International Series in Operations Research & Management Science
Volume 159
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International Series in Operations Research & Management Science

Volume 159

Series Editor: Frederick S. Hillier Stanford University, CA, USA Special Editorial Consultant: Camille C. Price Stephen F. Austin, State University, TX, USA

For further volumes: http://www.springer.com/series/6161

Masatoshi Sakawa • Ichiro Nishizaki Hideki Katagiri

Fuzzy Stochastic Multiobjective Programming

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Prof. Dr. Masatoshi Sakawa Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

Dr. Hideki Katagiri Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

Prof. Dr. Ichiro Nishizaki Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

ISSN 0884-8289 ISBN 978-1-4419-8401-2 e-ISBN 978-1-4419-8402-9 DOI 10.1007/978-1-4419-8402-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011921329 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Our Parents and Families

Preface

The increasing complexity of modern-day society has brought new problems having multiple objectives including economic, environmental, social and technical ones. Hence, it seems that the consideration of many objectives in the actual decision making process requires multiobjective approaches rather than that of a single objective. One of the major systems-analytic multiobjective approaches to decision making under constraints is multiobjective programming as a generalization of traditional single objective programming. For such multiobjective programming problems, it is signiﬁcant to realize that multiple objectives are often noncommensurable and conﬂict with each other. With this observation, in multiobjective programming problems, the notion of Pareto optimality or efﬁciency has been introduced instead of the optimality concept for single-objective problems. However, decisions with Pareto optimality or efﬁciency are not uniquely determined; the ﬁnal decision must be selected by a decision maker (DM), which well represents the subjective judgments, from the set of Pareto optimal or efﬁcient solutions. For deriving a compromise or satisﬁcing solution through interactions with a decision maker, interactive methods for multiobjective programming have been developed. When formulating a multiobjective programming problem which closely describes and represents the actual decision making situations, the uncertainty inherent in real-world complex systems or human being, such as the randomness of events related to the systems or the fuzziness of human judgments, should be reﬂected in the description of the objective functions and constraints as well as in the representation of parameters involved in the formulated problems. For dealing with not only the multiobjectiveness but also the uncertainty in decision making, multiobjective stochastic programming and multiobjective fuzzy programming have been individually developed together with the introduction of various optimization models and corresponding solution techniques. However, recalling the vagueness or fuzziness inherent in human judgments for the objective functions and/or constraints involving randomness, it is signiﬁcant to realize that the uncertainty in real-world decision making problems is often expressed by a fusion of fuzziness and randomness rather than either fuzziness or randomness. For such decision making situations, there are two types of inaccura-

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cies to be incorporated into multiobjective stochastic programming problems. One is the fuzzy goal of a DM for each of the stochastic objective functions, and the other is the experts’ ambiguous understanding of random parameters in the problemformulation process. When we model actual decision making situations under uncertainty as multiobjective programming problems involving random variables, following a major conventional stochastic programming approach together with the introduction of probabilistic constraints, the original problems can be transformed into deterministic multiobjective programming problems by replacing the original stochastic objective functions with some deterministic ones such as their expectations or variances. However, considering the imprecise nature of DMs’ judgments in multiobjective problems, it is natural to assume that the DM has a fuzzy goal for each of the resulting deterministic objective functions which originally involve randomness. Furthermore, in multiobjective stochastic programming, it is implicitly assumed that the realized values of random parameters can be deﬁnitely expressed by real values. However, considering that the realized values may be only ambiguously known to or observed by the experts, it may be more appropriate to express theses values as fuzzy numerical data which can be represented by means of fuzzy subsets of the real line known as fuzzy numbers. For handling and tackling such a fusion of fuzziness and randomness in multiobjective decision making, it is not hard to imagine that conventional fuzzy multiobjective programming or multiobjective stochastic programming cannot be applied. Naturally, simultaneous considerations of multiobjectiveness, fuzziness and randomness involved in the real-world decision making problems lead us to the new ﬁeld of multiobjective mathematical programming under fuzzy stochastic environments. So far, we have restricted ourselves to mathematical programming problems where decisions are made by a single DM. However, decision making problems in hierarchical managerial or public organizations are often formulated as two-level programming problems, where there exist two DMs. When we deal with decision making problems in decentralized organizations such as a ﬁrm’s organizing administrative ofﬁce and its autonomous divisions, it is quite natural to suppose that there exists communication and some cooperative relationship among the DMs. For handling such cooperative hierarchical decision making situations, two-level programming with random variables or fuzzy random variables are introduced to derive a satisfactory solution taking a balance between the DMs’ satisfactory degrees. Furthermore, in order to resolving conﬂict in hierarchical noncooperative decision making situations, computational methods for Stackelberg solutions for two-level programming problems involving random variables or fuzzy random variables are presented. Concerning computational aspects of multiobjective programming under fuzzy stochastic environments, because real-world decision problems under uncertainty can be often formulated as difﬁcult classes of optimization problems such as combinatorial problems and nonconvex nonlinear problems, it is difﬁcult to obtain exact optimal solutions, and thus it is quite natural for DMs to require approximate optimal solutions instead. To meet this demand, genetic algorithms initially introduced by Holland in early the 1970’s and tabu search methods proposed by Glover et al.

Preface

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in the mid-1980’s, which have attracted considerable attention as efﬁcient metaheuristics, should be incorporated into interactive multiobjective programming with random variables or fuzzy random variables. In this book, after presenting basic concepts in conventional multiobjective programming, the authors intend to introduce the latest advances in the new ﬁeld of multiobjective programming involving fuzziness and randomness. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective optimization in order to derive a satisﬁcing solution for a DM. Extensions to two-level programming are also given to resolving conﬂict in hierarchical decision making problems under fuzzy stochastic environments. Hiroshima, October 2010

Masatoshi Sakawa Ichiro Nishizaki Hideki Katagiri

Contents

1

Introduction and Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Description of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7

2

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fuzzy programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fuzzy goals and Fuzzy constraints . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Linear programming problems with fuzzy parameters . . . . . . 2.2 Stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Two-stage programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Chance constraint programming . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiobjective programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Multiobjective programming problem . . . . . . . . . . . . . . . . . . . 2.3.2 Interactive multiobjective programming . . . . . . . . . . . . . . . . . 2.3.3 Fuzzy multiobjective programming . . . . . . . . . . . . . . . . . . . . . 2.4 Two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Fuzzy programming for two-level programming . . . . . . . . . . 2.4.2 Stackelberg solution to two-level programming problem . . . . 2.5 Genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Fundamental elements in genetic algorithms . . . . . . . . . . . . . . 2.5.2 Genetic algorithm for integer programming . . . . . . . . . . . . . .

11 11 11 14 16 20 20 21 24 26 26 27 28 32 32 36 40 40 44

3

Fuzzy Multiobjective Stochastic Programming . . . . . . . . . . . . . . . . . . . . 3.1 Fuzzy multiobjective stochastic linear programming . . . . . . . . . . . . . 3.1.1 Expectation and variance models . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Probability and fractile models . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Simple recourse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Expectation and variance models . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 62 77 83 84

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3.2.2 3.2.3

Probability and fractile models . . . . . . . . . . . . . . . . . . . . . . . . . 90 Simple recourse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4

Multiobjective Fuzzy Random Programming . . . . . . . . . . . . . . . . . . . . . . 101 4.1 Multiobjective fuzzy random linear programming . . . . . . . . . . . . . . . 101 4.1.1 Possibility-based expectation and variance models . . . . . . . . 104 4.1.2 Possibility-based probability and fractile models . . . . . . . . . . 118 4.1.3 Level set-based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2.1 Possibility-based expectation and variance models . . . . . . . . 150 4.2.2 Possibility-based probability and fractile models . . . . . . . . . . 154 4.2.3 Level set-based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5

Stochastic and Fuzzy Random Two-Level Programming . . . . . . . . . . . . 169 5.1 Cooperative two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.1.1 Stochastic two-level linear programming . . . . . . . . . . . . . . . . . 170 5.1.2 Fuzzy random two-level linear programming . . . . . . . . . . . . . 177 5.1.3 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . 188 5.2 Noncooperative two-level programming . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2.1 Stochastic two-level linear programming . . . . . . . . . . . . . . . . . 201 5.2.2 Fuzzy random two-level linear programming . . . . . . . . . . . . . 207 5.2.3 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . 215

6

Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.1 Random fuzzy variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2 Random fuzzy linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.2.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 228 6.2.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 231 6.3 Multiobjective random fuzzy programming . . . . . . . . . . . . . . . . . . . . . 232 6.3.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 233 6.3.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 235 6.4 Random fuzzy two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.4.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 238 6.4.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 242

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Chapter 1

Introduction and Historical Remarks

1.1 Background The main characteristics of the real-world decision making problems facing humans today are uncertain due to the increase in complexity of modern-day society as well as the diversiﬁcation of values among people. Hence, when formulating a mathematical programming problem which closely describes and represents a real decision situation, various factors of the real system should be reﬂected in the description of the objective functions and the constraints. Naturally, these objective functions and the constraints involve many parameters whose possible values may be estimated by the experts. In the most traditional approaches, such parameters are ﬁxed at some values in an experimental and/or subjective manner through the experts’ understanding of the nature of the parameters. However, in most practical decision making situations, it would be more appropriate to consider uncertainty of the parameters of the objectives and constraints in the problems. From a probabilistic point of view, in 1955, linear programming problems with random variable coefﬁcients, called stochastic programming problems (Tintner, 1955), were individually introduced by Beale (1955), Dantzig (1955), Babbar (1955) and Wagner (1955). The two-stage problem, ﬁrst introduced by Beale (1955) and Dantzig (1955) as a simple recourse problem, is a solution technique via two stages for stochastic programming problems; in the ﬁrst stage, the penalty for the violation of the constraints is minimized under the assumption that the values of decision variables are ﬁxed at some constants, and in the second stage, the decision variables are determined so as to minimize the weighted sum of the penalty and the original objective function. The two-stage problems were comprehensively discussed by Walkup and Wets (1967) and Wets (1974) together with the introduction of more general types of recourse problems, and further extended by Beale, Forrest and Taylor (1980) and Louveaux (1980) as the multi-stage problems. Furthermore, the problem for obtaining the probability distribution function of an optimal solution and/or optimal value, called distribution problem, was initiated by Tintner (1955) and Wagner (1955), and further developed by Bereanu (1967, 1980).

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_1, © Springer Science+Business Media, LLC 2011

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1 Introduction and Historical Remarks

Considering the fact that the stochastic constraints are not always satisﬁed, the notion of chance or probabilistic constraints, which means that the constraints involving random variables need to be satisﬁed with a certain probability or over, was introduced by Charnes and Cooper (1959), and generalized by Miller and Wagner (1965) and Prekopa (1970) through the joint probability distribution function. For handling the randomness of the objective function value, three types of models based on different optimization criteria were considered by Charnes and Cooper (1963): the expectation optimization model, the variance minimization model and the probability maximization model. In the expectation optimization model, called the E-model, the objective function with random variable coefﬁcients is replaced by its expected value. The variance minimization model, called the V-model, was presented to consider not only the expectation but also the variance. Moreover, the probability maximization model, called the P-model or minimum-risk model, was developed in order to maximize the probability that the objective function value is better than or equal to the so-called target value speciﬁed by a decision maker (DM) as a constant. Meanwhile, the target value is optimized under a given probability level in the fractile criterion optimization model independently proposed by Kataoka (1963) and Geoffrion (1967). A brief and uniﬁed survey of major approaches to stochastic programming proposed before 1975 including the models described above can be found in the paper by Stancu-Minasian and Wets (1976). Since the ﬁrst international conference on multiple criteria decision making, held at the University of South Carolina in 1972 (Cochrane and Zeleny, 1972), it has been increasingly recognized that most of the real-world decision making problems usually involve multiple, noncommensurable, and conﬂicting objectives which should be considered simultaneously. One of the major systems-analytic multiobjective approaches to decision making under constraints is multiobjective optimization as a generalization of traditional single-objective optimization. For such multiobjective optimization problems, it is signiﬁcant to realize that multiple objectives are often noncommensurable and cannot be combined into a single objective. Moreover, the objectives usually conﬂict with each other and any improvement of one objective can be achieved only at the expense of at least one of the others. With this observation, in multiobjective optimization, the notion of Pareto optimality or efﬁciency was introduced instead of the optimality concept for single-objective optimization. However, decisions with Pareto optimality or efﬁciency are not uniquely determined; the ﬁnal decision must be selected from among Pareto optimal or efﬁcient solution set. For deriving a compromise or satisﬁcing solution for a DM in multiobjective optimization problems, goal programming approaches and interactive programming approaches have been developed. On the assumption that the DM can specify the goals of the objective functions, the goal programming approach ﬁrst appeared in a 1961 text by Charnes and Cooper (1961). Subsequent works on goal programming approaches have been numerously developed including Lee (1972), Ignizio (1976, 1982) and Charnes and Cooper (1977). The interactive programming approaches, which assume that the DM is able to give some preference information on a local level to a particular solution, were ﬁrst initiated by Geoffrion, Dyer and Feinberg (1972) and further developed by many researchers (Choo and Atkins, 1980; Sakawa

1.1 Background

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and Seo, 1980; Wierzbicki, 1980; Sakawa, 1981; Changkong and Haimes, 1983; Steuer and Choo, 1983). By considering the randomness of parameters involved in multiobjective problems, stochastic programming with multiple objective functions were ﬁrst introduced by Contini (1968) as a goal programming approach to multiobjective stochastic programming, and further studied by Stancu-Minasian (1984). For deriving a compromise or satisﬁcing solution for the DM in multiobjective stochastic decision making situations, an interactive programming method for multiobjective stochastic programming with normal random variables were ﬁrst presented by Goicoecha, Hansen and Duckstein (1982) as a natural extension of the so-called STEP method (Benayoun, de Montgolﬁer, Tergny and Larichev, 1971) which is an interactive method for deterministic problems. An interactive method for multiobjective stochastic programming with discrete random variables, called STRANGE, was proposed by Teghem, Dufrane, Thauvoye and Kunsch (1986) and Slowinski and Teghem (1988). The subsequent works on interactive multiobjective stochastic programming have been accumulated (Klein, Moskowitz and Ravindran, 1990; Urli and Nadeau, 1990, 2004). There seems to be no explicit deﬁnitions of the extended Pareto optimality concepts for multiobjective stochastic programming, until White (1982) deﬁned the Pareto optimal solutions for the expectation optimization model and the variance minimization model. More comprehensive discussions were provided by Stancu-Minasian (1984) and Caballero et al. (2001) through the introduction of extended Pareto optimal solution concepts for the probability maximization model and the fractile criterion optimization model. An overview of models and solution techniques for multiobjective stochastic programming problems were summarized in the context of Stancu-Minasian (1990). Considering the vague nature of the DM’s judgments in multiobjective linear programming, a fuzzy programming approach was ﬁrst presented by Zimmermann (1978) and further studied by Leberling (1981), Hannan (1981), Sakawa (1983), Sakawa and Yano (1985a), Rommelfanger (1990) and Sakawa (1993). In these fuzzy approaches, it has been implicitly assumed that the fuzzy decision or the minimum operator by Bellman and Zadeh (1970) is the proper representation of the DM’s fuzzy preferences. However, considering the diversiﬁcation of preferences or values among people, the DM does not always feel that such representation is appropriate for combining the fuzzy goals and/or constraints, and consequently it becomes evident that an interaction process for the DM to derive a satisﬁcing solution is necessary. Assuming that the DM has a fuzzy goal for each of the objective functions in multiobjective programming problems, Sakawa and Yano (1985a) and Sakawa (1993) proposed several interactive fuzzy decision making methods by incorporating the desirable features of the interactive approaches into the fuzzy programming. Realizing the fact that the possible values of parameters involved in formulated problems are often only ambiguously known to experts, through the interpretation of the experts’ understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy subsets of the real line known as fuzzy number by Dubois and Prade (1978, 1980), Sakawa and Yano (1985b, 1989) introduced the

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1 Introduction and Historical Remarks

concept of α-multiobjective programming and (M-) α-Pareto optimality based on the α-level sets of the fuzzy numbers. In most practical situations, however, it is natural to consider that the uncertainty in real-world decision making problems is often expressed by a fusion of fuzziness and randomness rather than either fuzziness or randomness. For handling not only the DM’s vague judgments in multiobjective problems but also the randomness of the parameters involved in the objectives and/or constraints, a fuzzy programming approach to multiobjective stochastic linear programming problems was ﬁrst taken by Hulsurkar, Biswal and Sinha (1997) by implicitly assuming that the fuzzy decision or the minimum operator is the proper representation of the DM’s fuzzy preferences. Realizing the drawbacks of the model of Hulsurkar, Biswal and Sinha (1997), Sakawa and his colleagues incorporated the techniques of an interactive fuzzy satisﬁcing method which was originally developed for deterministic problems (Sakawa, 1993; Sakawa and Yano, 1985a) into multiobjective stochastic programming problems, through the introduction of several fuzzy multiobjective stochastic programming models based on different optimization criteria such as expectation optimization (Sakawa, Kato and Nishizaki, 2003), variance minimization (Sakawa, Kato and Katagiri, 2002), probability maximization (Sakawa and Kato, 2002; Sakawa, Kato and Katagiri, 2004) and fractile criterion optimization (Sakawa, Katagiri and Kato, 2001), to derive a satisﬁcing solution for a DM from the extended Pareto optimal solution sets. When we model actual decision making situations under uncertainty as multiobjective stochastic programming problems, it is implicitly assumed that uncertain parameters or coefﬁcients involved in formulated problems can be expressed as random variables in probability theory. This means that the realized values of random parameters under the occurrence of some event are assumed to be deﬁnitely represented with real values. However, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts’ ambiguous understanding of the realized values of random parameters under the occurrence of events as fuzzy numbers. From such a point of view, a fuzzy random variable was ﬁrst introduced by Kwakernaak (1978), and its mathematical basis was constructed by Puri and Ralescu (1986). An overview of the developments of fuzzy random variables was found in the article of Gil, Lopez-Diaz and Ralescu (2006). Studies on linear programming problems with fuzzy random variable coefﬁcients, called fuzzy random linear programming problems, were initiated by Wang and Qiao (1993) and Qiao, Zhang and Wang (1994) as seeking the probability distribution of the optimal solution and optimal value. Optimization models of fuzzy random linear programming were ﬁrst developed by Luhandjula (1996) and Luhandjula and Gupta (1996), and further studied by Liu (2001a,b) and Rommelfanger (2007). A brief survey of major fuzzy stochastic programming models including fuzzy random programming was found in the paper by Luhandjula (2006). On the basis of possibility measure, possibilistic programming approaches to fuzzy random linear programming problems were introduced (Katagiri, Ishii and Itoh, 1997; Katagiri, Ishii and Sakawa, 2000; Katagiri and Sakawa, 2003). Multiobjective ex-

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5

tensions to fuzzy random linear programming were also considered, and several possibility-based fuzzy random programming models through the combination of a concept of possibility measure and different stochastic programming models were further developed (Katagiri, Sakawa and Ishii, 2001; Katagiri, Sakawa and Ohsaki, 2003; Katagiri, Sakawa, Kato and Nishizaki, 2008a; Katagiri, Sakawa, Kato and Ohsaki, 2003). Extensions to multiobjective integer problems with fuzzy random variables were provided by incorporating the branch-and-bound method into the interactive methods (Katagiri, Sakawa, Kato and Nishizaki, 2004; Katagiri, Sakawa and Nishizaki, 2006; Perkgoz, Katagiri, Sakawa and Kato, 2004). As a natural extension of the concept of M-α-Pareto optimality (Sakawa, 1993), Katagiri et al. (2004) and Katagiri, Sakawa, Kato and Nishizaki (2008b) deﬁned the extended Pareto optimality concepts in multiobjective fuzzy random programming by combining the concept of the M-α-Pareto optimality and optimization criteria in stochastic programming. Assuming that a DM decides that the degrees of all the membership functions of the fuzzy random variables should be greater than or equal to some value α, through the interpretation of the original fuzzy random programming problems as stochastic programming problems which depend on the speciﬁed degree α, they provided an interactive algorithm to derive a satisﬁcing solution from among the set of the extended M-α-Pareto optimal solutions. Recently, from a viewpoint of ambiguity and randomness different from the aforementioned fuzzy random variables, by considering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and mathematical programming problems with random fuzzy variables were formulated together with the development of a simulationbased approximate solution method (Liu, 2002). By focusing on the case where the means of random variables are represented with fuzzy numbers, solution methods for random fuzzy programming problems were developed through nonlinear programming techniques (Katagiri, Ishii and Sakawa, 2002; Katagiri, Hasuike, Ishii and Nishizaki, 2008; Hasuike, Katagiri and Ishii, 2009). So far, we have restricted ourselves to mathematical programming problems where decisions are made by a single DM. However, decision making problems in hierarchical managerial or public organizations are often formulated as two-level mathematical programming problems where there are two DMs: the upper level DM and the lower level DM. Assuming that the lower level DM behaves rationally, that is, optimally responds to the decision of the upper level DM, the upper level DM also speciﬁes a strategy so as to optimize his/her own objective. Computational methods for a pair of such decisions or strategies of both DMs, called a Stackelberg solution, are classiﬁed roughly into three categories: the vertex enumeration approach based on a characteristic that an extreme point of a set of rational responses of the lower level DM is also an extreme point of the feasible region (Bialas and Karwan, 1984), the Kuhn-Tucker approach in which the upper level problem with constraints including optimality conditions of the lower level problem is solved (Bialas and Karwan, 1984; J´udice and Faustino, 1992; Bard and Falk, 1982; Bard and Moore, 1990; Bard, 1983), and the penalty function approach which adds a penalty term

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1 Introduction and Historical Remarks

to the upper level objective function so as to satisfy optimality of the lower level problem (White and Anandalingam, 1993; Hansen, Jaumard and Savard, 1992). In order to deal with multiobjective problems in hierarchical decision making, Nishizaki and Sakawa (1999) formulated two-level multiobjective linear programming problems and developed a computational method for obtaining the corresponding Stackelberg solution. Considering stochastic events related to hierarchical decision making situations, Nishizaki, Sakawa and Katagiri (2003) and Katagiri, Nishizaki, Sakawa and Kato (2007) formulated two-level programming problems with random variables and developed algorithms for deriving the Stackelberg solutions based on some optimization criteria such as the expectation optimization and the variance minimization. Noncooperative fuzzy random two-level linear programming problems were also considered, and computational methods for obtaining the corresponding Stackelberg solutions were developed (Sakawa and Kato, 2009c; Sakawa and Katagiri, 2010b; Sakawa, Katagiri and Matsui, 2010b) From the viewpoint of taking the possibility of coordination or bargaining between DMs into account, attempts to derive Pareto optimal solutions to two-level linear programming problems were made (Wen and Hsu, 1991; Wen and Lin, 1996). Furthermore, under the assumption of communication and a cooperative relationship between DMs, solution methods for obtaining a Pareto optimal solution to a multi-level linear programming problem were presented by Lai et al. (Lai, 1996; Shih, Lai and Lee, 1996). After removing some drawbacks of the methods of Lai et al., Sakawa, Nishizaki and Uemura (1998) presented interactive fuzzy programming for two-level linear programming problems, and extended their model to treat two-level linear programming problems with fuzzy parameters (Sakawa, Nishizaki and Uemura, 2000) and those with random variables (Sakawa, Kato, Katagiri and Wang, 2003; Perkgoz, Sakawa, Kato and Katagiri, 2003; Kato, Wang, Katagiri and Sakawa, 2004; Sakawa and Kato, 2009a; Sakawa and Katagiri, 2010a). Extension to two-level integer programming problems (Sakawa, Katagiri and Matsui, 2010a) and fuzzy random linear two-level programming problems (Katagiri, Sakawa, Kato and Nishizaki, 2007; Sakawa and Kato, 2009b) were also provided. Consider computational aspects of interactive fuzzy satisﬁcing methods for multiobjective programming problems under fuzzy stochastic environments. Because fuzzy multiobjective stochastic programming problems with discrete decision variables are difﬁcult to solve strictly, it becomes important to develop highly efﬁcient approximate computational methods. As an efﬁcient meta-heuristics, genetic algorithms initiated by Holland (1975) have attracted the attention of many researchers with applicability in optimization, as well as in search and learning. Furthermore, publications of books by Goldberg (1989) and Michalewicz (1996) brought heightened and increasing interest in applications of genetic algorithms to complex function optimization. From an interactive multiobjective programming perspective, Sakawa and his colleagues have been advancing genetic algorithms to derive satisﬁcing solutions to multiobjective optimization problems with discrete decision variables (Sakawa, Kato, Sunada and Shibano, 1997; Sakawa, 2001). As a generalization of results along this line, interactive fuzzy satisﬁcing methods for multiobjective stochastic integer programming problems for deriving satisﬁcing solutions

1.2 Description of contents

7

for a DM from among the set of the extended Pareto optimal solutions have also been developed (Perkgoz, Kato, Katagiri and Sakawa, 2004; Kato, Perkgoz, Katagiri and Sakawa, 2004; Perkgoz, Sakawa, Kato and Katagiri, 2005).

1.2 Description of contents Although studies on multiobjective mathematical programming under uncertainty have been accumulated and several books on multiobjective mathematical programming under uncertainty have been published (e.g. Stancu-Minasian (1984); Slowinski and Teghem (1990); Sakawa (1993); Lai and Hwang (1994); Sakawa (2000)), there seems to be no book which concerns both randomness of events related to environments and fuzziness of human judgments simultaneously in multiobjective decision making problems. In this book, the authors are concerned with introducing the latest advances in the ﬁeld of multiobjective optimization under both fuzziness and randomness on the basis of authors’ continuing research works. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective programming for human-centered systems under uncertainty in most realistic situations when dealing with both fuzziness and randomness. Organization of each chapter is brieﬂy summarized as follows. Chapter 2 is devoted to mathematical preliminaries, which will be used throughout the remainder of this book. Starting with basic notions in fuzzy set theory, linear programming problems with fuzzy goals and fuzzy constraints, as well as linear programming problems with fuzzy parameters, are introduced. From the probabilistic point of view, stochastic programming such as two-stage programming and chance constraint programming is outlined. After reviewing the solution concepts in multiobjective programming, interactive fuzzy multiobjective programming is presented. Several basic notions including Pareto optimal solutions and Stackelberg equilibrium in two-level programming under cooperative and noncooperative decision making situations are reviewed. Fundamentals of genetic algorithms are also brieﬂy discussed as efﬁcient meta-heuristics for solving difﬁcult classes of optimization problems with nonconvex nonlinear objective functions and/or constraints. In Chapter 3, by considering the imprecision of a decision maker’s (DM’s) judgment for stochastic objective functions and/or constraints in multiobjective problems, fuzzy multiobjective stochastic programming is developed. After reviewing the main ideas of conventional single-objective stochastic programming problems such as two-stage problems and chance constrained programming problems, multiobjective stochastic programming problems are formulated by assuming that the DM has a fuzzy goal for each of the expectation and the variance of the original stochastic objective functions. In order to reﬂect the diversity of criteria for optimizing the stochastic objective functions, different optimization criteria from the expectation and the variance are also provided. Namely, we present the probability model maximizing the probability of the objective functions being greater than or equal to target values and the fractile model optimizing the target values under a given

8

1 Introduction and Historical Remarks

probability. It is shown that the original stochastic problems can be transformed into deterministic multiobjective problems with linear, linear fractional or nonlinear objective functions and/or constraints. Through the introduction of extended Pareto optimal solution concepts, interactive fuzzy satisﬁcing methods using linear or convex programming techniques are presented to derive a satisﬁcing solution for the DM from among the extended Pareto optimal solution set. The presented interactive methods for the multiobjective stochastic programming with continuous variables are immediately extended to deal with the integer problems with multiple stochastic objective functions and/or constraints by utilizing some genetic algorithms (Sakawa, 2001). In Chapter 4, taking account of not only the randomness of parameters involved in objective functions and/or constraints but also the experts’ ambiguous understanding of the realized values of the random parameters, we formulate multiobjective programming problems with fuzzy random variables. By incorporating a concept of possibility measure into stochastic programming models discussed in the previous chapter, four types of optimization models for fuzzy random programming are developed. Through the introduction of the extended Pareto optimal solution concepts on the basis of possibility theory and probability theory, considerable efforts are devoted to development of interactive methods for fuzzy random multiobjective programming to derive a satisﬁcing solution for a DM. Furthermore, to take the level set approach by Sakawa (1993), some Pareto optimality concepts for fuzzy random multiobjective programming problems are deﬁned by combining the notions of the M-α-Pareto optimality and optimization criteria in stochastic programming, and then interactive satisﬁcing methods are presented for deriving a satisﬁcing solution for the DM from among the extended M-α-Pareto optimal solution set. The presented interactive methods are extended to deal with integer programming problems with fuzzy random variables. In Chapter 5, for resolving conﬂict of decision making problems in hierarchical managerial or public organizations where there exist two DMs who have different interests in making decisions, two-level programming problems are discussed. By considering the uncertainty involved in cooperative and noncooperative decision making situations, two-level programming problems with random variable and fuzzy random variable coefﬁcients are formulated. By employing the optimization models provided in Chapters 3 and 4, the original problems are transformed into the deterministic two-level programming problems. For deriving a well-balanced satisfactory solution between the two DMs’ satisfaction degrees, interactive fuzzy programming methods for cooperative stochastic and fuzzy random two-level programming problems are developed. Moreover, in order to obtain Stackelberg solutions to noncooperative stochastic and fuzzy random two-level programming problems, computational methods using the combination of convex programming techniques and the branch-and-bound method are presented. Using the genetic algorithms proposed by Sakawa (2001), the presented methods are extended to deal with two-level integer programming problems with random variables and fuzzy random variables. Finally, Chapter 6 outlines some future research directions. In contrast to the fuzzy random variables dealing with the ambiguity of realized values of random

1.2 Description of contents

9

variables discussed in Chapter 4, random fuzzy variables are introduced as parameters or coefﬁcients involved in linear programming problems by considering the ambiguity of means and variances of random variables. Attention is particulary focused on the case where the mean of each of random variables is represented by a fuzzy number. From the viewpoint of simultaneous maximization of possibility and probability, the original random fuzzy linear programming problem is equivalently transformed into a deterministic nonlinear programming problem. The presented decision making model is extended to deal with multiobjective and two-level programming problems with random fuzzy variables.

Chapter 2

Fundamentals

This chapter is devoted to reviewing optimization concepts and the related computational methods that will be used in the remaining chapters. In particular, we deal with three optimization concepts incorporating fuzziness and ambiguity of human judgments, uncertainty of events characterizing decision making problems, and multiplicity of evaluation criteria. Fuzzy programming which is developed in order to take into account fuzziness and ambiguity of human judgments, and this method is ﬁrst presented together with the basic concepts in fuzzy set theory. Fundamentals of stochastic programming are also provided for decision problems under probabilistic uncertainty. Speciﬁcally, two-stage programming and chance constraint programming are covered. To meet expectations of diversiﬁed evaluations, multiobjective programming has been investigated. After presenting interactive methods for multiobjective linear programming problems, we give some techniques to deal with not only multiple objectives but also fuzzy goals for the objectives. In an organization with a hierarchical structure, we often ﬁnd decision making with two or more decision makers (DMs) attempting to optimize their objective functions. To model such a decision making problem mathematically, a two-level programming problem is formulated, and cooperative and noncooperative solution methods are presented for two-level programming. Genetic algorithms are considered to be one of the most practical and proven meta heuristics for difﬁcult classes of optimization problems, and ﬁnally basic concepts of genetic algorithms are provided.

2.1 Fuzzy programming 2.1.1 Fuzzy sets Throughout this book, the concepts of fuzzy set theory are used together with those of stochastic theory. Before discussing fuzzy programming, we provide the fundamentals of fuzzy set theory.

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_2, © Springer Science+Business Media, LLC 2011

11

12

2 Fundamentals

A fuzzy set initiated by Zadeh (1965) is deﬁned as follows: Deﬁnition 2.1 (Fuzzy set). Let X denote a universal set. Then, a fuzzy set A˜ is deﬁned by its membership function μA˜ : X → [0, 1].

(2.1)

We deal with mathematical programming in this book, and therefore the set X in Deﬁnition 2.1 means the real line R generally. The membership function μA˜ assigns to each element x ∈ X a real number μA˜ (x) in the interval [0, 1], and the value of ˜ A fuzzy set A˜ is represented by μA˜ (x) represents the grade of membership of x in A. a pair of an element x and its grade μA˜ (x), and thus it is often written as A˜ = {(x, μA˜ (x)) | x ∈ X}. For a given ordinary set A, the characteristic function 1 if x ∈ A cA (x) = 0 if x ∈ A

(2.2)

(2.3)

deﬁnes the set A which is expressed by A = {x ∈ X | cA (x) = 1}.

(2.4)

From these deﬁnitions, one ﬁnds that a fuzzy set A˜ is a natural extension of an ordinal set A. The concept of α-level sets is very important to serve a transfer relation between a fuzzy set and an ordinary set. Deﬁnition 2.2 (α-level set). For a given α ∈ [0, 1], the α-level set of a fuzzy set A˜ is deﬁned as an ordinary set Aα of elements x such that the membership function value μA˜ (x) of x exceeds α, i.e., Aα = {x | μA˜ (x) ≥ α}.

(2.5)

The extension principle introduced by Zadeh (1965) is to provide a general way for extending nonfuzzy mathematical concepts to the fuzzy framework. Deﬁnition 2.3 (Extension principle). Let f : X → Y be a mapping from a set X to a set Y . Then, the extension principle allows us to deﬁne the fuzzy set B˜ in Y induced by the fuzzy set A˜ in X through f as follows: B˜ = {(y, μB˜ (y)) | y = f (x), x ∈ X} with μB˜ (y) μ f (A) ˜ (y) =

⎧ ⎨ sup μA˜ (x)

if f −1 (y) = 0/

⎩

/ if f −1 (y) = 0,

y= f (x)

0

(2.6)

(2.7)

2.1 Fuzzy programming

13

where f −1 (y) is the inverse image of y, and 0/ means the empty set. Because we deal with mathematical programming under uncertainty, among fuzzy sets, fuzzy numbers which are linguistically-expressed such as “approximately m” or “about n” play important roles. Before deﬁning fuzzy numbers, we give the deﬁnitions of convex and normalized fuzzy sets. A fuzzy set A˜ is said to be convex if any α-level set Aα of A˜ is convex, and a fuzzy set A˜ is said to be normal if there is x such that μA˜ (x) = 1. Deﬁnition 2.4 (Fuzzy number). A fuzzy number is a convex normalized fuzzy set of the real line R whose membership function is piecewise continuous. By using the extension principle by Zadeh, the binary operation “∗” in R can be extended to the binary operation “” of fuzzy numbers M˜ and N˜ as μM ˜ N˜ (z) = sup min{μM˜ (x), μN˜ (y)}.

(2.8)

z=x∗y

For example, consider an extension of addition “+” of two numbers. By using (2.8), we can deﬁne the extended addition “⊕” of two fuzzy numbers as follows: μM⊕ ˜ N˜ (z) = sup min{μM˜ (x), μN˜ (y)} z=x+y

= sup min{μM˜ (x), μN˜ (z − x)}. x∈R

To provide easy computation of fuzzy numbers, Dubois and Prade (1978) introduce the concept of L-R fuzzy numbers. Deﬁnition 2.5 (L-R fuzzy number). A fuzzy number M˜ is said to be an L-R fuzzy number if ⎧ m−x ⎪ ⎪ if x ≤ m ⎨L α (2.9) μM˜ (x) = x−m ⎪ ⎪ if x ≥ m, ⎩R β ˜ and α and β are positive numbers which represent where m is the mean value of M, left and right spreads of the fuzzy number, respectively; a function L is a left shape function satisfying (i) L(x) = L(−x), (ii) L(0) = 1, (iii) L(x) is nonincreasing on [0, ∞); and a right shape function R is similarly deﬁned as L. By using its mean, left and right spreads, and shape functions, an L-R fuzzy number M˜ is symbolically written as M˜ = (m, α, β)LR .

(2.10)

For two L-R fuzzy numbers M˜ = (m, α, β)LR and N˜ = (n, γ, δ)LR , the extended addition M˜ ⊕ N˜ is calculated as (m, α, β)LR ⊕ (n, γ, δ)LR = (m + n, α + γ, β + δ)LR ,

(2.11)

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2 Fundamentals

and scalar multiplication for an L-R fuzzy numbers M˜ = (m, α, β)LR and a scalar value λ is also given as if λ > 0 (λm, λα, λβ)LR (2.12) λ ⊗ (m, α, β)LR = (λm, −λα, −λβ)LR if λ < 0, where ⊗ means the extended multiplication. The other operations are similarly given, and for further information on this issue, refer to Dubois and Prade (1978).

2.1.2 Fuzzy goals and Fuzzy constraints Zimmermann (1976) introduces the concept of fuzzy set theory into linear programming. Assuming that the membership functions for fuzzy sets are linear, he shows that, by employing the principle of the fuzzy decision by Bellman and Zadeh (1970), a linear programming problem with a fuzzy goal and fuzzy constraints can be solved by using standard linear programming techniques. A linear programming problem is represented as ⎫ minimize z(x1 , . . . , xn ) = c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎬ ········· (2.13) ⎪ ⎪ am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n, where x j is a decision variable, and c j , ai j and bi are given coefﬁcients of the objective function and the constraints. Let x = (x1 , . . . , xn )T denote a column vector of the decision variables, and let c = (c1 , . . . , cn ), A = [ai j ], i = 1, . . . , m, j = 1, . . . , n and b = (b1 , . . . , bm )T denote an n dimensional row vector of the coefﬁcients of the objective function, an m × n matrix of the coefﬁcients of the left-hand side of the constraints, and an m dimensional column vector of the coefﬁcients of the righthand side of the constraints, respectively; the superscript T means transposition of a vector or a matrix. Then, (2.13) is simply rewritten in the following vector and matrix representation: ⎫ minimize z(x) = cx ⎬ subject to Ax ≤ b (2.14) ⎭ x ≥ 0. To a standard linear programming problem (2.14), taking into account the imprecision or fuzziness with respect to the judgment of a decision maker (DM), Zimmermann formulates the following linear programming problem with a fuzzy goal and fuzzy constraints:

2.1 Fuzzy programming

15

cx z0 Ax b

(2.15) (2.16)

x ≥ 0,

(2.17)

where the symbol denotes a relaxed or fuzzy version of the ordinary inequality ≤. From the DM’s preference, the fuzzy goal (2.15) and the fuzzy constraints (2.16) mean that the objective function cx should be essentially smaller than or equal to a certain level z0 , and that the values of the constraints Ax should be substantially smaller than or equal to b, respectively. Assuming that the fuzzy goal and the fuzzy constraints are equally important, he employs the following uniﬁed formulation: Bx b

(2.18) x ≥ 0, where B=

z c , b = 0 . b A

(2.19)

To express the imprecision or fuzziness of the DM’s judgment, the ith fuzzy constraint (Bx)i b i is interpreted as the following linear membership function: ⎧ 1 ⎪ ⎪ ⎨ (Bx)i − b i μi ((Bx)i ) = 1 − ⎪ di ⎪ ⎩ 0

if (Bx)i ≤ b i if b i < (Bx)i ≤ b i + di

(2.20)

if (Bx)i > b i + di ,

where di is a subjectively speciﬁed constant expressing the limit of the admissible violation of the ith constraint, and it is depicted in Fig. 2.1. μi ((Bx)i ) 1

0

b’ i

b’i + d i

(Bx) i

Fig. 2.1 Linear membership function for the ith fuzzy constraint.

On the basis of the principle of the fuzzy decision by Bellman and Zadeh (1970), a mathematical programming problem for ﬁnding the maximum decision is represented as maximize min μi ((Bx)i ) 0≤i≤m+1 (2.21) subject to x ≥ 0.

16

2 Fundamentals

With the variable transformation b

i = b i /di and (B x)i = (Bx)i /di , and an auxiliary variable λ, (2.21) can be transformed into the following conventional linear programming problem: ⎫ maximize λ ⎬ subject to λ ≤ 1 + b

i − (B x)i , i = 0, . . . , m + 1 (2.22) ⎭ x ≥ 0. Because the fuzzy decision is represented as min0≤i≤m+1 μi ((Bx)i ) in (2.21), it is often called the minimum operator.

2.1.3 Linear programming problems with fuzzy parameters To describe a real-world decision situation mathematically, the formulation of linear programming is often employed due to its simplicity and ease in computation, and naturally, possible values of the parameters in a linear programming problem may be assigned by experts for the decision situations. In most situations, however, it is observed that the experts may know the possible values of the parameters only imprecisely or ambiguously. From this reason, to express the imprecise judgments of the experts, the following linear programming problem involving fuzzy parameters is formulated: ⎫ minimize C˜1 x1 + · · · + C˜n xn ⎪ ⎪ ⎪ subject to A˜ 11 x1 + · · · + A˜ 1n xn ≤ B˜ 1 ⎪ ⎬ ········· (2.23) ⎪ A˜ m1 x1 + · · · + A˜ mn xn ≤ B˜ m ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n, where C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n are fuzzy parameters represented by fuzzy numbers. Let the membership function of the fuzzy parameters C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n be denoted by μC˜ j , μA˜ i j , and μB˜i , i = 1, . . . , m, j = 1, . . . , n, respectively. Here, we give two approaches for solving the linear programming problem with fuzzy numbers (2.23): one is an approach based on the possibility-based model (Dubois and Prade, 1980) and the other is the level set-based model (Sakawa, 1993).

2.1.3.1 Possibility-based model To deal with binary relations between a pair of fuzzy numbers M˜ and N˜ from the viewpoint of possibility, Dubois and Prade (1978, 1980) give the following index: ˜ ∞ = sup{min{μM˜ (u), μN˜ (v)} | u ≥ v}. Pos M˜ ≥ N˜ = ΠM˜ N, (2.24)

2.1 Fuzzy programming

17

The index (2.24) can be interpreted as the degree of possibility that M˜ is larger than ˜ or equal to N. For a given degree α, the α-level sets Mα = {u ∈ R | μM˜ (u) ≥ α} and Nα = {v ∈ R | μN˜ (v) ≥ α} of the fuzzy M˜ and N˜ are represented by closed intervals Lnumbers L R R Mα = mα , mα and Nα = nα , nα , respectively. Then, it is known that Pos M˜ ≥ N˜ ≥ α if and only if mRα ≥ nLα .

(2.25)

Using this binary relation from the viewpoint of possibility, if the DM speciﬁes a certain degree α of possibility, the constraints of (2.23) can be interpreted as x ∈ Xpos (α) x ≥ 0 | Pos A˜ i1 x1 + · · · + A˜ in xn ≤ B˜ i ≥ α, i = 1, . . . , m . (2.26) The α-level set of the fuzzy number B˜ i for the right-hand side constant in the ith constraint is represented by (2.27) Biα = {u ∈ R | μB˜ i (u) ≥ α} = bLiα , bRiα . Similarly, the α-level set of the vector of fuzzy numbers A˜ i = (a˜i1 , . . . , a˜in ) in the left-hand side coefﬁcients of the ith constraint is represented by (2.28) Aiα = (ai1 , . . . , ain ) | ai j ∈ aLijα , aRijα , j = 1, . . . , n , and, (2.28) is simply denoted by Aiα = aLiα , aRiα . From the proposition (2.25), (2.26) can be represented as x ∈ Xpos (α) = x ≥ 0 | aLiα x ≤ bRiα , i = 1, . . . , m .

(2.29)

(2.30)

For the sake of simplicity, assume that all the fuzzy parameters in (2.23) are represented by L-R fuzzy numbers: C˜ j = (c j , β j , γ j )LR , j = 1, . . . , n, A˜ i j = (ai j , δi j , εi j )LR , i = 1, . . . , m, j = 1, . . . , n, B˜i = (bi , ζi , ηi )LR , i = 1, . . . , m; and L(y) = R(y) = max{0, 1 − |y|}. Then, (2.30) can be rewritten by ordinary linear inequalities n x ∈ Xpos (α) = x ≥ 0 ∑ {ai j − (1 − α)δi j }x j ≤ bi + (1 − α)ηi , i = 1, . . . , m . j=1

(2.31) As for the objective function of (2.23), we introduce the following linear membership function of the fuzzy goal:

μG˜ (y) =

⎧ 1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1− 0

y − z0 z1 − z0

if y < z1 if z1 ≤ y ≤ z0 if y ≥ z0 ,

(2.32)

18

2 Fundamentals

where z0 and z1 denote the values of the objective function such that the degrees of the membership function μG˜ are 0 and 1, respectively. Because from the assumption of the fuzzy parameters the objective function C˜1 x1 +· · ·+ C˜n xn of (2.23) can be represented by (∑nj=1 c j x j , ∑nj=1 β j x j , ∑nj=1 γ j x j )LR , with appropriate values of the parameters z0 and z1 of the fuzzy goal (2.32), the degree of possibility with respect to the objective function is expressed as n

∑ (β j − c j )x j + z0

˜ = sup min{μ ˜ (y), μ ˜ (y)} = ΠCx ˜ (G) Cx G

j=1 n

∑ β jx j − z

1

,

(2.33)

+z

0

j=1

where μCx ˜ is a membership function of the objective function for a given vector x of the decision variables. Under the assumption of the fuzzy parameters, a linear programming problem with fuzzy parameters in the possibility-based model is formulated as n

∑ (β j − c j )x j + z0

j=1 n

maximize

∑ β j x j − z1 + z0

j=1 n

subject to

∑ {a1 j − (1 − α)δ1 j }x j ≤ b1 + (1 − α)η1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ········· ⎪ ⎪ n ⎪ ⎪ ⎪ {a − (1 − α)δ }x ≤ b + (1 − α)η mj j m m⎪ ⎪ ∑ mj ⎪ ⎪ j=1 ⎪ ⎭ x j ≥ 0, j = 1, . . . , n.

(2.34)

j=1

Because (2.34) is a linear fractional programming problem, it can be solved by using the Charnes and Cooper method (1962) or the Bitran and Novaes method (1973).

2.1.3.2 Level set-based model Consider the level set-based model for solving a linear programming problem involving fuzzy parameters (Sakawa, 1993). At the beginning, we give the formulation (2.23) of a linear programming problem involving fuzzy parameters once again. minimize C˜1 x1 + · · · + C˜n xn subject to A˜ 11 x1 + · · · + A˜ 1n xn ≤ B˜ 1 ········· A˜ m1 x1 + · · · + A˜ mn xn ≤ B˜ m x j ≥ 0, j = 1, . . . , n.

2.1 Fuzzy programming

19

For a ceratin degree α speciﬁed by the DM, the α-level set of all of the fuzzy numbers C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n in (2.23) is deﬁned as the ordinary set (C, A, B)α for which their membership function values exceed the degree α, i.e., (C, A, B)α = {(c, a, b) | μC˜ j ≥ α, μA˜ i j ≥ α, μB˜ i ≥ α, i = 1, . . . , m, j = 1, . . . , n}, (2.35) where c, a, and b are an n dimensional row constant coefﬁcient vector for the objective function, an m × n constant coefﬁcient matrix for the left-hand side of the constraints, and an m dimensional column right-hand side constant vector for the constraints, respectively. It should be noted here that the α-level sets have the following property: α1 ≤ α2 ⇔ (C, A, B)α1 ⊇ (C, A, B)α2 . (2.36) Suppose the DM thinks that the degree of all of the membership functions of the fuzzy numbers involved in (2.23) should be larger than or equal to a certain degree α. Then, for such a degree α, (2.23) can be interpreted as the following nonfuzzy linear programming problem which depends on the coefﬁcient vector (c, a, b) ∈ (C, A, B)α : ⎫ minimize c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎬ ········· (2.37) ⎪ am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n. Observe that there exist an inﬁnite number of linear programming problems such as (2.37) depending on the coefﬁcient vector (c, a, b) ∈ (C, A, B)α , and the values of (c, a, b) are arbitrary for any (c, a, b) ∈ (C, A, B)α in the sense that the degree of all of the membership functions for fuzzy numbers in the linear programming problem involving fuzzy parameters (2.23) exceeds the degree α speciﬁed by the DM. However, if possible, it would be desirable for the DM to choose (c, a, b) ∈ (C, A, B)α in (2.37) so as to minimize the objective function under the constraints. From such a point of view, for the given degree α, it seems to be quite natural to have the linear programming problem involving fuzzy parameters (2.23) as the following nonfuzzy α-linear programming problem: ⎫ minimize c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎪ ⎪ ⎪ ⎬ ········· (2.38) am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎪ ⎪ x j ≥ 0, j = 1, . . . , n ⎪ ⎪ ⎭ (c, a, b) ∈ (C, A, B)α . It should be emphasized here that in the nonfuzzy α-linear programming problem (2.38), the parameters (c, a, b) are treated as decision variables rather than constants of the coefﬁcients.

20

2 Fundamentals

From the property of the α-level set for the fuzzy numbers C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n, the feasible regions for c j , ai j , and bi can be denoted by the closed intervals [cLjα , cRjα ], [aLijα , aRijα ], and [bLiα , cRiα ], respectively. Therefore, we can obtain an optimal solution to the nonfuzzy α-linear programming problem (2.38) by solving the following linear programming problem: ⎫ minimize cL1 x1 + · · · + cLn xn ⎪ ⎪ ⎪ subject to aL11 x1 + · · · + aL1n xn ≤ bR1 ⎪ ⎬ ········· (2.39) ⎪ aLm1 x1 + · · · + aLmn xn ≤ bRm ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n. It is important to realize here that (2.39) is no more nonlinear but an ordinary linear programming problem, and consequently it is easy to solve it by using linear programming techniques such as the simplex method.

2.2 Stochastic programming 2.2.1 Random variables An outcome of a random trial such as coin tossing is called a sample point, which is denoted by ω, and the set of all of the possible outcomes of the random trial is called a sample space, which is denoted by Ω. An event is a subset of the sample space. For any events A and B, the union and the intersection are deﬁned as follows: (i) Union of A and B: A ∪ B = {ω | ω ∈ A or ω ∈ B}. (ii) Intersection of A and B: A ∩ B = {ω | ω ∈ A and ω ∈ B}. Furthermore, for any event A, the compliment is deﬁned as (iii) Compliment of A: Ac = {ω | ω ∈ A}. A family B of subsets of the sample space Ω which has the following properties is called a σ-ﬁeld: (i) Ω ∈ B; (ii) A ∈ B ⇒ Ac ∈ B; (iii) A1 , A2 , . . . ∈ B ⇒

∞ k=1

Ak ∈ B.

2.2 Stochastic programming

21

Let S be a family of subsets of the sample space Ω. The intersection of all σﬁelds containing S is called the σ-ﬁeld generated by S, and it is denoted by σ[S]. Let E = (E, d) be a metric space, and let O be a family of all open subsets of E. Then, a σ-ﬁeld σ[O] generated by O is called a family of Borel sets on E. Consider the d-dimensional Euclidean space Rd as a special case of E. A family of all open subsets of Rd is deﬁned as Jd {(a1 , b1 ] × · · · × (ad , bd ] ⊂ Rd | −∞ ≤ ak ≤ bk ≤ +∞, k = 1, . . . , d}. (2.40) Then, σ[Jd ] is a σ-ﬁeld in Rd , and it is called a family of d-dimensional Borel sets on Rd . Let Ω be a sample space, and let B be a σ-ﬁeld in the sample space Ω. A realvalued function P deﬁned on the σ-ﬁeld B in Ω satisfying the following conditions is called a probability measure: (i) 0 ≤ P(A) ≤ 1 for A ∈ B; (ii) P(Ω) = 1; (iii) If A1 , A2 , . . . ∈ B and A1 , A2 , . . . is a disjoint sequence, then ∞ ∞ Ak = ∑ P(Ak ). P k=1

k=1

If B be a σ-ﬁeld in a sample space Ω and P is a probability measure on B, the triple (Ω, B, P) is called a probability measure space or simply a probability space. For an arbitrary probability space (Ω, B, P), let X(ω) be a real-valued function on Ω. Then, X(ω) is a random variable if {ω | X(ω) ≤ x} ∈ B

(2.41)

holds for each real value x. For the random variable X(ω), the function F(x) = P({ω | X(ω) ≤ x})

(2.42)

is called a distribution function, where P({ω | X(ω) ≤ x}) is often simply denoted by P(X ≤ x). The distribution function has the following properties: (i) lim F(x) = 0 and lim F(x) = 1; x→−∞

x→+∞

(ii) If x < y, then F(x) ≤ F(y). In this book, henceforth to discriminate a random variable from the other variables or parameters, we attach a bar “–” to a character like d.¯

2.2.2 Two-stage programming In the real-world decision making problems, some stochastic events may inﬂuence elements characterizing decision making problems such as demands of products,

22

2 Fundamentals

the amount of available resources and so forth. When such a decision making problem under uncertainty is formulated as a linear programming problem, it may be difﬁcult that the constraints of the problem always hold completely. Then, a shortage or an excess comes from the violation of the constraints, and the corresponding penalties are imposed as the occasion demands. From this point of view, two-stage programming had been investigated from the beginning of the development of linear programming (Beale, 1955; Dantzig, 1955; Wets, 1966; Everitt and Ziemba, 1978). To understand the framework of two-stage programming, consider a decision problem of a manufacturing company. Let x = (x1 , . . . , xn )T denote activity levels in a production plant of the company, and then w = T x denotes the amount of products, where T is an m × n matrix transforming the n kinds of activity levels into the m types of products. Let d¯i be the demand for the ith product which is only known in probability, and assume that the random variable d¯i of the demand is indicated by the probability distribution function Fi (di ) P(d¯i ≤ di ). Suppose that the DM selects the activity levels, say x = xˆ , and after the occurrence of the random event ˆ Then, if yˆ+ dˆi − wˆ i ≥ 0, the shortage the demands d¯ = (d¯1 , . . . , d¯m )T are ﬁxed at d. i − + ˆ of the ith product is yˆi , and if yˆi wˆ i − di ≥ 0, the excess of the ith product is yˆ− i . This situation can be formulated as ¯ T x + Iy+ − Iy− = d, where y+ and y− represent the errors for estimating the demands, and I is the m dimensional identity matrix. Let q+ and q− denote the penalty costs for making these errors, and let c be the original costs for the activities in the production plant. Then, the objective function to be minimized may be the expectation of cx + q+ y+ + q− y− . Adding the constraints Ax ≤ b for the activity levels such as the capacity, budget, technology, etc., we can formulate the standard form of the two-stage programming problem as ⎫ minimize cx + E q+ y+ + q− y− ⎪ ⎪ ⎬ subject to T x + Iy+ − Iy− = d¯ (2.43) Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0, where E means the function of expectation. In particular, a two-stage programming problem such that the coefﬁcient vector of y+ and y− is the identity matrix such as (2.43) is called a simple recourse problem. ˆ the following solution minimizes the For the selected x and realized values d, + objective function of (2.43) unless qi + q− i is negative, and therefore we assume − + q ≥ 0, i = 1, . . . , m: that q+ i i

2.2 Stochastic programming

23

⎫ n ⎪ ˆ if di ≥ ∑ ti j x j ⎪ ⎪ ⎬

n

− ˆ y+ i = di − ∑ ti j x j , yi = 0 j=1

y+ i = 0,

y− i =

n

∑ ti j x j − dˆi

if dˆi

z0i ,

where z0i and z1i denote the values of the ith objective function zi (x) such that the degrees of the membership function are 0 and 1, respectively, and it is depicted in Fig. 2.3. μi (zi (x)) 1

0

z 0i

z i1

zi (x)

Fig. 2.3 Linear membership function for the ith fuzzy goal.

Zimmermann (1978) also suggests a method for assessing these parameters z0i and z1i of the membership function. In his method, the parameter z1i is determined as z1i = min zi (x), x∈X

(2.66)

and the parameter z0i is speciﬁed as z0i = max zi (x jo ), j=i

(2.67)

where x jo is a feasible solution minimizing z j (x), i.e., x jo ∈ arg minx∈X z j (x). In this setting, z1i is set at the minimum of the ith objective function, and z0i is set at the maximum among the values of the ith objective function with respect to solutions minimizing z j (x), j = i. By setting the parameters as described above, the linear membership functions (2.65) can be identiﬁed. Following the principle of the fuzzy decision by Bellman and Zadeh (1970), the multiobjective linear programming problem (2.61) can be reformulated as the following maximin problem: maximize min {μi (zi (x))} 1≤i≤k (2.68) subject to Ax ≤ b, x ≥ 0. By introducing an auxiliary variable λ, the maximin problem (2.68) is equivalently rewritten as a standard linear programming problem

30

2 Fundamentals

⎫ ⎬

maximize λ subject to μi (zi (x)) ≥ λ, i = 1, . . . , k ⎭ Ax ≤ b, x ≥ 0.

(2.69)

In the formulation by Zimmermann, it is implicitly assumed that the DM feels that the fuzzy decision is appropriate for combining the fuzzy goals. However, in real-world decision situations, such an assumption is not always well-suited, and then an interactive method provides an alternative approach. Sakawa, Yano and Yumine (1987) propose an interactive method for a multiobjective linear programming problem with fuzzy goals. Because they incorporate not only “fuzzy min zi (x)” or “fuzzy max zi (x)” which is a fuzzy goal of the DM such as “zi (x) should be substantially less than or equal to pi , or greater than or equal to qi ” but also “fuzzy equal zi (x)” which is a fuzzy goal such as “zi (x) should be in the vicinity of ri ,” and thus the concept of Pareto optimality cannot be applied, they introduce the following concept of M-Pareto optimal solutions deﬁned in terms of membership functions instead of objective functions (Sakawa, 1993): Deﬁnition 2.7 (M-Pareto optimal solution). A point x∗ ∈ X is said to be an MPareto optimal solution if and only if there does not exist another x ∈ X such that μi (zi (x)) ≥ μi (zi (x∗ )) for all i ∈ {1, . . . , k} and μ j (z j (x)) = μ j (z j (x∗ )) for at least one j ∈ {1, . . . , k}. After identifying the membership functions μi (zi (x)), i = 1, . . . , k for the fuzzy goals of the objective functions zi (x), i = 1, . . . , k, the DM is asked to specify the reference membership values which are the aspiration levels of achievement for the values of the membership functions. It follows that a vector of the reference membership values is a natural extension of the reference point in the reference point method by Wierzbicki (1980). Let μˆ = (ˆμ1 , . . . , μˆ k )T denote the reference membership values for the membership functions μ (z(x)) = (μ1 (z1 (x)), . . . , μk (zk (x)))T . Then, by solving the minimax problem minimize max {ˆμi − μi (zi (x))} 1≤i≤k (2.70) subject to Ax ≤ b, x ≥ 0, an M-Pareto optimal solution closest to the vector of the reference membership values in the L∞ norm can be obtained. The maximin problem (2.70) is equivalently expressed as ⎫ minimize v ⎬ subject to μˆ i − μi (zi (x)) ≤ v, i = 1, . . . , k (2.71) ⎭ Ax ≤ b, x ≥ 0. As we discussed linear programming with fuzzy parameters in the subsection 2.1.3, also in multiobjective situations, the formulation involving fuzzy parameters is still important. Sakawa and Yano (1990) extend the concept of Pareto optimality in order to deal with multiobjective linear programming problems with fuzzy parameters characterized by fuzzy numbers in the level set-based model. For given

2.3 Multiobjective programming

31

parameters A and b in the constraints, let X(a, b) denote the corresponding feasible region, i.e., X(a, b) {x ∈ Rn | Ax ≤ b, x ≥ 0}. Deﬁnition 2.8 (M-α-Pareto optimal solution). For a certain degree α speciﬁed by the DM, a point x∗ ∈ X(a∗ , b∗ ) is said to be an M-α-Pareto optimal solution if and only if there does not exist another x ∈ X(a, b), (c, a, b) ∈ (C, A, B)α such that μi (zi (x)) ≥ μi (zi (x∗ )) for all i ∈ {1, . . . , k} and μ j (z j (x)) = μ j (z j (x∗ )) for at least one j ∈ {1, . . . , k}, where the corresponding values of the parameters (c∗ , a∗ , b∗ ) ∈ (C, A, B)α are called the α-level optimal parameters. To derive a satisﬁcing solution to a multiobjective linear programming problem with fuzzy parameters for a certain degree α speciﬁed by the DM, the minimax problem utilized in the interactive method can be formulated as ⎫ minimize v ⎪ ⎪ ⎬ subject to μˆ i − μi (zi (x)) ≤ v, i = 1, . . . , k (2.72) Ax ≤ b, x ≥ 0 ⎪ ⎪ ⎭ (c, a, b) ∈ (C, A, B)α . Let I1 , I2 , and I3 be the index sets of the fuzzy goals for “fuzzy min zi (x),” “fuzzy max zi (x),” and “fuzzy equal zi (x),” respectively. For notational convenience, denote the strictly monotone decreasing functions of μi , i ∈ I1 and the right-hand side functions of μi , i ∈ I3 by diR , i ∈ I1 ∪ I3 , and the strictly monotone increasing functions of μi , i ∈ I2 and the left-hand side functions of μi , i ∈ I3 by diL , i ∈ I2 ∪ I3 . From the properties of the α-level sets, (2.72) can be transformed into ⎫ minimize v ⎪ ⎪ ⎪ −1 ⎬ subject to cLiα x ≤ diR (ˆμi − v), i ∈ I1 ∪ I3 (2.73) −1 R ⎪ ciα x ≥ diL (ˆμi − v), i ∈ I2 ∪ I3 ⎪ ⎪ ⎭ aLi1 x1 + · · · + aLin xn ≤ bRi , i = 1, . . . , m, x ≥ 0, −1 −1 and diL are pseudo-inverse functions deﬁned by where diR −1 diR (h) = sup{y | diR (y) ≥ h},

(2.74)

−1 (h) = diL

(2.75)

inf{y | diL (y) ≥ h}.

It should be noted that (2.73) can be solved by using the combined use of the two-phase simplex method and the bisection method. For readers interested in this issue, refer to Sakawa (1993).

32

2 Fundamentals

2.4 Two-level programming 2.4.1 Fuzzy programming for two-level programming In the real world, we often encounter situations where there are two or more DMs in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. In this section, we consider a case where there are two DMs; one of the DMs ﬁrst makes a decision, and then after acknowledging the decision of the ﬁrst DM, the other DM chooses a decision. Such a situation is formulated as a two-level programming problem. A linear programming problem with two DMs is formulated as follows. For the sake of simplicity, we call the two DMs DM1 and DM2 in this subsection. Let x1 and x2 denote the column vectors of the decision variables of DM1 and DM2, respectively, and let z1 (x1 , x2 ) = c11 x1 + c12 x2 and z2 (x1 , x2 ) = c21 x1 + c22 x2 denote the objective functions of DM1 and DM2, respectively, where ci1 , i = 1, 2 are n1 dimensional coefﬁcient row vector, and ci2 , i = 1, 2 are n2 dimensional coefﬁcient row vector. Assume that A1 x1 + A2 x2 ≤ b are the common constraints of DM1 and DM2, where A1 is an m × n1 coefﬁcient matrix, A2 is an m × n2 coefﬁcient matrix, b is an m dimensional constant column vector. Then, for a given x2 , DM1 deals with the following linear programming problem: ⎫ minimize z1 (x1 , x2 ) = c11 x1 + c12 x2 ⎪ ⎬ x1 (2.76) subject to A1 x1 ≤ b − A2 x2 ⎪ ⎭ x1 ≥ 0. Similarly, for a given x1 , the linear programming problem for DM2 is formulated as ⎫ minimize z2 (x1 , x2 ) = c21 x1 + c22 x2 ⎪ ⎬ x2 (2.77) subject to A2 x2 ≤ b − A1 x1 ⎪ ⎭ x2 ≥ 0. Combining the two problems (2.76) and (2.77) into one problem, we formulate the following linear programming problem with DM1 and DM2: ⎫ minimize z1 (x1 , x2 ) = c11 x1 + c12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c21 x1 + c22 x2 (2.78) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x2 ≥ 0, x1 ≥ 0, where “minimize” and “minimize” mean that DM1 and DM2 are minimizers for for DM1

for DM2

their objective functions. Assume that DM1 ﬁrst makes a decision and then DM2 chooses a decision later, and to be better off each other, they can coordinate their decisions. Namely, we as-

2.4 Two-level programming

33

sume that DM1 and DM2 make a binding agreement to select actions cooperatively, and then it is predicted that the selected actions should be in the set of Pareto optimal solutions. It should be noted here that Pareto optimality in two-level programming is deﬁned for the two objective functions: z1 of DM1 and z2 of DM2. For two-level linear programming problems with cooperative DMs, fuzzy programming approaches have been developed (Lai, 1996; Shih, Lai and Lee, 1996; Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009). In fuzzy programming for two-level linear programming by Sakawa et al. (Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009), for each of the objective functions zi (x), i = 1, 2 of (2.78), it is assumed that the DMs have fuzzy goals such as “the objective function zi (x) should be substantially less than or equal to some speciﬁc value pi .” Although the membership function does not always need to be linear, for the sake of simplicity, assume that a membership function μi (zi ) which characterizes the fuzzy goal of each DM is linear and it is speciﬁed as ⎧ ⎪ 1 if zi (x) ≤ z1i ⎪ ⎪ ⎨ zi (x) − z0i (2.79) if z1i < zi (x) ≤ z0i μi (zi (x)) = 1 − z0 ⎪ z ⎪ i i ⎪ ⎩0 if zi (x) > z0i . By identifying the membership functions μ1 (z1 (x)) and μ2 (z2 (x)) for the objective functions z1 (x) and z2 (x), the original two-level linear programming problem (2.78) can be interpreted as the membership function maximization problem deﬁned by ⎫ minimize μ1 (z1 (x)) ⎪ ⎪ for DM1 ⎪ ⎬ minimize μ2 (z2 (x)) (2.80) for DM2 ⎪ subject to Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. In (2.80), x ∈ Rn is an n dimensional decision variable vector, and it is divided into two vectors x1 and x2 which are n1 and n2 dimensional decision variable vectors of DM1 and DM2, respectively, i.e., n = n1 + n2 . Because the two DMs make decisions cooperatively, the decision variable vector is represented simply by x without partition. To derive an overall satisfactory solution to the membership function maximization problem (2.80), we ﬁrst ﬁnd the maximizing decision of the fuzzy decision proposed by Bellman and Zadeh (1970). Namely, the following problem is solved for obtaining a solution which maximizes the smaller degree of satisfaction between those of the two DMs: ⎫ maximize min{μ1 (z1 (x)), μ2 (z2 (x))} ⎬ subject to Ax ≤ b (2.81) ⎭ x ≥ 0.

34

2 Fundamentals

By introducing an auxiliary variable λ, this problem can be transformed into the following equivalent problem: ⎫ maximize λ ⎪ ⎪ ⎪ subject to μ1 (z1 (x)) ≥ λ ⎪ ⎬ μ2 (z2 (x))} ≥ λ (2.82) ⎪ ⎪ Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. Solving (2.82), we can obtain a solution which maximizes the smaller satisfactory degree between those of both DMs. It should be noted that if the membership functions μi (zi (x)), i = 1, 2 are linear membership functions such as (2.79), (2.82) becomes a linear programming problem. Let x∗ denote an optimal solution to problem (2.82). Then, we deﬁne the satisfactory degree of both DMs under the constraints as (2.83) λ∗ = min{μ1 (z1 (x∗ )), μ2 (z2 (x∗ ))}. If DM1 is satisﬁed with the optimal solution x∗ , it follows that the optimal solution x∗ becomes a satisfactory solution; however, DM1 is not always satisﬁed with the solution x∗ . If DM1 is not satisﬁed with the solution x∗ , it is quite natural to assume that DM1 speciﬁes the minimal satisfactory level δ ∈ [0, 1] for the membership function μ1 (z1 (x)) subjectively. Then, the following problem is formulated: ⎫ maximize μ2 (z2 (x)) ⎪ ⎪ ⎬ subject to μ1 (z1 (x)) ≥ δ (2.84) Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0, where DM2’s membership function is maximized under the condition that DM1’s membership function μ1 (z1 (x)) is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1. It should be also noted that if the membership functions μi (zi (x)), i = 1, 2 are linear membership functions such as (2.79), then (2.84) also becomes a linear programming problem. If there exists an optimal solution to (2.84), it follows that DM1 obtains a satisfactory solution having a satisfactory degree larger than or equal to the minimal satisfactory level speciﬁed by DM1’s self. However, the larger the minimal satisfactory level δ is assessed, the smaller the DM2’s satisfactory degree becomes when the objective functions of DM1 and DM2 conﬂict with each other. Consequently, a relative difference between the satisfactory degrees of DM1 and DM2 becomes larger, and it follows that the overall satisfactory balance between both DMs is not appropriate. In order to take account of the overall satisfactory balance between both DMs, DM1 needs to compromise with DM2 on DM1’s own minimal satisfactory level. To do so, the following ratio of the satisfactory degree of DM2 to that of DM1 is helpful:

2.4 Two-level programming

35

Δ=

μ2 (z2 (x)) , μ1 (z1 (x))

(2.85)

which is originally introduced by Lai (1996). DM1 is guaranteed to have a satisfactory degree larger than or equal to the minimal satisfactory level for the fuzzy goal because the corresponding constraint μ1 (z1 (x)) ≥ δ is involved in (2.84). To take into account the overall satisfactory balance between both DMs, DM1 speciﬁes the lower bound Δmin and the upper bound Δmax of the ratio, and then it is veriﬁed whether the ratio Δ is in the interval [Δmin , Δmax ] or not. The condition that the overall satisfactory balance is appropriate is represented by (2.86) Δ ∈ [Δmin , Δmax ]. At the iteration l, let μ1 (zl1 ), μ2 (zl2 ), λl and Δl = μ2 (zl2 )/μ1 (zl1 ) denote DM1’s and DM2’s satisfactory degrees, a satisfactory degree of both DMs, and the ratio of satisfactory degrees of the two DMs, respectively. Let the solution be xl at the iteration l. The interactive process terminates if the following two conditions are satisﬁed and if DM1 concludes the solution as an overall satisfactory solution. Termination conditions of the interactive process Condition 1: DM1’s satisfactory degree is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1’s self, i.e., μ1 (zl1 ) ≥ δ. Condition 2: The ratio Δl of satisfactory degrees lies in the closed interval between the lower and the upper bounds speciﬁed by DM1, i.e., Δl ∈ [Δmin , Δmax ]. Condition 1 ensures the minimal satisfaction to DM1 in the sense of the attainment of the fuzzy goal, and condition 2 is provided in order to keep overall satisfactory balance between both DMs. If these two conditions are not satisﬁed simultaneously, DM1 needs to update the minimal satisfactory level δ. The updating procedures are summarized as follows: Procedure for updating the minimal satisfactory level δ Case 1: If condition 1 is not satisﬁed, then DM1 decreases the minimal satisfactory level δ. Case 2: If the ratio Δl exceeds its upper bound, then DM1 increases the minimal satisfactory level δ. Conversely, if the ratio Δl is below its lower bound, then DM1 decreases the minimal satisfactory level δ. Case 3: Although conditions 1 and 2 are satisﬁed, if DM1 is not satisﬁed with the obtained solution and judges that it is desirable to increase the satisfactory degree of DM1 at the expense of the satisfactory degree of DM2, then DM1 increases the minimal satisfactory level δ. Conversely, if DM1 judges that it is desirable to increase the satisfactory degree of DM2 at the expense of the satisfactory degree of DM1, then DM1 decreases the minimal satisfactory level δ. In particular, if condition 1 is not satisﬁed, it follows that there does not exist any feasible solution for (2.84), and therefore DM1 has to moderate the minimal satisfactory level.

36

2 Fundamentals

We are now ready to give a procedure of interactive fuzzy programming for deriving an overall satisfactory solution to (2.78), which is summarized in the following. Algorithm of interactive fuzzy programming Step 1: Ask DM1 to identify the membership function μ1 (z1 ) of the fuzzy goal for the objective function z1 (x). Similarly, ask DM2 to identify the membership function μ2 (z2 ) of the fuzzy goal for the objective function z2 (x). Step 2: Set l := 1 and solve (2.82), in which a smaller satisfactory degree between those of DM1 and DM2 is maximized. If DM1 is satisﬁed with the obtained optimal solution, the solution becomes a satisfactory solution. Otherwise, ask DM1 to specify the minimal satisfactory level δ together with the lower and the upper bounds [Δmin , Δmax ] of the ratio of satisfactory degrees Δl with the satisfactory degree λ∗ of both DMs and the related information about the solution in mind. Step 3: Set l := l + 1. Solve (2.84), in which the satisfactory degree of DM2 is maximized under the condition that the satisfactory degree of DM1 is larger than or equal to the minimal satisfactory level δ, and then an optimal solution xl to (2.84) is proposed to DM1 together with λl , μ1 (zl1 ), μ2 (zl2 ) and Δl . Step 4: If the solution xl satisﬁes the termination conditions and DM1 accepts it, then the procedure stops, and the solution xl is determined to be a satisfactory solution. Step 5: Ask DM1 to revise the minimal satisfactory level δ in accordance with the procedure of updating minimal satisfactory level. Return to step 3.

2.4.2 Stackelberg solution to two-level programming problem In the previous subsection, we considered a two-level linear programming problem in which two DMs can coordinate their decisions. However, for example, if the two DMs are corporate managers in different companies and the companies have business with each other, it is natural to suppose that there should be conﬂict interests between them. If the two DMs have conﬂicting interests, they cannot always cooperate in their decisions. Then, it is difﬁcult to employ a cooperative solution method such as the fuzzy programming approach in the previous subsection. For noncooperative decision making in a two-level programming problem, a DM who ﬁrst makes a decision is called the leader and the other DM is called the follower conventionally. The leader ﬁrst speciﬁes a decision and then the follower determines a decision so as to optimize the objective function of the follower with full knowledge of the decision of the leader. Anticipating this, the leader also makes a decision so as to optimize the objective function of self. The solution deﬁned as the above mentioned procedure is a Stackelberg equilibrium solution, and we call it a Stackelberg solution shortly. A two-level linear programming problem for obtaining the Stackelberg solution is formulated as

2.4 Two-level programming

37

⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ x ⎪ ⎪ ⎪ where y solves ⎬ minimize z2 (x, y) = c2 x + d2 y y ⎪ ⎪ ⎪ ⎪ subject to Ax + By ≤ b ⎪ ⎭ x ≥ 0, y ≥ 0,

(2.87)

where ci , i = 1, 2 are n1 dimensional coefﬁcient row vector, di , i = 1, 2 are n2 dimensional coefﬁcient row vector, A is an m × n1 coefﬁcient matrix, B is an m × n2 coefﬁcient matrix, b is an m-dimensional constant column vector. In the two-level linear programming problem (2.87), z1 (x, y) and z2 (x, y) represent the objective functions of the leader and the follower, respectively, and x and y represent the decision variables of the leader and the follower, respectively. Each DM knows the objective function of the opponent as well as the objective function of self and the constraints. The leader ﬁrst makes a decision, and then the follower makes a decision so as to minimize the objective function with full knowledge of the decision of the leader. Namely, after the leader chooses x, the follower solves the following linear programming problem: ⎫ minimize z2 (x, y) = d2 y + c2 x ⎪ ⎬ y (2.88) subject to By ≤ b − Ax ⎪ ⎭ y ≥ 0, and chooses an optimal solution y(x) to (2.88) as a rational response. Assuming that the follower chooses the rational response, the leader also makes a decision such that the objective function z1 (x, y(x)) is minimized. Then, the solution deﬁned as the above mentioned procedure is a Stackelberg solution. Computational methods for obtaining Stackelberg solutions to two-level linear programming problems are classiﬁed roughly into three categories: the vertex enumeration approach (Bialas and Karwan, 1984), the Kuhn-Tucker approach (Bard and Falk, 1982; Bard and Moore, 1990; Bialas and Karwan, 1984; Hansen, Jaumard and Savard, 1992), and the penalty function approach (White and Anandalingam, 1993). The vertex enumeration approach takes advantage of the property that there exists a Stackelberg solution in a set of extreme points of the feasible region. In the Kuhn-Tucker approach, the leader’s problem with constraints involving the optimality conditions of the follower’s problem is solved. In the penalty function approach, a penalty term is appended to the objective function of the leader so as to satisfy the optimality of the follower’s problem. It is well-known that a two-level linear programming problem is an NP-hard problem (Shimizu, Ishizuka and Bard, 1997). In the following, we outline a couple of conventional computational methods for obtaining Stackelberg solutions. The kth best method proposed by Bialas and Karwan (1984) can be thought of as the vertex enumeration approach, and it is based on a very simple idea. The solution search procedure of the method starts from a point which is an optimal solution to the problem of the leader and checks whether it is also an optimal solution to

38

2 Fundamentals

the problem of the follower or not. If the ﬁrst point is not the Stackelberg solution, the procedure continues to examine the second best solution to the problem of the leader, and so forth. At the beginning, the following linear programming problem is solved: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎬ x (2.89) subject to Ax + By ≤ b ⎭ x ≥ 0, y ≥ 0. Assuming that the feasible region of (2.89) is not empty and there are N extreme points, i.e., N basic solutions of (2.89). Let (ˆx[1] , yˆ [1] ) denote an optimal solution to (2.89), and (ˆx[2] , yˆ [2] ), . . . , (ˆx[N] , yˆ [N] ) be the rest of N − 1 basic feasible solutions such that z1 (x[ j] , y[ j] ) ≤ z1 (x[ j+1] , y[ j+1] ), j = 1, . . . , N −1. It is veriﬁed if the solution (ˆx[ j] , yˆ [ j] ) is optimal to (2.88) of the follower from j = 1 to j = N in turn. Then, the ﬁrst solution found to be optimal to (2.88) is the Stackelberg solution. In the Kuhn-Tucker approach, the leader’s problem with constraints involving the optimality conditions of the follower’s problem (2.88) is solved. The Kuhn-Tucker conditions for (2.88) are shown as follows: ⎫ uB − v = −d2 ⎪ ⎪ ⎬ u(Ax + By − b) − vy = 0 (2.90) Ax + By ≤ b ⎪ ⎪ ⎭ y ≥ 0, uT ≥ 0, vT ≥ 0, where u is an m dimensional row vector and v is an n2 dimensional row vector. Then, the follower’s problem (2.88) for a two-level linear programming problem can be replaced by the above conditions (2.90), and (2.87) is rewritten as the following equivalent single-level mathematical programming problem: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ ⎪ ⎪ subject to uB − v = −d2 ⎬ u(Ax + By − b) − vy = 0 (2.91) ⎪ ⎪ Ax + By ≤ b ⎪ ⎪ ⎭ x ≥ 0, y ≥ 0, uT ≥ 0, vT ≥ 0. From the equality constraint uB − v = −d2 of (2.91), v is eliminated and the equality constraint u(Ax + By − b) − vy = 0 is transformed into u(b − Ax − By) + (uB + d2 )y = 0.

(2.92)

Moreover, the complementarity condition (2.92) implies that b − Ax − By ≥ 0, uT ≥ 0, (uB+d2 )T ≥ 0, y ≥ 0. Let Ai and Bi be the ith row vector of the matrix A and the matrix B, respectively, and let B j and d2 j be the jth column vector of the matrix B and the jth element of the vector d2 , respectively. Then, the condition of either ui = 0 or bi − Ai x − Bi y = 0 for i = 1, . . . , m and the condition of either uB j + d2 j = 0 or y j = 0 for j = 1, . . . , n2 must be satisﬁed simultaneously. By introducing zero-one

2.4 Two-level programming

39

vectors w1 = (w11 , . . . , w1m ) and w2 = (w21 , . . . , w2n2 ), the equality constraint (2.92) can be expressed as follows (Fortuny-Amat and McCarl, 1981): ⎫ u ≤ Mw1 ⎪ ⎪ ⎬ b − Ax − By ≤ M(e − wT1 ) (2.93) uB + d2 ≤ Mw2 ⎪ ⎪ ⎭ T y ≤ M(e − w2 ), where e is an m dimensional vector of ones, and M is a large positive constant. Therefore, the mathematical programming problem (2.91) is equivalent to the following mixed zero-one programming problem, and it can be solved by a zeroone mixed integer solver: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ ⎪ ⎪ subject to 0 ≤ uT ≤ MwT1 ⎪ ⎪ ⎬ T 0 ≤ b − Ax − By ≤ M(e − w1 ) (2.94) 0 ≤ (uB + d2 )T ≤ MwT2 ⎪ ⎪ ⎪ ⎪ 0 ≤ y ≤ M(e − wT2 ) ⎪ ⎪ ⎭ x ≥ 0. In the penalty function approach, the duality gap of the follower’s problem (2.88) is appended to the objective function of the leader. The dual problem to (2.88) ignoring the constant term c2 x is written as ⎫ minimize u(Ax − b) ⎬ subject to −uB ≤ d2 (2.95) ⎭ uT ≥ 0, where u is an m-dimensional row vector. Because the duality gap d2 y − u(Ax − b) is zero if y is a rational response of the follower with respect to a choice x of the leader, the following mathematical programming problem is formulated: ⎫ minimize c1 x + d1 y + Ku(Ax − b) ⎪ ⎪ ⎬ subject to Ax + By ≤ b (2.96) −uB ≤ d2 ⎪ ⎪ ⎭ T x ≥ 0, y ≥ 0, u ≥ 0, where K is a constant value. By repeatedly solving (2.96) for updated values of K and u, (2.96) yields an optimal solution to (2.87), i.e., the Stackelberg solution.

40

2 Fundamentals

2.5 Genetic algorithms 2.5.1 Fundamental elements in genetic algorithms It is hard to obtain exact optimal solutions of difﬁcult classes of optimization problems such as combinatorial problems and nonconvex nonlinear problems, and thus it is quite natural for DMs to require approximate optimal solutions instead. To meet this demand, recently several meta-heuristics have been developed and their effectiveness is demonstrated. Among them, genetic algorithms are known to be one of the most practical and proven methods, and in this section we present the basic concepts of the genetic algorithms. A computational framework of genetic algorithms initiated by Holland (1975) has been attracted attention of many researchers with applicability in optimization, as well as in search and learning. Furthermore, publications of books by Goldberg (1989) and Michalewicz (1996) bring heightened and increasing interests in applications of genetic algorithms to complex function optimization. start Step 0 generating the initial population Step 1 calculating the fitness Step 2 reproducting the next generation Step 3 executing crossover operation Step 4 executing mutation operation Step 5 No

termination condition is satisfyed? Yes end

Fig. 2.4 Flowchart of genetic algorithms.

2.5 Genetic algorithms

41

The fundamental procedure of genetic algorithms is shown as a ﬂowchart in Fig. 2.4, and it is summarized as follows: Step 0: Initialization. Generate a given number of individuals randomly to form the initial population. Step 1: Evaluation. Calculate the ﬁtness value of each individual in the population. Step 2: Reproduction. According to the ﬁtness values and a reproduction rule speciﬁed in advance, select individuals from the current population to form the next population. Step 3: Crossover. Select two individuals randomly from the population, and exchange some part of the string of one individual for the corresponding part of the other individual with a given probability for crossover. Step 4: Mutation. Alter one or more genes in the string of an individual with a given probability of mutation. Step 5: Termination. Stop the procedure if the condition of termination is satisﬁed, and an individual with the maximum ﬁtness value is determined as an approximate optimal solution. Otherwise, return to step 1.

2.5.1.1 Representation of individuals When genetic algorithms are applied to optimization problems, a vector of decision variables corresponds to an individual in the population, which is represented by a string as in Fig. 2.5. 1 0 1 1 .... 1 0 1 0 0 n-bit string for a vector x of decision variables

Fig. 2.5 Individual represented by a string.

As seen in Fig. 2.5, each element of the string is either 1 or 0 usually, but real numbers, integers, alphabets, or some other symbols can also be used to represent individuals. Let s and x denote an individual represented by a string and a vector of the decision variables, respectively. The string s which means a chromosome in the context of biology is called the genotype of an individual, and the decision variables x is called the phenotype. The mapping from phenotypes to genotypes is called coding, and the reverse mapping is called decoding.

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2 Fundamentals

2.5.1.2 Fitness function and scaling In an optimization problem where the objective function is minimized or maximized, a solution with the smallest objective function value or the largest objective function value is searched. When a genetic algorithm is applied to solving an optimization problem, a solution to the optimization problem is associated with an individual in the genetic algorithm, and the objective function value of the solution corresponds to the ﬁtness of the individual. Thus, an individual with a larger ﬁtness value has a higher probability of surviving in the next generation. Let z(x) denote an objective function to be minimized in an optimization problem. The corresponding ﬁtness function in genetic algorithms is commonly deﬁned as (Goldberg, 1989) Cmax − z(x) if z(x) < Cmax (2.97) f (si ) = 0 otherwise, where si denotes the ith individual in the population, and Cmax is a given constant. For example, the value of Cmax is determined as the largest objective function value z(x) observed thus far, the largest value z(x) in the current population, or the largest value z(x) in the last t generations. Similarly, in maximization problems, to prevent the ﬁtness value from being negative, the constant Cmin is introduced, and the following ﬁtness function is often used: z(x) +Cmin if z(x) +Cmin > 0 (2.98) f (si ) = 0 otherwise. The absolute value of the smallest z(x) in the current population or in the last t generations is often used for the value of Cmin . To properly distribute ﬁtness values in the population, ﬁtness scaling is employed. The linear scaling, which is a simple and useful procedure, is represented by (2.99) f (si ) = a f (si ) + b, where f and f are the raw ﬁtness value and the scaled ﬁtness value, respectively; a and b are coefﬁcients. To perform the operation of reproduction appropriately, the coefﬁcients a and b may be chosen in such a way that the average scaled ﬁtness

is equal to the average raw ﬁtness value f , and the maximum scaled value fave ave

ﬁtness value is determined as fmax = Cmult fave , where Cmult is a given constant.

2.5.1.3 Genetic operators The three genetic operators, reproduction, crossover, and mutation, are outlined below. Individuals are copied into the next generation according to their ﬁtness values by a reproduction operator. The roulette wheel selection is one of the most popular

2.5 Genetic algorithms

43

reproduction operators, and in this method, each individual in the current population has a roulette wheel slot sized in proportion to its ﬁtness value. Let pop size be the number of individuals in the population. The percentage of the roulette wheel given pop size f (sl )%. Namely, the individual si is reproto an individual si is 100 f (si )/ ∑l=1 pop size f (sl ) each spin of the roulette duced with the probability p(si ) = f (si )/ ∑l=1 wheel. An example of the roulette wheel is given in Fig. 2.6; the numbers in the wheel are ﬁtness values of individuals, and the decimal numbers outside of the wheel are the corresponding probabilities. 0.04 s6 16

0.26 s5

0.14 s1 56

104 76

s4 0.12

48

s2

0.19

100 s3

p(si ) f (si )

s1

s2

s3

s4

s5

s6

0.14

0.19

0.25

0.12

0.26

0.04

56

76

100

48

104

16

0.25

Fig. 2.6 Biased roulette wheel.

Crossover creates offsprings into the next population by combining the genetic material of two parents. The variation caused by the crossover process may bring offsprings better ﬁtness values, and thus it is thought that crossover plays an important role in genetic algorithms. Although there are many different types of crossover, we provide a simple example here: a single-point crossover operator is the most simple operator of crossover. In this operation, two parent strings s1 and s2 are randomly chosen from a mating pool in which newly reproduced individuals are entered temporarily, and then one crossover point in the strings is chosen at random. Two offsprings are made by exchanging the substrings which are parts of the left-hand side of the parent strings s1 and s2 from the crossover point. The crossover operation is illustrated in Fig. 2.7. parent 1 1 1 0 0 1 0 1 0 1

offspring 1

10111 0 0 01

parent 2 1 0 1 1 1 0 0 1 0

offspring 2

11001 0 1 10

crossover point

exchange

Fig. 2.7 Crossover operation.

With a small probability, an operation of mutation provides the string of an individual with a randomly tiny alteration, and it is recognized that mutation serves as local search. In the representation of the 0-1 bit strings, mutation means changing a

44

2 Fundamentals

1 to a 0 and vice versa. A simple version of mutation operator is illustrated in Fig. 2.8. parent 1 0 1 1 1 0 1 0 0

offspring 1 0 0 1 1 0 1 0 0

Fig. 2.8 Mutation operation.

2.5.2 Genetic algorithm for integer programming So far we have discussed fundamental elements in genetic algorithms. In the remaining chapters of this book, we will deal with mathematical programming problems not only with continuous decision variables but also with discrete decision variables, and for solving mathematical programming problems with discrete decision variables, i.e., integer programming problems, we give a computational method based on the framework of genetic algorithms, which is called GADSLPRRSU by Sakawa (2001). GADSLPRRSU is an abbreviation for a genetic algorithm with double strings based on linear programming relaxation and reference solution updating. This method includes three key ideas: double strings (DS), linear programming relaxation (LPR), and reference solution updating (RSU). These ideas were introduced one by one in the development of this method. Sakawa, Kato, Sunada and Shibano (1997) ﬁrst attempt to solve a multidimensional 0-1 knapsack problem by applying a genetic algorithm. For a given set of n kinds of items, the aim of the multidimensional knapsack problem is to select a subset of the items so as to maximize the total values of the selected items under the multiple constraints such as capacities, budgets, and so forth. Let x j be a decision variable which indicates whether the jth item is selected or not. Namely, the decision variable x j is 1 if the jth item is selected, and otherwise it is 0. Then, the general form of multidimensional knapsack problems is given as ⎫ minimize cx ⎬ subject to Ax ≤ b (2.100) ⎭ x j ∈ {0, 1}, j = 1, . . . , n, where c, A and b are an n dimensional row vector of positive coefﬁcients in the objective function, an m × n matrix of positive coefﬁcients in the left-hand side constraints and an m dimensional column vector of positive constants in the righthand side constraints. If a simple genetic algorithm described above is applied to a multidimensional 0-1 knapsack problem, an individual in the population is not always decoded into a feasible solution to the multidimensional 0-1 knapsack problem. Sakawa, Kato,

2.5 Genetic algorithms

45

Sunada and Shibano (1997) resolve this difﬁculty by introducing the double strings representation of individuals and the corresponding decoding algorithm. Furthermore, to apply this method to a multidimensional integer knapsack problem in which the domain of a decision variable is extended to a set of nonnegative integers, they utilize information of optimal solutions to linear programming relaxation problems (Sakawa et al., 2000). By using the fact that the zero solution, x = 0, is feasible to a multidimensional 0-1 or integer knapsack problem, these methods decode any individual into a feasible solution to the problem. However, in an integer programming problem which includes a multidimensional 0-1 or integer knapsack problem as a special case, the zero solution is not always feasible, and therefore these methods cannot directly apply to integer programming problems. An integer programming problem is formally given as ⎫ minimize cx ⎬ subject to Ax ≤ b (2.101) ⎭ x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n, where A = [p1 , . . . , pn ] is an m × n coefﬁcient matrix, b = (b1 , . . . , bm )T is an m dimensional constant column vector and c = (c1 , . . . , cn ) is an n dimensional coefﬁcient row vector. In contrast to knapsack problems such as (2.100), any coefﬁcient of (2.101) is not always positive. To overcome this problem, they propose a revised version (GADSLPRRSU) of their methods with a decoding algorithm with reference solutions which are found in advance and are updated if necessary (Sakawa, 2001). GADSLPRRSU for solving integer programming problems (2.101) is summarized as follows. As we pointed out, GADSLPRRSU employs the double string representation of individuals depicted in Fig. 2.9. s(1)s(2) s(3)s(4)

s( j)

s(n -1) s(n )

3 6 2 1 ... 4 .... 5 8 2 0 0 1 ... 3 .... 1 4 gs(1) gs(2) gs(3)gs(4)

gs( j)

x 3 = 2, x 6= 0, x 2 = 0, x 1 = 1, .... , x 4 = 3, ... , x 5 = 1, x 8 = 4

gs(n -1) gs(n )

Fig. 2.9 Double string representation.

In the ﬁgure, each of s( j), j = 1, . . . , n is the index of an element in a solution vector and each of gs( j) ∈ {0, 1, . . . , νs( j) }, j = 1, . . . , n is the value of the element, respectively. For example, a pair (s( j), gs( j) ) means that the value of the s( j)th element xs( j) is gs( j) , i.e., xs( j) = gs( j) . A decoding algorithm for the double string representation with reference points generates feasible solutions from any individuals based on a certain reference solution x∗ which is used as the origin of decoding. In the following algorithm, b+ denotes a column vector of a part of the right-hand side constants in (2.101) which

46

2 Fundamentals

are positive, and the corresponding coefﬁcient matrix of the left-hand side of the + constraints is denoted by A+ = [p+ 1 , . . . , pn ]. Decoding algorithm using a reference solution Step 1: Let j := 1 and psum := 0. Step 2: If gs( j) = 0, set qs( j) := 0 and j := j + 1, and go to step 4. + + Step 3: If psum + gs( j) p+ s( j) ≤ b , set qs( j) := gs( j) , psum := psum + gs( j) ps( j) and j := j + 1. Otherwise, set qs( j) := 0 and j := j + 1. Step 4: If j ≤ n, return to step 2. Step 5: Let j := 1, l := 0 and sum := 0. Step 6: If gs( j) = 0, set j := j + 1 and go to step 8. If gs( j) = 0, set sum := sum + gs( j) ps( j) . Step 7: If sum ≤ b, set l := j, j := j + 1. Otherwise, set j := j + 1. Step 8: If j ≤ n, return to step 6. Step 9: If l ≤ 0, go to step 11. Step 10: For xs( j) satisfying 1 ≤ j ≤ l, let xs( j) := gs( j) . For xs( j) satisfying l + 1 ≤ j ≤ n, let xs( j) := 0, and the algorithm terminates. ∗ p Step 11: Let sum := ∑nk=1 xs(k) s(k) and j := 1. ∗ , let x Step 12: If gs( j) = xs( s( j) := gs( j) and j := j + 1, and go to step 16. j) ∗ p ∗ Step 13: If sum−xs( +g s( j) s( j) ps( j) ≤ b, set sum := sum−xs( j) ps( j) +gs( j) ps( j) j) and xs( j) := gs( j) , and go to step 16. ∗ Step 14: Let ts( j) := 0.5(xs( j) + gs( j) ). ∗ ∗ Step 15: If sum−xs( j) ps( j) +ts( j) ps( j) ≤ b, set sum := sum−xs( j) ps( j) +ts( j) ps( j) , ∗ . gs( j) := ts( j) and xs( j) := ts( j) . Otherwise, set xs( j) := xs( j) Step 16: If j > n, the algorithm terminates. Otherwise, return to step 12. For general integer programming problems involving positive and negative coefﬁcients in the constraints, this decoding algorithm produces feasible solution from any individuals in the population. However, the diversity of generated feasible solutions depends on the reference solution used in the decoding algorithm. To overcome this difﬁculty, GADSLPRRSU adopts the reference solution updating procedure in which the current reference solution is updated by another feasible solution if the diversity of generated solutions seems to be lost. It is expected that an optimal solution to the following linear programming relaxation problem becomes a good approximate optimal solution of the original integer programming problem (2.101): ⎫ minimize cx ⎬ subject to Ax ≤ b (2.102) ⎭ 0 ≤ x j ≤ ν j , j = 1, . . . , n. From this observation, GADSLPRRSU exploits the information about the optimal solution to the continuous relaxation problem to ﬁnd an approximate optimal solution with high accuracy in reasonable time. The computational procedure of GADSLPRRSU is summarized as follows:

2.5 Genetic algorithms

47

Computational procedure of GADSLPRRSU Step 0: Determine values of the parameters used in the genetic algorithm. Step 1: Generate the initial population consisting of N individuals based on the information of an optimal solution to the continuous relaxation problem (2.102). Step 2: Decode each individual (genotype) in the current population and calculate its ﬁtness based on the corresponding solution (phenotype). Step 3: If the termination condition is fulﬁlled, the procedure stops. Otherwise, let t := t + 1. Step 4: Apply the reproduction operator using the elitist expected value selection after linear scaling. Step 5: Apply the crossover operator, called PMX (Partially Matched Crossover) for a double string. Step 6: Apply the mutation based on the information of an optimal solution to the continuous relaxation problem (2.102). Step 7: Apply the inversion operator, and return to step 2. For more information about GADSLPRRSU, it is recommended to read Sakawa (2001).

Chapter 3

Fuzzy Multiobjective Stochastic Programming

In this chapter, by considering the imprecision of a decision maker’s (DM’s) judgments for stochastic objective functions and/or constraints in multiobjective problems, fuzzy multiobjective stochastic programming is developed. Assuming that the DM has a fuzzy goal for each of expectations and variances of the original stochastic objective functions, multiobjective stochastic programming problems are formulated. For reﬂecting the diversity of criteria for optimizing the stochastic objective functions, optimization criteria different from expectation and variance are also provided to maximize the probability of the objective functions being greater than or equal to target values as well as to optimize the target values under a given probability. For stochastic programming problems, when a shortage or an excess comes from the violation of the constraints, it is often observed that some penalties are imposed. We also deal with the simple recourse model, which is developed to formulate such stochastic situations. Through the introduction of extended Pareto optimal solution concepts, interactive fuzzy satisﬁcing methods using linear or convex programming techniques are presented to derive a satisﬁcing solution for the DM from among the extended Pareto optimal solution set. The proposed interactive methods for the multiobjective stochastic programming with continuous variables are immediately extended to deal with the problems with integer decision variables by utilizing some genetic algorithms.

