences are incompletely specified. Utility is one side of the economic problem, but production is the other side. Chapter 7 deals entirely with production analysis. The main characteristic throughout the Chapter refers to a standard joint production scenario, analysed from the point of view of MCDM schemes. Several general theorems on to shadow and market prices are expounded. One of these properties, which we call ''the three optima theorem", clearly shows that in a competitive economy, the best technological mix, the maximum profit point and the consumer' s utility optimum tend to coincide under weak assumptions. In our opinion, the three optima theorem can substantiate the principle of competitive markets such as those guaranteeing efficiency and welfare. Finally, Chapter 8 focuses on the utility specification problem in the n-arguments space within a risk aversion context. A link between Arrows' risk aversion coefficient and CP utility permits this task. The book is intended for postgraduate students and researchers in economics with an ORIMS orientation or in ORIMS with an economic orientation. In short, it attempts to fruitfully link economics and MCDM. The work reported in this study has evolved gradually through ten years of collaboration between the two authors. The publication of this book has meant both fears and joys. As the authors are not native English speakers, they fear that the following pages are not written in the best of styles but hope that readers are more interested in the rigour of the analysis than in the brilliance of the words. Both authors wish to acknowledge the beneficial intellectual influence of such leading figures in MCDM as: Abraham Charnes, William Cooper, James P. Ignizio, Ralph E. Steuer, Po-Lung Yu, Stanley Zionts, Milan Zeleny, among others. Comments raised by Francisco Amador, James P. Ignizio, Dylan F. Jones, Manuel A. Mor6n and Mehrdad Tamiz have been greatly appreciated. Finally, thanks are also given to Christine Mendez, who checked the English language and to Luis Dfaz-Balteiro for his suggestions and technical help and David Plä-Santamaria for the editing. This research was undertaken with the financial support of the Spanish "Comisi6n Interministerial de Ciencia y Tecnologfa (CICYT)" and the "Consejeria de Educaci6n y Cultura, Comunidad Aut6noma de Madrid". We would also like to thank the editors ofthe followingjournals for allowing us to draw upon our previous publications: European Journal o/Operational Research, Journal 0/ Multi-Criteria Decision Analysis, Journal 0/ the Operational Research Society, Lecture Notes in Economics and Mathematical Systems, Operations Research Letters and Theory and Decision. Madrid, May 1998 Enrique Ballestero and Carlos Romero CONTENTS Preface v Chapter 1. Multiple Criteria Decision Making: An Introduction 1-10 1. Traditional Paradigm for Decision Making: Comments and Criticisms 1 2. An Illustrative Example 3 3. Some Basic Definitions 5 4. Two Intermediate Concepts: Pareto Optimality and Trade-Offs amongst Criteria 7 5. Multiple Criteria Decision Making: A Historical Sketch 9 Chapter 2. Multiobjective Optimisation Methods 11-30 1. Basic Aspects 11 2. Techniques for the Generation ofthe Efficient Set 13 3. An Illustrative Example 15 4. Compromise Programming: Methodological Aspects 19 5. The Concept ofCompromise Set: Yu's Theorem 24 6. Two Economic Examples of Compromise Models 25 6.1 Equilibrium of a Monopolist 25 6.2 The" Leisure-Work" Dilemma 26 Appendix 29 Chapter 3. Satisficing MCDM Approaches: Goal Programming 31-49 1. Basic Aspects 31 2. Weighted Goal Programming (WGP) 33 3. Lexicographic Goal Programming (LGP) 34 4. The Sequential Method for Lexicographic Optimisation 36 5. Goal Programming Extensions 41 6. Some Critical Issues in Goal Programming 42 7. Two Economic Examples of GP Models 46 7.1 Satisficing Monopolist Equilibrium 46 7.2 A Satisficing Worker's Enterprise Equilibrium 48 Chapter 4. Multiattribute Utility Approaches 51-62 1. The Concept of Multiattribute Utility Function 51 2. Utility Decomposition: Preferential and Utility Independence Conditions 52 3. Determination of Multiattribute Utility Functions 55 4. A MAUT Application 56 5. A Final Reflection 61 Chapter 5. Miscellaneous Questions 63-75 1. Purpose 63 2. Some Comments on other MCDM Approaches 63 3. Links between Compromise Programming and Goal Programming 65 4. A Utility Interpretation of Compromise Programming and Goal Programming 68 5. Choosing a MCDM Technique: Some Considerations 73 viii Multiple Criteria Decision Making and its Applications to Economic Problems Chapter 6. A First Linkage: CP and Bi-Attribute Utility 77-101 1. Introduction 77 