Multicriteria Decision Making PDF
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INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 211 MULTICRITERIA DECISION MAKING EDITED BY G. LEITMANN UNIVERSITY OF C.\LlfORNIA ,BERKELEY A. MARZOLLO UNIVERSITY OF TRIESTE SPRINGER-VERLAG WIEN GMBH This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1975 by Springer-VerlagWien Originally published by Springer-Verlag Wien·New York in 1975 ISBN 978-3-211-81340-9 ISBN 978-3-7091-2438-3 (eBook) DOI 10.1007/978-3-7091-2438-3 LIST OF CONTRIBUTORS A. Blaquiere Laboratoire d'Automatique Theorique, Universite de Paris VI, Paris, France G. Castellani Department of Mathematics, Ca' Foscari University, Venice, Italy G. Leitmann Mechanical Engineering Department, University of California at Berkeley, California, USA Y. Medanic Mihailo Pupin Institute, Belgrade, Yugoslavia A. Marzollo Electrical Engineering Department, University of Trieste and International Centre for Mechanical Sciences, Udine, Italy W. Stadler Mechanical Engineering Department, University of California at Berkeley, California, USA W. Ukovich Electrical Engineering Department, University of Trieste, Italy M. Volpato Department of Mathematics, Ca' Foscari University, Venice, Italy P.L. Yu Graduate School of Business, University of Texas, Austin, Texas, USA. CONTENTS List of Contributors . 1 Preface. . . . . . 5 Cooperative and Non-cooperative Differential Games, G. Leitmann. . . . . . . . . . . . . 7 Vector Valued Optimization in Multi-player Quantitative Games, A. Blaqui~re. . . . . . . . . . . . . . . . . 33 Minimax Pareto Optimal Solutions with Application to Linear Quadratic Problems, J. Medanic. . . . . . . . . . . . . . . . . . . . . . . 55 Preference Optimality and Applications of Pareto Optimality, W. Stadler. . . . . . . . . . . . . . 125 Domination Structures and Non-dominated Solutions, P.L. Yu. ............. . 227 On Some Broad Classes of Vector Optimal Decisions and their Characterization A. Marzollo, W. Ukovich.. . . . . . . . . . . . . . . . . 281 Estimating the Common Cost of a Good When the Local Costs are Known in the Countries of a Community, M. Volpato. . . . . . . . . . . . . . . . . . . . . . . 325 Explicit Solution for a Class of Allocation Problems, G. Castellani. 351 Erratum of W. Stadler.. 3R7 PREFACE A considerable amount of research has been devoted recently to Multicriteria Decision Making, stimulated by the vast number of real problems, for example in industrial, urban and agricultural economics, in the social sciences, and in the design of complex engineering systems, where many decision makers are present or many, possibly conflicting objectives should be taken into account in order to reach some form of optimality. A rough division into two classes may be made among the approaches to Multicriteria Decision Making problems. The first one deals mainly with the empirical determination of preference structures in some specific problems, and seeks methods for their meaningful aggregation in order to arrive, often by ad hoc and iterative procedures, at practically reasonable solutions. The lines of thought which are followed and the used methods may be looked upon as modern developments of operations research. The second one, predominantly treated by researches whose background is often rooted in systems and control theory, or in mathematical programming and in its applications, seems more directed toward general and rigorous formulations in order to reduce Multicriteria Decision Problems conceptually to clearly defined classes of optimization problems for which definite solutions algorithms are sought. The contributions to the present volume follow mainly the latter line of thought, although references and comparisons are made to other, sometimes non-empirical methods, for example by P. L. Yu and, in general, algorithmic solutions are proposed to specific problems as a result of the conceptual methods used. The reader is introduced to the large class of multicriteria, multiagent problems which may be treated in the framework of game theory, both for static and dynamical systems, in the first two contributions by G. Leitman and A. Blaqui!ne. The formulation of the latter is so general as to encompass as specific cases the great majority of multi-objective, multiplayer problems that one may think of in cooperative, non-cooperative or mixed situations. J. Medanic gives an exhaustive solution, both in deterministic and stochastic cases, to the optimal regulator problem with vector valued quadratic performance, and applies the developed concepts to a multiplant cooperative control problem. The paper by W. Stadler imbeds both the static and dynamical vector optimization problem in the framework of preference optimality by borrowing techniques which have been developed mainly by mathematical economists, and so is able to give interesting sufficient and