3.1 Fuzzy multiobjective stochastic linear programming In actual decision making situations, we must often make a decision on the basis of imprecise information or uncertain data. As described in Chapter 2, stochastic programming and fuzzy programming have been developed for solving such decision making problems involving uncertainty. For multiobjective stochastic linear programming problems, Stancu-Minasian (1984, 1990) considered the minimum risk approach, while Leclercq (1982) and Teghem, Dufrane, Thauvoye and Kunsch (1986) proposed interactive methods. M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_3, © Springer Science+Business Media, LLC 2011

49

50

3 Fuzzy Multiobjective Stochastic Programming

Fuzzy multiobjective linear programming, ﬁrst proposed by Zimmermann (1978), has been also developed by numerous researchers, and an increasing number of successful applications have been appearing (Sakawa, Yano and Yumine, 1987; Luhandjula, 1987; Kacprzyk and Orlovski, 1987; Zimmermann, 1987; Verdegay and Delgado, 1989; Slowinski and Teghem, 1990; Sakawa, 1993; Lai and Hwang, 1994; Delgado, Kacprzyk, Verdegay and Vila, 1994; Slowinski, 1998; Sakawa, 2000, 2001). Although these studies focused on either fuzziness or randomness in multiobjective programming problems, it is important to simultaneously consider fuzziness and randomness in order to deal with more realistic decision making problems. In this section, we consider fuzzy programming approaches to multiobjective linear programming problems with random variable coefﬁcients in objective functions and/or constraints. In several stochastic models including the expectation model, the variance model, the probability model, the fractile model and the simple recourse model together with chance constrained programming techniques or two-stage problem formulations, the stochastic programming problems are transformed into deterministic ones. Assuming that a decision maker (DM) has a fuzzy goal for each of the objective functions, we present several interactive fuzzy satisﬁcing methods to derive a satisﬁcing solution for the DM by updating the reference membership levels for fuzzy goals represented by the membership functions. Some numerical examples are provided to demonstrate the feasibility and efﬁciency of the interactive fuzzy satisﬁcing method.

3.1.1 Expectation and variance models Throughout this subsection, assuming that the coefﬁcients in objective functions and right-hand side constants of constraints are random variables, we deal with multiobjective linear programming problems formulated as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ ⎪ ··· ⎬ minimize zk (x) = c¯ k x (3.1) ⎪ ⎪ subject to Ax ≤ b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, and A is an m × n coefﬁcient matrix, c¯ l , l = 1, . . . , k are n dimensional Gaussian random variable row vectors with ﬁnite means E[¯cl ] and n × n positive-deﬁnite variance-covariance matrices Vl = (vljh ) = (Cov{cl j , clh }), l = 1, . . . , k, and b¯ is an n dimensional vector whose elements are mutually independent random variables with continuous and increasing probability distribution functions. Multiobjective linear programming problems with random variable coefﬁcients are said to be multiobjective stochastic linear programming problems, which are of-

3.1 Fuzzy multiobjective stochastic linear programming

51

ten seen in actual decision making situations. For example, consider a production planning problem to optimize the gross proﬁt and the production cost simultaneously under the condition that unit proﬁts of the products, unit production costs of them and the maximal available amounts of the resources depend on seasonal factors or market prices. Such a production planning problem can be formulated as a multiobjective stochastic programming problem expressed by (3.1). Observing that (3.1) contains random variable coefﬁcients, the deﬁnitions and solution methods for ordinary mathematical programming problems not depending on stochastic events cannot be directly applied. Realizing such difﬁculty, we deal with the constraints in (3.1) as chance constrained conditions (Charnes and Cooper, 1959) which permit constraint violations up to speciﬁed probability limits. Let ηi , i = 1, . . . , m denote the probability that the ith constraint should be satisﬁed, and we call the probability the satisﬁcing probability level. Then, replacing the constraints in (3.1) with chance constrained conditions with satisﬁcing probability levels ηi , i = 1, . . . , m, we can reformulate the chance constrained problem ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ ⎪ ······ ⎪ ⎪ ⎪ ⎪ minimize zk (x) = c¯ k x ⎬ subject to P(ω | a1 x ≤ b1 (ω)) ≥ η1 (3.2) ⎪ ⎪ ········· ⎪ ⎪ ⎪ P(ω | am x ≤ bm (ω)) ≥ ηm ⎪ ⎪ ⎪ ⎭ x ≥ 0, where ai is the ith row vector of A, and bi (ω) is the ith element of the realized vector ¯ b(ω) for an elementary event ω. Denoting continuous and increasing distribution functions of the random variables b¯ i , i = 1, . . . , m by Fi (r) = P(ω | bi (ω) ≤ r), the ith constraint in (3.2) can be rewritten as P(ω | ai x ≤ bi (ω)) ≥ ηi ⇔ 1 − P(ω | bi (ω) ≤ ai x) ≥ ηi ⇔ 1 − Fi (ai x) ≥ ηi ⇔ ai x ≤ Fi−1 (1 − ηi ).

(3.3)

From (3.3), for the speciﬁed vector of the satisﬁcing probability levels η = (η1 , . . . , ηm )T , (3.2) can be equivalently transformed as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎬ ······ (3.4) minimize zk (x) = c¯ k x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where

η) x | ai x ≤ Fi−1 (1 − ηi ), i = 1, . . . , m, x ≥ 0 . X(η

52

3 Fuzzy Multiobjective Stochastic Programming

3.1.1.1 Expectation model As a ﬁrst attempt to deal with multiobjective stochastic programming problems, assuming that the DM would like to simply optimize the expected objective functions, we introduce the expectation model (Sakawa and Kato, 2002; Sakawa, Kato and Nishizaki, 2003). By replacing the objective functions zl (x) = c¯ l x, l = 1, . . . , k in (3.4) with their expectations, the multiobjective stochastic programming problem can be reformulated as ⎫ minimize zE1 (x) E[z1 (x)] = E[¯c1 ]x ⎪ ⎪ ⎬ ······ (3.5) minimize zEk (x) E[zk (x)] = E[¯ck ]x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where E[zl ] denotes the expectation of zl , and E[¯cl ] = (E[c¯l1 ], . . . , E[c¯ln ]). In order to consider the imprecise nature of the DM’s judgments for each expectation of the objective function zEl (x) = E[¯cl ]x, l = 1, . . . , k in (3.5), if we introduce the fuzzy goals such as “zEl (x) should be substantially less than or equal to a certain value”, (3.5) can be interpreted as maximize (μ1 (zE1 (x)), . . . , μk (zEk (x))),

(3.6)

η) x∈X(η

where μl is a membership function to quantify the fuzzy goal for the lth objective function in (3.5). To be more speciﬁc, if the DM feels that zEl (x) should be smaller than or equal to at most zEl,0 and zEl (x) ≤ zEl,1 (< zEl,0 ) is satisfactory, the shape of a typical membership function can be shown as in Fig. 3.1.

μl (zEl (x)) 1

0

z l,E 1

z l,E 0

z El (x)

Fig. 3.1 Example of the membership function μl (zEl (x)).

As a possible way to help the DM determine zEl,0 and zEl,1 , it is recommended to calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k obtained by solving the linear programming problems

3.1 Fuzzy multiobjective stochastic linear programming

53

⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η

maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭

(3.7)

η) x∈X(η

It should be noted here that (3.6) is regarded as a multiobjective decision making problem, and that there rarely exists a complete optimal solution that simultaneously optimizes all the objective functions. As discussed in Chapter 2, by directly extending Pareto optimality in ordinary multiobjective programming problems, Sakawa et al. deﬁned M-Pareto optimality on the basis of membership function values as a reasonable solution concept for the fuzzy multiobjective decision making problem (Sakawa and Yano, 1985a; Sakawa, Yano and Yumine, 1987; Sakawa and Yano, 1990; Sakawa, 1993). Introducing an aggregation function μD (x) for the k membership functions in (3.6), the fuzzy multiobjective programming problem can be rewritten as maximize μD (x) (3.8) η). subject to x ∈ X(η Following the conventional fuzzy approaches, to aggregate the multiple membership functions, Hulsurkar, Biswal and Sinha (1997) adopted the minimum operator of Bellman and Zadeh (1970) deﬁned by μD (x) = min μl (zEl (x)) 1≤l≤k

and the product operator of Zimmermann (1978) deﬁned as k

μD (x) = ∏ μl (zEl (x)). l=1

However, it should be emphasized here that such approaches are preferable only when the DM feels that the minimum operator or the product operator is appropriate. In other words, in general decision situations, the DM does not always use the minimum operator or the product operator for combining the fuzzy goals. The most crucial problem in (3.8) is probably the identiﬁcation of an appropriate aggregation function which well represents the DM’s fuzzy preference. If μD (x) can be explicitly identiﬁed, then (3.8) can be reduced to a standard mathematical programming problem. However, this rarely happens, and then as an alternative way, it is recommended for the DM to examine multiple possible solutions in the set of M-Pareto optimal solutions through an interactive solution procedure to ﬁnd a satisﬁcing solution to (3.8). In such an interactive fuzzy satisﬁcing method, to generate a candidate for a satisﬁcing solution which is also M-Pareto optimal, the DM is asked to specify the aspiration levels of achievement for all the membership function values, called the reference membership levels (Sakawa and Yano, 1985a; Sakawa, Yano and Yumine, 1987; Sakawa and Yano, 1989, 1990; Sakawa, 1993).

54

3 Fuzzy Multiobjective Stochastic Programming

For the DM’s reference membership levels μˆ l , l = 1, . . . , k, the corresponding MPareto optimal solution, which is the nearest to the vector of the given reference membership levels in the minimax sense or better than it if the reference membership levels are attainable, is obtained by solving the minimax problem minimize max {ˆμl − μl (zEl (x))} 1≤l≤k (3.9) η). subject to x ∈ X(η By introducing the auxiliary variable v, (3.9) can be equivalently transformed as ⎫ minimize v ⎪ ⎪ ⎪ ⎪ subject to μˆ 1 − μ1 (zE1 (x)) ≤ v ⎬ ········· (3.10) ⎪ μˆ k − μk (zEk (x)) ≤ v ⎪ ⎪ ⎪ ⎭ η). x ∈ X(η It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (3.10) can be reduced to a set of linear inequalities. Obtaining the optimal value v∗ to (3.10) is equivalent to determining the minimum value of v so that there exists a feasible solution x satisfying the constraints in (3.10). Since v satisﬁes zˆmax − max zEl,max ≤ v ≤ zˆmax − min zEl,min , 1≤l≤k

1≤l≤k

where zˆmax = max zˆl , 1≤l≤k

zEl,max = max zEl (x), x∈X

(x∗ , v∗ )

zEl,min = min zEl (x), x∈X

v∗

we can ﬁnd an optimal solution corresponding to by using the combined use of the two-phase simplex method and the bisection method. To be more speciﬁc, after calculating the optimal value v∗ to the problem ⎫ minimize v ⎪ ⎪ ⎪ subject to zE1 (x) ≤ μ−1 μ1 − v) ⎪ ⎬ 1 (ˆ ········· (3.11) ⎪ −1 E ⎪ ⎪ zk (x) ≤ μk (ˆμk − v) ⎪ ⎭ η) x ∈ X(η by the combined use of the phase one of the two-phase simplex method and the bisection method, we solve the linear programming problem ⎫ minimize zE1 (x) ⎪ ⎪ ⎪ subject to zE2 (x) ≤ μ−1 μ2 − v∗ ) ⎪ ⎬ 2 (ˆ (3.12) ········· ∗ ⎪ ⎪ ⎪ (ˆ μ − v ) zEk (x) ≤ μ−1 k ⎪ k ⎭ η), x ∈ X(η

3.1 Fuzzy multiobjective stochastic linear programming

55

where the ﬁrst objective function z1 (x) in (3.1) is supposed to be the most important to the DM. For the obtained optimal solution x∗ of (3.12), if there are inactive constraints in the ﬁrst (k − 1) constraints, after replacing μˆ l for inactive constraints with μl (zEl (x∗ )) + v∗ , we resolve the corresponding problem. Furthermore, if the obtained x∗ is not unique, the M-Pareto optimality test is performed by solving the linear programming problem maximize w = subject to

k

∑

εl l=1 μl (zE1 (x)) − ε1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

= μ1 (zE1 (x∗ )) ⎪ ········· ⎪ ⎪ ⎪ ⎪ μk (zEk (x)) − εk = μk (zEk (x∗ )) ⎪ ⎭ T η), ε = (ε1 , . . . , εk ) ≥ 0. x ∈ X(η

(3.13)

For the optimal solution xo and ε o to (3.13), (a) if w = 0, i.e., εl = 0 for l = 1, . . . , k, x∗ is Pareto optimal, and (b) if w > 0, i.e., εl > 0 for at least one l, not x∗ but xo is M-Pareto optimal. The DM should be either satisﬁed with the current M-Pareto optimal solution or continue to examine another solution by updating the reference membership levels. In order to help the DM express a degree of preference, trade-off rates between a standing membership function μl (zE1 (x)) and each of the other membership functions is quite useful. Such the trade-off rates are easily obtainable and are expressed as μ (zE (x∗ )) ∂μ1 (zE1 (x∗ )) = πl 1 E1 ∗ , l = 2, . . . , k, − E ∗ ∂μl (zl (x )) μl (zl (x )) where μ l denotes the differential coefﬁcient of μl , and πl , l = 1, . . . , k are the simplex multipliers in (3.12). Following the preceding discussions, we summarize the interactive algorithm in order to derive the satisﬁcing solution for the DM from among the M-Pareto optimal solution set. Interactive fuzzy satisﬁcing method for the expectation model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k by solving the linear programming problems (3.7). Step 3: Ask the DM to specify the membership functions μl , l = 1, . . . , k by taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 5: For the current reference membership levels μˆ l , solve the corresponding minimax problem (3.9). For the obtained optimal solution x∗ , if there are the inactive constraints in the ﬁrst (k − 1) constraints of (3.9), replace zˆl for inactive constraints with zEl (x∗ ) + v∗ and resolve the revised problem. Furthermore, if the solution x∗ is not unique, perform the M-Pareto optimality test.

56

3 Fuzzy Multiobjective Stochastic Programming

Step 6: The DM is supplied with the corresponding M-Pareto optimal solution and the trade-off rates between the membership functions. If the DM is satisﬁed with the current membership function values zEl (x), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by taking into account the current membership function values μl (zEl (x∗ )) together with the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k, and return to step 5. Observing that the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k in step 6 indicate the decrement value of the membership function μ1 with a unit increment of value of the membership function μl , the information of trade-off rates can be used to estimate the local shape of (μ1 (zE1 (x∗ )), . . . , μk (zEk (x∗ ))) around x∗ . Here it should be stressed for the DM that any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the given satisﬁcing probability levels ηi , i = 1, . . . , m.

3.1.1.2 Variance model When the DM would like to simply optimize the expected objective function values without concerning about the ﬂuctuation of the realized values, the expectation model is appropriate. On the other hand, if the DM prefers to decrease the ﬂuctuation of the objective function values from the viewpoint of the stability of the obtained values, by minimizing the variance of the objective function, the multiobjective stochastic programming problem can be reformulated as ⎫ minimize zV1 (x) = Var [z1 (x)] = xT V1 x ⎪ ⎪ ⎬ ········· (3.14) minimize zVk (x) = Var [zk (x)] = xT Vk x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where Var [zl (x)] denotes the variance of zl (x). Observing that (3.14) does not consider the expected objective function values, the original variance model may give a poor solution yielding undesirable values of the objective functions even if the ﬂuctuation of the objective function values is minimized. Realizing this difﬁculty, by incorporating the constraints with respect to the expectations into the original variance model, we formulate the modiﬁed variance model for the multiobjective stochastic programming problem represented as ⎫ minimize zV1 (x) = Var [z1 (x)] = xT V1 x ⎪ ⎪ ⎪ ⎪ ········· ⎬ V T minimize zk (x) = Var [zk (x)] = x Vk x (3.15) ⎪ ⎪ ¯ ≤γ ⎪ subject to E[C]x ⎪ ⎭ η), x ∈ X(η

3.1 Fuzzy multiobjective stochastic linear programming

57

¯ = (E[¯c1 ]T , . . . , E[¯ck ]T )T and γ = (γ1 , . . . , γk )T . Each γl is a permissible where E[C] expectation level speciﬁed by the DM taking into account the individual minima and maxima of zEl (x) calculated by solving the linear programming problems ⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η

maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭

(3.16)

η) x∈X(η

η, γ ) be the feasible region of (3.15), namely For notational convenience, let X(η ¯ ≤ γ , x ∈ X(η η, γ ) x E[C]x η) . X(η Quite similar to the expectation model, in order to consider the imprecise nature of the DM’s judgments for each objective function in (3.15), if we introduce the fuzzy goals such as “zVl (x) should be substantially less than or equal to a certain value”, (3.15) can be interpreted as maximize (μ1 (zV1 (x)), . . . , μk (zVk (x))), η,γγ) x∈X(η

(3.17)

where μl is a nonincreasing membership function to quantify the fuzzy goal for the lth objective function. To help the DM specify the membership function, it is desirable to calculate the individual minima of the variances of the objective functions by solving the quadratic programming problems minimize zVl (x) = xT Vl x, l = 1, . . . , k, η,γγ) x∈X(η

(3.18)

where Vl is an n × n positive-deﬁnite variance-covariance matrix for the coefﬁcient vector cl . For the DM’s reference membership levels μˆ l , i = 1, . . . , k, the corresponding M-Pareto optimal solution is obtained by solving the minimax problem minimize max {ˆμl − μl (zVl (x))} 1≤l≤k (3.19) η, γ ). subject to x ∈ X(η If an optimal solution (x∗ , v∗ ) to (3.19) is not unique, M-Pareto optimality of x∗ is not guaranteed. As discussed in the expectation model, we can obtain the tradeoff rates in the expectation model through the M-Pareto optimality test using linear programming techniques. Observing that (3.19) is not a linear programming problem, it is not easy to calculate the trade-off rates in the variance model. Therefore, to obtain an M-Pareto optimal solution with less effort, we consider the augmented minimax problem

58

3 Fuzzy Multiobjective Stochastic Programming

⎫ ⎪ ⎬ minimize max μˆ l − μl (zVl (x)) + ρ ∑ (ˆμi − μi (zVi (x))) 1≤l≤k i=1 ⎪ ⎭ η, γ ). subject to x ∈ X(η

k

(3.20)

By introducing the auxiliary variable v, this problem can be equivalently transformed as ⎫ minimize v ⎪ ⎪ ⎪ k ⎪ ⎪ V V ⎪ subject to μˆ 1 − μ1 (z1 (x)) + ρ ∑ μˆ i − μi (zi (x)) ≤ v ⎪ ⎪ ⎪ ⎬ i=1 ········· (3.21) ⎪ ⎪ k ⎪ ⎪ μˆ k − μk (zVk (x)) + ρ ∑ μˆ i − μi (zVi (x)) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎭ η, γ ), x ∈ X(η where ρ is a sufﬁciently small positive number. We assume that the membership function μl , l = 1, . . . , k are nonincreasing and η, γ ) and any concave. Then, letting gl (x) = μˆ l − μl (zVl (x)), for any x1 and x2 in X(η λ ∈ [0, 1], the convexity property of gl (x) is shown as gl (λx1 + (1 − λ)x2 ) = μˆ l − μl (λzVl (x)) + (1 − λ)zVl (x2 ) ≤ μˆ l − λμl (zVl (x1 )) − (1 − λ)zVl (x2 ) ≤ μˆ l − λμl (zVl (x1 )) + (1 − λ)μ( zVl (x2 )) = λ(ˆμl − μl (zVl (x2 ))) + (1 − λ)(ˆμl − μl (zVl (x2 )))) = λgl (x1 ) + (1 − λ)gl (x2 ).

(3.22)

Recalling that the sum of convex functions is also convex, one ﬁnds that each of the left-hand side functions of the constraints in (3.21), expressed as gl (x) + ρ ∑ki=1 gi (x), is also convex, which implies that (3.21) can be solved by a traditional convex programming technique such as the sequential quadratic programming method (Fletcher, 1980; Gill, Murray and Wright, 1981; Powell, 1983). In order to derive a satisﬁcing solution for the DM from among the M-Pareto optimal solution set, we present an interactive fuzzy satisﬁcing method where the reference membership levels are repeatedly updated until the DM obtains a satisﬁcing solution. Interactive fuzzy satisﬁcing method for the variance model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima and maxima of E[zl (x)], l = 1, . . . , k by solving linear programming problems (3.16). Step 3: Ask the DM to specify the permissible expectation levels γl , l = 1, . . . , k. Step 4: Calculate the individual minima of zVl (x), l = 1, . . . , k by solving the quadratic programming problems (3.18).

3.1 Fuzzy multiobjective stochastic linear programming

59

Step 5: Ask the DM to specify the membership functions μl , l = 1, . . . , k with the individual minima zVl,min obtained in step 4 in mind. Step 6: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 7: For the current reference membership levels μˆ l , l = 1, . . . , k, solve the augmented minimax problem (3.20). Step 8: The DM is supplied with the corresponding M-Pareto optimal solution. If the DM is satisﬁed with the current membership function values zVl (x), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k taking into account the current membership function values μl (zVl (x∗ )), and return to step 7. It is signiﬁcant to emphasize that, in the interactive process, any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the ﬁxed satisﬁcing probability levels ηi , i = 1, . . . , m and permissible expectation levels γl , l = 1, . . . , k.

3.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive fuzzy satisﬁcing method for the variance model, as a numerical example of (3.14), consider the multiobjective stochastic linear programming problem formulated as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ minimize z2 (x) = c¯ 2 x ⎪ ⎪ ⎪ ⎪ minimize z3 (x) = c¯ 3 x ⎪ ⎬ subject to a1 x ≤ b¯ 1 (3.23) ⎪ a2 x ≤ b¯ 2 ⎪ ⎪ ⎪ ⎪ a3 x ≤ b¯ 3 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x8 )T and the random variables b¯ i , i = 1, 2, 3 are Gaussian random variables N(220, 42 ), N(145, 32 ) and N(−18, 52 ). The coefﬁcient vectors c¯ l , l = 1, 2, 3 are also Gaussian random variables vectors with mean vectors E[¯cl ], l = 1, 2, 3 as shown in Table 3.1 and the variance-covariance matrices Vl , l = 1, 2, 3 are given as Table 3.1 Mean vectors E[¯cl ] of c¯ l , l = 1, 2, 3. E[¯c1 ] E[¯c2 ] E[¯c3 ]

5 10 −8

2 −7 −1

1 1 −7

2 −2 −4

1 −5 −2

6 3 −3

1 −4 −1

3 6 −1

60

3 Fuzzy Multiobjective Stochastic Programming ⎛

16.0 ⎜ −1.6 ⎜ ⎜ 1.8 ⎜ ⎜ −3.5 V1 = ⎜ ⎜ 1.3 ⎜ ⎜ −2.0 ⎝ 4.0 −1.4 ⎛ 4.0 ⎜ −1.4 ⎜ ⎜ 0.8 ⎜ ⎜ 0.2 V2 = ⎜ ⎜ 1.6 ⎜ 1.0 ⎜ ⎝ 1.2 2.0 ⎛ 4.0 ⎜ −0.9 ⎜ ⎜ 1.1 ⎜ ⎜ −1.8 V3 = ⎜ ⎜ 3.0 ⎜ 2.2 ⎜ ⎝ 3.3 1.0

−1.6 25.0 −2.2 1.6 −0.7 0.5 −1.3 2.0

−3.5 1.6 −2.0 16.0 −2.0 3.0 2.2 2.8

1.3 −0.7 5.0 −2.0 4.0 −1.0 0.8 −2.0

−1.4 0.8 0.2 4.0 0.2 −1.0 0.2 9.0 0.2 −1.0 0.2 36.0 −2.2 −1.5 0.8 0.8 1.5 0.4 0.9 1.0 −1.5 1.8 0.6 0.7

1.6 −2.2 −1.5 0.8 25.0 1.2 −0.2 2.0

1.8 −2.2 25.0 −2.0 5.0 −2.4 1.2 −2.1

−0.9 1.1 −1.8 3.0 1.0 0.8 0.6 −2.4 0.8 9.0 −2.5 2.8 0.6 −2.5 4.0 −1.4 −2.4 2.8 −1.4 16.0 1.2 3.5 2.1 0.5 −2.3 1.2 0.8 1.1 2.0 −2.9 2.3 −1.3

⎞ −2.0 4.0 1.4 0.5 −1.3 2.0 ⎟ ⎟ −2.4 1.2 −2.1 ⎟ ⎟ 3.0 2.2 2.8 ⎟ ⎟ −1.0 0.8 −2.0 ⎟ 1.0 −1.5 0.6 ⎟ ⎟ −1.5 4.0 −2.3 ⎠ 0.6 −2.3 4.0 ⎞ 1.0 1.2 2.0 0.8 0.9 1.8 ⎟ ⎟ 1.5 1.0 0.6 ⎟ ⎟ 0.4 −1.5 0.7 ⎟ ⎟ 1.2 −0.2 2.0 ⎟ ⎟ 25.0 0.5 1.4 ⎟ 0.5 9.0 0.8 ⎠ 1.4 0.8 16.0 ⎞ 2.2 3.3 1.0 1.2 −2.3 2.0 ⎟ ⎟ 3.5 1.2 −2.9 ⎟ ⎟ 2.1 0.8 2.3 ⎟ ⎟ 0.5 1.1 −1.3 ⎟ ⎟ 25.0 −2.6 3.0 ⎟ −2.6 36.0 0.8 ⎠ 3.0 0.8 25.0

The coefﬁcient vectors ai , i = 1, 2, 3 are given as Table 3.2. Table 3.2 Value of each element of ai , i = 1, 2, 3. a1 a2 a3

7 5 −4

2 6 −7

6 −4 −2

9 3 −6

11 3 −8

4 −7 −3

3 −1 −5

8 −3 −6

Suppose that the DM speciﬁes the satisﬁcing probability levels ηi , i = 1, 2, 3 as (η1 , η2 , η3 )T = (0.85, 0.95, 0.90)T . Then, the individual minima zEl,min and maxima zEl,max of zEl (x), l = 1, 2, 3 are calculated through the simplex method as zE1,min = 3.051, zE1,max = 323.781, zE2,min = −425.617, zE2,max = 304.476, zE3,min = −251.830 and zE3,max = −3.487. With these values in mind, assume that the DM subjectively speciﬁes permissible expectation levels γl , l = 1, 2, 3 as γ1 = 60.000, γ2 = 35.000, γ3 = −16.000. Using the sequential quadratic programming method, the individual minima zVl,min of zVl (x), l = 1, 2, 3 are calculated as zV1,min = 1.370, zV2,min = 19.640 and zV3,min = 6.413. By taking account of these values, assume that the DM determines the linear membership functions as

3.1 Fuzzy multiobjective stochastic linear programming

μl (zVl (x)) =

⎧ 1 ⎪ ⎪ ⎪ ⎨ zV (x) − zV l l,0 ⎪ zV − zVl,0 ⎪ ⎪ ⎩ l,1 0

61

if zVl (x) ≤ zVl,1 if zVl,1 < zVl (x) ≤ zVl,0 if zVl,0 < zVl (x),

where zVl,1 and zVl,0 , l = 1, 2, 3 are calculated as zV1,1 = 1.370, zV1,0 = 130.939, zV2,1 = 19.640, zV2,0 = 358.271, zV3,1 = 6.413, and zV3,0 = 401.016 by using Zimmermann’s method (Zimmermann, 1978). μl (zVl (x)) 1

0

z l,V 1

z l,V 0

z Vl (x)

Fig. 3.2 The shape of a linear membership function.

For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.000, 1.000, 1.000), the augmented minimax problem (3.20) is solved by the sequential quadratic programming method, and the DM is supplied with the corresponding membership function values in the ﬁrst iteration shown in Table 3.3. Assume that the DM is not satisﬁed with the ﬁrst membership function values, and the DM updates the reference membership levels to (0.900, 1.000, 1.000) for improving the satisfaction levels for μ2 and μ3 at the sacriﬁce of μ1 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the corresponding membership function values in the second iteration are calculated as shown in Table 3.3. Suppose that the DM is not satisﬁed with the second membership function values, and the DM updates the reference membership levels as (0.900, 0.950, 1.000) for improving the satisfaction levels for μ1 and μ3 at the expense of μ2 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the corresponding membership function values are calculated. After obtaining the result of the third iteration shown in Table 3.3, assume that the DM is satisﬁed with this membership function values. Then, it follows that the satisﬁcing solution for the DM is derived.

62

3 Fuzzy Multiobjective Stochastic Programming

Table 3.3 Process of interaction. Iteration 1st μˆ 1 1.000 1.000 μˆ 2 μˆ 3 1.000

2nd 0.900 1.000 1.000

3rd 0.900 0.950 1.000

μ1 (zV1 (x)) μ2 (zV2 (x)) μ3 (zV3 (x))

0.847 0.847 0.847

0.793 0.893 0.893

0.804 0.854 0.904

zV1 (x) zV2 (x) zV3 (x)

21.250 71.596 66.957

28.142 55.744 48.485

26.768 69.087 44.303

3.1.2 Probability and fractile models 3.1.2.1 Probability model In contrast to the expectation and variance models discussed in the previous subsection, a certain target value is introduced for each of the objective function values in the probability model, and the probability that the objective function value is smaller than or equal to the target value is maximized. In this subsection, along the same line as the problem formulation introduced by Stancu-Minasian (1984), by adding a random parameter υ¯ l to the lth objective functions to deal with more general problems than (3.1), we formulate a multiobjective stochastic programming problem ⎫ minimize z1 (x) = c¯ 1 x + υ¯ 1 ⎪ ⎪ ⎪ ⎪ ······ ⎬ minimize zk (x) = c¯ k x + υ¯ k (3.24) ⎪ ⎪ subject to Ax ≤ b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, c¯ l , l = 1, . . . , k are n dimensional random variable row vectors, υ¯ l , l = 1, . . . , k are the random parameters, A is an m × n coefﬁcient matrix, and b¯ is an n dimensional vector whose elements are mutually independent random variables with continuous and increasing probability distribution functions Fl . Furthermore, it is assumed that the random vector c¯ l and the random parameter υ¯ l are expressed by c¯ l = c1l + t¯l c2l and υ¯ l = υ1l + t¯l υ2l , respectively, where t¯l , l = 1, . . . , k are mutually independent random variables with the continuous and strictly increasing probability distribution functions Tl . As discussed in the previous subsection, by introducing the idea of chance constrained condition, (3.24) can be interpreted as

3.1 Fuzzy multiobjective stochastic linear programming

63

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

minimize z1 (x) = c¯ 1 x + υ¯ 1 ······ minimize zk (x) = c¯ k x + υ¯ k subject to P(ω | a1 x ≤ b1 (ω)) ≥ η1 ⎪ ⎪ ······ ⎪ ⎪ ⎪ P(ω | am x ≤ bm (ω)) ≥ ηm ⎪ ⎪ ⎪ ⎭ x ≥ 0,

(3.25)

where ai is the ith row vector of A, and bi (ω) is the ith element of the realized vector b(ω) for an elementary event ω. Recalling (3.3), it follows that P(ω | ai x ≤ bi (ω)) ≥ ηi ⇔ ai x ≤ Fi−1 (1 − ηi ), η) denote the feasible region of (3.26), i.e., and for notational convenience, let X(η η) {x | ai x ≤ Fi−1 (1 − ηi ), i = 1, . . . , m, x ≥ 0}. X(η Assuming that the DM maximizes the probability that each objective function zl (x) = c¯ l x + υ¯ l in (3.24) is smaller than or equal to a certain target value fl , in the probability model we consider a multiobjective stochastic programming problem ⎫ maximize zP1 (x) = P (ω | c1 (ω)x + υ1 (ω) ≤ f 1 ) ⎪ ⎪ ⎬ ······ (3.26) P maximize zk (x) = P (ω | ck (ω)x + υk (ω) ≤ fk ) ⎪ ⎪ ⎭ η), subject to x ∈ X(η where cl (ω) and υl (ω) denote the realization of c¯ l and υ¯ l , respectively. To help the DM specify the target values hl , l = 1, . . . , k, it is recommended to calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k obtained by solving the linear programming problems ⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η (3.27) maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭ η) x∈X(η

If we assume that c2l x + υ2l > 0, l = 1, . . . , k for any x ∈ X, from the property of the distribution functions Tl of the random variables t¯l , it is evident that the lth objective function in (3.26) is calculated as zPl (x) = P ω | (c1l + tl (ω)c2l )x + (υ1l + tl (ω)υ2l ) ≤ f l fl − (c1l x + υ1l ) = P ω tl (ω) ≤ (c2l x + υ2l ) f l − c1l x − υ1l . = Tl c2l x + υ2l

64

3 Fuzzy Multiobjective Stochastic Programming

Hence, (3.26) can be equivalently transformed into ⎫ f1 − c11 x − υ11 ⎪ P ⎪ maximize z1 (x) = T1 ⎪ ⎪ c21 x + υ21 ⎪ ⎪ ⎬ ········· fk − c1k x − υ1k ⎪ ⎪ ⎪ maximize zPk (x) = Tk ⎪ ⎪ c2k x + υ2k ⎪ ⎭ η). subject to x ∈ X(η

(3.28)

Considering the imprecise nature of the DM’s judgments for each objective function in (3.28), if we introduce the fuzzy goals such as “zPl (x) should be substantially greater than or equal to a certain value”, (3.28) can be interpreted as maximize (μ1 (zP1 (x)), . . . , μk (zPk (x))),

(3.29)

η) x∈X(η

where μl is a membership function to quantify a fuzzy goal for the lth objective function in (3.28). To elicit the membership function from the DM for each of the fuzzy goals, the DM is asked to assess an unacceptable objective function value zPl,0 , and a desirable objective function value zPl,1 as shown in Fig. 3.3. μl (zPl (x)) 1

0

z l,P 0

z l,P 1

z lP(x)

Fig. 3.3 Example of a membership function μl (zPl (x)).

In order to help the DM specify zPl,0 and zPl,1 , it serves as a useful reference to calculate the individual minima of zPl (x) by solving the ordinary single-objective linear fractional programming problems ⎫ fl − c1l x − υ1l ⎪ , l = 1, . . . , k, ⎪ minimize ⎬ x∈X c2l x + υ2l f l − c1l x − υ1l ⎪ ⎭ maximize , l = 1, . . . , k. ⎪ x∈X c2l x + υ2l

(3.30)

For the DM’s reference membership levels μˆ l , i = 1, . . . , k, the corresponding MPareto optimal solution, which is the nearest to the vector of the reference levels

3.1 Fuzzy multiobjective stochastic linear programming

65

in the minimax sense or is better than it if the reference membership levels are attainable, is obtained by solving the minimax problem minimize max {ˆμl − μl (zPl (x))} 1≤l≤k (3.31) η), subject to x ∈ X(η or equivalently

⎫ minimize v ⎪ ⎪ ⎪ subject to μˆ 1 − μ1 (zP1 (x)) ≤ v ⎪ ⎬ ········· ⎪ μˆ k − μk (zPk (x)) ≤ v ⎪ ⎪ ⎪ ⎭ η). x ∈ X(η

(3.32)

Recalling that each membership function μl is continuous and strictly increasing, it follows that (3.32) is equivalently transformed into ⎫ minimize v ⎪ ⎪ ⎪ μ1 − v) ⎪ subject to zP1 (x) ≥ μ−1 ⎬ 1 (ˆ ········· (3.33) ⎪ ⎪ ⎪ zPk (x) ≥ μ−1 (ˆ μ − v) k ⎪ k ⎭ η). x ∈ X(η From the continuity and strictly increasing property of the distribution function Tl , it follows that fl − c1l x − υ1l zPl (x) ≥ μ−1 (ˆ μ − v) ⇔ T μl − v) ≥ μ−1 l l l l (ˆ c2l x + υ2l ⇔

fl − c1l x − υ1l ≥ Tl−1 (μ−1 μl − v)). l (ˆ c2l x + υ2l

Consequently, (3.33) can be equivalently rewritten as ⎫ minimize v ⎪ ⎪ 1 1 ⎪ ⎪ f1 − c1 x − υ1 ⎪ −1 −1 ⎪ subject to ≥ T (μ (ˆ μ − v)) 1 ⎪ 1 1 2 2 ⎪ c1 x + υ1 ⎬ ········· ⎪ ⎪ fk − c1k x − υ1k ⎪ −1 −1 ⎪ ⎪ ≥ T (μ (ˆ μ − v)) k ⎪ k k 2 2 ⎪ ck x + υk ⎪ ⎭ η). x ∈ X(η

(3.34)

It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (3.34) can be reduced to a set of linear inequalities. Obtaining the optimal value v∗ to the above problem is equivalent to ﬁnding the minimum value of v so that there exists a feasible solution x satisfying the constraints of (3.34). From the fact that v satisﬁes

66

3 Fuzzy Multiobjective Stochastic Programming

μˆ max − max μl,max ≤ v ≤ μˆ max − min μl,min , 1≤l≤k

1≤l≤k

where μˆ max = max μˆ l , 1≤l≤k

μl,max = max μl (zPl (x)), η) x∈X(η

μl,min = min μl (zPl (x)), η) x∈X(η

we can obtain the minimum value of v by the combined use of the phase one of the two-phase simplex method and the bisection method. After ﬁnding the minimum value v∗ , in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem ⎫ c11 x + υ11 − f1 ⎪ ⎪ ⎪ minimize ⎪ 2 2 ⎪ c1 x + υ1 ⎪ ⎪ ⎪ 1 1 ⎪ f2 − c2 x − υ2 −1 −1 ∗ ⎪ ⎪ subject to ≥ T (μ (ˆ μ − v )) ⎬ 2 2 2 2 2 c2 x + υ2 (3.35) ⎪ ········· ⎪ ⎪ ⎪ fk − c1k x − υ1k ⎪ −1 −1 ∗ ⎪ ⎪ ≥ T (μ (ˆ μ − v )) k ⎪ k k ⎪ c2k x + υ2k ⎪ ⎪ ⎭ η x ∈ X(η ), where z1 (x) in (3.24) is supposed to be the most important to the DM. For notational convenience, let b = (b1 , . . . , bm )T where bi = Fi−1 (1 − ηi ). Then, using the variable transformation introduced by Charnes and Cooper (1962) s = 1/(c21 x + υ21 ), y = sx, s > 0,

(3.36)

the linear fractional programming problem (3.35) is equivalently transformed as ⎫ minimize c11 y + (υ11 − f 1 )s ⎪ ⎪ ⎪ subject to τ2 (c22 y + υ22 s) + c12 y + (υ12 − f2 )s ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ········· ⎪ ⎪ ⎪ τk (c2k y + υ2k s) + c1k y + (υ1k − fk )s ≤ 0 ⎪ ⎬ (3.37) Ay − sb ≤ 0 ⎪ ⎪ c21 y + υ21 s = 1 ⎪ ⎪ ⎪ ⎪ −s ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎪ ⎭ s ≥ 0, where τl = Tl−1 (μ−1 μl − v∗ )), and δ is a sufﬁciently small positive number introl (ˆ duced for the condition s > 0 in (3.36). If the optimal solution (y∗ , s∗ ) to (3.37) is not unique, M-Pareto optimality of ∗ x = s∗ y∗ is not always guaranteed. Observing that μl is a strictly increasing membership function, it should be noted here that the Pareto optimality test can be used in place of the M-Pareto optimality test. With this observation in mind, M-Pareto optimality of x∗ can be tested by solving the linear programming problem

3.1 Fuzzy multiobjective stochastic linear programming

maximize

w=

k

∑ εl

l=1

subject to

q1 (x∗ ) r1 (x) r1 (x∗ ) ········· qk (x∗ ) rk (x) qk (x) − εk = rk (x∗ ) x ∈ X, ε = (ε1 , . . . , εk )T ≥ 0, q1 (x) − ε1 =

67

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3.38)

where ql (x) = fl − c1l x − υ1l , rl (x) = c2l x + υ2l , l = 1, . . . , k. For the optimal solution to (3.38), (a) if w = 0, i.e., εl = 0 for l = 1, . . . , k, x∗ is M-Pareto optimal. On the other hand, (b) if w > 0, i.e., εl > 0 for at least one l, x∗ is not Pareto optimal. Then, we can ﬁnd an M-Pareto optimal solution according to the following algorithm. Step 1: For the optimal solution (xo , ε o ) to (3.38), after arbitrarily selecting j such as εoj > 0, through the combined use of the Charnes-Cooper variable transformation and the simplex method, solve the linear fractional programming problem ⎫ f j − c1j x − υ1j ⎪ ⎪ ⎪ maximize ⎪ ⎪ c2j x + υ2j ⎪ ⎪ ⎪ 1 1 1 o 1 ⎪ fl − cl x − υl fl − cl x − υl ⎬ o subject to = , ∀l ∈ {l | ε = 0} l 2 2 2 2 o (3.39) cl x + υl cl x + υl ⎪ ⎪ 1 1 1 o 1 ⎪ fl − cl x − υl fl − cl x − υl ⎪ ≥ , ∀l ∈ {l | εol > 0} ⎪ ⎪ ⎪ 2 2 2 2 o ⎪ cl x + υl cl x + υl ⎪ ⎭ η). x ∈ X(η Step 2: Let (x , ε ) be an optimal solution of (3.38). To test Pareto optimality of the optimal solution x´ to (3.39), solve (3.38) for x∗ := x´ . Step 3: If w = 0, stop the algorithm. Otherwise, i.e., if w > 0, set (xo , εo ) := (x , ε ), and return to step 1. By repeating this process at least k − 1 iterations, an M-Pareto optimal solution can be obtained. The DM should either be satisﬁed with the current Pareto optimal solution or continue to examine another solution by updating the reference membership levels. In order to help the DM express a degree of preference, the trade-off information between a standing membership function and each of the other membership functions is very useful. In a manner similar to the expectation model in 3.1.1.1, such trade-off information is easily obtainable because of the linearity of (3.37). To be more speciﬁc, for deriving the trade-off information, we introduce the Lagrange function L deﬁned by

68

3 Fuzzy Multiobjective Stochastic Programming k

L(y, s, π , ζ , φ ) = c11 y + (υ11 − f 1 )s + ∑ πl [τl (c2l y + υ2l s) l=2

m

+ {c1l y + (υ1l − f l )s}] + ∑ ζi (ai y − sbi ) i=1

+ ζm+1 (c21 y + υ21 s − 1) + ζm+2 (−s + δ) n

− ∑ φ j y j − φn+1 s,

(3.40)

j=1

where πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers of (3.37). Then, the partial derivative of L(y, s, π , ζ , φ ) with respect to τl is given as ∂L(y, s, π , ζ , φ ) = πl (c2l y + υ2l s), l = 2, . . . , k. ∂τl

(3.41)

On the other hand, for the optimal solution (y∗ , s∗ ) to (3.37) and the corresponding π∗ , ζ ∗ , φ ∗ ), from the Kuhn-Tucker necessity theorem (Sakawa, simplex multipliers (π 1993), it holds that ∗

L(y∗ , s∗ , π ∗ , ζ , φ ∗ ) = c11 y∗ + (υ11 − f 1 )s∗ .

(3.42)

If the ﬁrst (k − 1) constraints to (3.37) are active, τl is calculated as τl = −

c1l y∗ + (υ1l − f l )s∗ , l = 2, . . . , k. c2l y∗ + υ2l s∗

(3.43)

From (3.41), (3.42) and (3.43), one ﬁnds that ∂(c1 y∗ + (υ11 − f1 )s∗ ) 11 ∗ = π∗l (c2l y∗ + υ2l s∗ ), l = 2, . . . , k. cl y + (υ1l − f l )s∗ ∂ − c2l y∗ + υ2l s∗ By substituting x∗ for y∗ and s∗ in (3.44), the equation is rewritten as f1 − c11 x∗ − υ11 ∂ c2 x∗ + υ2l c2 x∗ + υ2 , l = 2, . . . , k. − 1 1 ∗ 1 1 = π∗l l2 ∗ c1 x + υ21 fl − cl x − υl ∂ c2l x∗ + υ2l Using the chain rule, it follows that

(3.44)

(3.45)

3.1 Fuzzy multiobjective stochastic linear programming

69

f1 − c11 x∗ − υ11 fl − c11 x∗ − υ11

T 2 ∗ 2 1 c21 x∗ + υ21 c21 x∗ + υ21 ∗ cl x + υl = π − , l c21 x∗ + υ21 fl − c1l x∗ − υ1l fl − c1l x∗ − υ1l ∂Tl T l c2l x∗ + υ2l c2l x∗ + υ2l ∂T1

l = 2, . . . , k,

(3.46)

where T denotes the differential coefﬁcient of T . Equivalently, we have

−

2 ∗ 2 P ∗ ∂zP1 (x∗ ) ∗ cl x + υl z1 (x ) , l = 2, . . . , k, = π

l ∂zPl (x∗ ) c21 x∗ + υ21 zPl (x∗ )

(3.47)

where zPl denotes the differential coefﬁcient of zPl . Once again, using the chain rule, we obtain 2 ∗ 2 P ∗

P ∗ ∂μ1 (zP1 (x∗ )) ∗ cl x + υl z1 (x ) μ1 (z1 (x )) = π , (3.48) −

l 2 ∗ ∂μl (zPl (x∗ )) c1 x + υ21 zPl (x∗ ) μ l (zPl (x∗ )) where μ l denotes the differential coefﬁcient of μl . It should be stressed here that in order to obtain the trade-off information from (3.48), the ﬁrst (k − 1) constraints in (3.37) must be active. Therefore, if there are inactive constraints, it is necessary to replace μˆ l for inactive constraints with μl (zPl (x∗ )) + v∗ and solve the corresponding problem to obtain the simplex multipliers. Now we are ready to summarize an interactive algorithm for deriving the satisﬁcing solution for the DM from among the M-Pareto optimal solution set. Interactive fuzzy satisﬁcing method for the probability model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x), l = 1, . . . , k by solving the linear programming problems (3.27). Step 3: Ask the DM to specify the target values fl , l = 1, . . . , k by taking into account the individual minima and maxima obtained in step 3. Step 4: Calculate the individual minima zPl,min and maxima zPl,max of zPl (x), l = 1, . . . , k in (3.28) by solving (3.30). Step 5: Ask the DM to specify the membership functions μl (zPl (x)), l = 1, . . . , k by considering the individual minima and maxima obtained in step 4. Step 6: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 7: For the current reference membership levels μˆ l , l = 1, . . . , k, solve the minimax problem (3.34) and the corresponding problem (3.37). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (3.37), replace μˆ l for the inactive constraints with μl (zPl (x∗ )) + v∗ and resolve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the M-Pareto optimality test. Step 8: The DM is supplied with the corresponding M-Pareto optimal solution x∗ and the trade-off rates between the membership functions. If the DM is sat-

70

3 Fuzzy Multiobjective Stochastic Programming

isﬁed with the current membership function values zPl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by considering the current membership function values μl (zPl (x∗ )), l = 1, . . . , k together with the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k, and return to step 7. Observing that the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k in step 8 indicate the decrement value of the membership function μ1 with a unit increment of the value of the membership function μl , such trade-off information is used to estimate the local shape of (μ1 (zP1 (x∗ )), . . ., μk (zPk (x∗ ))) around x∗ . 3.1.2.2 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy satisﬁcing method for the probability model, as a numerical example of (3.24), consider the multiobjective stochastic linear programming problem formulated as ⎫ minimize (c11 + t¯1 c21 )x + (υ11 + t¯1 υ21 ) ⎪ ⎪ ⎪ minimize (c12 + t¯2 c22 )x + (υ12 + t¯2 υ22 ) ⎪ ⎬ 1 2 1 2 (3.49) minimize (c3 + t¯3 c3 )x + (υ3 + t¯3 υ3 ) ⎪ ⎪ subject to ai x ≤ b¯ i , i = 1, . . . , 7 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x10 )T , and the random parameters t¯1 , t¯2 and t¯3 are assumed to be Gaussian random variables N(4, 22 ), N(3, 32 ) and N(3, 22 ), respectively. The righthand side random variables b¯ i , i = 1, . . . , 7 are assumed to be Gaussian random variables N(164, 302 ), N(−190, 202 ), N(−184, 152 ), N(99, 222 ), N(−150, 172 ), N(154, 352 ) and N(142, 422 ), respectively. The constant coefﬁcients involved in the objective functions and the constraints are shown in Tables 3.4 and 3.5. Suppose that the DM speciﬁes the satisﬁcing probability levels at (η1 , η2 , η3 , η4 , η5 , η6 , η7 ) = (0.85, 0.95, 0.80, 0.90, 0.85, 0.80, 0.90). Then, the individual minima and maxima of zEl (x), l = 1, . . . , k, are calculated as zE1,min = 1819.571, zE1,max = 4221.883, zE2,min = 286.617, zE2,max = 1380.041, zE3,min = −1087.249 and zE3,max = −919.647. In order to determine the membership function of fuzzy goals, the individual minima and of zPl (x), l = 1, . . . , k in the multiobjective problem (3.28) are calculated as zP1,min = 0.002, zP1,max = 0.880, zP2,min = 0.328, zP2,max = 0.783, zP3,min = 0.002 and zP3,max = 0.664. Furthermore, assume that the DM speciﬁes the linear membership functions to quantify fuzzy goals for objective functions ⎧ 1 ⎪ ⎪ ⎪ ⎨ zP (x) − zP l l,0 μl (zPl (x)) = P − zP ⎪ z ⎪ l,1 l,0 ⎪ ⎩ 0

if zPl (x) ≤ zPl,1 if zPl,1 < zPl (x) ≤ zPl,0 if zPl,0 < zPl (x),

3.1 Fuzzy multiobjective stochastic linear programming

71

Table 3.4 Constant coefﬁcients of objective functions. c11 c21 c12 c22 c13 c23

19 3 12 1 −18 2

48 2 −46 2 −26 1

21 2 −23 4 −22 3

10 1 −38 2 −28 2

18 4 −33 2 −15 1

35 3 −48 1 −29 2

46 1 12 2 −10 3

11 2 8 1 −19 3

24 4 19 2 −17 2

33 2 20 1 −28 1

υ11 υ21 υ12 υ22 υ13 υ23

−18 5 −27 6 −10 4

Table 3.5 Constant coefﬁcients of constraints. a1 a2 a3 a4 a5 a6 a7

12 −2 3 −11 −4 5 −3

−2 5 −16 6 7 −3 −4

4 3 −4 −5 −6 14 −6

−7 16 −8 9 −5 −3 9

13 6 −8 −1 13 −9 6

−1 −12 2 8 6 −7 18

−6 12 −12 −4 −2 4 11

6 4 −12 6 −5 −4 −9

11 −7 4 −9 14 −5 −4

−8 −10 −3 6 −6 9 7

where zPl,1 and zPl,0 are calculated as zP1,1 = 0.880, zP1,0 = 0.502, zP2,1 = 0.783, zP2,0 = 0.060, zP3,1 = 0.664 and zP3,0 = 0.446 by using the Zimmermann method (Zimmermann, 1978). For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.00, 1.00, 1.00), the minimax problem (3.31) is solved, and the DM is supplied with the corresponding membership function values as shown at the second column in Table 3.6. Assume that the DM is not satisﬁed with the ﬁrst membership function values, the DM updates the reference membership levels to (1.00, 1.00, 0.90) for improving μ1 and μ2 at the sacriﬁce of μ3 . For the updated reference membership levels, the corresponding minimax problem is solved, and the corresponding membership function values are calculated as shown at the third column in Table 3.14. Assume that the DM is not satisﬁed with the second membership function values, by considering the trade-off information −∂μ1 /∂μ2 = 0.060 and −∂μ1 /∂μ3 = 0.801, and the DM updates the reference membership levels to (0.95, 1.00, 0.90) for improving μ2 and μ3 at the expense of μ1 . In the third iteration, assume that the DM is satisﬁed with the third membership function values. Then, it follows that the satisﬁcing solution for the DM is derived.

3.1.2.3 Fractile model For a decision situation where the DM prefers to maximize the probability that each of the objective functions is smaller than or equal to a target value speciﬁed by the DM, the probability model is recommended. In contrast, if the DM rather wants to minimize the target value after the DM speciﬁes a certain permissible level for the probability that each of the objective functions is smaller than or equal to the target value which is regarded as not a constant value of the goal but a variable to

72

3 Fuzzy Multiobjective Stochastic Programming

Table 3.6 Process of interaction. Iteration μˆ 1 μˆ 2 μˆ 3

1st 1.000 1.000 1.000

2nd 1.000 1.000 0.900

3rd 0.950 1.000 0.900

μ1 (zP1 (x)) μ2 (zP2 (x)) μ3 (zP3 (x))

0.5747 0.5732 0.5733

0.6177 0.6172 0.5170

0.5948 0.6436 0.5435

zP1 (x) zP2 (x) zP3 (x)

0.719 0.474 0.571

0.736 0.506 0.559

0.727 0.525 0.565

−∂μ1 /∂μ2 −∂μ1 /∂μ3

0.060 0.831

0.060 0.801

0.060 0.816

be minimized, the fractile model is thought to be appropriate (Sakawa, Katagiri and Kato, 2001). Replacing the minimization of the objective functions zl (x), l = 1, . . . , k in (3.4) with the minimization of the target variables fl , l = 1, . . . , k under the probabilistic constraints with the permissible levels θl ∈ (1/2, 1) speciﬁed by the DM, we consider the fractile model for the multiobjective stochastic programming problems formulated as ⎫ minimize f1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ minimize fk (3.50) subject to P (ω | c1 (ω)x ≤ f1 ) ≥ θ1 ⎪ ⎪ ⎪ ········· ⎪ ⎪ ⎪ P (ω | ck (ω)x ≤ fk ) ≥ θk ⎪ ⎪ ⎪ ⎭ η), x ∈ X(η where each c¯ l is a Gaussian random variable vector with mean vector E[¯cl ] and the positive-deﬁnite variance-covariance matrix Vl . From the fact that the random variables c¯ l x − E[¯cl ]x ' , l = 1, . . . , k xT Vl x

(3.51)

are the standard Gaussian random variables with mean 0 and variance 12 , it follows that " # c (ω)x − E[¯c ]x f l − E[¯cl ]x l l ' P (ω | cl (ω)x ≤ f l ) ⇔ P ω ≤ ' xT Vl x xT Vl x " # fl − E[¯cl ]x ⇔ Φ ' , (3.52) xT Vl x

3.1 Fuzzy multiobjective stochastic linear programming

73

where Φ is the distribution function of the standard Gaussian random variable, deﬁned as y 1 2 1 Φ(y) = √ e− 2 x dx. 2π −∞ Then, in (3.50), the probabilistic constraints P (ω | cl (ω)x ≤ fl ) ≥ θl , l = 1, . . . , k can be transformed as # " fl − E[¯cl ]x fl − E[¯cl ]x ≥ Φ−1 (θl ) ≥ θl ⇔ ' Φ ' xT Vl x xT Vl x ' ⇔ fl ≥ E[¯cl ]x + Φ−1 (θl ) xT Vl x, where Φ−1 is the inverse function of Φ. By substituting (3.54) for (3.53), (3.50) can be transformed into ⎫ minimize f1 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ minimize fk ' −1 T subject to E[¯c1 ]x + Φ (θ1 ) x V1 x ≤ f1 ⎪ ⎪ ⎪ ········' · ⎪ ⎪ ⎪ −1 T ⎪ E[¯ck ]x + Φ (θk ) x Vk x ≤ fk ⎪ ⎪ ⎭ η). x ∈ X(η

(3.53)

(3.54)

(3.55)

Because (3.55) minimizes each fl under the constraint that each fl must be greater than or equal to ' (3.56) E[¯cl ]x + Φ−1 (θl ) xT Vl x, it becomes obvious that minimizing fl is equivalent to minimizing (3.56). Hence, (3.55) can be rewritten as ' ⎫ minimize zF1 (x) = E[¯c1 ]x + Φ−1 (θ1 ) xT V1 x ⎪ ⎪ ⎬ ········· ' (3.57) minimize zFk (x) = E[¯ck ]x + Φ−1 (θk ) xT Vk x ⎪ ⎪ ⎭ η). subject to x ∈ X(η Noting that each of the objective functions in (3.57) is convex from Φ−1 (θl ) ≥ 0 for any θl ∈ [1/2, 1), one ﬁnds that (3.57) is a multiobjective convex programming problem. In a way similar to the previous models, taking into account the imprecise nature of the DM’s judgments for each objective function, by introducing the fuzzy goals such as “zFl (x) should be substantially less than or equal to a certain value,” in place of (3.57), we consider the minimization problem of fuzzy goals

74

3 Fuzzy Multiobjective Stochastic Programming

maximize (μ1 (zF1 (x)), . . . , μk (zFk (x))), η) x∈X(η

(3.58)

where μl is assumed to be a concave membership function to quantify a fuzzy goal for the lth objective function in (3.57), as shown in Fig. 3.4. μl (zFl (x)) 1

0

z l,F 1

z l,F 0 z Fl (x)

Fig. 3.4 Example of a membership function μl (zFl (x)).

In order to ﬁnd a candidate for the satisﬁcing solution, the DM speciﬁes the reference membership levels μˆ l , i = 1, . . . , k, and then by solving the augmented minimax problem ⎫ k ⎪ F F minimize max μˆ l − μl (zl (x)) + ρ ∑ (ˆμi − μi (zi (x))) ⎬ 1≤l≤k (3.59) i=1 ⎪ ⎭ η), subject to x ∈ X(η an M-Pareto optimal solution corresponding to μˆ l , i = 1, . . . , k is obtained, and (3.59) is equivalently expressed as ⎫ minimize v ⎪ ⎪ ⎪ k ⎪ ⎪ F F ⎪ subject to μˆ 1 − μ1 (z1 (x)) + ρ ∑ μˆ i − μi (zi (x)) ≤ v ⎪ ⎪ ⎪ ⎬ i=1 ··············· (3.60) ⎪ ⎪ k ⎪ ⎪ μˆ k − μk (zFk (x)) + ρ ∑ μˆ i − μi (zFi (x)) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎭ η), x ∈ X(η where ρ is a sufﬁciently small positive number. As we illustrated that (3.21) is a convex programming problem, it can be shown that (3.60) is also convex under the assumption that each of membership functions μl , l = 1, . . . , k is nonincreasing and concave. Due to the convexity, an optimal solution to (3.60) can be found by using some convex programming technique such as the sequential quadratic programming method. We can now present the interactive algorithm for deriving a satisﬁcing solution for the DM from among the M-Pareto optimal solution set.

3.1 Fuzzy multiobjective stochastic linear programming

75

Interactive fuzzy satisﬁcing method for the fractile model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m and the permissible probability levels θl ∈ [1/2, 1), l = 1, . . . , k. Step 2: Calculate the individual minima zFl,min of zFl (x), l = 1, . . . , k. Step 3: Ask the DM to specify the nonincreasing and concave membership functions μl taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 5: For the reference membership levels μˆ l , l = 1, . . . , k, solve the augmented minimax problem (3.60). Step 6: The DM is supplied with the corresponding M-Pareto optimal solution x∗ . If the DM is satisﬁed with the current membership function values zFl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by considering the current membership function values, and return to step 5. It should be stressed for the DM that, in an interactive process, any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the satisﬁcing probability levels ηi , i = 1, . . . , m and the permissible probability levels θl , l = 1, . . . , k speciﬁed by the DM.

3.1.2.4 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy satisﬁcing method for the fractile model, as a numerical example of (3.50), consider the multiobjective stochastic linear programming problem ⎫ minimize c¯ 1 x ⎪ ⎪ ⎪ ⎪ minimize c¯ 2 x ⎪ ⎪ ⎬ minimize c¯ 3 x (3.61) subject to a1 x ≤ b¯ 1 ⎪ ⎪ ⎪ a2 x ≤ b¯ 2 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x4 )T , and (b¯ 1 , b¯ 2 ) is a Gaussian random variable vector with mean (27, −15) and variance (36, 49). The coefﬁcient vectors c¯ l , l = 1, 2, 3 are Gaussian random variable vectors, and their mean vectors E[¯cl ], l = 1, 2, 3 are given in Table 3.7 and the variance-covariance matrices Vl , l = 1, 2, 3 are

76

3 Fuzzy Multiobjective Stochastic Programming ⎞ 25 −1 0.8 −2 ⎜ −1 4 −1.2 1.2 ⎟ ⎟, V1 = ⎜ ⎝ 0.8 −1.2 4 2⎠ −2 1.2 2 9 ⎛ ⎞ 16 1.4 −1.2 1.4 ⎜ 1.4 1 1.5 −0.8 ⎟ ⎟, V2 = ⎜ ⎝ −1.2 1.5 25 −0.6 ⎠ 1.4 −0.8 −0.6 4 ⎛ ⎞ 4 −1.9 1.5 1.8 ⎜ −1.9 25 0.8 −0.4 ⎟ ⎟ V3 = ⎜ ⎝ 1.5 0.8 9 2.5 ⎠ . 1.8 −0.4 2.5 36 ⎛

The coefﬁcient vectors ai , i = 1, 2 of the constraints are also given in Table 3.8. Table 3.7 Mean of each element of E[¯cl ], l = 1, 2, 3. E[¯c1 ] E[¯c2 ] E[¯c3 ]

2 10 −8

3 −7 −5

2 1 −7

4 −2 −14

4 −7

6 −9

Table 3.8 Value of each element of ai , i = 1, 2, 3. a1 a2

7 −5

3 −6

Suppose that the DM speciﬁes the permissible probability levels at (θ1 , θ2 , θ3 ) = (0.75, 0.80, 0.70) and the satisﬁcing probability levels η1 = η2 = 0.7. Then, the individual minima zFl,min of zFl (x), l = 1, 2, 3 are calculated as zF1,min = 8.9825, zF2,min = −48.6224, zF3,min = −42.7926, through the sequential quadratic programming method. By taking account of these values, assume that the DM identiﬁes the following linear membership function: ⎧ 1 ⎪ ⎪ ⎪ ⎨ zF (x) − zF l l,0 μl (zFl (x)) = F F ⎪ zl,1 − zl,0 ⎪ ⎪ ⎩ 0

if zFl (x) ≤ zFl,1 if zFl,1 < zFl (x) ≤ zFl,0 if zFl,0 < zFl (x),

where zFl,1 and zFl,0 are calculated as zF1,1 = 8.9825, zF1,0 = 31.7493, zF2,1 = −48.6224, zF2,0 = 12.0371, zF3,1 = −42.7926 and zF3,0 = −16.2347 by using the Zimmermann method (Zimmermann, 1978). For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.000, 1.000, 1.000), the augmented minimax problem (3.60) is solved by the sequential quadratic programming method, and the DM is supplied with the ﬁrst membership function values shown in Table 3.9.

3.1 Fuzzy multiobjective stochastic linear programming

77

Assume that the DM is not satisﬁed with the current membership function values, and the DM updates the reference membership levels from (1.000, 1.000, 1.000) to (0.800, 1.000, 1.000) for improving μ2 and μ3 at the sacriﬁce of μ1 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the second membership function values are calculated as shown in Table 3.9. Although a similar procedure continues in this manner until the DM is satisﬁed with the obtained objective function values, for simplicity, we suppose that the DM is satisﬁed with the objective function values in the third iteration, and then it follows that the satisﬁcing solution for the DM is derived. Table 3.9 Process of interaction. Iteration 1st μˆ 1 1.000 μˆ 2 1.000 1.000 μˆ 3 zF1 (x∗ ) 20.3634 zF2 (x∗ ) −18.3000 −29.5167 zF3 (x∗ ) μ1 (zF1 (x∗ )) 0.5001 μ2 (zF2 (x∗ )) 0.5001 0.5001 μ3 (zF3 (x∗ ))

2nd 0.800 1.000 1.000 23.1834 −22.9178 −31.5386 0.3762 0.5762 0.5762

3rd 0.800 1.000 0.850 22.6022 −24.4663 −28.2329 0.4018 0.6018 0.4518

3.1.3 Simple recourse model In the chance constrained problems which we dealt with in the previous subsections, for random data variations, a mathematical model is formulated such that the violation of the constraints is permitted up to speciﬁed probability levels. On the other hand, in a two-stage model including a simple recourse model as a special case, a shortage or an excess arising from the violation of the constraints is penalized, and then the expectation of the amount of the penalties for the constraint violation is minimized. We consider a simple recourse model for the multiobjective stochastic programming problems ⎫ minimize z1 (x) = c1 x ⎪ ⎪ ⎪ ⎪ ······ ⎬ minimize zk (x) = ck x (3.62) ⎪ ⎪ subject to Ax = b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, cl , l = 1, 2, . . . , k are n dimensional coefﬁcient row vectors, A is an m × n coefﬁcient matrix, and b¯ is

78

3 Fuzzy Multiobjective Stochastic Programming

an m dimensional random variable column vector. It is noted that, in a simple recourse model, the random variables are involved only in the right-hand side of the constraints. To understand an idea of the formulation in the simple recourse model, consider a decision problem of a manufacturing company. Suppose that the company makes m types of products which requires n kinds of working processes, and a DM in the company desires to optimize the total proﬁt and the total production cost simultaneously. Let x = (x1 , . . . , xn ) denote activity levels for the n kinds of working processes which are decision variables, and then T x denotes the amount of products, where T is an m × n matrix transforming the n kinds of activity levels of the working processes into the m types of products. The total proﬁt is expressed as c1 T x, where an m dimensional coefﬁcient vector c1 is a vector of unit product proﬁts for the m types of products, and the total production cost is represented by c2 x, where an n dimensional coefﬁcient vector c2 is a vector of unit costs for the n kinds of activity levels of the working processes. Assume that the demand coefﬁcients d¯ = (d¯1 , . . . , d¯m )T for the m products are uncertain, and they are represented by random variables. The demand ¯ where y+ and y− represent the constraints are expressed as T x + Iy+ − Iy− = d, errors for estimating the demands, and I is the m dimensional identity matrix. These two objectives are optimized under the demand constraints T x + Iy+ − Iy− = d¯ together with the ordinary constraints Bx ≤ e without uncertainty for the activity levels such as the capacity, budget, technology, etc. The constraints Ax = b¯ in (3.62) can be interpreted as the combined form of the demand constraints with random variables and the ordinary constraints without uncertainty. Let us return to a general case of the recourse model. It is assumed that, in this model, the DM must make a decision before the realized values of the random variables involved in (3.62) are observed, and the penalty of the violation of the constraints is incorporated into the objective function in order to consider the loss caused by random date variations. To be more speciﬁc, by expressing the difference between Ax and b¯ in (3.62) − + T − − T as two vectors y+ = (y+ 1 , . . . , ym ) and y = (y1 , . . . , ym ) , the expectation of a recourse for the lth objective function is represented by

− − + + − y y + q y ) − y = b(ω) − Ax , (3.63) Rl (x) = E min (q+ l l y+ , y−

− where q+ l and ql are m dimensional constant row vectors, and b(ω) is an m dimensional realization vector of b¯ for an elementary event ω. Thinking of each element − + T − − T of y+ = (y+ 1 , . . . , ym ) and y = (y1 , . . . , ym ) as a shortage and an excess of the − left-hand side, respectively, we can regard each element of q+ l and ql as the cost to compensate the shortage and the cost to dispose the excess, respectively. Then, for the multiobjective stochastic programming problem, the simple recourse problem is formulated as

3.1 Fuzzy multiobjective stochastic linear programming

79

⎫ minimize c1 x + R1 (x) ⎪ ⎪ ⎬ ······ minimize ck x + Rk (x) ⎪ ⎪ ⎭ subject to x ≥ 0.

(3.64)

− Because q+ l and ql are interpreted as penalty coefﬁcients for shortages and ex− cesses, it is quite natural to assume that q+ l ≥ 0 and ql ≥ 0, and then, it is evident that, for all i = 1, . . . , m, the complementary relations − yˆ+ i > 0 ⇒ yˆi = 0, + yˆ− i > 0 ⇒ yˆi = 0

should be satisﬁed for an optimal solution. With this observation in mind, we have n

− yˆ+ i = bi (ω) − ∑ ai j x j , yˆi = 0 if bi (ω) ≥ − yˆ+ i = 0, yˆi =

j=1 n

∑ ai j x j − bi (ω)

if bi (ω)

dl jsl , ⎩ max 1 − γl j where the value of dl jsl varies depending on which scenario sl ∈ {1, . . . , Sl } occurs, and βl j and γl j are not random parameters but constants. Fig. 4.2 illustrates an example of the membership function μC˜l js (τ). Formally, the membership function of l the fuzzy random variable C˜¯ is represented by lj

4.1 Multiobjective fuzzy random linear programming

105

⎧ d¯l j − τ ⎪ ⎪ max 1 − , 0 ⎪ ⎨ βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ,0 ⎩ max 1 − γl j

if τ ≤ d¯l j (4.4) if τ > d¯l j .

μC~ljs (τ) l

1

βlj

0

dljsl

γ lj

τ

Fig. 4.2 Example of the membership function μC˜l js . l

˜¯ x Through the extension principle given in Chapter 2, each objective function C l is represented by a single fuzzy random variable whose realized value for the scenario sl is a triangular fuzzy number C˜ lsl x characterized by the membership function ⎧ dlsl x − υ ⎪ ⎪ max 1 − , 0 if υ ≤ dlsl x ⎪ ⎨ βl x (4.5) μC˜ ls x (υ) = l ⎪ υ − dlsl x ⎪ ⎪ ,0 if υ > dlsl x, ⎩ max 1 − γl x where dlsl is an n dimensional column vector which is different from the other dl sˆl , sˆl ∈ {1, . . . , Sl }, sˆl = sl , and β l and γ l are n dimensional constant column vectors. Fig. 4.3 illustrates an example of the membership function μC˜ ls x (υ). Also for the l ¯˜ x, its membership function is formally expressed as lth objective function C l

⎧ d¯ l x − υ ⎪ ⎪ ⎪ , 0 max 1 − ⎨ βl x μC˜¯ x (υ) = l ⎪ υ − d¯ l x ⎪ ⎪ max 1 − , 0 ⎩ γl x

if υ ≤ d¯ l x (4.6) if υ > d¯ l x.

Considering the imprecise nature of human judgments, it is quite natural to assume that the decision maker (DM) may have a fuzzy goal for each of the objective ˜¯ x, and in a minimization problem, the DM speciﬁes the fuzzy functions zl (x) = C l goal such that “the objective function value should be substantially less than or equal to some value.” Such a fuzzy goal can be quantiﬁed by eliciting the corresponding

106

4 Multiobjective Fuzzy Random Programming

μC~

(υ)

ls l x

1

0

βl x

dlsl x

υ

γl x

Fig. 4.3 Example of the membership function μC˜ ls x . l

membership functions through some interaction process from the DM. Here, for simplicity, the linear membership function ⎧ ⎪ 0 if y > z0l ⎪ ⎪ ⎨ y − z0 l if z1l ≤ y ≤ z0l (4.7) μG˜ l (y) = 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 1 if y < z1l is assumed for representing the fuzzy goal of the DM, where z0l and z1l are determined as ⎫ n ⎪ ⎪ z0l = max max ∑ dl jsl x j , l = 1, . . . , k, ⎪ ⎬ sl ∈{1,...,Sl } x ∈ X j=1 (4.8) n ⎪ z1l = min min ∑ dl jsl x j , l = 1, . . . , k. ⎪ ⎪ ⎭ sl ∈{1,...,Sl } x ∈ X j=1

It should be noted here that z0l and z1l are obtained by solving linear programming problems. Fig. 4.4 illustrates an example of the membership function μG˜ l of a fuzzy goal G˜ l . μG~ (y) l

1

0

z l1

Fig. 4.4 Example of the membership function of a fuzzy goal.

z l0

y

4.1 Multiobjective fuzzy random linear programming

107

Recalling that the membership function is regarded as a possibility distribution as discussed in Chapter 2, the degree of possibility that the objective function value ˜¯ x attains the fuzzy goal G˜ is expressed as C l l (4.9) ΠC˜¯ x (G˜ l ) = sup min μC˜¯ x (y), μG˜ l (y) , l = 1, . . . , k, l

l

y

and for a given scenario sl ∈ {1, . . . , Sl }, (4.9) is reduced to ΠC˜ ls x (G˜ l ) = sup min μC˜ ls x (y), μG˜ l (y) , l = 1, . . . , k. l

(4.10)

l

y

Fig. 4.5 illustrates the degree of possibility that the fuzzy goal G˜ l is fulﬁlled under the possibility distribution μC˜ls x . l

μG~ (y)

μC~

l

ls l x

(y)

1 ~

ΠC~ x(Gl ) ls l

0

z 1l

0

(d lsl −βl ) x dlsl x

z 0l

y

Fig. 4.5 Degree of possibility ΠC˜ ls x (G˜ l ). l

Observing that the degrees of possibility vary randomly depending on which scenario occurs, it should be noted here that conventional possibilistic programming approaches cannot be directly applied to (4.2). With this observation in mind, realizing that (4.2) involves not only fuzziness but also randomness, we consider four types of fuzzy stochastic decision making models for multiobjective fuzzy random programming problems by introducing the concepts from possibility theory: the expectation, the variance, the probability and the fractile models.

4.1.1.1 Possibility-based expectation model Assuming that the DM intends to simply maximize the expected degree of possibility that each of the original objective functions involving fuzzy random variable coefﬁcients attains the fuzzy goals, the original multiobjective fuzzy random programming problems can be reformulated as

108

4 Multiobjective Fuzzy Random Programming

(

)⎫ maximize E ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ ⎪ 1 ⎪ ⎬ · · · · · · ( ) maximize E ΠC˜¯ x (G˜ k ) ⎪ ⎪ ⎪ k ⎪ ⎭ subject to x ∈ X,

(4.11)

where E denotes the expectation operator. From the viewpoint of possibility maximization and expectation optimization, as an extension of Pareto optimality discussed in Chapter 2, we deﬁne the concept of E-P-Pareto optimality. Deﬁnition 4.2 (E-P-Pareto Optimal Solution). A point x∗ ∈ X is said to be an EP-Pareto optimal solution ( ) to((4.2) if and) only if there does not exist (another x ∈ )X such that E ΠC˜¯ x (G˜ l ) ≥ E ΠC˜¯ x∗ (G˜ l ) for all l ∈ {1, . . . , k} and E ΠC˜¯ x (G˜ v ) = v l ) l ( E ΠC˜¯ x∗ (G˜ v ) for at least one v ∈ {1, . . . , k}. v

By substituting the triangular fuzzy random variable (4.5) and the linear fuzzy goal (4.7) into the degree of possibility (4.9), the degree of possibility is explicitly represented by n

∑ (βl j − dl jsl )x j + z0l

j=1 n

ΠC˜ ls x (G˜ l ) = l

∑ βl j x j − z1l + z0l

.

(4.12)

j=1

Recalling that the occurrence probability of scenario sl is plsl , the expectation of the degree of possibility is calculated as " # n

(

) E ΠC˜¯ x (G˜ l ) = l

Sl

∑ plsl ΠC˜ lsl x (G˜ l ) =

∑

βl j −

j=1

sl =1

n

∑

Sl

∑ plsl dl jsl

sl =1

x j + z0l .

βl j x j − z1l + z0l

j=1

Letting n

ZlΠ,E (x)

∑

" βl j −

j=1

Sl

#

∑ plsl dl jsl

sl =1

n

∑ β1 j x j − z11 + z01

x j + z0l ,

j=1

(4.11) is rewritten as

⎫ maximize Z1Π,E (x) ⎪ ⎪ ⎬ ··· Π,E maximize Zk (x) ⎪ ⎪ ⎭ subject to x ∈ X.

(4.13)

4.1 Multiobjective fuzzy random linear programming

109

To calculate a candidate for the satisﬁcing solution which is also E-P-Pareto optimal, in interactive multiobjective programming, the DM is asked to specify reference levels zˆl , l = 1, . . . , k of the objectives function values of (4.13), and it is called the reference expectation levels. For the DM’s reference expectation levels zˆl , l = 1, . . . , k, an E-P-Pareto optimal solution, which is the nearest to a vector of the reference expectation levels or better than it if the reference expectation levels are attainable in a sense of minimax, is obtained by solving the minimax problem ⎫ minimize max zˆl − ZlΠ,E (x) ⎬ 1≤l≤k (4.14) ⎭ subject to x ∈ X. By introducing an auxiliary variable v, (4.14) can be transformed into ⎫ minimize v ⎪ ⎪ ⎪ subject to zˆ1 − Z1Π,E (x) ≤ v ⎪ ⎬ ········· ⎪ ⎪ zˆk − ZkΠ,E (x) ≤ v ⎪ ⎪ ⎭ x ∈ X,

(4.15)

and (4.15) is equivalently expressed as ⎫ ⎪ ⎪ ⎪ S1 n ⎪ ⎪ 0 ⎪ ⎪ − p d + z β x ∑ 1 j ∑ 1s1 1 js1 j 1 ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ subject to ≥ z ˆ − v 1 ⎪ n ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎬ j=1 · · · · · · · · · · · · · · · " # ⎪ ⎪ Sk n ⎪ ⎪ 0 ⎪ β x − p d + z ⎪ ∑ k j ∑ ksk k jsk j k ⎪ ⎪ ⎪ j=1 sk =1 ⎪ ⎪ ≥ z ˆ − v k n ⎪ ⎪ ⎪ 1 0 ⎪ β x − z + z ⎪ ∑ kj j k k ⎪ ⎪ j=1 ⎪ ⎭ x ∈ X. minimize v

"

#

(4.16)

It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (4.16) can be reduced to a set of linear inequalities. Since v satisﬁes Π,E Π,E zˆmax − max Zl,max ≤ v ≤ zˆmax − min Zl,min , l∈{1,...,k}

l∈{1,...,k}

where zˆmax = max zˆl , l∈{1,...,k}

Π,E Zl,max = max ZlΠ,E (x), x∈X

Π,E Zl,min = min ZlΠ,E (x), x∈X

110

4 Multiobjective Fuzzy Random Programming

we can obtain the minimum value of v by the combined use of the phase one of the two-phase simplex method and the bisection method. After ﬁnding the minimum value v∗ , in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem " # ⎫ S1 n ⎪ 0 ⎪ ⎪ ∑ ∑ p1s1 d1 js1 − β1 j x j − z1 ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ minimize ⎪ n ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎪ j=1 ⎪ " # ⎪ ⎪ S2 n ⎪ ⎪ 0 ⎪ β x − p d + z ⎪ 2 j 2s 2 js j ∑ ∑ 2 2 2 ⎪ ⎪ ⎪ j=1 s2 =1 ⎪ ∗ ⎬ subject to ≥ z ˆ − v 2 n 1 0 (4.17) β x − z + z ∑ 2j j 2 2 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ "· · · · · · · · · · · · · · · # ⎪ Sk ⎪ n ⎪ ⎪ 0 ⎪ ⎪ ∑ βk j − ∑ pksk dk jsk x j + zk ⎪ ⎪ ⎪ j=1 sk =1 ∗ ⎪ ≥ zˆk − v ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ β x − z + z j k j ⎪ ∑ k k ⎪ ⎪ j=1 ⎪ ⎭ x ∈ X, where the ﬁrst objective function z1 (x) in (4.2) is supposed to be the most important to the DM. From the assumption of β1 j > 0, j = 1, . . . , n and z01 > z11 , one ﬁnds that n

∑ β1 j x j − z11 + z01 > 0.

j=1

Then, using the variable transformation (Charnes and Cooper, 1962) # " ς = 1/

n

∑ β1 j x j − z11 + z01

, y = ςx, ς > 0,

(4.18)

j=1

the linear fractional programming problem (4.17) is equivalently transformed into the linear programming problem

4.1 Multiobjective fuzzy random linear programming

111

⎫ ⎪ ⎪ ⎪ minimize ∑ ∑ p1s1 d1 js1 − β1 j y j − ςz01 ⎪ ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ ⎪ " # " # ⎪ ⎪ S2 n n ⎪ ⎪ 1 0 0 ⎪ subject to τ2 ∑ β2 j y j − ςz2 + ςz2 + ∑ ∑ p2s2 d2 js2 − β2 j y j − ςz2 ≤ 0 ⎪ ⎪ ⎪ ⎪ j=1 j=1 s2 =1 ⎪ ⎪ ⎪ ⎪ · · · · · · · · · · · · · · · ⎪ " # " # ⎪ ⎬ Sk n n 1 0 0 τk ∑ βk j y j − ςzk + ςzk + ∑ ∑ pksk dk jsk − βk j y j − ςzk ≤ 0 ⎪ ⎪ j=1 j=1 sk =1 ⎪ ⎪ ⎪ ⎪ Ay − ςb ≤ 0 ⎪ ⎪ ⎪ n ⎪ ⎪ 0 1 ⎪ β y + ς(z − z ) = 1 ⎪ ∑ 1j j 1 1 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎪ ⎭ ς ≥ 0, (4.19) where τl zˆl − v∗ , l = 2, . . . , k, and δ is a sufﬁciently small positive number given for satisfying the condition ς > 0 in (4.18). It should be noted here that if the optimal solution (y∗ , ς∗ ) to (4.19) is not unique, E-P-Pareto optimality of x∗ (= y∗ /ς∗ ) is not always guaranteed. Realizing such a situation, E-P-Pareto optimality of x∗ can be tested by solving the linear programming problem ⎫ k ⎪ ⎪ ⎪ maximize w = ∑ εl ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ∗ ⎪ q1 (x ) ⎪ ⎬ r subject to q1 (x) − ε1 = (x) 1 ∗ r1 (x ) (4.20) ⎪ ········· ⎪ ⎪ ⎪ ⎪ qk (x∗ ) ⎪ ⎪ r qk (x) − εk = (x) ⎪ k ⎪ rk (x∗ ) ⎪ ⎭ T x ∈ X, ε = (ε1 , . . . , εk ) ≥ 0, n

"

#

S1

where ql (x) rl (x)

n

∑

j=1 n

" βl j −

Sl

∑ plsl dl jsl

sl =1

# x j + z0l , l = 1, . . . , k

∑ βl j x j − z1l + z0l , l = 1, . . . , k.

j=1

Noting that (4.20) is the same type of problem as (3.38), it is evident that an E-PPareto optimal solution can be obtained by using a similar computational method to the given method in Chapter 3. The DM must either be satisﬁed with the current E-P-Pareto optimal solution or continue to examine another solution by updating the reference expectation levels. In order to help the DM update the reference expectation levels, the trade-off infor-

112

4 Multiobjective Fuzzy Random Programming

mation between an expected degree of possibility with respect to the most important objective function and each of the other ones is quite useful. Such trade-off information is easily obtainable by using the simplex multipliers of linear programming problem (4.19). To derive the trade-off information, the Lagrange function L for (4.19) is deﬁned as L(y, ς, π, ζ, φ) =

n

S1

j=1

s1 =1

∑ ∑

p1s1 d1 js1 − β1 j y j − ςz01

% "

k

+ ∑ πl τl l=2 n

+∑

#

n

∑

βl j y j − ςz1l + ςz0l

j=1

Sl

∑ plsl dl jsl − βl j

sl =1

j=1

+ζm+1

& y j − ςz0l

n

∑ β1 j y j + ς(z01 − z11 ) − 1

m

+ ∑ ζi (ai y − ςbi ) i=1

+ ζm+2 (−ς + δ)

j=1 n

− ∑ φ j y j − φn+1 ς,

(4.21)

j=1

where ai , i = 1, . . . , m are n dimensional coefﬁcient row vectors, and πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers of (4.19). In a manner similar to (3.41)–(3.45) in Chapter 3, the trade-off rates are calculated as n

−

∂Z1Π,E (x∗ ) ∂ZlΠ,E (x∗ )

= π∗l

∑ βl j x∗j − z1l + z0l

j=1 n

∑

β1 j x∗j − z11 + z01

, l = 2, . . . , k.

(4.22)

j=1

It should be stressed here that in order to obtain the trade-off information from (4.22), the ﬁrst (k − 1) constraints in (4.19) must be active. Therefore, if there are inactive constraints, it is necessary to replace zˆl for inactive constraints with ZlΠ,E (x∗ ) + v∗ and solve the corresponding problem to obtain the simplex multipliers. Following the preceding discussion, we can now present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based expectation model Step 1: Determine the linear membership functions μG˜ l , l = 1, . . . , k deﬁned as (4.7) by calculating z0l and z1l , l = 1, . . . , k. Step 2: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k.

4.1 Multiobjective fuzzy random linear programming

113

Step 3: For the current reference expectation levels zˆl , l = 1, . . . , k, solve the minimax problem (4.14). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (4.19), replace zˆl for inactive constraints with ZlΠ,E (x∗ ) + v∗ and resolve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the E-P-Pareto optimality test. Step 4: The DM is supplied with the corresponding E-P-Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the current objective function values ZlΠ,E (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k by considering the current objective function values together with the trade-off rates −∂Z1Π,E /∂ZlΠ,E , l = 2, . . . , k, and return to step 3. Observing that the trade-off rate −∂Z1Π,E /∂ZlΠ,E , l = 2, . . . , k in step 4 indicates the decrement value of the objective function Z1Π,E with a unit increment of value of Π,E information is used to estimate the local an objective function Zl , such trade-off

shape of Z1Π,E (x∗ ), . . . , ZkΠ,E (x∗ ) around x∗ . Here it should be stressed for the DM that any improvement of one expectation of the degree of possibility can be achieved only at the expense of at least one of the other expectations.

4.1.1.2 Possibility-based variance model As discussed in the previous section, the possibility-based expectation model would be appropriate if the DM intends to simply maximize the expected degrees of possibility without concerning about those ﬂuctuations. However, when the DM prefers to decrease the ﬂuctuation of the objective function values, the possibility-based expectation model is not relevant because some scenario yielding a very low possibility of good performance may occur even with a small probability. To avoid such risk, from the risk-averse point of view, by minimizing the variance of the degree of possibility under the constraints of feasibility together with the conditions for the expected degrees of possibility, the possibility-based variance model is formulated for the multiobjective fuzzy random programming problems as ⎫ ) ( ⎪ minimize Var ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ 1 ⎪ ⎪ ⎪ · ·(· · · · · · · ⎪ ⎬ ) ˜ (4.23) minimize Var ΠC˜¯ x (Gk ) k ⎪ ) ( ⎪ ⎪ subject to E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ l ⎪ ⎭ x ∈ X, where Var denotes the variance operator, and ξl , l = 1, . . . , k are permissible expectation levels for the expected degrees of possibility speciﬁed by the DM.

114

4 Multiobjective Fuzzy Random Programming

For notational convenience, let X(ξξ) be the feasible region of (4.23), namely ) ( X(ξξ) x ∈ X E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k . l From the viewpoint of possibility maximization and variance minimization, we deﬁne the concept of V-P-Pareto optimal solutions. Deﬁnition 4.3 (V-P-Pareto Optimal Solution). A point x∗ ∈ X(ξξ) is said to be a V-P-Pareto optimal solution to (4.2) does not exist another ( ) if and ( only if there ) ˜ ˜ x ∈ X(ξξ) such that Var ΠC˜¯ x (Gl ) ≤ Var ΠC˜¯ x∗ (Gl ) for all l ∈ {1, . . . , k} and l l ) ( ) ( ˜ ˜ Var ΠC˜¯ x (Gv ) = Var ΠC˜¯ x∗ (Gv ) for at least one v ∈ {1, . . . , k}. v

v

Recalling the degree of possibility (4.9), each of the objective functions in (4.23) is calculated as & % ( ) n 1 Var Π ˜¯ (G˜ l ) = " # Var ∑ d¯l j x j Cl x

n

∑

2

j=1

βl j x j − z1l + z0l

j=1

1

="

n

∑

T #2 x Vl x,

(4.24)

βl j x j − z1l + z0l

j=1

where Vl is the variance-covariance matrix of d¯ l expressed by ⎤ ⎡ l l v11 v12 · · · vl1n ⎢ vl vl · · · vl ⎥ 2n ⎥ ⎢ 21 22 Vl = ⎢ . . . . ⎥ , l = 1, . . . , k, ⎣ .. .. . . .. ⎦ vln1 vln2 · · · vlnn and vlj j

= Var[d¯l j ] =

Sl

∑ plsl {dl jsl }

2

sl =1

−

Sl

2

∑ plsl dl jsl

sl =1

, j = 1, . . . , n,

vljr = Cov[d¯lr , d¯lr ] = E[d¯l j , d¯lr ] − E[d¯l j ]E[d¯lr ] =

Sl

Sl

Sl

sl =1

sl =1

sl =1

∑ plsl dl jsl dlrsl − ∑ plsl dl jsl ∑ plsl dlrsl ,

j = r, r = 1, . . . , n.

Furthermore, ) from (4.13), the constraint of the expected degree of possibility ( ˜ E ΠC˜¯ x (Gl ) ≥ ξl is calculated as l

4.1 Multiobjective fuzzy random linear programming n

Sl

∑ ∑ plsl dl jsl − (1 − ξl )βl j

j=1

115

sl =1

x j ≤ z0l − ξl (z0l − z1l ).

(4.25)

By substituting (4.24) and (4.25) into (4.23), (4.23) is equivalently transformed as minimize Z1Π,V (x) = "

1 n

T #2 x V1 x

∑ β1 j x j − z11 + z01

j=1

minimize

·················· 1 "

ZkΠ,V (x) =

T #2 x Vk x

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ Sl n ⎪ ⎪ 0 0 1 subject to ∑ ∑ plsl dl jsl − (1 − ξl )βl j x j ≤ zl − ξl (zl − zl ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ j=1 sl =1 ⎪ ⎭ x ∈ X. (4.26) From the fact that n

∑ βk j x j − z1k + z0k

n

∑ βl j x j − z1l + z0l > 0

j=1

≥ 0 due to the positive-semideﬁnite property of Vl , by computing the and square root of the objective functions of (4.26), (4.26) is rewritten as ' ⎫ xT V1 x ⎪ ⎪ ⎪ minimize Z1Π,SD (x) = n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ··············· ⎪ ⎪ ⎪ ⎬ ' T x Vk x Π,SD minimize Zk (x) = n ⎪ ⎪ ⎪ ⎪ ⎪ ∑ βk j x j − z1k + z0k ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ S n l ⎪ ⎪ 0 0 1 ⎪ subject to ∑ ∑ plsl dl jsl − (1 − ξl )βl j x j ≤ zl − ξl (zl − zl ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ j=1 sl =1 ⎪ ⎭ x ∈ X, (4.27) where each of the objective functions represents the standard deviation of the degree of possibility. Realizing that the minimization of the variance is equivalent to the minimization of the standard deviation, it is signiﬁcant to emphasize that each of Pareto optimal solutions of (4.27) becomes a V-P-Pareto optimal solution. xT Vl x

116

4 Multiobjective Fuzzy Random Programming

To calculate a candidate for the satisﬁcing solution, the DM is asked to specify the reference levels zˆl , i = 1, . . . , k of the objective function values of (4.27), and it is called the reference standard deviation levels. For the DM’s reference standard deviation levels zˆl , i = 1, . . . , k, a V-P-Pareto optimal solution is obtained by solving the minimax problem ⎫ ⎪ minimize max ZlΠ,SD (x) − zˆl ⎪ ⎪ 1≤l≤k ⎪ ⎪ ⎬ subject to

n

Sl

j=1

sl =1

∑ ∑ plsl dl jsl − (1 − ξl )βl j

x j ≤ z0l − ξl (z0l − z1l ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

x ∈ X.

(4.28) For notational convenience, we introduce Nl (x) and Ql (x) such that Nl (x) , ZlΠ,SD (x) − zˆl Ql (x) where Nl (x) =

'

" xt Vl x − zˆl

n

∑

(4.29)

# βl j x j − z1l + z0l

j=1

Dl (x) =

n

∑ βl j x j − z1l + z0l .

j=1

Since the numerator Nl (x) is a convex function and the denominator Dl (x) is an afﬁne function, it follows that Nl (x)/Dl (x) is a quasi-convex function. From this property of (4.28), although it cannot be solved directly by using convex programming techniques because of the nonconvexity of the objective function, we can solve (4.28) by using the following extended Dinkelbach-type algorithm (Borde and Crouzeix, 1987): Extended Dinkelbach-type algorithm for solving (4.28) Step 1: Step 2:

Set r := 0 and ﬁnd a feasible solution xr ∈ X(ξξ). For a qr calculated by Nl (xr ) r q = max , 1≤l≤k Dl (xr )

ﬁnd an optimal solution xc to the convex programming problem

4.1 Multiobjective fuzzy random linear programming

117

minimize v subject to

1 {Dl (x) − qr Nl (x)} ≤ v, l = 1, . . . , k Dl (xr ) n

Sl

j=1

sl =1

∑ ∑ plsl dl jsl − (1 − ξl )βl j

x ∈ X.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ x j ≤ z0l − ξl (z0l − z1l ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.30) Step 3: For a sufﬁciently small positive number ε, if v < ε, stop the algorithm. Otherwise, set xr := xc , r := r + 1, and return to step 2. Now we are ready to summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based variance model Step 1: Determine the linear membership functions μG˜ l , l = 1, . . . , k with z0l and z1l , l = 1, . . . , k obtained by solving linear programming problems (4.8). Step 2: Calculate the individual minima and maxima of ZlΠ,E (x), l = 1, . . . , k. Step 3: Ask the DM to specify the permissible expectation levels ξl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference standard deviation levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 5: For the current reference standard deviation levels zˆl , l = 1, . . . , k, solve the minimax problem (4.28) by using the extended Dinkelbach-type algorithm. Step 6: The DM is supplied with the obtained V-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values ZlΠ,SD (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference standard deviation levels zˆl , l = 1, . . . , k, and return to step 5.

4.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing algorithm for the possibility-based expectation model, consider the following example of multiobjective fuzzy random programming problems: ⎫ ˜¯ x ⎪ minimize z1 (x) = C ⎪ 1 ⎪ ⎪ ˜ ⎪ ¯ ⎬ minimize z2 (x) = C2 x ˜ (4.31) ¯ 3x minimize z3 (x) = C ⎪ ⎪ subject to ai x ≤ bi , i = 1, . . . , 4 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T , and the values of parameters involved in the objective functions and the constraints are shown in Tables 4.1 and 4.2, respectively.

118

4 Multiobjective Fuzzy Random Programming

After determining the linear membership functions to quantify fuzzy goals for the objective functions, the initial reference expectation levels are set at the ideal values (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 1.0, 1.0). For the initial reference expectation levels, the minimax problem (4.28) is solved by using the combined use of the variable transformation and the simplex method. The DM is supplied with the objective function values of the ﬁrst iteration as shown in Table 4.3. Assume that the DM is not satisﬁed with the objective function values of the ﬁrst iteration, and the DM updates the reference expectation levels as (1.0, 0.9, 1.0) for improving Z1Π,E and Z3Π,E at the expense of Z2Π,E . For the updated reference expectation levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.3. Moreover, assume that the DM is not satisﬁed with the objective function values of the second iteration, and the DM updates the reference expectation levels as (1.0, 0.9, 0.9) for improving Z1Π,E and Z2Π,E at the expense of Z3Π,E . For the updated reference expectation levels, the corresponding minimax problem is solved, and the objective function values of the third iteration are calculated as shown in Table 4.3. If the DM is satisﬁed with the current objective function values, it follows that the satisﬁcing solution for the DM is derived.

4.1.2 Possibility-based probability and fractile models In contrast to the expectation and variance models which are regarded as optimization approaches, as satisﬁcing approaches to multiobjective fuzzy random programming problems, we discuss the possibility-based probability and fractile models by incorporating the ideas of possibilistic programming into the probability and fractile models in stochastic programming. Throughout this subsection, assuming that coefﬁcients of the objective functions are fuzzy random variables, we consider the following multiobjective fuzzy random programming problem: ˜¯ x ⎫ minimize z1 (x) = C ⎪ 1 ⎪ ⎬ ········· (4.32) ˜¯ x ⎪ minimize zk (x) = C k ⎪ ⎭ subject to Ax ≤ b, x ≥ 0, where x is an n dimensional decision variable column vector, A is an m × n co˜¯ = (C˜¯ , . . . , C˜¯ ), efﬁcient matrix, b is an m dimensional column vector, and C l l1 ln l = 1, . . . , k are n dimensional coefﬁcient row vectors of fuzzy random variables. For notational convenience, let X denote the feasible region of (4.32), namely X {x ∈ Rn | Ax ≤ b, x ≥ 0}.