2. Utility Meaning ofthe Ideal Point 78 3. Preferences and the Compromise Choice 84 4. Economic Meaning of Approximation to the Ideal 87 5. ABounding Model for Standard Individuals: The Case ofthe "Average" Investor 89 6. An Example ofPortfolio Selection 93 7. The Case of a Decision-Maker with Particular Preferences 97 8. Some Conclusions and Comments 100 Chapter 7. Joint Production Shadow Prices and the Three Optima Theorem 103-123 1. Introduction 103 2. Shadow Prices: A General Theorem 104 3. A Car-Truck Illustrative Example of Shadow Prices 109 4. Three Crucial Optima for an Industry 110 5. Anchor Values and Market Prices: An Introductory but Restrictive Link 111 6. A Less Restrictive Approach to the Anchor Value-Market Price Link 113 7. Industry's Equilibrium and the Best-Compromise Solutions 114 8. The Three Optima Theorem 117 9. Extensions 121 10.Conclusions 122 Chapter 8. A Further Linkage: Multi-Attribute Utility in a Risk Aversion Context 125-154 1. Introduction. 125 2. Notation and Definitions 126 3. AReminder on Utility Functions with Separable Variables and their Standard Optimisation 127 4. The CP Distance Form as a Utility Function 129 5. A Case of Health Care Management 134 6. Searching for the Structure of the Utility Functions 138 7. The Structure of Uni-Dimensional Utility Functions 140 8. Main Assumption 141 9. Economic Meaning of Parameters 142 10. A More Extensive Approach to Utility and the Compromise Linkage 144 11. Specification and Optimisation 145 12. The Illustrative Work-Leisure Dilemma Again 147 13. Selecting a Car from Utility Characteristics: A Multiattribute Case 151 14. Conclusions 153 References 155-158 Index 159-160 1 MULTIPLE CRITERIA DECISION MAKING: AN INTRODUCTION 1. Traditional paradigm for declsion-making: comments and criticisms The basic traditional structure underlying any decision-making problem can be summarised as folIows. The existence of limited resources (understanding the term resource in a broad sense), generates the constraints ofthe problem. The value 'Ofthe decision variables satisfying the constraints define what is known as the feasible or attainable set. This set can either be continuous (infinite number of solutions) or discrete (finite number of solutions). Once the feasible set is established, the subsequent step consists in defining a criterion function which suitably reflects the preferences of decision maker by associating a number to each feasible solution. Finally, by resorting to more or less sophisticated mathematical techniques, the "best" or "optimal" solution is obtained. It should be pointed out that the initial phase of the aforementioned decisional paradigm requires purely technical information. In other words, only non-preferential information is necessary to determine the feasible set. The actual preferences of the decision-maker appear in the next phase, when the criterion function is established. In short, the first phase defines from technical information what is possible (feasible set), whereas the second phase defmes what is "best" from the decision-maker's preferential judgements. The intersection ofboth phases yields what is "best" among all possible choices, that is, the "optimum solution". This paradigm underpins any decision-making problem economic or not. Let us consider, for instance, the decision-making problem faced by a monopolist. The decision variables include the price and the amount of output to be sold. The feasible set -which limits the field of possible choices- is defined by the market demand function and possibly by two maximum bounds: one for the output (production capacity) an the other for the price (administrative constraint). The introduction of a criterion function such as profits allows you to order the feasible set. In this way, the optimum solution is obtained; that is, the feasible pair price-amount of output which provides the maximum profit for the company. In consumer theory, the set of feasible baskets is determined by the budget constraint. The utility criterion is then used to order the attainable baskets. The feasible basket of maximum utility or optimum solution is obtained by resorting to a Lagrangean optimisation. 