necessary conditions for optimality; vector optimization concepts are then applied to the design of minimally disturbing measuring devices for optimally controlled systems, and for optimal structural design in mechanics. A complete treatment of Domination Structures and Non-Dominated Solutions with an example of application to stock market behaviour is given by P.L. Yu. A Marzollo and W. Ukovich discuss some basic principles underlying the concepts of Vector Optimality and then give precise conditions, using the weakest hypotheses on the involved functions, for the characterization of "weakly" "ordinarily" and "strictly" Non Dominated Decisions, in the global, local and "differential" version. Specific economic relevance is stressed in the contribution of M. Volpato, who deals with the 6 Preface optimal choice for the relative amounts of a specific good to be produced in various countries of a community in order to give the maximal community yield from the residual resources. Unlike the classical theory on the subject, non-linear conversion prices from one product to another are also considered. The problem may be solved explicitely for a rather general class of functions by using non·linear and dynamic programming techniques which are developed in the following contribution by G. Castellani. It is shown that in this framework the optimal production policy for the community is also economically optimal for each individual country. This sketch of the contents of this volume is by no means exhaustive of the various phylosophical approaches to Multicriteria Optimization contained therein, nor of the techniques which, as a consequence, are suggested for the solution of many varied problems. We express our hope that the volume as a whole will stimulate the reader to giving further thought to the conceptual and mathematical challenges offered by the present extension of optimality theory and provide him with some useful methods for solving problems in which different points of view are to be considered, or, to recall the title of this volume, to solve "Multicriteria Decision" problems. George leitmann Angelo Marzollo COOPERATIVE AND NON-COOPERATIVE DIFFERENTIAL GAMES G. Leitmann Department of Mechanical Engineering University of California, Berkeley ABSTRACT. Many player differential games are discussed for a cooperative mood of play in the sense of Pareto, and for a non cooperative one, in the sense of Nash. In the cooperative case, the results are equally applicable to the situation of a single decision maker with multi-criteria. Necessary as well as sufficient conditions for optimal play are considered. Some examples are presented to illustrate the theory. 8 G. Leitmann 1. INTRODUCTION 1.1 Problem Statement We consider games which involve a number of players. The rules of the game assign to each player a cost fUnction of all the player's decisions and the sets from which these decisions can be selected. Let there be N players. Let J.(•) and D. be the cost fUnction and l l decision set, respectively, for player i. Then. J.(•) : D-+ R1 i = 1,2, ... ,N (1.1) l N where D C 1T D .• i=l l Loosely speaking, each player wishes to attain the smallest possible cost to himself. Thus, if there is a duE D such that for all i E {1,2, ... ,N} Yd ED (1.2) then du is certainly a desirable decision N-tuple (joint decision). Un- fortunately, such a utopian (absolutely cooperative) decision rarely exists (e.g., Refs. 1- 3) and the players are faced with a dilemma: What mood of play should they adopt, that is, how should an "optimal" decision be defined? Here we consider only two moods of play, one cooperative and the other non-cooperative, in the sense of Pareto4 and Nash,5 respec- tively. 1.2 Cooperative Play If the players decide to "cooperate" in making their individual Differential Games 9 decisions, they can do so by adopting a joint decision as suggested by Pareto. There is more than one way of defining Pareto-optimality. One of these is given by N Definition 1.1. A decision N-tuple d0 ED, DC TI Di, is Pareto i=l optimal iff for every dE D either Yi E {1,2, .•. ,N} or .there is at least one i E {1,2, ••• ,N} such that J.(d) > J.(d0) ~ ~ In this definition of Pareto-optimality cooperation is embodied in a statement such as "I am willing to forego a gain (a decrease in my cost) if it is to be at the expense of one of the other players (an in- crease in his cost)." Alternatively, we can state the equivalent N Definition 1.2. A decision N-tuple d0 ED, DC TI D., is Pareto i=l ~ optimal iff for all d E D ViE {l,2, ... ,N} implies J.(d) = J.(d0) ViE {1,2, ... ,N} ~ ~ This way of defining Pareto-optimality leads to a statement such as "If a joint decision is not Pareto-optimal, then there is another decision that results in the decrease of at least one cost without in- creasing any of the others."
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