4.1 Multiobjective fuzzy random linear programming

119

Table 4.1 Value of each parameter of the fuzzy random variable coefﬁcients. d11s1 d12s1 d13s1 p1 d21s2 d22s2 d23s2 p2 d31 d32 d33 p3

s1 = 1 −2.5 −3.5 −2.25 0.25 s2 = 1 −2.5 −0.75 −2.5 0.3 s3 = 1 3.0 2.5 4.5 0.2

s1 = 2 −2.0 −3.0 −2.0 0.4 s2 = 2 −2.0 −0.5 −2.25 0.5 s3 = 2 3.25 2.75 4.75 0.45

s1 = 3 −1.5 −2.5 −1.75 0.35 s2 = 3 −1.5 −0.25 −2.0 0.2 s3 = 3 3.5 3.0 5.0 0.35

β 0.4 0.5 0.4 β 0.3 0.4 0.3 β 0.4 0.5 0.4

Table 4.2 Value of each element of ai , i = 1, . . . , 4. a1 a2 a3 a4

3 2 3 1

2 1 4 3

1 2 3 2

Table 4.3 Process of interaction. Iteration

1st

2nd

3rd

zˆ1 zˆ2 zˆ3

1.000 1.000 1.000

1.000 0.900 1.000

1.000 0.900 0.900

Z1Π,E (x) Z2Π,E (x) Z3Π,E (x)

0.540 0.540 0.540

0.574 0.474 0.574

0.617 0.517 0.517

Although we dealt with discrete fuzzy random variables in the previous subsections, we assume that each C˜¯l j is a continuous fuzzy random variable whose realized value C˜l j (ω) for each elementary event ω is characterized by ⎧ dl j (ω) − τ ⎪ ⎪ if τ ≤ dl j (ω) ⎪ ⎨L βl j μC˜l j (ω) (τ) = (4.33) ⎪ τ − dl j (ω) ⎪ ⎪ ⎩R if τ > dl j (ω), γl j where L and R are nonincreasing continuous functions, and each dl j (ω) is a realized value of the coefﬁcient of a continuous random variable d¯l j for an elementary event ω. If L(t) = R(t) = max{0, 1 − t}, the L-R fuzzy number can be viewed as a

120

4 Multiobjective Fuzzy Random Programming

triangular fuzzy number. With this observation in mind, it is evident that the fuzzy numbers characterized by (4.33) are more general than those by (4.3). An example of the membership function μC˜l j (ω) is given in Fig. 4.6. Formally, the membership function of the fuzzy random variable C˜¯l j is represented by ⎧ ¯ dl j − τ ⎪ ⎪ L ⎪ ⎨ βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ⎩R γl j

if τ ≤ d¯l j (4.34) if τ > d¯l j .

μC~lj(ω)(τ) 1

0

L

βlj

R

γ lj

τ

dlj (ω)

Fig. 4.6 Example of the membership function μC˜l j (ω) .

Since the coefﬁcients of the objective function are fuzzy random variables whose ˜¯ x is also reprerealized values are L-R fuzzy numbers, each objective function C l sented by a single fuzzy random variable, and its realized value for an elementary event ω can be expressed as an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x (4.35) μC˜ l (ω)x (υ) = ⎪ ⎪ ⎪R υ − dl (ω)x ⎩ if υ > dl (ω)x, γl x where dl (ω) is an n dimensional column vector of the realized values of the random variables d¯ l j , j = 1, . . . , n. Fig. 4.7 illustrates an example of the membership func˜¯ x, its membership function is tion μC˜ l (ω)x (υ). Also for the lth objective function C l formally expressed as

4.1 Multiobjective fuzzy random linear programming

121

⎧ ¯ dl x − υ ⎪ ⎪ L ⎪ ⎨ βl x μC˜¯ x (υ) = l ⎪ ⎪ υ − d¯ l x ⎪ ⎩R γl x

if υ ≤ d¯ l x (4.36) if υ > d¯ l x.

μC~l (ω)x(υ) 1

L

0

βl x

R

υ

γl x dl (ω) x

Fig. 4.7 Example of the membership function μC˜¯ l (ω)x .

In a way similar to the previous models, taking into account the imprecise nature of the DM’s judgments, we introduce the fuzzy goal G˜ l with the membership function μG˜ l . To elicit a membership function μl from the DM for each of the fuzzy goals, the DM is asked to assess an unacceptable objective function value z0l and a desirable objective function value z1l . An example of the membership function of the fuzzy goal is depicted in Fig. 4.8.

μG~ (y) l

1

0

z 1l

z l0

y

Fig. 4.8 Example of the membership function μG˜ l .

To help the DM determine z0l and z11 , it is recommended to calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k obtained by solving the linear programming problems

122

4 Multiobjective Fuzzy Random Programming

⎫ minimize E[d¯ l ]x ⎪ ⎪ ⎪ subject to x ∈ X ⎪ ⎬ ⎪ ⎪ maximize E[d¯ l ]x ⎪ ⎪ ⎭ subject to x ∈ X

, l = 1, . . . , k,

(4.37)

where E[d¯ l ] = (E[d¯l1 ], . . . , E[d¯ln ]), l = 1, . . . , k. From the viewpoint of possibilistic programming, we consider the degree of pos˜¯ x attains the fuzzy goal G˜ is expressed sibility that the objective function value C l l as (4.38) ΠC˜¯ x (G˜ l ) = sup min μC˜¯ x (y), μG˜ l (y) , l = 1, . . . , k, l

l

y

and for a given elementary event ω, (4.38) is reduced to ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, . . . , k.

(4.39)

y

Fig. 4.9 illustrates the degree of possibility ΠC˜ l (ω)x (G˜ l ).

μG~ (y) l

μC~l(ω) x (y)

1

~ Π C~l(ω) x(Gl )

0

y

Fig. 4.9 Degree of possibility ΠC˜ l (ω)x (G˜ l ).

It is signiﬁcant to emphasize that the degree ΠC˜ l (ω)x (G˜ l ) deﬁned by (4.39) varies randomly due to the randomness of occurrence of an elementary event ω. With this observation in mind, realizing such randomness of the degree, by introducing some ideas from probability theory as well as the fractile model in stochastic programming into possibilistic programming, the possibility-based probability and fractile models are developed.

4.1.2.1 Possibility-based probability model As a satisﬁcing approach to multiobjective fuzzy random programming, assuming that the DM intends to maximize the probabilities that the degrees of possibility are greater than or equal to certain target values, we consider the possibility-based probability model for the multiobjective fuzzy random programming problem which

4.1 Multiobjective fuzzy random linear programming

is formulated as

123

⎫ maximize Z1Π,P (x) = P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎪ ⎬ · · · ··············· maximize ZkΠ,P (x) = P ω ΠC˜ k (ω)x (G˜ k ) ≥ hk ⎪ ⎪ ⎪ ⎪ ⎭ subject to x ∈ X,

(4.40)

where P denotes a probability measure, and hl , l = 1, . . . , k are target values for the degree of possibility speciﬁed by the DM. From the viewpoint of maximization of possibility and probability, we deﬁne the concept of P-P-Pareto optimal solutions. Deﬁnition 4.4 (P-P-Pareto Optimal Solution). A point x∗ ∈ X is said to be a P-PPareto optimal solution to (4.32) if and only if there does not exist another x ∈ X such that ZlΠ,P (x) ≥ ZlΠ,P (x∗ ) for all l ∈ {1, . . . , k} and ZvΠ,P (x) = ZvΠ,P (x∗ ) for at least one v ∈ {1, . . . , k}. From the deﬁnition of the degree of possibility, as shown in (4.39), the following relation holds: ΠC˜ l (ω)x (G˜ l ) ≥ hl ⇔ ∃y : μC˜ l (ω)x (y) ≥ hl , μG˜ l (y) ≥ hl

βl (ω)}x ≤ y ≤ {dl (ω) + R (hl )γγl (ω)}x, y ≤ μG˜ (hl ) ⇔ ∃y : {dl (ω) − L (hl )β

βl (ω)}x ≤ μG˜ (hl ), ⇔ {dl (ω) − L (hl )β l

l

(4.41)

where L (hl ) and μG˜ (hl ) are pseudo inverse functions deﬁned by l

L (hl ) = sup{r | L(r) ≥ hl }, μG˜ (hl ) = sup{r | μG˜ l (r) ≥ hl }. l

(4.42)

Along the same line as the problem formulation employed by Stancu-Minasian (1984), assume that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where t¯l is a random variable with the mean ml and the variance σ2l , and dll j , l = 1, 2 are constants. If dl2 x > 0, l = 1, · · · , k for any x ∈ X, one ﬁnds that βl (ω)}x ≤ μG˜ (hl ) P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = P ω {dl (ω) − L (hl )β l " (h )β 1 }x + μ (h ) # β − d {L l l l G˜ l l = P ω tl (ω) ≤ dl2 x " # βl − dl1 }x + μG˜ (hl ) {L (hl )β l = Tl . (4.43) dl2 x Consequently, (4.40) is rewritten as

124

4 Multiobjective Fuzzy Random Programming

" maximize Z1Π,P (x) = T1

β1 − d11 }x + μG˜ (h1 ) {L (h1 )β 1

d12 x

#⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

·················· " # βk − dk1 }x + μG˜ (hk ) {L (hk )β Π,P k maximize Zk (x) = Tk dk2 x subject to x ∈ X.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.44)

In order to calculate a candidate for the satisﬁcing solution which is also P-PPareto optimal, the DM is asked to specify reference levels zˆl , l = 1, . . . , k, and it is called the reference probability levels. For the DM’s reference probability levels zˆl , l = 1, . . . , k, a P-P-Pareto optimal solution, which is the nearest to a vector of the reference probability levels or better than it if the reference levels are attainable in a sense of minimax, is obtained by solving the minimax problem ⎫ Π,P minimize max zˆl − Zl (x) ⎬ 1≤l≤k (4.45) ⎭ subject to x ∈ X or equivalently ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ β1 − d1 }x + μG˜ (h1 ) {L (h1 )β ⎪ ⎪ 1 ⎪ ≥ T subject to (ˆ z − v) ⎪ 1 1 2 ⎪ d1 x ⎪ ⎬ minimize v

··················

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βk − dk1 }x + μG˜ (hk ) {L (hk )β ⎪ ⎪ k ⎪ ≥ T (ˆ z − v) k ⎪ k 2 ⎪ dk x ⎪ ⎪ ⎭ x ∈ X,

(4.46)

where Tl (s) is a pseudo inverse function deﬁned by Tl (s) = inf{r | Tl (r) ≥ s}, l = 1, . . . , k. After ﬁnding the minimum value v∗ by using the combination of the phase one of the two-phase simplex method and the bisection method, in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem

4.1 Multiobjective fuzzy random linear programming

125

⎫ ⎪ ⎪ ⎪ minimize ⎪ 2 ⎪ d1 x ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β2 − d2 }x + μG˜ (h2 ) {L (h2 )β ⎪ ∗ ⎪ 2 ⎪ ⎪ ≥ T (ˆ z − v ) subject to 2 2 ⎬ d2 x β1 }x − μG˜ (h1 ) {d11 − L (h1 )β 1

2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β {L (hk )β k − dk }x + μG˜ (hk ) ⎪ ∗ ⎪ k ⎪ ⎪ ≥ T (ˆ z − v ) k ⎪ k 2 ⎪ dk x ⎪ ⎭ x ∈ X, ··················

(4.47)

where the ﬁrst objective function z1 (x) in (4.32) is supposed to be the most important to the DM. Employing the Charnes-Cooper variable transformation ς = 1/dl2 x, y = ςx, ς > 0,

(4.48)

we can transform (4.47) equivalently into ⎫ ⎪ ⎪ ⎪ ⎪ 2 1 β2 }y − ςμG˜ (h2 ) ≤ 0 ⎪ subject to τ2 d2 y + {d2 − L (h2 )β ⎪ ⎪ 2 ⎪ ⎪ ⎪ ·················· ⎪ ⎪ ⎪ 2 1 β τk dk y + {dk − L (hk )β k }y − ςμG˜ (hk ) ≤ 0 ⎬ k Ay − ςb ≤ 0 ⎪ ⎪ ⎪ ⎪ d12 y = 1 ⎪ ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎭ ς ≥ 0, β1 }y − ςμG˜ (h1 ) minimize {d11 − L (h1 )β 1

(4.49)

where τl Tl (ˆzl − v∗ ), l = 2, . . . , k, and δ is a sufﬁciently small positive number for satisfying the condition ς > 0. If the optimal solution (y∗ , ς∗ ) to (4.49) is not unique, P-P-Pareto optimality of ∗ x (= y∗ /ς∗ ) is not always guaranteed. By solving the P-P-Pareto optimality test problem similar to (4.20), P-P-Pareto optimality of x∗ is veriﬁed. If the DM is not satisﬁed with the P-P-Pareto optimal solution with respect to the reference probability levels, the DM must update them to obtain another P-P-Pareto optimal solution. Then, to help the DM update them, the trade-off information between a probability of the most important objective function and each of the other probabilities is quite useful. To derive the trade-off information, we deﬁne the Lagrange function L for (4.49) by

126

4 Multiobjective Fuzzy Random Programming

β1 }y − ςμG˜ (h1 ) L(y, ς, π, ζ, φ) = {d11 − L (h1 )β 1 ( ) k βl }y − ςμG˜ (hl ) + ∑ πl τl dl2 y + {dl1 − L (hl )β l

l=2 m

+ ∑ ζi (ai y − ςbi ) + ζm+1 d12 y − 1 i=1

n

+ζm+2 (−ς + δ) − ∑ φ j y j − φn+1 ς,

(4.50)

j=1

where πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers. Through a procedure similar to (3.41)–(3.47) in 3.1.2.1, the trade-off rates are calculated as − Π,P

∂Z1Π,P (x∗ )

∂ZlΠ,P (x∗ )

= π∗l

dl2 x∗ Z1Π,P (x∗ ) , l = 2, . . . , k,

d12 x∗ Z Π,P (x∗ )

(4.51)

l

Π,P

where Zl (x∗ ) denotes the differential coefﬁcient of Zl (x∗ ). To calculate the trade-off information from (4.51), the ﬁrst (k − 1) constraints in (4.49) must be active. Therefore, if there are inactive constraints, it is necessary to replace zˆl of inactive constraints with ZlΠ,P (x∗ )+v∗ and solve the revised problem again to obtain the simplex multipliers. Following the above discussions, we now present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target values hl , l = 1, . . . , k. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k, solve the minimax problem (4.46). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (4.49), replace zˆl of inactive constraints with ZlΠ,P (x∗ ) + v∗ and solve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the P-P-Pareto optimality test. Step 6: The DM is supplied with the corresponding P-P-Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k

4.1 Multiobjective fuzzy random linear programming

127

by considering the current values of objective functions ZlΠ,P , l = 1, . . . , k together with the trade-off rates −∂Z1Π,P /∂ZlΠ,P , l = 2, . . . , k, and return to step 5. The trade-off rate −Z1Π,P /∂ZlΠ,P , l = 2, . . . , k in step 5 indicates the decrement value of the membership function Z1Π,P with a unit increment of the value of the objective function ZlΠ,P , and they are used to estimate the local shape of Z1Π,P (x∗ ), . . ., ZkΠ,P (x∗ ) around x∗ . 4.1.2.2 Possibility-based fractile model The possibility-based probability model, discussed in the previous subsection, is recommended for the DM who intends to maximize the probability with respect to the degree of possibility. On the other hand, if the DM prefers to maximize the degree of possibility rather than maximize the probability, from the viewpoint of fractile optimization criterion, it would be appropriate to employ the possibilitybased fractile model. In this subsection, assuming that the DM aspires to maximize the degree of possibility under the condition that the probability with respect to the degree of possibility is greater than or equal to a certain permissible probability level, we present the possibility-based fractile model for the multiobjective fuzzy random programming problem formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎬ (4.52) maximize hk ⎪ ⎪ ⎪ ˜ subject to P ω ΠC˜l (ω)x (Gl ) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X, 0 ≤ hl ≤ 1, where θl , l = 1, . . . , k are permissible probability levels speciﬁed by the DM, satisfying θl ∈ (1/2, 1]. For notational convenience, let X(θθ) be the feasible region of (4.52), namely X(θθ) x ∈ X P ω ΠC˜l (ω)x (G˜ l ) ≥ hl ≥ θl , 0 ≤ hl ≤ 1, l = 1, . . . , k . From the viewpoint of possibility maximization together with fractile optimization, we deﬁne the concept of F-P-Pareto optimal solutions. Deﬁnition 4.5 (F-P-Pareto Optimal Solution). A point x∗ ∈ X(θθ) is said to be an F-P-Pareto optimal solution to (4.32) if and only if there does not exist another x ∈ X(θθ) such that hl ≥ h∗l for all l ∈ {1, . . . , k} and hv = h∗v for at least one v ∈ {1, . . . , k}. Assume that d¯ l is an n dimensional Gaussian random variable with a mean vector ml and a variance-covariance matrix Vl . By using a transformation similar to the equivalent transformation shown in (4.41), one ﬁnds the relation that

128

4 Multiobjective Fuzzy Random Programming

βl }x ≤ μG˜ (hl ) ΠC˜¯ (ω)x (G˜ l ) ≥ hl ⇔ {dl (ω) − L (hl )β l

l

βl x dl (ω)x − ml x μG˜ l (hl ) − ml x + L (hl )β ' ' ≤ , (4.53) ⇔ xT Vl x xT Vl x

where L (hl ) and μG˜ (hl ) are pseudo inverse functions deﬁned by (4.42). Moreover, l observing that the left-hand side of (4.53) is the standard Gaussian random variable, one ﬁnds that P ω ΠC˜¯ (ω)x (G˜ l ) ≥ hl ≥ θl l # " d (ω)x − m x μ˜ (hl ) − ml x + L (hl )β βl x Gl l l ' ≤ ≥ θl ⇔ P ω ' xT Vl x xT Vl x " # βl x μG˜ (hl ) − ml x + L (hl )β l ' ⇔ Φ ≥ θl xT Vl x βl x μG˜ (hl ) − ml x + L (hl )β l ' ⇔ ≥ Φ−1 (θl ) xT Vl x ' βl }x + Φ−1 (θl ) xT Vl x ≤ μG˜ (hl ), ⇔ {ml − L (hl )β

(4.54)

l

where Φ is a probability distribution function of the standard Gaussian random variable, and Φ−1 is the inverse function of Φ. By substituting (4.54) into (4.52), (4.52) is transformed into ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize h1 .. .

maximize hk ' ⎪ ⎪ βl }x + Φ−1 (θl ) xT Vl x ≤ μG˜ (hl ), l = 1, . . . , k ⎪ subject to {ml − L (hl )β ⎪ ⎪ l ⎭ x ∈ X, (4.55) where Φ−1 (θl ) > 0 from the assumption of θ > 1/2. Assuming that L(t) = max{1 −t, 0} and that μG˜ l is a linear membership function deﬁned by (4.7), (4.55) can be transformed again into maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' ⎪ βl − ml )x − Φ−1 (θl ) xT Vl x + z0l (β ⎪ ⎪ ⎪ subject to ≥ h , l = 1, . . . , k l ⎪ 0 1 ⎪ β l x − zl + zl ⎪ ⎭ x ∈ X, or equivalently

(4.56)

4.1 Multiobjective fuzzy random linear programming

' β1 − m1 )x − Φ−1 (θ1 ) xT V1 x + z01 (β Π,F maximize Z1 (x) = β 1 x − z11 + z01 ··················

maximize

Π,F Zk (x) =

subject to x ∈ X.

' βk − mk )x − Φ−1 (θk ) xT Vk x + z0k (β β k x − z1k + z0k

129

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.57)

To derive a satisﬁcing solution to (4.57), the DM is asked to specify reference levels zˆl , l = 1, . . . , k, which are called the reference fractile levels, and an F-P-Pareto optimal solution is obtained by solving the minimax problem min max zˆl − ZlΠ,F (x) . (4.58) x∈X 1≤l≤k

It is signiﬁcant to emphasize here that the numerator and the denominator of Π,F zˆl − Zl (x) in (4.58) are concave and convex, respectively, and this fact means that (4.58) is solved by the extended Dinkelbach-type algorithm in a way similar to the solution method for (4.28). We now summarize an interactive algorithm for deriving the satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model Step 1: Calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl ∈ [1/2, 1), l = 1, . . . , k. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels zˆl , l = 1, . . . , k, solve the corresponding minimax problem (4.58) using the extended Dinkelbach-type algorithm. Step 6: The DM is supplied with the F-P-Pareto optimal solution. If the DM is Π,F satisﬁed with the current objective function values Zl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k, and return to step 5. It should be stressed for the DM that any improvement of one target value for the degree of possibility can be achieved only at the expense of at least one of the other target values for the ﬁxed permissible probability levels θl , l = 1, . . . , k.

130

4 Multiobjective Fuzzy Random Programming

4.1.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing algorithm for the possibility-based fractile model, consider the following multiobjective fuzzy random programming problem: ⎫ ¯˜ 1 x minimize z1 (x) = C ⎪ ⎪ ⎬ ˜¯ x minimize z2 (x) = C 2 (4.59) subject to ai x ≤ bi , i = 1, . . . , 4 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T , and parameters involved in the objective functions and the constraints are shown in Tables 4.4 and 4.5, respectively. ˜¯ . ¯˜ 1 and C Table 4.4 Value of each parameter of the fuzzy random variable coefﬁcients C 2 m1 m2 β1 β2 γ1 γ2

6 −7 0.6 0.4 0.6 0.4

7 −6 0.6 0.5 0.6 0.5

8 −6 0.5 0.5 0.5 0.5

6 3 4 2

3 5 2 3

Table 4.5 Value of each element of ai , i = 1, . . . , 4. a1 a2 a3 a4

2 6 5 2

After eliciting the membership functions to quantify the fuzzy goals for the objective functions from the DM, assume that the DM speciﬁes the permissible probability levels at θ1 = θ2 = 0.70. For the initial reference fractile levels (ˆz1 , zˆ2 ) = (1.0, 1.0), the corresponding minimax problem is solved through the extended Dinkelbach-type algorithm, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.6. Assume that the DM is not satisﬁed with the objective function values in the ﬁrst iteration, and the DM updates the reference fractile levels as (ˆz1 , zˆ2 ) = (1.0, 0.8) for improving the value of the objective function Z1Π,F at the expense of Z2Π,F . For the updated reference fractile levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated shown in Table 4.6. A similar procedure continues until the DM is satisﬁed with the obtained objective function values. If, in the third iteration, the DM is satisﬁed with the objective function values, it follows that the DM ﬁnds a satisﬁcing solution of self.

4.1 Multiobjective fuzzy random linear programming Table 4.6 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 Z1F (x) Z2F (x)

0.629 0.629

131

2nd 1.000 0.800

3rd 0.900 0.800

0.748 0.546

0.688 0.588

4.1.3 Level set-based models In this subsection, for dealing with multiobjective fuzzy random programming problems, we consider level set-based models in which, for the given degree α for the membership functions of fuzzy numbers, four types of stochastic optimization criteria are introduced: the expectation, variance, probability and fractile criteria. In connection with these criteria, four types of extended M-α-Pareto optimality concepts for fuzzy random multiobjective programming problems are deﬁned as a natural extension of the M-α-Pareto optimality concept discussed in Chapter 2. We give interactive methods for deriving a satisﬁcing solution for the DM from among the extended M-α-Pareto optimal solution. Throughout this subsection, we consider the following multiobjective fuzzy random linear programming problem: ˜¯ x ⎫ minimize z1 (x) = C 1 ⎪ ⎪ ⎬ ······ (4.60) ˜ ¯ kx ⎪ minimize zk (x) = C ⎪ ⎭ x ∈ X, ˜¯ = (C˜¯ , . . . , C˜¯ ), where x is an n dimensional decision variable column vector, and C l l1 ln l = 1, . . . , k are coefﬁcient vectors of fuzzy random variables. Although we dealt with fuzzy random variables such that only the parameters of center values d¯l j , l = 1, . . . , k, j = 1, . . . , n are random variables in the possibilitybased probability and fractile models given in subsection 4.1.2, we consider multiobjective fuzzy random programming problems involving more general fuzzy random variables such that not only center values but also spread parameters are random variables. To be more speciﬁc, for a given elementary event ω, a realized value C˜l j (ω) of each coefﬁcient C˜¯l j is characterized by a membership function ⎧ dl j (ω) − τ ⎪ ⎪ if τ ≤ dl j (ω) ⎪ ⎨L βl j (ω) μC˜l j (ω) (τ) = (4.61) ⎪ τ − dl j (ω) ⎪ ⎪ if τ > dl j (ω), ⎩R γl j (ω)

132

4 Multiobjective Fuzzy Random Programming

where L and R are nonincreasing continuous functions, and dl j (ω), βl j (ω) and γl j (ω) are realized values of continuous random variable d¯l j , β¯ l j and γ¯ l j for an elementary event ω, respectively. Fig. 4.10 illustrates an example of the membership function μC˜l j (ω) . Formally, the membership function of the fuzzy random variable C˜¯l j is represented by

μC˜¯ (τ) = lj

# ⎧ " ⎪ d¯l j − τ ⎪ ⎪ ⎪ ⎨L β¯

if τ ≤ d¯l j

lj

⎪ ⎪ τ − d¯l j ⎪ ⎪ ⎩R γ¯ l j

(4.62) if τ > d¯l j .

It is noted that not only the center values d¯l j but also spread parameters γ¯ l j and β¯ l j are random variables in (4.62), while only the center values d¯l j are random variables in (4.34).

μC~lj(ω)(τ) 1

L

0

βlj (ω)

R

γ lj (ω)

τ

dlj (ω)

Fig. 4.10 Example of the membership function μC˜l j (ω) .

˜¯ x can By using the extension principle (Zadeh, 1965), each objective function C l be represented by a single fuzzy random variable whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x ⎪ ⎨L βl (ω)x (4.63) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γ l (ω)x where dl (ω), β l (ω) and γ l (ω) are an n dimensional column vectors whose elements are realized values of the random variables d¯ j , β¯ j , γ¯ j , j = 1, . . . , n, respectively. Also ˜¯ x, its membership function is formally expressed as for the lth objective function C l

4.1 Multiobjective fuzzy random linear programming

⎧ dl (ω)x − υ ⎪ ⎪ ⎪ ⎨L βl (ω)x μC˜¯ (ω)x (υ) = l ⎪ υ − dl (ω)x ⎪ ⎪ ⎩R γ l (ω)x

133

if υ ≤ dl (ω)x (4.64) if υ > dl (ω)x.

Similarly to the previous section, assume that d¯ l , β¯ l and γ¯ l are random variable vectors expressed as ⎫ d¯ l = dl1 + t¯l dl2 ⎬ (4.65) β¯ l = β 1l + t¯l β 2l ⎭ γ¯ l = γ 1l + t¯l γ 2l , where t¯l , l = 1, . . . , k are positive random variables with mean ml and variance σ2l , and dl1 , dl2 , β 1l , β 2l , γ 1l , γ 2l are constant vectors. An α-level set of the realized value C˜l j (ω) of the fuzzy random variable C˜¯l j is expressed by (4.66) Cl jα (ω) {τ | μC˜l j (ω) (τ) ≥ α}, and from (4.61), Cl jα (ω) can be represented by the closed interval L Cl jα (ω),ClRjα (ω) = dl j (ω) − L (α)βl j (ω), dl j (ω) + R (α)γl j (ω) . Thus, by introducing a parameter λl j which is a real number satisfying 0 ≤ λl j ≤ 1, each of elements in Cl jα (ω) can be represented by λl jClLjα (ω) + (1 − λl j )ClRjα (ω). From this property of the α-level set of the fuzzy number C˜l j (ω), we deﬁne an αlevel set of the fuzzy random variable C˜¯l j as follows: C¯l jα λl jC¯lLjα + (1 − λl j )C¯lRjα | 0 ≤ λl j ≤ 1 , (4.67) where C¯lLjα and C¯lRjα are random variables expressed as C¯lLjα = d¯l j − L (α)β¯ l j , C¯lRjα = d¯l j + R (α)¯γl j .

(4.68) (4.69)

It should be noted here that C¯l jα can be viewed as a set of random variables. Now suppose that the DM intends to minimize each of the objective functions in (4.60) under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67). Then, (4.60) can be interpreted as the following multiobjective (nonfuzzy) stochastic programming problem depending on the the degree α:

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4 Multiobjective Fuzzy Random Programming

⎫ minimize C¯ 1 x ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . ¯ minimize Ck x ⎪ ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k,

(4.70)

˜¯ , and where C¯ lα = (C¯l1α , . . . , C¯lnα ) is an α-level set of the fuzzy random variable C l ¯ ¯ ¯ ¯ Cl ∈ Clα means that Cl j ∈ Cl jα for all j = 1, . . . , n. It should be noted here that a random variable C¯l j in the α-level set C¯l jα of the fuzzy random variable is characterized by the parameter λl j ∈ [0, 1]. Then, it follows that not only x but also λ l are decision variable vectors since C¯ l must be selected in C¯ lα so that each of the objective functions is minimized.

μC~lj(ω)(τ) 1

α

0

L

R

α-level set R L Cljα Cljα (ω) (ω)

τ

Fig. 4.11 An α-level set for a realized value of a fuzzy random variable coefﬁcient.

For such a multiobjective fuzzy random programming problem (4.70), considering the imprecise nature of human judgments, it is quite natural to assume that the DM has a fuzzy goal for each of the objective functions. In a minimization problem, the fuzzy goal stated by the DM may be to achieve “substantially less than or equal to some value.” Such a fuzzy goal can be quantiﬁed by eliciting a membership function μl for the fuzzy goal from the DM, which is depicted in Fig. 4.12. The parameters z0l and z1l denote the values of the objective function such that the membership function are 0 and 1, respectively, i.e., μl (z0l ) = 0 and μl (z1l ) = 1. The objective functions in (4.70) vary randomly due to the randomness of the coefﬁcients C¯ l , and to deal with this randomness, we provide four types of level set-based models for multiobjective fuzzy random programming problems by incorporating the criteria of stochastic programming.

4.1.3.1 Level set-based expectation model Assuming that the DM intends to minimize or maximize the expectation of each objective function represented as a random variable in (4.70), by replacing the ob-

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135

μl (y) 1

0

z 1l

z l0

y

Fig. 4.12 Membership function μl of the fuzzy goal for the lth objective function.

jective functions C¯ l x, l = 1, . . . , k in (4.70) with their expectations of the membership functions μl , l = 1, . . . , k, we consider the following level set-based expectation model for multiobjective fuzzy random programming problems: ⎫ maximize E μ1 (C¯ l x) ⎪ ⎪ ⎪ ⎪ · · · · · · ⎬ ¯ (4.71) maximize E μl (Ck x) ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k. From the viewpoint of maximization of the expectation for a degree α of possibility of fuzzy numbers given by the DM, we deﬁne the E-M-α-Pareto optimal solution. Deﬁnition 4.6 (E-M-α-Pareto optimal solution). A point x∗ ∈ X is said to be an EM-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ X and C¯ l ∈ C¯ lα , l = 1, . . . , k such that E[μl (C¯ l x)] ≥ E[μl (C¯ l∗ x∗ )] for all l ∈ {1, . . . , k}, and E[μv (C¯ v x)] = E[μv (C¯ v∗ x∗ )] at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα is said to be α-level optimal parameters. From the fact that ClLjα (ω) ≤ λl jClLjα (ω) + (1 − λl j )ClRjα (ω), ∀ω ∈ Ω, ∀λl j ∈ [0, 1],

(4.72)

L x)] ≤ E[μ (C ¯ l x)] for any C¯ l ∈ C¯ lα , and then, α-level optiit is evident that E[μl (C¯ lα l ∗ L ¯ ¯ mal parameters are given as Cl = Clα which implies λ∗l j = 1. Recalling (4.65) and (4.68), one ﬁnds that L x = (d¯ l − L (α)β¯ l )x C¯ lα ( ) β1l } + t¯l {dl2 − L (α)β β2l } x, = {dl1 − L (α)β

and then, (4.71) is equivalently transformed into

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4 Multiobjective Fuzzy Random Programming

⎫ E (x) ¯L maximize Z1α E[μ1 (C1α x)] ⎪ ⎪ ⎬ ········· E L maximize Zkα (x) E[μk (C¯ kα x)] ⎪ ⎪ ⎭ subject to x ∈ X,

(4.73)

where L E[μl (C¯ lα x)] =

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β2l } x fl (t)dt, {dl1 − L (α)β

(4.74)

and fl (t) is a probability density function of t¯l . To calculate a candidate for the satisﬁcing solution from among the set of E-Mα-Pareto optimal solutions, the DM is asked to specify a degree α of the α-level set and reference levels of achievement of the objective functions, called reference expectation levels. To be more explicit, for the degree α and the reference expectation levels zˆl , l = 1, . . . , k speciﬁed by the DM, the corresponding E-M-α-Pareto optimal solution, which is the nearest to a vector of the reference expectation levels zˆl , l = 1, . . . , k in a sense of minimax or better than it if the reference expectation levels are attainable, is obtained by solving the minimax problem E (x) minimize max zˆl − Zlα 1≤l≤k (4.75) subject to x ∈ X. The uniqueness of the optimal solution to (4.75) is required to guarantee E-M-αPareto optimality. Fortunately, by appending the augmentation term, this cumbrance can be overcome easily. Namely, we can obtain an E-M-α-Pareto optimal solution by solving the augmented minimax problem ⎫ k ⎪ ⎬ E E minimize max zˆl − Zlα (x) + ρ ∑ {ˆzl − Zlα (x)} (4.76) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X, where ρ is a sufﬁciently small positive number. Since the objective functions in (4.73) is expressed in the form of integration, it is generally difﬁcult to solve (4.76) in a strict sense. Realizing such difﬁculty, for obtaining an approximate solution, we employ genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001). Now we are ready to give an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-M-α Pareto optimal solution set. Interactive satisﬁcing method for level set-based expectation model Step 1: Calculate the individual minima and maxima of E[d¯ l ], l = 1, . . . , k by solving the linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1.

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137

Step 3: Ask the DM to specify the initial degree α of the α-level set of C¯lα . Step 4: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference expectation levels zˆl , l = 1, . . . , k and the degree α, solve the corresponding augmented minimax problem (4.75). Step 6: The DM is supplied with the E-M-α Pareto optimal solution x∗ . If the DM E (x∗ ), l = 1, . . . , k, then stop the is satisﬁed with the current objective functions Zlα algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k or the degree α, and return to step 5. It should be stressed to the DM that (1) any improvement of one expectation of the objective function can be achieved only at the expense of at least one of the other expectations for some ﬁxed degree α and (2) the greater value of the α yields the worse values of the expected objective functions for some ﬁxed reference expectation levels.

4.1.3.2 Level set-based variance model The level set-based expectation model, discussed in the previous subsection, would be appropriate if the DM would like to simply maximize or minimize the expected objective function values without concerning about their ﬂuctuation of the membership values of the fuzzy goals for a degree of possibility of fuzzy numbers given by the DM. Rather, when the DM prefers to decrease the ﬂuctuation of the membership values of the fuzzy goals, it is appropriate to employ a model minimizing the variance of the membership function values. Realizing such a situation, we consider the level set-based variance model for multiobjective fuzzy random programming problems formulated as ⎫ minimize Var μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · ·· · · · · · · ⎪ ⎪ ⎬ ¯ minimize Var μk (Ck x) (4.77) subject to E μl (C¯ l x) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where κl , l = 1, . . . , k are permissible expectation levels speciﬁed by the DM, satisfying 0 ≤ κl ≤ 1. In order to help the DM specify κl , l = 1, . . . , k, it is recommended to calculate E (x), l = 1, . . . , k, which are obtained by the individual minima and maxima of Zlα solving the nonlinear programming problems

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4 Multiobjective Fuzzy Random Programming

E (x) = E μ (C ¯ L x) ⎫ minimize Zlα l ⎪ lα ⎪ ⎪ ⎪ subject to x ∈ X ⎬ ⎪ E (x) = E μ (C ¯ L x) ⎪ ⎪ maximize Zlα l ⎪ lα ⎭ subject to x ∈ X

, l = 1, . . . , k,

(4.78)

where L E[μ1 (C¯ lα x)] =

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β2l } x fl (t)dt. {dl1 − L (α)β

For notational convenience, let XV (κ) be the feasible region of (4.77), namely, XV (κ) {x ∈ Rn | x ∈ X, E μl (C¯ l x) ≥ κl , l = 1, . . . , k}. From the viewpoint of minimization of the variance for a degree of possibility of fuzzy numbers given by the DM, the concept of V-M-α-Pareto optimal solutions is deﬁned. Deﬁnition 4.7 (V-M-α-Pareto optimal solution). A point x∗ ∈ XV (κ) is said to be a V-M-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ XV (κ) and C¯ l ∈ C¯ lα , l = 1, . . . , k such that Var[μl (C¯ l x)] ≤ Var[μl (C¯ l∗ x∗ )] for all l ∈ {1, . . . , k}, and Var[μv (C¯ l x)] = Var[μv (C¯ v∗ x∗ )] for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα , k = 1, . . . , k are said to be α-level optimal parameters. Concerning the α-level optimal parameters, as discussed in the previous subsection, if we use the level set-based expectation model, maximizing the expectaL . Unfortutions of the membership functions in (4.71) can be achieved for C¯ l∗ = C¯ lα L nately, however, realizing that the variance is not always minimized at C¯ l∗ = C¯ lα or ∗ R ¯ ¯ ¯ Cl = Clα , we must search the α-level optimal parameters in the α-level set Clα . As shown in (4.67), by using the parameter λl j , any random variable C¯l j in the α-level set C¯l jα for the fuzzy random variable C˜¯l j can be expressed by C¯l j (λl j ) = λl jC¯lLjα + (1 − λl j )C¯lRjα = λl j d¯l j − L (α)β¯ l j + (1 − λl j ) d¯l j + R (α)¯γl j = d¯l j − λl j L (α)β¯ l j + (1 − λl j )R (α)¯γl j = dl1j − λl j L (α)β1l j + (1 − λl j )R (α)γ1l j +t¯l dl2j − λl j L (α)β2l j + (1 − λl j )R (α)γ2l j 1 2 Cl j (λl j ) + t¯l Cl j (λl j ),

where Cl1j (λl j ) = dl1j − λl j L (α)β1l j + (1 − λl j )R (α)γ1l j , Cl2j (λl j ) = dl2j − λl j L (α)β2l j + (1 − λl j )R (α)γ2l j .

4.1 Multiobjective fuzzy random linear programming

139

Then, by using the probability density function f l of the random variable t¯l , the variance and the expectation are calculated as λ)x + t¯l Cl2 (λ λ)x Var μl (C¯ l x) = Var μl Cl1 (λ ∞ λ)x + tl Cl2 (λ λ)x − E μl (C¯ l x) }2 fl (t)dt, {μl Cl1 (λ = 0 λ)x + t¯l Cl2 (λ λ)x E μl (C¯ l x) = E μl Cl1 (λ ∞ λ)x + tCl2 (λ λ)x fl (t)dt, = μl Cl1 (λ 0

λ1 , . . . , λ k ), λ l = (λl1 , . . . , λln ), l = 1, . . . , k and where λ = (λ λ)x Clr (λ

n

∑ Clrj (λl j )x j , r = 1, 2.

j=1

Letting E Zlα (x, λ ) = V (x, λ ) = Zlα

∞ 0 ∞ 0

λ)x + tCl2 (λ λ)x fl (t)dt μl Cl1 (λ

(4.79)

E λ)x + tCl2 (λ λ)x − Zlα {μl Cl1 (λ (x, λ )}2 fl (t)dt,

(4.80)

we can rewrite (4.77) as V (x, λ ) minimize Z1α ······ V (x, λ ) minimize Zkα

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

E (x, λ ) ≥ κ , l = 1, . . . , k ⎪ subject to Zlα l ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎪ ⎭ x ∈ X.

(4.81)

V (x, λ ), Considering that the values of λ must be chosen so as to minimize Zlα l = 1, . . . , k in (4.81), one ﬁnds that not only x but also λ are decision variable vectors. To generate a candidate for the satisﬁcing solution from among the set of V-Mα-Pareto optimal solutions, the DM is asked to specify the reference levels zˆl , l = 1, . . . , k for the variance of the objective functions, which are called the reference variance levels, and the following augmented minimax problem is solved: ⎫ k ⎪ V V ⎪ minimize max Zlα (x, λ ) − zˆl + ρ ∑ {Zlα (x, λ ) − zˆl } ⎪ ⎪ ⎪ 1≤l≤k ⎬ l=1 (4.82) E subject to Zlα (x, λ ) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎭ x ∈ X,

140

4 Multiobjective Fuzzy Random Programming

where ρ is a sufﬁciently small positive number. Observing that this problem is nonlinear and non-convex, it is generally difﬁcult to calculate a global optimal solution, and then some heuristic approach is needed to obtain an approximate optimal solution. In genetic algorithms, the decision variables E (x, λ ) and x and λ are given in a string representing an individual, and then Zlα V Zlα (x, λ ) in (4.82) are calculated easily. From this property of genetic algorithms, it seems to be appropriate to employ the GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001) for solving (4.82). Now we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based variance model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. E (x), l = 1, . . . , k by Step 3: Calculate the individual minima and maxima of Zlα solving (4.78). Step 4: Ask the DM to specify κl , l = 1, . . . , k in consideration of the individual minima and maxima obtained in step 3. Step 5: Ask the DM to specify the initial degree α. Step 6: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 7: For the current reference variance levels and the degree α, solve the corresponding augmented minimax problem (4.82). Step 8: The DM is supplied with the V-M-α Pareto optimal solution x∗ . If the DM V (x∗ , λ ∗ ), l = 1, . . . , k, is satisﬁed with the current objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k or the degree α, and return to step 7. It should be emphasized to the DM that (1) any improvement of one variance of the membership function value can be achieved only at the expense of at least one of the other variances for some ﬁxed degree α and permissible expectation levels κl , l = 1, . . . , k, and (2) the greater value of the degree α gives the worse values of the variances for some ﬁxed reference variance levels and permissible expectation levels κl , l = 1, . . . , k.

4.1.3.3 Level set-based probability model In this subsection, assuming that the DM intends to maximize the probability that the membership function values of the fuzzy goals for the objective functions involving coefﬁcients represented by fuzzy random variables are better than or equal to some target values, under the condition that each of the coefﬁcients represented by fuzzy

4.1 Multiobjective fuzzy random linear programming

141

random variables are in the α-level set, we consider the level set-based probability model for the multiobjective fuzzy random programming problem formulated as ⎫ maximize P (ω | μ1 (C1 (ω)x) ≥ h1 ) ⎪ ⎪ ⎪ ⎪ ············ ⎬ maximize P (ω | μk (Ck (ω)x) ≥ hk ) (4.83) ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where hl , l = 1, . . . , k are target values speciﬁed by the DM, and C¯ lα is an α-level ˜¯ . set of the fuzzy random variable vector C l In contrast to the previous subsection, assuming that the spread parameters of fuzzy random variables are constant, we consider multiobjective fuzzy random pro˜¯ x is represented by a fuzzy gramming problems in which each objective function C l random variable whose realized value is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x μC˜ l (ω)x (υ) = (4.84) ⎪ υ − dl (ω)x ⎪ ⎪ ⎩R if υ > dl (ω)x. γl x ˜¯ x, its membership function is formally exAlso for the lth objective function C l pressed as ⎧ ¯ ⎪ ⎪L dl x − υ if υ ≤ d¯ l x ⎪ ⎨ βl x μC˜¯ x (υ) = (4.85) l ⎪ υ − d¯ l x ⎪ ⎪ ⎩R if υ > d¯ l x. γl x From the viewpoint of maximization of the probability for a degree of probability of fuzzy numbers given by the DM, we deﬁne the concept of P-M-α-Pareto optimal solutions. Deﬁnition 4.8 (P-M-α-Pareto optimal solution). A point x∗ ∈ X is said to be a PM-α-Pareto optimal solution to (4.83) if and only if there does not exist another x ∈ X and C¯ l ∈ C¯ lα , l = 1, . . . , k such that P(ω | μl (Cl (ω)x) ≥ hl ) ≥ P(ω | μl (Cl∗ (ω)x∗ ) ≥ hl ) for all l ∈ {1, . . . , k}, and P(ω | μv (C¯ v (ω)x) ≥ hv ) = P(ω | μv (Cv∗ (ω)x∗ ) ≥ hv ) for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα , l = 1, . . . , k are said to be α-level optimal parameters. Because, from the property (4.72), for all C¯ l ∈ C¯ lα , the relation L (ω)x) ≥ hl ⊇ {ω | μl (Cl (ω)x) ≥ hl } ω | μl (Clα holds, one ﬁnds that, for all C¯ l ∈ C¯ lα ,

(4.86)

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4 Multiobjective Fuzzy Random Programming

L P ω | μl (Clα (ω)x) ≥ hl ≥ P (ω | μl (Cl (ω)x) ≥ hl ) ,

(4.87)

L . Consequently, which implies that α-level optimal parameters are given as C¯ l∗ = C¯ lα (4.83) can be rewritten as ⎫ L maximize P ω | μ1 (C1α (ω)x) ≥ h1 ⎪ ⎪ ⎬ · · · · · · ·L· · · · · (4.88) maximize P ω | μk (Ckα (ω)x) ≥ hk ⎪ ⎪ ⎭ subject to x ∈ X.

Similarly to the previous models, we assume that the random parameter d¯ l of ˜¯ is expressed as d¯ = d1 + t¯ d2 , where t¯ is a random variable fuzzy random variable C l l l l l l with the mean ml and the variance σ2l . Then, the lth objective function in (4.88) is equivalently transformed as L β l )x ≤ μl (hl ) P ω | μl (Clα (ω)x) ≥ hl = P ω | (d¯ l − L (α)β β l )x + μl (hl ) (−dl1 + L (α)β ¯ = P ω tl ≤ dl2 x β l )x + μl (hl ) (−dl1 + L (α)β = Tl , (4.89) dl2 x where Tl is a probability distribution of t¯l , and L and μl are pseudo inverse functions deﬁned by (4.42). Consequently, (4.88) can be rewritten as ⎫ β1 )x + μ1 (h1 ) ⎪ (−d11 + L (α)β P ⎪ maximize Z1α (x) T1 ⎪ ⎪ d12 x ⎪ ⎪ ⎬ ·················· (4.90) 1 βk )x + μk (hk ) ⎪ (−dk + L (α)β P (x) ⎪ ⎪ maximize Zkα T k ⎪ ⎪ dk2 x ⎪ ⎭ subject to x ∈ X. In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, after specifying zˆl , l = 1, . . . , k which are called reference probability revels, we solve the minimax problem P minimize max {ˆzl − Zlα (x)} 1≤l≤k (4.91) subject to x ∈ X or equivalently

4.1 Multiobjective fuzzy random linear programming

143

⎫ minimize v ⎪ ⎪ 1 ⎪ β1 − d1 }x + μG˜ (h1 ) ⎪ {L (α)β ⎪ 1 ⎪ ⎪ ≥ T subject to (ˆ z − v) 1 ⎪ 1 2 ⎪ d1 x ⎪ ⎬ ··················

βk − dk1 }x + μG˜ (hk ) {L (α)β k dk2 x

x ∈ X,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ≥ Tk (ˆzk − v) ⎪ ⎪ ⎪ ⎪ ⎭

(4.92)

where Tl (s) is a pseudo inverse function deﬁned by Tl (s) = inf{r | Tl (r) ≥ s}, l = 1, . . . , k. After ﬁnding the minimum value v∗ of the objective function in (4.92) by using the combination of the phase one of the two-phase simplex method and the bisection method, in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem ⎫ β1 }x − μG˜ (h1 ) {d11 − L (α)β ⎪ ⎪ 1 ⎪ minimize ⎪ ⎪ ⎪ d12 x ⎪ ⎪ ⎪ ⎪ 1 ⎪ β2 − d2 }x + μG˜ (h2 ) {L (α)β ⎪ ∗ ⎪ 2 ⎪ subject to ≥ T (ˆ z − v ) ⎬ 2 2 2 d2 x (4.93) ⎪ ⎪ ·················· ⎪ ⎪ ⎪ ⎪ ⎪ βk − dk1 }x + μG˜ (hk ) {L (α)β ⎪ ∗ ⎪ k ⎪ (ˆ z − v ) ≥ T ⎪ k k 2 ⎪ ⎪ dk x ⎪ ⎭ x ∈ X, where the ﬁrst objective function z1 (x) in (4.60) is supposed to be the most important to the DM. Using the Charnes-Cooper variable transformation ς = 1/dl2 x, y = ςx, ς > 0,

(4.94)

the linear fractional programming problem (4.93) is equivalently transformed as

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4 Multiobjective Fuzzy Random Programming

⎫ β1 }y − ςμG˜ (h1 ) minimize {d11 − L (α)β ⎪ ⎪ 1 ⎪ ⎪ β2 }y − ςμG˜ (h2 ) ≤ 0 ⎪ subject to τ2 d22 y + {d21 − L (α)β ⎪ ⎪ 2 ⎪ ⎪ ⎪ ·················· ⎪ ⎪ 2 1 ⎬ β τk dk y + {dk − L (α)β k }y − ςμG˜ (hk ) ≤ 0 ⎪ k Ay − ςb ≤ 0 ⎪ ⎪ ⎪ d12 y = 1 ⎪ ⎪ ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎭ ς ≥ 0,

(4.95)

where τl Tl (ˆzl − v∗ ), l = 2, . . . , k, and δ is a sufﬁciently small positive number for satisfying the condition ς > 0. Since (4.95) is essentially the same as (4.49) in the possibility-based probability model, the P-M-α Pareto optimality test can be performed in a manner similar to the P P P-P-Pareto optimality test. Furthermore, the trade-off rates −∂Z1α /∂Zlα , l = 2, . . . , k are calculated in the same way as (4.49)–(4.51). Now we are ready to summarize an interactive algorithm for the level set-based probability model to derive a satisﬁcing solution for the DM. Interactive satisﬁcing method for the level set-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target values hl , l = 1, . . . , k and the initial degree α. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k and the degree α, solve the corresponding minimax problem (4.91). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of P (x∗ ) + v∗ and solve the revised (4.95), replace zˆl of inactive constraints with Zlα ∗ problem. Furthermore, if the obtained x is not unique, perform the P-M-α Pareto optimality test. Step 6: The DM is supplied with the P-M-α Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherobjective function values Zlα wise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k or the P , l = 1, . . . , k degree α by considering the current values of objective functions Zlα P P together with the trade-off rates −∂Z1α /∂Zlα , l = 2, . . . , k, and return to step 5. It should be stressed to the DM that (1) an increase of one probability with respect to the membership function of the fuzzy goal can be achieved only at the expense of at least one of the other probabilities for some ﬁxed degree α and target values hl , l = 1, . . . , k, and (2) the greater value of the degree α gives decreases of the

4.1 Multiobjective fuzzy random linear programming

145

probabilities for some ﬁxed reference probability levels and target values hl , l = 1, . . . , k.

4.1.3.4 Level set-based fractile model Assuming that the DM intends to maximize a target value for the degree of possibility for given permissible probability levels, by replacing minimization of the objective functions zl (x), l = 1, . . . , k in (4.70) with maximization of the target values hl , l = 1, . . . , k such that the probability with respect to the fuzzy goal attainment level is greater than or equal to a certain permissible probability level θl speciﬁed by the DM, we consider the level set-based fractile model for fuzzy random multiobjective programming problems formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ maximize hk (4.96) ¯ subject to P ω | μl (Cl (ω)x) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ 0 ≤ hl ≤ 1, l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k. For notational convenience, let XF (θθ, α) denote the feasible region of (4.96), namely XF (θθ, α) {x ∈ Rn | x ∈ X, P(ω | μl (Cl (ω)x) ≥ hl ) ≥ θl , C¯ l ∈ C¯ lα , l = 1, . . . , k}. From the viewpoint of maximization of the target values hl , l = 1, . . . , k for a degree α of possibility of fuzzy numbers given by the DM, we deﬁne the concept of F-M-α-Pareto optimal solution. Deﬁnition 4.9 (F-M-α-Pareto optimal solution). A point x∗ ∈ XF (θθ, α) is said to be an F-M-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ XF (θθ, α) and C¯ l ∈ C¯ lα such that hl ≥ h∗l , l = 1, . . . , k for all l ∈ {1, . . . , k}, and hv > h∗v for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα are said to be α-level optimal parameters. be the maximum value of hl such that P(ω | μl (Cl (ω)x) ≥ For a given θl , let hmax l

¯ ¯ hl ) ≥ θl for any Cl ∈ Clα . Then, recalling the property (4.87), there exists an hl such

L (ω)x) ≥ h ) ≥ θ , which implies that α-level optimal that hl ≥ hmax and P(ω | μl (Clα l l l ∗ L parameters are given as C¯ l = C¯ lα . From the above fact, by using the pseudo inverse function μl , the following relation holds: L P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα (ω)x ≤ μl (hl ) ≥ θl .

146

4 Multiobjective Fuzzy Random Programming

Assume that d¯ l is an n dimensional Gaussian random variable row vector with a mean vector ml and a positive-deﬁnite variance-covariance matrix Vl , and that β l and γ l are positive constants. Then, the following holds: L βl )x ≤ μl (hl )) (ω)x ≤ μl (hl ) = P (ω | (dl (ω) − L (α)β P ω | C¯ lα # " d (ω)x − m x L (α)β βl x − ml x + μl (hl ) l l ' ≤ =P ω ' xT Vl x xT Vl x " # βl − ml )x + μl (hl ) (L (α)β ' =Φ , (4.97) xT Vl x where Φ is the probability distribution of the standard normal distribution with mean 0 and variance 1. Furthermore, one ﬁnds that " Φ

βl − ml )x + μl (hl ) (L (α)β ' xT Vl x

# ≥ θl

' T β l )x + Φ−1 ⇔ (ml − L (α)β l (θl ) x Vl x ≤ μl (hl ),

where Φ−1 l is the inverse function of Φl . Assuming that μl is a linear membership function deﬁned by ⎧ ⎪ 0 if y > z0l ⎪ ⎪ ⎨ y − z0 l μG˜ l (y) = if z1l ≤ y ≤ z0l ⎪ z1l − z0l ⎪ ⎪ ⎩ 1 if y < z1l ,

(4.98)

(4.99)

from (4.98), (4.96) can be transformed into maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' 0 T ⎪ βl )x + Φ−1 (ml − L (α)β ⎪ l (θl ) x Vl x − zl ⎪ ⎪ subject to ≥ h , l = 1, . . . , k l ⎪ 0 1 ⎪ zl − zl ⎪ ⎭ x ∈ X, (4.100) or equivalently

4.1 Multiobjective fuzzy random linear programming

' −1 (α)β β (m − L )x + Φ (θ ) xT V1 x − z01 1 1 1 1 F (x) = maximize Z1α 0 1 z1 − z1 ··················

' −1 (α)β β − L )x + Φ (θ ) xT Vk x − z0k (m k k k k F (x) = maximize Zkα 0 1 zk − zk subject to x ∈ X.

147

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.101)

In order to derive a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set, after specifying zˆl , l = 1, . . . , k which are called reference fractile levels, we solve the minimax problem F minimize max zˆl − Zlα (x) 1≤l≤k (4.102) subject to x ∈ X. As we discussed for the fractile model in Chapter 3, from the convexity property F (x), the minimax problem (4.102) can be solved by using some convex of zˆl − Zlα programming technique such as the sequential quadratic programming method. Now we are ready to summarize an interactive algorithm for the level set-based fractile model to derive a satisﬁcing solution for the DM. Interactive satisﬁcing method for the level set-based fractile model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl , l = 1, . . . , k and the initial degree α. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels and the degree α, solve the corresponding minimax problem (4.102). Step 6: The DM is supplied with the F-M-α Pareto optimal solution x∗ . If the DM F (x∗ ), l = 1, . . . , k, then stop the is satisﬁed with the objective function values Zlα algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k or the degree α, and return to step 5. It should be emphasized to the DM that (1) an increase of one target value for the degree of possibility can be achieved only at the expense of at least one of the other target values for some ﬁxed degree α and permissible probability levels θl , l = 1, . . . , k, and (2) the greater value of the degree α gives decreases of the target values for some ﬁxed reference fractile levels and permissible probability levels θl , l = 1, . . . , k.

148

4 Multiobjective Fuzzy Random Programming

4.1.3.5 Numerical example To demonstrate the feasibility and efﬁciency of the proposed interactive satisﬁcing method for the level set-based fractile model, consider the following numerical example of multiobjective fuzzy random programming problems: ⎫ ¯˜ 1 x minimize C ⎪ ⎪ ⎬ ˜¯ x minimize C 2 (4.103) subject to ai x ≤ bi , i = 1, 2, 3 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T . The parameters involved in (4.103) are shown in Tables 4.7 and 4.8. The right-hand side constants of bi , i = 1, 2, 3 are assumed to be given as (b1 , b2 , b3 ) = (140, 135, 100). Table 4.7 Value of each parameter of the fuzzy random variable coefﬁcients. m1 m2 β1 β2 γ1 γ2

−5 4 1.0 1.5 1.0 1.5

−3 2 1.5 1.0 1.5 1.0

−6 7 1.0 1.5 1.0 1.5

6 3 6

3 2 4

Table 4.8 Value of each element of ai , i = 1, 2, 3. a1 a2 a3

4 5 7

Suppose that after calculating the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k, the DM determines the linear membership functions by setting z01 = 0, z11 = −150, z02 = 175 and z12 = 0, and that the DM speciﬁes the permissible probability levels at θ1 = θ2 = 0.7 and the degree of α-level set α = 0.7. For the initial reference fractile levels (ˆz1 , zˆ2 ) = (1.0, 1.0), the corresponding minimax problem is solved through the sequential quadratic programming method, and the DM is supplied with the corresponding objective function values of the ﬁrst iteration shown in Table 4.9. Assume that the DM is not satisﬁed with the current objective function values, and the DM updates the reference fractile levels as (ˆz1 , zˆ2 ) = (1.0, 0.8) for improvF at the expense of Z F . For the updated ing the value of the objective function Z1α 2α reference fractile levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.9. Assume that the DM is still not satisﬁed with the objective function values, the DM F at the updates the reference fractile levels to (ˆz1 , zˆ2 ) = (0.9, 0.8) so as to improve Z2α F expense of Z1α , and the objective function values of the third iteration are calculated as shown in Table 4.9.

4.2 Extensions to integer programming

149

After a similar procedure, if the DM is satisﬁed with the objective function values of the fourth iteration, it follows that the DM obtains a satisﬁcing solution. Table 4.9 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 α 0.700 F Z1α (x) F Z2α (x)

0.544 0.544

2nd 1.000 0.800 0.700

3rd 0.900 0.800 0.700

4th 0.900 0.800 0.600

0.628 0.428

0.586 0.486

0.600 0.500

4.2 Extensions to integer programming In this section, as natural extensions of multiobjective fuzzy random programming with continuous variables, multiobjective fuzzy random integer programming are discussed. For dealing with large-scale nonlinear programming problems to be solved in an interactive satisﬁcing method, we employ genetic algorithms to search approximate optimal solutions. Realizing that the real-world decision making problems are often formulated as mathematical programming problems with integer decision variables, we consider the multiobjective fuzzy random integer programming problem formulated as ⎫ ˜¯ x minimize z1 (x) = C ⎪ 1 ⎪ ⎪ ⎪ ······ ⎬ ˜ ¯ (4.104) minimize zk (x) = Ck x ⎪ ⎪ ⎪ subject to Ax ≤ b ⎪ ⎭ x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n, where x is an n dimensional integer decision variable column vector, A is an m × n coefﬁcient matrix, b is m dimensional column constant vector, and we assume ˜¯ = (C˜¯ , . . . , C˜¯ ), l = 1, . . . , k of each of the objective that the coefﬁcient vector C l l1 ln functions is an n dimensional row vector represented by fuzzy random variables. In the following, for notational convenience, let X int be the feasible region of (4.104), namely X int = {x | Ax ≤ b, x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n}.

150

4 Multiobjective Fuzzy Random Programming

4.2.1 Possibility-based expectation and variance models 4.2.1.1 Possibility-based expectation model As discussed in the previous section, if the DM intends to simply maximize the expected degree of possibility that each of the original objective functions involving fuzzy random variables attains the fuzzy goals, the possibility-based expectation model is recommended as a reasonable decision making model for solving the multiobjective fuzzy random programming problems. ˜¯ = (C˜¯ , . . . , C˜¯ ) Also, we assume that each element C˜¯l j of the coefﬁcient vector C l l1 ln is fuzzy random variable whose realized value is a fuzzy number C˜l jsl depending on a scenario sl ∈ {1, . . . , Sl } which occurs with a probability plsl , satisfying S ∑sll=1 plsl = 1. By substituting maximization of the expected degrees of possibility with respect to the fuzzy goal attainment for the minimization of the objective functions in (4.104), we formulate )⎫ ( maximize E ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ ⎪ 1 ⎪ ⎬ · · · · · · ( ) (4.105) maximize E ΠC˜¯ x (G˜ k ) ⎪ ⎪ ⎪ k ⎪ ⎭ subject to x ∈ X int , where ΠC˜¯ x (G˜k ) denotes a degree of possibility that the fuzzy random objective k ˜¯ x attains the fuzzy goal G˜ characterized by the linear membership funcfunction C l l tion μG˜ l deﬁned as ⎧ ⎪ 0 ⎪ ⎪ ⎨ y − z0 l μG˜ l (y) = ⎪ z1l − z0l ⎪ ⎪ ⎩ 1

if y > z0l if z1l ≤ y ≤ z0l

(4.106)

if y < z1l .

Similarly to the previous section, the values of z0l and z1l are assumed to be determined by ⎫ n ⎪ 0 zl max max ∑ dl jsl x j , l = 1, . . . , k, ⎪ ⎪ ⎬ 1≤sl ≤Sl x∈X int j=1 (4.107) n ⎪ z1l min min ∑ dl jsl x j , l = 1, . . . , k. ⎪ ⎪ ⎭ 1≤sl ≤Sl x∈X int j=1

As we formulated the expectation model for the multiobjective fuzzy random programming problem with continuous decision variables in 4.1.1.1, from (4.13), (4.105) is equivalently transformed as

4.2 Extensions to integer programming n

maximize

Π,E Z1 (x)

∑

151

β1 j −

j=1

S1

∑

s1 =1

p1s1 d1 js1

n

∑ β1 j x j − z11 + z01

j=1

n

maximize ZkΠ,E (x) subject to x ∈ X int .

∑

j=1

··················

x j + z01

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ βk j − ∑ pksk dk jsl x j + z0k ⎪ ⎪ ⎪ ⎪ ⎪ sk =1 ⎪ ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ βk j x j − zk + zk ⎪ ⎪ j=1 ⎪ ⎭ Sk

(4.108)

It should be noted here that the difference between (4.13) and (4.108) is only the condition of the decision variables, namely, the difference is whether continuous decision variables or integer ones in the problem formulation. In order to derive a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set, for the reference expectation levels zˆl , l = 1, . . . , k speciﬁed by the DM, the following augmented minimax problem is iteratively solved: ⎫ k ⎪ ⎬ minimize max zˆl − ZlΠ,E (x) + ρ ∑ {ˆzl − ZlΠ,E (x) (4.109) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. From integrality of the decision variables in (4.109), it is evident that the solution techniques developed for the possibility-based expectation model with continuous decision variables, discussed in the previous section, cannot be directly applied. However, realizing that (4.109) involves a nonlinear objective function and linear constraints, as we used to similar problems in Chapter 3, we can employ the modiﬁed version of the genetic algorithm with double strings based on using linear programming relaxation and reference solution updating (GADSLPRRSU) (Sakawa, 2001) for solving it. Following the preceding discussions, we can now summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based expectation model with integer decision variables Step 1: Determine the linear membership functions μG˜ l , l = 1. . . . , k deﬁned as (4.106) by calculating z0l and z1l through GADSLPRRSU. Step 2: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k.

152

4 Multiobjective Fuzzy Random Programming

Step 3: For the current reference expectation levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.109) through the revised GADSLPRRSU. Step 4: The DM is supplied with the E-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,E (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k in consideration of the current objective function values, and return to step 3.

4.2.1.2 Possibility-based variance model In order to help a risk-averse DM with making a decision for multiobjective fuzzy random integer programming problems, from the viewpoint of variance minimization together with possibility maximization, we formulate the following integer version of (4.23): ⎫ ) ( ⎪ minimize Var ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ 1 ⎪ ⎪ ⎪ · · · · · · · · · ⎪ ⎪ ( ) ⎬ minimize Var ΠC˜¯ x (G˜ k ) (4.110) k ⎪ ) ( ⎪ ⎪ subject to E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ l ⎪ ⎪ ⎭ int x∈X , where Var denotes the variance operator, and ξl , l = 1, . . . , k are permissible expectation levels for the expected degrees of possibility speciﬁed by the DM. From the same idea as in 4.1.1.2, we can transform (4.110) into ⎫ 1 T ⎪ minimize Z1Π,V (x) " #2 x V1 x ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ·················· ⎬ (4.111) 1 ⎪ T minimize ZkΠ,V (x) " ⎪ #2 x Vk x ⎪ ⎪ ⎪ n ⎪ ⎪ 1 0 ⎪ β x − z + z ⎪ ∑ kj j k k ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ Π,E ⎪ ⎪ subject to Zl (x) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ int x∈X , where

4.2 Extensions to integer programming n

ZlΠ,E (x) =

∑

153

βl j −

j=1 n

Sl

∑ plsl dl jsl

sl =1

∑ βl j x j − z1l + z0l

x j + z0l .

j=1

For the DM’s reference variance levels zˆl , i = 1, . . . , k, the corresponding V-PPareto optimal solution is obtained by solving the augmented minimax problem ⎫ k ⎪ Π,V Π,V ⎪ ⎪ minimize max Zl (x) − zˆl + ρ ∑ Zl (x) − zˆl ⎪ ⎬ 1≤l≤k l=1 (4.112) Π,E ⎪ subject to Zl (x) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X int , where ρ is a sufﬁciently small positive number. Since (4.112) is a nonlinear integer programming problem with a nonlinear objective function and linear constraints, similarly to the possibility-based expectation model, we can employ the revised GADSLPRRSU to solve it. Now we can present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based variance model with integer decision variables Step 1: Determine the linear membership functions μG˜ , l = 1. . . . , k deﬁned as (4.106) by calculating z0l and z1l through GADSLPRRSU. Step 2: Calculate the individual minima and maxima of ZlΠ,E (x), l = 1, . . . , k. Step 3: Ask the DM to specify the permissible expectation levels ξl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 5: For the current reference variance levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.112) through the revised GADSLPRRSU. Step 6: The DM is supplied with the corresponding V-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values ZlΠ,V (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the expectation model with integer decision variables, as a numerical example of (4.104), consider the following multiobjective fuzzy random programming problem:

154

4 Multiobjective Fuzzy Random Programming

minimize minimize minimize subject to

⎫ ˜¯ x ⎪ C 1 ⎪ ⎪ ⎪ ˜C ⎪ ¯ 2x ⎬ ˜C ¯ 3x ⎪ ⎪ ⎪ ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ x j ∈ {0, 1, . . . , 20}, j = 1, . . . , 100. ⎭

(4.113)

For determining the membership functions μG˜ l , l = 1, 2, 3, through the DADSLPRRSU, the parameter values of z0l and z1l , l = 1, . . . , 3 are calculated as z01 = 12.90, z11 = 6.00, z02 = 13.80, z12 = 4.90, z03 = 14.00 and z13 = 6.43. After setting the initial reference expectation levels at (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 1.0, 1.0), the corresponding minimax problem is solved through GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.10. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference expectation levels as (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 0.9, 1.0) for improving the value of the objective function Z1Π,E and Z3Π,E at the expense of Z2Π,E . For the updated reference expectation levels, the augmented minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.10. Again, assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference expectation levels to (ˆz1 , zˆ2 ) = (1.0, 0.9, 0.9) so as to improve Z1Π,E and Z2Π,E at the expense of Z3Π,E , and the objective function values are calculated. In the third iteration, assume that the DM is satisﬁed with the objective function values. Then, it follows that a satisﬁcing solution for the DM is derived. Table 4.10 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 zˆ3 1.000 Z1Π,E (x) Z2Π,E (x) Z3Π,E (x)

0.541 0.531 0.534

2nd 1.000 0.900 1.000

3rd 1.000 0.900 0.900

0.574 0.475 0.568

0.614 0.516 0.516

4.2.2 Possibility-based probability and fractile models In a way similar to the possibility-based probability and fractile models with continuous decision variables discussed in 4.1.2, we assume that a fuzzy random variable ˜¯ x is assumed to be a continuC˜¯l j which is a coefﬁcient of the objective function C l ous fuzzy random variable whose realized value C˜l j (ω) for an elementary event ω is characterized by

4.2 Extensions to integer programming

⎧ dl j (ω) − τ ⎪ ⎪ ⎪ ⎨L βl j μC˜l j (ω) (τ) = ⎪ τ − dl j (ω) ⎪ ⎪ ⎩R γl j

155

if τ ≤ dl j (ω) (4.114) if τ > dl j (ω).

4.2.2.1 Possibility-based probability model ˜¯ x, l = 1, . . . , k in Replacing minimization of the objective functions zl (x) = C l (4.104) with maximization of the probability that the degree of possibility is greater than or equal to a certain target value hl , we consider the following possibility-based probability model for multiobjective fuzzy random integer programming problems: ⎫ maximize Z1Π,P (x) P ω | ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎪ ⎬ ··· ······ (4.115) maximize ZkΠ,P (x) P ω | ΠC˜ k (ω)x (G˜ k ) ≥ hk ⎪ ⎪ ⎪ ⎪ ⎭ subject to x ∈ X int , ˜¯ x where ΠC˜¯ x (G˜ l ) denotes the degree of possibility that the objective function C l l involving coefﬁcients represented by fuzzy random variables attains the fuzzy goal G˜ l characterized by the membership function μG˜ l . As a possible way to help the DM specify the membership function μG˜ l , it is recommended to calculate the individual maxima and minima of the expectation E[d¯ l ]x, where d¯ l is a vector of random variables representing the center values of the fuzzy ˜¯ . These values are obtained by solving the linear programming random variables C l problems ⎫ minimize E[d¯ l ]x ⎪ ⎪ ⎪ subject to x ∈ X int ⎪ ⎬ , l = 1, . . . , k. (4.116) ⎪ ⎪ maximize E[d¯ l ]x ⎪ ⎪ ⎭ subject to x ∈ X int Assume that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where the mean and the variance of the random variable t¯l are denoted by ml and σ2l , respectively, and dl1j and dl2j are constants. By using the equivalent transformation (4.43)

of the probability Z1Π,P (x), (4.115) is rewritten as

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4 Multiobjective Fuzzy Random Programming

"

#⎫ ⎪ ⎪ 1 ⎪ maximize Z1Π,P (x) = T1 ⎪ 2 ⎪ ⎪ d1 x ⎪ ⎪ ⎪ ⎬ ·················· " # βk − dk1 }x + μG˜ (hk ) ⎪ ⎪ {L (hk )β ⎪ Π,P k ⎪ ⎪ maximize Zk (x) = Tk ⎪ 2 ⎪ dk x ⎪ ⎪ ⎭ subject to x ∈ X int . β1 − d11 }x + μG˜ (h1 ) {L (h1 )β

(4.117)

In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, we give an interactive satisﬁcing method which involves the process of iteratively solving the augmented minimax problem ⎫ ⎪ k ⎪ Π,P Π,P ⎬ minimize max zˆl − Zl (x) + ρ∑ zˆl − Zl (x) 1≤l≤k (4.118) l=l ⎪ ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number, and zˆl , l = 1, . . . , k are reference probability levels given by the DM. Following the preceding discussion, we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.116) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to determine the target values hl , l = 1, . . . , k. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.118) through the revised GADSLPRRSU. Step 6: The DM is supplied with the P-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.2.2 Possibility-based fractile model Replacing minimization of the objective functions zl (x), l = 1, . . . , k in (4.104) with maximization of the target values hl , l = 1, . . . , k such that the probability with respect to the degree of possibility is greater than or equal to some permissible proba-

4.2 Extensions to integer programming

157

bility levels, we consider the possibility-based fractile model for the multiobjective fuzzy random integer programming problem, and it is formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎬ (4.119) maximize hk ⎪ ⎪ ⎪ subject to P ω | ΠC˜l (ω)x (G˜ l ) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X int , where θl , l = 1, . . . , k are permissible probability levels speciﬁed by the DM, satisfying θ ∈ [1/2, 1). Similarly to the possibility-based fractile model with continuous decision variable, we assume that a vector d¯ l of the center values represented by random variables ˜¯ in (4.114) is assumed to be an n dimensional for the fuzzy random variables C l Gaussian random variable row vector with the mean vector ml and the positivedeﬁnite variance-covariance matrix Vl . As we discussed in 4.1.2.2, form (4.54), (4.119) is equivalently transformed into ' ⎫ β1 − m1 )x − Φ−1 (θ1 ) xT V1 x + z01 ⎪ (β Π,F ⎪ maximize Z1 (x) ⎪ ⎪ ⎪ β 1 x − z11 + z01 ⎪ ⎪ ⎪ ⎬ ·················· (4.120) ' ⎪ βk − mk )x − Φ−1 (θk ) xT Vk x + z0k ⎪ (β ⎪ Π,F ⎪ ⎪ maximize Zk (x) ⎪ ⎪ β k x − z1k + z0k ⎪ ⎭ int subject to x ∈ X . In order to calculate a candidate for the satisﬁcing solution to (4.120), the DM is asked to specify the reference fractile levels zˆl , i = 1, . . . , k, and the following augmented minimax problem is solved: ⎫ ⎪ k ⎬ Π,F Π,F minimize max zˆl − Zl (x) + ρ ∑ zˆl − Zi (x) (4.121) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. It is noted that an optimal solution to (4.121) is an F-P-Pareto optimal solution with respect to (4.120). We now summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.116) through GADSLPRRSU.

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4 Multiobjective Fuzzy Random Programming

Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl l = 1, . . . , k. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.121) using the revised GADSLPRRSU. Step 6: The DM is supplied with the F-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,F (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the possibility-based probability model with integer decision variables, consider the following numerical example of multiobjective fuzzy random programming problems: ⎫ ˜¯ x ⎪ minimize C ⎪ 1 ⎪ ⎪ ˜ ⎪ ¯ 2x ⎬ minimize C ˜¯ x (4.122) minimize C 3 ⎪ ⎪ ⎪ subject to ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ ⎭ x j ∈ {0, 1, . . . , 30}, j = 1, . . . , 100. Assume that the DM determines the linear membership functions μG˜ l , l = 1, 2, 3, by specifying the parameters z0l and z1l , l = 1, . . . , 3 as z01 = 379740, z11 = −382920, z02 = 359760, z12 = −402724, z03 = 414450 and z13 = −388170, and that the DM speciﬁes the target levels at (h1 , h2 , h3 ) = (0.65, 0.65, 0.65). After setting the initial reference probability levels at (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 1.00), the augmented minimax problem is solved through the revised GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration as shown in Table 4.11. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference probability levels as (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 0.90) for improving the value of the objective function Z1Π,P and Z2Π,P at the expense of Z3Π,P . For the updated reference probability levels, the revised augmented minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.11. If, in the third interaction, the DM is satisﬁed with the objective function values, it follows that the satisﬁcing solution for the DM is derived.

4.2 Extensions to integer programming Table 4.11 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 zˆ3 1.000 Z1Π,P (x) Z2Π,P (x) Z3Π,P (x)

0.741 0.740 0.742

159

2nd 1.000 1.000 0.900

3rd 1.000 0.950 0.900

0.766 0.763 0.664

0.777 0.730 0.682

4.2.3 Level set-based models In this subsection, we consider level set-based models for dealing with multiobjective fuzzy random integer programming problems. In particular, for the given degree α for the membership functions of fuzzy numbers, four types of stochastic optimization criteria are introduced. For deriving a satisﬁcing solution among from the extended M-α-Pareto optimal solution set, interactive satisﬁcing methods for the multiobjective fuzzy random integer programming problems are presented. Throughout this subsection, we consider the following multiobjective fuzzy random integer programming problem: ˜¯ x ⎫ minimize z1 (x) = C 1 ⎪ ⎪ ⎬ ······ (4.123) ˜¯ x ⎪ minimize zk (x) = C k ⎪ ⎭ x ∈ X int , ˜¯ = (C˜¯ , . . . , C˜¯ ), where x is an n dimensional decision variable column vector, and C l l1 ln l = 1, . . . , k are coefﬁcient vector represented by fuzzy random variables. As in the level set-based models for multiobjective fuzzy random programming problems with continuous decision variables discussed in 4.1.3, we assume that the membership function of fuzzy random variable C˜¯l j is formally represented by # ⎧ " ⎪ d¯l j − τ ⎪ ⎪ if τ ≤ d¯l j ⎪L ⎨ β¯ l j μC˜¯ (τ) = (4.124) lj ⎪ ¯l j ⎪ τ − d ⎪ ⎪ if τ > d¯l j . ⎩R γ¯ l j Furthermore, we also assume that the parameters d¯ l = (d¯l1 , . . . , d¯ln ), β¯ l = (β¯ l1 , . . ., ˜¯ are random variβ¯ ln ) and γ¯ l = (¯γl1 , . . . , γ¯ ln ) of the fuzzy random variable vector C l able vectors expressed as ⎫ d¯ l = dl1 + t¯l dl2 ⎬ (4.125) β¯ l = β 1l + t¯l β 2l ⎭ γ¯ l = γ 1l + t¯l γ 2l ,

160

4 Multiobjective Fuzzy Random Programming

where t¯l , l = 1, . . . , k are positive random variables with the mean ml and the variance σl , and dl1 , dl2 , β 1l , β 2l , γ 1l , γ 2l are constant vectors. According to the idea of the level set-based models, suppose that the DM intends to minimize each of the objective functions in (4.123) under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67). Then, (4.123) can be interpreted as the following multiobjective stochastic programming problem without fuzzy random variables depending on the the degree α: ⎫ minimize C¯ 1 x ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (4.126) ¯ minimize Ck x ⎪ ⎪ int ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, ˜¯ , and where C¯ lα = (C¯l1α , . . . , C¯lnα ) is an α-level set of the fuzzy random variable C l ¯ ¯ ¯ ¯ Cl ∈ Clα implies that Cl j ∈ Cl jα for all j = 1, . . . , n. It should be noted that not only x but also λ l characterizing the α-level sets C¯ lα are decision variable vectors since C¯ l must be selected in C¯ lα so that each of the objective functions is minimized. The objective functions in (4.126) vary randomly due to the randomness of the coefﬁcients C¯ l , and to deal with this randomness, we provide four types of level set-based models based on the criteria of stochastic programming for multiobjective fuzzy random integer programming problems.

4.2.3.1 Level set-based expectation model In the level set-based expectation model, it is assumed that the DM intends to maximize the expectations of the membership functions of the fuzzy goals for the objective functions in (4.126) for the speciﬁed degree α for the membership functions of fuzzy numbers with respect to the coefﬁcients, and then we consider the following multiobjective fuzzy random integer programming problems: ⎫ maximize E μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · · · · · · ⎬ maximize E μk (C¯ k x) (4.127) ⎪ ⎪ subject to x ∈ X int ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where μl is a membership function of fuzzy goal G˜ l for the lth objective function. As a possible way to help the DM determine the membership function μl of the fuzzy goal for the lth objective function in (4.127), it is recommended to calculate the individual maxima and minima of E[d¯ l ]x, l = 1, . . . , k obtained by solving the linear programming problems

4.2 Extensions to integer programming

minimize E[d¯ l ]x subject to x ∈ X int maximize E[d¯ l ]x subject to x ∈ X int

161

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

, l = 1, . . . , k,

(4.128)

˜¯ , which are deﬁned where d¯ l is a vector center values of fuzzy random variables C l by (4.124) and (4.125). As we discussed in the level set-based expectation model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.1, L . Hence, (4.127) is rewritten as the α-level optimal parameters are given as C¯ l∗ = C¯ lα ⎫ E (x) maximize Z1α ⎪ ⎪ ⎬ ··· (4.129) E (x) maximize Zkα ⎪ ⎪ ⎭ subject to x ∈ X int , where E (x) = Zlα

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β 2l } x f l (t)dt. {dl1 − L (α)β

(4.130)

To calculate a candidate for the satisﬁcing solution from among the set of E-Mα-Pareto optimal solutions, for the speciﬁed degree α and reference values zˆl , l = 1, . . . , k, the corresponding E-M-α-Pareto optimal solution is obtained by solving the augmented minimax problem ⎫ k ⎪ ⎬ E E minimize max zˆl − Zlα (x) + ρ ∑ zˆl − Zlα (x) (4.131) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. Realizing that it is generally difﬁcult to calculate an E-M-α-Pareto optimal solution of (4.131) exactly due to the integrality of decision variables together with the nonlinearity and nonconvexity properties of the objective functions expressed in the form of integration (4.130), we employ genetic algorithms to obtain a desirable approximate optimal solution. Now we are ready to present an interactive algorithm for deriving the satisﬁcing solution for the DM from among the E-M-α Pareto optimal solution set. Interactive satisﬁcing method for level set-based expectation model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1.

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4 Multiobjective Fuzzy Random Programming

Step 3: Ask the DM to specify the initial degree α for the α-level set. Step 4: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the speciﬁed reference expectation levels and the degree α, solve the corresponding augmented minimax problem (4.131) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate E-M-α Pareto optimal solution E (x∗ ), l = 1, . . . , k, then x∗ . If the DM is satisﬁed with the objective functions Zlα stop the algorithm. Otherwise, ask the DM to update the reference expectation levels or the degree α, and return to step 5.

4.2.3.2 Level set-based variance model Due to the occurrence of an undesirable scenario, there is a possibility that the goal attainment level is quite low in the level set-based expectation model. For the DM who expects to avoid such a risk, it is appropriate to employ the level set-based variance model for multiobjective fuzzy random integer programming problems. By substituting minimization of the variances of the membership values of the fuzzy goals with respect to the objective functions for minimization of objective functions in (4.126) and adding the conditions for the expectations, we formulate the multiobjective variance minimization problem ⎫ minimize Var μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · ·· · · · ⎪ ⎪ ⎬ ¯ minimize Var μk (Ck x) (4.132) subject to E μl (C¯ l x) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where κl , l = 1, . . . , k are permissible expectation levels speciﬁed by the DM. As a possible way to help the DM to specify the permissible expectation levels, E (x) deﬁned by it is convenient to calculate the individual maxima and minima of Zlα (4.130). To obtain these values, we solve the integer programming problems E (x) ⎫ minimize Zlα ⎪ ⎪ ⎪ subject to x ∈ X int ⎪ ⎬ , l = 1, . . . , k (4.133) ⎪ E ⎪ maximize Zlα (x) ⎪ ⎪ ⎭ subject to x ∈ X int by directly using GADSLPRRSU. Recalling the discussion for the level set-based variance model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.2, we can rewrite (4.132) as follows:

4.2 Extensions to integer programming V (x, λ ) minimize Z1α ······ V (x, λ ) minimize Zkα

163

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

E (x, λ ) ≥ κ , l = 1, . . . , k ⎪ subject to Zlα l ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎪ ⎭ x ∈ X int ,

(4.134)

where E Zlα (x, λ ) = V (x, λ) = Zlα

∞ 0 ∞ 0

λ)x + tCl2 (λ λ)x fl (t)dt, μl Cl1 (λ E λ)x + tl Cl2 (λ λ)x − Zlα {μl Cl1 (λ (x, λ)}2 fl (t)dt.

To calculate a candidate for the satisﬁcing solution from among the set of V-Mα-Pareto optimal solutions, we solve the augmented minimax problem &⎫ % k ⎪ V V ⎪ minimize max Zlα (x, λ ) − zˆl + ρ ∑ Zlα (x, λ ) − zˆl ⎪ ⎪ ⎪ 1≤l≤k ⎬ l=1 (4.135) E subject to Zlα (x, λ ) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎭ x ∈ X int , where ρ is a sufﬁciently small positive number. Noting that (4.135) is non-convex nonlinear integer programming problem, it would be appropriate to employ some heuristic approach in order to obtain an approximate optimal solution, and then we employ the revised GADSLPRRSU to solve (4.135). Now we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based variance model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. E (x), l = 1, . . . , k by Step 3: Calculate the individual minima and maxima of Zlα solving (4.133) through GADSLPRRSU. Step 4: Ask the DM to specify the permissible expectation levels κl , l = 1, . . . , k by considering the individual minima and maxima obtained in step 3. Step 5: Ask the DM to specify the initial degree α for the α-level set. Step 6: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 7: For the speciﬁed reference variance levels and the degree α, solve the augmented minimax problem (4.135) through the revised GADSLPRRSU.

164

4 Multiobjective Fuzzy Random Programming

Step 8: The DM is supplied with the approximate V-M-α Pareto optimal solu∗ V tion x∗ . If the DM is satisﬁed with the objective function values Zlα (x∗ , λ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k or the degree α, and return to step 7.

4.2.3.3 Level set-based probability model We deal with the level set-based probability model in which the probabilities that the membership function values of the fuzzy goals for the objective functions involving coefﬁcients represented by fuzzy random variables are greater than or equal to certain target values are minimized, and formulate the following multiobjective fuzzy random integer programming problems: ⎫ maximize P (ω | μ1 (C1 (ω)x) ≥ h1 ) ⎪ ⎪ ⎪ ⎪ ········· ⎬ maximize P (ω | μk (Ck (ω)x) ≥ hk ) (4.136) ⎪ ⎪ int ⎪ subject to x ∈ X ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where hl , l = 1, . . . , k are target values for the membership function values of the fuzzy goals. In the level set-based probability model, as in 4.1.3.3, we assume that the membership function of the fuzzy random variable C˜¯l j is represented by ⎧ ¯ dl j − τ ⎪ ⎪ ⎪ ⎨L βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ⎩R γl j

if τ ≤ d¯l j (4.137) if τ > d¯l j .

It is noted here that the spread parameters βl j and γl j of the fuzzy random variables are constant. From the discussion for the level set-based probability model for multiobjective fuzzy random linear programming problems with continuous decision variables in L . 4.1.3.3, one ﬁnds that the α-level optimal parameters C¯ l∗ is determined as C¯ l∗ = C¯ lα Hence, (4.136) can be rewritten as ⎫ L maximize P ω | μ1 (C1α (ω)x) ≥ h1 ⎪ ⎪ ⎬ · · · · · · ·L· · · · · · · · (4.138) maximize P ω | μk (Ckα (ω)x) ≥ hk ⎪ ⎪ ⎭ subject to x ∈ X int . Furthermore, the objective functions in (4.138) are transformed into

4.2 Extensions to integer programming

L P ω | μl (Clα (ω)x) ≥ hl = Tl

165

β l )x + μl (hl ) (−dl1 + L (α)β , dl2 x

where Tl is a probability distribution function of the random variable t¯l . Using the relation (4.139), we can rewrite (4.138) as follows: ⎫ β1 )x + μ∗1 (h1 ) ⎪ (−d11 + L (α)β P (x) ⎪ T maximize Z1α ⎪ 1 ⎪ d12 x ⎪ ⎪ ⎪ ⎪ ⎬ ·················· ⎪ βk )x + μk (hk ) ⎪ (−dk1 + L (α)β ⎪ P ⎪ ⎪ maximize Zkα (x) Tk ⎪ 2 ⎪ dk x ⎪ ⎭ int subject to x ∈ X .

(4.139)

(4.140)

In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, for the speciﬁed reference probability levels zˆl , l = 1, . . . , k, the following augmented minimax problem is formulated: ⎫ ⎪ k ⎬ P P minimize max zˆl − Zlα (x) + ρ ∑ (ˆzl − Zlα (x)) (4.141) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X int , where ρ is a sufﬁciently small positive number. Now we are ready to present an interactive algorithm for deriving the satisﬁcing solution for the DM from among the P-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target levels hl , l = 1, . . . , k and the initial degree α of the α-level set. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current probability variance levels and the degree α, solve the corresponding augmented minimax problem (4.141) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate P-M-α Pareto optimal solution P (x∗ ), l = 1, . . . , k, x∗ . If the DM is satisﬁed with the objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k or the degree α, and return to step 5.

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4 Multiobjective Fuzzy Random Programming

4.2.3.4 Level set-based fractile model We consider the level set-based fractile model which is the complementary model to the probability model. In this model, the target values hl , l = 1, . . . , k are maximized under the condition the probabilities with respect to the fuzzy goal attainment levels are greater than or equal to the speciﬁed permissible probability levels θl , l = 1, . . . , k, and then the level set-based fractile model for the multiobjective fuzzy random integer programming problems is formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎬ maximize hk (4.142) subject to P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where the membership function of the fuzzy random variable C˜¯l j is represented by (4.137). Recalling the discussion for the level set-based fractile model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.4, (4.142) is equivalently transformed into ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (4.143) maximize hk ⎪ ⎪ L ⎪ subject to P ω | μl (Clα (ω)x) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎭ x ∈ X int . Finally, we obtain the following problem: maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' 0 T ⎪ βl )x + Φ−1 (ml − L (α)β ⎪ l (θl ) x Vl x − zl ⎪ subject to ≥ hl , l = 1, . . . , k ⎪ ⎪ 0 1 ⎪ zl − zl ⎪ ⎭ int x∈X , (4.144) which is equivalently expressed as

4.2 Extensions to integer programming

' −1 (α)β β − L )x + Φ (θ ) xT V1 x − z01 (m 1 1 1 1 F maximize Zlα (x) 0 1 z1 − z1 ··················

' 0 T βk )x + Φ−1 (mk − L (α)β k (θk ) x Vk x − zk F maximize Zkα (x) z1k − z0k int subject to x ∈ X .

167

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.145)

In order to derive a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set, for the speciﬁed reference fractile levelsˆzl , l = 1, . . . , k, we solve the augmented minimax problem ⎫ ⎪ k ⎬ F F (x) + ρ ∑ (ˆzl − Zlα (x)) minimize max zˆl − Zlα (4.146) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X. Now we are ready to summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the F-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl , l = 1, . . . , k, and the initial degree α of the α-level set. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels and the degree α, solve the corresponding augmented minimax problem (4.146) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate F-M-α Pareto optimal solution F (x∗ ), l = 1, . . . , k, x∗ . If the DM is satisﬁed with the objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k or the degree α, and return to step 5.

4.2.3.5 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the level set-based probability model for multiobjective fuzzy random integer programming problems, consider the following numerical example:

168

4 Multiobjective Fuzzy Random Programming

minimize minimize minimize subject to

⎫ ˜¯ x ⎪ C 1 ⎪ ⎪ ⎪ ˜C ⎪ ¯ 2x ⎬ ˜C ¯ 3x ⎪ ⎪ ⎪ ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ x j ∈ {0, 1, . . . , 30}, j = 1, . . . , 100. ⎭

(4.147)

After calculating the individual minima and maxima of E[d¯ l ]x, l = 1, 2, 3 , assume that the DM determines the linear membership functions by setting z01 = 379740, z11 = −382920, z02 = 359760, z12 = −390635, z03 = 414450 and z13 = −388170, and that the DM speciﬁes the target values at (h1 , h2 , h3 ) = (0.65, 0.65, 0.65) and an initial degree of the α-level set α = 0.7. For the initial reference probability levels (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 1.00), the minimax problem is solved through GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.12. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference fractile levels as F (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 0.90, 1.00) for improving the value of the objective function Z1α F at the expense of Z F . For the updated reference probability levels, the corand Z3α 2α responding minimax problem is solved by using the revised GADSLPRRSU, and the objective function values of the second iteration are calculated as shown in Table 4.12. Furthermore, assume that the DM is not satisﬁed with the objective function values, and the DM updates the degree α from 0.7 to 0.6. The objective function values of the third iteration as shown in Table 4.12. After another revision of the reference probability levels, in the forth interaction, if the DM is satisﬁed with the objective function values, it follows that a satisﬁcing solution for the DM is derived. Table 4.12 Process of interaction. Iteration 1st zˆ1 1.000 zˆ2 1.000 1.000 zˆ3 α 0.700 P Z1α (x) P Z2α (x) P (x) Z3α

0.744 0.745 0.746

2nd 1.000 0.900 1.000 0.700

3rd 1.000 0.900 1.000 0.600

4th 1.000 0.900 0.950 0.600

0.761 0.666 0.766

0.786 0.686 0.787

0.809 0.708 0.758

Chapter 5

Stochastic and Fuzzy Random Two-Level Programming

In this chapter, for resolving the conﬂict between two decision makers (DMs) who have different interests in hierarchical managerial or public organizations, two-level programming is discussed. In particular, we consider two-level programming problems with random variable and fuzzy random variable coefﬁcients to deal with the uncertainty involved in cooperative and noncooperative decision making situations. By using optimization models provided in Chapters 3 and 4, the original stochastic and fuzzy random two-level programming problems are transformed into the deterministic two-level programming problems. Under situations where the two DMs can coordinate their decisions, to derive a satisfactory solution for the DM who have a higher priority in making decision, interactive fuzzy programming methods for stochastic and fuzzy random two-level programming problems are developed by considering a balance between the DMs’ satisfaction degrees. By contrast, in noncooperative decision situations, we also consider stochastic and fuzzy random two-level programming problems, and provide some computational methods using the combination of convex programming techniques and the branch-and-bound method to obtain Stackelberg equilibria or solutions. Furthermore, using genetic algorithms, these methods are extended to deal with two-level integer programming problems with random variables and fuzzy random variables.

5.1 Cooperative two-level programming In the real world, we often encounter situations where there are two decision makers (DMs) in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. Such decision making situations can be formulated as a two-level programming problem (Sakawa and Nishizaki, 2009); one of the DMs ﬁrst makes a decision, and then the other who knows the decision of the opponent makes a decision.

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_5, © Springer Science+Business Media, LLC 2011

169

170

5 Stochastic and Fuzzy Random Two-Level Programming

In the context of two-level programming, the DM at the upper level ﬁrst speciﬁes a strategy, and then the DM at the lower level speciﬁes a strategy so as to optimize the objective with full knowledge of the action of the DM at the upper level. In conventional multi-level mathematical programming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communication among DMs, or they do not make any binding agreement even if there exists such communication (Simaan and Cruz, 1973; Bialas and Karwan, 1984; Nishizaki and Sakawa, 2000). However, for decision making problems in such as decentralized large ﬁrms with divisional independence, it is quite natural to suppose that there exists communication and some cooperative relationship among the DMs. In order to deal with such cooperative two-level programming problems, Lai (1996) and Shih, Lai and Lee (1996) proposed solution concepts for two-level linear programming problems or multi-level ones such that decisions of DMs in all levels are sequential and all of the DMs essentially cooperate with each other. In their methods, the DMs identify membership functions of the fuzzy goals for their objective functions, and in particular, the DM at the upper level also speciﬁes those of the fuzzy goals for the decision variables. The DM at the lower level solves a fuzzy programming problem with a constraint with respect to a satisfactory degree of the DM at the upper level. Unfortunately, there is a possibility that their method leads a ﬁnal solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those of the decision variables. To overcome such a problem in their methods, by eliminating the fuzzy goals for the decision variables, Sakawa et al. proposed interactive fuzzy programming for two-level or multi-level linear programming problems to obtain a satisfactory solution for the DMs (Sakawa, Nishizaki and Uemura, 1998, 2000). The subsequent works on two-level or multi-level programming have been appearing (Lee, 2001; Sakawa and Nishizaki, 2002a,b; Sakawa, Nishizaki and Uemura, 2002; Sinha, 2003; Pramanik and Roy, 2007; Abo-Sinna and Baky, 2007; Roghanian, Sadjadi and Aryanezhad, 2007; Sakawa and Nishizaki, 2009; Sakawa and Kato, 2009a,b; Sakawa and Katagiri, 2010a; Sakawa, Katagiri and Matsui, 2010a). In this section, assuming cooperative behavior of the DMs, we consider solution methods for stochastic and fuzzy random two-level linear programming problems. Interactive fuzzy programming to obtain a satisfactory solution for the DM at the upper level in consideration of the cooperative relation between the DMs is presented.

5.1.1 Stochastic two-level linear programming From a similar viewpoint to in Chapter 3, we deal with two-level linear programming problems involving random variable coefﬁcients in objective functions and the right-hand side of constraints formulated as

5.1 Cooperative two-level programming

⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

171

(5.1)

where x1 is an n1 dimensional decision variable column vector for the DM at the upper level, x2 is an n2 dimensional decision variable column vector for the DM at the lower level, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors, and b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. For notational convenience, let DM1 and DM2 denote the DMs at the upper and the lower levels, respectively, and “minimize” and “minimize” mean that DM1 and DM2 are minimizers for their for DM1

for DM2

objective functions. By introducing the chance constrained conditions (Charnes and Cooper, 1959), (5.1) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 (5.2) for DM2 ⎪ subject to P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi , i = 1, . . . , m ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, ¯ and ηi , where ai j is the i th row vector of A j , j = 1, 2, b¯ i is the i th element of b, i = 1, . . . , m are satisﬁcing probability levels speciﬁed by DM1 with respect to the chance constraint conditions. From the continuity and the nondecreasingness of each distribution function Fi (r) = P(ω | bi (ω) ≤ r) for the random variable b¯ i , as we discussed in 3.1.1 of Chapter 3, the ith constraint in (5.2) can be equivalently transformed as P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi ⇔ ai1 x1 + ai2 x2 ≤ Fi−1 (1 − ηi ) bi (ηi ), (5.3) where Fi−1 is the inverse function of Fi . Then, (5.2) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.4)

η) = (b1 (η1 ), . . . , bm (ηm ))T . where b(η η) be the feasible region of In the following, for notational convenience, let X(η (5.4), namely, η) {(xT1 , xT2 )T | A1 x1 + A2 x2 ≤ b(η η), x1 ≥ 0, x2 ≥ 0}. X(η

172

5 Stochastic and Fuzzy Random Two-Level Programming

As we developed the four types of optimization models for fuzzy multiobjective stochastic programming problems in Chapter 3, we can consider extensions of these models so as to deal with cooperative stochastic two-level linear programming problems. However, to avoid unnecessary duplication in successive sections of this chapter, we will present the expectation and the variance models for stochastic two-level linear programming problems, and as for the probability and the fractile models, we will present them in the next section dealing with fuzzy random two-level linear programming.