2 Multiple Criteria Decision Making: An Introduction All decision-making problems addressed by mathematical programming approaches obey the same paradigm. In fact, under this context, feasible solutions are those that satisfy the constraints of the problem. These feasible solutions are ordered according to a criterion function called objective function. By resorting to more 0 less sophisticated mathematical techniques (e.g. the "Simplex" when the constraints and the objective function are linear) the "optimum solution" can be found. The above decisional paradigm has considerable logical soundness and underpins all traditional theories in decision analysis and in economics. In few words, the internal coherence of this paradigm is perfect. However from the standpoint of external coherence or empirical support, the theoretical framework displays certain weaknesses which may deviates its functioning from actual' decision making processes. In fact, in many real life situations, the decision-makers do not order the feasible solutions according to a single criterion but taken into consideration several criteria which reflect their own preferences. Thus, when we look for a new car or a new flat, the feasible set (all the cars and flats within our budgetary possibilities) is assessed according to different criteria such as power, mileage, comfort, etc, in the case of cars and number of rooms, neighbourhood, etc, in the case of flats. Decision-makers choose the "best" among all possibilities available. However, the concept of "best" is ambiguous and in many situations involve more than one criterion of choice. As Zeleny(1982), one of the leading figures in the multicriteria movement states: "multiple objectives (criteria) are all around us". Table 1.1 illustrates some problems already addressed within a multicriteria framework to emphasise the wide range of decisional problems that can be more suitably formulated by recognising the existence of several criteria. Another empirical problem related to this paradigm is that it considers the constraints defming the feasible set as rigid bonds which cannot be violated if infeasible solutions are to be avoided. This consideration is not in general realistic, since a certain relaxation of the constraints would not seriously affect the real framework where the problem is modelIed but could markedly improve the performance of certain criteria. The above examples and considerations clearly show that most decision makers do not take their decisions based on a single criterion but rather on several criteria. Moreover, the feasible sets are not so rigid as the traditional paradigm assumes. In conclusion, the decision makers whose rationality is weIl explained by this paradigm, are in general abstract entities whose behaviour has little to do with the behaviour followed by real decision makers made out of bone and flesh. Consequently, researchers from different disciplines -especially those from Management Science/Operational Research fields- have developed alternative decisional paradigms in the last thirty years in order to accommodate the real decision making processes more accurately. Multiple Criteria Decision Making anti its Applications to Economic Problems 3 Table 1.1. Some Illustrative Examples of Multi-Criteria Problems in Management, Engineering, Finance and Economics Livestock Ration Financial Planning Design 0/a n Extended Formulation Octagonal Ring -Cost ofthe ration -Finn's expansion -Sensitivity -Nutritional imbalance -Dividends -Rigidity -Bulk ofthe ration -Solvency FisIreries Management Water Reservoir Designing River Basin Planning -Cost oftishing -Risk offlooding -National income benefits -Employment -Energy production -Equity -Sustainable yield -Water supply ForestManagement Farm Management BehaviQur 0/B ig Firms -Timber production -Gross margin -Profits -Recreation -Risk -Sales revenue -Hunting -Seasonallabour -Stock prices -Wildlife -Environmental impacts (nitrates, pesticides, etc.) Port/oZio Selection Labour Supply Functions Capital Budgeting -Returns -Income -Net present value -Risk -Leisure -Internal rate ofreturn -Annual operating expenses 2. An illustrative example Let us insist upon these ideas by resorting to a simple example fonnulated within the classic monopolistic equilibrium problem proposed by Cournot in 1838. The demand and cost functions faced by the monopolist are, respectively: Q= 10-P C= 1+3Q where P is the market price, Q the amount of output demanded and C the production costs. Let us assume that the maximum price authorised by the govemment is 8 m.u. and the maximum production capacity ofthe different factories in the company is 9 units. Under this context, the monopolist equilibrium (i.e., maximum profit point) is obtained by solving the following optimisation problem: Max B = PQ-3Q-I subject to: Q~IO-P O~P~8 O~Q~9 By elementary differential calculus, the following optimum solution is obtained: P = 6.5 and Q = 3.5. The maximum profit (Cournot point) corresponding to this solution is B" = 11.5. This problem has been solved within the traditional decision
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