5.1.1.1 Expectation model When the DMs intend to simply minimize the expected value of the objective functions, by substituting the expectation of zl (x1 , x2 ) = c¯ l1 x1 + c¯ l2 x2 , l = 1, 2 for the original objective functions involving random variables in (5.4), we consider the expectation model for stochastic two-level programming problem formulated as ⎫ minimize zE1 (x1 , x2 ) E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎬ for DM1 minimize zE2 (x1 , x2 ) E[¯c21 ]x1 + E[¯c22 ]x2 (5.5) for DM2 ⎪ ⎭ T T T η), subject to (x1 , x2 ) ∈ X(η where E[¯ci j ] = (E[c¯i j1 ], . . . , E[c¯i jn j ]), i, j = 1, 2, and E[c¯i jk ] is the expectation of the random variable c¯i jk . Observing that E[¯cl1 ] and E[¯cl2 ], l = 1, 2 are constant vectors, it is evident that (5.5) is an ordinary two-level linear programming problem described in Chapter 2, and the interactive fuzzy programming techniques for two-level linear programming problems are directly applicable for deriving a satisfactory solution for DM1 taking into account the overall satisfactory balance between DM1 and DM2.

5.1.1.2 Variance model As discussed in the previous chapters, if the DMs are willing to simply minimize the expected objective function values without concerning about the ﬂuctuation of the realized values, the expectation model is appropriate. In contrast, when the DMs prefer to decrease the ﬂuctuation of the objective function values from the viewpoint of leveling of the minimized objective function values, they should employ the variance model in which the variances are minimized subject to the condition of the expectations. For a stochastic two-level programming problem, this model is formulated as follows:

5.1 Cooperative two-level programming

173

⎫ minimize zV1 (x1 , x2 ) (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ for DM1 ⎪ ⎪ minimize zV2 (x1 , x2 ) (xT1 , xT2 )V2 (xT1 , xT2 )T ⎪ ⎬ for DM2 subject to E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ T T T η), (x1 , x2 ) ∈ X(η

(5.6)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are permissible expectation levels speciﬁed by the DMs. η, γ ) denote the feasible region of (5.6), For notational convenience, let X(η namely, η) | E[¯cl1 ]x1 + E[¯cl2 ]x2 ≤ γl , l = 1, 2}. η, γ ) {(xT1 , xT2 )T ∈ X(η X(η To help each DM to specify the permissible expectation level, it is recommended to calculate the individual minima and maxima of the expected values of the objective functions obtained by solving the linear programming problems minimize zEl (x1 , x2 ) = E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2, (5.7) η) subject to (xT1 , xT2 )T ∈ X(η maximize zEl (x1 , x2 ) = E[¯cl1 ]x1 + E[¯cl2 ]x2 η) subject to (xT1 , xT2 )T ∈ X(η

, l = 1, 2.

(5.8)

It is natural that the DMs have fuzzy goals for their objective functions when they take fuzziness of human judgments into consideration. For each of the objective functions zVl (x1 , x2 ), l = 1, 2 of (5.6), assume that the DMs have fuzzy goals such as “the objective function zVl (x1 , x2 ) should be substantially less than or equal to some speciﬁc value pl ,” and let μl denote the membership function of the fuzzy goal for the DMl. Then, (5.6) can be interpreted as the fuzzy two-level programming problem ⎫ maximize μ1 (zV1 (x1 , x2 )) ⎪ ⎬ for DM1 V maximize μ2 (z2 (x1 , x2 )) (5.9) for DM2 ⎪ ⎭ T T T η γ subject to (x1 , x2 ) ∈ X(η , ). In order to help the DMs to specify the membership functions μl , l = 1, 2, it is recommended to calculate the individual minima of zVl (x1 , x2 ), l = 1, 2 by solving the quadratic programming problems minimize zVl (x1 , x2 ) = (xT1 , xT2 )Vl (xT1 , xT2 )T , l = 1, 2. (5.10) η, γ ) subject to (xT1 , xT2 )T ∈ X(η As an initial candidate for an overall satisfactory solution to the DMs, it would be useful for DM1 to obtain a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem

174

5 Stochastic and Fuzzy Random Two-Level Programming

maximize min{μ1 (zV1 (x1 , x2 )), μ2 (zV2 (x1 , x2 ))} η, γ ) subject to (xT1 , xT2 )T ∈ X(η or equivalently

⎫ minimize −v ⎪ ⎪ ⎪ subject to −μ1 (zV1 (x1 , x2 )) + v ≤ 0 ⎬ −μ2 (zV2 (x1 , x2 )) + v ≤ 0 ⎪ ⎪ ⎪ ⎭ η, γ ). (xT1 , xT2 )T ∈ X(η

(5.11)

(5.12)

The convexity of −μl (zVl (x1 , x2 )) + v can be easily shown in a manner similar to (3.22) of Chapter 3, and therefore, (5.12) can be solved by some convex programming technique such as the sequential quadratic programming method. If DM1 is satisﬁed with the membership function values μl (zVl (x∗1 , x∗2 )) of (5.12), it follows that the corresponding optimal solution (x∗1 , x∗2 ) becomes a satisfactory solution; however, DM1 is not always satisﬁed with the membership function values. It is quite natural to assume that DM1 speciﬁes the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )) subjectively. Consequently, if DM1 is not satisﬁed with μl (zVl (x∗1 , x∗2 )) of (5.12), the following problem is formulated: ⎫ maximize μ2 (zV2 (x1 , x2 )) ⎪ ⎬ subject to μ1 (zV1 (x1 , x2 )) ≥ δ (5.13) ⎪ ⎭ T T T η, γ ), (x1 , x2 ) ∈ X(η where DM2’s membership function is maximized under the condition that DM1’s membership function μ1 (zV1 (x1 , x2 )) is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1. From the assumption that μl , l = 1, 2 are nonincreasing, (5.13) is rewritten as the convex programming problem ⎫ minimize zV2 (x1 , x2 ) ⎪ ⎬ subject to zV1 (x1 , x2 ) ≤ μ1 (δ) (5.14) ⎪ ⎭ T T T η, γ ), (x1 , x2 ) ∈ X(η where μ1 is a pseudo inverse function deﬁned by μ1 (δ) sup{r | μ1 (r) ≥ δ}. If there exists an optimal solution to (5.13), it follows that DM1 obtains a satisfactory solution having a satisfactory degree larger than or equal to the minimal factory level speciﬁed by DM1. However, it is signiﬁcant to realize that the larger the minimal satisfactory level δ for μ1 is accessed, the smaller the DM’s satisfactory degree μ2 becomes when the objective functions of DM1 and DM2 conﬂict with each other. Consequently, a relative difference between the satisfactory degrees of DM1 and DM2 becomes larger, and it follows that the overall satisfactory balance between both DMs is not appropriate. In order to take account of the overall satisfactory balance between both DMs, realizing that DM1 needs to compromise with DM2 on DM1’s own minimal satisfactory level, we introduce the ratio Δ of the satisfactory degree of DM2 to that of

5.1 Cooperative two-level programming

DM1 deﬁned as Δ=

175

μ2 (zV2 (x1 , x2 )) . μ1 (zV1 (x1 , x2 ))

(5.15)

DM1 is guaranteed to have a satisfactory degree larger than or equal to the minimal satisfactory level for the fuzzy goal because the corresponding constraint is involved in (5.14). To take into account the overall satisfactory balance between both DMs, DM1 speciﬁes the lower bound Δmin and the upper bound Δmax for the ratio, and the ratio Δ is evaluated by verifying whether or not it is in the interval [Δmin , Δmax ]. This condition is represented by Δ ∈ [Δmin , Δmax ]. Now we are ready to present a procedure of interactive fuzzy programming for the variance model for deriving an overall satisfactory solution. Interactive fuzzy programming in the variance model Step 1: Ask DM1 to specify the satisﬁcing probability levels ηi , i = 1, . . . , m in (5.2). Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ), l = 1, 2 by solving the linear programming problems (5.7) and (5.8). Step 3: Ask each DM to specify the permissible expectation level γl by considering the individual minima and maxima obtained in step 2. Step 4: Calculate the individual minima zVl,min of zVl (x1 , x2 ) in (5.6) by solving the quadratic programming problems (5.10). Step 5: Ask each DM to specify the membership function μl (zVl (x1 , x2 )) by considering the individual minima calculated in step 4. Step 6: Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of the ratio Δ deﬁned by (5.15). Step 7: Solve the maximin problem (5.11), and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ corresponding to the optimal solution (x∗1 , x∗2 ) to (5.11). If DM1 is satisﬁed with the current membership function values, then stop. Otherwise, ask DM1 to specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )). Step 8: For the current minimal satisfactory level δ, solve the convex programming problem (5.14), and calculate the corresponding membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 9: If DM1 is satisﬁed with the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 8.

5.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the variance model, consider the following stochastic two-level linear

176

5 Stochastic and Fuzzy Random Two-Level Programming

programming problem: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 for DM1

minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 subject to a11 x1 + a12 x2 ≤ b¯ 1 ⎪ ⎪ a21 x1 + a22 x2 ≤ b¯ 2 ⎪ ⎪ ⎪ ¯ ⎪ a31 x1 + a32 x2 ≤ b3 ⎪ ⎭ T T x1 = (x11 , . . . , x15 ) ≥ 0, x2 = (x21 , . . . , x23 ) ≥ 0,

(5.16)

where b¯ 1 , b¯ 2 and b¯ 3 are Gaussian random variables N(220, 42 ), N(145, 32 ) and N(−18, 52 ), respectively; the mean vectors of c¯ l1 and c¯ l2 , l = 1, 2, and the coefﬁcient vectors ai1 and ai2 , i = 1, 2, 3 are shown as in Tables 5.1 and 5.2, respectively. Table 5.1 Mean vector E[¯cl j ] of c¯ l j . E[¯c11 ] E[¯c12 ] E[¯c21 ] E[¯c22 ]

5 6 10 3

2 1 −7 −4

1 3 1 6

2

1

−2

−5

9

11

3

3

−6

−8

Table 5.2 Each element of ai j in (5.16). a11 a12 a21 a22 a31 a32

7 4 5 −7 −4 −3

2 3 6 −1 −7 −5

6 8 −4 −3 −2 −6

The variance-covariance matrices Vl , l = 1, 2 are given as ⎡

−3.5 1.6 −2.0 16.0 −2.0 3.0 2.2 2.8

1.3 −0.7 5.0 −2.0 4.0 −1.0 0.8 −2.0

⎤ −2.0 4.0 −1.4 0.5 −1.3 2.0 ⎥ ⎥ −2.4 1.2 −2.1 ⎥ ⎥ 3.0 2.2 2.8 ⎥ ⎥, −1.0 0.8 −2.0 ⎥ 1.0 −1.5 0.6 ⎥ ⎥ −1.5 4.0 −2.3 ⎦ 0.6 −2.3 4.0

4.0 −1.4 0.8 0.2 ⎢ −1.4 4.0 0.2 −1.0 ⎢ ⎢ 0.8 0.2 9.0 0.2 ⎢ ⎢ 0.2 −1.0 0.2 36.0 V2 = ⎢ ⎢ 1.6 −2.2 −1.5 0.8 ⎢ 1.0 0.8 1.5 0.4 ⎢ ⎣ 1.2 0.9 1.0 −1.5 2.0 1.8 0.6 0.7

1.6 −2.2 −1.5 0.8 25.0 1.2 −0.2 2.0

⎤ 1.0 1.2 2.0 0.8 0.9 1.8 ⎥ ⎥ 1.5 1.0 0.6 ⎥ ⎥ 0.4 −1.5 0.7 ⎥ ⎥. 1.2 −0.2 2.0 ⎥ 25.0 0.5 1.4 ⎥ ⎥ 0.5 9.0 0.8 ⎦ 1.4 0.8 16.0

16.0 ⎢ −1.6 ⎢ ⎢ 1.8 ⎢ ⎢ −3.5 V1 = ⎢ ⎢ 1.3 ⎢ ⎢ −2.0 ⎣ 4.0 −1.4 ⎡

−1.6 25.0 −2.2 1.6 −0.7 0.5 −1.3 2.0

1.8 −2.2 25.0 −2.0 5.0 −2.4 1.2 −2.1

5.1 Cooperative two-level programming

177

For the speciﬁed satisﬁcing probability levels (γ1 , γ2 , γ3 ) = (0.85, 0.95, 0.90), the individual minima and maxima of zEl (x1 , x2 ), l = 1, 2 are calculated by solving linear programming problems (5.7) and (5.8): zE1,min = 3.053, zE2,min = −425.621, zE1,max = 323.784 and zE2,max = 304.479. Considering these values, suppose that each DM subjectively speciﬁes the permissible expectation levels γ1 = 60.0 and γ2 = 35.0. By solving the quadratic programming problems (5.10), the individual minima of zVl (x1 , x2 ), l = 1, 2 are calculated as zV1,min = 0.818 and zV2,min = 19.524. Taking account of these values, assume that each DM determines the linear membership function ⎧ 1 if zVl (x1 , x2 ) ≤ zVl,1 ⎪ ⎪ ⎪ ⎨ zV (x1 , x2 ) − zV l l,0 if zVl,1 < zVl (x1 , x2 ) ≤ zVl,0 μl (zVl (x1 , x2 )) = V V ⎪ zl,1 − zl,0 ⎪ ⎪ ⎩ 0 if zVl,0 < zVl (x1 , x2 ), where the parameters zVl,1 and zVl,0 , l = 1, 2 are determined as zV1,1 = 0.818, zV1,0 = 129.623, zV2,1 = 19.524 and zV2,0 = 206.620 by using Zimmermann’s method (Zimmermann, 1978). For the upper bound Δmax = 0.75 and the lower bound Δmin = 0.65 speciﬁed by MD1, the maximin problem (5.12) is solved, and DM1 is supplied with the corresponding membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ of the ﬁrst iteration as shown in Table 5.3. Assume that DM1 is not satisﬁed with the membership function values, and DM1 speciﬁes the minimal satisfactory level δ for μ1 (zV1 (x1 , x2 )) as 0.900. For the speciﬁed δ, the convex programming problem (5.14) is solved, and the corresponding membership function values and the ratio of the second iteration are calculated as shown in Table 5.3. Since the ratio Δ is greater than Δmax = 0.75, suppose that DM1 updates the minimal satisfactory level δ from 0.900 to 0.950. For the updated δ, the convex programming problem (5.14) is solved, and the corresponding membership function values and the ratio of the third iteration are shown in Table 5.3. After a similar procedure, in the fourth iteration, the condition for the ratio holds, i.e., Δ ∈ [0.65, 0.75]. Then, if DM1 is satisﬁed with the membership function values of the fourth iteration, it follows that a satisfactory solution is obtained.

5.1.2 Fuzzy random two-level linear programming We discussed the two approaches for dealing with multiobjective linear programming problems involving coefﬁcients represented by fuzzy random variables in Chapter 4, and for fuzzy random two-level linear programming, the two approaches, the possibility-based model and the level set-based model, are also developed. Al-

178

5 Stochastic and Fuzzy Random Two-Level Programming

Table 5.3 Process of interaction. Iteration 1st δ

2nd

3rd

4th

0.900

0.950

0.960

zV1 (x∗1 , x∗2 )

20.654

13.699

7.258

5.970

zV2 (x∗1 , x∗2 )

48.355

57.881

73.170

77.725

μ1 (zV1 (x∗1 , x∗2 ))

0.846

0.900

0.950

0.960

μ2 (zV2 (x∗1 , x∗2 ))

0.846

0.795

0.714

0.689

Δ

1.000

0.884

0.751

0.718

though several models can be considered by combining the optimization criteria of stochastic programming, taking into account the connection to the previous section, we will give the possibility-based probability model and the level set-based fractile model for fuzzy random two-level linear programming.

5.1.2.1 Possibility-based probability model In previous subsections, it is implicitly assumed that uncertain parameters or coefﬁcients involved in the formulated two-level programming problems can be expressed as random variables. This means that realized values of the random parameters by the occurrence of some event are assumed to be deﬁnitely represented by real values. However, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to express the experts’ ambiguous understanding of the realized values of random parameters by the occurrence of events as fuzzy numbers. To handle such a situation in a hierarchical decision making problem, we consider two-level linear programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ˜¯ x + C ˜¯ x ⎬ minimize z2 (x1 , x2 ) = C 21 1 22 2 (5.17) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector of DM1, x2 is an n2 dimensional decision variable column vector of DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2. Each element ˜¯ , l = 1, 2, j = 1, 2 is a fuzzy random C˜¯l jk , k = 1, 2, . . . , n j of coefﬁcient vectors C lj variable whose realized value for an elementary event ω ∈ Ω is an L-R fuzzy number characterized by the membership function

5.1 Cooperative two-level programming

⎧ dl jk (ω) − τ ⎪ ⎪ ⎪ ⎨L βl jk μC˜l jk (ω) (τ) = ⎪ τ − dl jk (ω) ⎪ ⎪ ⎩R γl jk

179

if τ ≤ dl jk (ω) (5.18) if τ > dl jk (ω),

where the shape functions L and R are nonincreasing continuous functions from [0, ∞) to [0, 1], and the parameters βl jk and γl jk , representing the left and right spreads, are positive numbers. For notational convenience, let X denote the feasi˜¯ = (C ˜¯ , C ˜¯ T T T ble region of (5.17) and let C l l1 l2 ), x = (x1 , x2 ) . ˜ ¯ Since each coefﬁcient Cl jk is a fuzzy random variable whose realized values are ˜¯ x = C ˜¯ x + C ˜¯ x , l = 1, 2 is L-R fuzzy numbers, each of objective functions C l l1 1 l2 2 also a fuzzy random variable. Then, their realized values are represented as fuzzy numbers which are characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x (5.19) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β In order to consider the imprecise nature of each DMs’ judgments for the objective functions zl (x1 , x2 ) in (5.17), we introduce a fuzzy goal such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value.” Let μG˜ l denote the membership function of the fuzzy goal G˜ l with respect to the objective function zl (x1 , x2 ). To elicit the membership function μG˜ l from each DM for the objective function zl (x1 , x2 ), it is recommended to calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the linear programming problems minimize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2, (5.20) subject to x ∈ X maximize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2. (5.21) subject to x ∈ X Recalling that the membership function is regarded as a possibility distribution as discussed in the previous chapters, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.22) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2, y

Observing that the degree of possibility ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of an elementary event ω, it is evident that the interactive fuzzy programming for deterministic two-level programming problems cannot be directly applied.

180

5 Stochastic and Fuzzy Random Two-Level Programming

Along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs prefer to maximize the probabilities with respect to the degrees of possibility, the original fuzzy random two-level programming problem (5.17) can be reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎬ for DM1 (5.23) maximize P ω ΠC˜ 2 (ω)x (G˜ 2 ) ≥ h2 ⎪ ⎪ for DM2 ⎪ ⎭ subject to x ∈ X, where hl , l = 1, 2 are target values for the degrees of possibility speciﬁed by the DMs. Recalling the discussion in 4.1.2.1 of Chapter 4, the following relation holds: βl )x ≤ μG˜ (hl ), ΠC˜ l (ω)x (G˜ l ) ≥ hl ⇔ (dl (ω) − L (hl )β l

where

L

and

μG˜ l

are pseudo inverse functions of L and μG˜ l deﬁned as (4.42). Employing the assumption in Chapter 4 that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where t¯l is a random variable with the mean ml and the variance σ2l , and dll j , l = 1, 2 are constants. If dl2 x > 0, l = 1, · · · , k for any x ∈ X, one ﬁnds that the objective functions in (5.23) can be rewritten as # " βl − dl1 )x + μG˜ (hl ) (L (hl )β l , P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl dl2 x where Tl is a probability distribution function of t¯l . Consequently, (5.23) can be equivalently transformed into #⎫ " β1 − d11 )x + μG˜ (h1 ) ⎪ (h )β (L 1 ⎪ 1 ⎪ ⎪ maximize Z1Π,P (x) T1 ⎪ 2 ⎪ for DM1 d1 x ⎪ ⎬ " # 1 β2 − d2 )x + μG˜ (h2 ) ⎪ (L (h )β 2 2 ⎪ maximize Z2Π,P (x) T2 ⎪ ⎪ for DM2 ⎪ d22 x ⎪ ⎪ ⎭ subject to x ∈ X.

(5.24)

In order to obtain an initial candidate for an overall satisfactory solution to (5.24), it would be useful for DM1 to ﬁnd a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem maximize min Z1Π,P (x), Z2Π,P (x) (5.25) subject to x ∈ X, or equivalently

5.1 Cooperative two-level programming

181

⎫ maximize v ⎪ ⎪ ⎪ Π,P subject to Z1 (x) ≥ v ⎬ Z2Π,P (x) ≥ v ⎪ ⎪ ⎪ ⎭ x ∈ X.

(5.26)

From the fact that the distribution functions Tl , l = 1, 2 in (5.24) are nondecreasing, without loss of generality, (5.26) is equivalently transformed into ⎫ maximize v ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β1 )x + μ1 (h1 ) (−d1 + L (h1 )β ⎪ ⎪ ≥ T (v) subject to ⎬ 1 2 d1 x (5.27) ⎪ ⎪ β2 )x + μ2 (h2 ) (−d21 + L (h2 )β ⎪ ⎪ ≥ T2 (v) ⎪ ⎪ ⎪ d22 x ⎪ ⎭ x ∈ X, where Tl is the pseudo inverse function of Tl , which is deﬁned by Tl (v) = inf{r | Tl (r) ≥ v}. Although (5.27) is a nonlinear programming problem, observing that the constraints of (5.27) are linear if v is ﬁxed, we can easily ﬁnd the maximum of v by using the combined use of the simplex method and the bisection method. If DM1 is satisﬁed with the membership function values ZlΠ,P (x∗ ), l = 1, 2, it follows that the corresponding optimal solution x∗ to (5.27) becomes a satisfactory solution; however, DM1 is not always satisﬁed with the membership function values. As we discussed for the variance model in 5.1.1.2, assume that DM1 speciﬁes the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )) subjectively, and the following problem is formulated: ⎫ ⎪ maximize Z2Π,P (x) ⎬ Π,P (5.28) subject to Z1 (x) ≥ δ ⎪ ⎭ x ∈ X, or equivalently maximize T2

β2 )x + μ2 (h2 ) (−d21 + L (h2 )β d22 x

β1 )x + μ1 (h1 ) (−d11 + L (h1 )β 2 d1 x x ∈ X.

subject to T1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ≥δ⎪ ⎪ ⎪ ⎪ ⎭

(5.29)

From the assumption that dl2 x > 0, l = 1, 2 together with the nonincreasing property of Tl , (5.29) can be equivalently transformed into

182

5 Stochastic and Fuzzy Random Two-Level Programming

⎫ β2 )x (−d21 + L (h2 )β ⎪ ⎪ ⎪ maximize ⎬ 2 d2 x β1 x ≤ μ1 (h1 ) ⎪ subject to T1 (δ)d12 + d11 − L (h1 )β ⎪ ⎪ ⎭ x ∈ X.

(5.30)

Observing that (5.30) is a linear fractional programming problem, in the same way as discussed in the previous chapters, (5.30) can be solved through the combined use of the linear programming techniques and the variable transformation method. By using the ratio Δ = Z2Π,P (x)/Z1Π,P (x) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the possibility-based probability model for deriving an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the linear programming problems (5.20) and (5.21). Step 2: Ask each DM to specify the membership function μl by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify the target values hl , l = 1, 2. Step 4: For the target values hl , l = 1, 2, solve the maximin problem (5.25). Step 5: DM1 is supplied with the objective function values Z1Π,P (x∗ ) and Z2Π,P (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed with them and prefers updating hl , l = 1, 2, ask DM1 to update hl , and return to step 4. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 6: For the minimal satisfactory level δ, solve the linear fractional programming problem (5.30). Step 7: DM1 is supplied with the values of Z1Π,P (x∗ ), Z2Π,P (x∗ ) and the ratio Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed with them and prefers updating hl , l = 1, 2, ask DM1 to update hl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.1.2.2 Level set-based fractile model As in the level set-based models in the previous chapters, for the given original fuzzy random two-level programming problem (5.17), where a realized value of each coefﬁcient C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j for an elementary event ω is assumed to be a fuzzy number characterized by the membership function (5.18), suppose that the DMs intend to minimize their objective functions under the condition that all the coefﬁcient vector of the objective functions in (5.17) belong to the α-level set of fuzzy random variable deﬁned in Chapter 4. Then, we consider the following multiobjective stochastic programming problem depending on the the degree α:

5.1 Cooperative two-level programming

minimize z1 (x1 , x2 ) = C¯ 1 x for DM1 minimize z2 (x1 , x2 ) = C¯ 2 x

183

⎫ ⎪ ⎪ ⎪ ⎬ (5.31)

for DM2

⎪ ⎪ subject to x ∈ X ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α ,

where x = (xT1 , xT2 )T , and C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned by (4.67). In order to consider the imprecise nature of the DMs’ judgments for the objective functions C¯ l x, l = 1, 2, we introduce the fuzzy goals which can be quantiﬁed by eliciting the membership functions μl , l = 1, 2 from DM1 and DM2, respectively. Then, observing that the values of μl (Cl (ω)x) vary randomly due to the stochastic occurrence of an elementary event ω, it is evident that the solution methods for ordinary deterministic two-level linear programming problems cannot be directly applied. As we discussed in Chapter 4, assuming that the DMs intend to maximize their target values for the degrees of possibility for given permissible probability levels, by replacing minimization of the objective functions zl (x1 , x2 ), l = 1, 2 in (5.31) with maximization of the target values hl , l = 1, 2 such that the probabilities with respect to the fuzzy goal attainment levels are greater than or equal to certain permissible probability levels θl , l = 1, 2 speciﬁed by the DMs, we consider the level set-based fractile model for fuzzy random two-level programming problems formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ for DM2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.32) ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values, and θl , l = 1, 2 are permissible probability levels speciﬁed by the DMs, respectively. As we discussed in 4.1.3.4 of Chapter 4, from the relation L (ω)x ≤ μl (hl ) ≥ θl , P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα (5.32) can be transformed into maximize h1 for DM1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize h2 for DM2 L (ω)x) ≥ h ≥ θ subject to P ω | μ1 (C1α 1 1⎪ ⎪ ⎪ L P ω | μ2 (C2α (ω)x) ≥ h2 ≥ θ2 ⎪ ⎪ ⎪ ⎭ x ∈ X.

(5.33)

184

5 Stochastic and Fuzzy Random Two-Level Programming

Similarly to the previous model, we assume that the random parameter d¯ l of ˜¯ is expressed as d¯ = d1 + t¯ d2 , where t¯ is a random variable fuzzy random variable C l l l l l l with the mean ml and the variance σ2l , and dl2 x > 0, l = 1, 2. Then, the following for the constraint in (5.33) holds: β l )x + μl (hl ) (−dl1 + L (α)β L (ω)x) ≥ hl = Tl P ω | μl (Clα . dl2 x Then, it follows that βl )x + μl (hl ) (−dl1 + L (α)β βl )x ≥ hl , ≥ θl ⇔ μl (Tl (θl )dl2 + dl1 − L (α)β Tl 2 dl x where Tl is a pseudo inverse function of Tl which is deﬁned by Tl (θl ) = inf{r | Tl (r) ≥ θl }. Let F βl )x, (x) = (Tl (θl )dl2 + dl1 − L (α)β (5.34) Zlα and consequently, (5.33) can be transformed into maximize h1 for DM1

maximize h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

for DM2 F subject to μ1 Z1α (x) ≥ h1 ⎪ ⎪ F ⎪ μ2 Z2α (x) ≥ h2 ⎪ ⎪ ⎪ ⎭ x ∈ X,

or equivalently

⎫ F (x) ⎪ maximize μ1 Z1α for DM1 F ⎬ maximize μ2 Z2α (x) ⎪ for DM2 ⎭ subject to x ∈ X.

(5.35)

(5.36)

To derive an overall satisfactory solution to (5.36), we ﬁrst solve the following problem maximizing the smaller degree of satisfaction between those of the two DMs: F F (x) , μ2 Z2α (x) maximize min μ1 Z1α (5.37) subject to x ∈ X. By introducing an auxiliary variable v, from (5.34), this problem is rewritten as ⎫ maximize v ⎪ ⎪ β1 )x ≤ μ1 (v) ⎬ subject to (T1 (θ1 )d12 + d11 − L (α)β (5.38) β2 )x ≤ μ2 (v) ⎪ (T2 (θ2 )d22 + d21 − L (α)β ⎪ ⎭ x ∈ X.

5.1 Cooperative two-level programming

185

It is noted that (5.38) can be solved by the combined use of the simplex method and the bisection method, in a similar way to solving (5.27) in the possibility-based probability model. F ∗ If DM1 is satisﬁed with the membership function values μl Zlα (x ) , l = 1, 2, the corresponding optimal solution x∗ to (5.38) is regarded as a satisfactory solution. F (x)) is larger than or equal to the Otherwise, by introducing the constraint that μ1 (Z1α minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the following F (x)): problem maximizing the membership function μ2 (Z2α F ⎫ (x) maximize μ2 Z2α ⎬ F (5.39) subject to μ1 Z1α (x) ≥ δ ⎭ x ∈ X, which can be transformed into the linear programming problem ⎫ β2 )x minimize (T2 (θ2 )d22 + d21 − L (α)β ⎬ β1 )x ≤ μ1 (δ) subject to (T1 (θ1 )d12 + d11 − L (α)β ⎭ x ∈ X.

(5.40)

F (x))/μ (Z F (x)) and its permissible range [Δ By using the ratio Δ = μ2 (Z2α 1 1α min , Δmax ], we present a procedure of interactive fuzzy programming for the level-based fractile model in order to derive a satisfactory solution.

Interactive fuzzy programming in the level set-based fractile model Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2 by solving the linear programming problems (5.20) and (5.21). Step 2: Ask DMs to specify membership functions μl , l = 1, 2 by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify an initial degree α of the α-level set and the permissible probability levels θl , l = 1, 2. Step 4: For the current values of α and θl , l = 1, 2, solve the maximin problem (5.37). F ∗ (x ) and Step 5: is supplied with the membership function values μ1 Z1α F DM1 μ2 Z2α (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers to update α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , and return step4. Otherwise, ask DM1 to specify the minimal satisfactory level δ for to F (x) and the permissible range [Δ μ1 Z1α min , Δmax ] of the ratio Δ. Step 6: For the minimal satisfactory level δ, solve the linear programming problem (5.40). F ∗ F ∗ (x ) , μ2 Z2α (x ) and Δ. If Δ ∈ [Δmin , Δmax ] Step 7: DM1 is supplied with μ1 Z1α and DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers to update α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

186

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the level set-based fractile model, consider the following fuzzy random two-level linear programming problem: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ ˜ ˜ ¯ ¯ ⎬ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 for DM2 (5.41) subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 26 ⎪ ⎪ ⎪ T ⎪ x1 = (x11 , x12 , x13 , x14 , x15 ) ≥ 0 ⎪ ⎪ ⎭ x2 = (x21 , x22 , x23 , x24 , x25 )T ≥ 0, ˜¯ , l = 1, 2, j = 1, 2 are shown in Table 5.4, and t¯ , l = 1, 2 where the parameters of C lj l are given as Gaussian random variables N(10, 22 ) and N(10.0, 42 ), respectively. After solving linear programming problems (5.20) and (5.21), assume that the DMs determine the linear membership functions ⎧ ⎪ 1 if z ≤ z1l ⎪ ⎪ ⎨ z − z0 l if z1l < z ≤ z0l μl (z) = 1 − z0 ⎪ z ⎪ l ⎪ ⎩ l 0 if z > z0l by setting z11 = −4620.067, z01 = −3944.688, z12 = −2042.5 and z02 = −1503.8 in accordance with Zimmermann’s method (Zimmermann, 1978). Assume that DM1 speciﬁes an initial degree of α-level set at α = 0.9 and the permissible probability levels (θ1 , θ2 ) = (0.9, 0.9), respectively. Then, the maximin problem (5.37) is solved, and the corresponding objective function values and the ratio of the ﬁrst iteration are calculated as shown in Table 5.6. Suppose that DM1 is not satisﬁed with the objective function values and updates the permissible probability levels (θ1 , θ2 ) from (0.9, 0.9) to (0.8, 0.8). For the updated (θ1 , θ2 ), the corresponding maximin problem (5.37) is solved, and the result of the second iteration is shown in Table 5.6. Assume that DM1 is not satisﬁed with the objective function values and prefers to update neither the degree α nor the permissible probability levels θl . Then, suppose that DM1 speciﬁes the minimal satisfactory level at δ = 0.80 and the upper and the lower bounds of the ratio Δ at Δmax = 0.8 and Δmin = 0.7. For the speciﬁed δ = 0.80, the linear programming problem (5.40) is solved, and the result of the third iteration is shown in Table 5.6. A similar procedure continues in this manner until DM1 is satisﬁed with the membership function values. In the ﬁfth iteration, the ratio Δ exists in the interval [0.70, 0.80], and if DM1 is satisﬁed with the membership function values, it follows that a satisfactory solution is derived.

5.1 Cooperative two-level programming

187

Table 5.4 Value of each element of dl , β l and γ l , l = 1, 2. x11

x12

x13

x14

x15

x21

x22

x23

x24

x25

d11 d12 d21 d22

−300 3 −20 1

−270 2 −20 2

−200 3 −20 3

−150 2 −20 2

−100 1 −20 4

−15 2 −130 3

−15 2 −120 3

−15 2 −95 2

−15 2 −75 2

−15 2 −60 3

β1 β2 γ1 γ2

3 2 2.5 1.4

3 1 2.5 1.4

2 1 1.5 1.2

2 1 1 2.3

3 1 3 2.3

1 3 4 2.3

1.5 2 2 2.3

1.3 6 1.5 2.1

1.2 4 1.5 2

2.4 2 1.5 1.5

x22 −40 −80 −80 −150 0 7 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0

x23 −30 −70 −70 −120 0 6 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0

Table 5.5 Value of each element of ai , i = 1, . . . , 26.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26

x11 −100 −50 −250 −100 12 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

x12 −90 −50 −200 −90 11 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

x13 −80 −50 −200 −80 10 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

x14 −70 −40 −150 −70 9 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

x15 −60 −30 −100 −60 8 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

x21 −50 −100 −90 −200 0 8 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

x24 −20 −60 −60 −90 0 5 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0

x25 −10 −50 −50 −50 0 4 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1

b −2000 −2000 −5500 −4500 220 160 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 8 8 8 8 8 8 8 8 8 8

Table 5.6 Process of interaction. Iteration

1st

2nd

3rd

4th

5th

α θ1 θ2

0.900 0.900 0.900

0.900 0.800 0.800

0.900 0.800 0.800

0.900 0.800 0.700

0.900 0.800 0.700

δ F ∗ μ1 Z1α (x ) F (x∗ ) μ2 Z2α

—

—

0.800

0.700

0.750

0.298 0.298

0.584 0.584

0.800 0.556

0.700 0.579

0.750 0.578

Δ

1.000

1.000

0.695

0.828

0.770

188

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.3 Extensions to integer programming So far, we have discussed stochastic and fuzzy random two-level programming with continuous decision variables, and in this subsection, as we considered in Chapters 3 and 4, we deal with stochastic and fuzzy random two-level programming with not continuous but discrete decision variables. For dealing with large-scale two-level nonlinear integer programming problems to be solved in the interactive fuzzy programming, we employ genetic algorithms to search approximate optimal solutions. In the same composition as 5.1.1, we will give the four models for stochastic and fuzzy random two-level integer programming.

5.1.3.1 Expectation model for stochastic two-level integer programming problems Stochastic two-level integer programming problems are generally formulated as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.42) ¯ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional integer decision variable column vector for DM1, x2 is an n2 dimensional integer decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional random variable row vectors with the mean vectors c¯ l j . The elements of vector b¯ are mutually independent random variables with continuous and increasing probability distribution functions, and νl jl , l = 1, 2, jl = 1, . . . , nl are positive integer values of the upper bounds of the decision variables. In a similar way to the two-level problems with continuous decision variables, by replacing the original stochastic constraints with the chance constrained conditions with satisﬁcing probability levels ηi , i = 1, . . . , m, (5.42) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.43) subject to P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi , i = 1, . . . , m ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where ai j is the ith row vector of A j , j = 1, 2, and bi (ω) is a realized value of b¯ i for an elementary event ω. In a similar transformation in 5.1.1, (5.43) can be rewritten as

5.1 Cooperative two-level programming

189

⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎬ for DM1 minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎭ η), subject to (xT1 , xT2 )T ∈ X int (η

(5.44)

where η) {(xT1 , xT2 )T | A1 x1 + A2 x2 ≤ b(η η), xl jl ∈ {0, 1, . . . , νl jl }, X int (η jl = 1, . . . , nl , l = 1, 2}. Assuming that the DMs are willing to simply minimize the expectation of their own objective functions, by substituting minimization of the expected objective functions for minimization of the original stochastic objective functions in (5.44), we consider the expectation model for the stochastic two-level integer programming problem formulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎬ for DM1 minimize zE2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.45) for DM2 ⎪ ⎭ T T T int η). subject to (x1 , x2 ) ∈ X (η In order to consider the imprecise nature of the DMs’ judgments for the objective functions zEl (x1 , x2 ), l = 1, 2 in (5.45), if we introduce a fuzzy goal such as “zEl (x1 , x2 ) should be substantially less than or equal to a certain value,” (5.45) can be rewritten as the fuzzy two-level integer programming problem ⎫ maximize μ1 (zE1 (x1 , x2 )) ⎪ ⎬ for DM1 maximize μ2 (zE2 (x1 , x2 )) (5.46) ⎪ for DM2 ⎭ η), subject to (xT1 , xT2 )T ∈ X int (η where μl , l = 1, 2 are membership functions to quantify fuzzy goals speciﬁed by DMl, l = 1, 2. To specify the membership functions of the fuzzy goals, the DMs refer to the individual minima and maxima of E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2 obtained by solving the integer programming problems minimize E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2, (5.47) η) subject to (xT1 , xT2 )T ∈ X int (η maximize E[¯cl1 ]x1 + E[¯cl2 ]x2 η) subject to (xT1 , xT2 )T ∈ X int (η

, l = 1, 2.

(5.48)

Since (5.47) and (5.48) are deterministic integer programming problems with linear objective functions and constraints, to obtain (approximate) optimal solutions, GADSLPRRSU given in Chapter 2 can be employed. To derive an overall satisfactory solution to (5.46), the ﬁrst step to obtain a satisfactory solution is to ﬁnd a solution maximizing the smaller degree of satisfaction between those of the two DMs by solving the maximin problem

190

5 Stochastic and Fuzzy Random Two-Level Programming

maximize min{μ1 (zE1 (x1 , x2 )), μ2 (zE2 (x1 , x2 ))} η). subject to (xT1 , xT2 )T ∈ X int (η

(5.49)

From the fact that this problem includes a nonlinear objective function and linear constraints, (5.49) can be solved by the revised GADSLPRRSU given in Chapter 2. If DM1 is satisﬁed with the membership function values μl (zEl (x∗1 , x∗2 )), l = 1, 2 of (5.49), the corresponding optimal solution (x∗1 , x∗2 ) can be viewed as a satisfactory solution. However, if DM1 is not satisﬁed with them, assuming that DM1 subjectively speciﬁes the minimal satisfactory level for the DM1’s own membership function value μ1 (zE1 (x1 , x2 )), we consider the fuzzy integer programming problem ⎫ maximize μ2 (zE2 (x1 , x2 )) ⎪ ⎬ (5.50) subject to μ1 (zE1 (x1 , x2 )) ≥ δ ⎪ ⎭ T T T int η). (x1 , x2 ) ∈ X (η From the nonincreasing property of the membership function μl , l = 1, 2, (5.50) can be equivalently transformed into ⎫ minimize zE2 (x1 , x2 ) ⎪ ⎬ E subject to z1 (x1 , x2 ) ≤ μ1 (δ) (5.51) ⎪ η), ⎭ (xT1 , xT2 )T ∈ X int (η where μ1 is a pseudo inverse function of μ1 deﬁned by μ1 (δ) = sup{t | μ1 (t) ≥ δ}. Since both the objective function and the constraints in (5.51) are linear, to solve (5.51), we can directly use GADSLPRRSU. By using the ratio Δ = μ2 (zE2 (x1 , x2 ))/μ1 (zE1 (x1 , x2 )) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the expectation model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the expectation model with integer decision variables Step 1: Ask DM1 to subjectively specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ), by solving (5.47) and (5.48) through GADSLPRRSU. Step 3: Ask each DM to specify the membership function μl (zEl (x1 , x2 )) by considering the individual minima and maxima obtained in step 2. Step 4: Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of the ratio Δ. Step 5: Solve the maximin problem (5.49) through the revised GADSLPRRSU, and calculate the membership function values μl (zEl (x∗1 , x2 )∗ ), l = 1, 2 and the ratio Δ. If DM1 is satisﬁed with the membership function values, then stop. Otherwise, ask DM1 to subjectively specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zE1 (x1 , x2 )).

5.1 Cooperative two-level programming

191

Step 6: For the minimal satisfactory level δ, solve (5.51) through GADSLPRRSU, and calculate the corresponding membership function values μl (zEl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 7: If DM1 is satisﬁed with the current membership function values and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.1.3.2 Variance model for stochastic two-level integer programming problems In the variance model, from the viewpoint of risk-aversion, the variances of the DMs’ objective functions with random variable coefﬁcients are minimized, and the following problem with the constraints for the expectations of the objective functions is formulated: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ DM1 ⎪ ⎪ minimize zV2 (x1 , x2 ) = (xT1 , xT2 )V2 (xT1 , xT2 )T ⎪ ⎪ ⎬ DM2 (5.52) subject to E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ T T T int η), (x1 , x2 ) ∈ X (η where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and γl , l = 1, 2 are permissible expectation levels speciﬁed by the DMs. η, γ ) denote the feasible region of (5.52), For notational convenience, let X int (η namely, η, γ ) {(xT1 , xT2 )T ∈ X int (η η) | E[¯cl1 ]x1 + E[¯cl2 ]x2 ≤ γl , l = 1, 2}. X int (η Considering the imprecise nature of each DM’s judgments for each objective function zVl (x1 , x2 ) in (5.52), if we introduce a fuzzy goal such as “zVl (x1 , x2 ) should be substantially less than or equal to a certain value,” (5.52) can be rewritten as ⎫ maximize μ1 (zV1 (x1 , x2 )) ⎪ ⎬ DM1 V maximize μ2 (z2 (x1 , x2 )) (5.53) DM2 ⎪ ⎭ T T T int η γ subject to (x1 , x2 ) ∈ X (η , ), where μl is a nonincreasing membership function to quantify the fuzzy goal. To help the DM to specify the membership functions μl , it is recommended to calculate the individual minima of zVl (x1 , x2 ) by solving the quadratic integer programming problems minimize zVl (x1 , x2 ) = (xT1 , xT2 )Vl (xT1 , xT2 )T , l = 1, 2. (5.54) η, γ ) subject to (xT1 , xT2 )T ∈ X int (η

192

5 Stochastic and Fuzzy Random Two-Level Programming

To solve (5.54) with the nonlinear objective function and the linear constraints, we can directly apply the revised GADSLPRRSU. In order to obtain an initial candidate for an overall satisfactory solution to the DMs, using the revised GADSLPRRSU, we solve the maximin problem maximize min{μ1 (zV1 (x1 , x2 )), μ2 (zV2 (x1 , x2 ))} (5.55) η, γ ). subject to (xT1 , xT2 )T ∈ X int (η If DM1 is satisﬁed with the membership function values of (5.55), the corresponding optimal solution is regarded as a satisfactory solution. Otherwise, it would be quite natural to assume that DM1 speciﬁes the minimal satisfactory level for the membership function μ1 (zV1 (x1 , x2 )). To calculate a candidate for the satisfactory solution, by introducing the constraint that μ1 (zV1 (x1 , x2 )) is larger than or equal to the speciﬁed minimal satisfactory level δ ∈ (0, 1), we consider the following problem maximizing DM2’s membership function μ2 (zV2 (x1 , x2 )): ⎫ maximize μ2 (zV2 (x1 , x2 )) ⎪ ⎬ subject to μ1 (zV1 (x1 , x2 )) ≥ δ (5.56) ⎪ ⎭ T T T int η, γ ). (x1 , x2 ) ∈ X (η Because, unlike (5.54) and (5.55), (5.56) involves a nonlinear constraint, to solve it, we employ the revised GADSLPRRSU. By using the ratio Δ = μ2 (zV2 (x1 , x2 ))/μ1 (zV1 (x1 , x2 )) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the variance model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the variance model with integer decision variables Step 1: Ask DM1 to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ) by solving (5.47) and (5.48) through GADSLPRRSU. Step 3: Ask each DM to specify the permissible expectation level γl by considering the individual minima and maxima obtained in step 2. Step 4: Calculate the individual minima zVl,min of zVl (x1 , x2 ) by solving (5.54) through the revised GADSLPRRSU. Step 5: Ask each DM to specify the membership function μl (zVl (x1 , x2 )) by considering the individual minima calculated in step 4. Step 6 Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of Δ. Step 7: Solve the maximin problem (5.55) through the revised GADSLPRRSU, and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. If DM1 is satisﬁed with the membership function values, then stop. Otherwise, ask DM1 to specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )).

5.1 Cooperative two-level programming

193

Step 8: For the minimal satisfactory level δ, solve (5.56) through the revised GADSLPRRSU, and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 9: If DM1 is satisﬁed with the membership function values and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 8.

5.1.3.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the variance model with integer decision variables, consider the following numerical example to stochastic two-level integer programming problems: ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.57) ¯ subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 10 ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , 30}, j1 = 1, . . . , 15 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , 30}, j2 = 1, . . . , 15, where x1 = (x11 , . . . , x115 )T , x2 = (x21 , . . . , x215 )T , and b¯ i , i = 1, . . . , 10 is a Gaussian random variable with the mean si and the variance σ2i given in Table 5.7. The coefﬁcients c¯ l1 and c¯ l2 , l = 1, 2 are Gaussian random variables, and their means are given in Table 5.8. ¯ Table 5.7 Means and variances of b. bi b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 si 78166 150930 231496 293906 369243 423722 497184 588256 667464 731073 σ2i 3502 12002 18002 9802 15002 25202 17502 28002 32002 25002 Table 5.8 Means of coefﬁcients of objective functions. E[¯c11 ] 22 46 −12 38 −41 −2 −50 3 45 22 43 −21 −31 17 38 E[¯c12 ] −5 37 20 45 −8 −19 18 4 8 39 30 47 −46 47 45 13 −13 7 −30 35 −42 18 −36 23 −50 42 −1 27 −2 22 E[¯c21 ] E[¯c22 ] −50 19 −42 24 −11 14 −24 7 −48 46 −6 49 −39 11 −28

Assume that DM1 speciﬁes the satisﬁcing probability levels as η1 = 0.95, η2 = 0.80, η3 = 0.85, η4 = 0.90, η5 = 0.90, η6 = 0.85, η7 = 0.85, η8 = 0.95, η9 = 0.80 and η10 = 0.95, and the individual minima and maxima of zEl (x1 , x2 ) are calculated as zE1,min = −6972, zE2,min = −12006, zE1,max = 15113 and zE2,max = 9745. Taking account of these values, suppose that the DMs subjectively specify the permissible expectation levels at γ1 = −1000 and γ2 = −5000.

194

5 Stochastic and Fuzzy Random Two-Level Programming

By using the revised GASDLPRRSU, the individual minima of zVl (x1 , x2 ) are calculated as zV1,min = 260912 and zV2,min = 853492. Assume that based on these values, the DMs subjectively determine the linear membership functions to quantify fuzzy goals as

μl (zVl (x1 , x2 )) =

⎧ 1 ⎪ ⎪ ⎪ ⎨ zV (x1 , x2 ) − zV l l,0 ⎪ ⎪ ⎪ ⎩

zVl,1 − zVl,0

if zVl (x1 , x2 ) ≤ zVl,1 if zVl,1 < zVl (x1 , x2 ) < zVl,0 if zVl (x1 , x2 ) ≥ zVl,0 ,

0

where zVl,1 and zVl,0 , l = 1, 2 are calculated as zV1,1 = 260912, zV1,0 = 2131786, zV2,1 = 853492 and zV2,0 = 2543661 by using Zimmermann’s method (Zimmermann, 1978). For the speciﬁed upper and lower bounds Δmax = 0.85 and Δmin = 0.70 of the ratio Δ, by solving the maximin problem (5.55) through the revised GADSLPRRSU, the membership function values and the ratio are calculated as μ1 (zV1 (x∗1 , x∗2 )) = 0.7573, μ2 (zV2 (x∗1 , x∗2 )) = 0.7663, and Δ = 1.0119. Suppose that DM1 is not satisﬁed with the membership function values and speciﬁes the minimal satisfactory level at δ = 0.80. For the speciﬁed δ = 0.80, by solving (5.56) through the revised GADSLPRRSU, the membership function values and the ratio are calculated as μ1 (zV1 (x∗1 , x∗2 )) = 0.8044, μ2 (zV2 (x∗1 , x∗2 )) = 0.7481 and Δ = 0.9299. Since the ratio Δ is greater than Δmax = 0.85, suppose that DM1 updates the minimal satisfactory level δ from 0.80 to 0.90. A similar procedure continues in this manner until the DM1 is satisﬁed with the membership function values. In the forth interaction, the ratio Δ exists in the interval [0.70, 0.85], and if DM1 is satisﬁed with the membership function values, it follows that the satisfactory solution is derived. Table 5.9 Process of interaction. Iteration 1st δ — μ1 (zV1 (x∗1 , x∗2 )) 0.7573 μ2 (zV2 (x∗1 , x∗2 )) 0.7663 zV1 (x∗1 , x∗2 ) 714946 zV2 (x∗1 , x∗2 ) 1248412 Δ 1.0119

2nd 0.80 0.8044 0.7481 626827 1279327 0.9299

3rd 0.90 0.9077 0.5946 433605 1538687 0.6551

4th 0.85 0.8535 0.6942 535088 1370382 0.8134

5.1.3.4 Possibility-based probability model for fuzzy random two-level integer programming problems As we discussed in 4.1.1, considering that possible realized values of the random parameters involved in two-level integer programming problems are often only am-

5.1 Cooperative two-level programming

195

biguously known to the experts, we deal with two-level linear programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ ˜ ˜ ¯ ¯ ⎬ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 for DM2 (5.58) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, and x2 is an n2 dimensional decision variable column vector for DM2. Each element C˜¯l jk , ˜¯ , l = 1, 2, j = 1, 2 is a fuzzy random variable k = 1, 2, . . . , n j of coefﬁcient vectors C lj whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl jk (ω) − τ ⎪ ⎪ if τ ≤ dl jk (ω) ⎪ ⎨L βl jk μC˜l jk (ω) (τ) = (5.59) ⎪ τ − dl jk (ω) ⎪ ⎪ if τ > dl jk (ω). ⎩R γl jk For notational convenience, let X int denote the feasible region of (5.58) and let ˜¯ ), x = (xT , xT )T . Then, it follows that each objective function C ˜¯ x = ˜C ¯˜ l1 , C ¯ l = (C l2 l 1 2 ˜C ˜ ¯ l1 x1 + C ¯ l2 x2 is a fuzzy random variable whose realized values are fuzzy numbers characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ L if υ ≤ dl (ω)x ⎪ ⎨ βl x (5.60) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β In order to consider the imprecise nature of each DMs’ judgments for the objective functions zl (x1 , x2 ) in (5.58), we introduce a fuzzy goal such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value.” Let μG˜ l denote the nonincreasing membership function of the fuzzy goal G˜ l with respect to the objective function zl (x1 , x2 ). To elicit the membership function μG˜ l from each DM for the objective function zl (x1 , x2 ), it is recommended to calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the integer programming problems minimize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2, (5.61) subject to x ∈ X int

196

5 Stochastic and Fuzzy Random Two-Level Programming

maximize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 subject to x ∈ X int

, l = 1, 2.

(5.62)

To solve (5.61) and (5.62) with the linear objective functions and constraints, we employ GADSLPRRSU. Recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. (5.63) y

Along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs prefer to maximize the probabilities with respect to the degrees of possibility, the original fuzzy random two-level integer programming problem (5.58) can be reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎬ for DM1 (5.64) ˜ maximize P ω ΠC˜ 2 (ω)x (G2 ) ≥ h2 ⎪ ⎪ for DM2 ⎪ ⎭ subject to x ∈ X int , where hl , l = 1, 2 are target values for the degrees of possibility speciﬁed by the DMs. As we discussed in 5.1.2.1 for the same model with continuous decision variables, (5.64) can be transformed into #⎫ " β1 − d11 )x + μG˜ (h1 ) ⎪ (h )β (L 1 ⎪ 1 ⎪ ⎪ maximize Z1Π,P (x) T1 ⎪ 2 ⎪ for DM1 d1 x ⎪ ⎬ " # 1 (5.65) β2 − d2 )x + μG˜ (h2 ) ⎪ (L (h )β 2 2 ⎪ maximize Z2Π,P (x) T2 ⎪ ⎪ for DM2 ⎪ d22 x ⎪ ⎪ ⎭ int subject to x ∈ X . In order to obtain an initial candidate for an overall satisfactory solution to (5.65), we solve the maximin problem ⎫ maximize min Z1Π,P (x), Z2Π,P (x) ⎬ (5.66) ⎭ subject to x ∈ X int , which can be solved by using the revised GADSLPRRSU. If DM1 is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, 2, the corresponding optimal solution x∗ to (5.66) is regarded as a satisfactory solution. However, if DM1 is not satisﬁed, by introducing the constraint that Z1Π,P (x) is larger than or equal to the speciﬁed minimal satisfactory level δ ∈ (0, 1), we consider the

5.1 Cooperative two-level programming

following problem maximizing DM2’s objective function: ⎫ ⎪ maximize Z2Π,P (x) ⎬ Π,P subject to Z1 (x) ≥ δ ⎪ ⎭ x ∈ X int .

197

(5.67)

From the assumption that dl2 x > 0, l = 1, 2 together with the nonincreasing property of Tl , (5.67) can be equivalently transformed into ⎫ β2 )x (−d21 + L (h2 )β ⎪ ⎪ maximize ⎪ ⎬ d22 x (5.68) β1 x ≤ μ1 (h1 ) ⎪ subject to T1 (δ)d12 + d11 − L (h1 )β ⎪ ⎪ ⎭ x ∈ X int , which can be solved through the revised GADSLPRRSU. By using the ratio Δ = Z2Π,P (x)/Z1Π,P (x) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the possibility-based probability model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving (5.61) and (5.62) through GADSLPRRSU. Step 2: Ask each DM to specify the membership function μl by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify the target values hl , l = 1, 2. Step 4: For the speciﬁed target values hl , l = 1, 2, solve the maximin problem (5.66) through the revised GADSLPRRSU. Step 5: DM1 is supplied with the objective function values Z1Π,P (x∗ ) and Z2Π,P (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed and prefers updating hl , l = 1, 2, ask DM1 to update hl , and return to step 4. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 6: For the speciﬁed minimal satisfactory δ, solve (5.68) by uisng the revised GADSLPRRSU. Step 7: DM1 is supplied with the objective function values of Z1Π,P (x∗ ), Z2Π,P (x∗ ) and the ratio Δ. If Δ ∈ [Δmin , Δmax ] holds and DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed and prefers updating hl , l = 1, 2, ask DM1 to update hl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

198

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.3.5 Level set-based fractile model for fuzzy random two-level programming According to the idea of the level set-based models, suppose that the DMs intend to minimize their objective functions under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67) in Chapter 4. Then, (5.58) can be interpreted as the following stochastic two-level integer programming problem without fuzzy random variables depending on the the degree α: ⎫ ⎪ ⎪ ⎪ ⎬

minimize z1 (x1 , x2 ) = C¯ 11 x1 + C¯ 12 x2 for DM1 minimize z2 (x1 , x2 ) = C¯ 21 x1 + C¯ 22 x2 for DM2

⎪ subject to x = (xT1 , xT2 )T ∈ X int ⎪ ⎪ ⎭ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ C1 = (C11 , C12 ) ∈ C1α , C2 = (C21 , C22 ) ∈ C2α ,

(5.69)

where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . Considering the imprecise nature of the DMs’ judgments, we assume that each DM has a fuzzy goal for the objective function in (5.69), and the fuzzy goals are determined by the membership functions μl , l = 1, 2 for DM1 and DM2. In the level set-based fractile model, assuming that the DMs prefer to maximize the target values for the satisfaction degrees of the DMs for the objective function values, we consider the following fuzzy random two-level integer programming problem: ⎫ maximize h1 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ for DM2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.70) ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values, and θl , l = 1, 2 are permissible probability levels speciﬁed by the DMs, respectively. As we considered in 5.1.2.2 for the same model with continuous decision variables, letting F βl )x, Zlα (x) (Tl (θl )dl2 + dl1 − L (α)β (5.70) can be transformed into maximize h1 for DM1

maximize h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

for DM2 F subject to μ1 Z1α (x) ≥ h1 ⎪ F ⎪ ⎪ μ2 Z2α (x) ≥ h2 ⎪ ⎪ ⎪ ⎭ x ∈ X int ,

(5.71)

5.1 Cooperative two-level programming

or equivalently

F ⎫ (x) ⎪ maximize μ1 Z1α for DM1 F ⎬ maximize μ2 Z2α (x) ⎪ for DM2 ⎭ subject to x ∈ X int .

199

(5.72)

To derive an overall satisfactory solution to (5.72), it would be useful for DM1 to obtain a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem F F (x) , μ2 Z2α (x) maximize min μ1 Z1α (5.73) subject to x ∈ X int , which can be solved by using the revised GADSLPRRSU. F ∗ (x ) , l = 1, 2, If DM1 is satisﬁed with the membership function values μl Zlα the corresponding optimal solution x∗ to (5.73) is regarded as a satisfactory solution. F (x)) is larger than or equal to the Otherwise, introducing the constraint that μ1 (Z1α minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the following F (x)): problem maximizing the membership function μ2 (Z2α ⎫ F maximize μ2 Z2α (x) ⎪ ⎬ F (5.74) (x) ≥ δ subject to μ1 Z1α ⎪ ⎭ int x∈X , which is equivalently expressed as ⎫ β2 )x minimize (T2 (θ2 )d22 + d21 − L (α)β ⎬ β1 )x ≤ μ1 (δ) subject to (T1 (θ1 )d12 + d11 − L (α)β ⎭ x ∈ X int .

(5.75)

It is noted that (5.75) can be solved by GADSLPRRSU. In order to derive the satisfactory solution with well-balanced membership function values between both DMs, we ratio Fintroduce the of the satisfactory degree F of DM2 to that of DM1 Δ = μ2 Z2α (x) /μ1 Z1α (x) , and its permissible range [Δmin , Δmax ]. The, we summarize a procedure of interactive fuzzy programming for the level-based fractile model with integer decision variables in order to derive a satisfactory solution. Interactive fuzzy programming in the level set-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2 by solving (5.61) and (5.62) through GADSLPRRSU. Step 2: Ask DMs to specify membership functions μl , l = 1, 2 by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify an initial degree α of the α-level set and the permissible probability levels θl , l = 1, 2.

200

5 Stochastic and Fuzzy Random Two-Level Programming

Step 4: For the speciﬁed values of α and θl , l = 1, 2, solve the maximin problem (5.73) through the revised GADSLPRRSU. F (x∗ ) and μ Z F (x∗ ) for the Step 5: DM1 is supplied with the values of μ1 Z1α 2 2α optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers updating α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , and returnto step 4. Otherwise, ask F (x) and the permisDM1 to specify the minimal satisfactory level δ for μ1 Z1α sible range [Δmin , Δmax ] of the ratio Δ. Step 6: For the minimal satisfactory level δ, solve through (5.75) GADSLPRRSU. F (x∗ ) , μ Z F (x∗ ) and Δ. If Step 7: DM1 is supplied with the values of μ1 Z1α 2 2α Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers updating α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.2 Noncooperative two-level programming In the previous section, from the viewpoint of taking the possibility of coordination or bargaining between two DMs into account in hierarchical decision making situations, it is assumed that there exist communication and a cooperative relationship between the two DMs. However, in the real-world hierarchical managerial or public organizations, observing the decision making situations where the DMs do not make any binding agreement even if there exists such communication, we deal with noncooperative two-level programming problems. As discussed in Chapter 2, the concept of Stackelberg solution can be viewed as one of reasonable solution concepts of noncooperative two-level programming problems, in which the DM at upper level (DM1) makes a decision to optimize the DM1’s own objective function under the assumption that the DM at lower level (DM2) makes a decision to optimize the DM2’s own objective function for a given decision of DM1. Computational methods for obtaining Stackelberg solutions to two-level linear programming problems are classiﬁed roughly into three categories: the vertex enumeration approach (Bialas and Karwan, 1984), the Kuhn-Tucker approach (Bard and Falk, 1982; Bard and Moore, 1990; Bialas and Karwan, 1984; Hansen, Jaumard and Savard, 1992), and the penalty function approach (White and Anandalingam, 1993). The subsequent works on two-level programming problems under noncooperative behavior of the DMs have been appearing (Nishizaki and Sakawa, 1999, 2000; G¨um¨us¸ and Floudas, 2001; Nishizaki, Sakawa and Katagiri, 2003; Colson, Marcotte and Savard, 2005; Faisca et al., 2007) including some applications: aluminum production process (Nicholls, 1996), pollution control policy determination (Amouzegar and Moshirvaziri, 1999), tax credits determination for biofuel producers (Dempe and BardZ, 2001), pricing in competitive electricity markets (Fampa

5.2 Noncooperative two-level programming

201

et al., 2008), supply chain planning (Roghanian, Sadjadi and Aryanezhad, 2007) and so forth. Fuzzy random programming approaches to two-level programming problems were also considered together with the computational methods for obtaining the corresponding Stackelberg solutions (Sakawa and Kato, 2009c; Sakawa and Katagiri, 2010b; Sakawa, Katagiri and Matsui, 2010b) In this section, through the four models as discussed in the previous section, we consider noncooperative two-level programming problems with random variables and fuzzy random variables. Under some assumptions, it is shown that the formulated stochastic and fuzzy random two-level programming problems are transformed into deterministic ones. Computational methods for obtaining Stackelberg solutions for the presented models are developed, and some numerical examples are provided to demonstrate the feasibility and efﬁciency of the computational methods.

5.2.1 Stochastic two-level linear programming In order to take into account the randomness of the parameters in hierarchical decision making problems under noncooperative environments, we consider the following two-level linear programming problems where coefﬁcients in objective functions and the right-hand side of constraints are represented by random variables: ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.76)

where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors, and b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. By replacing the chance constrained conditions (Charnes and Cooper, 1959) with the original constraints, (5.76) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.77)

202

5 Stochastic and Fuzzy Random Two-Level Programming

where η = (η1 , . . . , ηm )T is a vector of the satisﬁcing probability levels for holding η) = (b1 (η1 ), . . . , bm (ηm ))T is deﬁned by (5.3). the constraints, and b(η

5.2.1.1 Expectation model Assuming that the DMs intend to simply minimize the expectation of their own objective functions, the stochastic two-level linear programming problem can be reformulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ E minimize z2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.78) x2 ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where E denotes the expectation operator. Observing that E[¯cl1 ] and E[¯cl2 ], l = 1, 2 are constant vectors, it is evident that this problem is an ordinary noncooperative two-level linear programming problem. Thus, the computational techniques for obtaining Stackelberg solutions given in Chapter 2 are directly applicable.

5.2.1.2 Variance model In order to take account of each DM’s concern about the ﬂuctuation of the realized objective function values, to stochastic two-level linear programming problems, we consider the following variance minimization model with the constraints for the expectations of the objective functions: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ V T T T T T minimize z2 (x1 , x2 ) = (x1 , x2 )V2 (x1 , x2 ) ⎪ ⎬ x2

η) subject to A1 x1 + A2 x2 ≤ b(η E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 x1 ≥ 0, x2 ≥ 0,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5.79)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of Gaussian random variable vectors (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are the permissible expectation levels speciﬁed by DM1 and DM2, respectively. For a decision xˆ 1 by DM1, a rational response of DM2 is an optimal solution x∗2 of the quadratic programming problem

5.2 Noncooperative two-level programming

⎫ minimize zV2 (ˆx1 , x2 ) = (ˆxT1 , xT2 )V2 (ˆxT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]ˆx1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ E[¯c21 ]ˆx1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ x2 ≥ 0.

203

(5.80)

Observing that the objective function of (5.80) is strictly convex from the assumption that V2 is positive-deﬁnite, it is evident that (5.80) can be solved by a conventional convex programming techniques such as the sequential quadratic programming method. Although a rational response is not always unique generally, the optimal solution x∗2 to (5.80) is unique due to the strictly convexity property of the objective function, and it follows that, for a given x1 , the set of rational responses denoted by R(x1 ) is a singleton. Then, the Stackelberg solution for the variance model is an optimal solution (x1 , x2 ) to the following problem: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎬ subject to A 1 x1 + A 2 x2 ≤ b

(5.81) ⎭ x1 ≥ 0, x2 ∈ R(x1 ), where, for the sake of simplicity, we use coefﬁcient matrices A 1 , A 2 and a coefﬁcient η), γ1 and γ2 in (5.80). vector b instead of E[¯c11 ], E[¯c12 ], E[¯c21 ], E[¯c22 ], b(η Realizing that the constraint x2 ∈ R(x1 ) in (5.81) can be replaced by the KuhnTucker conditions for (5.80), for a DM1’s decision x∗1 , we introduce the Lagrange function deﬁned as T ∗T T T

∗

L(x2 , λ , ζ; x∗1 ) = (x∗T 1 , x2 )V2 (x1 , x2 ) + λ (A1 x1 + A2 x2 − b ) − ζx2 ,

where λ and ζ are Lagrange multiplier vectors. Then, the Kuhn-Tucker conditions are given as ⎫ n1 n1 +n2 ⎪ ∗

⎪ 2 ∑ v2(n1 +i) j x1 j + 2 ∑ v2(n1 +i) j x2, j−n1 + λ A2,·i − ζi = 0, i = 1, . . . , n2 ⎪ ⎪ ⎪ ⎬ j=1 j=n1 +1

∗

A1 x1 + A2 x2 − b ≤ 0 ⎪ ⎪ ⎪ λ (A 1 x∗1 + A 2 x2 − b ) = 0, ζx2 = 0 ⎪ ⎪ ⎭ x2 ≥ 0, λ ≥ 0, ζ ≥ 0, (5.82) where v2i j is the i j-element of V2 , and A 2,·i is an ith column vector of A 2 . Substituting the Kuhn-Tucker conditions (5.82) for the constraint x2 ∈ R(x1 ) in (5.81), (5.81) can be transformed into the following single-level quadratic programming problem with linear complementarity constraints:

204

5 Stochastic and Fuzzy Random Two-Level Programming

minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T n1

n1 +n2

j=1

j=n1 +1

subject to 2 ∑ v2(n1 +i) j x1 j + 2

∑

v2(n1 +i) j x2, j−n1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

λA 2,·i − ζi = 0, i = 1, . . . , n2 ⎪ +λ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0, ζx2 = 0 ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, λ ≥ 0, ζ ≥ 0.

(5.83)

A 1 x1 + A 2 x2 − b ≤ 0 λ (A 1 x1 + A 2 x2 − b ) =

Although (5.83) cannot be solved by a conventional convex programming technique due to the nonconvexity of the constraints, it is signiﬁcant to realize that if the linear complementarity constraints are eliminated, this problem can be viewed as an ordinary quadratic programming problem. With this observation in mind, we solve (5.83) by using the branching technique with respect to the linear complementarity constraints like the Bard-Moore method (Bard and Moore, 1990). Namely, after branching with respect to the linear complementarity constraints, by solving subproblems generated by the branching operation using a conventional convex programming technique, we can obtain a Stackelberg solution for the variance model, i.e., an optimal solution to the deterministic problem (5.83). To improve the readability, in the description of the following algorithm for solving (5.79), let m+n +2 ∑i=1 2 ui gi = 0 denote the linear complementarity constraint in (5.83). Now we are ready to summarize the algorithm for obtaining a Stackelberg solution for the variance model. Algorithm for obtaining a Stackelberg solution for the variance model / Sk− := 0, / Sk0 := W and V c := ∞. Step 1: Set k := 0, Sk+ := 0, + Step 2: Let ui = 0 for Sk and gi = 0 for Sk− . Solve (5.83) without the linear complementarity constraint. If the problem is infeasible, then go to step 6. Otherwise, set k := k + 1 and let (xk1 , xk2 , λ k , ζk ) denote the obtained solution. Step 3: If zV1 (xk1 , xk2 ) ≥ V c , then go to step 6. Step 4: If the linear complementarity constraint is satisﬁed, i.e., ui gi = 0, i = 1, . . . , m + n2 + 2, then go to step 5. Otherwise, ﬁnd i∗ such that the amount of violation of linear complementarity constraint ui∗ gi∗ is the largest, and let Sk+ := Sk+ ∪ i∗ , Sk0 := Sk0 \i∗ , append i∗ to Pk , and return to step 2. Step 5: Set V c := zV1 (xk1 , xk2 ). Step 6: If there exists no unexplored node, then go to step 7. Otherwise, branch to the newest unexplored node, and update Sk+ , Sk− , Sk0 and Pk . Return to step 2. Step 7: Stop the algorithm: if V c = ∞, there is no feasible solution; otherwise, the solution corresponding to the current V c is a Stackelberg solution for the variance model.

5.2.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented variance and expectation models, consider the following numerical examples to stochastic two-level

5.2 Noncooperative two-level programming

205

linear programming problems: minimize z1 (x1 , x2 ) = c¯11 x1 + c¯12 x2 x1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

where x2 solves minimize z2 (x1 , x2 ) = c¯21 x1 + c¯22 x2 x2 ⎪ subject to −x1 + 3x2 ≤ b¯ 1 , 10x1 − x2 ≤ b¯ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 3x1 + x2 ≥ b¯ 3 , x1 + 2x2 ≥ b¯ 4 ⎪ ⎭ 3x1 + 2x2 ≥ b¯ 5 , x1 ≥ 0, x2 ≥ 0,

(5.84)

where the means of random variables are shown in Table 5.10, and the variancecovariance matrices of c¯ 1 = (c¯11 , c¯12 ) and c¯ 2 = (c¯21 , c¯22 ) are given as

2 1 1 −1 V1 = , V2 = . 1 3 −1 6 Assume that all the right-hand side constant of the constraints are represented by normal random variables, and the means and the variances of the random variables together with the satisﬁcing probability levels for the stochastic constraints are given in Table 5.11. Table 5.10 Means of random variable coefﬁcients in the objective functions. coefﬁcient mean

c¯11 −2.0

c¯12 −3.0

c¯21 2.0

c¯22 1.0

Table 5.11 Random variable coefﬁcients in the constraints and satisﬁcing probability levels. coefﬁcient mean variance probability

b¯ 1 50.11 9.0 0.85

b¯ 2 113.15 36.0 0.70

b¯ 3 15.16 9.0 0.90

b¯ 4 13.16 4.0 0.70

b¯ 5 25.63 16.0 0.80

Then, the problem formulation in the variance model is given as minimize 2x12 + 2x1 x2 + 3x22 x1

where x2 solves minimize x12 − 2x1 x2 + 6x22 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

subject to −x1 + 3x2 ≤ 47, 10x1 − x2 ≤ 110 ⎪ ⎪ ⎪ −3x1 − x2 ≤ −19, −x1 − 2x2 ≤ −15 ⎪ ⎪ ⎪ ⎪ −3x1 − 2x2 ≤ −29, −2x1 − 3x2 ≤ −31 ⎪ ⎪ ⎭ 2x1 + x2 ≤ 33, x1 ≥ 0, x2 ≥ 0.

(5.85)

Considering the Kuhn-Tucker conditions for the constraint that x2 is a rational response to x1 , (5.85) is transformed into the following single-level quadratic pro-

206

5 Stochastic and Fuzzy Random Two-Level Programming

gramming problem with linear complementarity constraints: ⎫ minimize 2x12 + 2x1 x2 + 3x22 ⎪ ⎪ ⎪ subject to 2x1 + 12x2 + 3λ1 − λ2 − λ3 − 2λ4 − 2λ5 − 3λ6 + λ7 − ζ = 0 ⎪ ⎪ ⎪ ⎪ ⎪ −x1 + 3x2 ≤ 47, 10x1 − x2 ≤ 110 ⎪ ⎪ ⎪ ⎪ −3x1 − x2 ≤ −19, −x1 − 2x2 ≤ −15 ⎪ ⎪ ⎪ ⎪ −3x1 − 2x2 ≤ −29, 2x1 + x2 ≤ 33 ⎪ ⎪ ⎬ −2x1 − 3x2 ≤ −31 (5.86) λ1 (−x1 + 3x2 − 47) = 0, λ2 (10x1 − x2 − 110) = 0 ⎪ ⎪ ⎪ ⎪ λ3 (−3x1 − x2 + 19) = 0, λ4 (−x1 − 2x2 + 15) = 0 ⎪ ⎪ ⎪ ⎪ λ5 (−3x1 − 2x2 + 29) = 0, λ6 (−2x1 − 3x2 + 31) = 0 ⎪ ⎪ ⎪ ⎪ λ7 (2x1 + x2 − 33) = 0, ζx2 = 0 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0, λi ≥ 0, i = 1, . . . , 7 ⎪ ⎪ ⎭ ζ ≥ 0. Fig. 5.1 illustrates the feasible region of (5.85) and the Stackelberg solutions for the variance and expectation models. The Stackelberg solution to (5.85) is denoted by the white circle at the point (5.17, 6.89). In order to illustrate the effect of considering the expectation of the objective function values, by solving (5.85) without the constraint for the expectation, the corresponding Stackelberg solution is denoted by the black circle at the point (7, 4). The black square at the point (1, 16) denotes the Stackelberg solution corresponding to the expectation model. Table 5.12 shows the means and the variances of the objective functions of DM1 and DM2. For simplicity, we denote the expectation and variance models by E-model and V-model, respectively. In particular, we denote the variance model without the constraint for the expectation by V-model without expectation. Table 5.12 Stackelberg solutions of the expectation and variance models. Model

Solution

V-model V-model without expectation E-model

(5.17, 6.89)

Mean of DM1 −31

Mean of DM2 17

Variance of DM1 266.94

Variance of DM2 240.25

(7, 4)

−26

18

202

89

(1, 16)

−50

18

802

1505

From Table 5.12, it is observed that the mean of the DM1’s objective function in the expectation model is smaller than that of the variance model, while the variance of the objective function of each DM in the expectation model is considerably large. In particular, although the mean of the DM2’s objective function in the variance model is almost the same as that in the expectation model, not only the variance of the objective function of DM2 but also that of DM1 are substantially reduced. It is found that the variance model yields the Stackelberg solution in which the variance is reduced adequately with the suitable levels of the mean speciﬁed by DM1.

5.2 Noncooperative two-level programming

207

Fig. 5.1 Feasible region and the Stackelberg solutions.

5.2.2 Fuzzy random two-level linear programming 5.2.2.1 Possibility-based probability model In previous subsections, uncertain parameters or coefﬁcients involved in formulated two-level linear programming problems are assumed to be random variables. However, it would be signiﬁcant to realize that we are faced with decision making situations where possible realized values of the random parameters are often only ambiguously known to the experts. In order to deal with such a hierarchical decision making problem, we formulate the following noncooperative two-level linear programming problem with fuzzy random variable coefﬁcients: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ ˜ ˜ ¯ ¯ (5.87) minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 ⎪ x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2.

208

5 Stochastic and Fuzzy Random Two-Level Programming

˜¯ = (C ˜¯ , C ˜¯ ) For simplicity, let X denote the feasible region of (5.87), and let C l l1 l2 T T T and x = (x1 , x2 ) . As in the possibility-based probability model for cooperative decision making situations described in 5.1.2.1, assume that elements C˜¯l jk , ˜¯ , l = 1, 2, j = 1, 2 are fuzzy random variables k = 1, 2, . . . , n j of coefﬁcient vectors C lj whose realized value for an elementary event ω is characterized by the membership ˜¯ x + C ˜¯ x is ˜¯ x = C function (5.18). Then, it follows that each objective function C l l1 1 l2 2 represented by a single fuzzy random variable whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x ⎪ ⎨L βl x μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ R if υ > dl (ω)x, ⎩ γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2, and an where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β element d¯l j of d¯ l is a random variable expressed as d¯l j = dl1j + t¯l dl2j . Considering the imprecise natures of DMs’ judgments, we introduce a fuzzy goal G˜ l such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value” which is assumed to be quantiﬁed by a nonincreasing membership function μG˜ l . Furthermore, recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.88) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. y

It should be noted here that the degree ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of each elementary event ω. With this observation in mind, along the same line as the possibility-based probability model for cooperative decision making situations in 5.1.2.1, assuming that the DMs intend to maximize the probability that ΠC˜ l (ω)x (G˜ l ) is greater than or equal to the speciﬁed target value hl , (5.87) is reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ (5.89) ˜ maximize P ω ΠC˜ 2 (ω)x (G2 ) ≥ h2 ⎪ x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Assuming that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, the objective functions in (5.89) can be equivalently transformed as

5.2 Noncooperative two-level programming

P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl

209

"

βl − dl1 )x + μG˜ (hl ) (L (hl )β l

dl2 x

# ,

where Tl is a probability distribution function of t¯l , and L and μG˜ are pseudo inverse l functions of L and μG˜ l , respectively. Furthermore, since the distribution functions Tl , l = 1, 2 are nondecreasing, (5.89) is equivalently transformed into ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β Π,P ⎪ 1 ⎪ maximize Z1 (x1 , x2 ) = ⎪ ⎪ x1 d12 x ⎪ ⎪ ⎪ ⎪ where x2 solves ⎬ 1 β (5.90) (h )β − d )x + μ (h ) (L 2 2 2 G˜ 2 2 ⎪ ⎪ maximize Z2Π,P (x1 , x2 ) = ⎪ ⎪ x2 d22 x ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. For any feasible decision xˆ 1 of DM1, a rational response of DM2 is given as an optimal solution to the single-level programming problem 1 )x + (L (h )β 1 x + μ (h ) ⎫ β22 − d22 (L (h2 )β 2 2 β 21 − d21 )ˆ 1 G˜ 2 2 ⎪ ⎪ ⎬ 2 2 x2 d22 x2 + d21 xˆ 1 ⎪ subject to A2 x2 ≤ b − A1 xˆ 1 ⎪ ⎭ x2 ≥ 0.

maximize

(5.91)

Let R(ˆx1 ) denote a set of rational responses of DM2 for a decision xˆ 1 by DM1 and the inducible region is deﬁned by IR = {(x1 , x2 ) | (x1 , x2 ) ∈ X, x2 ∈ R(x1 )}. Since DM1 should select a solution (x1 , x2 ) to maximize Z1Π,P (x1 , x2 ) from among the inducible region IR, the Stackelberg solution is obtained by solving ⎫ ⎪ maximize Z1Π,P (x1 , x2 ) ⎬ x1 (5.92) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ∈ R(x1 ). In contrast to the variance model discussed in the previous subsection, through the combined use of the variable transformation (Charnes and Cooper, 1962) and the kth best method for two-level linear programming problems (Bialas and Karwan, 1984) we give a computational method for the Stackelberg solution. As the ﬁrst step for obtaining the Stackelberg solution, by removing the objective function of DM2 from (5.90), we consider the single-level programming problem ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β ⎪ 1 ⎬ maximize x1 ,x2 d12 x ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Z1Π,P (x) =

(5.93)

210

5 Stochastic and Fuzzy Random Two-Level Programming

From the fact that the objective function in (5.93) is a linear fractional function with the positive denominator, by employing the variable transformation (Charnes and Cooper, 1962): 1 1 = 2 2 d12 x d11 x1 + d12 x2 y1 = tx1 , y2 = tx2 ,

t=

(5.93) can be equivalently transformed into the linear programming problem ⎫ 1 1 β11 − d11 β12 − d12 )y1 + (L (h1 )β )y2 + μG˜ (h1 )t ⎪ maximize (L (h1 )β ⎪ 1 ⎪ y1 , y2 , t ⎪ ⎬ subject to A1 y1 + A2 y2 − bt ≤ 0 (5.94) 2 y + d2 y = 1 ⎪ d11 ⎪ 1 12 2 ⎪ ⎪ ⎭ y1 ≥ 0, y2 ≥ 0, t ≥ 0. Let (yT1[1] , yT2[1] ,t[1] )T be an optimal solution to (5.94). Then, T (x1[1] , xT2[1] )T of the feasible region of (5.93) is expressed as

an extreme point

(xT1[1] , xT2[1] )T = (yT1[1] /t[1] , yT2[1] /t[1] )T . In order to check whether the extreme point (xT1[i] , xT2[i] )T is in the inducible region IR, i.e., x2[i] is a rational response to x1[i] or not, we solve the linear fractional programming problem ⎫ 1 )x + (L (h )β 1 β22 − d22 (L (h2 )β 2 2 β 21 − d21 )x1[i] + μG˜ (h2 ) ⎪ ⎪ 2 ⎪ ⎬ maximize 2 2 x2 d22 x2 + d21 x1[i] (5.95) ⎪ subject to A2 x2 ≤ b − A1 x1[i] ⎪ ⎪ ⎭ x2 ≥ 0. Using a similar variable transformation u=

1 2 2 d22 x2 + d21 x1[i]

w2 = ux2 , (5.95) can be equivalently written as the linear programming problem ⎫ 1 1 β22 − d22 β21 − d21 maximize (L (h2 )β )w2 +[(L (h2 )β )x1[i] + μG˜ (h2 )]u ⎪ ⎪ 2 ⎪ w2 , u ⎪ ⎬ subject to A2 w2 − (b − A1 x1[i] )u ≤ 0 (5.96) 2 2 ⎪ w2 + d21 x1[i] u = 1 d22 ⎪ ⎪ ⎪ ⎭ w2 ≥ 0, u ≥ 0.

5.2 Noncooperative two-level programming

211

Let (wT2[i] , u[i] )T be an optimal solution of (5.96). If w2[i] /u[i] = x2[i] holds, then the current extreme point (xT1[i] , xT2[i] )T exists in IR, and it follows that (xT1[i] , xT2[i] )T is a Stackelberg solution. Otherwise, we use a set W[i] of feasible extreme points which are adjacent to (xT1[i] , xT2[i] )T and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[i] , x2[i] ). After setting U := U ∪ {(xT1[i] , xT2[i] )T } and W := (W ∪W[i] )\U, let a feasible extreme point

(xT1[i] , xT2[i] )T in W such that Z1Π,P (x1 , x2 ) is the greatest be the next extreme point (xT1[i+1] , xT2[i+1] )T . By repeating this procedure, we can obtain a Stackelberg solution. Now we summarize the algorithm for obtaining a Stackelberg solution for the possibility-based probability model. Algorithm for obtaining a Stackelberg solution for the possibility-based probability model

Step 1: Set i := 1, and solve (5.94). By using the optimal solution (yT1[1] , yT2[1] ,t[1] )T to (5.94), generate the initial extreme point (xT1[1] , xT2[1] )T := (yT1[1] /t[1] , yT2[1] /t[1] )T . / Let W := {(xT1[1] , xT2[1] )T } and U := 0. Step 2: For the current extreme point (xT1[i] , xT2[i] )T , solve the corresponding linear programming problem (5.96). For the obtained optimal solution (wT2[i] , u[i] )T , if w2[i] /u[i] = x2[i] , then stop; then, (xT1[i] , xT2[i] )T is a Stackelberg solution. Otherwise, go to step 3. Step 3: Let W[i] be a set of feasible extreme points which are adjacent to (xT1[i] , xT2[i] )T and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[i] , x2[i] ). Set U := U ∪ {(xT1[i] , xT2[i] )T } and W := (W ∪W[i] )\U. Step 4: Set i := i + 1. Choose an extreme point (xT1[i] , xT2[i] )T such that Z1Π,P (x1[i] , x2[i] ) = max(xT ,xT )T ∈W Z1Π,P (x1 , x2 ), and return to step 2. 1

2

5.2.2.2 Level set-based fractile model Along the same line as the level set-based fractile model for cooperative decision making situations discussed in 5.1.2.2, we consider the level-set based fractile model for noncooperative fuzzy random two-level linear programming problems and present an algorithm for obtaining the Stackelberg solutions. If DM1 supposes that the degree of all of the membership functions of the fuzzy random variables involved in (5.87) should be greater than or equal to some value α, the original fuzzy random two-level linear programming problem (5.87) is reformulated as

212

5 Stochastic and Fuzzy Random Two-Level Programming

minimize C¯ 1 x = C¯ 11 x1 + C¯ 12 x2 x1 , C¯ 1

where x2 solves minimize C¯ 2 x = C¯ 21 x1 + C¯ 22 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

x2 , C¯ 2

⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0 ⎪ ⎪ ⎭ C¯ 1 = (C¯ 11 , C¯ 12 ) ∈ C¯ 1α , C¯ 2 = (C¯ 21 , C¯ 22 ) ∈ C¯ 2α ,

(5.97)

where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . To take into account the imprecise nature of the DMs’ judgments, we assume that DMs have fuzzy goals for their objective functions in the α-stochastic two-level linear programming problem (5.97), and the fuzzy goals are quantiﬁed by the nonincreasing and continuous membership functions μl , l = 1, 2. Then, the satisfaction degrees μl (C¯ l (ω)x), l = 1, 2 may vary randomly due to the stochastic occurrence of an elementary event ω. With this observation in mind, assuming that the DMs intend to maximize the target values for the satisfaction degrees under the probabilistic constraint with permissible probability levels, we consider the following level set-based fractile model for noncooperative two-level fuzzy random linear programming problem: ⎫ maximize h1 ⎪ ⎪ ¯ ⎪ x1 , h1 , C1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ ⎪ x2 , h2 , C¯ 2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.98) ⎪ ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0 ⎪ ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values and θl , l = 1, 2 are the permissible probability levels speciﬁed by DMl, l = 1, 2. From the discussion in 4.1.3.4 of Chapter 4, one ﬁnds that L (ω)x ≤ μl (hl ) ≥ θl , P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα and, from the assumption that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X and the property of the nonincreasing function μl and the nondecreasing function Tl , one ﬁnds that βl − dl1 )x + μl (hl ) (L (α)β L ¯ ≥ θl P ω | Clα x ≤ μl (hl ) ≥ θl ⇔ Tl dl2 x ⇔

βl − dl1 )x + μl (hl ) (L (α)β ≥ Tl (θl ) , dl2 x

5.2 Noncooperative two-level programming

213

where μl and Tl are pseudo inverse functions of μl and Tl , respectively. Furthermore, assuming that the fuzzy goals are quantiﬁed by the linear membership functions deﬁned as ⎧ ⎪ 1 if zl < z1l ⎪ ⎪ ⎨ z − z0 l l (5.99) μl (zl ) = if z1l ≤ zl ≤ z0l 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if zl > z0l , (5.98) can be rewritten as maximize h1 x1 , h1

where x2 solves maximize h2 x2 , h2

subject to

β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β ≥ h1 z01 − z11

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ β2 − d21 − T2 (θ2 )d22 )x + z02 (L (α)β ⎪ ⎪ ≥ h 2⎪ ⎪ 0 1 ⎪ z2 − z2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.100)

Because maximizing h1 is equivalent to maximizing β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β , z01 − z11 (5.100) can be equivalently transformed into ⎫ F β1 − d11 − T1 (θ1 )d12 x ⎪ maximize Z1α (x1 , x2 ) = L (α)β ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ F 1 2 β2 − d2 − T2 (θ2 )d2 x maximize Z2α (x1 , x2 ) = L (α)β x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.101)

(5.102)

It should be noted that (5.102) is an ordinary two-level linear programming problem, and therefore it can be solved by the solution technique for obtaining a Stackelberg solution given in Chapter 2.

5.2.2.3 Numerical example In order to demonstrate the feasibility and efﬁciency of the presented computational method for obtaining a Stackelberg solution for the possibility-based probability

214

5 Stochastic and Fuzzy Random Two-Level Programming

model, consider the following numerical example to fuzzy random two-level linear programming problems: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜ ˜ ¯ 21 x1 + C ¯ 22 x2 minimize z2 (x1 , x2 ) = C (5.103) x2 ⎪ ⎪ subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 5 ⎪ ⎪ ⎪ ⎪ ⎪ x1 = (x11 , x12 , x13 )T ≥ 0 ⎪ ⎭ T x2 = (x21 , x22 , x23 ) ≥ 0, ˜¯ , l = 1, 2, j = 1, 2 are vectors of fuzzy random variables C¯˜ , k = where C lj l jk ˜¯ is rep1, 2, . . . , n j which are characterized by (5.18). The parameters d¯l jk of C lj resented by d¯l jk = dl1jk + t¯l dl2jk , and t¯l is a Gaussian random variable with the mean 0 and the variance 12 . Values of coefﬁcients in constraints are shown in Table 5.13, and values of the parameters d¯ l1 , d¯ l2 , βl and γl , l = 1, 2 of the fuzzy random variables are shown in Table 5.14. Table 5.13 Values of coefﬁcients in constraints. x11 x12 x13 a1 4 3 1 a2 2 4 5 a3 3 2 3 0 0 0 a4 a5 1 1 1

x21 3 3 3 1 0

x22 2 4 3 1 0

x23 5 2 2 1 0

b 100 115 90 17 22

Table 5.14 Values of dl1 , dl2 , β l and γ l .

d11 d21 d12 d22

β1 β2 γ1 γ2

x11

x12

x13

x21

x22

x23

−1.7 −1.8 0.6 0.7 0.3 0.4 0.3 0.4

−2.4 −1.6 0.1 0.2 0.2 0.2 0.2 0.2

−2.9 −1.2 0.3 0.5 0.6 0.3 0.6 0.3

−1.5 −2.0 0.4 0.1 0.2 0.5 0.2 0.5

−1.0 −2.4 0.7 0.5 0.7 0.2 0.7 0.2

−1.3 −2.8 0.2 0.3 0.4 0.8 0.4 0.8

By solving the linear programming problem (5.94), the initial extreme point (xT1[1] , xT2[1] )T = (0.00, 22.0, 0.00, 0.00, 6.80)T is obtained. / and solve the linear programming problem Let W := {(xT1[1] , xT2[1] )T } and U := 0, (5.96) in order to obtain the rational response for x1[1] . Since the optimal solution w˜ 2[1] /u[1] = (9.00, 0.00, 0.00)T to (5.96) is not equal to x2[1] = (0.00, 0.00, 6.80)T , the current extreme point (xT1[1] , xT2[1] )T is not a Stackelberg solution. Then, we

5.2 Noncooperative two-level programming

215

enumerate feasible extreme points (xT1 , xT2 )T which are adjacent to (xT1[1] , xT2[1] )T

and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[1] , x2[1] ), and make W[1] . After setting U := U ∪(xT1[1] , xT2[1] )T and W := (W ∪W[1] )\U, we ﬁnd a feasible extreme point (xT1 , xT2 )T in W such that Z1Π,P (x1 , x2 ) is the greatest, and let the point be the next extreme point (xT1[i+1] , xT2[i+1] )T to be examined. By repeating this procedure, we ﬁnally obtain the following Stackelberg solution: (xT1 , xT2 )T = (0.00, 14.56, 7.44, 0.00, 0.00, 9.78)T .

5.2.3 Extensions to integer programming Considering that the real-world decision making problems are often formulated as integer programming problems, as natural extensions of noncooperative stochastic two-level programming problems with continuous variables discussed in the previous section, we present two-level integer programming problems involving random variable coefﬁcients in objective functions and the right-hand side of constraints formulated as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 (5.104) ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors with the mean vectors c¯ l j and the variance-covariance matrices. b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. By introducing the chance constrained conditions (Charnes and Cooper, 1959) instead of the original constraints, (5.104) can be rewritten as minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 x1

where x2 solves minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ,

(5.105)

216

5 Stochastic and Fuzzy Random Two-Level Programming

where η = (η1 , . . . , ηm )T is a vector of the satisﬁcing probability levels for holding η) = (b1 (η1 ), . . . , bm (ηm ))T is deﬁned by (5.3). the constraints, and b(η

5.2.3.1 Expectation model for stochastic two-level integer programming problems Assuming that the DMs intend to simply minimize the expectation of their own objective functions, the stochastic two-level integer programming problem (5.104) can be reformulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ E minimize z2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.106) x2 ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where E denotes the expectation operator. For a decision xˆ 1 by DM1, the rational response of DM2 is obtained by an optimal solution x2 of the following problem: ⎫ ⎪ minimize zE1 (ˆx1 , x2 ) = E[¯c11 ]ˆx1 + E[¯c12 ]x2 ⎬ x2 (5.107) η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . Since (5.107) is a single-objective integer programming problem with the linear objective function and constraints, we employ GADSLPRRSU to solve it. Let R(x1 ) be a set of rational responses of DM2, i.e., the set of optimal solutions x∗2 to (5.107). Then, the Stackelberg solution for the expectation model is an optimal solution (x∗1 , x∗2 ) to the following problem: ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎬ η) subject to A1 x1 + A2 x2 ≤ b(η (5.108) ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ). In order to obtain a Stackelberg solution, x1 must be selected so as to minimize zE1 (x1 , x2 ) under the condition that x2 is selected from among R(x1 ). With this observation in mind, the Stackelberg solution can be obtained by nesting use of the two genetic algorithms: one is a genetic algorithm for ﬁnding a rational response x2 of DM2 for a given decision x1 of DM1, and the other is a genetic algorithm for ﬁnding a decision x1 of DM1 so as to minimize zE1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ).

5.2 Noncooperative two-level programming

217

5.2.3.2 Variance model for stochastic two-level integer programming problems In order to consider each DM’s concern about the ﬂuctuation of the realized objective function values, from the viewpoint of leveling of the minimized objective function values, by introducing the constraints that the expectation of each objective function must be less than or equal to a certain value, we consider the variance minimization model for noncooperative stochastic two-level integer programming problems formulated as minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T x1

where x2 solves minimize zV2 (x1 , x2 ) = (xT1 , xT2 )V2 (xT1 , xT2 )T x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ,

(5.109)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are the permissible expectation levels speciﬁed by DM1 and DM2, respectively. For a decision xˆ 1 by DM1, the rational response of DM2 is obtained by an optimal solution x2 of the following problem: ⎫ minimize zV2 (ˆx1 , x2 ) = (ˆxT1 , xT2 )V2 (ˆxT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]ˆx1 + E[¯c12 ]x2 ≤ γ1 (5.110) ⎪ ⎪ E[¯c21 ]ˆx1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . To solve (5.110) with the quadratic objective function and linear constraints, we use the revised GADSLPRRSU. Let R(x1 ) be a set of rational responses of DM2, i.e., the set of optimal solutions x∗2 to (5.110). Then, the Stackelberg solution for the variance model is an optimal solution (x∗1 , x∗2 ) to the following problem: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ).

(5.111)

Similarly to solving (5.108) in the expectation model, to ﬁnd a Stackelberg solution, we nest two genetic algorithms: one searches a rational response x2 of DM2 for

218

5 Stochastic and Fuzzy Random Two-Level Programming

a given decision x1 of DM1, and the other ﬁnds a decision x1 of DM1 minimizing zV1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ). 5.2.3.3 Possibility-based probability model for fuzzy random two-level integer programming problems In previous subsections, it is assumed that uncertain parameters or coefﬁcients involved in the formulated two-level programming problems are given as random variables. However, it would be signiﬁcant to realize that the possible realized values of these random parameters are often only ambiguously known to the experts. For handling such a hierarchical decision making problem, we deal with two-level integer programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜¯ x + C ˜¯ x minimize z2 (x1 , x2 ) = C 21 1 22 2 (5.112) x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2. ˜¯ = For simplicity, let X int denote the feasible region of (5.112), and let C l ˜¯ ˜¯ , C T T T (C l1 l2 ) and x = (x1 , x2 ) . As we examined in the possibility-based probability model for two-level programming problems with continuous decision variables in 5.1.2.1, assume that el˜¯ , l = 1, 2, j = 1, 2 are fuzzy ements C˜¯l jk , k = 1, 2, . . . , n j of coefﬁcient vectors C lj random variables whose realized value for each elementary event ω is characterized by the membership function (5.18). Then, it follows that each objective function ˜¯ x = C ˜¯ x + C ˜¯ x is represented by a single fuzzy random variable whose realC l l1 1 l2 2 ized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ L if υ ≤ dl (ω)x ⎪ ⎨ βl x μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β

5.2 Noncooperative two-level programming

219

Considering the imprecise nature of DMs’ judgments, we introduce a fuzzy goal G˜ l such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value” which is quantiﬁed by a membership function μG˜ l . Furthermore, recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.113) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. y

It should be noted here that the degree of possibility ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of an elementary event ω. With this observation in mind, along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs are willing to maximize the probability that ΠC˜ l (ω)x (G˜ l ) is greater than or equal to target value hl , (5.112) is reformulated as maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 x1

where x2 solves maximize P ω ΠC˜ 2 (ω)x (G˜ 2 ) ≥ h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.114)

Assuming that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, where X denotes the feasible region of (5.114), the objective functions in (5.114) can be equivalently transformed as # " βl − dl1 )x + μG˜ (hl ) (L (hl )β l . P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl dl2 x Since the distribution functions Tl , l = 1, 2 are nondecreasing functions, (5.114) is equivalently transformed into ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β Π,P ⎪ 1 ⎪ maximize Z1 (x1 , x2 ) ⎪ ⎪ x1 d12 x ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ 1 β (h )β − d )x + μ (h ) (L 2 2 ˜ 2 2 G Π,P 2 (5.115) maximize Z2 (x1 , x2 ) ⎪ ⎪ x2 d22 x ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . For any feasible decision xˆ 1 of DM1, a rational response is given as an optimal solution to the single-level integer programming problem

220

5 Stochastic and Fuzzy Random Two-Level Programming 1 )x + (L (h )β 1 x + μ (h ) ⎫ β22 − d22 (L (h2 )β 2 2 β 21 − d21 )ˆ 1 G˜ 2 2 ⎪ ⎪ ⎬ maximize 2 2 x2 d22 x2 + d21 xˆ 1 ⎪ subject to A2 x2 ≤ b − A1 xˆ 1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.116)

Let R(ˆx1 ) denote a set of rational responses of DM2 for a decision xˆ 1 by DM1 and IR = {(x1 , x2 ) | (x1 , x2 ) ∈ X, x2 ∈ R(x1 )} the inducible region. Then, since DM1 selects a solution (x1 , x2 ) maximizing Z1Π,P (x1 , x2 ) from among the inducible region IR, the Stackelberg solution can be obtained by solving ⎫ ⎪ maximize Z1Π,P (x1 , x2 ) ⎬ x1 (5.117) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ). To solve (5.117), we employ two genetic algorithms which are nested: one searches a rational response x2 of DM2 for a given decision x1 of DM1, and the other seeks a decision x1 of DM1 maximizing zV1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ). 5.2.3.4 Level set-based fractile model for fuzzy random two-level integer programming problems Along the same line as the level set-based optimization models discussed in Chapter 4, assuming that DM1 decides that the degree of all of the membership functions of the fuzzy random variables involved in (5.112) should be greater than or equal to some value α, the original fuzzy random two-level programming problem (5.112) is reformulated as ⎫ minimize C¯ 1 x = C¯ 11 x1 + C¯ 12 x2 ⎪ ⎪ x1 , C¯ 1 ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ¯ ¯ ¯ ⎪ minimize C2 x = C21 x1 + C22 x2 ⎬ ¯ x2 , C2 (5.118) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎪ ⎪ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ⎪ ⎪ ⎭ C¯ 1 = (C¯ 11 , C¯ 12 ) ∈ C¯ 1α , C¯ 2 = (C¯ 21 , C¯ 22 ) ∈ C¯ 2α , where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C˜¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . In order to consider the imprecise nature of the DMs’ judgments, assuming that each DM has a fuzzy goal for the objective function in the α-stochastic two-level integer programming problem (5.118), we introduce the nonincreasing and continuous membership functions μ1 and μ2 to quantify the fuzzy goals of DM1 and DM2 for their own objective function values, respectively. Then, observing that the satisfaction degrees μl (Cl (ω)x), l = 1, 2 vary randomly due to the stochastic occurrence of

5.2 Noncooperative two-level programming

221

an elementary event ω, it is evident that computational techniques for conventional two-level programming problems cannot be applied. With this observation in mind, assuming that the DMs intend to maximize the target values for the satisfaction degrees under the probabilistic constraint with permissible probability levels, we consider the level set-based fractile model for noncooperative two-level fuzzy random integer programming problem formulated as ⎫ maximize h1 ⎪ ⎪ x1 , h1 , C¯ 1 ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ ¯ ⎪ x2 , h2 , C2 ⎪ ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.119) P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎪ ⎪ ⎪ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ⎪ ⎪ ⎭ ¯ ¯ ¯ ¯ C1 ∈ C1α , C2 ∈ C2α , where hl and θl , l = 1, 2 are decision variables representing the target values and the permissible probability levels speciﬁed by DMl, l = 1, 2, respectively. If we assume that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, where X denotes the feasible region of (5.119), and the property of the nonincreasing function μl and the nondecreasing function Tl , one ﬁnds that together with βl − dl1 )x + μl (hl ) (L (α)β L ≥ Tl (θl ) , P ω | μl (C¯ lα x) ≥ hl ≥ θl ⇔ dl2 x where μl and Tl are pseudo inverse functions of μl and Tl , respectively. Furthermore, assuming that the fuzzy goals are quantiﬁed by linear membership functions deﬁned as ⎧ ⎪ 1 if zl < z1l ⎪ ⎪ ⎨ z − z0 l l (5.120) if z1l ≤ zl ≤ z0l μl (zl ) = 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if zl > z0l , (5.119) can be rewritten as

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5 Stochastic and Fuzzy Random Two-Level Programming

maximize h1 x1 , h1

where x2 solves maximize h2 x2 , h2

subject to

β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β ≥ h1 z01 − z11

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ β2 − d21 − T2 (θ2 )d22 )x + z02 (L (α)β ⎪ ⎪ ≥ h 2⎪ ⎪ 0 1 ⎪ z2 − z2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.121)

Because maximizing h1 is equivalent to maximizing β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β , z01 − z11 (5.121) can be equivalently transformed as ⎫ F (x , x ) = L (α)β β1 − d11 − T1 (θ1 )d12 x ⎪ maximize Z1α 1 2 ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ F 1 2 β2 − d2 − T2 (θ2 )d2 x maximize Z2α (x1 , x2 ) = L (α)β x2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.122)

Observing that (5.122) is the same type of problem as (5.106), a Stackelberg solution of (5.122) can be obtained by using a computational method for (5.106).

5.2.3.5 Numerical example In order to demonstrate the feasibility and efﬁciency of the proposed computational methods for obtaining a Stackelberg solution for the possibility-based probability model, consider the following numerical example to fuzzy random two-level integer programming problems: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜ ˜ ¯ ¯ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 (5.123) x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , 30}, j1 = 1, . . . , 15 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , 30}, j2 = 1, . . . , 15,

5.2 Noncooperative two-level programming

223

˜¯ , l = 1, 2, j = 1, 2 are vectors whose elements C¯˜ , k = 1, 2, . . . , n are where C j lj l jk fuzzy random variables, and the random parameter t¯l of C˜¯l jk is a uniform random variable distributed at the closed interval [−10, 10]. Elements of dl1 , dl2 , β l and γ l , l = 1, 2 are randomly chosen from the closed intervals [−500, 499], [0, 50], [0, 25] and [0, 25], respectively. The values of bi , i = 1, . . . , 26 are determined by ⎞ ⎛ bi =

∑− 30ai j + 5 ⎝ ∑+ ai j − ∑− ai j ⎠ ,

j∈Jai

j∈Jai

j∈Jai

where Ja+i { j | ai j > 0, 1 ≤ j ≤ 30} and Ja−i { j | ai j < 0, 1 ≤ j ≤ 30}. Assume that the DM1 speciﬁes θ1 = θ2 = 0.7 and α = 0.75. Then, by the computational method using the genetic algorithm, the (approximate) Stackelberg solution x∗ for the level set-based fractile model is obtained, and the correspondF (x∗ ) = 0.838054 and ing optimal objective function values are calculated as Z1α F ∗ Z2α (x ) = 0.497792.

Chapter 6

Future Research Directions

This chapter outlines some future research directions. In contrast to the fuzzy random variables treating the ambiguity of realized values of the random parameters discussed in Chapter 4, by considering the ambiguity of means and/or variances characterizing random variables, the concept of random fuzzy variables is introduced to express the coefﬁcients involved in mathematical programming problems under fuzzy stochastic environments. In particular, attention is focused on the case where the mean of each of random variables is represented by a fuzzy number. From the viewpoint of simultaneous maximization of the possibility and the probability under some speciﬁed target value for the objective function, the linear programming problem involving random fuzzy variable coefﬁcients is transformed into a deterministic nonlinear programming problem. Furthermore, assuming that the decision maker intends to optimize the target variable under the permissible possibility level, it is shown that the original random fuzzy linear programming problem is equivalently transformed into a convex programming problem. The presented decision making models are extended to deal with multiobjective and two-level programming problems with random fuzzy variables.

6.1 Random fuzzy variable In the framework of stochastic programming, as discussed in Chapter 3, it is implicitly assumed that the uncertain parameter which well represents the stochastic factor of real systems can be deﬁnitely expressed as a single random variable. However, from the expert’s experimental point of view, the experts may think of a collection of random variables to be appropriate to express stochastic factors rather than only a single random variables. In this case, reﬂecting the expert’s conviction degree that each of random variables properly represents the stochastic factor, it would be quite reasonable to assign the different degrees of possibility to each of random variables. For handling such an uncertain parameter, a random fuzzy variable was deﬁned by Liu (2002) as a function from a possibility space to a collection of random variables, M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_6, © Springer Science+Business Media, LLC 2011

225

226

6 Future Research Directions

which is considered to be an extended concept of fuzzy variable (Nahmias, 1978). It should be noted here that the fuzzy variables can be viewed as another way of dealing with the imprecision which was originally represented by fuzzy sets. Although we can employ Liu’s deﬁnition, considering that we have consistently discussed various concepts in relation to the fuzzy sets in the previous chapters, we deﬁne the random fuzzy variables by extending not the fuzzy variables but the fuzzy sets. Deﬁnition 6.1 (Random fuzzy variable). Let Γ be a collection of random variables. Then, a random fuzzy variable C¯˜ is deﬁned by its membership function μC¯˜ : Γ → [0, 1].

(6.1)

In Deﬁnition 6.1, the membership function μC¯˜ assigns each random variable γ¯ ∈ Γ to a real number μC¯˜ (¯γ). It should be noted here that if Γ is deﬁned as R, then (6.1) becomes equivalent to the membership function of an ordinary fuzzy set discussed in (2.1). In this sense, a random fuzzy variable can be regarded as an extended concept of fuzzy sets. On the other hand, if Γ is deﬁned as a singleton Γ = {¯γ} and μC¯˜ (¯γ) = 1, then the corresponding random fuzzy variable C¯˜ can be viewed as an ordinary random variable. When taking account of the imprecise nature of the realized values of random variables, as discussed in Chapter 4, it would be appropriate to employ the concept of fuzzy random variables. However, it should be emphasized here that if mean and/or variance of random variables are speciﬁed by the expert as a set of real values or fuzzy sets, such uncertain parameters can be represented by not fuzzy random variables but random fuzzy variables. As a simple example of random fuzzy variables, we consider a Gaussian random variable whose mean value is not deﬁnitely speciﬁed as a constant. For example, when some random parameter γ¯ is represented by the Gaussian random variable N(si , 102 ) where the expert identiﬁes a set {s1 , s2 , s3 } of possible mean values as (s1 , s2 , s3 ) = (90, 100, 110), if the membership function μC¯˜ is deﬁned by ⎧ 0.5 ⎪ ⎪ ⎪ ⎨0.7 μC¯˜ (¯γ) = ⎪ 0.3 ⎪ ⎪ ⎩ 0

if γ¯ ∼ N(90, 102 ) if γ¯ ∼ N(100, 102 ) if γ¯ ∼ N(110, 102 ) otherwise,

then C¯˜ is a random fuzzy variable. More generally, when the mean values are expressed as fuzzy sets or fuzzy numbers, the corresponding random variable with the fuzzy mean is represented by a random fuzzy variable, as we will discuss in the following sections.

6.2 Random fuzzy linear programming

227

6.2 Random fuzzy linear programming In Chapter 4, considering the imprecise nature of human judgments for the realized values of random variable coefﬁcients, we have discussed mathematical programming problems with fuzzy random variable coefﬁcients. However, if the mean values of random variables are expressed as fuzzy numbers, it is not hard to imagine that the methodology of fuzzy random programming is not applicable. With this observation in mind, for handling such random variable coefﬁcients with fuzzy mean values involved in objective functions, we consider the random fuzzy version of linear programming problems formulated as ¯˜ minimize Cx (6.2) subject to Ax ≤ b, x ≥ 0, where x is an n-dimensional decision variable column vector, A is an m × n matrix, and b is an m-dimensional column constant vector. Considering that the real data with uncertainty are often distributed normally, from the practical point of view, we assume that each C¯˜ j of the coefﬁcient vector ¯˜ = (C¯˜ , . . . , C¯˜ ) is a Gaussian random variable whose mean value is represented C 1 n by a fuzzy number. To be more explicit, C¯˜ j , j = 1, . . . , n are mutually independent Gaussian random variable N(M˜ j , σ2j ) where the mean value M˜ j is expressed as a fuzzy set characterized by the membership function ⎧ mj −t ⎪ ⎪ if m j ≥ t ⎪ ⎨L αj μM˜ j (t) = (6.3) ⎪ t −mj ⎪ ⎪ ⎩R if m j < t, βj where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, the membership function of C¯˜ j is given as μC¯˜ (¯γ j ) = {μM˜ j (s j ) | γ¯ j ∼ N(s j , σ2j )}, ∀¯γ j ∈ Γ. j

(6.4)

¯˜ can be exThrough the Zadeh’s extension principle, the objective function Cx pressed as a random fuzzy variable with the membership function n (6.5) μCx ¯ = sup min μC¯˜ (¯γ j ) u¯ = ∑ γ¯ j x j , ∀u¯ ∈ Γ, ¯˜ (u) j 1≤ j≤n γ¯ j=1 where γ¯ = (¯γ1 , . . . , γ¯ n ). It should be noted here that if each γ¯ j is a Gaussian random variable and x j , j = 1, . . . , n are ﬁxed as real values, then u¯ = ∑nj=1 γ¯ j x j is also a Gaussian random variable due to its reproductive property.

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6 Future Research Directions

By substituting (6.4) into (6.5), it follows that n 2 μCx ¯ = sup min μM˜ j (s j ) γ¯ j ∼ N(s j , σ j ), u¯ = ∑ γ¯ j x j ¯˜ (u) 1≤ j≤n s j=1 # " n n 2 2 , = sup min μM˜ j (s j ) u¯ ∼ N ∑ s j x j , ∑ σ j x j 1≤ j≤n s j=1 j=1

(6.6)

where s = (s1 , . . . , sn ). ¯˜ is represented by a random fuzzy variable with the membership Realizing that Cx function μCx ¯˜ deﬁned as (6.6), it is evident that conventional fuzzy and stochastic programming approaches, even the fuzzy random programming approach discussed in Chapter 4, cannot be directly applicable.

6.2.1 Possibility-based probability model Assuming that the decision maker (DM) concerns about the probability that the objective function is smaller than or equal to a certain target value, we consider the ˜ probability P ω C(ω)x ≤ f , where P is a probability measure, and f is a target ¯˜ value for the objective function Cx. Through the Zadeh’s extension principle, the ˜ probability P ω C(ω)x ≤ f is expressed by a fuzzy set P˜ with the membership function ¯ | p = P (ω | u(ω) ≤ f ) . μP˜ (p) = sup μCx ¯˜ (u)

(6.7)

u¯

By substituting (6.6) into (6.7), the membership function μP˜ is rewritten as n n 2 2 μP˜ (p) = sup min μM˜ j (s j ) p = P (ω | u(ω) ≤ f ) , u¯ ∼ N ∑ s j x j , ∑ σ j x j . s 1≤ j≤n j=1 j=1 Considering the imprecise nature of the DM’s judgments for the probability with respect to the random fuzzy objective function value, we introduce a fuzzy goal such as “P˜ should be greater than or equal to a certain value,” which is assumed to be quantiﬁed by a nondecreasing continuous membership function μG˜ . Having determined the fuzzy goal of the DM, if we regard μP˜ (p) as a possibility distribution on the basis of the concept of possibility measure, the degree of possibility that the fuzzy goal G˜ is fulﬁlled under the possibility distribution μP˜ (p) is given as ˜ = sup min{μP˜ (p), μG˜ (p)}. (6.8) ΠP˜ (G) p

Then, the original random fuzzy linear programming problem (6.2) is interpreted as the possibility-based probability maximization problem

6.2 Random fuzzy linear programming

229

˜ maximize ΠP˜ (G) subject to Ax ≤ b, x ≥ 0 or equivalently

(6.9)

⎫ maximize h ⎬ ˜ ≥h subject to ΠP˜ (G) ⎭ Ax ≤ b, x ≥ 0.

(6.10)

˜ ≥ h in (6.10) is equivalently replaced by the From (6.8), the constraint ΠP˜ (G) condition that there exists a p such that μP˜ (p) ≥ h and μG˜ (p) ≥ h, namely, n n 2 2 ≥h sup min μM˜ j (s j ) p = P (ω | u(ω) ≤ f ) , u¯ ∼ N ∑ s j x j , ∑ σ j x j s 1≤ j≤n j=1 j=1 (6.11) and p ≥ μG˜ (h), where s = (s1 , . . . , sn ) and μG˜ (h) is a pseudo inverse function deﬁned ¯ such as μG˜ (h) = sup{p | μG˜ (p) ≥ h}. This implies that there exists a vector (p, s, u) that # " min μM˜ j (s j ) ≥ h, u¯ ∼ N

1≤ j≤n

n

n

j=1

j=1

∑ s j x j , ∑ σ2j x2j

, p = P (ω | u(ω) ≤ f ) , p ≥ μG˜ (h),

which can be equivalently transformed into the condition that there exists a vector (s, u) ¯ such that # " μM˜ j (s j ) ≥ h, j = 1, . . . , n, u¯ ∼ N

n

n

j=1

j=1

∑ s j x j , ∑ σ2j x2j

, P (ω | u(ω) ≤ f ) ≥ μG˜ (h). (6.12)

In view of (6.3), it follows that μM˜ j (s j ) ≥ h ⇔ s j ∈ [m j − L (h)α j , m j + R (h)β j ], where L (h) and R (h) are pseudo inverse functions deﬁned as L (h) = sup{t | L(t) ≥ h} and R (h) = sup{t | R(t) ≥ h}. Hence, (6.12) is rewritten as the equivalent condition that there exists a u¯ such that " # P (ω | u(ω) ≤ f ) ≥ μG˜ (h), u¯ ∼ N

n

n

j=1

j=1

∑ {m j − L (h)α j }x j , ∑ σ2j x2j

.

Since P (ω | u(ω) ≤ f ) is transformed into ⎛ n n f − ∑ {m j − L (h)α j }x j ⎜ u(ω) − ∑ {m j − L (h)α j }x j ⎜ j=1 j=1 0 0 ≤ P⎜ ⎜ω n n ⎝ 2 2 σjxj σ2j x2j ∑ ∑ j=1

j=1

(6.13)

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

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6 Future Research Directions

in consideration of n

u¯ − ∑ {m j − L (h)α j }x j j=1

0

∼ N(0, 1),

n

∑ σ2j x2j

j=1

(6.13) is equivalently transformed as ⎛ n

⎞

⎜ f − ∑ {m j − L (h)α j }x j ⎟ ⎟ ⎜ j=1 ⎟ ≥ μ˜ (h), 0 Φ⎜ ⎟ ⎜ G n ⎠ ⎝ 2 2 ∑ σjxj

(6.14)

j=1

where Φ is a probability distribution function of the standard Gaussian random variable N(0, 1). From the monotone increasingness of Φ, (6.14) is rewritten as 0 n

n

j=1

j=1

∑ {m j − L(h)α j }x j + Φ−1 (μG˜ (h)) ∑ σ2j x2j ≤ f ,

(6.15)

where Φ−1 is the inverse function of Φ. From (6.11)−(6.15), it holds that ˜ ≥h⇔ ΠP˜ (G)

0

n

∑ {m j − L (h)α j }x j + Φ

−1

j=1

n

∑ σ2j x2j ≤ f .

(μG˜ (h))

(6.16)

j=1

Consequently, (6.10) is equivalently transformed into maximize h n

subject to

0

∑ {m j − L (h)α j }x j + Φ

j=1

Ax ≤ b, x ≥ 0.

−1

(μG˜ (h))

n

⎫ ⎪ ⎪ ⎪ ⎬

∑ σ2j x2j ≤ f ⎪

j=1

⎪ ⎪ ⎭

(6.17)

Noting that this problem is a nonconvex programming problem, an (approximate) optimal solution of (6.17) is obtained through some nonlinear programming techniques or meta heuristic approaches such as genetic algorithms.

6.2 Random fuzzy linear programming

231

6.2.2 Possibility-based fractile model The possibility-based probability for the DM who prefers model is recommended ˜ to maximize the probability P ω C(ω)x ≤ f for the speciﬁed target value f . On ¯˜ under the other hand, if the DM would like to minimize the objective function Cx the condition that the degree of possibility with respect to the attained probability ¯˜ formulated as ˜ P ω C(ω)x ≤ f , we consider the problem of minimizing f for Cx ⎫ minimize f ⎬ ˜ ≥h subject to ΠP˜ (G) (6.18) ⎭ Ax ≤ b, x ≥ 0, where h is a permissible possibility level speciﬁed by the DM. Recalling the equivalent transformation (6.16), (6.18) is transformed into minimize f

0

n

∑ {m j − L (h)α j }x j + Φ

subject to

−1

j=1

(μG˜ (h))

n

∑ σ2j x2j ≤ f

j=1

Ax ≤ b, x ≥ 0 or equivalently n

minimize

0

∑ {m j − L (h)α j }x j + Φ

j=1

subject to Ax ≤ b, x ≥ 0.

−1

(μG˜ (h))

n

∑

j=1

σ2j x2j

⎫ ⎪ ⎬ ⎪ ⎭

(6.19)

From the viewpoint of reliability assurance, considering that the DM often feels that the attained probability p should be greater than or equal to at least 1/2, it would be natural to assume that μG˜ (p) ≥ 0 for any p ≥ 1/2, which implies that Φ−1 (μG˜ (h)) ≥ 0 holds for any h. Then, (6.19) is a convex programming problem, and it can be solved by convex programming techniques such as the sequential quadratic programming method. So far, we have discussed linear programming problems in which the coefﬁcients of the objective function are represented by random fuzzy variables. In the near future, we will extend the proposed models to more general cases where not only the objective function but also the constraints involve random fuzzy variables coefﬁcients. Furthermore, extensions to integer problems as well as developments of expectation and variance models will be considered elsewhere.

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6.3 Multiobjective random fuzzy programming In the previous section, we have dealt with random fuzzy linear programming problems with a single objective function. However, most of the real-world decision making problems usually involve multiple, noncommensurable, and conﬂicting objectives which should be considered simultaneously. In this section, as a multiobjective version of random fuzzy linear programming problems, we consider the multiobjective random fuzzy programming problems formulated as ⎫ ¯˜ x minimize C ⎪ 1 ⎪ ⎬ ··· (6.20) ¯˜ x ⎪ minimize C k ⎪ ⎭ subject to Ax ≤ b, x ≥ 0 , where x is an n-dimensional decision variable column vector, A is an m × n matrix, ¯˜ = (C¯˜ , . . . , C¯˜ ) is a random fuzzy b is an m × 1 constant column vector, and C l l1 ln variable coefﬁcient vector. Here, assume that C¯˜l j is a Gaussian random variable whose mean value is a fuzzy number M˜ l j characterized by the membership function ⎧ ml j − t ⎪ ⎪ if ml j ≥ t ⎪ ⎨L αl j (6.21) μM˜ l j (t) = ⎪ t − ml j ⎪ ⎪ ⎩R if ml j < t, βl j where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, the membership function of C¯˜l j is expressed as μC¯˜ (¯γl j ) = {μM˜ l j (sl j )|¯γl j ∼ N(sl j , σ2l j )}, ∀¯γl j ∈ Γ. lj

(6.22)

¯˜ x is expressed Using the Zadeh’s extension principle, each objective function C l as a random fuzzy variable characterized by the membership function n (6.23) μC¯˜ l x (u¯l ) sup min μC¯˜ (¯γl j ) u¯l = ∑ γ¯ l j x j , ∀u¯l ∈ Γ, 1≤ j≤n l j γ¯l j=1 where γ¯ l = (¯γ1 , . . . , γ¯ n ). By substituting (6.22) into (6.23), the membership function of a random fuzzy ¯˜ x in (6.20) is rewritten as variable corresponding to the objective function C l n n 2 2 μC¯˜ l x (u¯l ) = sup min μM˜ l j (sl j ) u¯l ∼ N ∑ sl j x j , ∑ σl j x j , (6.24) 1≤ j≤n sl j=1 j=1

6.3 Multiobjective random fuzzy programming

233

where sl = (sl1 , . . . , sln ). ¯˜ x is expressed as a random fuzzy variable with the membership Observing C l function μCx ¯˜ deﬁned by (6.24), it is signiﬁcant to realize that the fuzzy random programming models discussed in Chapter 4 cannot be applied.

6.3.1 Possibility-based probability model Assuming that the DM concerns about the probability that each of the objective ¯˜ x is smaller than or equal to a certain target values f , we introduce function values C l l the probability P ω C˜ l (ω)x ≤ fl which is expressed as a fuzzy set P˜l with the membership function (6.25) μP˜l (pl ) = sup μC¯˜ l x (u¯l ) pl = P (ω | ul (ω) ≤ f l ) , u¯l

where fl , l = 1, . . . , k are target values speciﬁed by the DM as constants. In order to consider the imprecise nature of the DM’s judgments for the probability with respect to the lth objective function, we introduce the fuzzy goal G˜ l with the nondecreasing continuous membership function μG˜ l . Recalling that the membership function μP˜l (pl ) can be viewed as a possibility distribution on the basis of the concept of possibility measure, the degree of possibility that the fuzzy goal G˜ l for the probability P˜l is satisﬁed is expressed by ΠP˜l (G˜ l ) sup min{μP˜l (pl ), μG˜ l (pl )}. pl

Then, we consider the possibility-based probability model for multiobjective random fuzzy programming problems formulated as ⎫ maximize ΠP˜1 (G˜ 1 ) ⎪ ⎪ ⎬ ······ (6.26) maximize ΠP˜k (G˜ k ) ⎪ ⎪ ⎭ subject to Ax ≤ b, x ≥ 0 or equivalently maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ⎪ ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , 0 ≤ hl ≤ 1, l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0.

(6.27)

To generate a candidate for a satisﬁcing solution of the DM for the reference levels hˆ l , l = 1, . . . , k speciﬁed by the DM, called reference possibility levels, the corresponding optimal solution is obtained by solving the augmented minimax problem

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6 Future Research Directions

minimize max

l=1,...,k

⎫ ⎪ ⎪ hˆ l − hl + ρ ∑ (hˆ l − hl ) ⎪ ⎬ k

l=1

subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k Ax ≤ b, x ≥ 0,

⎪ ⎪ ⎪ ⎭

(6.28)

where ρ is a sufﬁciently small positive number. Quite similar to the equivalent transformation (6.16), it holds that 0 n n −1 ˜ ΠP˜l (Gl ) ≥ hl ⇔ ∑ {ml j − L (hl )αl j }x j + Φ (μG˜ (hl )) ∑ σ2l j x2j ≤ fl , (6.29) j=1

l

j=1

where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (hl ) = sup{pl | μG˜ l (pl ) ≥ hl } l and L (hl ) = sup{t | L(t) ≥ hl } are pseudo inverse functions. Consequently, (6.28) is rewritten as ⎫ k ⎪ ⎪ ⎪ minimize max hˆ l − hl + ρ ∑ (hˆ l − hl ) ⎪ ⎪ ⎪ l=1,...,k ⎬ l=1 0 subject to

n

n

j=1

j=1

⎪ ∑ {ml j − L (hl )αl j }x j + Φ−1 (μG˜ l (hl )) ∑ σ2l j x2j ≤ fl , l = 1, . . . , k ⎪ ⎪ ⎪

Ax ≤ b, x ≥ 0.

⎪ ⎪ ⎭

(6.30) Observing that (6.30) is a nonconvex nonlinear programming problem, we can employ genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001) in order to obtain an approximate solution. Now we are ready to construct an interactive algorithm for deriving the satisﬁcing solution for the DM from among the Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model Step 1: Ask the DM to specify the membership functions μG˜ l and the target values fl , l = 1, . . . , k. Step 2: Set the initial reference possibility levels at 1s, which can be viewed as the ideal values, i.e., hˆ l = 1, l = 1, . . . , k. Step 3: For the current reference possibility levels hˆ l , l = 1, . . . , k, solve the augmented minimax problem (6.30). Step 4: The DM is supplied with the corresponding Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values h∗l , l = 1, . . . , k, then stop. Otherwise, ask the DM to update the reference possibility levels hˆ l , l = 1, . . . , k by considering the current values of objective functions hl , l = 1, . . . , k, and return to step 3.

6.3 Multiobjective random fuzzy programming

235

It should be noted for the DM that any improvement of one objective function value can be achieved only at the expense of at least one of the other objective function values for the ﬁxed target values fl , l = 1, . . . , k.

6.3.2 Possibility-based fractile model As discussed in the when the DM is willing to maximize the previous subsection, probability P ω C˜ l (ω)x ≤ fl for the speciﬁed target value fl , the possibilitybased probability model is appropriate. However, if we assume that the DM would ¯˜ x, l = 1, . . . , k under the condition that like to minimize the objective functions C l the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, we consider the multiobjective programming problem ⎫ minimize f1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (6.31) minimize fk ⎪ ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0, where hl , l = 1, . . . , k are permissible possibility levels speciﬁed by the DM. To consider the imprecise nature of human judgments for the target variables fl , l = 1, . . . , k, by introducing the fuzzy goals characterized by the nonincreasing concave and continuous membership functions μl , l = 1, . . . , k, we consider the fuzzy multiobjective programming problem ⎫ maximize μ1 ( f1 ) ⎪ ⎪ ⎪ ⎪ ··· ⎬ maximize μk ( fk ) (6.32) ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0. Recalling the equivalent transformation (6.16), it holds that 0 n n −1 ΠP˜l (G˜ l ) ≥ hl ⇔ ∑ {ml j − L (hl )αl j }x j + Φ (μG˜ (hl )) ∑ σ2l j x2j ≤ fl , (6.33) j=1

l

j=1

where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (hl ) = sup{pl | μG˜ l (pl ) ≥ hl } l and L (hl ) = sup{t | L(t) ≥ hl } are pseudo inverse functions. From (6.33), (6.32) is transformed into

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6 Future Research Directions

maximize μ1 ( f1 ) ··· maximize μk ( fk )

0

n

subject to

∑ {ml j − L (hl )αl j }x j + Φ

−1

j=1

(μG˜ (hl )) l

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ n

⎪ ⎪ ⎪ ⎭

⎪ ∑ σ2l j x2j ≤ fl , l = 1, . . . , k ⎪ ⎪ ⎪

j=1

Ax ≤ b, x ≥ 0

(6.34) or equivalently

⎫ maximize μ1 Z1Π,F (x) ⎪ ⎪ ⎪ ⎪ ⎬ ·· · · · · · · · maximize μk ZkΠ,F (x) ⎪ ⎪ ⎪ ⎪ ⎭ subject to Ax ≤ b, x ≥ 0,

(6.35)

where ZlΠ,F (x) =

n

0

∑ {ml j − L (hl )αl j }x j + Φ

−1

j=1

(μG˜ (hl )) l

n

∑ σ2l j x2j .

j=1

As mentioned in the previous section, assuming μG˜ l (pl ) ≥ 0 for any pl ≥ 1/2,

l = 1, . . . , k, each of the objective functions ZlΠ,F (x), l = 1, . . . , k is convex due to the property of Φ−1 (μG˜ (hk )) ≥ 0. k In order to ﬁnd a candidate for the satisﬁcing solution, through the introduction of the reference membership levels μˆ l , l = 1, . . . , k speciﬁed by the DM, a Pareto optimal solution is obtained by solving the augmented minimax problem minimize

max

l=1,...,k

k μˆ l − μl ZlΠ,F (x) + ρ ∑ μˆ l − μl ZlΠ,F (x)

subject to Ax ≤ b, x ≥ 0

⎫ ⎪ ⎬

l=1

⎪ ⎭

(6.36)

or equivalently ⎫ ⎪ ⎪ ⎪ k ⎪ ⎪ Π,F Π,F ⎪ subject to μˆ 1 − μ1 Z1 (x) + ρ ∑ μˆ l − μl Zl (x) ≤ v ⎪ ⎪ ⎪ ⎬ l=1 ··············· ⎪ ⎪ k ⎪ ⎪ μˆ k − μk ZkΠ,F (x) + ρ ∑ μˆ l − μl ZlΠ,F (x) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ l=1 ⎭ Ax ≤ b, x ≥ 0, minimize v

(6.37)

where ρ is a sufﬁciently small positive number. When μl is concave, similar to the convex property shown in (3.22), it is easily shown that (6.37) is a convex programming problem. Hence, an optimal solution of

6.4 Random fuzzy two-level programming

237

(6.37) can be obtained by using some convex programming techniques such as the sequential quadratic programming method. We now summarize an interactive algorithm in order to derive the satisﬁcing solution for the DM from among the Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model Step 1: Ask the DM to specify the membership functions μG˜ l and the permissible possibility levels hl , l = 1, . . . , k. Step 2: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 3: For the current reference fractile levels μˆ l , l = 1, . . . , k, solve the corresponding minimax problem (6.36). Step 4: The DM is supplied with the corresponding Pareto optimal solution x∗ . If Π,F ∗ the DM is satisﬁed with the current membership function values μl Zl (x ) , l = 1, . . . , k, then stop. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k, and return to step 3. It should be stressed for the DM that any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the ﬁxed permissible possibility levels hl , l = 1, . . . , k. In the future, along the same line as in Chapter 4, we will consider random fuzzy versions of possibility-based expectation and variance models. Extensions to more general problems where random fuzzy variables coefﬁcients are involved not only in the objective function but also in the constraints will be worth considering. Furthermore, extensions of the proposed models to integer problems will be discussed elsewhere.

6.4 Random fuzzy two-level programming Although we have restricted ourselves to the problems where decisions are made by a single DM in the previous section, it should be noted here that decision making problems in hierarchical managerial or public organizations are often formulated as two-level programming problems, where there exist two DMs. In contrast to the fuzzy random two-level programming discussed in Chapter 5, we consider the random fuzzy version of two-level linear programming problems formulated as ⎫ ¯˜ x + C ¯˜ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ¯˜ x + C ¯˜ x ⎬ minimize z2 (x1 , x2 ) = C 21 1 22 2 (6.38) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector for the DM at the upper level (DM1), x2 is an n2 dimensional decision variable column vector for the

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6 Future Research Directions

DM at the lower level (DM2), and zl (x1 , x2 ), l = 1, 2 are the objective functions ¯˜ , for DMl, l = 1, 2, respectively. Suppose that each of C¯˜l jk , k = 1, 2, . . . , n j of C lj l = 1, 2, j = 1, 2 is the Gaussian random variable with fuzzy mean value M˜ l jk which is represented by the fuzzy set with the membership function ⎧ ml jk − τ ⎪ ⎪ if ml jk ≥ τ ⎪ ⎨L αl jk (6.39) μM˜ l jk (τ) = ⎪ τ − ml jk ⎪ ⎪ ⎩R if ml jk < τ, βl jk where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, C¯˜l jk is expressed as a random fuzzy variable with the membership function μC¯˜ (¯γl jk ) = {μM˜ l jk (sl jk ) | γ¯ l jk ∼ N(sl jk , σ2l jk )}, ∀¯γl jk ∈ Γ. l jk

(6.40)

Through the extension principle, in view of (6.40), the membership function of a random fuzzy variable corresponding to each of objective functions zl (x1 , x2 ), l = 1, 2 is given as 2 nj μ ¯˜ (¯γl jk ) u¯l = ∑ ∑ γ¯ l jk x jk μC¯˜ l x (u¯l ) = sup min 1≤k≤n j , j=1,2 Cl jk γ¯ l j=1 k=1 " # 2 nj 2 nj 2 2 μM˜ l jk (sl jk ) u¯l ∼ N ∑ ∑ sl jk x jk , ∑ ∑ σl jk x jk = sup min , 1≤k≤n j , j=1,2 sl j=1 k=1 j=1 k=1 (6.41) where γ¯ l = (¯γl11 , . . . , γ¯ l1n1 , γ¯ l21 , . . . , γ¯ l2n2 ) and sl = (sl11 , . . . , sl1n1 , s121 , . . . , sl2n2 ).

6.4.1 Possibility-based probability model Assuming that the DMs concerns about the probabilities that their own objective ¯˜ x are smaller than or equal to certain target values f , l = 1, 2, we function values C l l introduce the probabilities P ω C˜ l (ω)x ≤ fl which are expressed as fuzzy sets P˜l with the membership functions μP˜l (pl ) = sup μC¯˜ l x (u¯l ) | pl = P (ω | ul (ω) ≤ fl ) , (6.42) u¯l

where fl , l = 1, 2 are target values speciﬁed by the DMs as constants. In order to consider the imprecise nature of the human judgments, we introduce the fuzzy goals G˜ l of the DMs for the probabilities P˜l , l = 1, 2, which is characterized by a nondecreasing membership function μG˜ l . Recalling that the membership

6.4 Random fuzzy two-level programming

239

function is regarded as a possibility distribution, the degree of possibility that the probability P˜l attains the fuzzy goal G˜ l is expressed as ΠP˜l (G˜ l ) = sup min{μP˜l (pl ), μG˜ l (pl )}. pl

Then, assuming that the DMs are willing to maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probability model for random fuzzy two-level programming problems formulated as ⎫ maximize Z1Π,P (x1 , x2 ) = ΠP˜1 (G1 ) ⎪ ⎪ ⎪ for DM1 ⎪ ⎬ maximize Z2Π,P (x1 , x2 ) = ΠP˜2 (G2 ) (6.43) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. As an initial candidate for an overall satisfactory solution to (6.43), it would be useful for DM1 to ﬁnd a solution which maximize the smaller degree of satisfaction between the two DMs by solving the maximin problem ⎫ maximize min Z1Π,P (x1 , x2 ), Z2Π,P (x1 , x2 ) ⎪ ⎬ (6.44) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently maximize v subject to Z1Π,P (x1 , x2 ) ≥ v

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

Z2Π,P (x2 , x2 ) ≥ v ⎪ ⎪ A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.45)

Similar to the equivalent transformation (6.16), in view of ZlΠ,P (x1 , x2 ) = ΠP˜l (Gl ), it holds that 1 2 2 nj 2 nj 2 Π,P −1 Zl (x1 , x2 ) ≥ v ⇔ ∑ ∑ {ml jk −L (v)αl jk }x jk +Φ (μG˜ (v))3 ∑ ∑ σ2l jk x2jk ≤ fl , j=1 k=1

l

j=1 k=1

(6.46) where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (v) = sup{pl | μG˜ l (pl ) ≥ v} and l L (v) = sup{t | L(t) ≥ v} are pseudo inverse functions. Consequently, (6.45) is equivalently transformed as

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6 Future Research Directions

maximize v 2

subject to

∑

1

j=1 k=1 2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ 2 2 ⎪ σ x ≤ f ⎪ ∑ 1 jk jk 1 ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎬

1 2 2 nj 2 −1 3 {m − L (v)α }x + Φ (μ (v)) 1 jk jk ∑ 1 jk ∑ G˜ 1 2 2 2 −1 3 {m − L (v)α }x + Φ (μ (v)) 2 jk jk ∑ 2 jk ∑ G˜ nj

2

j=1 k=1

nj

⎪

⎪ ⎪ ∑ σ22 jk x2jk ≤ f2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

(6.47) Although (6.47) is nonconvex, an approximate solution can be obtained by genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001). If DM1 is satisﬁed with the degrees ZlΠ,P (x∗1 , x∗2 ), l = 1, 2, the corresponding optimal solution (x∗1 , x∗2 ) to (6.47) is regarded as the satisfactory solution. However, if DM1 is not satisﬁed, by introducing the constraint that Z1Π,P (x1 , x2 ) is larger than or equal to the minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the maximization problem formulated as ⎫ ⎪ maximize Z2Π,P (x1 , x2 ) ⎪ ⎪ ⎬ Π,P subject to Z1 (x1 , x2 ) ≥ δ (6.48) A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently

⎫ maximize h ⎪ ⎪ ⎪ Π,P ⎪ subject to Z2 (x1 , x2 ) ≥ h ⎪ ⎬ Π,P

(6.49)

Z1 (x1 , x2 ) ≥ δ ⎪ ⎪ A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Similar to the equivalent transformation (6.16), it follows that Z2Π,P (x1 , x2 ) ≥ h ⇔

2

∑

j=1 k=1

2

Z1Π,P (x1 , x2 ) ≥ δ ⇔

2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ ⎪ 2 2 ⎪ σ x ≤ f , ⎪ 2 ∑ 2 jk jk ⎪ ⎬ j=1 k=1

1 2 2 nj 2 −1 ∑ {m2 jk − L (h)α2 jk }x jk + Φ (μG˜ (h))3 ∑ 1 2 2 2 −1 3∑ {m − L (δ)α }x + Φ (μ (δ)) 1 jk jk ∑ 1 jk G˜ nj

j=1 k=1

1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ σ x ≤ f . ⎪ ∑ 1 jk jk 1 ⎪ ⎭ nj

j=1 k=1

(6.50) Consequently, (6.49) is rewritten as

6.4 Random fuzzy two-level programming

maximize h 2

subject to

∑

241

2

j=1 k=1 2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ 2 2 ⎪ σ x ≤ f ⎪ ∑ 2 jk jk 2 ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎬

1 2 2 nj 2 −1 3 {m − L (h)α }x + Φ (μ (h)) 2 jk jk ∑ 2 jk ∑ G˜ 1 2 2 2 −1 3 {m − L (δ)α }x + Φ (μ (δ)) 1 jk jk ∑ 1 jk ∑ G˜ nj

1

j=1 k=1

nj

⎪

⎪ ⎪ ∑ σ21 jk x2jk ≤ f1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

(6.51) In view of the nonconvexity of (6.51), quite similar to (6.47), genetic algorithms can be applied for solving this problem. Realizing that the objective functions of DM1 and DM2 often conﬂict with each other, it should be noted here that the larger the minimal satisfactory level δ for the DM1’s objective function value is speciﬁed, the smaller the DM2’s objective function value becomes, which may lead to the unbalanced satisfactory degrees of DM1 and DM2 due to the large difference between the objective function values of both DMs. In order to derive the satisfactory solution which has well-balanced objective function values of DM1 and DM2, by introducing the ratio Δ between the satisfactory degrees of both DMs expressed as Δ=

Z2Π,P (x1 , x2 )

Z1Π,P (x1 , x2 )

,

(6.52)

we assume that DM1 speciﬁes the lower bound Δmin and the upper bound Δmax of Δ, which are used for evaluating the appropriateness of the ratio Δ. To be more speciﬁc, if it holds that Δ ∈ [Δmin , Δmax ], then DM1 regards the corresponding solution as a promising candidate for the satisfactory solution with well-balanced membership function values. Now we construct a procedure of interactive fuzzy programming for the possibilitybased probability model in order to derive an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model Step 1: Ask each DM to specify the membership function μG˜ l and the target values fl , l = 1, 2. Step 2: For the current target values fl , l = 1, 2, solve the maximin problem (6.44). Step 3: DM1 is supplied with the objective function values Z1Π,P (x∗1 , x∗2 ) and Z2Π,P (x∗1 , x∗2 ) for the optimal solution (x∗1 , x∗2 ) obtained in step 2. If DM1 is satisﬁed with the current membership function values, then stop. If DM1 is not satisﬁed and prefers to update fl , l = 1, 2, ask DM1 to update fl , and return to step 2. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 4: For the current minimal satisfactory δ, solve (6.48).

242

6 Future Research Directions

Step 5: DM1 is supplied with the current values of Z1Π,P (x∗1 , x∗2 ), Z2Π,P (x∗1 , x∗2 ) and Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the current objective function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 4.

6.4.2 Possibility-based fractile model When the DMs would like to maximize the probability for the speciﬁed target value, the possibility-based probability model is recommended for the DM1. On the other hand, if the DMs would like to minimize their own objective function values under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, we consider the two-level programming problem ⎫ maximize μ1 ( f1 ) ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize μ2 ( f2 ) ⎪ ⎪ for DM2 ⎬ subject to ΠP˜1 (G1 ) ≥ h1 (6.53) ⎪ ΠP˜2 (G2 ) ≥ h2 ⎪ ⎪ ⎪ ⎪ A1 x 1 + A 2 x 2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where h1 and h2 are permissible possibility levels speciﬁed by the DMs, and μ1 and μ2 are the membership functions of the fuzzy goals for the target variables f 1 and f 2 , respectively. Recalling the equivalent transformation (6.50), (6.53) is transformed into ⎫ maximize μ1 ( f1 ) ⎪ ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize μ2 ( f2 ) ⎪ ⎪ for DM2 ⎪ 1 ⎪ 2 ⎪ n ⎪ j 2 nj 2 2 ⎪ ⎪ −1 2 2 3 subject to ∑ ∑ {m1 jk − L (hl )α1 jk }x jk + Φ (μG˜ (h1 )) ∑ ∑ σ1 jk x jk ≤ f1 ⎪ ⎪ ⎬ 1 j=1 k=1 2

∑

j=1 k=1

1 2 2 nj 2 −1 ∑ {m2 jk − L (h2 )α2 jk }x jk + Φ (μG˜ (h2 ))3 ∑

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0

2

⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∑ σ2 jk x jk ≤ f2 ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ nj

(6.54) or equivalently

6.4 Random fuzzy two-level programming

243

⎫ maximize μ1 Z1Π,F (x1 , x2 ) ⎪ ⎪ for DM1 ⎪ ⎪ ⎬ maximize μ2 Z2Π,F (x1 , x2 ) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(6.55)

where ZlΠ,F (x1 , x2 ) =

2

∑

1 2 2 2 −1 3∑ {m − L (h )α }x + Φ (μ (h )) l l jk jk l ∑ l jk G˜ nj

j=1 k=1

l

nj

∑ σ2l jk x2jk ,

j=1 k=1

l = 1, 2. (6.56) are convex functions of (x1 , x2 ). As the ﬁrst step toward deriving an overall satisfactory solution to (6.55), in order to ﬁnd a solution which maximize the smaller degree of satisfaction between the two DMs, we solve the maximin problem ⎫ maximize min μ1 Z1Π,F (x1 , x2 ) , μ2 Z2Π,F (x1 , x2 ) ⎪ ⎬ (6.57) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. By introducing an auxiliary variable v, this problem is rewritten as ⎫ maximize v ⎪ ⎪ ⎪ ⎪ subject to μ1 Z1Π,F (x1 , x2 ) ≥ v ⎪ ⎪ ⎪ ⎬ Π,F μ2 Z2 (x1 , x2 ) ≥ v ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.58)

Although the membership function does not always need to be linear, for the sake of simplicity, we adopt a linear membership function which characterizes the fuzzy goal of each decision maker. The linear membership functions μl , l = 1, 2 are deﬁned as ⎧ ⎪ 1 if ZlΠ,F (x1 , x2 ) ≤ z1l ⎪ ⎪ ⎨ Π,F Zl (x1 , x2 ) − z0l μl ZlΠ,F (x1 , x2 ) = if z1l < ZlΠ,F (x1 , x2 ) < z0l 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if ZlΠ,F (x1 , x2 ) ≥ z0l . Then, (6.58) is equivalently transformed into the convex programming problem

244

6 Future Research Directions

⎫ ⎪ ⎪ ⎪ ⎪ 2 nj ⎪ ⎪ ⎪ subject to ∑ ∑ {m1 jk − L (h1 )α1 jk }x jk ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ 1 ⎪ ⎪ 2 2 nj ⎪ ⎪ 2 2 2 −1 1 0 0⎪ 3 +Φ (μG˜ (h1 )) ∑ ∑ σ1 jk x jk ≤ (z1 − z1 )v + z1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ j=1 k=1 ⎬

maximize v

nj

2

∑ ∑ {m2 jk − L (h2 )α2 jk }x jk

⎪ ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ 1 ⎪ 2 2 nj ⎪ ⎪ 2 ⎪ ⎪ 2 2 −1 1 0 0 ⎪ +Φ (μG˜ (h2 ))3 ∑ ∑ σ2 jk x jk ≤ (z2 − z2 )v + z2 ⎪ ⎪ 2 ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.59)

If DM1 is satisﬁed with the membership function values μl ZlΠ,F (x∗1 , x∗2 ) , l = 1, 2, the corresponding optimal solution x∗ to (6.58) is regarded as the satisfactory solution. Otherwise, by introducing the constraint that μ1 (Z1Π,F (x)) is larger than or equal to the minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the Π,F problem of maximizing the membership function μ2 (Z2 (x1 , x2 )) formulated as ⎫ Π,F ⎪ maximize μ2 Z2 (x1 , x2 ) ⎪ ⎪ ⎪ ⎪ ⎬ Π,F subject to μ1 Z1 (x1 , x2 ) ≥ δ (6.60) ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently, we obtain the convex programming problem 2

minimize

∑

1 2 2 2 −1 ∑ {m2 jk − L (h2)α2 jk }x jk + Φ (μG˜ (h2 ))3 ∑ nj

2

j=1 k=1 2

subject to

nj

∑ ∑ {m1 jk − L (h1)α1 jk }x jk

j=1 k=1

1 2 2 2 −1 +Φ (μG˜ (h1 ))3 ∑ 1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

⎫ ⎪ ⎪ ⎪ ⎪ ∑ σ22 jk x2jk ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ nj

nj

∑

j=1 k=1

σ21 jk x2jk

≤

(z11 − z01 )δ + z01

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(6.61) In general, when the objective functions of DM1 and DM2 conﬂict with each other, it should be noted here that the larger the minimal satisfactory level δ for μ1 is speciﬁed by DM1, the smaller the satisfactory degree for μ2 becomes, which may

6.4 Random fuzzy two-level programming

245

lead to the improper satisfactory balance between DM1 and DM2 due to the large difference between the membership function values of both DMs. In order to derive the satisfactory solution which has well-balanced membership function values between both DMs, by introducing the ratio Δ expressed as μ2 Z2Π,F (x1 , x2 ) , (6.62) Δ= μ1 Z1Π,F (x1 , x2 ) the lower bound Δmin and the upper bound Δmax of Δ, speciﬁed by DM1, are introduced to determine whether or not the ratio Δ is appropriate. To be more explicit, if it holds that Δ ∈ [Δmin , Δmax ], then DM1 regards the corresponding solution as a preferable candidate for the satisfactory solution with well-balanced membership function values. Now we summarize a procedure of interactive fuzzy programming for the possibilitybased fractile model in order to derive a satisfactory solution. Interactive fuzzy programming in the possibility-based fractile model Step 1: Ask DMs to specify the membership functions μl , l = 1, 2. Step 2: Ask DM1 to specify the permissible possibility levels hl , l = 1, 2. Step 3: For the current hl , l = 1, 2, solve the maximin problem (6.57). Step 4: DM1 is supplied with the current values of the membership functions μ1 and μ2 for the optimal solution obtained in step 3. If DM1 is satisﬁed with the current membership function values, then stop. If DM1 is not satisﬁed and prefers ask to update hl , l = 1, 2, ask DM1 to update hl , and return to step 3. Otherwise, Π,F DM1 to specify the minimal satisfactory level δ for μ1 Z1 (x1 , x2 ) and the permissible range [Δmin , Δmax ] of the ratio Δ. Step 5: For the current minimal satisfactory level δ, solve the convex programming problem (6.61). Step 6: DM1 is supplied with the current values of the membership function μ1 , μ2 and the ratio Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the current membership function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 5. In this section, from the viewpoint of taking the possibility of coordination or bargaining between the DMs into account, it is assumed that there exist communication and a cooperative relationship between the DMs. However, as discussed in Chapter 5, we are faced with the decision making situations where the DMs do not make any binding agreement even if there exists some communication. For dealing with such situations, extensions to noncooperative cases as well as the developments of the computational methods for obtaining Stackelberg solutions will be discussed elsewhere. Furthermore, an integer generalization along the same line as in Chapter 5 will be reported elsewhere.

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Index

L-R fuzzy number, 13, 119, 132, 141, 178, 195, 208, 218 L∞ norm, 27 α-level optimal parameter, 135, 138, 141 α-level set, 12, 102, 133, 136, 138, 141, 160 α-linear programming problem, 19 σ-ﬁeld, 20, 102 kth best method, 37, 209 algorithm of interactive fuzzy programming, 36 augmented minimax problem, 57, 87, 136, 233, 236 bisection method, 31, 54, 66, 110, 124, 143, 181, 185 Borel σ-ﬁeld, 102 Borel set, 21 Cartesian product, 198, 212, 220 chance constrained condition, 51, 171, 188, 215 chance constraint programming, 24 continuous fuzzy random variable, 119 crossover, 41 discrete fuzzy random variable, 102 distribution function, 21 double strings, 44 dual problem, 39 duality gap, 39 E-model, 25 E-P-Pareto optimal solution, 108, 151 E-P-Pareto optimality test, 113 extended Dinkelbach-type algorithm, 116, 129 extension principle, 105, 132, 227, 232, 238

F-P-Pareto optimal solution, 127, 147 feasible extreme point, 211 feasible region, 20, 27, 31, 37, 57, 104, 171 ﬁtness function, 42 ﬁtness scaling, 42 ﬁtness value, 41 fractile criterion model, 25 fuzzy decision, 15, 29 fuzzy goal, 52, 57, 105, 173, 228 fuzzy multiobjective programming, 28 fuzzy number, 13, 182 fuzzy programming for two-level linear programming, 33 fuzzy random variable, 101, 178, 182, 195, 198, 207, 211, 218, 220 fuzzy set, 12 fuzzy variable, 226 GADSLPRRSU, 44, 86, 87, 91, 95, 151, 162 Gaussian random variable, 50, 72, 84, 91, 127, 146, 157, 226, 227 genetic algorithm, 40 GENOCOPIII, 86, 136, 140, 234, 240 inducible region, 210, 220 integer programming problem, 44 interactive fuzzy multiobjective stochastic integer programming, 83 interactive fuzzy satisﬁcing method, 53, 55, 58, 69, 75, 92, 95 interactive satisﬁcing method, 112, 151 Kuhn-Tucker condition, 38, 203 Kuhn-Tucker necessity theorem, 68 Lagrange function, 67, 112, 125, 203 Lagrange multiplier, 203

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9, © Springer Science+Business Media, LLC 2011

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264 level set-based model, 18 linear fractional programming problem, 18, 66, 110, 182 linear membership function, 15, 28, 106, 128, 146, 150, 186, 213, 221, 243 linear programming problem involving fuzzy parameters, 16 linear programming problem with a fuzzy goal and fuzzy constraints, 14 linear programming relaxation, 44 M-Pareto optimal solution, 30, 53, 87 M-Pareto optimality, 53 Manhattan distance, 27 maximin problem, 173, 180, 186, 189, 192, 196, 199, 239 minimal satisfactory level, 34, 185, 186, 192, 196, 199, 240 minimax problem, 109, 124, 142, 147, 151 minimum operator, 53 mixed zero-one programming problem, 39 multiobjective stochastic integer programming problem, 84 multiobjective stochastic linear programming problem, 81 multiobjective stochastic programming problem, 51 mutation, 41 nonlinear integer programming problem, 91 NP-hard problem, 37 P-M-α Pareto optimality test, 144 P-model, 25 P-P-Pareto optimal solution, 123, 142 P-P-Pareto optimality test, 125 Pareto optimal solution, 27, 236 Pareto optimality in two-level programming, 33 permissible expectation level, 57, 85, 113 permissible possibility level, 231 permissible probability level, 75, 127 phase one of the two-phase simplex method, 54, 110, 124, 143 possibilistic programming, 107, 118 possibility distribution, 107, 179, 228, 233 power set, 102 probability density function, 23 probability measure, 21, 102 probability measure space, 21

Index probability space, 21, 102 product operator, 53 pseudo inverse function, 123, 143, 229 quadratic integer programming problem, 86 random variable, 21 ratio of the satisfactory degree, 34, 199 rational response, 37, 202, 209, 216, 217, 219 realized value, 102 reference membership level, 53 reference point method, 27 reference solution updating, 44 reproduction, 41 revised GADSLPRRSU, 86, 152, 153, 163 revised GENOCOPIII, 136, 140, 234, 240 roulette wheel selection, 42 sample space, 20, 102 satisfactory solution, 33, 172, 173, 180, 184, 189, 192, 196, 199, 239 satisﬁcing probability level, 51, 171 satisﬁcing solution, 85, 86, 109, 151, 236 sequential quadratic programming method, 24, 147, 148, 174, 203, 231 shape function, 13 simple recourse problem, 22, 78 simplex multiplier, 68, 112 Stackelberg equilibrium solution, 36 Stackelberg solution, 36, 202, 203, 209, 213, 216, 217 standard Gaussian random variable, 72, 230 target value, 62, 90, 122, 228 target variable, 72, 235 Tchebyshev norm, 27 trade-off rate, 55, 112 triangular fuzzy number, 105, 120 two-phase simplex method, 31, 54, 66 two-stage model, 77 two-stage programming, 22 V-model, 25 V-P-Pareto optimal solution, 114, 153 variable transformation, 66 variance-covariance matrix, 25, 50, 173 weak Pareto optimality, 27 Zimmermann method, 29, 61, 177

Volume 159

Series Editor: Frederick S. Hillier Stanford University, CA, USA Special Editorial Consultant: Camille C. Price Stephen F. Austin, State University, TX, USA

For further volumes: http://www.springer.com/series/6161

Masatoshi Sakawa • Ichiro Nishizaki Hideki Katagiri

Fuzzy Stochastic Multiobjective Programming

1C

Prof. Dr. Masatoshi Sakawa Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

Dr. Hideki Katagiri Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

Prof. Dr. Ichiro Nishizaki Hiroshima University Graduate School of Engineering Department of System Cybernetics Kagamiyama 1-4-1 739-8527 Higashi-Hiroshima Japan [email protected]

ISSN 0884-8289 ISBN 978-1-4419-8401-2 e-ISBN 978-1-4419-8402-9 DOI 10.1007/978-1-4419-8402-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011921329 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Our Parents and Families

Preface

The increasing complexity of modern-day society has brought new problems having multiple objectives including economic, environmental, social and technical ones. Hence, it seems that the consideration of many objectives in the actual decision making process requires multiobjective approaches rather than that of a single objective. One of the major systems-analytic multiobjective approaches to decision making under constraints is multiobjective programming as a generalization of traditional single objective programming. For such multiobjective programming problems, it is signiﬁcant to realize that multiple objectives are often noncommensurable and conﬂict with each other. With this observation, in multiobjective programming problems, the notion of Pareto optimality or efﬁciency has been introduced instead of the optimality concept for single-objective problems. However, decisions with Pareto optimality or efﬁciency are not uniquely determined; the ﬁnal decision must be selected by a decision maker (DM), which well represents the subjective judgments, from the set of Pareto optimal or efﬁcient solutions. For deriving a compromise or satisﬁcing solution through interactions with a decision maker, interactive methods for multiobjective programming have been developed. When formulating a multiobjective programming problem which closely describes and represents the actual decision making situations, the uncertainty inherent in real-world complex systems or human being, such as the randomness of events related to the systems or the fuzziness of human judgments, should be reﬂected in the description of the objective functions and constraints as well as in the representation of parameters involved in the formulated problems. For dealing with not only the multiobjectiveness but also the uncertainty in decision making, multiobjective stochastic programming and multiobjective fuzzy programming have been individually developed together with the introduction of various optimization models and corresponding solution techniques. However, recalling the vagueness or fuzziness inherent in human judgments for the objective functions and/or constraints involving randomness, it is signiﬁcant to realize that the uncertainty in real-world decision making problems is often expressed by a fusion of fuzziness and randomness rather than either fuzziness or randomness. For such decision making situations, there are two types of inaccura-

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cies to be incorporated into multiobjective stochastic programming problems. One is the fuzzy goal of a DM for each of the stochastic objective functions, and the other is the experts’ ambiguous understanding of random parameters in the problemformulation process. When we model actual decision making situations under uncertainty as multiobjective programming problems involving random variables, following a major conventional stochastic programming approach together with the introduction of probabilistic constraints, the original problems can be transformed into deterministic multiobjective programming problems by replacing the original stochastic objective functions with some deterministic ones such as their expectations or variances. However, considering the imprecise nature of DMs’ judgments in multiobjective problems, it is natural to assume that the DM has a fuzzy goal for each of the resulting deterministic objective functions which originally involve randomness. Furthermore, in multiobjective stochastic programming, it is implicitly assumed that the realized values of random parameters can be deﬁnitely expressed by real values. However, considering that the realized values may be only ambiguously known to or observed by the experts, it may be more appropriate to express theses values as fuzzy numerical data which can be represented by means of fuzzy subsets of the real line known as fuzzy numbers. For handling and tackling such a fusion of fuzziness and randomness in multiobjective decision making, it is not hard to imagine that conventional fuzzy multiobjective programming or multiobjective stochastic programming cannot be applied. Naturally, simultaneous considerations of multiobjectiveness, fuzziness and randomness involved in the real-world decision making problems lead us to the new ﬁeld of multiobjective mathematical programming under fuzzy stochastic environments. So far, we have restricted ourselves to mathematical programming problems where decisions are made by a single DM. However, decision making problems in hierarchical managerial or public organizations are often formulated as two-level programming problems, where there exist two DMs. When we deal with decision making problems in decentralized organizations such as a ﬁrm’s organizing administrative ofﬁce and its autonomous divisions, it is quite natural to suppose that there exists communication and some cooperative relationship among the DMs. For handling such cooperative hierarchical decision making situations, two-level programming with random variables or fuzzy random variables are introduced to derive a satisfactory solution taking a balance between the DMs’ satisfactory degrees. Furthermore, in order to resolving conﬂict in hierarchical noncooperative decision making situations, computational methods for Stackelberg solutions for two-level programming problems involving random variables or fuzzy random variables are presented. Concerning computational aspects of multiobjective programming under fuzzy stochastic environments, because real-world decision problems under uncertainty can be often formulated as difﬁcult classes of optimization problems such as combinatorial problems and nonconvex nonlinear problems, it is difﬁcult to obtain exact optimal solutions, and thus it is quite natural for DMs to require approximate optimal solutions instead. To meet this demand, genetic algorithms initially introduced by Holland in early the 1970’s and tabu search methods proposed by Glover et al.

Preface

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in the mid-1980’s, which have attracted considerable attention as efﬁcient metaheuristics, should be incorporated into interactive multiobjective programming with random variables or fuzzy random variables. In this book, after presenting basic concepts in conventional multiobjective programming, the authors intend to introduce the latest advances in the new ﬁeld of multiobjective programming involving fuzziness and randomness. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective optimization in order to derive a satisﬁcing solution for a DM. Extensions to two-level programming are also given to resolving conﬂict in hierarchical decision making problems under fuzzy stochastic environments. Hiroshima, October 2010

Masatoshi Sakawa Ichiro Nishizaki Hideki Katagiri

Contents

1

Introduction and Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Description of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7

2

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fuzzy programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fuzzy goals and Fuzzy constraints . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Linear programming problems with fuzzy parameters . . . . . . 2.2 Stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Two-stage programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Chance constraint programming . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiobjective programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Multiobjective programming problem . . . . . . . . . . . . . . . . . . . 2.3.2 Interactive multiobjective programming . . . . . . . . . . . . . . . . . 2.3.3 Fuzzy multiobjective programming . . . . . . . . . . . . . . . . . . . . . 2.4 Two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Fuzzy programming for two-level programming . . . . . . . . . . 2.4.2 Stackelberg solution to two-level programming problem . . . . 2.5 Genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Fundamental elements in genetic algorithms . . . . . . . . . . . . . . 2.5.2 Genetic algorithm for integer programming . . . . . . . . . . . . . .

11 11 11 14 16 20 20 21 24 26 26 27 28 32 32 36 40 40 44

3

Fuzzy Multiobjective Stochastic Programming . . . . . . . . . . . . . . . . . . . . 3.1 Fuzzy multiobjective stochastic linear programming . . . . . . . . . . . . . 3.1.1 Expectation and variance models . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Probability and fractile models . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Simple recourse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Expectation and variance models . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 62 77 83 84

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Contents

3.2.2 3.2.3

Probability and fractile models . . . . . . . . . . . . . . . . . . . . . . . . . 90 Simple recourse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4

Multiobjective Fuzzy Random Programming . . . . . . . . . . . . . . . . . . . . . . 101 4.1 Multiobjective fuzzy random linear programming . . . . . . . . . . . . . . . 101 4.1.1 Possibility-based expectation and variance models . . . . . . . . 104 4.1.2 Possibility-based probability and fractile models . . . . . . . . . . 118 4.1.3 Level set-based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2.1 Possibility-based expectation and variance models . . . . . . . . 150 4.2.2 Possibility-based probability and fractile models . . . . . . . . . . 154 4.2.3 Level set-based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5

Stochastic and Fuzzy Random Two-Level Programming . . . . . . . . . . . . 169 5.1 Cooperative two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.1.1 Stochastic two-level linear programming . . . . . . . . . . . . . . . . . 170 5.1.2 Fuzzy random two-level linear programming . . . . . . . . . . . . . 177 5.1.3 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . 188 5.2 Noncooperative two-level programming . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2.1 Stochastic two-level linear programming . . . . . . . . . . . . . . . . . 201 5.2.2 Fuzzy random two-level linear programming . . . . . . . . . . . . . 207 5.2.3 Extensions to integer programming . . . . . . . . . . . . . . . . . . . . . 215

6

Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.1 Random fuzzy variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2 Random fuzzy linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.2.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 228 6.2.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 231 6.3 Multiobjective random fuzzy programming . . . . . . . . . . . . . . . . . . . . . 232 6.3.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 233 6.3.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 235 6.4 Random fuzzy two-level programming . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.4.1 Possibility-based probability model . . . . . . . . . . . . . . . . . . . . . 238 6.4.2 Possibility-based fractile model . . . . . . . . . . . . . . . . . . . . . . . . 242

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Chapter 1

Introduction and Historical Remarks

1.1 Background The main characteristics of the real-world decision making problems facing humans today are uncertain due to the increase in complexity of modern-day society as well as the diversiﬁcation of values among people. Hence, when formulating a mathematical programming problem which closely describes and represents a real decision situation, various factors of the real system should be reﬂected in the description of the objective functions and the constraints. Naturally, these objective functions and the constraints involve many parameters whose possible values may be estimated by the experts. In the most traditional approaches, such parameters are ﬁxed at some values in an experimental and/or subjective manner through the experts’ understanding of the nature of the parameters. However, in most practical decision making situations, it would be more appropriate to consider uncertainty of the parameters of the objectives and constraints in the problems. From a probabilistic point of view, in 1955, linear programming problems with random variable coefﬁcients, called stochastic programming problems (Tintner, 1955), were individually introduced by Beale (1955), Dantzig (1955), Babbar (1955) and Wagner (1955). The two-stage problem, ﬁrst introduced by Beale (1955) and Dantzig (1955) as a simple recourse problem, is a solution technique via two stages for stochastic programming problems; in the ﬁrst stage, the penalty for the violation of the constraints is minimized under the assumption that the values of decision variables are ﬁxed at some constants, and in the second stage, the decision variables are determined so as to minimize the weighted sum of the penalty and the original objective function. The two-stage problems were comprehensively discussed by Walkup and Wets (1967) and Wets (1974) together with the introduction of more general types of recourse problems, and further extended by Beale, Forrest and Taylor (1980) and Louveaux (1980) as the multi-stage problems. Furthermore, the problem for obtaining the probability distribution function of an optimal solution and/or optimal value, called distribution problem, was initiated by Tintner (1955) and Wagner (1955), and further developed by Bereanu (1967, 1980).

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_1, © Springer Science+Business Media, LLC 2011

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1 Introduction and Historical Remarks

Considering the fact that the stochastic constraints are not always satisﬁed, the notion of chance or probabilistic constraints, which means that the constraints involving random variables need to be satisﬁed with a certain probability or over, was introduced by Charnes and Cooper (1959), and generalized by Miller and Wagner (1965) and Prekopa (1970) through the joint probability distribution function. For handling the randomness of the objective function value, three types of models based on different optimization criteria were considered by Charnes and Cooper (1963): the expectation optimization model, the variance minimization model and the probability maximization model. In the expectation optimization model, called the E-model, the objective function with random variable coefﬁcients is replaced by its expected value. The variance minimization model, called the V-model, was presented to consider not only the expectation but also the variance. Moreover, the probability maximization model, called the P-model or minimum-risk model, was developed in order to maximize the probability that the objective function value is better than or equal to the so-called target value speciﬁed by a decision maker (DM) as a constant. Meanwhile, the target value is optimized under a given probability level in the fractile criterion optimization model independently proposed by Kataoka (1963) and Geoffrion (1967). A brief and uniﬁed survey of major approaches to stochastic programming proposed before 1975 including the models described above can be found in the paper by Stancu-Minasian and Wets (1976). Since the ﬁrst international conference on multiple criteria decision making, held at the University of South Carolina in 1972 (Cochrane and Zeleny, 1972), it has been increasingly recognized that most of the real-world decision making problems usually involve multiple, noncommensurable, and conﬂicting objectives which should be considered simultaneously. One of the major systems-analytic multiobjective approaches to decision making under constraints is multiobjective optimization as a generalization of traditional single-objective optimization. For such multiobjective optimization problems, it is signiﬁcant to realize that multiple objectives are often noncommensurable and cannot be combined into a single objective. Moreover, the objectives usually conﬂict with each other and any improvement of one objective can be achieved only at the expense of at least one of the others. With this observation, in multiobjective optimization, the notion of Pareto optimality or efﬁciency was introduced instead of the optimality concept for single-objective optimization. However, decisions with Pareto optimality or efﬁciency are not uniquely determined; the ﬁnal decision must be selected from among Pareto optimal or efﬁcient solution set. For deriving a compromise or satisﬁcing solution for a DM in multiobjective optimization problems, goal programming approaches and interactive programming approaches have been developed. On the assumption that the DM can specify the goals of the objective functions, the goal programming approach ﬁrst appeared in a 1961 text by Charnes and Cooper (1961). Subsequent works on goal programming approaches have been numerously developed including Lee (1972), Ignizio (1976, 1982) and Charnes and Cooper (1977). The interactive programming approaches, which assume that the DM is able to give some preference information on a local level to a particular solution, were ﬁrst initiated by Geoffrion, Dyer and Feinberg (1972) and further developed by many researchers (Choo and Atkins, 1980; Sakawa

1.1 Background

3

and Seo, 1980; Wierzbicki, 1980; Sakawa, 1981; Changkong and Haimes, 1983; Steuer and Choo, 1983). By considering the randomness of parameters involved in multiobjective problems, stochastic programming with multiple objective functions were ﬁrst introduced by Contini (1968) as a goal programming approach to multiobjective stochastic programming, and further studied by Stancu-Minasian (1984). For deriving a compromise or satisﬁcing solution for the DM in multiobjective stochastic decision making situations, an interactive programming method for multiobjective stochastic programming with normal random variables were ﬁrst presented by Goicoecha, Hansen and Duckstein (1982) as a natural extension of the so-called STEP method (Benayoun, de Montgolﬁer, Tergny and Larichev, 1971) which is an interactive method for deterministic problems. An interactive method for multiobjective stochastic programming with discrete random variables, called STRANGE, was proposed by Teghem, Dufrane, Thauvoye and Kunsch (1986) and Slowinski and Teghem (1988). The subsequent works on interactive multiobjective stochastic programming have been accumulated (Klein, Moskowitz and Ravindran, 1990; Urli and Nadeau, 1990, 2004). There seems to be no explicit deﬁnitions of the extended Pareto optimality concepts for multiobjective stochastic programming, until White (1982) deﬁned the Pareto optimal solutions for the expectation optimization model and the variance minimization model. More comprehensive discussions were provided by Stancu-Minasian (1984) and Caballero et al. (2001) through the introduction of extended Pareto optimal solution concepts for the probability maximization model and the fractile criterion optimization model. An overview of models and solution techniques for multiobjective stochastic programming problems were summarized in the context of Stancu-Minasian (1990). Considering the vague nature of the DM’s judgments in multiobjective linear programming, a fuzzy programming approach was ﬁrst presented by Zimmermann (1978) and further studied by Leberling (1981), Hannan (1981), Sakawa (1983), Sakawa and Yano (1985a), Rommelfanger (1990) and Sakawa (1993). In these fuzzy approaches, it has been implicitly assumed that the fuzzy decision or the minimum operator by Bellman and Zadeh (1970) is the proper representation of the DM’s fuzzy preferences. However, considering the diversiﬁcation of preferences or values among people, the DM does not always feel that such representation is appropriate for combining the fuzzy goals and/or constraints, and consequently it becomes evident that an interaction process for the DM to derive a satisﬁcing solution is necessary. Assuming that the DM has a fuzzy goal for each of the objective functions in multiobjective programming problems, Sakawa and Yano (1985a) and Sakawa (1993) proposed several interactive fuzzy decision making methods by incorporating the desirable features of the interactive approaches into the fuzzy programming. Realizing the fact that the possible values of parameters involved in formulated problems are often only ambiguously known to experts, through the interpretation of the experts’ understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy subsets of the real line known as fuzzy number by Dubois and Prade (1978, 1980), Sakawa and Yano (1985b, 1989) introduced the

4

1 Introduction and Historical Remarks

concept of α-multiobjective programming and (M-) α-Pareto optimality based on the α-level sets of the fuzzy numbers. In most practical situations, however, it is natural to consider that the uncertainty in real-world decision making problems is often expressed by a fusion of fuzziness and randomness rather than either fuzziness or randomness. For handling not only the DM’s vague judgments in multiobjective problems but also the randomness of the parameters involved in the objectives and/or constraints, a fuzzy programming approach to multiobjective stochastic linear programming problems was ﬁrst taken by Hulsurkar, Biswal and Sinha (1997) by implicitly assuming that the fuzzy decision or the minimum operator is the proper representation of the DM’s fuzzy preferences. Realizing the drawbacks of the model of Hulsurkar, Biswal and Sinha (1997), Sakawa and his colleagues incorporated the techniques of an interactive fuzzy satisﬁcing method which was originally developed for deterministic problems (Sakawa, 1993; Sakawa and Yano, 1985a) into multiobjective stochastic programming problems, through the introduction of several fuzzy multiobjective stochastic programming models based on different optimization criteria such as expectation optimization (Sakawa, Kato and Nishizaki, 2003), variance minimization (Sakawa, Kato and Katagiri, 2002), probability maximization (Sakawa and Kato, 2002; Sakawa, Kato and Katagiri, 2004) and fractile criterion optimization (Sakawa, Katagiri and Kato, 2001), to derive a satisﬁcing solution for a DM from the extended Pareto optimal solution sets. When we model actual decision making situations under uncertainty as multiobjective stochastic programming problems, it is implicitly assumed that uncertain parameters or coefﬁcients involved in formulated problems can be expressed as random variables in probability theory. This means that the realized values of random parameters under the occurrence of some event are assumed to be deﬁnitely represented with real values. However, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts’ ambiguous understanding of the realized values of random parameters under the occurrence of events as fuzzy numbers. From such a point of view, a fuzzy random variable was ﬁrst introduced by Kwakernaak (1978), and its mathematical basis was constructed by Puri and Ralescu (1986). An overview of the developments of fuzzy random variables was found in the article of Gil, Lopez-Diaz and Ralescu (2006). Studies on linear programming problems with fuzzy random variable coefﬁcients, called fuzzy random linear programming problems, were initiated by Wang and Qiao (1993) and Qiao, Zhang and Wang (1994) as seeking the probability distribution of the optimal solution and optimal value. Optimization models of fuzzy random linear programming were ﬁrst developed by Luhandjula (1996) and Luhandjula and Gupta (1996), and further studied by Liu (2001a,b) and Rommelfanger (2007). A brief survey of major fuzzy stochastic programming models including fuzzy random programming was found in the paper by Luhandjula (2006). On the basis of possibility measure, possibilistic programming approaches to fuzzy random linear programming problems were introduced (Katagiri, Ishii and Itoh, 1997; Katagiri, Ishii and Sakawa, 2000; Katagiri and Sakawa, 2003). Multiobjective ex-

1.1 Background

5

tensions to fuzzy random linear programming were also considered, and several possibility-based fuzzy random programming models through the combination of a concept of possibility measure and different stochastic programming models were further developed (Katagiri, Sakawa and Ishii, 2001; Katagiri, Sakawa and Ohsaki, 2003; Katagiri, Sakawa, Kato and Nishizaki, 2008a; Katagiri, Sakawa, Kato and Ohsaki, 2003). Extensions to multiobjective integer problems with fuzzy random variables were provided by incorporating the branch-and-bound method into the interactive methods (Katagiri, Sakawa, Kato and Nishizaki, 2004; Katagiri, Sakawa and Nishizaki, 2006; Perkgoz, Katagiri, Sakawa and Kato, 2004). As a natural extension of the concept of M-α-Pareto optimality (Sakawa, 1993), Katagiri et al. (2004) and Katagiri, Sakawa, Kato and Nishizaki (2008b) deﬁned the extended Pareto optimality concepts in multiobjective fuzzy random programming by combining the concept of the M-α-Pareto optimality and optimization criteria in stochastic programming. Assuming that a DM decides that the degrees of all the membership functions of the fuzzy random variables should be greater than or equal to some value α, through the interpretation of the original fuzzy random programming problems as stochastic programming problems which depend on the speciﬁed degree α, they provided an interactive algorithm to derive a satisﬁcing solution from among the set of the extended M-α-Pareto optimal solutions. Recently, from a viewpoint of ambiguity and randomness different from the aforementioned fuzzy random variables, by considering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and mathematical programming problems with random fuzzy variables were formulated together with the development of a simulationbased approximate solution method (Liu, 2002). By focusing on the case where the means of random variables are represented with fuzzy numbers, solution methods for random fuzzy programming problems were developed through nonlinear programming techniques (Katagiri, Ishii and Sakawa, 2002; Katagiri, Hasuike, Ishii and Nishizaki, 2008; Hasuike, Katagiri and Ishii, 2009). So far, we have restricted ourselves to mathematical programming problems where decisions are made by a single DM. However, decision making problems in hierarchical managerial or public organizations are often formulated as two-level mathematical programming problems where there are two DMs: the upper level DM and the lower level DM. Assuming that the lower level DM behaves rationally, that is, optimally responds to the decision of the upper level DM, the upper level DM also speciﬁes a strategy so as to optimize his/her own objective. Computational methods for a pair of such decisions or strategies of both DMs, called a Stackelberg solution, are classiﬁed roughly into three categories: the vertex enumeration approach based on a characteristic that an extreme point of a set of rational responses of the lower level DM is also an extreme point of the feasible region (Bialas and Karwan, 1984), the Kuhn-Tucker approach in which the upper level problem with constraints including optimality conditions of the lower level problem is solved (Bialas and Karwan, 1984; J´udice and Faustino, 1992; Bard and Falk, 1982; Bard and Moore, 1990; Bard, 1983), and the penalty function approach which adds a penalty term

6

1 Introduction and Historical Remarks

to the upper level objective function so as to satisfy optimality of the lower level problem (White and Anandalingam, 1993; Hansen, Jaumard and Savard, 1992). In order to deal with multiobjective problems in hierarchical decision making, Nishizaki and Sakawa (1999) formulated two-level multiobjective linear programming problems and developed a computational method for obtaining the corresponding Stackelberg solution. Considering stochastic events related to hierarchical decision making situations, Nishizaki, Sakawa and Katagiri (2003) and Katagiri, Nishizaki, Sakawa and Kato (2007) formulated two-level programming problems with random variables and developed algorithms for deriving the Stackelberg solutions based on some optimization criteria such as the expectation optimization and the variance minimization. Noncooperative fuzzy random two-level linear programming problems were also considered, and computational methods for obtaining the corresponding Stackelberg solutions were developed (Sakawa and Kato, 2009c; Sakawa and Katagiri, 2010b; Sakawa, Katagiri and Matsui, 2010b) From the viewpoint of taking the possibility of coordination or bargaining between DMs into account, attempts to derive Pareto optimal solutions to two-level linear programming problems were made (Wen and Hsu, 1991; Wen and Lin, 1996). Furthermore, under the assumption of communication and a cooperative relationship between DMs, solution methods for obtaining a Pareto optimal solution to a multi-level linear programming problem were presented by Lai et al. (Lai, 1996; Shih, Lai and Lee, 1996). After removing some drawbacks of the methods of Lai et al., Sakawa, Nishizaki and Uemura (1998) presented interactive fuzzy programming for two-level linear programming problems, and extended their model to treat two-level linear programming problems with fuzzy parameters (Sakawa, Nishizaki and Uemura, 2000) and those with random variables (Sakawa, Kato, Katagiri and Wang, 2003; Perkgoz, Sakawa, Kato and Katagiri, 2003; Kato, Wang, Katagiri and Sakawa, 2004; Sakawa and Kato, 2009a; Sakawa and Katagiri, 2010a). Extension to two-level integer programming problems (Sakawa, Katagiri and Matsui, 2010a) and fuzzy random linear two-level programming problems (Katagiri, Sakawa, Kato and Nishizaki, 2007; Sakawa and Kato, 2009b) were also provided. Consider computational aspects of interactive fuzzy satisﬁcing methods for multiobjective programming problems under fuzzy stochastic environments. Because fuzzy multiobjective stochastic programming problems with discrete decision variables are difﬁcult to solve strictly, it becomes important to develop highly efﬁcient approximate computational methods. As an efﬁcient meta-heuristics, genetic algorithms initiated by Holland (1975) have attracted the attention of many researchers with applicability in optimization, as well as in search and learning. Furthermore, publications of books by Goldberg (1989) and Michalewicz (1996) brought heightened and increasing interest in applications of genetic algorithms to complex function optimization. From an interactive multiobjective programming perspective, Sakawa and his colleagues have been advancing genetic algorithms to derive satisﬁcing solutions to multiobjective optimization problems with discrete decision variables (Sakawa, Kato, Sunada and Shibano, 1997; Sakawa, 2001). As a generalization of results along this line, interactive fuzzy satisﬁcing methods for multiobjective stochastic integer programming problems for deriving satisﬁcing solutions

1.2 Description of contents

7

for a DM from among the set of the extended Pareto optimal solutions have also been developed (Perkgoz, Kato, Katagiri and Sakawa, 2004; Kato, Perkgoz, Katagiri and Sakawa, 2004; Perkgoz, Sakawa, Kato and Katagiri, 2005).

1.2 Description of contents Although studies on multiobjective mathematical programming under uncertainty have been accumulated and several books on multiobjective mathematical programming under uncertainty have been published (e.g. Stancu-Minasian (1984); Slowinski and Teghem (1990); Sakawa (1993); Lai and Hwang (1994); Sakawa (2000)), there seems to be no book which concerns both randomness of events related to environments and fuzziness of human judgments simultaneously in multiobjective decision making problems. In this book, the authors are concerned with introducing the latest advances in the ﬁeld of multiobjective optimization under both fuzziness and randomness on the basis of authors’ continuing research works. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective programming for human-centered systems under uncertainty in most realistic situations when dealing with both fuzziness and randomness. Organization of each chapter is brieﬂy summarized as follows. Chapter 2 is devoted to mathematical preliminaries, which will be used throughout the remainder of this book. Starting with basic notions in fuzzy set theory, linear programming problems with fuzzy goals and fuzzy constraints, as well as linear programming problems with fuzzy parameters, are introduced. From the probabilistic point of view, stochastic programming such as two-stage programming and chance constraint programming is outlined. After reviewing the solution concepts in multiobjective programming, interactive fuzzy multiobjective programming is presented. Several basic notions including Pareto optimal solutions and Stackelberg equilibrium in two-level programming under cooperative and noncooperative decision making situations are reviewed. Fundamentals of genetic algorithms are also brieﬂy discussed as efﬁcient meta-heuristics for solving difﬁcult classes of optimization problems with nonconvex nonlinear objective functions and/or constraints. In Chapter 3, by considering the imprecision of a decision maker’s (DM’s) judgment for stochastic objective functions and/or constraints in multiobjective problems, fuzzy multiobjective stochastic programming is developed. After reviewing the main ideas of conventional single-objective stochastic programming problems such as two-stage problems and chance constrained programming problems, multiobjective stochastic programming problems are formulated by assuming that the DM has a fuzzy goal for each of the expectation and the variance of the original stochastic objective functions. In order to reﬂect the diversity of criteria for optimizing the stochastic objective functions, different optimization criteria from the expectation and the variance are also provided. Namely, we present the probability model maximizing the probability of the objective functions being greater than or equal to target values and the fractile model optimizing the target values under a given

8

1 Introduction and Historical Remarks

probability. It is shown that the original stochastic problems can be transformed into deterministic multiobjective problems with linear, linear fractional or nonlinear objective functions and/or constraints. Through the introduction of extended Pareto optimal solution concepts, interactive fuzzy satisﬁcing methods using linear or convex programming techniques are presented to derive a satisﬁcing solution for the DM from among the extended Pareto optimal solution set. The presented interactive methods for the multiobjective stochastic programming with continuous variables are immediately extended to deal with the integer problems with multiple stochastic objective functions and/or constraints by utilizing some genetic algorithms (Sakawa, 2001). In Chapter 4, taking account of not only the randomness of parameters involved in objective functions and/or constraints but also the experts’ ambiguous understanding of the realized values of the random parameters, we formulate multiobjective programming problems with fuzzy random variables. By incorporating a concept of possibility measure into stochastic programming models discussed in the previous chapter, four types of optimization models for fuzzy random programming are developed. Through the introduction of the extended Pareto optimal solution concepts on the basis of possibility theory and probability theory, considerable efforts are devoted to development of interactive methods for fuzzy random multiobjective programming to derive a satisﬁcing solution for a DM. Furthermore, to take the level set approach by Sakawa (1993), some Pareto optimality concepts for fuzzy random multiobjective programming problems are deﬁned by combining the notions of the M-α-Pareto optimality and optimization criteria in stochastic programming, and then interactive satisﬁcing methods are presented for deriving a satisﬁcing solution for the DM from among the extended M-α-Pareto optimal solution set. The presented interactive methods are extended to deal with integer programming problems with fuzzy random variables. In Chapter 5, for resolving conﬂict of decision making problems in hierarchical managerial or public organizations where there exist two DMs who have different interests in making decisions, two-level programming problems are discussed. By considering the uncertainty involved in cooperative and noncooperative decision making situations, two-level programming problems with random variable and fuzzy random variable coefﬁcients are formulated. By employing the optimization models provided in Chapters 3 and 4, the original problems are transformed into the deterministic two-level programming problems. For deriving a well-balanced satisfactory solution between the two DMs’ satisfaction degrees, interactive fuzzy programming methods for cooperative stochastic and fuzzy random two-level programming problems are developed. Moreover, in order to obtain Stackelberg solutions to noncooperative stochastic and fuzzy random two-level programming problems, computational methods using the combination of convex programming techniques and the branch-and-bound method are presented. Using the genetic algorithms proposed by Sakawa (2001), the presented methods are extended to deal with two-level integer programming problems with random variables and fuzzy random variables. Finally, Chapter 6 outlines some future research directions. In contrast to the fuzzy random variables dealing with the ambiguity of realized values of random

1.2 Description of contents

9

variables discussed in Chapter 4, random fuzzy variables are introduced as parameters or coefﬁcients involved in linear programming problems by considering the ambiguity of means and variances of random variables. Attention is particulary focused on the case where the mean of each of random variables is represented by a fuzzy number. From the viewpoint of simultaneous maximization of possibility and probability, the original random fuzzy linear programming problem is equivalently transformed into a deterministic nonlinear programming problem. The presented decision making model is extended to deal with multiobjective and two-level programming problems with random fuzzy variables.

Chapter 2

Fundamentals

This chapter is devoted to reviewing optimization concepts and the related computational methods that will be used in the remaining chapters. In particular, we deal with three optimization concepts incorporating fuzziness and ambiguity of human judgments, uncertainty of events characterizing decision making problems, and multiplicity of evaluation criteria. Fuzzy programming which is developed in order to take into account fuzziness and ambiguity of human judgments, and this method is ﬁrst presented together with the basic concepts in fuzzy set theory. Fundamentals of stochastic programming are also provided for decision problems under probabilistic uncertainty. Speciﬁcally, two-stage programming and chance constraint programming are covered. To meet expectations of diversiﬁed evaluations, multiobjective programming has been investigated. After presenting interactive methods for multiobjective linear programming problems, we give some techniques to deal with not only multiple objectives but also fuzzy goals for the objectives. In an organization with a hierarchical structure, we often ﬁnd decision making with two or more decision makers (DMs) attempting to optimize their objective functions. To model such a decision making problem mathematically, a two-level programming problem is formulated, and cooperative and noncooperative solution methods are presented for two-level programming. Genetic algorithms are considered to be one of the most practical and proven meta heuristics for difﬁcult classes of optimization problems, and ﬁnally basic concepts of genetic algorithms are provided.

2.1 Fuzzy programming 2.1.1 Fuzzy sets Throughout this book, the concepts of fuzzy set theory are used together with those of stochastic theory. Before discussing fuzzy programming, we provide the fundamentals of fuzzy set theory.

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_2, © Springer Science+Business Media, LLC 2011

11

12

2 Fundamentals

A fuzzy set initiated by Zadeh (1965) is deﬁned as follows: Deﬁnition 2.1 (Fuzzy set). Let X denote a universal set. Then, a fuzzy set A˜ is deﬁned by its membership function μA˜ : X → [0, 1].

(2.1)

We deal with mathematical programming in this book, and therefore the set X in Deﬁnition 2.1 means the real line R generally. The membership function μA˜ assigns to each element x ∈ X a real number μA˜ (x) in the interval [0, 1], and the value of ˜ A fuzzy set A˜ is represented by μA˜ (x) represents the grade of membership of x in A. a pair of an element x and its grade μA˜ (x), and thus it is often written as A˜ = {(x, μA˜ (x)) | x ∈ X}. For a given ordinary set A, the characteristic function 1 if x ∈ A cA (x) = 0 if x ∈ A

(2.2)

(2.3)

deﬁnes the set A which is expressed by A = {x ∈ X | cA (x) = 1}.

(2.4)

From these deﬁnitions, one ﬁnds that a fuzzy set A˜ is a natural extension of an ordinal set A. The concept of α-level sets is very important to serve a transfer relation between a fuzzy set and an ordinary set. Deﬁnition 2.2 (α-level set). For a given α ∈ [0, 1], the α-level set of a fuzzy set A˜ is deﬁned as an ordinary set Aα of elements x such that the membership function value μA˜ (x) of x exceeds α, i.e., Aα = {x | μA˜ (x) ≥ α}.

(2.5)

The extension principle introduced by Zadeh (1965) is to provide a general way for extending nonfuzzy mathematical concepts to the fuzzy framework. Deﬁnition 2.3 (Extension principle). Let f : X → Y be a mapping from a set X to a set Y . Then, the extension principle allows us to deﬁne the fuzzy set B˜ in Y induced by the fuzzy set A˜ in X through f as follows: B˜ = {(y, μB˜ (y)) | y = f (x), x ∈ X} with μB˜ (y) μ f (A) ˜ (y) =

⎧ ⎨ sup μA˜ (x)

if f −1 (y) = 0/

⎩

/ if f −1 (y) = 0,

y= f (x)

0

(2.6)

(2.7)

2.1 Fuzzy programming

13

where f −1 (y) is the inverse image of y, and 0/ means the empty set. Because we deal with mathematical programming under uncertainty, among fuzzy sets, fuzzy numbers which are linguistically-expressed such as “approximately m” or “about n” play important roles. Before deﬁning fuzzy numbers, we give the deﬁnitions of convex and normalized fuzzy sets. A fuzzy set A˜ is said to be convex if any α-level set Aα of A˜ is convex, and a fuzzy set A˜ is said to be normal if there is x such that μA˜ (x) = 1. Deﬁnition 2.4 (Fuzzy number). A fuzzy number is a convex normalized fuzzy set of the real line R whose membership function is piecewise continuous. By using the extension principle by Zadeh, the binary operation “∗” in R can be extended to the binary operation “” of fuzzy numbers M˜ and N˜ as μM ˜ N˜ (z) = sup min{μM˜ (x), μN˜ (y)}.

(2.8)

z=x∗y

For example, consider an extension of addition “+” of two numbers. By using (2.8), we can deﬁne the extended addition “⊕” of two fuzzy numbers as follows: μM⊕ ˜ N˜ (z) = sup min{μM˜ (x), μN˜ (y)} z=x+y

= sup min{μM˜ (x), μN˜ (z − x)}. x∈R

To provide easy computation of fuzzy numbers, Dubois and Prade (1978) introduce the concept of L-R fuzzy numbers. Deﬁnition 2.5 (L-R fuzzy number). A fuzzy number M˜ is said to be an L-R fuzzy number if ⎧ m−x ⎪ ⎪ if x ≤ m ⎨L α (2.9) μM˜ (x) = x−m ⎪ ⎪ if x ≥ m, ⎩R β ˜ and α and β are positive numbers which represent where m is the mean value of M, left and right spreads of the fuzzy number, respectively; a function L is a left shape function satisfying (i) L(x) = L(−x), (ii) L(0) = 1, (iii) L(x) is nonincreasing on [0, ∞); and a right shape function R is similarly deﬁned as L. By using its mean, left and right spreads, and shape functions, an L-R fuzzy number M˜ is symbolically written as M˜ = (m, α, β)LR .

(2.10)

For two L-R fuzzy numbers M˜ = (m, α, β)LR and N˜ = (n, γ, δ)LR , the extended addition M˜ ⊕ N˜ is calculated as (m, α, β)LR ⊕ (n, γ, δ)LR = (m + n, α + γ, β + δ)LR ,

(2.11)

14

2 Fundamentals

and scalar multiplication for an L-R fuzzy numbers M˜ = (m, α, β)LR and a scalar value λ is also given as if λ > 0 (λm, λα, λβ)LR (2.12) λ ⊗ (m, α, β)LR = (λm, −λα, −λβ)LR if λ < 0, where ⊗ means the extended multiplication. The other operations are similarly given, and for further information on this issue, refer to Dubois and Prade (1978).

2.1.2 Fuzzy goals and Fuzzy constraints Zimmermann (1976) introduces the concept of fuzzy set theory into linear programming. Assuming that the membership functions for fuzzy sets are linear, he shows that, by employing the principle of the fuzzy decision by Bellman and Zadeh (1970), a linear programming problem with a fuzzy goal and fuzzy constraints can be solved by using standard linear programming techniques. A linear programming problem is represented as ⎫ minimize z(x1 , . . . , xn ) = c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎬ ········· (2.13) ⎪ ⎪ am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n, where x j is a decision variable, and c j , ai j and bi are given coefﬁcients of the objective function and the constraints. Let x = (x1 , . . . , xn )T denote a column vector of the decision variables, and let c = (c1 , . . . , cn ), A = [ai j ], i = 1, . . . , m, j = 1, . . . , n and b = (b1 , . . . , bm )T denote an n dimensional row vector of the coefﬁcients of the objective function, an m × n matrix of the coefﬁcients of the left-hand side of the constraints, and an m dimensional column vector of the coefﬁcients of the righthand side of the constraints, respectively; the superscript T means transposition of a vector or a matrix. Then, (2.13) is simply rewritten in the following vector and matrix representation: ⎫ minimize z(x) = cx ⎬ subject to Ax ≤ b (2.14) ⎭ x ≥ 0. To a standard linear programming problem (2.14), taking into account the imprecision or fuzziness with respect to the judgment of a decision maker (DM), Zimmermann formulates the following linear programming problem with a fuzzy goal and fuzzy constraints:

2.1 Fuzzy programming

15

cx z0 Ax b

(2.15) (2.16)

x ≥ 0,

(2.17)

where the symbol denotes a relaxed or fuzzy version of the ordinary inequality ≤. From the DM’s preference, the fuzzy goal (2.15) and the fuzzy constraints (2.16) mean that the objective function cx should be essentially smaller than or equal to a certain level z0 , and that the values of the constraints Ax should be substantially smaller than or equal to b, respectively. Assuming that the fuzzy goal and the fuzzy constraints are equally important, he employs the following uniﬁed formulation: Bx b

(2.18) x ≥ 0, where B=

z c , b = 0 . b A

(2.19)

To express the imprecision or fuzziness of the DM’s judgment, the ith fuzzy constraint (Bx)i b i is interpreted as the following linear membership function: ⎧ 1 ⎪ ⎪ ⎨ (Bx)i − b i μi ((Bx)i ) = 1 − ⎪ di ⎪ ⎩ 0

if (Bx)i ≤ b i if b i < (Bx)i ≤ b i + di

(2.20)

if (Bx)i > b i + di ,

where di is a subjectively speciﬁed constant expressing the limit of the admissible violation of the ith constraint, and it is depicted in Fig. 2.1. μi ((Bx)i ) 1

0

b’ i

b’i + d i

(Bx) i

Fig. 2.1 Linear membership function for the ith fuzzy constraint.

On the basis of the principle of the fuzzy decision by Bellman and Zadeh (1970), a mathematical programming problem for ﬁnding the maximum decision is represented as maximize min μi ((Bx)i ) 0≤i≤m+1 (2.21) subject to x ≥ 0.

16

2 Fundamentals

With the variable transformation b

i = b i /di and (B x)i = (Bx)i /di , and an auxiliary variable λ, (2.21) can be transformed into the following conventional linear programming problem: ⎫ maximize λ ⎬ subject to λ ≤ 1 + b

i − (B x)i , i = 0, . . . , m + 1 (2.22) ⎭ x ≥ 0. Because the fuzzy decision is represented as min0≤i≤m+1 μi ((Bx)i ) in (2.21), it is often called the minimum operator.

2.1.3 Linear programming problems with fuzzy parameters To describe a real-world decision situation mathematically, the formulation of linear programming is often employed due to its simplicity and ease in computation, and naturally, possible values of the parameters in a linear programming problem may be assigned by experts for the decision situations. In most situations, however, it is observed that the experts may know the possible values of the parameters only imprecisely or ambiguously. From this reason, to express the imprecise judgments of the experts, the following linear programming problem involving fuzzy parameters is formulated: ⎫ minimize C˜1 x1 + · · · + C˜n xn ⎪ ⎪ ⎪ subject to A˜ 11 x1 + · · · + A˜ 1n xn ≤ B˜ 1 ⎪ ⎬ ········· (2.23) ⎪ A˜ m1 x1 + · · · + A˜ mn xn ≤ B˜ m ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n, where C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n are fuzzy parameters represented by fuzzy numbers. Let the membership function of the fuzzy parameters C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n be denoted by μC˜ j , μA˜ i j , and μB˜i , i = 1, . . . , m, j = 1, . . . , n, respectively. Here, we give two approaches for solving the linear programming problem with fuzzy numbers (2.23): one is an approach based on the possibility-based model (Dubois and Prade, 1980) and the other is the level set-based model (Sakawa, 1993).

2.1.3.1 Possibility-based model To deal with binary relations between a pair of fuzzy numbers M˜ and N˜ from the viewpoint of possibility, Dubois and Prade (1978, 1980) give the following index: ˜ ∞ = sup{min{μM˜ (u), μN˜ (v)} | u ≥ v}. Pos M˜ ≥ N˜ = ΠM˜ N, (2.24)

2.1 Fuzzy programming

17

The index (2.24) can be interpreted as the degree of possibility that M˜ is larger than ˜ or equal to N. For a given degree α, the α-level sets Mα = {u ∈ R | μM˜ (u) ≥ α} and Nα = {v ∈ R | μN˜ (v) ≥ α} of the fuzzy M˜ and N˜ are represented by closed intervals Lnumbers L R R Mα = mα , mα and Nα = nα , nα , respectively. Then, it is known that Pos M˜ ≥ N˜ ≥ α if and only if mRα ≥ nLα .

(2.25)

Using this binary relation from the viewpoint of possibility, if the DM speciﬁes a certain degree α of possibility, the constraints of (2.23) can be interpreted as x ∈ Xpos (α) x ≥ 0 | Pos A˜ i1 x1 + · · · + A˜ in xn ≤ B˜ i ≥ α, i = 1, . . . , m . (2.26) The α-level set of the fuzzy number B˜ i for the right-hand side constant in the ith constraint is represented by (2.27) Biα = {u ∈ R | μB˜ i (u) ≥ α} = bLiα , bRiα . Similarly, the α-level set of the vector of fuzzy numbers A˜ i = (a˜i1 , . . . , a˜in ) in the left-hand side coefﬁcients of the ith constraint is represented by (2.28) Aiα = (ai1 , . . . , ain ) | ai j ∈ aLijα , aRijα , j = 1, . . . , n , and, (2.28) is simply denoted by Aiα = aLiα , aRiα . From the proposition (2.25), (2.26) can be represented as x ∈ Xpos (α) = x ≥ 0 | aLiα x ≤ bRiα , i = 1, . . . , m .

(2.29)

(2.30)

For the sake of simplicity, assume that all the fuzzy parameters in (2.23) are represented by L-R fuzzy numbers: C˜ j = (c j , β j , γ j )LR , j = 1, . . . , n, A˜ i j = (ai j , δi j , εi j )LR , i = 1, . . . , m, j = 1, . . . , n, B˜i = (bi , ζi , ηi )LR , i = 1, . . . , m; and L(y) = R(y) = max{0, 1 − |y|}. Then, (2.30) can be rewritten by ordinary linear inequalities n x ∈ Xpos (α) = x ≥ 0 ∑ {ai j − (1 − α)δi j }x j ≤ bi + (1 − α)ηi , i = 1, . . . , m . j=1

(2.31) As for the objective function of (2.23), we introduce the following linear membership function of the fuzzy goal:

μG˜ (y) =

⎧ 1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1− 0

y − z0 z1 − z0

if y < z1 if z1 ≤ y ≤ z0 if y ≥ z0 ,

(2.32)

18

2 Fundamentals

where z0 and z1 denote the values of the objective function such that the degrees of the membership function μG˜ are 0 and 1, respectively. Because from the assumption of the fuzzy parameters the objective function C˜1 x1 +· · ·+ C˜n xn of (2.23) can be represented by (∑nj=1 c j x j , ∑nj=1 β j x j , ∑nj=1 γ j x j )LR , with appropriate values of the parameters z0 and z1 of the fuzzy goal (2.32), the degree of possibility with respect to the objective function is expressed as n

∑ (β j − c j )x j + z0

˜ = sup min{μ ˜ (y), μ ˜ (y)} = ΠCx ˜ (G) Cx G

j=1 n

∑ β jx j − z

1

,

(2.33)

+z

0

j=1

where μCx ˜ is a membership function of the objective function for a given vector x of the decision variables. Under the assumption of the fuzzy parameters, a linear programming problem with fuzzy parameters in the possibility-based model is formulated as n

∑ (β j − c j )x j + z0

j=1 n

maximize

∑ β j x j − z1 + z0

j=1 n

subject to

∑ {a1 j − (1 − α)δ1 j }x j ≤ b1 + (1 − α)η1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ········· ⎪ ⎪ n ⎪ ⎪ ⎪ {a − (1 − α)δ }x ≤ b + (1 − α)η mj j m m⎪ ⎪ ∑ mj ⎪ ⎪ j=1 ⎪ ⎭ x j ≥ 0, j = 1, . . . , n.

(2.34)

j=1

Because (2.34) is a linear fractional programming problem, it can be solved by using the Charnes and Cooper method (1962) or the Bitran and Novaes method (1973).

2.1.3.2 Level set-based model Consider the level set-based model for solving a linear programming problem involving fuzzy parameters (Sakawa, 1993). At the beginning, we give the formulation (2.23) of a linear programming problem involving fuzzy parameters once again. minimize C˜1 x1 + · · · + C˜n xn subject to A˜ 11 x1 + · · · + A˜ 1n xn ≤ B˜ 1 ········· A˜ m1 x1 + · · · + A˜ mn xn ≤ B˜ m x j ≥ 0, j = 1, . . . , n.

2.1 Fuzzy programming

19

For a ceratin degree α speciﬁed by the DM, the α-level set of all of the fuzzy numbers C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n in (2.23) is deﬁned as the ordinary set (C, A, B)α for which their membership function values exceed the degree α, i.e., (C, A, B)α = {(c, a, b) | μC˜ j ≥ α, μA˜ i j ≥ α, μB˜ i ≥ α, i = 1, . . . , m, j = 1, . . . , n}, (2.35) where c, a, and b are an n dimensional row constant coefﬁcient vector for the objective function, an m × n constant coefﬁcient matrix for the left-hand side of the constraints, and an m dimensional column right-hand side constant vector for the constraints, respectively. It should be noted here that the α-level sets have the following property: α1 ≤ α2 ⇔ (C, A, B)α1 ⊇ (C, A, B)α2 . (2.36) Suppose the DM thinks that the degree of all of the membership functions of the fuzzy numbers involved in (2.23) should be larger than or equal to a certain degree α. Then, for such a degree α, (2.23) can be interpreted as the following nonfuzzy linear programming problem which depends on the coefﬁcient vector (c, a, b) ∈ (C, A, B)α : ⎫ minimize c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎬ ········· (2.37) ⎪ am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n. Observe that there exist an inﬁnite number of linear programming problems such as (2.37) depending on the coefﬁcient vector (c, a, b) ∈ (C, A, B)α , and the values of (c, a, b) are arbitrary for any (c, a, b) ∈ (C, A, B)α in the sense that the degree of all of the membership functions for fuzzy numbers in the linear programming problem involving fuzzy parameters (2.23) exceeds the degree α speciﬁed by the DM. However, if possible, it would be desirable for the DM to choose (c, a, b) ∈ (C, A, B)α in (2.37) so as to minimize the objective function under the constraints. From such a point of view, for the given degree α, it seems to be quite natural to have the linear programming problem involving fuzzy parameters (2.23) as the following nonfuzzy α-linear programming problem: ⎫ minimize c1 x1 + · · · + cn xn ⎪ ⎪ ⎪ subject to a11 x1 + · · · + a1n xn ≤ b1 ⎪ ⎪ ⎪ ⎬ ········· (2.38) am1 x1 + · · · + amn xn ≤ bm ⎪ ⎪ ⎪ ⎪ x j ≥ 0, j = 1, . . . , n ⎪ ⎪ ⎭ (c, a, b) ∈ (C, A, B)α . It should be emphasized here that in the nonfuzzy α-linear programming problem (2.38), the parameters (c, a, b) are treated as decision variables rather than constants of the coefﬁcients.

20

2 Fundamentals

From the property of the α-level set for the fuzzy numbers C˜ j , A˜ i j , and B˜ i , i = 1, . . . , m, j = 1, . . . , n, the feasible regions for c j , ai j , and bi can be denoted by the closed intervals [cLjα , cRjα ], [aLijα , aRijα ], and [bLiα , cRiα ], respectively. Therefore, we can obtain an optimal solution to the nonfuzzy α-linear programming problem (2.38) by solving the following linear programming problem: ⎫ minimize cL1 x1 + · · · + cLn xn ⎪ ⎪ ⎪ subject to aL11 x1 + · · · + aL1n xn ≤ bR1 ⎪ ⎬ ········· (2.39) ⎪ aLm1 x1 + · · · + aLmn xn ≤ bRm ⎪ ⎪ ⎪ ⎭ x j ≥ 0, j = 1, . . . , n. It is important to realize here that (2.39) is no more nonlinear but an ordinary linear programming problem, and consequently it is easy to solve it by using linear programming techniques such as the simplex method.

2.2 Stochastic programming 2.2.1 Random variables An outcome of a random trial such as coin tossing is called a sample point, which is denoted by ω, and the set of all of the possible outcomes of the random trial is called a sample space, which is denoted by Ω. An event is a subset of the sample space. For any events A and B, the union and the intersection are deﬁned as follows: (i) Union of A and B: A ∪ B = {ω | ω ∈ A or ω ∈ B}. (ii) Intersection of A and B: A ∩ B = {ω | ω ∈ A and ω ∈ B}. Furthermore, for any event A, the compliment is deﬁned as (iii) Compliment of A: Ac = {ω | ω ∈ A}. A family B of subsets of the sample space Ω which has the following properties is called a σ-ﬁeld: (i) Ω ∈ B; (ii) A ∈ B ⇒ Ac ∈ B; (iii) A1 , A2 , . . . ∈ B ⇒

∞ k=1

Ak ∈ B.

2.2 Stochastic programming

21

Let S be a family of subsets of the sample space Ω. The intersection of all σﬁelds containing S is called the σ-ﬁeld generated by S, and it is denoted by σ[S]. Let E = (E, d) be a metric space, and let O be a family of all open subsets of E. Then, a σ-ﬁeld σ[O] generated by O is called a family of Borel sets on E. Consider the d-dimensional Euclidean space Rd as a special case of E. A family of all open subsets of Rd is deﬁned as Jd {(a1 , b1 ] × · · · × (ad , bd ] ⊂ Rd | −∞ ≤ ak ≤ bk ≤ +∞, k = 1, . . . , d}. (2.40) Then, σ[Jd ] is a σ-ﬁeld in Rd , and it is called a family of d-dimensional Borel sets on Rd . Let Ω be a sample space, and let B be a σ-ﬁeld in the sample space Ω. A realvalued function P deﬁned on the σ-ﬁeld B in Ω satisfying the following conditions is called a probability measure: (i) 0 ≤ P(A) ≤ 1 for A ∈ B; (ii) P(Ω) = 1; (iii) If A1 , A2 , . . . ∈ B and A1 , A2 , . . . is a disjoint sequence, then ∞ ∞ Ak = ∑ P(Ak ). P k=1

k=1

If B be a σ-ﬁeld in a sample space Ω and P is a probability measure on B, the triple (Ω, B, P) is called a probability measure space or simply a probability space. For an arbitrary probability space (Ω, B, P), let X(ω) be a real-valued function on Ω. Then, X(ω) is a random variable if {ω | X(ω) ≤ x} ∈ B

(2.41)

holds for each real value x. For the random variable X(ω), the function F(x) = P({ω | X(ω) ≤ x})

(2.42)

is called a distribution function, where P({ω | X(ω) ≤ x}) is often simply denoted by P(X ≤ x). The distribution function has the following properties: (i) lim F(x) = 0 and lim F(x) = 1; x→−∞

x→+∞

(ii) If x < y, then F(x) ≤ F(y). In this book, henceforth to discriminate a random variable from the other variables or parameters, we attach a bar “–” to a character like d.¯

2.2.2 Two-stage programming In the real-world decision making problems, some stochastic events may inﬂuence elements characterizing decision making problems such as demands of products,

22

2 Fundamentals

the amount of available resources and so forth. When such a decision making problem under uncertainty is formulated as a linear programming problem, it may be difﬁcult that the constraints of the problem always hold completely. Then, a shortage or an excess comes from the violation of the constraints, and the corresponding penalties are imposed as the occasion demands. From this point of view, two-stage programming had been investigated from the beginning of the development of linear programming (Beale, 1955; Dantzig, 1955; Wets, 1966; Everitt and Ziemba, 1978). To understand the framework of two-stage programming, consider a decision problem of a manufacturing company. Let x = (x1 , . . . , xn )T denote activity levels in a production plant of the company, and then w = T x denotes the amount of products, where T is an m × n matrix transforming the n kinds of activity levels into the m types of products. Let d¯i be the demand for the ith product which is only known in probability, and assume that the random variable d¯i of the demand is indicated by the probability distribution function Fi (di ) P(d¯i ≤ di ). Suppose that the DM selects the activity levels, say x = xˆ , and after the occurrence of the random event ˆ Then, if yˆ+ dˆi − wˆ i ≥ 0, the shortage the demands d¯ = (d¯1 , . . . , d¯m )T are ﬁxed at d. i − + ˆ of the ith product is yˆi , and if yˆi wˆ i − di ≥ 0, the excess of the ith product is yˆ− i . This situation can be formulated as ¯ T x + Iy+ − Iy− = d, where y+ and y− represent the errors for estimating the demands, and I is the m dimensional identity matrix. Let q+ and q− denote the penalty costs for making these errors, and let c be the original costs for the activities in the production plant. Then, the objective function to be minimized may be the expectation of cx + q+ y+ + q− y− . Adding the constraints Ax ≤ b for the activity levels such as the capacity, budget, technology, etc., we can formulate the standard form of the two-stage programming problem as ⎫ minimize cx + E q+ y+ + q− y− ⎪ ⎪ ⎬ subject to T x + Iy+ − Iy− = d¯ (2.43) Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0, where E means the function of expectation. In particular, a two-stage programming problem such that the coefﬁcient vector of y+ and y− is the identity matrix such as (2.43) is called a simple recourse problem. ˆ the following solution minimizes the For the selected x and realized values d, + objective function of (2.43) unless qi + q− i is negative, and therefore we assume − + q ≥ 0, i = 1, . . . , m: that q+ i i

2.2 Stochastic programming

23

⎫ n ⎪ ˆ if di ≥ ∑ ti j x j ⎪ ⎪ ⎬

n

− ˆ y+ i = di − ∑ ti j x j , yi = 0 j=1

y+ i = 0,

y− i =

n

∑ ti j x j − dˆi

if dˆi

z0i ,

where z0i and z1i denote the values of the ith objective function zi (x) such that the degrees of the membership function are 0 and 1, respectively, and it is depicted in Fig. 2.3. μi (zi (x)) 1

0

z 0i

z i1

zi (x)

Fig. 2.3 Linear membership function for the ith fuzzy goal.

Zimmermann (1978) also suggests a method for assessing these parameters z0i and z1i of the membership function. In his method, the parameter z1i is determined as z1i = min zi (x), x∈X

(2.66)

and the parameter z0i is speciﬁed as z0i = max zi (x jo ), j=i

(2.67)

where x jo is a feasible solution minimizing z j (x), i.e., x jo ∈ arg minx∈X z j (x). In this setting, z1i is set at the minimum of the ith objective function, and z0i is set at the maximum among the values of the ith objective function with respect to solutions minimizing z j (x), j = i. By setting the parameters as described above, the linear membership functions (2.65) can be identiﬁed. Following the principle of the fuzzy decision by Bellman and Zadeh (1970), the multiobjective linear programming problem (2.61) can be reformulated as the following maximin problem: maximize min {μi (zi (x))} 1≤i≤k (2.68) subject to Ax ≤ b, x ≥ 0. By introducing an auxiliary variable λ, the maximin problem (2.68) is equivalently rewritten as a standard linear programming problem

30

2 Fundamentals

⎫ ⎬

maximize λ subject to μi (zi (x)) ≥ λ, i = 1, . . . , k ⎭ Ax ≤ b, x ≥ 0.

(2.69)

In the formulation by Zimmermann, it is implicitly assumed that the DM feels that the fuzzy decision is appropriate for combining the fuzzy goals. However, in real-world decision situations, such an assumption is not always well-suited, and then an interactive method provides an alternative approach. Sakawa, Yano and Yumine (1987) propose an interactive method for a multiobjective linear programming problem with fuzzy goals. Because they incorporate not only “fuzzy min zi (x)” or “fuzzy max zi (x)” which is a fuzzy goal of the DM such as “zi (x) should be substantially less than or equal to pi , or greater than or equal to qi ” but also “fuzzy equal zi (x)” which is a fuzzy goal such as “zi (x) should be in the vicinity of ri ,” and thus the concept of Pareto optimality cannot be applied, they introduce the following concept of M-Pareto optimal solutions deﬁned in terms of membership functions instead of objective functions (Sakawa, 1993): Deﬁnition 2.7 (M-Pareto optimal solution). A point x∗ ∈ X is said to be an MPareto optimal solution if and only if there does not exist another x ∈ X such that μi (zi (x)) ≥ μi (zi (x∗ )) for all i ∈ {1, . . . , k} and μ j (z j (x)) = μ j (z j (x∗ )) for at least one j ∈ {1, . . . , k}. After identifying the membership functions μi (zi (x)), i = 1, . . . , k for the fuzzy goals of the objective functions zi (x), i = 1, . . . , k, the DM is asked to specify the reference membership values which are the aspiration levels of achievement for the values of the membership functions. It follows that a vector of the reference membership values is a natural extension of the reference point in the reference point method by Wierzbicki (1980). Let μˆ = (ˆμ1 , . . . , μˆ k )T denote the reference membership values for the membership functions μ (z(x)) = (μ1 (z1 (x)), . . . , μk (zk (x)))T . Then, by solving the minimax problem minimize max {ˆμi − μi (zi (x))} 1≤i≤k (2.70) subject to Ax ≤ b, x ≥ 0, an M-Pareto optimal solution closest to the vector of the reference membership values in the L∞ norm can be obtained. The maximin problem (2.70) is equivalently expressed as ⎫ minimize v ⎬ subject to μˆ i − μi (zi (x)) ≤ v, i = 1, . . . , k (2.71) ⎭ Ax ≤ b, x ≥ 0. As we discussed linear programming with fuzzy parameters in the subsection 2.1.3, also in multiobjective situations, the formulation involving fuzzy parameters is still important. Sakawa and Yano (1990) extend the concept of Pareto optimality in order to deal with multiobjective linear programming problems with fuzzy parameters characterized by fuzzy numbers in the level set-based model. For given

2.3 Multiobjective programming

31

parameters A and b in the constraints, let X(a, b) denote the corresponding feasible region, i.e., X(a, b) {x ∈ Rn | Ax ≤ b, x ≥ 0}. Deﬁnition 2.8 (M-α-Pareto optimal solution). For a certain degree α speciﬁed by the DM, a point x∗ ∈ X(a∗ , b∗ ) is said to be an M-α-Pareto optimal solution if and only if there does not exist another x ∈ X(a, b), (c, a, b) ∈ (C, A, B)α such that μi (zi (x)) ≥ μi (zi (x∗ )) for all i ∈ {1, . . . , k} and μ j (z j (x)) = μ j (z j (x∗ )) for at least one j ∈ {1, . . . , k}, where the corresponding values of the parameters (c∗ , a∗ , b∗ ) ∈ (C, A, B)α are called the α-level optimal parameters. To derive a satisﬁcing solution to a multiobjective linear programming problem with fuzzy parameters for a certain degree α speciﬁed by the DM, the minimax problem utilized in the interactive method can be formulated as ⎫ minimize v ⎪ ⎪ ⎬ subject to μˆ i − μi (zi (x)) ≤ v, i = 1, . . . , k (2.72) Ax ≤ b, x ≥ 0 ⎪ ⎪ ⎭ (c, a, b) ∈ (C, A, B)α . Let I1 , I2 , and I3 be the index sets of the fuzzy goals for “fuzzy min zi (x),” “fuzzy max zi (x),” and “fuzzy equal zi (x),” respectively. For notational convenience, denote the strictly monotone decreasing functions of μi , i ∈ I1 and the right-hand side functions of μi , i ∈ I3 by diR , i ∈ I1 ∪ I3 , and the strictly monotone increasing functions of μi , i ∈ I2 and the left-hand side functions of μi , i ∈ I3 by diL , i ∈ I2 ∪ I3 . From the properties of the α-level sets, (2.72) can be transformed into ⎫ minimize v ⎪ ⎪ ⎪ −1 ⎬ subject to cLiα x ≤ diR (ˆμi − v), i ∈ I1 ∪ I3 (2.73) −1 R ⎪ ciα x ≥ diL (ˆμi − v), i ∈ I2 ∪ I3 ⎪ ⎪ ⎭ aLi1 x1 + · · · + aLin xn ≤ bRi , i = 1, . . . , m, x ≥ 0, −1 −1 and diL are pseudo-inverse functions deﬁned by where diR −1 diR (h) = sup{y | diR (y) ≥ h},

(2.74)

−1 (h) = diL

(2.75)

inf{y | diL (y) ≥ h}.

It should be noted that (2.73) can be solved by using the combined use of the two-phase simplex method and the bisection method. For readers interested in this issue, refer to Sakawa (1993).

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2 Fundamentals

2.4 Two-level programming 2.4.1 Fuzzy programming for two-level programming In the real world, we often encounter situations where there are two or more DMs in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. In this section, we consider a case where there are two DMs; one of the DMs ﬁrst makes a decision, and then after acknowledging the decision of the ﬁrst DM, the other DM chooses a decision. Such a situation is formulated as a two-level programming problem. A linear programming problem with two DMs is formulated as follows. For the sake of simplicity, we call the two DMs DM1 and DM2 in this subsection. Let x1 and x2 denote the column vectors of the decision variables of DM1 and DM2, respectively, and let z1 (x1 , x2 ) = c11 x1 + c12 x2 and z2 (x1 , x2 ) = c21 x1 + c22 x2 denote the objective functions of DM1 and DM2, respectively, where ci1 , i = 1, 2 are n1 dimensional coefﬁcient row vector, and ci2 , i = 1, 2 are n2 dimensional coefﬁcient row vector. Assume that A1 x1 + A2 x2 ≤ b are the common constraints of DM1 and DM2, where A1 is an m × n1 coefﬁcient matrix, A2 is an m × n2 coefﬁcient matrix, b is an m dimensional constant column vector. Then, for a given x2 , DM1 deals with the following linear programming problem: ⎫ minimize z1 (x1 , x2 ) = c11 x1 + c12 x2 ⎪ ⎬ x1 (2.76) subject to A1 x1 ≤ b − A2 x2 ⎪ ⎭ x1 ≥ 0. Similarly, for a given x1 , the linear programming problem for DM2 is formulated as ⎫ minimize z2 (x1 , x2 ) = c21 x1 + c22 x2 ⎪ ⎬ x2 (2.77) subject to A2 x2 ≤ b − A1 x1 ⎪ ⎭ x2 ≥ 0. Combining the two problems (2.76) and (2.77) into one problem, we formulate the following linear programming problem with DM1 and DM2: ⎫ minimize z1 (x1 , x2 ) = c11 x1 + c12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c21 x1 + c22 x2 (2.78) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x2 ≥ 0, x1 ≥ 0, where “minimize” and “minimize” mean that DM1 and DM2 are minimizers for for DM1

for DM2

their objective functions. Assume that DM1 ﬁrst makes a decision and then DM2 chooses a decision later, and to be better off each other, they can coordinate their decisions. Namely, we as-

2.4 Two-level programming

33

sume that DM1 and DM2 make a binding agreement to select actions cooperatively, and then it is predicted that the selected actions should be in the set of Pareto optimal solutions. It should be noted here that Pareto optimality in two-level programming is deﬁned for the two objective functions: z1 of DM1 and z2 of DM2. For two-level linear programming problems with cooperative DMs, fuzzy programming approaches have been developed (Lai, 1996; Shih, Lai and Lee, 1996; Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009). In fuzzy programming for two-level linear programming by Sakawa et al. (Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009), for each of the objective functions zi (x), i = 1, 2 of (2.78), it is assumed that the DMs have fuzzy goals such as “the objective function zi (x) should be substantially less than or equal to some speciﬁc value pi .” Although the membership function does not always need to be linear, for the sake of simplicity, assume that a membership function μi (zi ) which characterizes the fuzzy goal of each DM is linear and it is speciﬁed as ⎧ ⎪ 1 if zi (x) ≤ z1i ⎪ ⎪ ⎨ zi (x) − z0i (2.79) if z1i < zi (x) ≤ z0i μi (zi (x)) = 1 − z0 ⎪ z ⎪ i i ⎪ ⎩0 if zi (x) > z0i . By identifying the membership functions μ1 (z1 (x)) and μ2 (z2 (x)) for the objective functions z1 (x) and z2 (x), the original two-level linear programming problem (2.78) can be interpreted as the membership function maximization problem deﬁned by ⎫ minimize μ1 (z1 (x)) ⎪ ⎪ for DM1 ⎪ ⎬ minimize μ2 (z2 (x)) (2.80) for DM2 ⎪ subject to Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. In (2.80), x ∈ Rn is an n dimensional decision variable vector, and it is divided into two vectors x1 and x2 which are n1 and n2 dimensional decision variable vectors of DM1 and DM2, respectively, i.e., n = n1 + n2 . Because the two DMs make decisions cooperatively, the decision variable vector is represented simply by x without partition. To derive an overall satisfactory solution to the membership function maximization problem (2.80), we ﬁrst ﬁnd the maximizing decision of the fuzzy decision proposed by Bellman and Zadeh (1970). Namely, the following problem is solved for obtaining a solution which maximizes the smaller degree of satisfaction between those of the two DMs: ⎫ maximize min{μ1 (z1 (x)), μ2 (z2 (x))} ⎬ subject to Ax ≤ b (2.81) ⎭ x ≥ 0.

34

2 Fundamentals

By introducing an auxiliary variable λ, this problem can be transformed into the following equivalent problem: ⎫ maximize λ ⎪ ⎪ ⎪ subject to μ1 (z1 (x)) ≥ λ ⎪ ⎬ μ2 (z2 (x))} ≥ λ (2.82) ⎪ ⎪ Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. Solving (2.82), we can obtain a solution which maximizes the smaller satisfactory degree between those of both DMs. It should be noted that if the membership functions μi (zi (x)), i = 1, 2 are linear membership functions such as (2.79), (2.82) becomes a linear programming problem. Let x∗ denote an optimal solution to problem (2.82). Then, we deﬁne the satisfactory degree of both DMs under the constraints as (2.83) λ∗ = min{μ1 (z1 (x∗ )), μ2 (z2 (x∗ ))}. If DM1 is satisﬁed with the optimal solution x∗ , it follows that the optimal solution x∗ becomes a satisfactory solution; however, DM1 is not always satisﬁed with the solution x∗ . If DM1 is not satisﬁed with the solution x∗ , it is quite natural to assume that DM1 speciﬁes the minimal satisfactory level δ ∈ [0, 1] for the membership function μ1 (z1 (x)) subjectively. Then, the following problem is formulated: ⎫ maximize μ2 (z2 (x)) ⎪ ⎪ ⎬ subject to μ1 (z1 (x)) ≥ δ (2.84) Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0, where DM2’s membership function is maximized under the condition that DM1’s membership function μ1 (z1 (x)) is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1. It should be also noted that if the membership functions μi (zi (x)), i = 1, 2 are linear membership functions such as (2.79), then (2.84) also becomes a linear programming problem. If there exists an optimal solution to (2.84), it follows that DM1 obtains a satisfactory solution having a satisfactory degree larger than or equal to the minimal satisfactory level speciﬁed by DM1’s self. However, the larger the minimal satisfactory level δ is assessed, the smaller the DM2’s satisfactory degree becomes when the objective functions of DM1 and DM2 conﬂict with each other. Consequently, a relative difference between the satisfactory degrees of DM1 and DM2 becomes larger, and it follows that the overall satisfactory balance between both DMs is not appropriate. In order to take account of the overall satisfactory balance between both DMs, DM1 needs to compromise with DM2 on DM1’s own minimal satisfactory level. To do so, the following ratio of the satisfactory degree of DM2 to that of DM1 is helpful:

2.4 Two-level programming

35

Δ=

μ2 (z2 (x)) , μ1 (z1 (x))

(2.85)

which is originally introduced by Lai (1996). DM1 is guaranteed to have a satisfactory degree larger than or equal to the minimal satisfactory level for the fuzzy goal because the corresponding constraint μ1 (z1 (x)) ≥ δ is involved in (2.84). To take into account the overall satisfactory balance between both DMs, DM1 speciﬁes the lower bound Δmin and the upper bound Δmax of the ratio, and then it is veriﬁed whether the ratio Δ is in the interval [Δmin , Δmax ] or not. The condition that the overall satisfactory balance is appropriate is represented by (2.86) Δ ∈ [Δmin , Δmax ]. At the iteration l, let μ1 (zl1 ), μ2 (zl2 ), λl and Δl = μ2 (zl2 )/μ1 (zl1 ) denote DM1’s and DM2’s satisfactory degrees, a satisfactory degree of both DMs, and the ratio of satisfactory degrees of the two DMs, respectively. Let the solution be xl at the iteration l. The interactive process terminates if the following two conditions are satisﬁed and if DM1 concludes the solution as an overall satisfactory solution. Termination conditions of the interactive process Condition 1: DM1’s satisfactory degree is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1’s self, i.e., μ1 (zl1 ) ≥ δ. Condition 2: The ratio Δl of satisfactory degrees lies in the closed interval between the lower and the upper bounds speciﬁed by DM1, i.e., Δl ∈ [Δmin , Δmax ]. Condition 1 ensures the minimal satisfaction to DM1 in the sense of the attainment of the fuzzy goal, and condition 2 is provided in order to keep overall satisfactory balance between both DMs. If these two conditions are not satisﬁed simultaneously, DM1 needs to update the minimal satisfactory level δ. The updating procedures are summarized as follows: Procedure for updating the minimal satisfactory level δ Case 1: If condition 1 is not satisﬁed, then DM1 decreases the minimal satisfactory level δ. Case 2: If the ratio Δl exceeds its upper bound, then DM1 increases the minimal satisfactory level δ. Conversely, if the ratio Δl is below its lower bound, then DM1 decreases the minimal satisfactory level δ. Case 3: Although conditions 1 and 2 are satisﬁed, if DM1 is not satisﬁed with the obtained solution and judges that it is desirable to increase the satisfactory degree of DM1 at the expense of the satisfactory degree of DM2, then DM1 increases the minimal satisfactory level δ. Conversely, if DM1 judges that it is desirable to increase the satisfactory degree of DM2 at the expense of the satisfactory degree of DM1, then DM1 decreases the minimal satisfactory level δ. In particular, if condition 1 is not satisﬁed, it follows that there does not exist any feasible solution for (2.84), and therefore DM1 has to moderate the minimal satisfactory level.

36

2 Fundamentals

We are now ready to give a procedure of interactive fuzzy programming for deriving an overall satisfactory solution to (2.78), which is summarized in the following. Algorithm of interactive fuzzy programming Step 1: Ask DM1 to identify the membership function μ1 (z1 ) of the fuzzy goal for the objective function z1 (x). Similarly, ask DM2 to identify the membership function μ2 (z2 ) of the fuzzy goal for the objective function z2 (x). Step 2: Set l := 1 and solve (2.82), in which a smaller satisfactory degree between those of DM1 and DM2 is maximized. If DM1 is satisﬁed with the obtained optimal solution, the solution becomes a satisfactory solution. Otherwise, ask DM1 to specify the minimal satisfactory level δ together with the lower and the upper bounds [Δmin , Δmax ] of the ratio of satisfactory degrees Δl with the satisfactory degree λ∗ of both DMs and the related information about the solution in mind. Step 3: Set l := l + 1. Solve (2.84), in which the satisfactory degree of DM2 is maximized under the condition that the satisfactory degree of DM1 is larger than or equal to the minimal satisfactory level δ, and then an optimal solution xl to (2.84) is proposed to DM1 together with λl , μ1 (zl1 ), μ2 (zl2 ) and Δl . Step 4: If the solution xl satisﬁes the termination conditions and DM1 accepts it, then the procedure stops, and the solution xl is determined to be a satisfactory solution. Step 5: Ask DM1 to revise the minimal satisfactory level δ in accordance with the procedure of updating minimal satisfactory level. Return to step 3.

2.4.2 Stackelberg solution to two-level programming problem In the previous subsection, we considered a two-level linear programming problem in which two DMs can coordinate their decisions. However, for example, if the two DMs are corporate managers in different companies and the companies have business with each other, it is natural to suppose that there should be conﬂict interests between them. If the two DMs have conﬂicting interests, they cannot always cooperate in their decisions. Then, it is difﬁcult to employ a cooperative solution method such as the fuzzy programming approach in the previous subsection. For noncooperative decision making in a two-level programming problem, a DM who ﬁrst makes a decision is called the leader and the other DM is called the follower conventionally. The leader ﬁrst speciﬁes a decision and then the follower determines a decision so as to optimize the objective function of the follower with full knowledge of the decision of the leader. Anticipating this, the leader also makes a decision so as to optimize the objective function of self. The solution deﬁned as the above mentioned procedure is a Stackelberg equilibrium solution, and we call it a Stackelberg solution shortly. A two-level linear programming problem for obtaining the Stackelberg solution is formulated as

2.4 Two-level programming

37

⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ x ⎪ ⎪ ⎪ where y solves ⎬ minimize z2 (x, y) = c2 x + d2 y y ⎪ ⎪ ⎪ ⎪ subject to Ax + By ≤ b ⎪ ⎭ x ≥ 0, y ≥ 0,

(2.87)

where ci , i = 1, 2 are n1 dimensional coefﬁcient row vector, di , i = 1, 2 are n2 dimensional coefﬁcient row vector, A is an m × n1 coefﬁcient matrix, B is an m × n2 coefﬁcient matrix, b is an m-dimensional constant column vector. In the two-level linear programming problem (2.87), z1 (x, y) and z2 (x, y) represent the objective functions of the leader and the follower, respectively, and x and y represent the decision variables of the leader and the follower, respectively. Each DM knows the objective function of the opponent as well as the objective function of self and the constraints. The leader ﬁrst makes a decision, and then the follower makes a decision so as to minimize the objective function with full knowledge of the decision of the leader. Namely, after the leader chooses x, the follower solves the following linear programming problem: ⎫ minimize z2 (x, y) = d2 y + c2 x ⎪ ⎬ y (2.88) subject to By ≤ b − Ax ⎪ ⎭ y ≥ 0, and chooses an optimal solution y(x) to (2.88) as a rational response. Assuming that the follower chooses the rational response, the leader also makes a decision such that the objective function z1 (x, y(x)) is minimized. Then, the solution deﬁned as the above mentioned procedure is a Stackelberg solution. Computational methods for obtaining Stackelberg solutions to two-level linear programming problems are classiﬁed roughly into three categories: the vertex enumeration approach (Bialas and Karwan, 1984), the Kuhn-Tucker approach (Bard and Falk, 1982; Bard and Moore, 1990; Bialas and Karwan, 1984; Hansen, Jaumard and Savard, 1992), and the penalty function approach (White and Anandalingam, 1993). The vertex enumeration approach takes advantage of the property that there exists a Stackelberg solution in a set of extreme points of the feasible region. In the Kuhn-Tucker approach, the leader’s problem with constraints involving the optimality conditions of the follower’s problem is solved. In the penalty function approach, a penalty term is appended to the objective function of the leader so as to satisfy the optimality of the follower’s problem. It is well-known that a two-level linear programming problem is an NP-hard problem (Shimizu, Ishizuka and Bard, 1997). In the following, we outline a couple of conventional computational methods for obtaining Stackelberg solutions. The kth best method proposed by Bialas and Karwan (1984) can be thought of as the vertex enumeration approach, and it is based on a very simple idea. The solution search procedure of the method starts from a point which is an optimal solution to the problem of the leader and checks whether it is also an optimal solution to

38

2 Fundamentals

the problem of the follower or not. If the ﬁrst point is not the Stackelberg solution, the procedure continues to examine the second best solution to the problem of the leader, and so forth. At the beginning, the following linear programming problem is solved: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎬ x (2.89) subject to Ax + By ≤ b ⎭ x ≥ 0, y ≥ 0. Assuming that the feasible region of (2.89) is not empty and there are N extreme points, i.e., N basic solutions of (2.89). Let (ˆx[1] , yˆ [1] ) denote an optimal solution to (2.89), and (ˆx[2] , yˆ [2] ), . . . , (ˆx[N] , yˆ [N] ) be the rest of N − 1 basic feasible solutions such that z1 (x[ j] , y[ j] ) ≤ z1 (x[ j+1] , y[ j+1] ), j = 1, . . . , N −1. It is veriﬁed if the solution (ˆx[ j] , yˆ [ j] ) is optimal to (2.88) of the follower from j = 1 to j = N in turn. Then, the ﬁrst solution found to be optimal to (2.88) is the Stackelberg solution. In the Kuhn-Tucker approach, the leader’s problem with constraints involving the optimality conditions of the follower’s problem (2.88) is solved. The Kuhn-Tucker conditions for (2.88) are shown as follows: ⎫ uB − v = −d2 ⎪ ⎪ ⎬ u(Ax + By − b) − vy = 0 (2.90) Ax + By ≤ b ⎪ ⎪ ⎭ y ≥ 0, uT ≥ 0, vT ≥ 0, where u is an m dimensional row vector and v is an n2 dimensional row vector. Then, the follower’s problem (2.88) for a two-level linear programming problem can be replaced by the above conditions (2.90), and (2.87) is rewritten as the following equivalent single-level mathematical programming problem: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ ⎪ ⎪ subject to uB − v = −d2 ⎬ u(Ax + By − b) − vy = 0 (2.91) ⎪ ⎪ Ax + By ≤ b ⎪ ⎪ ⎭ x ≥ 0, y ≥ 0, uT ≥ 0, vT ≥ 0. From the equality constraint uB − v = −d2 of (2.91), v is eliminated and the equality constraint u(Ax + By − b) − vy = 0 is transformed into u(b − Ax − By) + (uB + d2 )y = 0.

(2.92)

Moreover, the complementarity condition (2.92) implies that b − Ax − By ≥ 0, uT ≥ 0, (uB+d2 )T ≥ 0, y ≥ 0. Let Ai and Bi be the ith row vector of the matrix A and the matrix B, respectively, and let B j and d2 j be the jth column vector of the matrix B and the jth element of the vector d2 , respectively. Then, the condition of either ui = 0 or bi − Ai x − Bi y = 0 for i = 1, . . . , m and the condition of either uB j + d2 j = 0 or y j = 0 for j = 1, . . . , n2 must be satisﬁed simultaneously. By introducing zero-one

2.4 Two-level programming

39

vectors w1 = (w11 , . . . , w1m ) and w2 = (w21 , . . . , w2n2 ), the equality constraint (2.92) can be expressed as follows (Fortuny-Amat and McCarl, 1981): ⎫ u ≤ Mw1 ⎪ ⎪ ⎬ b − Ax − By ≤ M(e − wT1 ) (2.93) uB + d2 ≤ Mw2 ⎪ ⎪ ⎭ T y ≤ M(e − w2 ), where e is an m dimensional vector of ones, and M is a large positive constant. Therefore, the mathematical programming problem (2.91) is equivalent to the following mixed zero-one programming problem, and it can be solved by a zeroone mixed integer solver: ⎫ minimize z1 (x, y) = c1 x + d1 y ⎪ ⎪ ⎪ ⎪ subject to 0 ≤ uT ≤ MwT1 ⎪ ⎪ ⎬ T 0 ≤ b − Ax − By ≤ M(e − w1 ) (2.94) 0 ≤ (uB + d2 )T ≤ MwT2 ⎪ ⎪ ⎪ ⎪ 0 ≤ y ≤ M(e − wT2 ) ⎪ ⎪ ⎭ x ≥ 0. In the penalty function approach, the duality gap of the follower’s problem (2.88) is appended to the objective function of the leader. The dual problem to (2.88) ignoring the constant term c2 x is written as ⎫ minimize u(Ax − b) ⎬ subject to −uB ≤ d2 (2.95) ⎭ uT ≥ 0, where u is an m-dimensional row vector. Because the duality gap d2 y − u(Ax − b) is zero if y is a rational response of the follower with respect to a choice x of the leader, the following mathematical programming problem is formulated: ⎫ minimize c1 x + d1 y + Ku(Ax − b) ⎪ ⎪ ⎬ subject to Ax + By ≤ b (2.96) −uB ≤ d2 ⎪ ⎪ ⎭ T x ≥ 0, y ≥ 0, u ≥ 0, where K is a constant value. By repeatedly solving (2.96) for updated values of K and u, (2.96) yields an optimal solution to (2.87), i.e., the Stackelberg solution.

40

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2.5 Genetic algorithms 2.5.1 Fundamental elements in genetic algorithms It is hard to obtain exact optimal solutions of difﬁcult classes of optimization problems such as combinatorial problems and nonconvex nonlinear problems, and thus it is quite natural for DMs to require approximate optimal solutions instead. To meet this demand, recently several meta-heuristics have been developed and their effectiveness is demonstrated. Among them, genetic algorithms are known to be one of the most practical and proven methods, and in this section we present the basic concepts of the genetic algorithms. A computational framework of genetic algorithms initiated by Holland (1975) has been attracted attention of many researchers with applicability in optimization, as well as in search and learning. Furthermore, publications of books by Goldberg (1989) and Michalewicz (1996) bring heightened and increasing interests in applications of genetic algorithms to complex function optimization. start Step 0 generating the initial population Step 1 calculating the fitness Step 2 reproducting the next generation Step 3 executing crossover operation Step 4 executing mutation operation Step 5 No

termination condition is satisfyed? Yes end

Fig. 2.4 Flowchart of genetic algorithms.

2.5 Genetic algorithms

41

The fundamental procedure of genetic algorithms is shown as a ﬂowchart in Fig. 2.4, and it is summarized as follows: Step 0: Initialization. Generate a given number of individuals randomly to form the initial population. Step 1: Evaluation. Calculate the ﬁtness value of each individual in the population. Step 2: Reproduction. According to the ﬁtness values and a reproduction rule speciﬁed in advance, select individuals from the current population to form the next population. Step 3: Crossover. Select two individuals randomly from the population, and exchange some part of the string of one individual for the corresponding part of the other individual with a given probability for crossover. Step 4: Mutation. Alter one or more genes in the string of an individual with a given probability of mutation. Step 5: Termination. Stop the procedure if the condition of termination is satisﬁed, and an individual with the maximum ﬁtness value is determined as an approximate optimal solution. Otherwise, return to step 1.

2.5.1.1 Representation of individuals When genetic algorithms are applied to optimization problems, a vector of decision variables corresponds to an individual in the population, which is represented by a string as in Fig. 2.5. 1 0 1 1 .... 1 0 1 0 0 n-bit string for a vector x of decision variables

Fig. 2.5 Individual represented by a string.

As seen in Fig. 2.5, each element of the string is either 1 or 0 usually, but real numbers, integers, alphabets, or some other symbols can also be used to represent individuals. Let s and x denote an individual represented by a string and a vector of the decision variables, respectively. The string s which means a chromosome in the context of biology is called the genotype of an individual, and the decision variables x is called the phenotype. The mapping from phenotypes to genotypes is called coding, and the reverse mapping is called decoding.

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2.5.1.2 Fitness function and scaling In an optimization problem where the objective function is minimized or maximized, a solution with the smallest objective function value or the largest objective function value is searched. When a genetic algorithm is applied to solving an optimization problem, a solution to the optimization problem is associated with an individual in the genetic algorithm, and the objective function value of the solution corresponds to the ﬁtness of the individual. Thus, an individual with a larger ﬁtness value has a higher probability of surviving in the next generation. Let z(x) denote an objective function to be minimized in an optimization problem. The corresponding ﬁtness function in genetic algorithms is commonly deﬁned as (Goldberg, 1989) Cmax − z(x) if z(x) < Cmax (2.97) f (si ) = 0 otherwise, where si denotes the ith individual in the population, and Cmax is a given constant. For example, the value of Cmax is determined as the largest objective function value z(x) observed thus far, the largest value z(x) in the current population, or the largest value z(x) in the last t generations. Similarly, in maximization problems, to prevent the ﬁtness value from being negative, the constant Cmin is introduced, and the following ﬁtness function is often used: z(x) +Cmin if z(x) +Cmin > 0 (2.98) f (si ) = 0 otherwise. The absolute value of the smallest z(x) in the current population or in the last t generations is often used for the value of Cmin . To properly distribute ﬁtness values in the population, ﬁtness scaling is employed. The linear scaling, which is a simple and useful procedure, is represented by (2.99) f (si ) = a f (si ) + b, where f and f are the raw ﬁtness value and the scaled ﬁtness value, respectively; a and b are coefﬁcients. To perform the operation of reproduction appropriately, the coefﬁcients a and b may be chosen in such a way that the average scaled ﬁtness

is equal to the average raw ﬁtness value f , and the maximum scaled value fave ave

ﬁtness value is determined as fmax = Cmult fave , where Cmult is a given constant.

2.5.1.3 Genetic operators The three genetic operators, reproduction, crossover, and mutation, are outlined below. Individuals are copied into the next generation according to their ﬁtness values by a reproduction operator. The roulette wheel selection is one of the most popular

2.5 Genetic algorithms

43

reproduction operators, and in this method, each individual in the current population has a roulette wheel slot sized in proportion to its ﬁtness value. Let pop size be the number of individuals in the population. The percentage of the roulette wheel given pop size f (sl )%. Namely, the individual si is reproto an individual si is 100 f (si )/ ∑l=1 pop size f (sl ) each spin of the roulette duced with the probability p(si ) = f (si )/ ∑l=1 wheel. An example of the roulette wheel is given in Fig. 2.6; the numbers in the wheel are ﬁtness values of individuals, and the decimal numbers outside of the wheel are the corresponding probabilities. 0.04 s6 16

0.26 s5

0.14 s1 56

104 76

s4 0.12

48

s2

0.19

100 s3

p(si ) f (si )

s1

s2

s3

s4

s5

s6

0.14

0.19

0.25

0.12

0.26

0.04

56

76

100

48

104

16

0.25

Fig. 2.6 Biased roulette wheel.

Crossover creates offsprings into the next population by combining the genetic material of two parents. The variation caused by the crossover process may bring offsprings better ﬁtness values, and thus it is thought that crossover plays an important role in genetic algorithms. Although there are many different types of crossover, we provide a simple example here: a single-point crossover operator is the most simple operator of crossover. In this operation, two parent strings s1 and s2 are randomly chosen from a mating pool in which newly reproduced individuals are entered temporarily, and then one crossover point in the strings is chosen at random. Two offsprings are made by exchanging the substrings which are parts of the left-hand side of the parent strings s1 and s2 from the crossover point. The crossover operation is illustrated in Fig. 2.7. parent 1 1 1 0 0 1 0 1 0 1

offspring 1

10111 0 0 01

parent 2 1 0 1 1 1 0 0 1 0

offspring 2

11001 0 1 10

crossover point

exchange

Fig. 2.7 Crossover operation.

With a small probability, an operation of mutation provides the string of an individual with a randomly tiny alteration, and it is recognized that mutation serves as local search. In the representation of the 0-1 bit strings, mutation means changing a

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2 Fundamentals

1 to a 0 and vice versa. A simple version of mutation operator is illustrated in Fig. 2.8. parent 1 0 1 1 1 0 1 0 0

offspring 1 0 0 1 1 0 1 0 0

Fig. 2.8 Mutation operation.

2.5.2 Genetic algorithm for integer programming So far we have discussed fundamental elements in genetic algorithms. In the remaining chapters of this book, we will deal with mathematical programming problems not only with continuous decision variables but also with discrete decision variables, and for solving mathematical programming problems with discrete decision variables, i.e., integer programming problems, we give a computational method based on the framework of genetic algorithms, which is called GADSLPRRSU by Sakawa (2001). GADSLPRRSU is an abbreviation for a genetic algorithm with double strings based on linear programming relaxation and reference solution updating. This method includes three key ideas: double strings (DS), linear programming relaxation (LPR), and reference solution updating (RSU). These ideas were introduced one by one in the development of this method. Sakawa, Kato, Sunada and Shibano (1997) ﬁrst attempt to solve a multidimensional 0-1 knapsack problem by applying a genetic algorithm. For a given set of n kinds of items, the aim of the multidimensional knapsack problem is to select a subset of the items so as to maximize the total values of the selected items under the multiple constraints such as capacities, budgets, and so forth. Let x j be a decision variable which indicates whether the jth item is selected or not. Namely, the decision variable x j is 1 if the jth item is selected, and otherwise it is 0. Then, the general form of multidimensional knapsack problems is given as ⎫ minimize cx ⎬ subject to Ax ≤ b (2.100) ⎭ x j ∈ {0, 1}, j = 1, . . . , n, where c, A and b are an n dimensional row vector of positive coefﬁcients in the objective function, an m × n matrix of positive coefﬁcients in the left-hand side constraints and an m dimensional column vector of positive constants in the righthand side constraints. If a simple genetic algorithm described above is applied to a multidimensional 0-1 knapsack problem, an individual in the population is not always decoded into a feasible solution to the multidimensional 0-1 knapsack problem. Sakawa, Kato,

2.5 Genetic algorithms

45

Sunada and Shibano (1997) resolve this difﬁculty by introducing the double strings representation of individuals and the corresponding decoding algorithm. Furthermore, to apply this method to a multidimensional integer knapsack problem in which the domain of a decision variable is extended to a set of nonnegative integers, they utilize information of optimal solutions to linear programming relaxation problems (Sakawa et al., 2000). By using the fact that the zero solution, x = 0, is feasible to a multidimensional 0-1 or integer knapsack problem, these methods decode any individual into a feasible solution to the problem. However, in an integer programming problem which includes a multidimensional 0-1 or integer knapsack problem as a special case, the zero solution is not always feasible, and therefore these methods cannot directly apply to integer programming problems. An integer programming problem is formally given as ⎫ minimize cx ⎬ subject to Ax ≤ b (2.101) ⎭ x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n, where A = [p1 , . . . , pn ] is an m × n coefﬁcient matrix, b = (b1 , . . . , bm )T is an m dimensional constant column vector and c = (c1 , . . . , cn ) is an n dimensional coefﬁcient row vector. In contrast to knapsack problems such as (2.100), any coefﬁcient of (2.101) is not always positive. To overcome this problem, they propose a revised version (GADSLPRRSU) of their methods with a decoding algorithm with reference solutions which are found in advance and are updated if necessary (Sakawa, 2001). GADSLPRRSU for solving integer programming problems (2.101) is summarized as follows. As we pointed out, GADSLPRRSU employs the double string representation of individuals depicted in Fig. 2.9. s(1)s(2) s(3)s(4)

s( j)

s(n -1) s(n )

3 6 2 1 ... 4 .... 5 8 2 0 0 1 ... 3 .... 1 4 gs(1) gs(2) gs(3)gs(4)

gs( j)

x 3 = 2, x 6= 0, x 2 = 0, x 1 = 1, .... , x 4 = 3, ... , x 5 = 1, x 8 = 4

gs(n -1) gs(n )

Fig. 2.9 Double string representation.

In the ﬁgure, each of s( j), j = 1, . . . , n is the index of an element in a solution vector and each of gs( j) ∈ {0, 1, . . . , νs( j) }, j = 1, . . . , n is the value of the element, respectively. For example, a pair (s( j), gs( j) ) means that the value of the s( j)th element xs( j) is gs( j) , i.e., xs( j) = gs( j) . A decoding algorithm for the double string representation with reference points generates feasible solutions from any individuals based on a certain reference solution x∗ which is used as the origin of decoding. In the following algorithm, b+ denotes a column vector of a part of the right-hand side constants in (2.101) which

46

2 Fundamentals

are positive, and the corresponding coefﬁcient matrix of the left-hand side of the + constraints is denoted by A+ = [p+ 1 , . . . , pn ]. Decoding algorithm using a reference solution Step 1: Let j := 1 and psum := 0. Step 2: If gs( j) = 0, set qs( j) := 0 and j := j + 1, and go to step 4. + + Step 3: If psum + gs( j) p+ s( j) ≤ b , set qs( j) := gs( j) , psum := psum + gs( j) ps( j) and j := j + 1. Otherwise, set qs( j) := 0 and j := j + 1. Step 4: If j ≤ n, return to step 2. Step 5: Let j := 1, l := 0 and sum := 0. Step 6: If gs( j) = 0, set j := j + 1 and go to step 8. If gs( j) = 0, set sum := sum + gs( j) ps( j) . Step 7: If sum ≤ b, set l := j, j := j + 1. Otherwise, set j := j + 1. Step 8: If j ≤ n, return to step 6. Step 9: If l ≤ 0, go to step 11. Step 10: For xs( j) satisfying 1 ≤ j ≤ l, let xs( j) := gs( j) . For xs( j) satisfying l + 1 ≤ j ≤ n, let xs( j) := 0, and the algorithm terminates. ∗ p Step 11: Let sum := ∑nk=1 xs(k) s(k) and j := 1. ∗ , let x Step 12: If gs( j) = xs( s( j) := gs( j) and j := j + 1, and go to step 16. j) ∗ p ∗ Step 13: If sum−xs( +g s( j) s( j) ps( j) ≤ b, set sum := sum−xs( j) ps( j) +gs( j) ps( j) j) and xs( j) := gs( j) , and go to step 16. ∗ Step 14: Let ts( j) := 0.5(xs( j) + gs( j) ). ∗ ∗ Step 15: If sum−xs( j) ps( j) +ts( j) ps( j) ≤ b, set sum := sum−xs( j) ps( j) +ts( j) ps( j) , ∗ . gs( j) := ts( j) and xs( j) := ts( j) . Otherwise, set xs( j) := xs( j) Step 16: If j > n, the algorithm terminates. Otherwise, return to step 12. For general integer programming problems involving positive and negative coefﬁcients in the constraints, this decoding algorithm produces feasible solution from any individuals in the population. However, the diversity of generated feasible solutions depends on the reference solution used in the decoding algorithm. To overcome this difﬁculty, GADSLPRRSU adopts the reference solution updating procedure in which the current reference solution is updated by another feasible solution if the diversity of generated solutions seems to be lost. It is expected that an optimal solution to the following linear programming relaxation problem becomes a good approximate optimal solution of the original integer programming problem (2.101): ⎫ minimize cx ⎬ subject to Ax ≤ b (2.102) ⎭ 0 ≤ x j ≤ ν j , j = 1, . . . , n. From this observation, GADSLPRRSU exploits the information about the optimal solution to the continuous relaxation problem to ﬁnd an approximate optimal solution with high accuracy in reasonable time. The computational procedure of GADSLPRRSU is summarized as follows:

2.5 Genetic algorithms

47

Computational procedure of GADSLPRRSU Step 0: Determine values of the parameters used in the genetic algorithm. Step 1: Generate the initial population consisting of N individuals based on the information of an optimal solution to the continuous relaxation problem (2.102). Step 2: Decode each individual (genotype) in the current population and calculate its ﬁtness based on the corresponding solution (phenotype). Step 3: If the termination condition is fulﬁlled, the procedure stops. Otherwise, let t := t + 1. Step 4: Apply the reproduction operator using the elitist expected value selection after linear scaling. Step 5: Apply the crossover operator, called PMX (Partially Matched Crossover) for a double string. Step 6: Apply the mutation based on the information of an optimal solution to the continuous relaxation problem (2.102). Step 7: Apply the inversion operator, and return to step 2. For more information about GADSLPRRSU, it is recommended to read Sakawa (2001).

Chapter 3

Fuzzy Multiobjective Stochastic Programming

In this chapter, by considering the imprecision of a decision maker’s (DM’s) judgments for stochastic objective functions and/or constraints in multiobjective problems, fuzzy multiobjective stochastic programming is developed. Assuming that the DM has a fuzzy goal for each of expectations and variances of the original stochastic objective functions, multiobjective stochastic programming problems are formulated. For reﬂecting the diversity of criteria for optimizing the stochastic objective functions, optimization criteria different from expectation and variance are also provided to maximize the probability of the objective functions being greater than or equal to target values as well as to optimize the target values under a given probability. For stochastic programming problems, when a shortage or an excess comes from the violation of the constraints, it is often observed that some penalties are imposed. We also deal with the simple recourse model, which is developed to formulate such stochastic situations. Through the introduction of extended Pareto optimal solution concepts, interactive fuzzy satisﬁcing methods using linear or convex programming techniques are presented to derive a satisﬁcing solution for the DM from among the extended Pareto optimal solution set. The proposed interactive methods for the multiobjective stochastic programming with continuous variables are immediately extended to deal with the problems with integer decision variables by utilizing some genetic algorithms.

3.1 Fuzzy multiobjective stochastic linear programming In actual decision making situations, we must often make a decision on the basis of imprecise information or uncertain data. As described in Chapter 2, stochastic programming and fuzzy programming have been developed for solving such decision making problems involving uncertainty. For multiobjective stochastic linear programming problems, Stancu-Minasian (1984, 1990) considered the minimum risk approach, while Leclercq (1982) and Teghem, Dufrane, Thauvoye and Kunsch (1986) proposed interactive methods. M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_3, © Springer Science+Business Media, LLC 2011

49

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3 Fuzzy Multiobjective Stochastic Programming

Fuzzy multiobjective linear programming, ﬁrst proposed by Zimmermann (1978), has been also developed by numerous researchers, and an increasing number of successful applications have been appearing (Sakawa, Yano and Yumine, 1987; Luhandjula, 1987; Kacprzyk and Orlovski, 1987; Zimmermann, 1987; Verdegay and Delgado, 1989; Slowinski and Teghem, 1990; Sakawa, 1993; Lai and Hwang, 1994; Delgado, Kacprzyk, Verdegay and Vila, 1994; Slowinski, 1998; Sakawa, 2000, 2001). Although these studies focused on either fuzziness or randomness in multiobjective programming problems, it is important to simultaneously consider fuzziness and randomness in order to deal with more realistic decision making problems. In this section, we consider fuzzy programming approaches to multiobjective linear programming problems with random variable coefﬁcients in objective functions and/or constraints. In several stochastic models including the expectation model, the variance model, the probability model, the fractile model and the simple recourse model together with chance constrained programming techniques or two-stage problem formulations, the stochastic programming problems are transformed into deterministic ones. Assuming that a decision maker (DM) has a fuzzy goal for each of the objective functions, we present several interactive fuzzy satisﬁcing methods to derive a satisﬁcing solution for the DM by updating the reference membership levels for fuzzy goals represented by the membership functions. Some numerical examples are provided to demonstrate the feasibility and efﬁciency of the interactive fuzzy satisﬁcing method.

3.1.1 Expectation and variance models Throughout this subsection, assuming that the coefﬁcients in objective functions and right-hand side constants of constraints are random variables, we deal with multiobjective linear programming problems formulated as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ ⎪ ··· ⎬ minimize zk (x) = c¯ k x (3.1) ⎪ ⎪ subject to Ax ≤ b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, and A is an m × n coefﬁcient matrix, c¯ l , l = 1, . . . , k are n dimensional Gaussian random variable row vectors with ﬁnite means E[¯cl ] and n × n positive-deﬁnite variance-covariance matrices Vl = (vljh ) = (Cov{cl j , clh }), l = 1, . . . , k, and b¯ is an n dimensional vector whose elements are mutually independent random variables with continuous and increasing probability distribution functions. Multiobjective linear programming problems with random variable coefﬁcients are said to be multiobjective stochastic linear programming problems, which are of-

3.1 Fuzzy multiobjective stochastic linear programming

51

ten seen in actual decision making situations. For example, consider a production planning problem to optimize the gross proﬁt and the production cost simultaneously under the condition that unit proﬁts of the products, unit production costs of them and the maximal available amounts of the resources depend on seasonal factors or market prices. Such a production planning problem can be formulated as a multiobjective stochastic programming problem expressed by (3.1). Observing that (3.1) contains random variable coefﬁcients, the deﬁnitions and solution methods for ordinary mathematical programming problems not depending on stochastic events cannot be directly applied. Realizing such difﬁculty, we deal with the constraints in (3.1) as chance constrained conditions (Charnes and Cooper, 1959) which permit constraint violations up to speciﬁed probability limits. Let ηi , i = 1, . . . , m denote the probability that the ith constraint should be satisﬁed, and we call the probability the satisﬁcing probability level. Then, replacing the constraints in (3.1) with chance constrained conditions with satisﬁcing probability levels ηi , i = 1, . . . , m, we can reformulate the chance constrained problem ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ ⎪ ······ ⎪ ⎪ ⎪ ⎪ minimize zk (x) = c¯ k x ⎬ subject to P(ω | a1 x ≤ b1 (ω)) ≥ η1 (3.2) ⎪ ⎪ ········· ⎪ ⎪ ⎪ P(ω | am x ≤ bm (ω)) ≥ ηm ⎪ ⎪ ⎪ ⎭ x ≥ 0, where ai is the ith row vector of A, and bi (ω) is the ith element of the realized vector ¯ b(ω) for an elementary event ω. Denoting continuous and increasing distribution functions of the random variables b¯ i , i = 1, . . . , m by Fi (r) = P(ω | bi (ω) ≤ r), the ith constraint in (3.2) can be rewritten as P(ω | ai x ≤ bi (ω)) ≥ ηi ⇔ 1 − P(ω | bi (ω) ≤ ai x) ≥ ηi ⇔ 1 − Fi (ai x) ≥ ηi ⇔ ai x ≤ Fi−1 (1 − ηi ).

(3.3)

From (3.3), for the speciﬁed vector of the satisﬁcing probability levels η = (η1 , . . . , ηm )T , (3.2) can be equivalently transformed as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎬ ······ (3.4) minimize zk (x) = c¯ k x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where

η) x | ai x ≤ Fi−1 (1 − ηi ), i = 1, . . . , m, x ≥ 0 . X(η

52

3 Fuzzy Multiobjective Stochastic Programming

3.1.1.1 Expectation model As a ﬁrst attempt to deal with multiobjective stochastic programming problems, assuming that the DM would like to simply optimize the expected objective functions, we introduce the expectation model (Sakawa and Kato, 2002; Sakawa, Kato and Nishizaki, 2003). By replacing the objective functions zl (x) = c¯ l x, l = 1, . . . , k in (3.4) with their expectations, the multiobjective stochastic programming problem can be reformulated as ⎫ minimize zE1 (x) E[z1 (x)] = E[¯c1 ]x ⎪ ⎪ ⎬ ······ (3.5) minimize zEk (x) E[zk (x)] = E[¯ck ]x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where E[zl ] denotes the expectation of zl , and E[¯cl ] = (E[c¯l1 ], . . . , E[c¯ln ]). In order to consider the imprecise nature of the DM’s judgments for each expectation of the objective function zEl (x) = E[¯cl ]x, l = 1, . . . , k in (3.5), if we introduce the fuzzy goals such as “zEl (x) should be substantially less than or equal to a certain value”, (3.5) can be interpreted as maximize (μ1 (zE1 (x)), . . . , μk (zEk (x))),

(3.6)

η) x∈X(η

where μl is a membership function to quantify the fuzzy goal for the lth objective function in (3.5). To be more speciﬁc, if the DM feels that zEl (x) should be smaller than or equal to at most zEl,0 and zEl (x) ≤ zEl,1 (< zEl,0 ) is satisfactory, the shape of a typical membership function can be shown as in Fig. 3.1.

μl (zEl (x)) 1

0

z l,E 1

z l,E 0

z El (x)

Fig. 3.1 Example of the membership function μl (zEl (x)).

As a possible way to help the DM determine zEl,0 and zEl,1 , it is recommended to calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k obtained by solving the linear programming problems

3.1 Fuzzy multiobjective stochastic linear programming

53

⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η

maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭

(3.7)

η) x∈X(η

It should be noted here that (3.6) is regarded as a multiobjective decision making problem, and that there rarely exists a complete optimal solution that simultaneously optimizes all the objective functions. As discussed in Chapter 2, by directly extending Pareto optimality in ordinary multiobjective programming problems, Sakawa et al. deﬁned M-Pareto optimality on the basis of membership function values as a reasonable solution concept for the fuzzy multiobjective decision making problem (Sakawa and Yano, 1985a; Sakawa, Yano and Yumine, 1987; Sakawa and Yano, 1990; Sakawa, 1993). Introducing an aggregation function μD (x) for the k membership functions in (3.6), the fuzzy multiobjective programming problem can be rewritten as maximize μD (x) (3.8) η). subject to x ∈ X(η Following the conventional fuzzy approaches, to aggregate the multiple membership functions, Hulsurkar, Biswal and Sinha (1997) adopted the minimum operator of Bellman and Zadeh (1970) deﬁned by μD (x) = min μl (zEl (x)) 1≤l≤k

and the product operator of Zimmermann (1978) deﬁned as k

μD (x) = ∏ μl (zEl (x)). l=1

However, it should be emphasized here that such approaches are preferable only when the DM feels that the minimum operator or the product operator is appropriate. In other words, in general decision situations, the DM does not always use the minimum operator or the product operator for combining the fuzzy goals. The most crucial problem in (3.8) is probably the identiﬁcation of an appropriate aggregation function which well represents the DM’s fuzzy preference. If μD (x) can be explicitly identiﬁed, then (3.8) can be reduced to a standard mathematical programming problem. However, this rarely happens, and then as an alternative way, it is recommended for the DM to examine multiple possible solutions in the set of M-Pareto optimal solutions through an interactive solution procedure to ﬁnd a satisﬁcing solution to (3.8). In such an interactive fuzzy satisﬁcing method, to generate a candidate for a satisﬁcing solution which is also M-Pareto optimal, the DM is asked to specify the aspiration levels of achievement for all the membership function values, called the reference membership levels (Sakawa and Yano, 1985a; Sakawa, Yano and Yumine, 1987; Sakawa and Yano, 1989, 1990; Sakawa, 1993).

54

3 Fuzzy Multiobjective Stochastic Programming

For the DM’s reference membership levels μˆ l , l = 1, . . . , k, the corresponding MPareto optimal solution, which is the nearest to the vector of the given reference membership levels in the minimax sense or better than it if the reference membership levels are attainable, is obtained by solving the minimax problem minimize max {ˆμl − μl (zEl (x))} 1≤l≤k (3.9) η). subject to x ∈ X(η By introducing the auxiliary variable v, (3.9) can be equivalently transformed as ⎫ minimize v ⎪ ⎪ ⎪ ⎪ subject to μˆ 1 − μ1 (zE1 (x)) ≤ v ⎬ ········· (3.10) ⎪ μˆ k − μk (zEk (x)) ≤ v ⎪ ⎪ ⎪ ⎭ η). x ∈ X(η It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (3.10) can be reduced to a set of linear inequalities. Obtaining the optimal value v∗ to (3.10) is equivalent to determining the minimum value of v so that there exists a feasible solution x satisfying the constraints in (3.10). Since v satisﬁes zˆmax − max zEl,max ≤ v ≤ zˆmax − min zEl,min , 1≤l≤k

1≤l≤k

where zˆmax = max zˆl , 1≤l≤k

zEl,max = max zEl (x), x∈X

(x∗ , v∗ )

zEl,min = min zEl (x), x∈X

v∗

we can ﬁnd an optimal solution corresponding to by using the combined use of the two-phase simplex method and the bisection method. To be more speciﬁc, after calculating the optimal value v∗ to the problem ⎫ minimize v ⎪ ⎪ ⎪ subject to zE1 (x) ≤ μ−1 μ1 − v) ⎪ ⎬ 1 (ˆ ········· (3.11) ⎪ −1 E ⎪ ⎪ zk (x) ≤ μk (ˆμk − v) ⎪ ⎭ η) x ∈ X(η by the combined use of the phase one of the two-phase simplex method and the bisection method, we solve the linear programming problem ⎫ minimize zE1 (x) ⎪ ⎪ ⎪ subject to zE2 (x) ≤ μ−1 μ2 − v∗ ) ⎪ ⎬ 2 (ˆ (3.12) ········· ∗ ⎪ ⎪ ⎪ (ˆ μ − v ) zEk (x) ≤ μ−1 k ⎪ k ⎭ η), x ∈ X(η

3.1 Fuzzy multiobjective stochastic linear programming

55

where the ﬁrst objective function z1 (x) in (3.1) is supposed to be the most important to the DM. For the obtained optimal solution x∗ of (3.12), if there are inactive constraints in the ﬁrst (k − 1) constraints, after replacing μˆ l for inactive constraints with μl (zEl (x∗ )) + v∗ , we resolve the corresponding problem. Furthermore, if the obtained x∗ is not unique, the M-Pareto optimality test is performed by solving the linear programming problem maximize w = subject to

k

∑

εl l=1 μl (zE1 (x)) − ε1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

= μ1 (zE1 (x∗ )) ⎪ ········· ⎪ ⎪ ⎪ ⎪ μk (zEk (x)) − εk = μk (zEk (x∗ )) ⎪ ⎭ T η), ε = (ε1 , . . . , εk ) ≥ 0. x ∈ X(η

(3.13)

For the optimal solution xo and ε o to (3.13), (a) if w = 0, i.e., εl = 0 for l = 1, . . . , k, x∗ is Pareto optimal, and (b) if w > 0, i.e., εl > 0 for at least one l, not x∗ but xo is M-Pareto optimal. The DM should be either satisﬁed with the current M-Pareto optimal solution or continue to examine another solution by updating the reference membership levels. In order to help the DM express a degree of preference, trade-off rates between a standing membership function μl (zE1 (x)) and each of the other membership functions is quite useful. Such the trade-off rates are easily obtainable and are expressed as μ (zE (x∗ )) ∂μ1 (zE1 (x∗ )) = πl 1 E1 ∗ , l = 2, . . . , k, − E ∗ ∂μl (zl (x )) μl (zl (x )) where μ l denotes the differential coefﬁcient of μl , and πl , l = 1, . . . , k are the simplex multipliers in (3.12). Following the preceding discussions, we summarize the interactive algorithm in order to derive the satisﬁcing solution for the DM from among the M-Pareto optimal solution set. Interactive fuzzy satisﬁcing method for the expectation model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k by solving the linear programming problems (3.7). Step 3: Ask the DM to specify the membership functions μl , l = 1, . . . , k by taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 5: For the current reference membership levels μˆ l , solve the corresponding minimax problem (3.9). For the obtained optimal solution x∗ , if there are the inactive constraints in the ﬁrst (k − 1) constraints of (3.9), replace zˆl for inactive constraints with zEl (x∗ ) + v∗ and resolve the revised problem. Furthermore, if the solution x∗ is not unique, perform the M-Pareto optimality test.

56

3 Fuzzy Multiobjective Stochastic Programming

Step 6: The DM is supplied with the corresponding M-Pareto optimal solution and the trade-off rates between the membership functions. If the DM is satisﬁed with the current membership function values zEl (x), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by taking into account the current membership function values μl (zEl (x∗ )) together with the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k, and return to step 5. Observing that the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k in step 6 indicate the decrement value of the membership function μ1 with a unit increment of value of the membership function μl , the information of trade-off rates can be used to estimate the local shape of (μ1 (zE1 (x∗ )), . . . , μk (zEk (x∗ ))) around x∗ . Here it should be stressed for the DM that any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the given satisﬁcing probability levels ηi , i = 1, . . . , m.

3.1.1.2 Variance model When the DM would like to simply optimize the expected objective function values without concerning about the ﬂuctuation of the realized values, the expectation model is appropriate. On the other hand, if the DM prefers to decrease the ﬂuctuation of the objective function values from the viewpoint of the stability of the obtained values, by minimizing the variance of the objective function, the multiobjective stochastic programming problem can be reformulated as ⎫ minimize zV1 (x) = Var [z1 (x)] = xT V1 x ⎪ ⎪ ⎬ ········· (3.14) minimize zVk (x) = Var [zk (x)] = xT Vk x ⎪ ⎪ ⎭ η), subject to x ∈ X(η where Var [zl (x)] denotes the variance of zl (x). Observing that (3.14) does not consider the expected objective function values, the original variance model may give a poor solution yielding undesirable values of the objective functions even if the ﬂuctuation of the objective function values is minimized. Realizing this difﬁculty, by incorporating the constraints with respect to the expectations into the original variance model, we formulate the modiﬁed variance model for the multiobjective stochastic programming problem represented as ⎫ minimize zV1 (x) = Var [z1 (x)] = xT V1 x ⎪ ⎪ ⎪ ⎪ ········· ⎬ V T minimize zk (x) = Var [zk (x)] = x Vk x (3.15) ⎪ ⎪ ¯ ≤γ ⎪ subject to E[C]x ⎪ ⎭ η), x ∈ X(η

3.1 Fuzzy multiobjective stochastic linear programming

57

¯ = (E[¯c1 ]T , . . . , E[¯ck ]T )T and γ = (γ1 , . . . , γk )T . Each γl is a permissible where E[C] expectation level speciﬁed by the DM taking into account the individual minima and maxima of zEl (x) calculated by solving the linear programming problems ⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η

maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭

(3.16)

η) x∈X(η

η, γ ) be the feasible region of (3.15), namely For notational convenience, let X(η ¯ ≤ γ , x ∈ X(η η, γ ) x E[C]x η) . X(η Quite similar to the expectation model, in order to consider the imprecise nature of the DM’s judgments for each objective function in (3.15), if we introduce the fuzzy goals such as “zVl (x) should be substantially less than or equal to a certain value”, (3.15) can be interpreted as maximize (μ1 (zV1 (x)), . . . , μk (zVk (x))), η,γγ) x∈X(η

(3.17)

where μl is a nonincreasing membership function to quantify the fuzzy goal for the lth objective function. To help the DM specify the membership function, it is desirable to calculate the individual minima of the variances of the objective functions by solving the quadratic programming problems minimize zVl (x) = xT Vl x, l = 1, . . . , k, η,γγ) x∈X(η

(3.18)

where Vl is an n × n positive-deﬁnite variance-covariance matrix for the coefﬁcient vector cl . For the DM’s reference membership levels μˆ l , i = 1, . . . , k, the corresponding M-Pareto optimal solution is obtained by solving the minimax problem minimize max {ˆμl − μl (zVl (x))} 1≤l≤k (3.19) η, γ ). subject to x ∈ X(η If an optimal solution (x∗ , v∗ ) to (3.19) is not unique, M-Pareto optimality of x∗ is not guaranteed. As discussed in the expectation model, we can obtain the tradeoff rates in the expectation model through the M-Pareto optimality test using linear programming techniques. Observing that (3.19) is not a linear programming problem, it is not easy to calculate the trade-off rates in the variance model. Therefore, to obtain an M-Pareto optimal solution with less effort, we consider the augmented minimax problem

58

3 Fuzzy Multiobjective Stochastic Programming

⎫ ⎪ ⎬ minimize max μˆ l − μl (zVl (x)) + ρ ∑ (ˆμi − μi (zVi (x))) 1≤l≤k i=1 ⎪ ⎭ η, γ ). subject to x ∈ X(η

k

(3.20)

By introducing the auxiliary variable v, this problem can be equivalently transformed as ⎫ minimize v ⎪ ⎪ ⎪ k ⎪ ⎪ V V ⎪ subject to μˆ 1 − μ1 (z1 (x)) + ρ ∑ μˆ i − μi (zi (x)) ≤ v ⎪ ⎪ ⎪ ⎬ i=1 ········· (3.21) ⎪ ⎪ k ⎪ ⎪ μˆ k − μk (zVk (x)) + ρ ∑ μˆ i − μi (zVi (x)) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎭ η, γ ), x ∈ X(η where ρ is a sufﬁciently small positive number. We assume that the membership function μl , l = 1, . . . , k are nonincreasing and η, γ ) and any concave. Then, letting gl (x) = μˆ l − μl (zVl (x)), for any x1 and x2 in X(η λ ∈ [0, 1], the convexity property of gl (x) is shown as gl (λx1 + (1 − λ)x2 ) = μˆ l − μl (λzVl (x)) + (1 − λ)zVl (x2 ) ≤ μˆ l − λμl (zVl (x1 )) − (1 − λ)zVl (x2 ) ≤ μˆ l − λμl (zVl (x1 )) + (1 − λ)μ( zVl (x2 )) = λ(ˆμl − μl (zVl (x2 ))) + (1 − λ)(ˆμl − μl (zVl (x2 )))) = λgl (x1 ) + (1 − λ)gl (x2 ).

(3.22)

Recalling that the sum of convex functions is also convex, one ﬁnds that each of the left-hand side functions of the constraints in (3.21), expressed as gl (x) + ρ ∑ki=1 gi (x), is also convex, which implies that (3.21) can be solved by a traditional convex programming technique such as the sequential quadratic programming method (Fletcher, 1980; Gill, Murray and Wright, 1981; Powell, 1983). In order to derive a satisﬁcing solution for the DM from among the M-Pareto optimal solution set, we present an interactive fuzzy satisﬁcing method where the reference membership levels are repeatedly updated until the DM obtains a satisﬁcing solution. Interactive fuzzy satisﬁcing method for the variance model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima and maxima of E[zl (x)], l = 1, . . . , k by solving linear programming problems (3.16). Step 3: Ask the DM to specify the permissible expectation levels γl , l = 1, . . . , k. Step 4: Calculate the individual minima of zVl (x), l = 1, . . . , k by solving the quadratic programming problems (3.18).

3.1 Fuzzy multiobjective stochastic linear programming

59

Step 5: Ask the DM to specify the membership functions μl , l = 1, . . . , k with the individual minima zVl,min obtained in step 4 in mind. Step 6: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 7: For the current reference membership levels μˆ l , l = 1, . . . , k, solve the augmented minimax problem (3.20). Step 8: The DM is supplied with the corresponding M-Pareto optimal solution. If the DM is satisﬁed with the current membership function values zVl (x), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k taking into account the current membership function values μl (zVl (x∗ )), and return to step 7. It is signiﬁcant to emphasize that, in the interactive process, any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the ﬁxed satisﬁcing probability levels ηi , i = 1, . . . , m and permissible expectation levels γl , l = 1, . . . , k.

3.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive fuzzy satisﬁcing method for the variance model, as a numerical example of (3.14), consider the multiobjective stochastic linear programming problem formulated as ⎫ minimize z1 (x) = c¯ 1 x ⎪ ⎪ ⎪ minimize z2 (x) = c¯ 2 x ⎪ ⎪ ⎪ ⎪ minimize z3 (x) = c¯ 3 x ⎪ ⎬ subject to a1 x ≤ b¯ 1 (3.23) ⎪ a2 x ≤ b¯ 2 ⎪ ⎪ ⎪ ⎪ a3 x ≤ b¯ 3 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x8 )T and the random variables b¯ i , i = 1, 2, 3 are Gaussian random variables N(220, 42 ), N(145, 32 ) and N(−18, 52 ). The coefﬁcient vectors c¯ l , l = 1, 2, 3 are also Gaussian random variables vectors with mean vectors E[¯cl ], l = 1, 2, 3 as shown in Table 3.1 and the variance-covariance matrices Vl , l = 1, 2, 3 are given as Table 3.1 Mean vectors E[¯cl ] of c¯ l , l = 1, 2, 3. E[¯c1 ] E[¯c2 ] E[¯c3 ]

5 10 −8

2 −7 −1

1 1 −7

2 −2 −4

1 −5 −2

6 3 −3

1 −4 −1

3 6 −1

60

3 Fuzzy Multiobjective Stochastic Programming ⎛

16.0 ⎜ −1.6 ⎜ ⎜ 1.8 ⎜ ⎜ −3.5 V1 = ⎜ ⎜ 1.3 ⎜ ⎜ −2.0 ⎝ 4.0 −1.4 ⎛ 4.0 ⎜ −1.4 ⎜ ⎜ 0.8 ⎜ ⎜ 0.2 V2 = ⎜ ⎜ 1.6 ⎜ 1.0 ⎜ ⎝ 1.2 2.0 ⎛ 4.0 ⎜ −0.9 ⎜ ⎜ 1.1 ⎜ ⎜ −1.8 V3 = ⎜ ⎜ 3.0 ⎜ 2.2 ⎜ ⎝ 3.3 1.0

−1.6 25.0 −2.2 1.6 −0.7 0.5 −1.3 2.0

−3.5 1.6 −2.0 16.0 −2.0 3.0 2.2 2.8

1.3 −0.7 5.0 −2.0 4.0 −1.0 0.8 −2.0

−1.4 0.8 0.2 4.0 0.2 −1.0 0.2 9.0 0.2 −1.0 0.2 36.0 −2.2 −1.5 0.8 0.8 1.5 0.4 0.9 1.0 −1.5 1.8 0.6 0.7

1.6 −2.2 −1.5 0.8 25.0 1.2 −0.2 2.0

1.8 −2.2 25.0 −2.0 5.0 −2.4 1.2 −2.1

−0.9 1.1 −1.8 3.0 1.0 0.8 0.6 −2.4 0.8 9.0 −2.5 2.8 0.6 −2.5 4.0 −1.4 −2.4 2.8 −1.4 16.0 1.2 3.5 2.1 0.5 −2.3 1.2 0.8 1.1 2.0 −2.9 2.3 −1.3

⎞ −2.0 4.0 1.4 0.5 −1.3 2.0 ⎟ ⎟ −2.4 1.2 −2.1 ⎟ ⎟ 3.0 2.2 2.8 ⎟ ⎟ −1.0 0.8 −2.0 ⎟ 1.0 −1.5 0.6 ⎟ ⎟ −1.5 4.0 −2.3 ⎠ 0.6 −2.3 4.0 ⎞ 1.0 1.2 2.0 0.8 0.9 1.8 ⎟ ⎟ 1.5 1.0 0.6 ⎟ ⎟ 0.4 −1.5 0.7 ⎟ ⎟ 1.2 −0.2 2.0 ⎟ ⎟ 25.0 0.5 1.4 ⎟ 0.5 9.0 0.8 ⎠ 1.4 0.8 16.0 ⎞ 2.2 3.3 1.0 1.2 −2.3 2.0 ⎟ ⎟ 3.5 1.2 −2.9 ⎟ ⎟ 2.1 0.8 2.3 ⎟ ⎟ 0.5 1.1 −1.3 ⎟ ⎟ 25.0 −2.6 3.0 ⎟ −2.6 36.0 0.8 ⎠ 3.0 0.8 25.0

The coefﬁcient vectors ai , i = 1, 2, 3 are given as Table 3.2. Table 3.2 Value of each element of ai , i = 1, 2, 3. a1 a2 a3

7 5 −4

2 6 −7

6 −4 −2

9 3 −6

11 3 −8

4 −7 −3

3 −1 −5

8 −3 −6

Suppose that the DM speciﬁes the satisﬁcing probability levels ηi , i = 1, 2, 3 as (η1 , η2 , η3 )T = (0.85, 0.95, 0.90)T . Then, the individual minima zEl,min and maxima zEl,max of zEl (x), l = 1, 2, 3 are calculated through the simplex method as zE1,min = 3.051, zE1,max = 323.781, zE2,min = −425.617, zE2,max = 304.476, zE3,min = −251.830 and zE3,max = −3.487. With these values in mind, assume that the DM subjectively speciﬁes permissible expectation levels γl , l = 1, 2, 3 as γ1 = 60.000, γ2 = 35.000, γ3 = −16.000. Using the sequential quadratic programming method, the individual minima zVl,min of zVl (x), l = 1, 2, 3 are calculated as zV1,min = 1.370, zV2,min = 19.640 and zV3,min = 6.413. By taking account of these values, assume that the DM determines the linear membership functions as

3.1 Fuzzy multiobjective stochastic linear programming

μl (zVl (x)) =

⎧ 1 ⎪ ⎪ ⎪ ⎨ zV (x) − zV l l,0 ⎪ zV − zVl,0 ⎪ ⎪ ⎩ l,1 0

61

if zVl (x) ≤ zVl,1 if zVl,1 < zVl (x) ≤ zVl,0 if zVl,0 < zVl (x),

where zVl,1 and zVl,0 , l = 1, 2, 3 are calculated as zV1,1 = 1.370, zV1,0 = 130.939, zV2,1 = 19.640, zV2,0 = 358.271, zV3,1 = 6.413, and zV3,0 = 401.016 by using Zimmermann’s method (Zimmermann, 1978). μl (zVl (x)) 1

0

z l,V 1

z l,V 0

z Vl (x)

Fig. 3.2 The shape of a linear membership function.

For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.000, 1.000, 1.000), the augmented minimax problem (3.20) is solved by the sequential quadratic programming method, and the DM is supplied with the corresponding membership function values in the ﬁrst iteration shown in Table 3.3. Assume that the DM is not satisﬁed with the ﬁrst membership function values, and the DM updates the reference membership levels to (0.900, 1.000, 1.000) for improving the satisfaction levels for μ2 and μ3 at the sacriﬁce of μ1 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the corresponding membership function values in the second iteration are calculated as shown in Table 3.3. Suppose that the DM is not satisﬁed with the second membership function values, and the DM updates the reference membership levels as (0.900, 0.950, 1.000) for improving the satisfaction levels for μ1 and μ3 at the expense of μ2 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the corresponding membership function values are calculated. After obtaining the result of the third iteration shown in Table 3.3, assume that the DM is satisﬁed with this membership function values. Then, it follows that the satisﬁcing solution for the DM is derived.

62

3 Fuzzy Multiobjective Stochastic Programming

Table 3.3 Process of interaction. Iteration 1st μˆ 1 1.000 1.000 μˆ 2 μˆ 3 1.000

2nd 0.900 1.000 1.000

3rd 0.900 0.950 1.000

μ1 (zV1 (x)) μ2 (zV2 (x)) μ3 (zV3 (x))

0.847 0.847 0.847

0.793 0.893 0.893

0.804 0.854 0.904

zV1 (x) zV2 (x) zV3 (x)

21.250 71.596 66.957

28.142 55.744 48.485

26.768 69.087 44.303

3.1.2 Probability and fractile models 3.1.2.1 Probability model In contrast to the expectation and variance models discussed in the previous subsection, a certain target value is introduced for each of the objective function values in the probability model, and the probability that the objective function value is smaller than or equal to the target value is maximized. In this subsection, along the same line as the problem formulation introduced by Stancu-Minasian (1984), by adding a random parameter υ¯ l to the lth objective functions to deal with more general problems than (3.1), we formulate a multiobjective stochastic programming problem ⎫ minimize z1 (x) = c¯ 1 x + υ¯ 1 ⎪ ⎪ ⎪ ⎪ ······ ⎬ minimize zk (x) = c¯ k x + υ¯ k (3.24) ⎪ ⎪ subject to Ax ≤ b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, c¯ l , l = 1, . . . , k are n dimensional random variable row vectors, υ¯ l , l = 1, . . . , k are the random parameters, A is an m × n coefﬁcient matrix, and b¯ is an n dimensional vector whose elements are mutually independent random variables with continuous and increasing probability distribution functions Fl . Furthermore, it is assumed that the random vector c¯ l and the random parameter υ¯ l are expressed by c¯ l = c1l + t¯l c2l and υ¯ l = υ1l + t¯l υ2l , respectively, where t¯l , l = 1, . . . , k are mutually independent random variables with the continuous and strictly increasing probability distribution functions Tl . As discussed in the previous subsection, by introducing the idea of chance constrained condition, (3.24) can be interpreted as

3.1 Fuzzy multiobjective stochastic linear programming

63

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

minimize z1 (x) = c¯ 1 x + υ¯ 1 ······ minimize zk (x) = c¯ k x + υ¯ k subject to P(ω | a1 x ≤ b1 (ω)) ≥ η1 ⎪ ⎪ ······ ⎪ ⎪ ⎪ P(ω | am x ≤ bm (ω)) ≥ ηm ⎪ ⎪ ⎪ ⎭ x ≥ 0,

(3.25)

where ai is the ith row vector of A, and bi (ω) is the ith element of the realized vector b(ω) for an elementary event ω. Recalling (3.3), it follows that P(ω | ai x ≤ bi (ω)) ≥ ηi ⇔ ai x ≤ Fi−1 (1 − ηi ), η) denote the feasible region of (3.26), i.e., and for notational convenience, let X(η η) {x | ai x ≤ Fi−1 (1 − ηi ), i = 1, . . . , m, x ≥ 0}. X(η Assuming that the DM maximizes the probability that each objective function zl (x) = c¯ l x + υ¯ l in (3.24) is smaller than or equal to a certain target value fl , in the probability model we consider a multiobjective stochastic programming problem ⎫ maximize zP1 (x) = P (ω | c1 (ω)x + υ1 (ω) ≤ f 1 ) ⎪ ⎪ ⎬ ······ (3.26) P maximize zk (x) = P (ω | ck (ω)x + υk (ω) ≤ fk ) ⎪ ⎪ ⎭ η), subject to x ∈ X(η where cl (ω) and υl (ω) denote the realization of c¯ l and υ¯ l , respectively. To help the DM specify the target values hl , l = 1, . . . , k, it is recommended to calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k obtained by solving the linear programming problems ⎫ minimize zEl (x) = E[¯cl ]x, l = 1, . . . , k, ⎬ η) x∈X(η (3.27) maximize zEl (x) = E[¯cl ]x, l = 1, . . . , k. ⎭ η) x∈X(η

If we assume that c2l x + υ2l > 0, l = 1, . . . , k for any x ∈ X, from the property of the distribution functions Tl of the random variables t¯l , it is evident that the lth objective function in (3.26) is calculated as zPl (x) = P ω | (c1l + tl (ω)c2l )x + (υ1l + tl (ω)υ2l ) ≤ f l fl − (c1l x + υ1l ) = P ω tl (ω) ≤ (c2l x + υ2l ) f l − c1l x − υ1l . = Tl c2l x + υ2l

64

3 Fuzzy Multiobjective Stochastic Programming

Hence, (3.26) can be equivalently transformed into ⎫ f1 − c11 x − υ11 ⎪ P ⎪ maximize z1 (x) = T1 ⎪ ⎪ c21 x + υ21 ⎪ ⎪ ⎬ ········· fk − c1k x − υ1k ⎪ ⎪ ⎪ maximize zPk (x) = Tk ⎪ ⎪ c2k x + υ2k ⎪ ⎭ η). subject to x ∈ X(η

(3.28)

Considering the imprecise nature of the DM’s judgments for each objective function in (3.28), if we introduce the fuzzy goals such as “zPl (x) should be substantially greater than or equal to a certain value”, (3.28) can be interpreted as maximize (μ1 (zP1 (x)), . . . , μk (zPk (x))),

(3.29)

η) x∈X(η

where μl is a membership function to quantify a fuzzy goal for the lth objective function in (3.28). To elicit the membership function from the DM for each of the fuzzy goals, the DM is asked to assess an unacceptable objective function value zPl,0 , and a desirable objective function value zPl,1 as shown in Fig. 3.3. μl (zPl (x)) 1

0

z l,P 0

z l,P 1

z lP(x)

Fig. 3.3 Example of a membership function μl (zPl (x)).

In order to help the DM specify zPl,0 and zPl,1 , it serves as a useful reference to calculate the individual minima of zPl (x) by solving the ordinary single-objective linear fractional programming problems ⎫ fl − c1l x − υ1l ⎪ , l = 1, . . . , k, ⎪ minimize ⎬ x∈X c2l x + υ2l f l − c1l x − υ1l ⎪ ⎭ maximize , l = 1, . . . , k. ⎪ x∈X c2l x + υ2l

(3.30)

For the DM’s reference membership levels μˆ l , i = 1, . . . , k, the corresponding MPareto optimal solution, which is the nearest to the vector of the reference levels

3.1 Fuzzy multiobjective stochastic linear programming

65

in the minimax sense or is better than it if the reference membership levels are attainable, is obtained by solving the minimax problem minimize max {ˆμl − μl (zPl (x))} 1≤l≤k (3.31) η), subject to x ∈ X(η or equivalently

⎫ minimize v ⎪ ⎪ ⎪ subject to μˆ 1 − μ1 (zP1 (x)) ≤ v ⎪ ⎬ ········· ⎪ μˆ k − μk (zPk (x)) ≤ v ⎪ ⎪ ⎪ ⎭ η). x ∈ X(η

(3.32)

Recalling that each membership function μl is continuous and strictly increasing, it follows that (3.32) is equivalently transformed into ⎫ minimize v ⎪ ⎪ ⎪ μ1 − v) ⎪ subject to zP1 (x) ≥ μ−1 ⎬ 1 (ˆ ········· (3.33) ⎪ ⎪ ⎪ zPk (x) ≥ μ−1 (ˆ μ − v) k ⎪ k ⎭ η). x ∈ X(η From the continuity and strictly increasing property of the distribution function Tl , it follows that fl − c1l x − υ1l zPl (x) ≥ μ−1 (ˆ μ − v) ⇔ T μl − v) ≥ μ−1 l l l l (ˆ c2l x + υ2l ⇔

fl − c1l x − υ1l ≥ Tl−1 (μ−1 μl − v)). l (ˆ c2l x + υ2l

Consequently, (3.33) can be equivalently rewritten as ⎫ minimize v ⎪ ⎪ 1 1 ⎪ ⎪ f1 − c1 x − υ1 ⎪ −1 −1 ⎪ subject to ≥ T (μ (ˆ μ − v)) 1 ⎪ 1 1 2 2 ⎪ c1 x + υ1 ⎬ ········· ⎪ ⎪ fk − c1k x − υ1k ⎪ −1 −1 ⎪ ⎪ ≥ T (μ (ˆ μ − v)) k ⎪ k k 2 2 ⎪ ck x + υk ⎪ ⎭ η). x ∈ X(η

(3.34)

It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (3.34) can be reduced to a set of linear inequalities. Obtaining the optimal value v∗ to the above problem is equivalent to ﬁnding the minimum value of v so that there exists a feasible solution x satisfying the constraints of (3.34). From the fact that v satisﬁes

66

3 Fuzzy Multiobjective Stochastic Programming

μˆ max − max μl,max ≤ v ≤ μˆ max − min μl,min , 1≤l≤k

1≤l≤k

where μˆ max = max μˆ l , 1≤l≤k

μl,max = max μl (zPl (x)), η) x∈X(η

μl,min = min μl (zPl (x)), η) x∈X(η

we can obtain the minimum value of v by the combined use of the phase one of the two-phase simplex method and the bisection method. After ﬁnding the minimum value v∗ , in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem ⎫ c11 x + υ11 − f1 ⎪ ⎪ ⎪ minimize ⎪ 2 2 ⎪ c1 x + υ1 ⎪ ⎪ ⎪ 1 1 ⎪ f2 − c2 x − υ2 −1 −1 ∗ ⎪ ⎪ subject to ≥ T (μ (ˆ μ − v )) ⎬ 2 2 2 2 2 c2 x + υ2 (3.35) ⎪ ········· ⎪ ⎪ ⎪ fk − c1k x − υ1k ⎪ −1 −1 ∗ ⎪ ⎪ ≥ T (μ (ˆ μ − v )) k ⎪ k k ⎪ c2k x + υ2k ⎪ ⎪ ⎭ η x ∈ X(η ), where z1 (x) in (3.24) is supposed to be the most important to the DM. For notational convenience, let b = (b1 , . . . , bm )T where bi = Fi−1 (1 − ηi ). Then, using the variable transformation introduced by Charnes and Cooper (1962) s = 1/(c21 x + υ21 ), y = sx, s > 0,

(3.36)

the linear fractional programming problem (3.35) is equivalently transformed as ⎫ minimize c11 y + (υ11 − f 1 )s ⎪ ⎪ ⎪ subject to τ2 (c22 y + υ22 s) + c12 y + (υ12 − f2 )s ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ········· ⎪ ⎪ ⎪ τk (c2k y + υ2k s) + c1k y + (υ1k − fk )s ≤ 0 ⎪ ⎬ (3.37) Ay − sb ≤ 0 ⎪ ⎪ c21 y + υ21 s = 1 ⎪ ⎪ ⎪ ⎪ −s ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎪ ⎭ s ≥ 0, where τl = Tl−1 (μ−1 μl − v∗ )), and δ is a sufﬁciently small positive number introl (ˆ duced for the condition s > 0 in (3.36). If the optimal solution (y∗ , s∗ ) to (3.37) is not unique, M-Pareto optimality of ∗ x = s∗ y∗ is not always guaranteed. Observing that μl is a strictly increasing membership function, it should be noted here that the Pareto optimality test can be used in place of the M-Pareto optimality test. With this observation in mind, M-Pareto optimality of x∗ can be tested by solving the linear programming problem

3.1 Fuzzy multiobjective stochastic linear programming

maximize

w=

k

∑ εl

l=1

subject to

q1 (x∗ ) r1 (x) r1 (x∗ ) ········· qk (x∗ ) rk (x) qk (x) − εk = rk (x∗ ) x ∈ X, ε = (ε1 , . . . , εk )T ≥ 0, q1 (x) − ε1 =

67

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3.38)

where ql (x) = fl − c1l x − υ1l , rl (x) = c2l x + υ2l , l = 1, . . . , k. For the optimal solution to (3.38), (a) if w = 0, i.e., εl = 0 for l = 1, . . . , k, x∗ is M-Pareto optimal. On the other hand, (b) if w > 0, i.e., εl > 0 for at least one l, x∗ is not Pareto optimal. Then, we can ﬁnd an M-Pareto optimal solution according to the following algorithm. Step 1: For the optimal solution (xo , ε o ) to (3.38), after arbitrarily selecting j such as εoj > 0, through the combined use of the Charnes-Cooper variable transformation and the simplex method, solve the linear fractional programming problem ⎫ f j − c1j x − υ1j ⎪ ⎪ ⎪ maximize ⎪ ⎪ c2j x + υ2j ⎪ ⎪ ⎪ 1 1 1 o 1 ⎪ fl − cl x − υl fl − cl x − υl ⎬ o subject to = , ∀l ∈ {l | ε = 0} l 2 2 2 2 o (3.39) cl x + υl cl x + υl ⎪ ⎪ 1 1 1 o 1 ⎪ fl − cl x − υl fl − cl x − υl ⎪ ≥ , ∀l ∈ {l | εol > 0} ⎪ ⎪ ⎪ 2 2 2 2 o ⎪ cl x + υl cl x + υl ⎪ ⎭ η). x ∈ X(η Step 2: Let (x , ε ) be an optimal solution of (3.38). To test Pareto optimality of the optimal solution x´ to (3.39), solve (3.38) for x∗ := x´ . Step 3: If w = 0, stop the algorithm. Otherwise, i.e., if w > 0, set (xo , εo ) := (x , ε ), and return to step 1. By repeating this process at least k − 1 iterations, an M-Pareto optimal solution can be obtained. The DM should either be satisﬁed with the current Pareto optimal solution or continue to examine another solution by updating the reference membership levels. In order to help the DM express a degree of preference, the trade-off information between a standing membership function and each of the other membership functions is very useful. In a manner similar to the expectation model in 3.1.1.1, such trade-off information is easily obtainable because of the linearity of (3.37). To be more speciﬁc, for deriving the trade-off information, we introduce the Lagrange function L deﬁned by

68

3 Fuzzy Multiobjective Stochastic Programming k

L(y, s, π , ζ , φ ) = c11 y + (υ11 − f 1 )s + ∑ πl [τl (c2l y + υ2l s) l=2

m

+ {c1l y + (υ1l − f l )s}] + ∑ ζi (ai y − sbi ) i=1

+ ζm+1 (c21 y + υ21 s − 1) + ζm+2 (−s + δ) n

− ∑ φ j y j − φn+1 s,

(3.40)

j=1

where πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers of (3.37). Then, the partial derivative of L(y, s, π , ζ , φ ) with respect to τl is given as ∂L(y, s, π , ζ , φ ) = πl (c2l y + υ2l s), l = 2, . . . , k. ∂τl

(3.41)

On the other hand, for the optimal solution (y∗ , s∗ ) to (3.37) and the corresponding π∗ , ζ ∗ , φ ∗ ), from the Kuhn-Tucker necessity theorem (Sakawa, simplex multipliers (π 1993), it holds that ∗

L(y∗ , s∗ , π ∗ , ζ , φ ∗ ) = c11 y∗ + (υ11 − f 1 )s∗ .

(3.42)

If the ﬁrst (k − 1) constraints to (3.37) are active, τl is calculated as τl = −

c1l y∗ + (υ1l − f l )s∗ , l = 2, . . . , k. c2l y∗ + υ2l s∗

(3.43)

From (3.41), (3.42) and (3.43), one ﬁnds that ∂(c1 y∗ + (υ11 − f1 )s∗ ) 11 ∗ = π∗l (c2l y∗ + υ2l s∗ ), l = 2, . . . , k. cl y + (υ1l − f l )s∗ ∂ − c2l y∗ + υ2l s∗ By substituting x∗ for y∗ and s∗ in (3.44), the equation is rewritten as f1 − c11 x∗ − υ11 ∂ c2 x∗ + υ2l c2 x∗ + υ2 , l = 2, . . . , k. − 1 1 ∗ 1 1 = π∗l l2 ∗ c1 x + υ21 fl − cl x − υl ∂ c2l x∗ + υ2l Using the chain rule, it follows that

(3.44)

(3.45)

3.1 Fuzzy multiobjective stochastic linear programming

69

f1 − c11 x∗ − υ11 fl − c11 x∗ − υ11

T 2 ∗ 2 1 c21 x∗ + υ21 c21 x∗ + υ21 ∗ cl x + υl = π − , l c21 x∗ + υ21 fl − c1l x∗ − υ1l fl − c1l x∗ − υ1l ∂Tl T l c2l x∗ + υ2l c2l x∗ + υ2l ∂T1

l = 2, . . . , k,

(3.46)

where T denotes the differential coefﬁcient of T . Equivalently, we have

−

2 ∗ 2 P ∗ ∂zP1 (x∗ ) ∗ cl x + υl z1 (x ) , l = 2, . . . , k, = π

l ∂zPl (x∗ ) c21 x∗ + υ21 zPl (x∗ )

(3.47)

where zPl denotes the differential coefﬁcient of zPl . Once again, using the chain rule, we obtain 2 ∗ 2 P ∗

P ∗ ∂μ1 (zP1 (x∗ )) ∗ cl x + υl z1 (x ) μ1 (z1 (x )) = π , (3.48) −

l 2 ∗ ∂μl (zPl (x∗ )) c1 x + υ21 zPl (x∗ ) μ l (zPl (x∗ )) where μ l denotes the differential coefﬁcient of μl . It should be stressed here that in order to obtain the trade-off information from (3.48), the ﬁrst (k − 1) constraints in (3.37) must be active. Therefore, if there are inactive constraints, it is necessary to replace μˆ l for inactive constraints with μl (zPl (x∗ )) + v∗ and solve the corresponding problem to obtain the simplex multipliers. Now we are ready to summarize an interactive algorithm for deriving the satisﬁcing solution for the DM from among the M-Pareto optimal solution set. Interactive fuzzy satisﬁcing method for the probability model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x), l = 1, . . . , k by solving the linear programming problems (3.27). Step 3: Ask the DM to specify the target values fl , l = 1, . . . , k by taking into account the individual minima and maxima obtained in step 3. Step 4: Calculate the individual minima zPl,min and maxima zPl,max of zPl (x), l = 1, . . . , k in (3.28) by solving (3.30). Step 5: Ask the DM to specify the membership functions μl (zPl (x)), l = 1, . . . , k by considering the individual minima and maxima obtained in step 4. Step 6: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 7: For the current reference membership levels μˆ l , l = 1, . . . , k, solve the minimax problem (3.34) and the corresponding problem (3.37). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (3.37), replace μˆ l for the inactive constraints with μl (zPl (x∗ )) + v∗ and resolve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the M-Pareto optimality test. Step 8: The DM is supplied with the corresponding M-Pareto optimal solution x∗ and the trade-off rates between the membership functions. If the DM is sat-

70

3 Fuzzy Multiobjective Stochastic Programming

isﬁed with the current membership function values zPl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by considering the current membership function values μl (zPl (x∗ )), l = 1, . . . , k together with the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k, and return to step 7. Observing that the trade-off rates −∂μ1 /∂μl , l = 2, . . . , k in step 8 indicate the decrement value of the membership function μ1 with a unit increment of the value of the membership function μl , such trade-off information is used to estimate the local shape of (μ1 (zP1 (x∗ )), . . ., μk (zPk (x∗ ))) around x∗ . 3.1.2.2 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy satisﬁcing method for the probability model, as a numerical example of (3.24), consider the multiobjective stochastic linear programming problem formulated as ⎫ minimize (c11 + t¯1 c21 )x + (υ11 + t¯1 υ21 ) ⎪ ⎪ ⎪ minimize (c12 + t¯2 c22 )x + (υ12 + t¯2 υ22 ) ⎪ ⎬ 1 2 1 2 (3.49) minimize (c3 + t¯3 c3 )x + (υ3 + t¯3 υ3 ) ⎪ ⎪ subject to ai x ≤ b¯ i , i = 1, . . . , 7 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x10 )T , and the random parameters t¯1 , t¯2 and t¯3 are assumed to be Gaussian random variables N(4, 22 ), N(3, 32 ) and N(3, 22 ), respectively. The righthand side random variables b¯ i , i = 1, . . . , 7 are assumed to be Gaussian random variables N(164, 302 ), N(−190, 202 ), N(−184, 152 ), N(99, 222 ), N(−150, 172 ), N(154, 352 ) and N(142, 422 ), respectively. The constant coefﬁcients involved in the objective functions and the constraints are shown in Tables 3.4 and 3.5. Suppose that the DM speciﬁes the satisﬁcing probability levels at (η1 , η2 , η3 , η4 , η5 , η6 , η7 ) = (0.85, 0.95, 0.80, 0.90, 0.85, 0.80, 0.90). Then, the individual minima and maxima of zEl (x), l = 1, . . . , k, are calculated as zE1,min = 1819.571, zE1,max = 4221.883, zE2,min = 286.617, zE2,max = 1380.041, zE3,min = −1087.249 and zE3,max = −919.647. In order to determine the membership function of fuzzy goals, the individual minima and of zPl (x), l = 1, . . . , k in the multiobjective problem (3.28) are calculated as zP1,min = 0.002, zP1,max = 0.880, zP2,min = 0.328, zP2,max = 0.783, zP3,min = 0.002 and zP3,max = 0.664. Furthermore, assume that the DM speciﬁes the linear membership functions to quantify fuzzy goals for objective functions ⎧ 1 ⎪ ⎪ ⎪ ⎨ zP (x) − zP l l,0 μl (zPl (x)) = P − zP ⎪ z ⎪ l,1 l,0 ⎪ ⎩ 0

if zPl (x) ≤ zPl,1 if zPl,1 < zPl (x) ≤ zPl,0 if zPl,0 < zPl (x),

3.1 Fuzzy multiobjective stochastic linear programming

71

Table 3.4 Constant coefﬁcients of objective functions. c11 c21 c12 c22 c13 c23

19 3 12 1 −18 2

48 2 −46 2 −26 1

21 2 −23 4 −22 3

10 1 −38 2 −28 2

18 4 −33 2 −15 1

35 3 −48 1 −29 2

46 1 12 2 −10 3

11 2 8 1 −19 3

24 4 19 2 −17 2

33 2 20 1 −28 1

υ11 υ21 υ12 υ22 υ13 υ23

−18 5 −27 6 −10 4

Table 3.5 Constant coefﬁcients of constraints. a1 a2 a3 a4 a5 a6 a7

12 −2 3 −11 −4 5 −3

−2 5 −16 6 7 −3 −4

4 3 −4 −5 −6 14 −6

−7 16 −8 9 −5 −3 9

13 6 −8 −1 13 −9 6

−1 −12 2 8 6 −7 18

−6 12 −12 −4 −2 4 11

6 4 −12 6 −5 −4 −9

11 −7 4 −9 14 −5 −4

−8 −10 −3 6 −6 9 7

where zPl,1 and zPl,0 are calculated as zP1,1 = 0.880, zP1,0 = 0.502, zP2,1 = 0.783, zP2,0 = 0.060, zP3,1 = 0.664 and zP3,0 = 0.446 by using the Zimmermann method (Zimmermann, 1978). For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.00, 1.00, 1.00), the minimax problem (3.31) is solved, and the DM is supplied with the corresponding membership function values as shown at the second column in Table 3.6. Assume that the DM is not satisﬁed with the ﬁrst membership function values, the DM updates the reference membership levels to (1.00, 1.00, 0.90) for improving μ1 and μ2 at the sacriﬁce of μ3 . For the updated reference membership levels, the corresponding minimax problem is solved, and the corresponding membership function values are calculated as shown at the third column in Table 3.14. Assume that the DM is not satisﬁed with the second membership function values, by considering the trade-off information −∂μ1 /∂μ2 = 0.060 and −∂μ1 /∂μ3 = 0.801, and the DM updates the reference membership levels to (0.95, 1.00, 0.90) for improving μ2 and μ3 at the expense of μ1 . In the third iteration, assume that the DM is satisﬁed with the third membership function values. Then, it follows that the satisﬁcing solution for the DM is derived.

3.1.2.3 Fractile model For a decision situation where the DM prefers to maximize the probability that each of the objective functions is smaller than or equal to a target value speciﬁed by the DM, the probability model is recommended. In contrast, if the DM rather wants to minimize the target value after the DM speciﬁes a certain permissible level for the probability that each of the objective functions is smaller than or equal to the target value which is regarded as not a constant value of the goal but a variable to

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3 Fuzzy Multiobjective Stochastic Programming

Table 3.6 Process of interaction. Iteration μˆ 1 μˆ 2 μˆ 3

1st 1.000 1.000 1.000

2nd 1.000 1.000 0.900

3rd 0.950 1.000 0.900

μ1 (zP1 (x)) μ2 (zP2 (x)) μ3 (zP3 (x))

0.5747 0.5732 0.5733

0.6177 0.6172 0.5170

0.5948 0.6436 0.5435

zP1 (x) zP2 (x) zP3 (x)

0.719 0.474 0.571

0.736 0.506 0.559

0.727 0.525 0.565

−∂μ1 /∂μ2 −∂μ1 /∂μ3

0.060 0.831

0.060 0.801

0.060 0.816

be minimized, the fractile model is thought to be appropriate (Sakawa, Katagiri and Kato, 2001). Replacing the minimization of the objective functions zl (x), l = 1, . . . , k in (3.4) with the minimization of the target variables fl , l = 1, . . . , k under the probabilistic constraints with the permissible levels θl ∈ (1/2, 1) speciﬁed by the DM, we consider the fractile model for the multiobjective stochastic programming problems formulated as ⎫ minimize f1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ minimize fk (3.50) subject to P (ω | c1 (ω)x ≤ f1 ) ≥ θ1 ⎪ ⎪ ⎪ ········· ⎪ ⎪ ⎪ P (ω | ck (ω)x ≤ fk ) ≥ θk ⎪ ⎪ ⎪ ⎭ η), x ∈ X(η where each c¯ l is a Gaussian random variable vector with mean vector E[¯cl ] and the positive-deﬁnite variance-covariance matrix Vl . From the fact that the random variables c¯ l x − E[¯cl ]x ' , l = 1, . . . , k xT Vl x

(3.51)

are the standard Gaussian random variables with mean 0 and variance 12 , it follows that " # c (ω)x − E[¯c ]x f l − E[¯cl ]x l l ' P (ω | cl (ω)x ≤ f l ) ⇔ P ω ≤ ' xT Vl x xT Vl x " # fl − E[¯cl ]x ⇔ Φ ' , (3.52) xT Vl x

3.1 Fuzzy multiobjective stochastic linear programming

73

where Φ is the distribution function of the standard Gaussian random variable, deﬁned as y 1 2 1 Φ(y) = √ e− 2 x dx. 2π −∞ Then, in (3.50), the probabilistic constraints P (ω | cl (ω)x ≤ fl ) ≥ θl , l = 1, . . . , k can be transformed as # " fl − E[¯cl ]x fl − E[¯cl ]x ≥ Φ−1 (θl ) ≥ θl ⇔ ' Φ ' xT Vl x xT Vl x ' ⇔ fl ≥ E[¯cl ]x + Φ−1 (θl ) xT Vl x, where Φ−1 is the inverse function of Φ. By substituting (3.54) for (3.53), (3.50) can be transformed into ⎫ minimize f1 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ minimize fk ' −1 T subject to E[¯c1 ]x + Φ (θ1 ) x V1 x ≤ f1 ⎪ ⎪ ⎪ ········' · ⎪ ⎪ ⎪ −1 T ⎪ E[¯ck ]x + Φ (θk ) x Vk x ≤ fk ⎪ ⎪ ⎭ η). x ∈ X(η

(3.53)

(3.54)

(3.55)

Because (3.55) minimizes each fl under the constraint that each fl must be greater than or equal to ' (3.56) E[¯cl ]x + Φ−1 (θl ) xT Vl x, it becomes obvious that minimizing fl is equivalent to minimizing (3.56). Hence, (3.55) can be rewritten as ' ⎫ minimize zF1 (x) = E[¯c1 ]x + Φ−1 (θ1 ) xT V1 x ⎪ ⎪ ⎬ ········· ' (3.57) minimize zFk (x) = E[¯ck ]x + Φ−1 (θk ) xT Vk x ⎪ ⎪ ⎭ η). subject to x ∈ X(η Noting that each of the objective functions in (3.57) is convex from Φ−1 (θl ) ≥ 0 for any θl ∈ [1/2, 1), one ﬁnds that (3.57) is a multiobjective convex programming problem. In a way similar to the previous models, taking into account the imprecise nature of the DM’s judgments for each objective function, by introducing the fuzzy goals such as “zFl (x) should be substantially less than or equal to a certain value,” in place of (3.57), we consider the minimization problem of fuzzy goals

74

3 Fuzzy Multiobjective Stochastic Programming

maximize (μ1 (zF1 (x)), . . . , μk (zFk (x))), η) x∈X(η

(3.58)

where μl is assumed to be a concave membership function to quantify a fuzzy goal for the lth objective function in (3.57), as shown in Fig. 3.4. μl (zFl (x)) 1

0

z l,F 1

z l,F 0 z Fl (x)

Fig. 3.4 Example of a membership function μl (zFl (x)).

In order to ﬁnd a candidate for the satisﬁcing solution, the DM speciﬁes the reference membership levels μˆ l , i = 1, . . . , k, and then by solving the augmented minimax problem ⎫ k ⎪ F F minimize max μˆ l − μl (zl (x)) + ρ ∑ (ˆμi − μi (zi (x))) ⎬ 1≤l≤k (3.59) i=1 ⎪ ⎭ η), subject to x ∈ X(η an M-Pareto optimal solution corresponding to μˆ l , i = 1, . . . , k is obtained, and (3.59) is equivalently expressed as ⎫ minimize v ⎪ ⎪ ⎪ k ⎪ ⎪ F F ⎪ subject to μˆ 1 − μ1 (z1 (x)) + ρ ∑ μˆ i − μi (zi (x)) ≤ v ⎪ ⎪ ⎪ ⎬ i=1 ··············· (3.60) ⎪ ⎪ k ⎪ ⎪ μˆ k − μk (zFk (x)) + ρ ∑ μˆ i − μi (zFi (x)) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎭ η), x ∈ X(η where ρ is a sufﬁciently small positive number. As we illustrated that (3.21) is a convex programming problem, it can be shown that (3.60) is also convex under the assumption that each of membership functions μl , l = 1, . . . , k is nonincreasing and concave. Due to the convexity, an optimal solution to (3.60) can be found by using some convex programming technique such as the sequential quadratic programming method. We can now present the interactive algorithm for deriving a satisﬁcing solution for the DM from among the M-Pareto optimal solution set.

3.1 Fuzzy multiobjective stochastic linear programming

75

Interactive fuzzy satisﬁcing method for the fractile model Step 1: Ask the DM to specify the satisﬁcing probability levels ηi , i = 1, . . . , m and the permissible probability levels θl ∈ [1/2, 1), l = 1, . . . , k. Step 2: Calculate the individual minima zFl,min of zFl (x), l = 1, . . . , k. Step 3: Ask the DM to specify the nonincreasing and concave membership functions μl taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 5: For the reference membership levels μˆ l , l = 1, . . . , k, solve the augmented minimax problem (3.60). Step 6: The DM is supplied with the corresponding M-Pareto optimal solution x∗ . If the DM is satisﬁed with the current membership function values zFl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k by considering the current membership function values, and return to step 5. It should be stressed for the DM that, in an interactive process, any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the satisﬁcing probability levels ηi , i = 1, . . . , m and the permissible probability levels θl , l = 1, . . . , k speciﬁed by the DM.

3.1.2.4 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy satisﬁcing method for the fractile model, as a numerical example of (3.50), consider the multiobjective stochastic linear programming problem ⎫ minimize c¯ 1 x ⎪ ⎪ ⎪ ⎪ minimize c¯ 2 x ⎪ ⎪ ⎬ minimize c¯ 3 x (3.61) subject to a1 x ≤ b¯ 1 ⎪ ⎪ ⎪ a2 x ≤ b¯ 2 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , . . . , x4 )T , and (b¯ 1 , b¯ 2 ) is a Gaussian random variable vector with mean (27, −15) and variance (36, 49). The coefﬁcient vectors c¯ l , l = 1, 2, 3 are Gaussian random variable vectors, and their mean vectors E[¯cl ], l = 1, 2, 3 are given in Table 3.7 and the variance-covariance matrices Vl , l = 1, 2, 3 are

76

3 Fuzzy Multiobjective Stochastic Programming ⎞ 25 −1 0.8 −2 ⎜ −1 4 −1.2 1.2 ⎟ ⎟, V1 = ⎜ ⎝ 0.8 −1.2 4 2⎠ −2 1.2 2 9 ⎛ ⎞ 16 1.4 −1.2 1.4 ⎜ 1.4 1 1.5 −0.8 ⎟ ⎟, V2 = ⎜ ⎝ −1.2 1.5 25 −0.6 ⎠ 1.4 −0.8 −0.6 4 ⎛ ⎞ 4 −1.9 1.5 1.8 ⎜ −1.9 25 0.8 −0.4 ⎟ ⎟ V3 = ⎜ ⎝ 1.5 0.8 9 2.5 ⎠ . 1.8 −0.4 2.5 36 ⎛

The coefﬁcient vectors ai , i = 1, 2 of the constraints are also given in Table 3.8. Table 3.7 Mean of each element of E[¯cl ], l = 1, 2, 3. E[¯c1 ] E[¯c2 ] E[¯c3 ]

2 10 −8

3 −7 −5

2 1 −7

4 −2 −14

4 −7

6 −9

Table 3.8 Value of each element of ai , i = 1, 2, 3. a1 a2

7 −5

3 −6

Suppose that the DM speciﬁes the permissible probability levels at (θ1 , θ2 , θ3 ) = (0.75, 0.80, 0.70) and the satisﬁcing probability levels η1 = η2 = 0.7. Then, the individual minima zFl,min of zFl (x), l = 1, 2, 3 are calculated as zF1,min = 8.9825, zF2,min = −48.6224, zF3,min = −42.7926, through the sequential quadratic programming method. By taking account of these values, assume that the DM identiﬁes the following linear membership function: ⎧ 1 ⎪ ⎪ ⎪ ⎨ zF (x) − zF l l,0 μl (zFl (x)) = F F ⎪ zl,1 − zl,0 ⎪ ⎪ ⎩ 0

if zFl (x) ≤ zFl,1 if zFl,1 < zFl (x) ≤ zFl,0 if zFl,0 < zFl (x),

where zFl,1 and zFl,0 are calculated as zF1,1 = 8.9825, zF1,0 = 31.7493, zF2,1 = −48.6224, zF2,0 = 12.0371, zF3,1 = −42.7926 and zF3,0 = −16.2347 by using the Zimmermann method (Zimmermann, 1978). For the initial reference membership levels (ˆμ1 , μˆ 2 , μˆ 3 ) = (1.000, 1.000, 1.000), the augmented minimax problem (3.60) is solved by the sequential quadratic programming method, and the DM is supplied with the ﬁrst membership function values shown in Table 3.9.

3.1 Fuzzy multiobjective stochastic linear programming

77

Assume that the DM is not satisﬁed with the current membership function values, and the DM updates the reference membership levels from (1.000, 1.000, 1.000) to (0.800, 1.000, 1.000) for improving μ2 and μ3 at the sacriﬁce of μ1 . For the updated reference membership levels, the corresponding augmented minimax problem is solved, and the second membership function values are calculated as shown in Table 3.9. Although a similar procedure continues in this manner until the DM is satisﬁed with the obtained objective function values, for simplicity, we suppose that the DM is satisﬁed with the objective function values in the third iteration, and then it follows that the satisﬁcing solution for the DM is derived. Table 3.9 Process of interaction. Iteration 1st μˆ 1 1.000 μˆ 2 1.000 1.000 μˆ 3 zF1 (x∗ ) 20.3634 zF2 (x∗ ) −18.3000 −29.5167 zF3 (x∗ ) μ1 (zF1 (x∗ )) 0.5001 μ2 (zF2 (x∗ )) 0.5001 0.5001 μ3 (zF3 (x∗ ))

2nd 0.800 1.000 1.000 23.1834 −22.9178 −31.5386 0.3762 0.5762 0.5762

3rd 0.800 1.000 0.850 22.6022 −24.4663 −28.2329 0.4018 0.6018 0.4518

3.1.3 Simple recourse model In the chance constrained problems which we dealt with in the previous subsections, for random data variations, a mathematical model is formulated such that the violation of the constraints is permitted up to speciﬁed probability levels. On the other hand, in a two-stage model including a simple recourse model as a special case, a shortage or an excess arising from the violation of the constraints is penalized, and then the expectation of the amount of the penalties for the constraint violation is minimized. We consider a simple recourse model for the multiobjective stochastic programming problems ⎫ minimize z1 (x) = c1 x ⎪ ⎪ ⎪ ⎪ ······ ⎬ minimize zk (x) = ck x (3.62) ⎪ ⎪ subject to Ax = b¯ ⎪ ⎪ ⎭ x ≥ 0, where x is an n dimensional decision variable column vector, cl , l = 1, 2, . . . , k are n dimensional coefﬁcient row vectors, A is an m × n coefﬁcient matrix, and b¯ is

78

3 Fuzzy Multiobjective Stochastic Programming

an m dimensional random variable column vector. It is noted that, in a simple recourse model, the random variables are involved only in the right-hand side of the constraints. To understand an idea of the formulation in the simple recourse model, consider a decision problem of a manufacturing company. Suppose that the company makes m types of products which requires n kinds of working processes, and a DM in the company desires to optimize the total proﬁt and the total production cost simultaneously. Let x = (x1 , . . . , xn ) denote activity levels for the n kinds of working processes which are decision variables, and then T x denotes the amount of products, where T is an m × n matrix transforming the n kinds of activity levels of the working processes into the m types of products. The total proﬁt is expressed as c1 T x, where an m dimensional coefﬁcient vector c1 is a vector of unit product proﬁts for the m types of products, and the total production cost is represented by c2 x, where an n dimensional coefﬁcient vector c2 is a vector of unit costs for the n kinds of activity levels of the working processes. Assume that the demand coefﬁcients d¯ = (d¯1 , . . . , d¯m )T for the m products are uncertain, and they are represented by random variables. The demand ¯ where y+ and y− represent the constraints are expressed as T x + Iy+ − Iy− = d, errors for estimating the demands, and I is the m dimensional identity matrix. These two objectives are optimized under the demand constraints T x + Iy+ − Iy− = d¯ together with the ordinary constraints Bx ≤ e without uncertainty for the activity levels such as the capacity, budget, technology, etc. The constraints Ax = b¯ in (3.62) can be interpreted as the combined form of the demand constraints with random variables and the ordinary constraints without uncertainty. Let us return to a general case of the recourse model. It is assumed that, in this model, the DM must make a decision before the realized values of the random variables involved in (3.62) are observed, and the penalty of the violation of the constraints is incorporated into the objective function in order to consider the loss caused by random date variations. To be more speciﬁc, by expressing the difference between Ax and b¯ in (3.62) − + T − − T as two vectors y+ = (y+ 1 , . . . , ym ) and y = (y1 , . . . , ym ) , the expectation of a recourse for the lth objective function is represented by

− − + + − y y + q y ) − y = b(ω) − Ax , (3.63) Rl (x) = E min (q+ l l y+ , y−

− where q+ l and ql are m dimensional constant row vectors, and b(ω) is an m dimensional realization vector of b¯ for an elementary event ω. Thinking of each element − + T − − T of y+ = (y+ 1 , . . . , ym ) and y = (y1 , . . . , ym ) as a shortage and an excess of the − left-hand side, respectively, we can regard each element of q+ l and ql as the cost to compensate the shortage and the cost to dispose the excess, respectively. Then, for the multiobjective stochastic programming problem, the simple recourse problem is formulated as

3.1 Fuzzy multiobjective stochastic linear programming

79

⎫ minimize c1 x + R1 (x) ⎪ ⎪ ⎬ ······ minimize ck x + Rk (x) ⎪ ⎪ ⎭ subject to x ≥ 0.

(3.64)

− Because q+ l and ql are interpreted as penalty coefﬁcients for shortages and ex− cesses, it is quite natural to assume that q+ l ≥ 0 and ql ≥ 0, and then, it is evident that, for all i = 1, . . . , m, the complementary relations − yˆ+ i > 0 ⇒ yˆi = 0, + yˆ− i > 0 ⇒ yˆi = 0

should be satisﬁed for an optimal solution. With this observation in mind, we have n

− yˆ+ i = bi (ω) − ∑ ai j x j , yˆi = 0 if bi (ω) ≥ − yˆ+ i = 0, yˆi =

j=1 n

∑ ai j x j − bi (ω)

if bi (ω)

dl jsl , ⎩ max 1 − γl j where the value of dl jsl varies depending on which scenario sl ∈ {1, . . . , Sl } occurs, and βl j and γl j are not random parameters but constants. Fig. 4.2 illustrates an example of the membership function μC˜l js (τ). Formally, the membership function of l the fuzzy random variable C˜¯ is represented by lj

4.1 Multiobjective fuzzy random linear programming

105

⎧ d¯l j − τ ⎪ ⎪ max 1 − , 0 ⎪ ⎨ βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ,0 ⎩ max 1 − γl j

if τ ≤ d¯l j (4.4) if τ > d¯l j .

μC~ljs (τ) l

1

βlj

0

dljsl

γ lj

τ

Fig. 4.2 Example of the membership function μC˜l js . l

˜¯ x Through the extension principle given in Chapter 2, each objective function C l is represented by a single fuzzy random variable whose realized value for the scenario sl is a triangular fuzzy number C˜ lsl x characterized by the membership function ⎧ dlsl x − υ ⎪ ⎪ max 1 − , 0 if υ ≤ dlsl x ⎪ ⎨ βl x (4.5) μC˜ ls x (υ) = l ⎪ υ − dlsl x ⎪ ⎪ ,0 if υ > dlsl x, ⎩ max 1 − γl x where dlsl is an n dimensional column vector which is different from the other dl sˆl , sˆl ∈ {1, . . . , Sl }, sˆl = sl , and β l and γ l are n dimensional constant column vectors. Fig. 4.3 illustrates an example of the membership function μC˜ ls x (υ). Also for the l ¯˜ x, its membership function is formally expressed as lth objective function C l

⎧ d¯ l x − υ ⎪ ⎪ ⎪ , 0 max 1 − ⎨ βl x μC˜¯ x (υ) = l ⎪ υ − d¯ l x ⎪ ⎪ max 1 − , 0 ⎩ γl x

if υ ≤ d¯ l x (4.6) if υ > d¯ l x.

Considering the imprecise nature of human judgments, it is quite natural to assume that the decision maker (DM) may have a fuzzy goal for each of the objective ˜¯ x, and in a minimization problem, the DM speciﬁes the fuzzy functions zl (x) = C l goal such that “the objective function value should be substantially less than or equal to some value.” Such a fuzzy goal can be quantiﬁed by eliciting the corresponding

106

4 Multiobjective Fuzzy Random Programming

μC~

(υ)

ls l x

1

0

βl x

dlsl x

υ

γl x

Fig. 4.3 Example of the membership function μC˜ ls x . l

membership functions through some interaction process from the DM. Here, for simplicity, the linear membership function ⎧ ⎪ 0 if y > z0l ⎪ ⎪ ⎨ y − z0 l if z1l ≤ y ≤ z0l (4.7) μG˜ l (y) = 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 1 if y < z1l is assumed for representing the fuzzy goal of the DM, where z0l and z1l are determined as ⎫ n ⎪ ⎪ z0l = max max ∑ dl jsl x j , l = 1, . . . , k, ⎪ ⎬ sl ∈{1,...,Sl } x ∈ X j=1 (4.8) n ⎪ z1l = min min ∑ dl jsl x j , l = 1, . . . , k. ⎪ ⎪ ⎭ sl ∈{1,...,Sl } x ∈ X j=1

It should be noted here that z0l and z1l are obtained by solving linear programming problems. Fig. 4.4 illustrates an example of the membership function μG˜ l of a fuzzy goal G˜ l . μG~ (y) l

1

0

z l1

Fig. 4.4 Example of the membership function of a fuzzy goal.

z l0

y

4.1 Multiobjective fuzzy random linear programming

107

Recalling that the membership function is regarded as a possibility distribution as discussed in Chapter 2, the degree of possibility that the objective function value ˜¯ x attains the fuzzy goal G˜ is expressed as C l l (4.9) ΠC˜¯ x (G˜ l ) = sup min μC˜¯ x (y), μG˜ l (y) , l = 1, . . . , k, l

l

y

and for a given scenario sl ∈ {1, . . . , Sl }, (4.9) is reduced to ΠC˜ ls x (G˜ l ) = sup min μC˜ ls x (y), μG˜ l (y) , l = 1, . . . , k. l

(4.10)

l

y

Fig. 4.5 illustrates the degree of possibility that the fuzzy goal G˜ l is fulﬁlled under the possibility distribution μC˜ls x . l

μG~ (y)

μC~

l

ls l x

(y)

1 ~

ΠC~ x(Gl ) ls l

0

z 1l

0

(d lsl −βl ) x dlsl x

z 0l

y

Fig. 4.5 Degree of possibility ΠC˜ ls x (G˜ l ). l

Observing that the degrees of possibility vary randomly depending on which scenario occurs, it should be noted here that conventional possibilistic programming approaches cannot be directly applied to (4.2). With this observation in mind, realizing that (4.2) involves not only fuzziness but also randomness, we consider four types of fuzzy stochastic decision making models for multiobjective fuzzy random programming problems by introducing the concepts from possibility theory: the expectation, the variance, the probability and the fractile models.

4.1.1.1 Possibility-based expectation model Assuming that the DM intends to simply maximize the expected degree of possibility that each of the original objective functions involving fuzzy random variable coefﬁcients attains the fuzzy goals, the original multiobjective fuzzy random programming problems can be reformulated as

108

4 Multiobjective Fuzzy Random Programming

(

)⎫ maximize E ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ ⎪ 1 ⎪ ⎬ · · · · · · ( ) maximize E ΠC˜¯ x (G˜ k ) ⎪ ⎪ ⎪ k ⎪ ⎭ subject to x ∈ X,

(4.11)

where E denotes the expectation operator. From the viewpoint of possibility maximization and expectation optimization, as an extension of Pareto optimality discussed in Chapter 2, we deﬁne the concept of E-P-Pareto optimality. Deﬁnition 4.2 (E-P-Pareto Optimal Solution). A point x∗ ∈ X is said to be an EP-Pareto optimal solution ( ) to((4.2) if and) only if there does not exist (another x ∈ )X such that E ΠC˜¯ x (G˜ l ) ≥ E ΠC˜¯ x∗ (G˜ l ) for all l ∈ {1, . . . , k} and E ΠC˜¯ x (G˜ v ) = v l ) l ( E ΠC˜¯ x∗ (G˜ v ) for at least one v ∈ {1, . . . , k}. v

By substituting the triangular fuzzy random variable (4.5) and the linear fuzzy goal (4.7) into the degree of possibility (4.9), the degree of possibility is explicitly represented by n

∑ (βl j − dl jsl )x j + z0l

j=1 n

ΠC˜ ls x (G˜ l ) = l

∑ βl j x j − z1l + z0l

.

(4.12)

j=1

Recalling that the occurrence probability of scenario sl is plsl , the expectation of the degree of possibility is calculated as " # n

(

) E ΠC˜¯ x (G˜ l ) = l

Sl

∑ plsl ΠC˜ lsl x (G˜ l ) =

∑

βl j −

j=1

sl =1

n

∑

Sl

∑ plsl dl jsl

sl =1

x j + z0l .

βl j x j − z1l + z0l

j=1

Letting n

ZlΠ,E (x)

∑

" βl j −

j=1

Sl

#

∑ plsl dl jsl

sl =1

n

∑ β1 j x j − z11 + z01

x j + z0l ,

j=1

(4.11) is rewritten as

⎫ maximize Z1Π,E (x) ⎪ ⎪ ⎬ ··· Π,E maximize Zk (x) ⎪ ⎪ ⎭ subject to x ∈ X.

(4.13)

4.1 Multiobjective fuzzy random linear programming

109

To calculate a candidate for the satisﬁcing solution which is also E-P-Pareto optimal, in interactive multiobjective programming, the DM is asked to specify reference levels zˆl , l = 1, . . . , k of the objectives function values of (4.13), and it is called the reference expectation levels. For the DM’s reference expectation levels zˆl , l = 1, . . . , k, an E-P-Pareto optimal solution, which is the nearest to a vector of the reference expectation levels or better than it if the reference expectation levels are attainable in a sense of minimax, is obtained by solving the minimax problem ⎫ minimize max zˆl − ZlΠ,E (x) ⎬ 1≤l≤k (4.14) ⎭ subject to x ∈ X. By introducing an auxiliary variable v, (4.14) can be transformed into ⎫ minimize v ⎪ ⎪ ⎪ subject to zˆ1 − Z1Π,E (x) ≤ v ⎪ ⎬ ········· ⎪ ⎪ zˆk − ZkΠ,E (x) ≤ v ⎪ ⎪ ⎭ x ∈ X,

(4.15)

and (4.15) is equivalently expressed as ⎫ ⎪ ⎪ ⎪ S1 n ⎪ ⎪ 0 ⎪ ⎪ − p d + z β x ∑ 1 j ∑ 1s1 1 js1 j 1 ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ subject to ≥ z ˆ − v 1 ⎪ n ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎬ j=1 · · · · · · · · · · · · · · · " # ⎪ ⎪ Sk n ⎪ ⎪ 0 ⎪ β x − p d + z ⎪ ∑ k j ∑ ksk k jsk j k ⎪ ⎪ ⎪ j=1 sk =1 ⎪ ⎪ ≥ z ˆ − v k n ⎪ ⎪ ⎪ 1 0 ⎪ β x − z + z ⎪ ∑ kj j k k ⎪ ⎪ j=1 ⎪ ⎭ x ∈ X. minimize v

"

#

(4.16)

It is important to note here that, in this formulation, if the value of v is ﬁxed, the constraints of (4.16) can be reduced to a set of linear inequalities. Since v satisﬁes Π,E Π,E zˆmax − max Zl,max ≤ v ≤ zˆmax − min Zl,min , l∈{1,...,k}

l∈{1,...,k}

where zˆmax = max zˆl , l∈{1,...,k}

Π,E Zl,max = max ZlΠ,E (x), x∈X

Π,E Zl,min = min ZlΠ,E (x), x∈X

110

4 Multiobjective Fuzzy Random Programming

we can obtain the minimum value of v by the combined use of the phase one of the two-phase simplex method and the bisection method. After ﬁnding the minimum value v∗ , in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem " # ⎫ S1 n ⎪ 0 ⎪ ⎪ ∑ ∑ p1s1 d1 js1 − β1 j x j − z1 ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ minimize ⎪ n ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎪ j=1 ⎪ " # ⎪ ⎪ S2 n ⎪ ⎪ 0 ⎪ β x − p d + z ⎪ 2 j 2s 2 js j ∑ ∑ 2 2 2 ⎪ ⎪ ⎪ j=1 s2 =1 ⎪ ∗ ⎬ subject to ≥ z ˆ − v 2 n 1 0 (4.17) β x − z + z ∑ 2j j 2 2 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ "· · · · · · · · · · · · · · · # ⎪ Sk ⎪ n ⎪ ⎪ 0 ⎪ ⎪ ∑ βk j − ∑ pksk dk jsk x j + zk ⎪ ⎪ ⎪ j=1 sk =1 ∗ ⎪ ≥ zˆk − v ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ β x − z + z j k j ⎪ ∑ k k ⎪ ⎪ j=1 ⎪ ⎭ x ∈ X, where the ﬁrst objective function z1 (x) in (4.2) is supposed to be the most important to the DM. From the assumption of β1 j > 0, j = 1, . . . , n and z01 > z11 , one ﬁnds that n

∑ β1 j x j − z11 + z01 > 0.

j=1

Then, using the variable transformation (Charnes and Cooper, 1962) # " ς = 1/

n

∑ β1 j x j − z11 + z01

, y = ςx, ς > 0,

(4.18)

j=1

the linear fractional programming problem (4.17) is equivalently transformed into the linear programming problem

4.1 Multiobjective fuzzy random linear programming

111

⎫ ⎪ ⎪ ⎪ minimize ∑ ∑ p1s1 d1 js1 − β1 j y j − ςz01 ⎪ ⎪ ⎪ ⎪ j=1 s1 =1 ⎪ ⎪ ⎪ " # " # ⎪ ⎪ S2 n n ⎪ ⎪ 1 0 0 ⎪ subject to τ2 ∑ β2 j y j − ςz2 + ςz2 + ∑ ∑ p2s2 d2 js2 − β2 j y j − ςz2 ≤ 0 ⎪ ⎪ ⎪ ⎪ j=1 j=1 s2 =1 ⎪ ⎪ ⎪ ⎪ · · · · · · · · · · · · · · · ⎪ " # " # ⎪ ⎬ Sk n n 1 0 0 τk ∑ βk j y j − ςzk + ςzk + ∑ ∑ pksk dk jsk − βk j y j − ςzk ≤ 0 ⎪ ⎪ j=1 j=1 sk =1 ⎪ ⎪ ⎪ ⎪ Ay − ςb ≤ 0 ⎪ ⎪ ⎪ n ⎪ ⎪ 0 1 ⎪ β y + ς(z − z ) = 1 ⎪ ∑ 1j j 1 1 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎪ ⎭ ς ≥ 0, (4.19) where τl zˆl − v∗ , l = 2, . . . , k, and δ is a sufﬁciently small positive number given for satisfying the condition ς > 0 in (4.18). It should be noted here that if the optimal solution (y∗ , ς∗ ) to (4.19) is not unique, E-P-Pareto optimality of x∗ (= y∗ /ς∗ ) is not always guaranteed. Realizing such a situation, E-P-Pareto optimality of x∗ can be tested by solving the linear programming problem ⎫ k ⎪ ⎪ ⎪ maximize w = ∑ εl ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ∗ ⎪ q1 (x ) ⎪ ⎬ r subject to q1 (x) − ε1 = (x) 1 ∗ r1 (x ) (4.20) ⎪ ········· ⎪ ⎪ ⎪ ⎪ qk (x∗ ) ⎪ ⎪ r qk (x) − εk = (x) ⎪ k ⎪ rk (x∗ ) ⎪ ⎭ T x ∈ X, ε = (ε1 , . . . , εk ) ≥ 0, n

"

#

S1

where ql (x) rl (x)

n

∑

j=1 n

" βl j −

Sl

∑ plsl dl jsl

sl =1

# x j + z0l , l = 1, . . . , k

∑ βl j x j − z1l + z0l , l = 1, . . . , k.

j=1

Noting that (4.20) is the same type of problem as (3.38), it is evident that an E-PPareto optimal solution can be obtained by using a similar computational method to the given method in Chapter 3. The DM must either be satisﬁed with the current E-P-Pareto optimal solution or continue to examine another solution by updating the reference expectation levels. In order to help the DM update the reference expectation levels, the trade-off infor-

112

4 Multiobjective Fuzzy Random Programming

mation between an expected degree of possibility with respect to the most important objective function and each of the other ones is quite useful. Such trade-off information is easily obtainable by using the simplex multipliers of linear programming problem (4.19). To derive the trade-off information, the Lagrange function L for (4.19) is deﬁned as L(y, ς, π, ζ, φ) =

n

S1

j=1

s1 =1

∑ ∑

p1s1 d1 js1 − β1 j y j − ςz01

% "

k

+ ∑ πl τl l=2 n

+∑

#

n

∑

βl j y j − ςz1l + ςz0l

j=1

Sl

∑ plsl dl jsl − βl j

sl =1

j=1

+ζm+1

& y j − ςz0l

n

∑ β1 j y j + ς(z01 − z11 ) − 1

m

+ ∑ ζi (ai y − ςbi ) i=1

+ ζm+2 (−ς + δ)

j=1 n

− ∑ φ j y j − φn+1 ς,

(4.21)

j=1

where ai , i = 1, . . . , m are n dimensional coefﬁcient row vectors, and πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers of (4.19). In a manner similar to (3.41)–(3.45) in Chapter 3, the trade-off rates are calculated as n

−

∂Z1Π,E (x∗ ) ∂ZlΠ,E (x∗ )

= π∗l

∑ βl j x∗j − z1l + z0l

j=1 n

∑

β1 j x∗j − z11 + z01

, l = 2, . . . , k.

(4.22)

j=1

It should be stressed here that in order to obtain the trade-off information from (4.22), the ﬁrst (k − 1) constraints in (4.19) must be active. Therefore, if there are inactive constraints, it is necessary to replace zˆl for inactive constraints with ZlΠ,E (x∗ ) + v∗ and solve the corresponding problem to obtain the simplex multipliers. Following the preceding discussion, we can now present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based expectation model Step 1: Determine the linear membership functions μG˜ l , l = 1, . . . , k deﬁned as (4.7) by calculating z0l and z1l , l = 1, . . . , k. Step 2: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k.

4.1 Multiobjective fuzzy random linear programming

113

Step 3: For the current reference expectation levels zˆl , l = 1, . . . , k, solve the minimax problem (4.14). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (4.19), replace zˆl for inactive constraints with ZlΠ,E (x∗ ) + v∗ and resolve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the E-P-Pareto optimality test. Step 4: The DM is supplied with the corresponding E-P-Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the current objective function values ZlΠ,E (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k by considering the current objective function values together with the trade-off rates −∂Z1Π,E /∂ZlΠ,E , l = 2, . . . , k, and return to step 3. Observing that the trade-off rate −∂Z1Π,E /∂ZlΠ,E , l = 2, . . . , k in step 4 indicates the decrement value of the objective function Z1Π,E with a unit increment of value of Π,E information is used to estimate the local an objective function Zl , such trade-off

shape of Z1Π,E (x∗ ), . . . , ZkΠ,E (x∗ ) around x∗ . Here it should be stressed for the DM that any improvement of one expectation of the degree of possibility can be achieved only at the expense of at least one of the other expectations.

4.1.1.2 Possibility-based variance model As discussed in the previous section, the possibility-based expectation model would be appropriate if the DM intends to simply maximize the expected degrees of possibility without concerning about those ﬂuctuations. However, when the DM prefers to decrease the ﬂuctuation of the objective function values, the possibility-based expectation model is not relevant because some scenario yielding a very low possibility of good performance may occur even with a small probability. To avoid such risk, from the risk-averse point of view, by minimizing the variance of the degree of possibility under the constraints of feasibility together with the conditions for the expected degrees of possibility, the possibility-based variance model is formulated for the multiobjective fuzzy random programming problems as ⎫ ) ( ⎪ minimize Var ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ 1 ⎪ ⎪ ⎪ · ·(· · · · · · · ⎪ ⎬ ) ˜ (4.23) minimize Var ΠC˜¯ x (Gk ) k ⎪ ) ( ⎪ ⎪ subject to E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ l ⎪ ⎭ x ∈ X, where Var denotes the variance operator, and ξl , l = 1, . . . , k are permissible expectation levels for the expected degrees of possibility speciﬁed by the DM.

114

4 Multiobjective Fuzzy Random Programming

For notational convenience, let X(ξξ) be the feasible region of (4.23), namely ) ( X(ξξ) x ∈ X E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k . l From the viewpoint of possibility maximization and variance minimization, we deﬁne the concept of V-P-Pareto optimal solutions. Deﬁnition 4.3 (V-P-Pareto Optimal Solution). A point x∗ ∈ X(ξξ) is said to be a V-P-Pareto optimal solution to (4.2) does not exist another ( ) if and ( only if there ) ˜ ˜ x ∈ X(ξξ) such that Var ΠC˜¯ x (Gl ) ≤ Var ΠC˜¯ x∗ (Gl ) for all l ∈ {1, . . . , k} and l l ) ( ) ( ˜ ˜ Var ΠC˜¯ x (Gv ) = Var ΠC˜¯ x∗ (Gv ) for at least one v ∈ {1, . . . , k}. v

v

Recalling the degree of possibility (4.9), each of the objective functions in (4.23) is calculated as & % ( ) n 1 Var Π ˜¯ (G˜ l ) = " # Var ∑ d¯l j x j Cl x

n

∑

2

j=1

βl j x j − z1l + z0l

j=1

1

="

n

∑

T #2 x Vl x,

(4.24)

βl j x j − z1l + z0l

j=1

where Vl is the variance-covariance matrix of d¯ l expressed by ⎤ ⎡ l l v11 v12 · · · vl1n ⎢ vl vl · · · vl ⎥ 2n ⎥ ⎢ 21 22 Vl = ⎢ . . . . ⎥ , l = 1, . . . , k, ⎣ .. .. . . .. ⎦ vln1 vln2 · · · vlnn and vlj j

= Var[d¯l j ] =

Sl

∑ plsl {dl jsl }

2

sl =1

−

Sl

2

∑ plsl dl jsl

sl =1

, j = 1, . . . , n,

vljr = Cov[d¯lr , d¯lr ] = E[d¯l j , d¯lr ] − E[d¯l j ]E[d¯lr ] =

Sl

Sl

Sl

sl =1

sl =1

sl =1

∑ plsl dl jsl dlrsl − ∑ plsl dl jsl ∑ plsl dlrsl ,

j = r, r = 1, . . . , n.

Furthermore, ) from (4.13), the constraint of the expected degree of possibility ( ˜ E ΠC˜¯ x (Gl ) ≥ ξl is calculated as l

4.1 Multiobjective fuzzy random linear programming n

Sl

∑ ∑ plsl dl jsl − (1 − ξl )βl j

j=1

115

sl =1

x j ≤ z0l − ξl (z0l − z1l ).

(4.25)

By substituting (4.24) and (4.25) into (4.23), (4.23) is equivalently transformed as minimize Z1Π,V (x) = "

1 n

T #2 x V1 x

∑ β1 j x j − z11 + z01

j=1

minimize

·················· 1 "

ZkΠ,V (x) =

T #2 x Vk x

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ Sl n ⎪ ⎪ 0 0 1 subject to ∑ ∑ plsl dl jsl − (1 − ξl )βl j x j ≤ zl − ξl (zl − zl ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ j=1 sl =1 ⎪ ⎭ x ∈ X. (4.26) From the fact that n

∑ βk j x j − z1k + z0k

n

∑ βl j x j − z1l + z0l > 0

j=1

≥ 0 due to the positive-semideﬁnite property of Vl , by computing the and square root of the objective functions of (4.26), (4.26) is rewritten as ' ⎫ xT V1 x ⎪ ⎪ ⎪ minimize Z1Π,SD (x) = n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ··············· ⎪ ⎪ ⎪ ⎬ ' T x Vk x Π,SD minimize Zk (x) = n ⎪ ⎪ ⎪ ⎪ ⎪ ∑ βk j x j − z1k + z0k ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ S n l ⎪ ⎪ 0 0 1 ⎪ subject to ∑ ∑ plsl dl jsl − (1 − ξl )βl j x j ≤ zl − ξl (zl − zl ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ j=1 sl =1 ⎪ ⎭ x ∈ X, (4.27) where each of the objective functions represents the standard deviation of the degree of possibility. Realizing that the minimization of the variance is equivalent to the minimization of the standard deviation, it is signiﬁcant to emphasize that each of Pareto optimal solutions of (4.27) becomes a V-P-Pareto optimal solution. xT Vl x

116

4 Multiobjective Fuzzy Random Programming

To calculate a candidate for the satisﬁcing solution, the DM is asked to specify the reference levels zˆl , i = 1, . . . , k of the objective function values of (4.27), and it is called the reference standard deviation levels. For the DM’s reference standard deviation levels zˆl , i = 1, . . . , k, a V-P-Pareto optimal solution is obtained by solving the minimax problem ⎫ ⎪ minimize max ZlΠ,SD (x) − zˆl ⎪ ⎪ 1≤l≤k ⎪ ⎪ ⎬ subject to

n

Sl

j=1

sl =1

∑ ∑ plsl dl jsl − (1 − ξl )βl j

x j ≤ z0l − ξl (z0l − z1l ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

x ∈ X.

(4.28) For notational convenience, we introduce Nl (x) and Ql (x) such that Nl (x) , ZlΠ,SD (x) − zˆl Ql (x) where Nl (x) =

'

" xt Vl x − zˆl

n

∑

(4.29)

# βl j x j − z1l + z0l

j=1

Dl (x) =

n

∑ βl j x j − z1l + z0l .

j=1

Since the numerator Nl (x) is a convex function and the denominator Dl (x) is an afﬁne function, it follows that Nl (x)/Dl (x) is a quasi-convex function. From this property of (4.28), although it cannot be solved directly by using convex programming techniques because of the nonconvexity of the objective function, we can solve (4.28) by using the following extended Dinkelbach-type algorithm (Borde and Crouzeix, 1987): Extended Dinkelbach-type algorithm for solving (4.28) Step 1: Step 2:

Set r := 0 and ﬁnd a feasible solution xr ∈ X(ξξ). For a qr calculated by Nl (xr ) r q = max , 1≤l≤k Dl (xr )

ﬁnd an optimal solution xc to the convex programming problem

4.1 Multiobjective fuzzy random linear programming

117

minimize v subject to

1 {Dl (x) − qr Nl (x)} ≤ v, l = 1, . . . , k Dl (xr ) n

Sl

j=1

sl =1

∑ ∑ plsl dl jsl − (1 − ξl )βl j

x ∈ X.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ x j ≤ z0l − ξl (z0l − z1l ), l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.30) Step 3: For a sufﬁciently small positive number ε, if v < ε, stop the algorithm. Otherwise, set xr := xc , r := r + 1, and return to step 2. Now we are ready to summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based variance model Step 1: Determine the linear membership functions μG˜ l , l = 1, . . . , k with z0l and z1l , l = 1, . . . , k obtained by solving linear programming problems (4.8). Step 2: Calculate the individual minima and maxima of ZlΠ,E (x), l = 1, . . . , k. Step 3: Ask the DM to specify the permissible expectation levels ξl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference standard deviation levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 5: For the current reference standard deviation levels zˆl , l = 1, . . . , k, solve the minimax problem (4.28) by using the extended Dinkelbach-type algorithm. Step 6: The DM is supplied with the obtained V-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values ZlΠ,SD (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference standard deviation levels zˆl , l = 1, . . . , k, and return to step 5.

4.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing algorithm for the possibility-based expectation model, consider the following example of multiobjective fuzzy random programming problems: ⎫ ˜¯ x ⎪ minimize z1 (x) = C ⎪ 1 ⎪ ⎪ ˜ ⎪ ¯ ⎬ minimize z2 (x) = C2 x ˜ (4.31) ¯ 3x minimize z3 (x) = C ⎪ ⎪ subject to ai x ≤ bi , i = 1, . . . , 4 ⎪ ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T , and the values of parameters involved in the objective functions and the constraints are shown in Tables 4.1 and 4.2, respectively.

118

4 Multiobjective Fuzzy Random Programming

After determining the linear membership functions to quantify fuzzy goals for the objective functions, the initial reference expectation levels are set at the ideal values (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 1.0, 1.0). For the initial reference expectation levels, the minimax problem (4.28) is solved by using the combined use of the variable transformation and the simplex method. The DM is supplied with the objective function values of the ﬁrst iteration as shown in Table 4.3. Assume that the DM is not satisﬁed with the objective function values of the ﬁrst iteration, and the DM updates the reference expectation levels as (1.0, 0.9, 1.0) for improving Z1Π,E and Z3Π,E at the expense of Z2Π,E . For the updated reference expectation levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.3. Moreover, assume that the DM is not satisﬁed with the objective function values of the second iteration, and the DM updates the reference expectation levels as (1.0, 0.9, 0.9) for improving Z1Π,E and Z2Π,E at the expense of Z3Π,E . For the updated reference expectation levels, the corresponding minimax problem is solved, and the objective function values of the third iteration are calculated as shown in Table 4.3. If the DM is satisﬁed with the current objective function values, it follows that the satisﬁcing solution for the DM is derived.

4.1.2 Possibility-based probability and fractile models In contrast to the expectation and variance models which are regarded as optimization approaches, as satisﬁcing approaches to multiobjective fuzzy random programming problems, we discuss the possibility-based probability and fractile models by incorporating the ideas of possibilistic programming into the probability and fractile models in stochastic programming. Throughout this subsection, assuming that coefﬁcients of the objective functions are fuzzy random variables, we consider the following multiobjective fuzzy random programming problem: ˜¯ x ⎫ minimize z1 (x) = C ⎪ 1 ⎪ ⎬ ········· (4.32) ˜¯ x ⎪ minimize zk (x) = C k ⎪ ⎭ subject to Ax ≤ b, x ≥ 0, where x is an n dimensional decision variable column vector, A is an m × n co˜¯ = (C˜¯ , . . . , C˜¯ ), efﬁcient matrix, b is an m dimensional column vector, and C l l1 ln l = 1, . . . , k are n dimensional coefﬁcient row vectors of fuzzy random variables. For notational convenience, let X denote the feasible region of (4.32), namely X {x ∈ Rn | Ax ≤ b, x ≥ 0}.

4.1 Multiobjective fuzzy random linear programming

119

Table 4.1 Value of each parameter of the fuzzy random variable coefﬁcients. d11s1 d12s1 d13s1 p1 d21s2 d22s2 d23s2 p2 d31 d32 d33 p3

s1 = 1 −2.5 −3.5 −2.25 0.25 s2 = 1 −2.5 −0.75 −2.5 0.3 s3 = 1 3.0 2.5 4.5 0.2

s1 = 2 −2.0 −3.0 −2.0 0.4 s2 = 2 −2.0 −0.5 −2.25 0.5 s3 = 2 3.25 2.75 4.75 0.45

s1 = 3 −1.5 −2.5 −1.75 0.35 s2 = 3 −1.5 −0.25 −2.0 0.2 s3 = 3 3.5 3.0 5.0 0.35

β 0.4 0.5 0.4 β 0.3 0.4 0.3 β 0.4 0.5 0.4

Table 4.2 Value of each element of ai , i = 1, . . . , 4. a1 a2 a3 a4

3 2 3 1

2 1 4 3

1 2 3 2

Table 4.3 Process of interaction. Iteration

1st

2nd

3rd

zˆ1 zˆ2 zˆ3

1.000 1.000 1.000

1.000 0.900 1.000

1.000 0.900 0.900

Z1Π,E (x) Z2Π,E (x) Z3Π,E (x)

0.540 0.540 0.540

0.574 0.474 0.574

0.617 0.517 0.517

Although we dealt with discrete fuzzy random variables in the previous subsections, we assume that each C˜¯l j is a continuous fuzzy random variable whose realized value C˜l j (ω) for each elementary event ω is characterized by ⎧ dl j (ω) − τ ⎪ ⎪ if τ ≤ dl j (ω) ⎪ ⎨L βl j μC˜l j (ω) (τ) = (4.33) ⎪ τ − dl j (ω) ⎪ ⎪ ⎩R if τ > dl j (ω), γl j where L and R are nonincreasing continuous functions, and each dl j (ω) is a realized value of the coefﬁcient of a continuous random variable d¯l j for an elementary event ω. If L(t) = R(t) = max{0, 1 − t}, the L-R fuzzy number can be viewed as a

120

4 Multiobjective Fuzzy Random Programming

triangular fuzzy number. With this observation in mind, it is evident that the fuzzy numbers characterized by (4.33) are more general than those by (4.3). An example of the membership function μC˜l j (ω) is given in Fig. 4.6. Formally, the membership function of the fuzzy random variable C˜¯l j is represented by ⎧ ¯ dl j − τ ⎪ ⎪ L ⎪ ⎨ βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ⎩R γl j

if τ ≤ d¯l j (4.34) if τ > d¯l j .

μC~lj(ω)(τ) 1

0

L

βlj

R

γ lj

τ

dlj (ω)

Fig. 4.6 Example of the membership function μC˜l j (ω) .

Since the coefﬁcients of the objective function are fuzzy random variables whose ˜¯ x is also reprerealized values are L-R fuzzy numbers, each objective function C l sented by a single fuzzy random variable, and its realized value for an elementary event ω can be expressed as an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x (4.35) μC˜ l (ω)x (υ) = ⎪ ⎪ ⎪R υ − dl (ω)x ⎩ if υ > dl (ω)x, γl x where dl (ω) is an n dimensional column vector of the realized values of the random variables d¯ l j , j = 1, . . . , n. Fig. 4.7 illustrates an example of the membership func˜¯ x, its membership function is tion μC˜ l (ω)x (υ). Also for the lth objective function C l formally expressed as

4.1 Multiobjective fuzzy random linear programming

121

⎧ ¯ dl x − υ ⎪ ⎪ L ⎪ ⎨ βl x μC˜¯ x (υ) = l ⎪ ⎪ υ − d¯ l x ⎪ ⎩R γl x

if υ ≤ d¯ l x (4.36) if υ > d¯ l x.

μC~l (ω)x(υ) 1

L

0

βl x

R

υ

γl x dl (ω) x

Fig. 4.7 Example of the membership function μC˜¯ l (ω)x .

In a way similar to the previous models, taking into account the imprecise nature of the DM’s judgments, we introduce the fuzzy goal G˜ l with the membership function μG˜ l . To elicit a membership function μl from the DM for each of the fuzzy goals, the DM is asked to assess an unacceptable objective function value z0l and a desirable objective function value z1l . An example of the membership function of the fuzzy goal is depicted in Fig. 4.8.

μG~ (y) l

1

0

z 1l

z l0

y

Fig. 4.8 Example of the membership function μG˜ l .

To help the DM determine z0l and z11 , it is recommended to calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k obtained by solving the linear programming problems

122

4 Multiobjective Fuzzy Random Programming

⎫ minimize E[d¯ l ]x ⎪ ⎪ ⎪ subject to x ∈ X ⎪ ⎬ ⎪ ⎪ maximize E[d¯ l ]x ⎪ ⎪ ⎭ subject to x ∈ X

, l = 1, . . . , k,

(4.37)

where E[d¯ l ] = (E[d¯l1 ], . . . , E[d¯ln ]), l = 1, . . . , k. From the viewpoint of possibilistic programming, we consider the degree of pos˜¯ x attains the fuzzy goal G˜ is expressed sibility that the objective function value C l l as (4.38) ΠC˜¯ x (G˜ l ) = sup min μC˜¯ x (y), μG˜ l (y) , l = 1, . . . , k, l

l

y

and for a given elementary event ω, (4.38) is reduced to ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, . . . , k.

(4.39)

y

Fig. 4.9 illustrates the degree of possibility ΠC˜ l (ω)x (G˜ l ).

μG~ (y) l

μC~l(ω) x (y)

1

~ Π C~l(ω) x(Gl )

0

y

Fig. 4.9 Degree of possibility ΠC˜ l (ω)x (G˜ l ).

It is signiﬁcant to emphasize that the degree ΠC˜ l (ω)x (G˜ l ) deﬁned by (4.39) varies randomly due to the randomness of occurrence of an elementary event ω. With this observation in mind, realizing such randomness of the degree, by introducing some ideas from probability theory as well as the fractile model in stochastic programming into possibilistic programming, the possibility-based probability and fractile models are developed.

4.1.2.1 Possibility-based probability model As a satisﬁcing approach to multiobjective fuzzy random programming, assuming that the DM intends to maximize the probabilities that the degrees of possibility are greater than or equal to certain target values, we consider the possibility-based probability model for the multiobjective fuzzy random programming problem which

4.1 Multiobjective fuzzy random linear programming

is formulated as

123

⎫ maximize Z1Π,P (x) = P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎪ ⎬ · · · ··············· maximize ZkΠ,P (x) = P ω ΠC˜ k (ω)x (G˜ k ) ≥ hk ⎪ ⎪ ⎪ ⎪ ⎭ subject to x ∈ X,

(4.40)

where P denotes a probability measure, and hl , l = 1, . . . , k are target values for the degree of possibility speciﬁed by the DM. From the viewpoint of maximization of possibility and probability, we deﬁne the concept of P-P-Pareto optimal solutions. Deﬁnition 4.4 (P-P-Pareto Optimal Solution). A point x∗ ∈ X is said to be a P-PPareto optimal solution to (4.32) if and only if there does not exist another x ∈ X such that ZlΠ,P (x) ≥ ZlΠ,P (x∗ ) for all l ∈ {1, . . . , k} and ZvΠ,P (x) = ZvΠ,P (x∗ ) for at least one v ∈ {1, . . . , k}. From the deﬁnition of the degree of possibility, as shown in (4.39), the following relation holds: ΠC˜ l (ω)x (G˜ l ) ≥ hl ⇔ ∃y : μC˜ l (ω)x (y) ≥ hl , μG˜ l (y) ≥ hl

βl (ω)}x ≤ y ≤ {dl (ω) + R (hl )γγl (ω)}x, y ≤ μG˜ (hl ) ⇔ ∃y : {dl (ω) − L (hl )β

βl (ω)}x ≤ μG˜ (hl ), ⇔ {dl (ω) − L (hl )β l

l

(4.41)

where L (hl ) and μG˜ (hl ) are pseudo inverse functions deﬁned by l

L (hl ) = sup{r | L(r) ≥ hl }, μG˜ (hl ) = sup{r | μG˜ l (r) ≥ hl }. l

(4.42)

Along the same line as the problem formulation employed by Stancu-Minasian (1984), assume that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where t¯l is a random variable with the mean ml and the variance σ2l , and dll j , l = 1, 2 are constants. If dl2 x > 0, l = 1, · · · , k for any x ∈ X, one ﬁnds that βl (ω)}x ≤ μG˜ (hl ) P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = P ω {dl (ω) − L (hl )β l " (h )β 1 }x + μ (h ) # β − d {L l l l G˜ l l = P ω tl (ω) ≤ dl2 x " # βl − dl1 }x + μG˜ (hl ) {L (hl )β l = Tl . (4.43) dl2 x Consequently, (4.40) is rewritten as

124

4 Multiobjective Fuzzy Random Programming

" maximize Z1Π,P (x) = T1

β1 − d11 }x + μG˜ (h1 ) {L (h1 )β 1

d12 x

#⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

·················· " # βk − dk1 }x + μG˜ (hk ) {L (hk )β Π,P k maximize Zk (x) = Tk dk2 x subject to x ∈ X.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.44)

In order to calculate a candidate for the satisﬁcing solution which is also P-PPareto optimal, the DM is asked to specify reference levels zˆl , l = 1, . . . , k, and it is called the reference probability levels. For the DM’s reference probability levels zˆl , l = 1, . . . , k, a P-P-Pareto optimal solution, which is the nearest to a vector of the reference probability levels or better than it if the reference levels are attainable in a sense of minimax, is obtained by solving the minimax problem ⎫ Π,P minimize max zˆl − Zl (x) ⎬ 1≤l≤k (4.45) ⎭ subject to x ∈ X or equivalently ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ β1 − d1 }x + μG˜ (h1 ) {L (h1 )β ⎪ ⎪ 1 ⎪ ≥ T subject to (ˆ z − v) ⎪ 1 1 2 ⎪ d1 x ⎪ ⎬ minimize v

··················

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βk − dk1 }x + μG˜ (hk ) {L (hk )β ⎪ ⎪ k ⎪ ≥ T (ˆ z − v) k ⎪ k 2 ⎪ dk x ⎪ ⎪ ⎭ x ∈ X,

(4.46)

where Tl (s) is a pseudo inverse function deﬁned by Tl (s) = inf{r | Tl (r) ≥ s}, l = 1, . . . , k. After ﬁnding the minimum value v∗ by using the combination of the phase one of the two-phase simplex method and the bisection method, in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem

4.1 Multiobjective fuzzy random linear programming

125

⎫ ⎪ ⎪ ⎪ minimize ⎪ 2 ⎪ d1 x ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β2 − d2 }x + μG˜ (h2 ) {L (h2 )β ⎪ ∗ ⎪ 2 ⎪ ⎪ ≥ T (ˆ z − v ) subject to 2 2 ⎬ d2 x β1 }x − μG˜ (h1 ) {d11 − L (h1 )β 1

2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β {L (hk )β k − dk }x + μG˜ (hk ) ⎪ ∗ ⎪ k ⎪ ⎪ ≥ T (ˆ z − v ) k ⎪ k 2 ⎪ dk x ⎪ ⎭ x ∈ X, ··················

(4.47)

where the ﬁrst objective function z1 (x) in (4.32) is supposed to be the most important to the DM. Employing the Charnes-Cooper variable transformation ς = 1/dl2 x, y = ςx, ς > 0,

(4.48)

we can transform (4.47) equivalently into ⎫ ⎪ ⎪ ⎪ ⎪ 2 1 β2 }y − ςμG˜ (h2 ) ≤ 0 ⎪ subject to τ2 d2 y + {d2 − L (h2 )β ⎪ ⎪ 2 ⎪ ⎪ ⎪ ·················· ⎪ ⎪ ⎪ 2 1 β τk dk y + {dk − L (hk )β k }y − ςμG˜ (hk ) ≤ 0 ⎬ k Ay − ςb ≤ 0 ⎪ ⎪ ⎪ ⎪ d12 y = 1 ⎪ ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎭ ς ≥ 0, β1 }y − ςμG˜ (h1 ) minimize {d11 − L (h1 )β 1

(4.49)

where τl Tl (ˆzl − v∗ ), l = 2, . . . , k, and δ is a sufﬁciently small positive number for satisfying the condition ς > 0. If the optimal solution (y∗ , ς∗ ) to (4.49) is not unique, P-P-Pareto optimality of ∗ x (= y∗ /ς∗ ) is not always guaranteed. By solving the P-P-Pareto optimality test problem similar to (4.20), P-P-Pareto optimality of x∗ is veriﬁed. If the DM is not satisﬁed with the P-P-Pareto optimal solution with respect to the reference probability levels, the DM must update them to obtain another P-P-Pareto optimal solution. Then, to help the DM update them, the trade-off information between a probability of the most important objective function and each of the other probabilities is quite useful. To derive the trade-off information, we deﬁne the Lagrange function L for (4.49) by

126

4 Multiobjective Fuzzy Random Programming

β1 }y − ςμG˜ (h1 ) L(y, ς, π, ζ, φ) = {d11 − L (h1 )β 1 ( ) k βl }y − ςμG˜ (hl ) + ∑ πl τl dl2 y + {dl1 − L (hl )β l

l=2 m

+ ∑ ζi (ai y − ςbi ) + ζm+1 d12 y − 1 i=1

n

+ζm+2 (−ς + δ) − ∑ φ j y j − φn+1 ς,

(4.50)

j=1

where πl , l = 1, . . . , k, ζi , i = 1, . . . , m + 2, φ j , j = 1, . . . , n + 1 are simplex multipliers. Through a procedure similar to (3.41)–(3.47) in 3.1.2.1, the trade-off rates are calculated as − Π,P

∂Z1Π,P (x∗ )

∂ZlΠ,P (x∗ )

= π∗l

dl2 x∗ Z1Π,P (x∗ ) , l = 2, . . . , k,

d12 x∗ Z Π,P (x∗ )

(4.51)

l

Π,P

where Zl (x∗ ) denotes the differential coefﬁcient of Zl (x∗ ). To calculate the trade-off information from (4.51), the ﬁrst (k − 1) constraints in (4.49) must be active. Therefore, if there are inactive constraints, it is necessary to replace zˆl of inactive constraints with ZlΠ,P (x∗ )+v∗ and solve the revised problem again to obtain the simplex multipliers. Following the above discussions, we now present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target values hl , l = 1, . . . , k. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k, solve the minimax problem (4.46). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of (4.49), replace zˆl of inactive constraints with ZlΠ,P (x∗ ) + v∗ and solve the revised problem. Furthermore, if the obtained x∗ is not unique, perform the P-P-Pareto optimality test. Step 6: The DM is supplied with the corresponding P-P-Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k

4.1 Multiobjective fuzzy random linear programming

127

by considering the current values of objective functions ZlΠ,P , l = 1, . . . , k together with the trade-off rates −∂Z1Π,P /∂ZlΠ,P , l = 2, . . . , k, and return to step 5. The trade-off rate −Z1Π,P /∂ZlΠ,P , l = 2, . . . , k in step 5 indicates the decrement value of the membership function Z1Π,P with a unit increment of the value of the objective function ZlΠ,P , and they are used to estimate the local shape of Z1Π,P (x∗ ), . . ., ZkΠ,P (x∗ ) around x∗ . 4.1.2.2 Possibility-based fractile model The possibility-based probability model, discussed in the previous subsection, is recommended for the DM who intends to maximize the probability with respect to the degree of possibility. On the other hand, if the DM prefers to maximize the degree of possibility rather than maximize the probability, from the viewpoint of fractile optimization criterion, it would be appropriate to employ the possibilitybased fractile model. In this subsection, assuming that the DM aspires to maximize the degree of possibility under the condition that the probability with respect to the degree of possibility is greater than or equal to a certain permissible probability level, we present the possibility-based fractile model for the multiobjective fuzzy random programming problem formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎬ (4.52) maximize hk ⎪ ⎪ ⎪ ˜ subject to P ω ΠC˜l (ω)x (Gl ) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X, 0 ≤ hl ≤ 1, where θl , l = 1, . . . , k are permissible probability levels speciﬁed by the DM, satisfying θl ∈ (1/2, 1]. For notational convenience, let X(θθ) be the feasible region of (4.52), namely X(θθ) x ∈ X P ω ΠC˜l (ω)x (G˜ l ) ≥ hl ≥ θl , 0 ≤ hl ≤ 1, l = 1, . . . , k . From the viewpoint of possibility maximization together with fractile optimization, we deﬁne the concept of F-P-Pareto optimal solutions. Deﬁnition 4.5 (F-P-Pareto Optimal Solution). A point x∗ ∈ X(θθ) is said to be an F-P-Pareto optimal solution to (4.32) if and only if there does not exist another x ∈ X(θθ) such that hl ≥ h∗l for all l ∈ {1, . . . , k} and hv = h∗v for at least one v ∈ {1, . . . , k}. Assume that d¯ l is an n dimensional Gaussian random variable with a mean vector ml and a variance-covariance matrix Vl . By using a transformation similar to the equivalent transformation shown in (4.41), one ﬁnds the relation that

128

4 Multiobjective Fuzzy Random Programming

βl }x ≤ μG˜ (hl ) ΠC˜¯ (ω)x (G˜ l ) ≥ hl ⇔ {dl (ω) − L (hl )β l

l

βl x dl (ω)x − ml x μG˜ l (hl ) − ml x + L (hl )β ' ' ≤ , (4.53) ⇔ xT Vl x xT Vl x

where L (hl ) and μG˜ (hl ) are pseudo inverse functions deﬁned by (4.42). Moreover, l observing that the left-hand side of (4.53) is the standard Gaussian random variable, one ﬁnds that P ω ΠC˜¯ (ω)x (G˜ l ) ≥ hl ≥ θl l # " d (ω)x − m x μ˜ (hl ) − ml x + L (hl )β βl x Gl l l ' ≤ ≥ θl ⇔ P ω ' xT Vl x xT Vl x " # βl x μG˜ (hl ) − ml x + L (hl )β l ' ⇔ Φ ≥ θl xT Vl x βl x μG˜ (hl ) − ml x + L (hl )β l ' ⇔ ≥ Φ−1 (θl ) xT Vl x ' βl }x + Φ−1 (θl ) xT Vl x ≤ μG˜ (hl ), ⇔ {ml − L (hl )β

(4.54)

l

where Φ is a probability distribution function of the standard Gaussian random variable, and Φ−1 is the inverse function of Φ. By substituting (4.54) into (4.52), (4.52) is transformed into ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize h1 .. .

maximize hk ' ⎪ ⎪ βl }x + Φ−1 (θl ) xT Vl x ≤ μG˜ (hl ), l = 1, . . . , k ⎪ subject to {ml − L (hl )β ⎪ ⎪ l ⎭ x ∈ X, (4.55) where Φ−1 (θl ) > 0 from the assumption of θ > 1/2. Assuming that L(t) = max{1 −t, 0} and that μG˜ l is a linear membership function deﬁned by (4.7), (4.55) can be transformed again into maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' ⎪ βl − ml )x − Φ−1 (θl ) xT Vl x + z0l (β ⎪ ⎪ ⎪ subject to ≥ h , l = 1, . . . , k l ⎪ 0 1 ⎪ β l x − zl + zl ⎪ ⎭ x ∈ X, or equivalently

(4.56)

4.1 Multiobjective fuzzy random linear programming

' β1 − m1 )x − Φ−1 (θ1 ) xT V1 x + z01 (β Π,F maximize Z1 (x) = β 1 x − z11 + z01 ··················

maximize

Π,F Zk (x) =

subject to x ∈ X.

' βk − mk )x − Φ−1 (θk ) xT Vk x + z0k (β β k x − z1k + z0k

129

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.57)

To derive a satisﬁcing solution to (4.57), the DM is asked to specify reference levels zˆl , l = 1, . . . , k, which are called the reference fractile levels, and an F-P-Pareto optimal solution is obtained by solving the minimax problem min max zˆl − ZlΠ,F (x) . (4.58) x∈X 1≤l≤k

It is signiﬁcant to emphasize here that the numerator and the denominator of Π,F zˆl − Zl (x) in (4.58) are concave and convex, respectively, and this fact means that (4.58) is solved by the extended Dinkelbach-type algorithm in a way similar to the solution method for (4.28). We now summarize an interactive algorithm for deriving the satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model Step 1: Calculate the individual minima and maxima of E[¯cl ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl ∈ [1/2, 1), l = 1, . . . , k. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels zˆl , l = 1, . . . , k, solve the corresponding minimax problem (4.58) using the extended Dinkelbach-type algorithm. Step 6: The DM is supplied with the F-P-Pareto optimal solution. If the DM is Π,F satisﬁed with the current objective function values Zl (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k, and return to step 5. It should be stressed for the DM that any improvement of one target value for the degree of possibility can be achieved only at the expense of at least one of the other target values for the ﬁxed permissible probability levels θl , l = 1, . . . , k.

130

4 Multiobjective Fuzzy Random Programming

4.1.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing algorithm for the possibility-based fractile model, consider the following multiobjective fuzzy random programming problem: ⎫ ¯˜ 1 x minimize z1 (x) = C ⎪ ⎪ ⎬ ˜¯ x minimize z2 (x) = C 2 (4.59) subject to ai x ≤ bi , i = 1, . . . , 4 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T , and parameters involved in the objective functions and the constraints are shown in Tables 4.4 and 4.5, respectively. ˜¯ . ¯˜ 1 and C Table 4.4 Value of each parameter of the fuzzy random variable coefﬁcients C 2 m1 m2 β1 β2 γ1 γ2

6 −7 0.6 0.4 0.6 0.4

7 −6 0.6 0.5 0.6 0.5

8 −6 0.5 0.5 0.5 0.5

6 3 4 2

3 5 2 3

Table 4.5 Value of each element of ai , i = 1, . . . , 4. a1 a2 a3 a4

2 6 5 2

After eliciting the membership functions to quantify the fuzzy goals for the objective functions from the DM, assume that the DM speciﬁes the permissible probability levels at θ1 = θ2 = 0.70. For the initial reference fractile levels (ˆz1 , zˆ2 ) = (1.0, 1.0), the corresponding minimax problem is solved through the extended Dinkelbach-type algorithm, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.6. Assume that the DM is not satisﬁed with the objective function values in the ﬁrst iteration, and the DM updates the reference fractile levels as (ˆz1 , zˆ2 ) = (1.0, 0.8) for improving the value of the objective function Z1Π,F at the expense of Z2Π,F . For the updated reference fractile levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated shown in Table 4.6. A similar procedure continues until the DM is satisﬁed with the obtained objective function values. If, in the third iteration, the DM is satisﬁed with the objective function values, it follows that the DM ﬁnds a satisﬁcing solution of self.

4.1 Multiobjective fuzzy random linear programming Table 4.6 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 Z1F (x) Z2F (x)

0.629 0.629

131

2nd 1.000 0.800

3rd 0.900 0.800

0.748 0.546

0.688 0.588

4.1.3 Level set-based models In this subsection, for dealing with multiobjective fuzzy random programming problems, we consider level set-based models in which, for the given degree α for the membership functions of fuzzy numbers, four types of stochastic optimization criteria are introduced: the expectation, variance, probability and fractile criteria. In connection with these criteria, four types of extended M-α-Pareto optimality concepts for fuzzy random multiobjective programming problems are deﬁned as a natural extension of the M-α-Pareto optimality concept discussed in Chapter 2. We give interactive methods for deriving a satisﬁcing solution for the DM from among the extended M-α-Pareto optimal solution. Throughout this subsection, we consider the following multiobjective fuzzy random linear programming problem: ˜¯ x ⎫ minimize z1 (x) = C 1 ⎪ ⎪ ⎬ ······ (4.60) ˜ ¯ kx ⎪ minimize zk (x) = C ⎪ ⎭ x ∈ X, ˜¯ = (C˜¯ , . . . , C˜¯ ), where x is an n dimensional decision variable column vector, and C l l1 ln l = 1, . . . , k are coefﬁcient vectors of fuzzy random variables. Although we dealt with fuzzy random variables such that only the parameters of center values d¯l j , l = 1, . . . , k, j = 1, . . . , n are random variables in the possibilitybased probability and fractile models given in subsection 4.1.2, we consider multiobjective fuzzy random programming problems involving more general fuzzy random variables such that not only center values but also spread parameters are random variables. To be more speciﬁc, for a given elementary event ω, a realized value C˜l j (ω) of each coefﬁcient C˜¯l j is characterized by a membership function ⎧ dl j (ω) − τ ⎪ ⎪ if τ ≤ dl j (ω) ⎪ ⎨L βl j (ω) μC˜l j (ω) (τ) = (4.61) ⎪ τ − dl j (ω) ⎪ ⎪ if τ > dl j (ω), ⎩R γl j (ω)

132

4 Multiobjective Fuzzy Random Programming

where L and R are nonincreasing continuous functions, and dl j (ω), βl j (ω) and γl j (ω) are realized values of continuous random variable d¯l j , β¯ l j and γ¯ l j for an elementary event ω, respectively. Fig. 4.10 illustrates an example of the membership function μC˜l j (ω) . Formally, the membership function of the fuzzy random variable C˜¯l j is represented by

μC˜¯ (τ) = lj

# ⎧ " ⎪ d¯l j − τ ⎪ ⎪ ⎪ ⎨L β¯

if τ ≤ d¯l j

lj

⎪ ⎪ τ − d¯l j ⎪ ⎪ ⎩R γ¯ l j

(4.62) if τ > d¯l j .

It is noted that not only the center values d¯l j but also spread parameters γ¯ l j and β¯ l j are random variables in (4.62), while only the center values d¯l j are random variables in (4.34).

μC~lj(ω)(τ) 1

L

0

βlj (ω)

R

γ lj (ω)

τ

dlj (ω)

Fig. 4.10 Example of the membership function μC˜l j (ω) .

˜¯ x can By using the extension principle (Zadeh, 1965), each objective function C l be represented by a single fuzzy random variable whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x ⎪ ⎨L βl (ω)x (4.63) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γ l (ω)x where dl (ω), β l (ω) and γ l (ω) are an n dimensional column vectors whose elements are realized values of the random variables d¯ j , β¯ j , γ¯ j , j = 1, . . . , n, respectively. Also ˜¯ x, its membership function is formally expressed as for the lth objective function C l

4.1 Multiobjective fuzzy random linear programming

⎧ dl (ω)x − υ ⎪ ⎪ ⎪ ⎨L βl (ω)x μC˜¯ (ω)x (υ) = l ⎪ υ − dl (ω)x ⎪ ⎪ ⎩R γ l (ω)x

133

if υ ≤ dl (ω)x (4.64) if υ > dl (ω)x.

Similarly to the previous section, assume that d¯ l , β¯ l and γ¯ l are random variable vectors expressed as ⎫ d¯ l = dl1 + t¯l dl2 ⎬ (4.65) β¯ l = β 1l + t¯l β 2l ⎭ γ¯ l = γ 1l + t¯l γ 2l , where t¯l , l = 1, . . . , k are positive random variables with mean ml and variance σ2l , and dl1 , dl2 , β 1l , β 2l , γ 1l , γ 2l are constant vectors. An α-level set of the realized value C˜l j (ω) of the fuzzy random variable C˜¯l j is expressed by (4.66) Cl jα (ω) {τ | μC˜l j (ω) (τ) ≥ α}, and from (4.61), Cl jα (ω) can be represented by the closed interval L Cl jα (ω),ClRjα (ω) = dl j (ω) − L (α)βl j (ω), dl j (ω) + R (α)γl j (ω) . Thus, by introducing a parameter λl j which is a real number satisfying 0 ≤ λl j ≤ 1, each of elements in Cl jα (ω) can be represented by λl jClLjα (ω) + (1 − λl j )ClRjα (ω). From this property of the α-level set of the fuzzy number C˜l j (ω), we deﬁne an αlevel set of the fuzzy random variable C˜¯l j as follows: C¯l jα λl jC¯lLjα + (1 − λl j )C¯lRjα | 0 ≤ λl j ≤ 1 , (4.67) where C¯lLjα and C¯lRjα are random variables expressed as C¯lLjα = d¯l j − L (α)β¯ l j , C¯lRjα = d¯l j + R (α)¯γl j .

(4.68) (4.69)

It should be noted here that C¯l jα can be viewed as a set of random variables. Now suppose that the DM intends to minimize each of the objective functions in (4.60) under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67). Then, (4.60) can be interpreted as the following multiobjective (nonfuzzy) stochastic programming problem depending on the the degree α:

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4 Multiobjective Fuzzy Random Programming

⎫ minimize C¯ 1 x ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . ¯ minimize Ck x ⎪ ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k,

(4.70)

˜¯ , and where C¯ lα = (C¯l1α , . . . , C¯lnα ) is an α-level set of the fuzzy random variable C l ¯ ¯ ¯ ¯ Cl ∈ Clα means that Cl j ∈ Cl jα for all j = 1, . . . , n. It should be noted here that a random variable C¯l j in the α-level set C¯l jα of the fuzzy random variable is characterized by the parameter λl j ∈ [0, 1]. Then, it follows that not only x but also λ l are decision variable vectors since C¯ l must be selected in C¯ lα so that each of the objective functions is minimized.

μC~lj(ω)(τ) 1

α

0

L

R

α-level set R L Cljα Cljα (ω) (ω)

τ

Fig. 4.11 An α-level set for a realized value of a fuzzy random variable coefﬁcient.

For such a multiobjective fuzzy random programming problem (4.70), considering the imprecise nature of human judgments, it is quite natural to assume that the DM has a fuzzy goal for each of the objective functions. In a minimization problem, the fuzzy goal stated by the DM may be to achieve “substantially less than or equal to some value.” Such a fuzzy goal can be quantiﬁed by eliciting a membership function μl for the fuzzy goal from the DM, which is depicted in Fig. 4.12. The parameters z0l and z1l denote the values of the objective function such that the membership function are 0 and 1, respectively, i.e., μl (z0l ) = 0 and μl (z1l ) = 1. The objective functions in (4.70) vary randomly due to the randomness of the coefﬁcients C¯ l , and to deal with this randomness, we provide four types of level set-based models for multiobjective fuzzy random programming problems by incorporating the criteria of stochastic programming.

4.1.3.1 Level set-based expectation model Assuming that the DM intends to minimize or maximize the expectation of each objective function represented as a random variable in (4.70), by replacing the ob-

4.1 Multiobjective fuzzy random linear programming

135

μl (y) 1

0

z 1l

z l0

y

Fig. 4.12 Membership function μl of the fuzzy goal for the lth objective function.

jective functions C¯ l x, l = 1, . . . , k in (4.70) with their expectations of the membership functions μl , l = 1, . . . , k, we consider the following level set-based expectation model for multiobjective fuzzy random programming problems: ⎫ maximize E μ1 (C¯ l x) ⎪ ⎪ ⎪ ⎪ · · · · · · ⎬ ¯ (4.71) maximize E μl (Ck x) ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k. From the viewpoint of maximization of the expectation for a degree α of possibility of fuzzy numbers given by the DM, we deﬁne the E-M-α-Pareto optimal solution. Deﬁnition 4.6 (E-M-α-Pareto optimal solution). A point x∗ ∈ X is said to be an EM-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ X and C¯ l ∈ C¯ lα , l = 1, . . . , k such that E[μl (C¯ l x)] ≥ E[μl (C¯ l∗ x∗ )] for all l ∈ {1, . . . , k}, and E[μv (C¯ v x)] = E[μv (C¯ v∗ x∗ )] at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα is said to be α-level optimal parameters. From the fact that ClLjα (ω) ≤ λl jClLjα (ω) + (1 − λl j )ClRjα (ω), ∀ω ∈ Ω, ∀λl j ∈ [0, 1],

(4.72)

L x)] ≤ E[μ (C ¯ l x)] for any C¯ l ∈ C¯ lα , and then, α-level optiit is evident that E[μl (C¯ lα l ∗ L ¯ ¯ mal parameters are given as Cl = Clα which implies λ∗l j = 1. Recalling (4.65) and (4.68), one ﬁnds that L x = (d¯ l − L (α)β¯ l )x C¯ lα ( ) β1l } + t¯l {dl2 − L (α)β β2l } x, = {dl1 − L (α)β

and then, (4.71) is equivalently transformed into

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4 Multiobjective Fuzzy Random Programming

⎫ E (x) ¯L maximize Z1α E[μ1 (C1α x)] ⎪ ⎪ ⎬ ········· E L maximize Zkα (x) E[μk (C¯ kα x)] ⎪ ⎪ ⎭ subject to x ∈ X,

(4.73)

where L E[μl (C¯ lα x)] =

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β2l } x fl (t)dt, {dl1 − L (α)β

(4.74)

and fl (t) is a probability density function of t¯l . To calculate a candidate for the satisﬁcing solution from among the set of E-Mα-Pareto optimal solutions, the DM is asked to specify a degree α of the α-level set and reference levels of achievement of the objective functions, called reference expectation levels. To be more explicit, for the degree α and the reference expectation levels zˆl , l = 1, . . . , k speciﬁed by the DM, the corresponding E-M-α-Pareto optimal solution, which is the nearest to a vector of the reference expectation levels zˆl , l = 1, . . . , k in a sense of minimax or better than it if the reference expectation levels are attainable, is obtained by solving the minimax problem E (x) minimize max zˆl − Zlα 1≤l≤k (4.75) subject to x ∈ X. The uniqueness of the optimal solution to (4.75) is required to guarantee E-M-αPareto optimality. Fortunately, by appending the augmentation term, this cumbrance can be overcome easily. Namely, we can obtain an E-M-α-Pareto optimal solution by solving the augmented minimax problem ⎫ k ⎪ ⎬ E E minimize max zˆl − Zlα (x) + ρ ∑ {ˆzl − Zlα (x)} (4.76) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X, where ρ is a sufﬁciently small positive number. Since the objective functions in (4.73) is expressed in the form of integration, it is generally difﬁcult to solve (4.76) in a strict sense. Realizing such difﬁculty, for obtaining an approximate solution, we employ genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001). Now we are ready to give an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-M-α Pareto optimal solution set. Interactive satisﬁcing method for level set-based expectation model Step 1: Calculate the individual minima and maxima of E[d¯ l ], l = 1, . . . , k by solving the linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1.

4.1 Multiobjective fuzzy random linear programming

137

Step 3: Ask the DM to specify the initial degree α of the α-level set of C¯lα . Step 4: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference expectation levels zˆl , l = 1, . . . , k and the degree α, solve the corresponding augmented minimax problem (4.75). Step 6: The DM is supplied with the E-M-α Pareto optimal solution x∗ . If the DM E (x∗ ), l = 1, . . . , k, then stop the is satisﬁed with the current objective functions Zlα algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k or the degree α, and return to step 5. It should be stressed to the DM that (1) any improvement of one expectation of the objective function can be achieved only at the expense of at least one of the other expectations for some ﬁxed degree α and (2) the greater value of the α yields the worse values of the expected objective functions for some ﬁxed reference expectation levels.

4.1.3.2 Level set-based variance model The level set-based expectation model, discussed in the previous subsection, would be appropriate if the DM would like to simply maximize or minimize the expected objective function values without concerning about their ﬂuctuation of the membership values of the fuzzy goals for a degree of possibility of fuzzy numbers given by the DM. Rather, when the DM prefers to decrease the ﬂuctuation of the membership values of the fuzzy goals, it is appropriate to employ a model minimizing the variance of the membership function values. Realizing such a situation, we consider the level set-based variance model for multiobjective fuzzy random programming problems formulated as ⎫ minimize Var μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · ·· · · · · · · ⎪ ⎪ ⎬ ¯ minimize Var μk (Ck x) (4.77) subject to E μl (C¯ l x) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where κl , l = 1, . . . , k are permissible expectation levels speciﬁed by the DM, satisfying 0 ≤ κl ≤ 1. In order to help the DM specify κl , l = 1, . . . , k, it is recommended to calculate E (x), l = 1, . . . , k, which are obtained by the individual minima and maxima of Zlα solving the nonlinear programming problems

138

4 Multiobjective Fuzzy Random Programming

E (x) = E μ (C ¯ L x) ⎫ minimize Zlα l ⎪ lα ⎪ ⎪ ⎪ subject to x ∈ X ⎬ ⎪ E (x) = E μ (C ¯ L x) ⎪ ⎪ maximize Zlα l ⎪ lα ⎭ subject to x ∈ X

, l = 1, . . . , k,

(4.78)

where L E[μ1 (C¯ lα x)] =

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β2l } x fl (t)dt. {dl1 − L (α)β

For notational convenience, let XV (κ) be the feasible region of (4.77), namely, XV (κ) {x ∈ Rn | x ∈ X, E μl (C¯ l x) ≥ κl , l = 1, . . . , k}. From the viewpoint of minimization of the variance for a degree of possibility of fuzzy numbers given by the DM, the concept of V-M-α-Pareto optimal solutions is deﬁned. Deﬁnition 4.7 (V-M-α-Pareto optimal solution). A point x∗ ∈ XV (κ) is said to be a V-M-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ XV (κ) and C¯ l ∈ C¯ lα , l = 1, . . . , k such that Var[μl (C¯ l x)] ≤ Var[μl (C¯ l∗ x∗ )] for all l ∈ {1, . . . , k}, and Var[μv (C¯ l x)] = Var[μv (C¯ v∗ x∗ )] for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα , k = 1, . . . , k are said to be α-level optimal parameters. Concerning the α-level optimal parameters, as discussed in the previous subsection, if we use the level set-based expectation model, maximizing the expectaL . Unfortutions of the membership functions in (4.71) can be achieved for C¯ l∗ = C¯ lα L nately, however, realizing that the variance is not always minimized at C¯ l∗ = C¯ lα or ∗ R ¯ ¯ ¯ Cl = Clα , we must search the α-level optimal parameters in the α-level set Clα . As shown in (4.67), by using the parameter λl j , any random variable C¯l j in the α-level set C¯l jα for the fuzzy random variable C˜¯l j can be expressed by C¯l j (λl j ) = λl jC¯lLjα + (1 − λl j )C¯lRjα = λl j d¯l j − L (α)β¯ l j + (1 − λl j ) d¯l j + R (α)¯γl j = d¯l j − λl j L (α)β¯ l j + (1 − λl j )R (α)¯γl j = dl1j − λl j L (α)β1l j + (1 − λl j )R (α)γ1l j +t¯l dl2j − λl j L (α)β2l j + (1 − λl j )R (α)γ2l j 1 2 Cl j (λl j ) + t¯l Cl j (λl j ),

where Cl1j (λl j ) = dl1j − λl j L (α)β1l j + (1 − λl j )R (α)γ1l j , Cl2j (λl j ) = dl2j − λl j L (α)β2l j + (1 − λl j )R (α)γ2l j .

4.1 Multiobjective fuzzy random linear programming

139

Then, by using the probability density function f l of the random variable t¯l , the variance and the expectation are calculated as λ)x + t¯l Cl2 (λ λ)x Var μl (C¯ l x) = Var μl Cl1 (λ ∞ λ)x + tl Cl2 (λ λ)x − E μl (C¯ l x) }2 fl (t)dt, {μl Cl1 (λ = 0 λ)x + t¯l Cl2 (λ λ)x E μl (C¯ l x) = E μl Cl1 (λ ∞ λ)x + tCl2 (λ λ)x fl (t)dt, = μl Cl1 (λ 0

λ1 , . . . , λ k ), λ l = (λl1 , . . . , λln ), l = 1, . . . , k and where λ = (λ λ)x Clr (λ

n

∑ Clrj (λl j )x j , r = 1, 2.

j=1

Letting E Zlα (x, λ ) = V (x, λ ) = Zlα

∞ 0 ∞ 0

λ)x + tCl2 (λ λ)x fl (t)dt μl Cl1 (λ

(4.79)

E λ)x + tCl2 (λ λ)x − Zlα {μl Cl1 (λ (x, λ )}2 fl (t)dt,

(4.80)

we can rewrite (4.77) as V (x, λ ) minimize Z1α ······ V (x, λ ) minimize Zkα

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

E (x, λ ) ≥ κ , l = 1, . . . , k ⎪ subject to Zlα l ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎪ ⎭ x ∈ X.

(4.81)

V (x, λ ), Considering that the values of λ must be chosen so as to minimize Zlα l = 1, . . . , k in (4.81), one ﬁnds that not only x but also λ are decision variable vectors. To generate a candidate for the satisﬁcing solution from among the set of V-Mα-Pareto optimal solutions, the DM is asked to specify the reference levels zˆl , l = 1, . . . , k for the variance of the objective functions, which are called the reference variance levels, and the following augmented minimax problem is solved: ⎫ k ⎪ V V ⎪ minimize max Zlα (x, λ ) − zˆl + ρ ∑ {Zlα (x, λ ) − zˆl } ⎪ ⎪ ⎪ 1≤l≤k ⎬ l=1 (4.82) E subject to Zlα (x, λ ) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎭ x ∈ X,

140

4 Multiobjective Fuzzy Random Programming

where ρ is a sufﬁciently small positive number. Observing that this problem is nonlinear and non-convex, it is generally difﬁcult to calculate a global optimal solution, and then some heuristic approach is needed to obtain an approximate optimal solution. In genetic algorithms, the decision variables E (x, λ ) and x and λ are given in a string representing an individual, and then Zlα V Zlα (x, λ ) in (4.82) are calculated easily. From this property of genetic algorithms, it seems to be appropriate to employ the GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001) for solving (4.82). Now we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based variance model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. E (x), l = 1, . . . , k by Step 3: Calculate the individual minima and maxima of Zlα solving (4.78). Step 4: Ask the DM to specify κl , l = 1, . . . , k in consideration of the individual minima and maxima obtained in step 3. Step 5: Ask the DM to specify the initial degree α. Step 6: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 7: For the current reference variance levels and the degree α, solve the corresponding augmented minimax problem (4.82). Step 8: The DM is supplied with the V-M-α Pareto optimal solution x∗ . If the DM V (x∗ , λ ∗ ), l = 1, . . . , k, is satisﬁed with the current objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k or the degree α, and return to step 7. It should be emphasized to the DM that (1) any improvement of one variance of the membership function value can be achieved only at the expense of at least one of the other variances for some ﬁxed degree α and permissible expectation levels κl , l = 1, . . . , k, and (2) the greater value of the degree α gives the worse values of the variances for some ﬁxed reference variance levels and permissible expectation levels κl , l = 1, . . . , k.

4.1.3.3 Level set-based probability model In this subsection, assuming that the DM intends to maximize the probability that the membership function values of the fuzzy goals for the objective functions involving coefﬁcients represented by fuzzy random variables are better than or equal to some target values, under the condition that each of the coefﬁcients represented by fuzzy

4.1 Multiobjective fuzzy random linear programming

141

random variables are in the α-level set, we consider the level set-based probability model for the multiobjective fuzzy random programming problem formulated as ⎫ maximize P (ω | μ1 (C1 (ω)x) ≥ h1 ) ⎪ ⎪ ⎪ ⎪ ············ ⎬ maximize P (ω | μk (Ck (ω)x) ≥ hk ) (4.83) ⎪ ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where hl , l = 1, . . . , k are target values speciﬁed by the DM, and C¯ lα is an α-level ˜¯ . set of the fuzzy random variable vector C l In contrast to the previous subsection, assuming that the spread parameters of fuzzy random variables are constant, we consider multiobjective fuzzy random pro˜¯ x is represented by a fuzzy gramming problems in which each objective function C l random variable whose realized value is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x μC˜ l (ω)x (υ) = (4.84) ⎪ υ − dl (ω)x ⎪ ⎪ ⎩R if υ > dl (ω)x. γl x ˜¯ x, its membership function is formally exAlso for the lth objective function C l pressed as ⎧ ¯ ⎪ ⎪L dl x − υ if υ ≤ d¯ l x ⎪ ⎨ βl x μC˜¯ x (υ) = (4.85) l ⎪ υ − d¯ l x ⎪ ⎪ ⎩R if υ > d¯ l x. γl x From the viewpoint of maximization of the probability for a degree of probability of fuzzy numbers given by the DM, we deﬁne the concept of P-M-α-Pareto optimal solutions. Deﬁnition 4.8 (P-M-α-Pareto optimal solution). A point x∗ ∈ X is said to be a PM-α-Pareto optimal solution to (4.83) if and only if there does not exist another x ∈ X and C¯ l ∈ C¯ lα , l = 1, . . . , k such that P(ω | μl (Cl (ω)x) ≥ hl ) ≥ P(ω | μl (Cl∗ (ω)x∗ ) ≥ hl ) for all l ∈ {1, . . . , k}, and P(ω | μv (C¯ v (ω)x) ≥ hv ) = P(ω | μv (Cv∗ (ω)x∗ ) ≥ hv ) for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα , l = 1, . . . , k are said to be α-level optimal parameters. Because, from the property (4.72), for all C¯ l ∈ C¯ lα , the relation L (ω)x) ≥ hl ⊇ {ω | μl (Cl (ω)x) ≥ hl } ω | μl (Clα holds, one ﬁnds that, for all C¯ l ∈ C¯ lα ,

(4.86)

142

4 Multiobjective Fuzzy Random Programming

L P ω | μl (Clα (ω)x) ≥ hl ≥ P (ω | μl (Cl (ω)x) ≥ hl ) ,

(4.87)

L . Consequently, which implies that α-level optimal parameters are given as C¯ l∗ = C¯ lα (4.83) can be rewritten as ⎫ L maximize P ω | μ1 (C1α (ω)x) ≥ h1 ⎪ ⎪ ⎬ · · · · · · ·L· · · · · (4.88) maximize P ω | μk (Ckα (ω)x) ≥ hk ⎪ ⎪ ⎭ subject to x ∈ X.

Similarly to the previous models, we assume that the random parameter d¯ l of ˜¯ is expressed as d¯ = d1 + t¯ d2 , where t¯ is a random variable fuzzy random variable C l l l l l l with the mean ml and the variance σ2l . Then, the lth objective function in (4.88) is equivalently transformed as L β l )x ≤ μl (hl ) P ω | μl (Clα (ω)x) ≥ hl = P ω | (d¯ l − L (α)β β l )x + μl (hl ) (−dl1 + L (α)β ¯ = P ω tl ≤ dl2 x β l )x + μl (hl ) (−dl1 + L (α)β = Tl , (4.89) dl2 x where Tl is a probability distribution of t¯l , and L and μl are pseudo inverse functions deﬁned by (4.42). Consequently, (4.88) can be rewritten as ⎫ β1 )x + μ1 (h1 ) ⎪ (−d11 + L (α)β P ⎪ maximize Z1α (x) T1 ⎪ ⎪ d12 x ⎪ ⎪ ⎬ ·················· (4.90) 1 βk )x + μk (hk ) ⎪ (−dk + L (α)β P (x) ⎪ ⎪ maximize Zkα T k ⎪ ⎪ dk2 x ⎪ ⎭ subject to x ∈ X. In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, after specifying zˆl , l = 1, . . . , k which are called reference probability revels, we solve the minimax problem P minimize max {ˆzl − Zlα (x)} 1≤l≤k (4.91) subject to x ∈ X or equivalently

4.1 Multiobjective fuzzy random linear programming

143

⎫ minimize v ⎪ ⎪ 1 ⎪ β1 − d1 }x + μG˜ (h1 ) ⎪ {L (α)β ⎪ 1 ⎪ ⎪ ≥ T subject to (ˆ z − v) 1 ⎪ 1 2 ⎪ d1 x ⎪ ⎬ ··················

βk − dk1 }x + μG˜ (hk ) {L (α)β k dk2 x

x ∈ X,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ≥ Tk (ˆzk − v) ⎪ ⎪ ⎪ ⎪ ⎭

(4.92)

where Tl (s) is a pseudo inverse function deﬁned by Tl (s) = inf{r | Tl (r) ≥ s}, l = 1, . . . , k. After ﬁnding the minimum value v∗ of the objective function in (4.92) by using the combination of the phase one of the two-phase simplex method and the bisection method, in order to uniquely determine x∗ corresponding to v∗ , we solve the linear fractional programming problem ⎫ β1 }x − μG˜ (h1 ) {d11 − L (α)β ⎪ ⎪ 1 ⎪ minimize ⎪ ⎪ ⎪ d12 x ⎪ ⎪ ⎪ ⎪ 1 ⎪ β2 − d2 }x + μG˜ (h2 ) {L (α)β ⎪ ∗ ⎪ 2 ⎪ subject to ≥ T (ˆ z − v ) ⎬ 2 2 2 d2 x (4.93) ⎪ ⎪ ·················· ⎪ ⎪ ⎪ ⎪ ⎪ βk − dk1 }x + μG˜ (hk ) {L (α)β ⎪ ∗ ⎪ k ⎪ (ˆ z − v ) ≥ T ⎪ k k 2 ⎪ ⎪ dk x ⎪ ⎭ x ∈ X, where the ﬁrst objective function z1 (x) in (4.60) is supposed to be the most important to the DM. Using the Charnes-Cooper variable transformation ς = 1/dl2 x, y = ςx, ς > 0,

(4.94)

the linear fractional programming problem (4.93) is equivalently transformed as

144

4 Multiobjective Fuzzy Random Programming

⎫ β1 }y − ςμG˜ (h1 ) minimize {d11 − L (α)β ⎪ ⎪ 1 ⎪ ⎪ β2 }y − ςμG˜ (h2 ) ≤ 0 ⎪ subject to τ2 d22 y + {d21 − L (α)β ⎪ ⎪ 2 ⎪ ⎪ ⎪ ·················· ⎪ ⎪ 2 1 ⎬ β τk dk y + {dk − L (α)β k }y − ςμG˜ (hk ) ≤ 0 ⎪ k Ay − ςb ≤ 0 ⎪ ⎪ ⎪ d12 y = 1 ⎪ ⎪ ⎪ ⎪ ⎪ −ς ≤ −δ ⎪ ⎪ ⎪ ⎪ y≥0 ⎪ ⎭ ς ≥ 0,

(4.95)

where τl Tl (ˆzl − v∗ ), l = 2, . . . , k, and δ is a sufﬁciently small positive number for satisfying the condition ς > 0. Since (4.95) is essentially the same as (4.49) in the possibility-based probability model, the P-M-α Pareto optimality test can be performed in a manner similar to the P P P-P-Pareto optimality test. Furthermore, the trade-off rates −∂Z1α /∂Zlα , l = 2, . . . , k are calculated in the same way as (4.49)–(4.51). Now we are ready to summarize an interactive algorithm for the level set-based probability model to derive a satisﬁcing solution for the DM. Interactive satisﬁcing method for the level set-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target values hl , l = 1, . . . , k and the initial degree α. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k and the degree α, solve the corresponding minimax problem (4.91). For the obtained optimal solution x∗ , if there are inactive constraints in the ﬁrst (k − 1) constraints of P (x∗ ) + v∗ and solve the revised (4.95), replace zˆl of inactive constraints with Zlα ∗ problem. Furthermore, if the obtained x is not unique, perform the P-M-α Pareto optimality test. Step 6: The DM is supplied with the P-M-α Pareto optimal solution x∗ and the trade-off rates between the objective functions. If the DM is satisﬁed with the P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherobjective function values Zlα wise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k or the P , l = 1, . . . , k degree α by considering the current values of objective functions Zlα P P together with the trade-off rates −∂Z1α /∂Zlα , l = 2, . . . , k, and return to step 5. It should be stressed to the DM that (1) an increase of one probability with respect to the membership function of the fuzzy goal can be achieved only at the expense of at least one of the other probabilities for some ﬁxed degree α and target values hl , l = 1, . . . , k, and (2) the greater value of the degree α gives decreases of the

4.1 Multiobjective fuzzy random linear programming

145

probabilities for some ﬁxed reference probability levels and target values hl , l = 1, . . . , k.

4.1.3.4 Level set-based fractile model Assuming that the DM intends to maximize a target value for the degree of possibility for given permissible probability levels, by replacing minimization of the objective functions zl (x), l = 1, . . . , k in (4.70) with maximization of the target values hl , l = 1, . . . , k such that the probability with respect to the fuzzy goal attainment level is greater than or equal to a certain permissible probability level θl speciﬁed by the DM, we consider the level set-based fractile model for fuzzy random multiobjective programming problems formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎬ maximize hk (4.96) ¯ subject to P ω | μl (Cl (ω)x) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ 0 ≤ hl ≤ 1, l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k. For notational convenience, let XF (θθ, α) denote the feasible region of (4.96), namely XF (θθ, α) {x ∈ Rn | x ∈ X, P(ω | μl (Cl (ω)x) ≥ hl ) ≥ θl , C¯ l ∈ C¯ lα , l = 1, . . . , k}. From the viewpoint of maximization of the target values hl , l = 1, . . . , k for a degree α of possibility of fuzzy numbers given by the DM, we deﬁne the concept of F-M-α-Pareto optimal solution. Deﬁnition 4.9 (F-M-α-Pareto optimal solution). A point x∗ ∈ XF (θθ, α) is said to be an F-M-α-Pareto optimal solution to (4.60) if and only if there does not exist another x ∈ XF (θθ, α) and C¯ l ∈ C¯ lα such that hl ≥ h∗l , l = 1, . . . , k for all l ∈ {1, . . . , k}, and hv > h∗v for at least one v ∈ {1, . . . , k}, where the corresponding parameters of the vector C¯ l∗ ∈ C¯ lα are said to be α-level optimal parameters. be the maximum value of hl such that P(ω | μl (Cl (ω)x) ≥ For a given θl , let hmax l

¯ ¯ hl ) ≥ θl for any Cl ∈ Clα . Then, recalling the property (4.87), there exists an hl such

L (ω)x) ≥ h ) ≥ θ , which implies that α-level optimal that hl ≥ hmax and P(ω | μl (Clα l l l ∗ L parameters are given as C¯ l = C¯ lα . From the above fact, by using the pseudo inverse function μl , the following relation holds: L P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα (ω)x ≤ μl (hl ) ≥ θl .

146

4 Multiobjective Fuzzy Random Programming

Assume that d¯ l is an n dimensional Gaussian random variable row vector with a mean vector ml and a positive-deﬁnite variance-covariance matrix Vl , and that β l and γ l are positive constants. Then, the following holds: L βl )x ≤ μl (hl )) (ω)x ≤ μl (hl ) = P (ω | (dl (ω) − L (α)β P ω | C¯ lα # " d (ω)x − m x L (α)β βl x − ml x + μl (hl ) l l ' ≤ =P ω ' xT Vl x xT Vl x " # βl − ml )x + μl (hl ) (L (α)β ' =Φ , (4.97) xT Vl x where Φ is the probability distribution of the standard normal distribution with mean 0 and variance 1. Furthermore, one ﬁnds that " Φ

βl − ml )x + μl (hl ) (L (α)β ' xT Vl x

# ≥ θl

' T β l )x + Φ−1 ⇔ (ml − L (α)β l (θl ) x Vl x ≤ μl (hl ),

where Φ−1 l is the inverse function of Φl . Assuming that μl is a linear membership function deﬁned by ⎧ ⎪ 0 if y > z0l ⎪ ⎪ ⎨ y − z0 l μG˜ l (y) = if z1l ≤ y ≤ z0l ⎪ z1l − z0l ⎪ ⎪ ⎩ 1 if y < z1l ,

(4.98)

(4.99)

from (4.98), (4.96) can be transformed into maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' 0 T ⎪ βl )x + Φ−1 (ml − L (α)β ⎪ l (θl ) x Vl x − zl ⎪ ⎪ subject to ≥ h , l = 1, . . . , k l ⎪ 0 1 ⎪ zl − zl ⎪ ⎭ x ∈ X, (4.100) or equivalently

4.1 Multiobjective fuzzy random linear programming

' −1 (α)β β (m − L )x + Φ (θ ) xT V1 x − z01 1 1 1 1 F (x) = maximize Z1α 0 1 z1 − z1 ··················

' −1 (α)β β − L )x + Φ (θ ) xT Vk x − z0k (m k k k k F (x) = maximize Zkα 0 1 zk − zk subject to x ∈ X.

147

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.101)

In order to derive a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set, after specifying zˆl , l = 1, . . . , k which are called reference fractile levels, we solve the minimax problem F minimize max zˆl − Zlα (x) 1≤l≤k (4.102) subject to x ∈ X. As we discussed for the fractile model in Chapter 3, from the convexity property F (x), the minimax problem (4.102) can be solved by using some convex of zˆl − Zlα programming technique such as the sequential quadratic programming method. Now we are ready to summarize an interactive algorithm for the level set-based fractile model to derive a satisﬁcing solution for the DM. Interactive satisﬁcing method for the level set-based fractile model Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.37). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl , l = 1, . . . , k and the initial degree α. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels and the degree α, solve the corresponding minimax problem (4.102). Step 6: The DM is supplied with the F-M-α Pareto optimal solution x∗ . If the DM F (x∗ ), l = 1, . . . , k, then stop the is satisﬁed with the objective function values Zlα algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k or the degree α, and return to step 5. It should be emphasized to the DM that (1) an increase of one target value for the degree of possibility can be achieved only at the expense of at least one of the other target values for some ﬁxed degree α and permissible probability levels θl , l = 1, . . . , k, and (2) the greater value of the degree α gives decreases of the target values for some ﬁxed reference fractile levels and permissible probability levels θl , l = 1, . . . , k.

148

4 Multiobjective Fuzzy Random Programming

4.1.3.5 Numerical example To demonstrate the feasibility and efﬁciency of the proposed interactive satisﬁcing method for the level set-based fractile model, consider the following numerical example of multiobjective fuzzy random programming problems: ⎫ ¯˜ 1 x minimize C ⎪ ⎪ ⎬ ˜¯ x minimize C 2 (4.103) subject to ai x ≤ bi , i = 1, 2, 3 ⎪ ⎪ ⎭ x ≥ 0, where x = (x1 , x2 , x3 )T . The parameters involved in (4.103) are shown in Tables 4.7 and 4.8. The right-hand side constants of bi , i = 1, 2, 3 are assumed to be given as (b1 , b2 , b3 ) = (140, 135, 100). Table 4.7 Value of each parameter of the fuzzy random variable coefﬁcients. m1 m2 β1 β2 γ1 γ2

−5 4 1.0 1.5 1.0 1.5

−3 2 1.5 1.0 1.5 1.0

−6 7 1.0 1.5 1.0 1.5

6 3 6

3 2 4

Table 4.8 Value of each element of ai , i = 1, 2, 3. a1 a2 a3

4 5 7

Suppose that after calculating the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k, the DM determines the linear membership functions by setting z01 = 0, z11 = −150, z02 = 175 and z12 = 0, and that the DM speciﬁes the permissible probability levels at θ1 = θ2 = 0.7 and the degree of α-level set α = 0.7. For the initial reference fractile levels (ˆz1 , zˆ2 ) = (1.0, 1.0), the corresponding minimax problem is solved through the sequential quadratic programming method, and the DM is supplied with the corresponding objective function values of the ﬁrst iteration shown in Table 4.9. Assume that the DM is not satisﬁed with the current objective function values, and the DM updates the reference fractile levels as (ˆz1 , zˆ2 ) = (1.0, 0.8) for improvF at the expense of Z F . For the updated ing the value of the objective function Z1α 2α reference fractile levels, the corresponding minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.9. Assume that the DM is still not satisﬁed with the objective function values, the DM F at the updates the reference fractile levels to (ˆz1 , zˆ2 ) = (0.9, 0.8) so as to improve Z2α F expense of Z1α , and the objective function values of the third iteration are calculated as shown in Table 4.9.

4.2 Extensions to integer programming

149

After a similar procedure, if the DM is satisﬁed with the objective function values of the fourth iteration, it follows that the DM obtains a satisﬁcing solution. Table 4.9 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 α 0.700 F Z1α (x) F Z2α (x)

0.544 0.544

2nd 1.000 0.800 0.700

3rd 0.900 0.800 0.700

4th 0.900 0.800 0.600

0.628 0.428

0.586 0.486

0.600 0.500

4.2 Extensions to integer programming In this section, as natural extensions of multiobjective fuzzy random programming with continuous variables, multiobjective fuzzy random integer programming are discussed. For dealing with large-scale nonlinear programming problems to be solved in an interactive satisﬁcing method, we employ genetic algorithms to search approximate optimal solutions. Realizing that the real-world decision making problems are often formulated as mathematical programming problems with integer decision variables, we consider the multiobjective fuzzy random integer programming problem formulated as ⎫ ˜¯ x minimize z1 (x) = C ⎪ 1 ⎪ ⎪ ⎪ ······ ⎬ ˜ ¯ (4.104) minimize zk (x) = Ck x ⎪ ⎪ ⎪ subject to Ax ≤ b ⎪ ⎭ x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n, where x is an n dimensional integer decision variable column vector, A is an m × n coefﬁcient matrix, b is m dimensional column constant vector, and we assume ˜¯ = (C˜¯ , . . . , C˜¯ ), l = 1, . . . , k of each of the objective that the coefﬁcient vector C l l1 ln functions is an n dimensional row vector represented by fuzzy random variables. In the following, for notational convenience, let X int be the feasible region of (4.104), namely X int = {x | Ax ≤ b, x j ∈ {0, 1, . . . , ν j }, j = 1, . . . , n}.

150

4 Multiobjective Fuzzy Random Programming

4.2.1 Possibility-based expectation and variance models 4.2.1.1 Possibility-based expectation model As discussed in the previous section, if the DM intends to simply maximize the expected degree of possibility that each of the original objective functions involving fuzzy random variables attains the fuzzy goals, the possibility-based expectation model is recommended as a reasonable decision making model for solving the multiobjective fuzzy random programming problems. ˜¯ = (C˜¯ , . . . , C˜¯ ) Also, we assume that each element C˜¯l j of the coefﬁcient vector C l l1 ln is fuzzy random variable whose realized value is a fuzzy number C˜l jsl depending on a scenario sl ∈ {1, . . . , Sl } which occurs with a probability plsl , satisfying S ∑sll=1 plsl = 1. By substituting maximization of the expected degrees of possibility with respect to the fuzzy goal attainment for the minimization of the objective functions in (4.104), we formulate )⎫ ( maximize E ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ ⎪ 1 ⎪ ⎬ · · · · · · ( ) (4.105) maximize E ΠC˜¯ x (G˜ k ) ⎪ ⎪ ⎪ k ⎪ ⎭ subject to x ∈ X int , where ΠC˜¯ x (G˜k ) denotes a degree of possibility that the fuzzy random objective k ˜¯ x attains the fuzzy goal G˜ characterized by the linear membership funcfunction C l l tion μG˜ l deﬁned as ⎧ ⎪ 0 ⎪ ⎪ ⎨ y − z0 l μG˜ l (y) = ⎪ z1l − z0l ⎪ ⎪ ⎩ 1

if y > z0l if z1l ≤ y ≤ z0l

(4.106)

if y < z1l .

Similarly to the previous section, the values of z0l and z1l are assumed to be determined by ⎫ n ⎪ 0 zl max max ∑ dl jsl x j , l = 1, . . . , k, ⎪ ⎪ ⎬ 1≤sl ≤Sl x∈X int j=1 (4.107) n ⎪ z1l min min ∑ dl jsl x j , l = 1, . . . , k. ⎪ ⎪ ⎭ 1≤sl ≤Sl x∈X int j=1

As we formulated the expectation model for the multiobjective fuzzy random programming problem with continuous decision variables in 4.1.1.1, from (4.13), (4.105) is equivalently transformed as

4.2 Extensions to integer programming n

maximize

Π,E Z1 (x)

∑

151

β1 j −

j=1

S1

∑

s1 =1

p1s1 d1 js1

n

∑ β1 j x j − z11 + z01

j=1

n

maximize ZkΠ,E (x) subject to x ∈ X int .

∑

j=1

··················

x j + z01

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ βk j − ∑ pksk dk jsl x j + z0k ⎪ ⎪ ⎪ ⎪ ⎪ sk =1 ⎪ ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ βk j x j − zk + zk ⎪ ⎪ j=1 ⎪ ⎭ Sk

(4.108)

It should be noted here that the difference between (4.13) and (4.108) is only the condition of the decision variables, namely, the difference is whether continuous decision variables or integer ones in the problem formulation. In order to derive a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set, for the reference expectation levels zˆl , l = 1, . . . , k speciﬁed by the DM, the following augmented minimax problem is iteratively solved: ⎫ k ⎪ ⎬ minimize max zˆl − ZlΠ,E (x) + ρ ∑ {ˆzl − ZlΠ,E (x) (4.109) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. From integrality of the decision variables in (4.109), it is evident that the solution techniques developed for the possibility-based expectation model with continuous decision variables, discussed in the previous section, cannot be directly applied. However, realizing that (4.109) involves a nonlinear objective function and linear constraints, as we used to similar problems in Chapter 3, we can employ the modiﬁed version of the genetic algorithm with double strings based on using linear programming relaxation and reference solution updating (GADSLPRRSU) (Sakawa, 2001) for solving it. Following the preceding discussions, we can now summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the E-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based expectation model with integer decision variables Step 1: Determine the linear membership functions μG˜ l , l = 1. . . . , k deﬁned as (4.106) by calculating z0l and z1l through GADSLPRRSU. Step 2: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k.

152

4 Multiobjective Fuzzy Random Programming

Step 3: For the current reference expectation levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.109) through the revised GADSLPRRSU. Step 4: The DM is supplied with the E-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,E (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference expectation levels zˆl , l = 1, . . . , k in consideration of the current objective function values, and return to step 3.

4.2.1.2 Possibility-based variance model In order to help a risk-averse DM with making a decision for multiobjective fuzzy random integer programming problems, from the viewpoint of variance minimization together with possibility maximization, we formulate the following integer version of (4.23): ⎫ ) ( ⎪ minimize Var ΠC˜¯ x (G˜ 1 ) ⎪ ⎪ 1 ⎪ ⎪ ⎪ · · · · · · · · · ⎪ ⎪ ( ) ⎬ minimize Var ΠC˜¯ x (G˜ k ) (4.110) k ⎪ ) ( ⎪ ⎪ subject to E ΠC˜¯ x (G˜ l ) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ l ⎪ ⎪ ⎭ int x∈X , where Var denotes the variance operator, and ξl , l = 1, . . . , k are permissible expectation levels for the expected degrees of possibility speciﬁed by the DM. From the same idea as in 4.1.1.2, we can transform (4.110) into ⎫ 1 T ⎪ minimize Z1Π,V (x) " #2 x V1 x ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ∑ β1 j x j − z1 + z1 ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ·················· ⎬ (4.111) 1 ⎪ T minimize ZkΠ,V (x) " ⎪ #2 x Vk x ⎪ ⎪ ⎪ n ⎪ ⎪ 1 0 ⎪ β x − z + z ⎪ ∑ kj j k k ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ Π,E ⎪ ⎪ subject to Zl (x) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ int x∈X , where

4.2 Extensions to integer programming n

ZlΠ,E (x) =

∑

153

βl j −

j=1 n

Sl

∑ plsl dl jsl

sl =1

∑ βl j x j − z1l + z0l

x j + z0l .

j=1

For the DM’s reference variance levels zˆl , i = 1, . . . , k, the corresponding V-PPareto optimal solution is obtained by solving the augmented minimax problem ⎫ k ⎪ Π,V Π,V ⎪ ⎪ minimize max Zl (x) − zˆl + ρ ∑ Zl (x) − zˆl ⎪ ⎬ 1≤l≤k l=1 (4.112) Π,E ⎪ subject to Zl (x) ≥ ξl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X int , where ρ is a sufﬁciently small positive number. Since (4.112) is a nonlinear integer programming problem with a nonlinear objective function and linear constraints, similarly to the possibility-based expectation model, we can employ the revised GADSLPRRSU to solve it. Now we can present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based variance model with integer decision variables Step 1: Determine the linear membership functions μG˜ , l = 1. . . . , k deﬁned as (4.106) by calculating z0l and z1l through GADSLPRRSU. Step 2: Calculate the individual minima and maxima of ZlΠ,E (x), l = 1, . . . , k. Step 3: Ask the DM to specify the permissible expectation levels ξl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 2. Step 4: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 5: For the current reference variance levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.112) through the revised GADSLPRRSU. Step 6: The DM is supplied with the corresponding V-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values ZlΠ,V (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the expectation model with integer decision variables, as a numerical example of (4.104), consider the following multiobjective fuzzy random programming problem:

154

4 Multiobjective Fuzzy Random Programming

minimize minimize minimize subject to

⎫ ˜¯ x ⎪ C 1 ⎪ ⎪ ⎪ ˜C ⎪ ¯ 2x ⎬ ˜C ¯ 3x ⎪ ⎪ ⎪ ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ x j ∈ {0, 1, . . . , 20}, j = 1, . . . , 100. ⎭

(4.113)

For determining the membership functions μG˜ l , l = 1, 2, 3, through the DADSLPRRSU, the parameter values of z0l and z1l , l = 1, . . . , 3 are calculated as z01 = 12.90, z11 = 6.00, z02 = 13.80, z12 = 4.90, z03 = 14.00 and z13 = 6.43. After setting the initial reference expectation levels at (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 1.0, 1.0), the corresponding minimax problem is solved through GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.10. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference expectation levels as (ˆz1 , zˆ2 , zˆ3 ) = (1.0, 0.9, 1.0) for improving the value of the objective function Z1Π,E and Z3Π,E at the expense of Z2Π,E . For the updated reference expectation levels, the augmented minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.10. Again, assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference expectation levels to (ˆz1 , zˆ2 ) = (1.0, 0.9, 0.9) so as to improve Z1Π,E and Z2Π,E at the expense of Z3Π,E , and the objective function values are calculated. In the third iteration, assume that the DM is satisﬁed with the objective function values. Then, it follows that a satisﬁcing solution for the DM is derived. Table 4.10 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 zˆ3 1.000 Z1Π,E (x) Z2Π,E (x) Z3Π,E (x)

0.541 0.531 0.534

2nd 1.000 0.900 1.000

3rd 1.000 0.900 0.900

0.574 0.475 0.568

0.614 0.516 0.516

4.2.2 Possibility-based probability and fractile models In a way similar to the possibility-based probability and fractile models with continuous decision variables discussed in 4.1.2, we assume that a fuzzy random variable ˜¯ x is assumed to be a continuC˜¯l j which is a coefﬁcient of the objective function C l ous fuzzy random variable whose realized value C˜l j (ω) for an elementary event ω is characterized by

4.2 Extensions to integer programming

⎧ dl j (ω) − τ ⎪ ⎪ ⎪ ⎨L βl j μC˜l j (ω) (τ) = ⎪ τ − dl j (ω) ⎪ ⎪ ⎩R γl j

155

if τ ≤ dl j (ω) (4.114) if τ > dl j (ω).

4.2.2.1 Possibility-based probability model ˜¯ x, l = 1, . . . , k in Replacing minimization of the objective functions zl (x) = C l (4.104) with maximization of the probability that the degree of possibility is greater than or equal to a certain target value hl , we consider the following possibility-based probability model for multiobjective fuzzy random integer programming problems: ⎫ maximize Z1Π,P (x) P ω | ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎪ ⎬ ··· ······ (4.115) maximize ZkΠ,P (x) P ω | ΠC˜ k (ω)x (G˜ k ) ≥ hk ⎪ ⎪ ⎪ ⎪ ⎭ subject to x ∈ X int , ˜¯ x where ΠC˜¯ x (G˜ l ) denotes the degree of possibility that the objective function C l l involving coefﬁcients represented by fuzzy random variables attains the fuzzy goal G˜ l characterized by the membership function μG˜ l . As a possible way to help the DM specify the membership function μG˜ l , it is recommended to calculate the individual maxima and minima of the expectation E[d¯ l ]x, where d¯ l is a vector of random variables representing the center values of the fuzzy ˜¯ . These values are obtained by solving the linear programming random variables C l problems ⎫ minimize E[d¯ l ]x ⎪ ⎪ ⎪ subject to x ∈ X int ⎪ ⎬ , l = 1, . . . , k. (4.116) ⎪ ⎪ maximize E[d¯ l ]x ⎪ ⎪ ⎭ subject to x ∈ X int Assume that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where the mean and the variance of the random variable t¯l are denoted by ml and σ2l , respectively, and dl1j and dl2j are constants. By using the equivalent transformation (4.43)

of the probability Z1Π,P (x), (4.115) is rewritten as

156

4 Multiobjective Fuzzy Random Programming

"

#⎫ ⎪ ⎪ 1 ⎪ maximize Z1Π,P (x) = T1 ⎪ 2 ⎪ ⎪ d1 x ⎪ ⎪ ⎪ ⎬ ·················· " # βk − dk1 }x + μG˜ (hk ) ⎪ ⎪ {L (hk )β ⎪ Π,P k ⎪ ⎪ maximize Zk (x) = Tk ⎪ 2 ⎪ dk x ⎪ ⎪ ⎭ subject to x ∈ X int . β1 − d11 }x + μG˜ (h1 ) {L (h1 )β

(4.117)

In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, we give an interactive satisﬁcing method which involves the process of iteratively solving the augmented minimax problem ⎫ ⎪ k ⎪ Π,P Π,P ⎬ minimize max zˆl − Zl (x) + ρ∑ zˆl − Zl (x) 1≤l≤k (4.118) l=l ⎪ ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number, and zˆl , l = 1, . . . , k are reference probability levels given by the DM. Following the preceding discussion, we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving linear programming problems (4.116) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to determine the target values hl , l = 1, . . . , k. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference probability levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.118) through the revised GADSLPRRSU. Step 6: The DM is supplied with the P-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.2.2 Possibility-based fractile model Replacing minimization of the objective functions zl (x), l = 1, . . . , k in (4.104) with maximization of the target values hl , l = 1, . . . , k such that the probability with respect to the degree of possibility is greater than or equal to some permissible proba-

4.2 Extensions to integer programming

157

bility levels, we consider the possibility-based fractile model for the multiobjective fuzzy random integer programming problem, and it is formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎬ (4.119) maximize hk ⎪ ⎪ ⎪ subject to P ω | ΠC˜l (ω)x (G˜ l ) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ x ∈ X int , where θl , l = 1, . . . , k are permissible probability levels speciﬁed by the DM, satisfying θ ∈ [1/2, 1). Similarly to the possibility-based fractile model with continuous decision variable, we assume that a vector d¯ l of the center values represented by random variables ˜¯ in (4.114) is assumed to be an n dimensional for the fuzzy random variables C l Gaussian random variable row vector with the mean vector ml and the positivedeﬁnite variance-covariance matrix Vl . As we discussed in 4.1.2.2, form (4.54), (4.119) is equivalently transformed into ' ⎫ β1 − m1 )x − Φ−1 (θ1 ) xT V1 x + z01 ⎪ (β Π,F ⎪ maximize Z1 (x) ⎪ ⎪ ⎪ β 1 x − z11 + z01 ⎪ ⎪ ⎪ ⎬ ·················· (4.120) ' ⎪ βk − mk )x − Φ−1 (θk ) xT Vk x + z0k ⎪ (β ⎪ Π,F ⎪ ⎪ maximize Zk (x) ⎪ ⎪ β k x − z1k + z0k ⎪ ⎭ int subject to x ∈ X . In order to calculate a candidate for the satisﬁcing solution to (4.120), the DM is asked to specify the reference fractile levels zˆl , i = 1, . . . , k, and the following augmented minimax problem is solved: ⎫ ⎪ k ⎬ Π,F Π,F minimize max zˆl − Zl (x) + ρ ∑ zˆl − Zi (x) (4.121) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. It is noted that an optimal solution to (4.121) is an F-P-Pareto optimal solution with respect to (4.120). We now summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.116) through GADSLPRRSU.

158

4 Multiobjective Fuzzy Random Programming

Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl l = 1, . . . , k. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels zˆl , l = 1, . . . , k, solve the augmented minimax problem (4.121) using the revised GADSLPRRSU. Step 6: The DM is supplied with the F-P-Pareto optimal solution x∗ . If the DM is satisﬁed with the objective function values ZlΠ,F (x∗ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k, and return to step 5.

4.2.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the possibility-based probability model with integer decision variables, consider the following numerical example of multiobjective fuzzy random programming problems: ⎫ ˜¯ x ⎪ minimize C ⎪ 1 ⎪ ⎪ ˜ ⎪ ¯ 2x ⎬ minimize C ˜¯ x (4.122) minimize C 3 ⎪ ⎪ ⎪ subject to ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ ⎭ x j ∈ {0, 1, . . . , 30}, j = 1, . . . , 100. Assume that the DM determines the linear membership functions μG˜ l , l = 1, 2, 3, by specifying the parameters z0l and z1l , l = 1, . . . , 3 as z01 = 379740, z11 = −382920, z02 = 359760, z12 = −402724, z03 = 414450 and z13 = −388170, and that the DM speciﬁes the target levels at (h1 , h2 , h3 ) = (0.65, 0.65, 0.65). After setting the initial reference probability levels at (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 1.00), the augmented minimax problem is solved through the revised GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration as shown in Table 4.11. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference probability levels as (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 0.90) for improving the value of the objective function Z1Π,P and Z2Π,P at the expense of Z3Π,P . For the updated reference probability levels, the revised augmented minimax problem is solved, and the objective function values of the second iteration are calculated as shown in Table 4.11. If, in the third interaction, the DM is satisﬁed with the objective function values, it follows that the satisﬁcing solution for the DM is derived.

4.2 Extensions to integer programming Table 4.11 Process of interaction. Iteration 1st zˆ1 1.000 1.000 zˆ2 zˆ3 1.000 Z1Π,P (x) Z2Π,P (x) Z3Π,P (x)

0.741 0.740 0.742

159

2nd 1.000 1.000 0.900

3rd 1.000 0.950 0.900

0.766 0.763 0.664

0.777 0.730 0.682

4.2.3 Level set-based models In this subsection, we consider level set-based models for dealing with multiobjective fuzzy random integer programming problems. In particular, for the given degree α for the membership functions of fuzzy numbers, four types of stochastic optimization criteria are introduced. For deriving a satisﬁcing solution among from the extended M-α-Pareto optimal solution set, interactive satisﬁcing methods for the multiobjective fuzzy random integer programming problems are presented. Throughout this subsection, we consider the following multiobjective fuzzy random integer programming problem: ˜¯ x ⎫ minimize z1 (x) = C 1 ⎪ ⎪ ⎬ ······ (4.123) ˜¯ x ⎪ minimize zk (x) = C k ⎪ ⎭ x ∈ X int , ˜¯ = (C˜¯ , . . . , C˜¯ ), where x is an n dimensional decision variable column vector, and C l l1 ln l = 1, . . . , k are coefﬁcient vector represented by fuzzy random variables. As in the level set-based models for multiobjective fuzzy random programming problems with continuous decision variables discussed in 4.1.3, we assume that the membership function of fuzzy random variable C˜¯l j is formally represented by # ⎧ " ⎪ d¯l j − τ ⎪ ⎪ if τ ≤ d¯l j ⎪L ⎨ β¯ l j μC˜¯ (τ) = (4.124) lj ⎪ ¯l j ⎪ τ − d ⎪ ⎪ if τ > d¯l j . ⎩R γ¯ l j Furthermore, we also assume that the parameters d¯ l = (d¯l1 , . . . , d¯ln ), β¯ l = (β¯ l1 , . . ., ˜¯ are random variβ¯ ln ) and γ¯ l = (¯γl1 , . . . , γ¯ ln ) of the fuzzy random variable vector C l able vectors expressed as ⎫ d¯ l = dl1 + t¯l dl2 ⎬ (4.125) β¯ l = β 1l + t¯l β 2l ⎭ γ¯ l = γ 1l + t¯l γ 2l ,

160

4 Multiobjective Fuzzy Random Programming

where t¯l , l = 1, . . . , k are positive random variables with the mean ml and the variance σl , and dl1 , dl2 , β 1l , β 2l , γ 1l , γ 2l are constant vectors. According to the idea of the level set-based models, suppose that the DM intends to minimize each of the objective functions in (4.123) under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67). Then, (4.123) can be interpreted as the following multiobjective stochastic programming problem without fuzzy random variables depending on the the degree α: ⎫ minimize C¯ 1 x ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (4.126) ¯ minimize Ck x ⎪ ⎪ int ⎪ subject to x ∈ X ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, ˜¯ , and where C¯ lα = (C¯l1α , . . . , C¯lnα ) is an α-level set of the fuzzy random variable C l ¯ ¯ ¯ ¯ Cl ∈ Clα implies that Cl j ∈ Cl jα for all j = 1, . . . , n. It should be noted that not only x but also λ l characterizing the α-level sets C¯ lα are decision variable vectors since C¯ l must be selected in C¯ lα so that each of the objective functions is minimized. The objective functions in (4.126) vary randomly due to the randomness of the coefﬁcients C¯ l , and to deal with this randomness, we provide four types of level set-based models based on the criteria of stochastic programming for multiobjective fuzzy random integer programming problems.

4.2.3.1 Level set-based expectation model In the level set-based expectation model, it is assumed that the DM intends to maximize the expectations of the membership functions of the fuzzy goals for the objective functions in (4.126) for the speciﬁed degree α for the membership functions of fuzzy numbers with respect to the coefﬁcients, and then we consider the following multiobjective fuzzy random integer programming problems: ⎫ maximize E μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · · · · · · ⎬ maximize E μk (C¯ k x) (4.127) ⎪ ⎪ subject to x ∈ X int ⎪ ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where μl is a membership function of fuzzy goal G˜ l for the lth objective function. As a possible way to help the DM determine the membership function μl of the fuzzy goal for the lth objective function in (4.127), it is recommended to calculate the individual maxima and minima of E[d¯ l ]x, l = 1, . . . , k obtained by solving the linear programming problems

4.2 Extensions to integer programming

minimize E[d¯ l ]x subject to x ∈ X int maximize E[d¯ l ]x subject to x ∈ X int

161

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

, l = 1, . . . , k,

(4.128)

˜¯ , which are deﬁned where d¯ l is a vector center values of fuzzy random variables C l by (4.124) and (4.125). As we discussed in the level set-based expectation model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.1, L . Hence, (4.127) is rewritten as the α-level optimal parameters are given as C¯ l∗ = C¯ lα ⎫ E (x) maximize Z1α ⎪ ⎪ ⎬ ··· (4.129) E (x) maximize Zkα ⎪ ⎪ ⎭ subject to x ∈ X int , where E (x) = Zlα

∞ 0

μl

( ) β1l } + t{dl2 − L (α)β β 2l } x f l (t)dt. {dl1 − L (α)β

(4.130)

To calculate a candidate for the satisﬁcing solution from among the set of E-Mα-Pareto optimal solutions, for the speciﬁed degree α and reference values zˆl , l = 1, . . . , k, the corresponding E-M-α-Pareto optimal solution is obtained by solving the augmented minimax problem ⎫ k ⎪ ⎬ E E minimize max zˆl − Zlα (x) + ρ ∑ zˆl − Zlα (x) (4.131) 1≤l≤k l=1 ⎪ ⎭ int subject to x ∈ X , where ρ is a sufﬁciently small positive number. Realizing that it is generally difﬁcult to calculate an E-M-α-Pareto optimal solution of (4.131) exactly due to the integrality of decision variables together with the nonlinearity and nonconvexity properties of the objective functions expressed in the form of integration (4.130), we employ genetic algorithms to obtain a desirable approximate optimal solution. Now we are ready to present an interactive algorithm for deriving the satisﬁcing solution for the DM from among the E-M-α Pareto optimal solution set. Interactive satisﬁcing method for level set-based expectation model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μG˜ l , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1.

162

4 Multiobjective Fuzzy Random Programming

Step 3: Ask the DM to specify the initial degree α for the α-level set. Step 4: Set the initial reference expectation levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the speciﬁed reference expectation levels and the degree α, solve the corresponding augmented minimax problem (4.131) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate E-M-α Pareto optimal solution E (x∗ ), l = 1, . . . , k, then x∗ . If the DM is satisﬁed with the objective functions Zlα stop the algorithm. Otherwise, ask the DM to update the reference expectation levels or the degree α, and return to step 5.

4.2.3.2 Level set-based variance model Due to the occurrence of an undesirable scenario, there is a possibility that the goal attainment level is quite low in the level set-based expectation model. For the DM who expects to avoid such a risk, it is appropriate to employ the level set-based variance model for multiobjective fuzzy random integer programming problems. By substituting minimization of the variances of the membership values of the fuzzy goals with respect to the objective functions for minimization of objective functions in (4.126) and adding the conditions for the expectations, we formulate the multiobjective variance minimization problem ⎫ minimize Var μ1 (C¯ 1 x) ⎪ ⎪ ⎪ ⎪ · ·· · · · ⎪ ⎪ ⎬ ¯ minimize Var μk (Ck x) (4.132) subject to E μl (C¯ l x) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where κl , l = 1, . . . , k are permissible expectation levels speciﬁed by the DM. As a possible way to help the DM to specify the permissible expectation levels, E (x) deﬁned by it is convenient to calculate the individual maxima and minima of Zlα (4.130). To obtain these values, we solve the integer programming problems E (x) ⎫ minimize Zlα ⎪ ⎪ ⎪ subject to x ∈ X int ⎪ ⎬ , l = 1, . . . , k (4.133) ⎪ E ⎪ maximize Zlα (x) ⎪ ⎪ ⎭ subject to x ∈ X int by directly using GADSLPRRSU. Recalling the discussion for the level set-based variance model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.2, we can rewrite (4.132) as follows:

4.2 Extensions to integer programming V (x, λ ) minimize Z1α ······ V (x, λ ) minimize Zkα

163

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

E (x, λ ) ≥ κ , l = 1, . . . , k ⎪ subject to Zlα l ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎪ ⎭ x ∈ X int ,

(4.134)

where E Zlα (x, λ ) = V (x, λ) = Zlα

∞ 0 ∞ 0

λ)x + tCl2 (λ λ)x fl (t)dt, μl Cl1 (λ E λ)x + tl Cl2 (λ λ)x − Zlα {μl Cl1 (λ (x, λ)}2 fl (t)dt.

To calculate a candidate for the satisﬁcing solution from among the set of V-Mα-Pareto optimal solutions, we solve the augmented minimax problem &⎫ % k ⎪ V V ⎪ minimize max Zlα (x, λ ) − zˆl + ρ ∑ Zlα (x, λ ) − zˆl ⎪ ⎪ ⎪ 1≤l≤k ⎬ l=1 (4.135) E subject to Zlα (x, λ ) ≥ κl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ 0 ≤ λl j ≤ 1, l = 1, . . . , k, j = 1, . . . , n ⎪ ⎭ x ∈ X int , where ρ is a sufﬁciently small positive number. Noting that (4.135) is non-convex nonlinear integer programming problem, it would be appropriate to employ some heuristic approach in order to obtain an approximate optimal solution, and then we employ the revised GADSLPRRSU to solve (4.135). Now we are ready to present an interactive algorithm for deriving a satisﬁcing solution for the DM from among the V-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based variance model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128) through GADSLPRRSU. Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. E (x), l = 1, . . . , k by Step 3: Calculate the individual minima and maxima of Zlα solving (4.133) through GADSLPRRSU. Step 4: Ask the DM to specify the permissible expectation levels κl , l = 1, . . . , k by considering the individual minima and maxima obtained in step 3. Step 5: Ask the DM to specify the initial degree α for the α-level set. Step 6: Set the initial reference variance levels at 0s, which can be viewed as the ideal values, i.e., zˆl = 0, l = 1, . . . , k. Step 7: For the speciﬁed reference variance levels and the degree α, solve the augmented minimax problem (4.135) through the revised GADSLPRRSU.

164

4 Multiobjective Fuzzy Random Programming

Step 8: The DM is supplied with the approximate V-M-α Pareto optimal solu∗ V tion x∗ . If the DM is satisﬁed with the objective function values Zlα (x∗ , λ ), l = 1, . . . , k, then stop the algorithm. Otherwise, ask the DM to update the reference variance levels zˆl , l = 1, . . . , k or the degree α, and return to step 7.

4.2.3.3 Level set-based probability model We deal with the level set-based probability model in which the probabilities that the membership function values of the fuzzy goals for the objective functions involving coefﬁcients represented by fuzzy random variables are greater than or equal to certain target values are minimized, and formulate the following multiobjective fuzzy random integer programming problems: ⎫ maximize P (ω | μ1 (C1 (ω)x) ≥ h1 ) ⎪ ⎪ ⎪ ⎪ ········· ⎬ maximize P (ω | μk (Ck (ω)x) ≥ hk ) (4.136) ⎪ ⎪ int ⎪ subject to x ∈ X ⎪ ⎭ C¯ l ∈ C¯ lα , l = 1, . . . , k, where hl , l = 1, . . . , k are target values for the membership function values of the fuzzy goals. In the level set-based probability model, as in 4.1.3.3, we assume that the membership function of the fuzzy random variable C˜¯l j is represented by ⎧ ¯ dl j − τ ⎪ ⎪ ⎪ ⎨L βl j μC˜¯ (τ) = lj ⎪ τ − d¯l j ⎪ ⎪ ⎩R γl j

if τ ≤ d¯l j (4.137) if τ > d¯l j .

It is noted here that the spread parameters βl j and γl j of the fuzzy random variables are constant. From the discussion for the level set-based probability model for multiobjective fuzzy random linear programming problems with continuous decision variables in L . 4.1.3.3, one ﬁnds that the α-level optimal parameters C¯ l∗ is determined as C¯ l∗ = C¯ lα Hence, (4.136) can be rewritten as ⎫ L maximize P ω | μ1 (C1α (ω)x) ≥ h1 ⎪ ⎪ ⎬ · · · · · · ·L· · · · · · · · (4.138) maximize P ω | μk (Ckα (ω)x) ≥ hk ⎪ ⎪ ⎭ subject to x ∈ X int . Furthermore, the objective functions in (4.138) are transformed into

4.2 Extensions to integer programming

L P ω | μl (Clα (ω)x) ≥ hl = Tl

165

β l )x + μl (hl ) (−dl1 + L (α)β , dl2 x

where Tl is a probability distribution function of the random variable t¯l . Using the relation (4.139), we can rewrite (4.138) as follows: ⎫ β1 )x + μ∗1 (h1 ) ⎪ (−d11 + L (α)β P (x) ⎪ T maximize Z1α ⎪ 1 ⎪ d12 x ⎪ ⎪ ⎪ ⎪ ⎬ ·················· ⎪ βk )x + μk (hk ) ⎪ (−dk1 + L (α)β ⎪ P ⎪ ⎪ maximize Zkα (x) Tk ⎪ 2 ⎪ dk x ⎪ ⎭ int subject to x ∈ X .

(4.139)

(4.140)

In order to derive a satisﬁcing solution for the DM from among the P-P-Pareto optimal solution set, for the speciﬁed reference probability levels zˆl , l = 1, . . . , k, the following augmented minimax problem is formulated: ⎫ ⎪ k ⎬ P P minimize max zˆl − Zlα (x) + ρ ∑ (ˆzl − Zlα (x)) (4.141) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X int , where ρ is a sufﬁciently small positive number. Now we are ready to present an interactive algorithm for deriving the satisﬁcing solution for the DM from among the P-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the target levels hl , l = 1, . . . , k and the initial degree α of the α-level set. Step 4: Set the initial reference probability levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current probability variance levels and the degree α, solve the corresponding augmented minimax problem (4.141) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate P-M-α Pareto optimal solution P (x∗ ), l = 1, . . . , k, x∗ . If the DM is satisﬁed with the objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference probability levels zˆl , l = 1, . . . , k or the degree α, and return to step 5.

166

4 Multiobjective Fuzzy Random Programming

4.2.3.4 Level set-based fractile model We consider the level set-based fractile model which is the complementary model to the probability model. In this model, the target values hl , l = 1, . . . , k are maximized under the condition the probabilities with respect to the fuzzy goal attainment levels are greater than or equal to the speciﬁed permissible probability levels θl , l = 1, . . . , k, and then the level set-based fractile model for the multiobjective fuzzy random integer programming problems is formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎬ maximize hk (4.142) subject to P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎪ ⎭ ¯ ¯ Cl ∈ Clα , l = 1, . . . , k, where the membership function of the fuzzy random variable C˜¯l j is represented by (4.137). Recalling the discussion for the level set-based fractile model for multiobjective fuzzy random linear programming problems with continuous decision variables in 4.1.3.4, (4.142) is equivalently transformed into ⎫ maximize h1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (4.143) maximize hk ⎪ ⎪ L ⎪ subject to P ω | μl (Clα (ω)x) ≥ hl ≥ θl , l = 1, . . . , k ⎪ ⎪ ⎭ x ∈ X int . Finally, we obtain the following problem: maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ' 0 T ⎪ βl )x + Φ−1 (ml − L (α)β ⎪ l (θl ) x Vl x − zl ⎪ subject to ≥ hl , l = 1, . . . , k ⎪ ⎪ 0 1 ⎪ zl − zl ⎪ ⎭ int x∈X , (4.144) which is equivalently expressed as

4.2 Extensions to integer programming

' −1 (α)β β − L )x + Φ (θ ) xT V1 x − z01 (m 1 1 1 1 F maximize Zlα (x) 0 1 z1 − z1 ··················

' 0 T βk )x + Φ−1 (mk − L (α)β k (θk ) x Vk x − zk F maximize Zkα (x) z1k − z0k int subject to x ∈ X .

167

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.145)

In order to derive a satisﬁcing solution for the DM from among the F-P-Pareto optimal solution set, for the speciﬁed reference fractile levelsˆzl , l = 1, . . . , k, we solve the augmented minimax problem ⎫ ⎪ k ⎬ F F (x) + ρ ∑ (ˆzl − Zlα (x)) minimize max zˆl − Zlα (4.146) 1≤l≤k l=1 ⎪ ⎭ subject to x ∈ X. Now we are ready to summarize an interactive algorithm for deriving a satisﬁcing solution for the DM from among the F-M-α Pareto optimal solution set. Interactive satisﬁcing method for the level set-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l ]x, l = 1, . . . , k by solving the linear programming problems (4.128). Step 2: Ask the DM to specify the membership functions μl , l = 1, . . . , k taking into account the individual minima and maxima obtained in step 1. Step 3: Ask the DM to specify the permissible probability levels θl , l = 1, . . . , k, and the initial degree α of the α-level set. Step 4: Set the initial reference fractile levels at 1s, which can be viewed as the ideal values, i.e., zˆl = 1, l = 1, . . . , k. Step 5: For the current reference fractile levels and the degree α, solve the corresponding augmented minimax problem (4.146) through the revised GADSLPRRSU. Step 6: The DM is supplied with the approximate F-M-α Pareto optimal solution F (x∗ ), l = 1, . . . , k, x∗ . If the DM is satisﬁed with the objective function values Zlα then stop the algorithm. Otherwise, ask the DM to update the reference fractile levels zˆl , l = 1, . . . , k or the degree α, and return to step 5.

4.2.3.5 Numerical example To demonstrate the feasibility and efﬁciency of the interactive satisﬁcing method for the level set-based probability model for multiobjective fuzzy random integer programming problems, consider the following numerical example:

168

4 Multiobjective Fuzzy Random Programming

minimize minimize minimize subject to

⎫ ˜¯ x ⎪ C 1 ⎪ ⎪ ⎪ ˜C ⎪ ¯ 2x ⎬ ˜C ¯ 3x ⎪ ⎪ ⎪ ai x ≤ bi , i = 1, . . . , 10 ⎪ ⎪ x j ∈ {0, 1, . . . , 30}, j = 1, . . . , 100. ⎭

(4.147)

After calculating the individual minima and maxima of E[d¯ l ]x, l = 1, 2, 3 , assume that the DM determines the linear membership functions by setting z01 = 379740, z11 = −382920, z02 = 359760, z12 = −390635, z03 = 414450 and z13 = −388170, and that the DM speciﬁes the target values at (h1 , h2 , h3 ) = (0.65, 0.65, 0.65) and an initial degree of the α-level set α = 0.7. For the initial reference probability levels (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 1.00, 1.00), the minimax problem is solved through GADSLPRRSU, and the DM is supplied with the objective function values of the ﬁrst iteration shown in Table 4.12. Assume that the DM is not satisﬁed with the objective function values, and the DM updates the reference fractile levels as F (ˆz1 , zˆ2 , zˆ3 ) = (1.00, 0.90, 1.00) for improving the value of the objective function Z1α F at the expense of Z F . For the updated reference probability levels, the corand Z3α 2α responding minimax problem is solved by using the revised GADSLPRRSU, and the objective function values of the second iteration are calculated as shown in Table 4.12. Furthermore, assume that the DM is not satisﬁed with the objective function values, and the DM updates the degree α from 0.7 to 0.6. The objective function values of the third iteration as shown in Table 4.12. After another revision of the reference probability levels, in the forth interaction, if the DM is satisﬁed with the objective function values, it follows that a satisﬁcing solution for the DM is derived. Table 4.12 Process of interaction. Iteration 1st zˆ1 1.000 zˆ2 1.000 1.000 zˆ3 α 0.700 P Z1α (x) P Z2α (x) P (x) Z3α

0.744 0.745 0.746

2nd 1.000 0.900 1.000 0.700

3rd 1.000 0.900 1.000 0.600

4th 1.000 0.900 0.950 0.600

0.761 0.666 0.766

0.786 0.686 0.787

0.809 0.708 0.758

Chapter 5

Stochastic and Fuzzy Random Two-Level Programming

In this chapter, for resolving the conﬂict between two decision makers (DMs) who have different interests in hierarchical managerial or public organizations, two-level programming is discussed. In particular, we consider two-level programming problems with random variable and fuzzy random variable coefﬁcients to deal with the uncertainty involved in cooperative and noncooperative decision making situations. By using optimization models provided in Chapters 3 and 4, the original stochastic and fuzzy random two-level programming problems are transformed into the deterministic two-level programming problems. Under situations where the two DMs can coordinate their decisions, to derive a satisfactory solution for the DM who have a higher priority in making decision, interactive fuzzy programming methods for stochastic and fuzzy random two-level programming problems are developed by considering a balance between the DMs’ satisfaction degrees. By contrast, in noncooperative decision situations, we also consider stochastic and fuzzy random two-level programming problems, and provide some computational methods using the combination of convex programming techniques and the branch-and-bound method to obtain Stackelberg equilibria or solutions. Furthermore, using genetic algorithms, these methods are extended to deal with two-level integer programming problems with random variables and fuzzy random variables.

5.1 Cooperative two-level programming In the real world, we often encounter situations where there are two decision makers (DMs) in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. Such decision making situations can be formulated as a two-level programming problem (Sakawa and Nishizaki, 2009); one of the DMs ﬁrst makes a decision, and then the other who knows the decision of the opponent makes a decision.

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_5, © Springer Science+Business Media, LLC 2011

169

170

5 Stochastic and Fuzzy Random Two-Level Programming

In the context of two-level programming, the DM at the upper level ﬁrst speciﬁes a strategy, and then the DM at the lower level speciﬁes a strategy so as to optimize the objective with full knowledge of the action of the DM at the upper level. In conventional multi-level mathematical programming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communication among DMs, or they do not make any binding agreement even if there exists such communication (Simaan and Cruz, 1973; Bialas and Karwan, 1984; Nishizaki and Sakawa, 2000). However, for decision making problems in such as decentralized large ﬁrms with divisional independence, it is quite natural to suppose that there exists communication and some cooperative relationship among the DMs. In order to deal with such cooperative two-level programming problems, Lai (1996) and Shih, Lai and Lee (1996) proposed solution concepts for two-level linear programming problems or multi-level ones such that decisions of DMs in all levels are sequential and all of the DMs essentially cooperate with each other. In their methods, the DMs identify membership functions of the fuzzy goals for their objective functions, and in particular, the DM at the upper level also speciﬁes those of the fuzzy goals for the decision variables. The DM at the lower level solves a fuzzy programming problem with a constraint with respect to a satisfactory degree of the DM at the upper level. Unfortunately, there is a possibility that their method leads a ﬁnal solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those of the decision variables. To overcome such a problem in their methods, by eliminating the fuzzy goals for the decision variables, Sakawa et al. proposed interactive fuzzy programming for two-level or multi-level linear programming problems to obtain a satisfactory solution for the DMs (Sakawa, Nishizaki and Uemura, 1998, 2000). The subsequent works on two-level or multi-level programming have been appearing (Lee, 2001; Sakawa and Nishizaki, 2002a,b; Sakawa, Nishizaki and Uemura, 2002; Sinha, 2003; Pramanik and Roy, 2007; Abo-Sinna and Baky, 2007; Roghanian, Sadjadi and Aryanezhad, 2007; Sakawa and Nishizaki, 2009; Sakawa and Kato, 2009a,b; Sakawa and Katagiri, 2010a; Sakawa, Katagiri and Matsui, 2010a). In this section, assuming cooperative behavior of the DMs, we consider solution methods for stochastic and fuzzy random two-level linear programming problems. Interactive fuzzy programming to obtain a satisfactory solution for the DM at the upper level in consideration of the cooperative relation between the DMs is presented.

5.1.1 Stochastic two-level linear programming From a similar viewpoint to in Chapter 3, we deal with two-level linear programming problems involving random variable coefﬁcients in objective functions and the right-hand side of constraints formulated as

5.1 Cooperative two-level programming

⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

171

(5.1)

where x1 is an n1 dimensional decision variable column vector for the DM at the upper level, x2 is an n2 dimensional decision variable column vector for the DM at the lower level, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors, and b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. For notational convenience, let DM1 and DM2 denote the DMs at the upper and the lower levels, respectively, and “minimize” and “minimize” mean that DM1 and DM2 are minimizers for their for DM1

for DM2

objective functions. By introducing the chance constrained conditions (Charnes and Cooper, 1959), (5.1) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 (5.2) for DM2 ⎪ subject to P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi , i = 1, . . . , m ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, ¯ and ηi , where ai j is the i th row vector of A j , j = 1, 2, b¯ i is the i th element of b, i = 1, . . . , m are satisﬁcing probability levels speciﬁed by DM1 with respect to the chance constraint conditions. From the continuity and the nondecreasingness of each distribution function Fi (r) = P(ω | bi (ω) ≤ r) for the random variable b¯ i , as we discussed in 3.1.1 of Chapter 3, the ith constraint in (5.2) can be equivalently transformed as P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi ⇔ ai1 x1 + ai2 x2 ≤ Fi−1 (1 − ηi ) bi (ηi ), (5.3) where Fi−1 is the inverse function of Fi . Then, (5.2) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.4)

η) = (b1 (η1 ), . . . , bm (ηm ))T . where b(η η) be the feasible region of In the following, for notational convenience, let X(η (5.4), namely, η) {(xT1 , xT2 )T | A1 x1 + A2 x2 ≤ b(η η), x1 ≥ 0, x2 ≥ 0}. X(η

172

5 Stochastic and Fuzzy Random Two-Level Programming

As we developed the four types of optimization models for fuzzy multiobjective stochastic programming problems in Chapter 3, we can consider extensions of these models so as to deal with cooperative stochastic two-level linear programming problems. However, to avoid unnecessary duplication in successive sections of this chapter, we will present the expectation and the variance models for stochastic two-level linear programming problems, and as for the probability and the fractile models, we will present them in the next section dealing with fuzzy random two-level linear programming.

5.1.1.1 Expectation model When the DMs intend to simply minimize the expected value of the objective functions, by substituting the expectation of zl (x1 , x2 ) = c¯ l1 x1 + c¯ l2 x2 , l = 1, 2 for the original objective functions involving random variables in (5.4), we consider the expectation model for stochastic two-level programming problem formulated as ⎫ minimize zE1 (x1 , x2 ) E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎬ for DM1 minimize zE2 (x1 , x2 ) E[¯c21 ]x1 + E[¯c22 ]x2 (5.5) for DM2 ⎪ ⎭ T T T η), subject to (x1 , x2 ) ∈ X(η where E[¯ci j ] = (E[c¯i j1 ], . . . , E[c¯i jn j ]), i, j = 1, 2, and E[c¯i jk ] is the expectation of the random variable c¯i jk . Observing that E[¯cl1 ] and E[¯cl2 ], l = 1, 2 are constant vectors, it is evident that (5.5) is an ordinary two-level linear programming problem described in Chapter 2, and the interactive fuzzy programming techniques for two-level linear programming problems are directly applicable for deriving a satisfactory solution for DM1 taking into account the overall satisfactory balance between DM1 and DM2.

5.1.1.2 Variance model As discussed in the previous chapters, if the DMs are willing to simply minimize the expected objective function values without concerning about the ﬂuctuation of the realized values, the expectation model is appropriate. In contrast, when the DMs prefer to decrease the ﬂuctuation of the objective function values from the viewpoint of leveling of the minimized objective function values, they should employ the variance model in which the variances are minimized subject to the condition of the expectations. For a stochastic two-level programming problem, this model is formulated as follows:

5.1 Cooperative two-level programming

173

⎫ minimize zV1 (x1 , x2 ) (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ for DM1 ⎪ ⎪ minimize zV2 (x1 , x2 ) (xT1 , xT2 )V2 (xT1 , xT2 )T ⎪ ⎬ for DM2 subject to E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ T T T η), (x1 , x2 ) ∈ X(η

(5.6)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are permissible expectation levels speciﬁed by the DMs. η, γ ) denote the feasible region of (5.6), For notational convenience, let X(η namely, η) | E[¯cl1 ]x1 + E[¯cl2 ]x2 ≤ γl , l = 1, 2}. η, γ ) {(xT1 , xT2 )T ∈ X(η X(η To help each DM to specify the permissible expectation level, it is recommended to calculate the individual minima and maxima of the expected values of the objective functions obtained by solving the linear programming problems minimize zEl (x1 , x2 ) = E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2, (5.7) η) subject to (xT1 , xT2 )T ∈ X(η maximize zEl (x1 , x2 ) = E[¯cl1 ]x1 + E[¯cl2 ]x2 η) subject to (xT1 , xT2 )T ∈ X(η

, l = 1, 2.

(5.8)

It is natural that the DMs have fuzzy goals for their objective functions when they take fuzziness of human judgments into consideration. For each of the objective functions zVl (x1 , x2 ), l = 1, 2 of (5.6), assume that the DMs have fuzzy goals such as “the objective function zVl (x1 , x2 ) should be substantially less than or equal to some speciﬁc value pl ,” and let μl denote the membership function of the fuzzy goal for the DMl. Then, (5.6) can be interpreted as the fuzzy two-level programming problem ⎫ maximize μ1 (zV1 (x1 , x2 )) ⎪ ⎬ for DM1 V maximize μ2 (z2 (x1 , x2 )) (5.9) for DM2 ⎪ ⎭ T T T η γ subject to (x1 , x2 ) ∈ X(η , ). In order to help the DMs to specify the membership functions μl , l = 1, 2, it is recommended to calculate the individual minima of zVl (x1 , x2 ), l = 1, 2 by solving the quadratic programming problems minimize zVl (x1 , x2 ) = (xT1 , xT2 )Vl (xT1 , xT2 )T , l = 1, 2. (5.10) η, γ ) subject to (xT1 , xT2 )T ∈ X(η As an initial candidate for an overall satisfactory solution to the DMs, it would be useful for DM1 to obtain a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem

174

5 Stochastic and Fuzzy Random Two-Level Programming

maximize min{μ1 (zV1 (x1 , x2 )), μ2 (zV2 (x1 , x2 ))} η, γ ) subject to (xT1 , xT2 )T ∈ X(η or equivalently

⎫ minimize −v ⎪ ⎪ ⎪ subject to −μ1 (zV1 (x1 , x2 )) + v ≤ 0 ⎬ −μ2 (zV2 (x1 , x2 )) + v ≤ 0 ⎪ ⎪ ⎪ ⎭ η, γ ). (xT1 , xT2 )T ∈ X(η

(5.11)

(5.12)

The convexity of −μl (zVl (x1 , x2 )) + v can be easily shown in a manner similar to (3.22) of Chapter 3, and therefore, (5.12) can be solved by some convex programming technique such as the sequential quadratic programming method. If DM1 is satisﬁed with the membership function values μl (zVl (x∗1 , x∗2 )) of (5.12), it follows that the corresponding optimal solution (x∗1 , x∗2 ) becomes a satisfactory solution; however, DM1 is not always satisﬁed with the membership function values. It is quite natural to assume that DM1 speciﬁes the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )) subjectively. Consequently, if DM1 is not satisﬁed with μl (zVl (x∗1 , x∗2 )) of (5.12), the following problem is formulated: ⎫ maximize μ2 (zV2 (x1 , x2 )) ⎪ ⎬ subject to μ1 (zV1 (x1 , x2 )) ≥ δ (5.13) ⎪ ⎭ T T T η, γ ), (x1 , x2 ) ∈ X(η where DM2’s membership function is maximized under the condition that DM1’s membership function μ1 (zV1 (x1 , x2 )) is larger than or equal to the minimal satisfactory level δ speciﬁed by DM1. From the assumption that μl , l = 1, 2 are nonincreasing, (5.13) is rewritten as the convex programming problem ⎫ minimize zV2 (x1 , x2 ) ⎪ ⎬ subject to zV1 (x1 , x2 ) ≤ μ1 (δ) (5.14) ⎪ ⎭ T T T η, γ ), (x1 , x2 ) ∈ X(η where μ1 is a pseudo inverse function deﬁned by μ1 (δ) sup{r | μ1 (r) ≥ δ}. If there exists an optimal solution to (5.13), it follows that DM1 obtains a satisfactory solution having a satisfactory degree larger than or equal to the minimal factory level speciﬁed by DM1. However, it is signiﬁcant to realize that the larger the minimal satisfactory level δ for μ1 is accessed, the smaller the DM’s satisfactory degree μ2 becomes when the objective functions of DM1 and DM2 conﬂict with each other. Consequently, a relative difference between the satisfactory degrees of DM1 and DM2 becomes larger, and it follows that the overall satisfactory balance between both DMs is not appropriate. In order to take account of the overall satisfactory balance between both DMs, realizing that DM1 needs to compromise with DM2 on DM1’s own minimal satisfactory level, we introduce the ratio Δ of the satisfactory degree of DM2 to that of

5.1 Cooperative two-level programming

DM1 deﬁned as Δ=

175

μ2 (zV2 (x1 , x2 )) . μ1 (zV1 (x1 , x2 ))

(5.15)

DM1 is guaranteed to have a satisfactory degree larger than or equal to the minimal satisfactory level for the fuzzy goal because the corresponding constraint is involved in (5.14). To take into account the overall satisfactory balance between both DMs, DM1 speciﬁes the lower bound Δmin and the upper bound Δmax for the ratio, and the ratio Δ is evaluated by verifying whether or not it is in the interval [Δmin , Δmax ]. This condition is represented by Δ ∈ [Δmin , Δmax ]. Now we are ready to present a procedure of interactive fuzzy programming for the variance model for deriving an overall satisfactory solution. Interactive fuzzy programming in the variance model Step 1: Ask DM1 to specify the satisﬁcing probability levels ηi , i = 1, . . . , m in (5.2). Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ), l = 1, 2 by solving the linear programming problems (5.7) and (5.8). Step 3: Ask each DM to specify the permissible expectation level γl by considering the individual minima and maxima obtained in step 2. Step 4: Calculate the individual minima zVl,min of zVl (x1 , x2 ) in (5.6) by solving the quadratic programming problems (5.10). Step 5: Ask each DM to specify the membership function μl (zVl (x1 , x2 )) by considering the individual minima calculated in step 4. Step 6: Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of the ratio Δ deﬁned by (5.15). Step 7: Solve the maximin problem (5.11), and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ corresponding to the optimal solution (x∗1 , x∗2 ) to (5.11). If DM1 is satisﬁed with the current membership function values, then stop. Otherwise, ask DM1 to specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )). Step 8: For the current minimal satisfactory level δ, solve the convex programming problem (5.14), and calculate the corresponding membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 9: If DM1 is satisﬁed with the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 8.

5.1.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the variance model, consider the following stochastic two-level linear

176

5 Stochastic and Fuzzy Random Two-Level Programming

programming problem: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 for DM1

minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 subject to a11 x1 + a12 x2 ≤ b¯ 1 ⎪ ⎪ a21 x1 + a22 x2 ≤ b¯ 2 ⎪ ⎪ ⎪ ¯ ⎪ a31 x1 + a32 x2 ≤ b3 ⎪ ⎭ T T x1 = (x11 , . . . , x15 ) ≥ 0, x2 = (x21 , . . . , x23 ) ≥ 0,

(5.16)

where b¯ 1 , b¯ 2 and b¯ 3 are Gaussian random variables N(220, 42 ), N(145, 32 ) and N(−18, 52 ), respectively; the mean vectors of c¯ l1 and c¯ l2 , l = 1, 2, and the coefﬁcient vectors ai1 and ai2 , i = 1, 2, 3 are shown as in Tables 5.1 and 5.2, respectively. Table 5.1 Mean vector E[¯cl j ] of c¯ l j . E[¯c11 ] E[¯c12 ] E[¯c21 ] E[¯c22 ]

5 6 10 3

2 1 −7 −4

1 3 1 6

2

1

−2

−5

9

11

3

3

−6

−8

Table 5.2 Each element of ai j in (5.16). a11 a12 a21 a22 a31 a32

7 4 5 −7 −4 −3

2 3 6 −1 −7 −5

6 8 −4 −3 −2 −6

The variance-covariance matrices Vl , l = 1, 2 are given as ⎡

−3.5 1.6 −2.0 16.0 −2.0 3.0 2.2 2.8

1.3 −0.7 5.0 −2.0 4.0 −1.0 0.8 −2.0

⎤ −2.0 4.0 −1.4 0.5 −1.3 2.0 ⎥ ⎥ −2.4 1.2 −2.1 ⎥ ⎥ 3.0 2.2 2.8 ⎥ ⎥, −1.0 0.8 −2.0 ⎥ 1.0 −1.5 0.6 ⎥ ⎥ −1.5 4.0 −2.3 ⎦ 0.6 −2.3 4.0

4.0 −1.4 0.8 0.2 ⎢ −1.4 4.0 0.2 −1.0 ⎢ ⎢ 0.8 0.2 9.0 0.2 ⎢ ⎢ 0.2 −1.0 0.2 36.0 V2 = ⎢ ⎢ 1.6 −2.2 −1.5 0.8 ⎢ 1.0 0.8 1.5 0.4 ⎢ ⎣ 1.2 0.9 1.0 −1.5 2.0 1.8 0.6 0.7

1.6 −2.2 −1.5 0.8 25.0 1.2 −0.2 2.0

⎤ 1.0 1.2 2.0 0.8 0.9 1.8 ⎥ ⎥ 1.5 1.0 0.6 ⎥ ⎥ 0.4 −1.5 0.7 ⎥ ⎥. 1.2 −0.2 2.0 ⎥ 25.0 0.5 1.4 ⎥ ⎥ 0.5 9.0 0.8 ⎦ 1.4 0.8 16.0

16.0 ⎢ −1.6 ⎢ ⎢ 1.8 ⎢ ⎢ −3.5 V1 = ⎢ ⎢ 1.3 ⎢ ⎢ −2.0 ⎣ 4.0 −1.4 ⎡

−1.6 25.0 −2.2 1.6 −0.7 0.5 −1.3 2.0

1.8 −2.2 25.0 −2.0 5.0 −2.4 1.2 −2.1

5.1 Cooperative two-level programming

177

For the speciﬁed satisﬁcing probability levels (γ1 , γ2 , γ3 ) = (0.85, 0.95, 0.90), the individual minima and maxima of zEl (x1 , x2 ), l = 1, 2 are calculated by solving linear programming problems (5.7) and (5.8): zE1,min = 3.053, zE2,min = −425.621, zE1,max = 323.784 and zE2,max = 304.479. Considering these values, suppose that each DM subjectively speciﬁes the permissible expectation levels γ1 = 60.0 and γ2 = 35.0. By solving the quadratic programming problems (5.10), the individual minima of zVl (x1 , x2 ), l = 1, 2 are calculated as zV1,min = 0.818 and zV2,min = 19.524. Taking account of these values, assume that each DM determines the linear membership function ⎧ 1 if zVl (x1 , x2 ) ≤ zVl,1 ⎪ ⎪ ⎪ ⎨ zV (x1 , x2 ) − zV l l,0 if zVl,1 < zVl (x1 , x2 ) ≤ zVl,0 μl (zVl (x1 , x2 )) = V V ⎪ zl,1 − zl,0 ⎪ ⎪ ⎩ 0 if zVl,0 < zVl (x1 , x2 ), where the parameters zVl,1 and zVl,0 , l = 1, 2 are determined as zV1,1 = 0.818, zV1,0 = 129.623, zV2,1 = 19.524 and zV2,0 = 206.620 by using Zimmermann’s method (Zimmermann, 1978). For the upper bound Δmax = 0.75 and the lower bound Δmin = 0.65 speciﬁed by MD1, the maximin problem (5.12) is solved, and DM1 is supplied with the corresponding membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ of the ﬁrst iteration as shown in Table 5.3. Assume that DM1 is not satisﬁed with the membership function values, and DM1 speciﬁes the minimal satisfactory level δ for μ1 (zV1 (x1 , x2 )) as 0.900. For the speciﬁed δ, the convex programming problem (5.14) is solved, and the corresponding membership function values and the ratio of the second iteration are calculated as shown in Table 5.3. Since the ratio Δ is greater than Δmax = 0.75, suppose that DM1 updates the minimal satisfactory level δ from 0.900 to 0.950. For the updated δ, the convex programming problem (5.14) is solved, and the corresponding membership function values and the ratio of the third iteration are shown in Table 5.3. After a similar procedure, in the fourth iteration, the condition for the ratio holds, i.e., Δ ∈ [0.65, 0.75]. Then, if DM1 is satisﬁed with the membership function values of the fourth iteration, it follows that a satisfactory solution is obtained.

5.1.2 Fuzzy random two-level linear programming We discussed the two approaches for dealing with multiobjective linear programming problems involving coefﬁcients represented by fuzzy random variables in Chapter 4, and for fuzzy random two-level linear programming, the two approaches, the possibility-based model and the level set-based model, are also developed. Al-

178

5 Stochastic and Fuzzy Random Two-Level Programming

Table 5.3 Process of interaction. Iteration 1st δ

2nd

3rd

4th

0.900

0.950

0.960

zV1 (x∗1 , x∗2 )

20.654

13.699

7.258

5.970

zV2 (x∗1 , x∗2 )

48.355

57.881

73.170

77.725

μ1 (zV1 (x∗1 , x∗2 ))

0.846

0.900

0.950

0.960

μ2 (zV2 (x∗1 , x∗2 ))

0.846

0.795

0.714

0.689

Δ

1.000

0.884

0.751

0.718

though several models can be considered by combining the optimization criteria of stochastic programming, taking into account the connection to the previous section, we will give the possibility-based probability model and the level set-based fractile model for fuzzy random two-level linear programming.

5.1.2.1 Possibility-based probability model In previous subsections, it is implicitly assumed that uncertain parameters or coefﬁcients involved in the formulated two-level programming problems can be expressed as random variables. This means that realized values of the random parameters by the occurrence of some event are assumed to be deﬁnitely represented by real values. However, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to express the experts’ ambiguous understanding of the realized values of random parameters by the occurrence of events as fuzzy numbers. To handle such a situation in a hierarchical decision making problem, we consider two-level linear programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ˜¯ x + C ˜¯ x ⎬ minimize z2 (x1 , x2 ) = C 21 1 22 2 (5.17) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector of DM1, x2 is an n2 dimensional decision variable column vector of DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2. Each element ˜¯ , l = 1, 2, j = 1, 2 is a fuzzy random C˜¯l jk , k = 1, 2, . . . , n j of coefﬁcient vectors C lj variable whose realized value for an elementary event ω ∈ Ω is an L-R fuzzy number characterized by the membership function

5.1 Cooperative two-level programming

⎧ dl jk (ω) − τ ⎪ ⎪ ⎪ ⎨L βl jk μC˜l jk (ω) (τ) = ⎪ τ − dl jk (ω) ⎪ ⎪ ⎩R γl jk

179

if τ ≤ dl jk (ω) (5.18) if τ > dl jk (ω),

where the shape functions L and R are nonincreasing continuous functions from [0, ∞) to [0, 1], and the parameters βl jk and γl jk , representing the left and right spreads, are positive numbers. For notational convenience, let X denote the feasi˜¯ = (C ˜¯ , C ˜¯ T T T ble region of (5.17) and let C l l1 l2 ), x = (x1 , x2 ) . ˜ ¯ Since each coefﬁcient Cl jk is a fuzzy random variable whose realized values are ˜¯ x = C ˜¯ x + C ˜¯ x , l = 1, 2 is L-R fuzzy numbers, each of objective functions C l l1 1 l2 2 also a fuzzy random variable. Then, their realized values are represented as fuzzy numbers which are characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x L ⎪ ⎨ βl x (5.19) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β In order to consider the imprecise nature of each DMs’ judgments for the objective functions zl (x1 , x2 ) in (5.17), we introduce a fuzzy goal such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value.” Let μG˜ l denote the membership function of the fuzzy goal G˜ l with respect to the objective function zl (x1 , x2 ). To elicit the membership function μG˜ l from each DM for the objective function zl (x1 , x2 ), it is recommended to calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the linear programming problems minimize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2, (5.20) subject to x ∈ X maximize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2. (5.21) subject to x ∈ X Recalling that the membership function is regarded as a possibility distribution as discussed in the previous chapters, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.22) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2, y

Observing that the degree of possibility ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of an elementary event ω, it is evident that the interactive fuzzy programming for deterministic two-level programming problems cannot be directly applied.

180

5 Stochastic and Fuzzy Random Two-Level Programming

Along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs prefer to maximize the probabilities with respect to the degrees of possibility, the original fuzzy random two-level programming problem (5.17) can be reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎬ for DM1 (5.23) maximize P ω ΠC˜ 2 (ω)x (G˜ 2 ) ≥ h2 ⎪ ⎪ for DM2 ⎪ ⎭ subject to x ∈ X, where hl , l = 1, 2 are target values for the degrees of possibility speciﬁed by the DMs. Recalling the discussion in 4.1.2.1 of Chapter 4, the following relation holds: βl )x ≤ μG˜ (hl ), ΠC˜ l (ω)x (G˜ l ) ≥ hl ⇔ (dl (ω) − L (hl )β l

where

L

and

μG˜ l

are pseudo inverse functions of L and μG˜ l deﬁned as (4.42). Employing the assumption in Chapter 4 that d¯l j is a random variable expressed as d¯l j = dl1j + t¯l dl2j , where t¯l is a random variable with the mean ml and the variance σ2l , and dll j , l = 1, 2 are constants. If dl2 x > 0, l = 1, · · · , k for any x ∈ X, one ﬁnds that the objective functions in (5.23) can be rewritten as # " βl − dl1 )x + μG˜ (hl ) (L (hl )β l , P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl dl2 x where Tl is a probability distribution function of t¯l . Consequently, (5.23) can be equivalently transformed into #⎫ " β1 − d11 )x + μG˜ (h1 ) ⎪ (h )β (L 1 ⎪ 1 ⎪ ⎪ maximize Z1Π,P (x) T1 ⎪ 2 ⎪ for DM1 d1 x ⎪ ⎬ " # 1 β2 − d2 )x + μG˜ (h2 ) ⎪ (L (h )β 2 2 ⎪ maximize Z2Π,P (x) T2 ⎪ ⎪ for DM2 ⎪ d22 x ⎪ ⎪ ⎭ subject to x ∈ X.

(5.24)

In order to obtain an initial candidate for an overall satisfactory solution to (5.24), it would be useful for DM1 to ﬁnd a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem maximize min Z1Π,P (x), Z2Π,P (x) (5.25) subject to x ∈ X, or equivalently

5.1 Cooperative two-level programming

181

⎫ maximize v ⎪ ⎪ ⎪ Π,P subject to Z1 (x) ≥ v ⎬ Z2Π,P (x) ≥ v ⎪ ⎪ ⎪ ⎭ x ∈ X.

(5.26)

From the fact that the distribution functions Tl , l = 1, 2 in (5.24) are nondecreasing, without loss of generality, (5.26) is equivalently transformed into ⎫ maximize v ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ β1 )x + μ1 (h1 ) (−d1 + L (h1 )β ⎪ ⎪ ≥ T (v) subject to ⎬ 1 2 d1 x (5.27) ⎪ ⎪ β2 )x + μ2 (h2 ) (−d21 + L (h2 )β ⎪ ⎪ ≥ T2 (v) ⎪ ⎪ ⎪ d22 x ⎪ ⎭ x ∈ X, where Tl is the pseudo inverse function of Tl , which is deﬁned by Tl (v) = inf{r | Tl (r) ≥ v}. Although (5.27) is a nonlinear programming problem, observing that the constraints of (5.27) are linear if v is ﬁxed, we can easily ﬁnd the maximum of v by using the combined use of the simplex method and the bisection method. If DM1 is satisﬁed with the membership function values ZlΠ,P (x∗ ), l = 1, 2, it follows that the corresponding optimal solution x∗ to (5.27) becomes a satisfactory solution; however, DM1 is not always satisﬁed with the membership function values. As we discussed for the variance model in 5.1.1.2, assume that DM1 speciﬁes the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )) subjectively, and the following problem is formulated: ⎫ ⎪ maximize Z2Π,P (x) ⎬ Π,P (5.28) subject to Z1 (x) ≥ δ ⎪ ⎭ x ∈ X, or equivalently maximize T2

β2 )x + μ2 (h2 ) (−d21 + L (h2 )β d22 x

β1 )x + μ1 (h1 ) (−d11 + L (h1 )β 2 d1 x x ∈ X.

subject to T1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ≥δ⎪ ⎪ ⎪ ⎪ ⎭

(5.29)

From the assumption that dl2 x > 0, l = 1, 2 together with the nonincreasing property of Tl , (5.29) can be equivalently transformed into

182

5 Stochastic and Fuzzy Random Two-Level Programming

⎫ β2 )x (−d21 + L (h2 )β ⎪ ⎪ ⎪ maximize ⎬ 2 d2 x β1 x ≤ μ1 (h1 ) ⎪ subject to T1 (δ)d12 + d11 − L (h1 )β ⎪ ⎪ ⎭ x ∈ X.

(5.30)

Observing that (5.30) is a linear fractional programming problem, in the same way as discussed in the previous chapters, (5.30) can be solved through the combined use of the linear programming techniques and the variable transformation method. By using the ratio Δ = Z2Π,P (x)/Z1Π,P (x) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the possibility-based probability model for deriving an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the linear programming problems (5.20) and (5.21). Step 2: Ask each DM to specify the membership function μl by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify the target values hl , l = 1, 2. Step 4: For the target values hl , l = 1, 2, solve the maximin problem (5.25). Step 5: DM1 is supplied with the objective function values Z1Π,P (x∗ ) and Z2Π,P (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed with them and prefers updating hl , l = 1, 2, ask DM1 to update hl , and return to step 4. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 6: For the minimal satisfactory level δ, solve the linear fractional programming problem (5.30). Step 7: DM1 is supplied with the values of Z1Π,P (x∗ ), Z2Π,P (x∗ ) and the ratio Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed with them and prefers updating hl , l = 1, 2, ask DM1 to update hl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.1.2.2 Level set-based fractile model As in the level set-based models in the previous chapters, for the given original fuzzy random two-level programming problem (5.17), where a realized value of each coefﬁcient C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j for an elementary event ω is assumed to be a fuzzy number characterized by the membership function (5.18), suppose that the DMs intend to minimize their objective functions under the condition that all the coefﬁcient vector of the objective functions in (5.17) belong to the α-level set of fuzzy random variable deﬁned in Chapter 4. Then, we consider the following multiobjective stochastic programming problem depending on the the degree α:

5.1 Cooperative two-level programming

minimize z1 (x1 , x2 ) = C¯ 1 x for DM1 minimize z2 (x1 , x2 ) = C¯ 2 x

183

⎫ ⎪ ⎪ ⎪ ⎬ (5.31)

for DM2

⎪ ⎪ subject to x ∈ X ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α ,

where x = (xT1 , xT2 )T , and C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned by (4.67). In order to consider the imprecise nature of the DMs’ judgments for the objective functions C¯ l x, l = 1, 2, we introduce the fuzzy goals which can be quantiﬁed by eliciting the membership functions μl , l = 1, 2 from DM1 and DM2, respectively. Then, observing that the values of μl (Cl (ω)x) vary randomly due to the stochastic occurrence of an elementary event ω, it is evident that the solution methods for ordinary deterministic two-level linear programming problems cannot be directly applied. As we discussed in Chapter 4, assuming that the DMs intend to maximize their target values for the degrees of possibility for given permissible probability levels, by replacing minimization of the objective functions zl (x1 , x2 ), l = 1, 2 in (5.31) with maximization of the target values hl , l = 1, 2 such that the probabilities with respect to the fuzzy goal attainment levels are greater than or equal to certain permissible probability levels θl , l = 1, 2 speciﬁed by the DMs, we consider the level set-based fractile model for fuzzy random two-level programming problems formulated as ⎫ maximize h1 ⎪ ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ for DM2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.32) ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ x∈X ⎪ ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values, and θl , l = 1, 2 are permissible probability levels speciﬁed by the DMs, respectively. As we discussed in 4.1.3.4 of Chapter 4, from the relation L (ω)x ≤ μl (hl ) ≥ θl , P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα (5.32) can be transformed into maximize h1 for DM1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize h2 for DM2 L (ω)x) ≥ h ≥ θ subject to P ω | μ1 (C1α 1 1⎪ ⎪ ⎪ L P ω | μ2 (C2α (ω)x) ≥ h2 ≥ θ2 ⎪ ⎪ ⎪ ⎭ x ∈ X.

(5.33)

184

5 Stochastic and Fuzzy Random Two-Level Programming

Similarly to the previous model, we assume that the random parameter d¯ l of ˜¯ is expressed as d¯ = d1 + t¯ d2 , where t¯ is a random variable fuzzy random variable C l l l l l l with the mean ml and the variance σ2l , and dl2 x > 0, l = 1, 2. Then, the following for the constraint in (5.33) holds: β l )x + μl (hl ) (−dl1 + L (α)β L (ω)x) ≥ hl = Tl P ω | μl (Clα . dl2 x Then, it follows that βl )x + μl (hl ) (−dl1 + L (α)β βl )x ≥ hl , ≥ θl ⇔ μl (Tl (θl )dl2 + dl1 − L (α)β Tl 2 dl x where Tl is a pseudo inverse function of Tl which is deﬁned by Tl (θl ) = inf{r | Tl (r) ≥ θl }. Let F βl )x, (x) = (Tl (θl )dl2 + dl1 − L (α)β (5.34) Zlα and consequently, (5.33) can be transformed into maximize h1 for DM1

maximize h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

for DM2 F subject to μ1 Z1α (x) ≥ h1 ⎪ ⎪ F ⎪ μ2 Z2α (x) ≥ h2 ⎪ ⎪ ⎪ ⎭ x ∈ X,

or equivalently

⎫ F (x) ⎪ maximize μ1 Z1α for DM1 F ⎬ maximize μ2 Z2α (x) ⎪ for DM2 ⎭ subject to x ∈ X.

(5.35)

(5.36)

To derive an overall satisfactory solution to (5.36), we ﬁrst solve the following problem maximizing the smaller degree of satisfaction between those of the two DMs: F F (x) , μ2 Z2α (x) maximize min μ1 Z1α (5.37) subject to x ∈ X. By introducing an auxiliary variable v, from (5.34), this problem is rewritten as ⎫ maximize v ⎪ ⎪ β1 )x ≤ μ1 (v) ⎬ subject to (T1 (θ1 )d12 + d11 − L (α)β (5.38) β2 )x ≤ μ2 (v) ⎪ (T2 (θ2 )d22 + d21 − L (α)β ⎪ ⎭ x ∈ X.

5.1 Cooperative two-level programming

185

It is noted that (5.38) can be solved by the combined use of the simplex method and the bisection method, in a similar way to solving (5.27) in the possibility-based probability model. F ∗ If DM1 is satisﬁed with the membership function values μl Zlα (x ) , l = 1, 2, the corresponding optimal solution x∗ to (5.38) is regarded as a satisfactory solution. F (x)) is larger than or equal to the Otherwise, by introducing the constraint that μ1 (Z1α minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the following F (x)): problem maximizing the membership function μ2 (Z2α F ⎫ (x) maximize μ2 Z2α ⎬ F (5.39) subject to μ1 Z1α (x) ≥ δ ⎭ x ∈ X, which can be transformed into the linear programming problem ⎫ β2 )x minimize (T2 (θ2 )d22 + d21 − L (α)β ⎬ β1 )x ≤ μ1 (δ) subject to (T1 (θ1 )d12 + d11 − L (α)β ⎭ x ∈ X.

(5.40)

F (x))/μ (Z F (x)) and its permissible range [Δ By using the ratio Δ = μ2 (Z2α 1 1α min , Δmax ], we present a procedure of interactive fuzzy programming for the level-based fractile model in order to derive a satisfactory solution.

Interactive fuzzy programming in the level set-based fractile model Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2 by solving the linear programming problems (5.20) and (5.21). Step 2: Ask DMs to specify membership functions μl , l = 1, 2 by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify an initial degree α of the α-level set and the permissible probability levels θl , l = 1, 2. Step 4: For the current values of α and θl , l = 1, 2, solve the maximin problem (5.37). F ∗ (x ) and Step 5: is supplied with the membership function values μ1 Z1α F DM1 μ2 Z2α (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers to update α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , and return step4. Otherwise, ask DM1 to specify the minimal satisfactory level δ for to F (x) and the permissible range [Δ μ1 Z1α min , Δmax ] of the ratio Δ. Step 6: For the minimal satisfactory level δ, solve the linear programming problem (5.40). F ∗ F ∗ (x ) , μ2 Z2α (x ) and Δ. If Δ ∈ [Δmin , Δmax ] Step 7: DM1 is supplied with μ1 Z1α and DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers to update α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

186

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.2.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the level set-based fractile model, consider the following fuzzy random two-level linear programming problem: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ ˜ ˜ ¯ ¯ ⎬ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 for DM2 (5.41) subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 26 ⎪ ⎪ ⎪ T ⎪ x1 = (x11 , x12 , x13 , x14 , x15 ) ≥ 0 ⎪ ⎪ ⎭ x2 = (x21 , x22 , x23 , x24 , x25 )T ≥ 0, ˜¯ , l = 1, 2, j = 1, 2 are shown in Table 5.4, and t¯ , l = 1, 2 where the parameters of C lj l are given as Gaussian random variables N(10, 22 ) and N(10.0, 42 ), respectively. After solving linear programming problems (5.20) and (5.21), assume that the DMs determine the linear membership functions ⎧ ⎪ 1 if z ≤ z1l ⎪ ⎪ ⎨ z − z0 l if z1l < z ≤ z0l μl (z) = 1 − z0 ⎪ z ⎪ l ⎪ ⎩ l 0 if z > z0l by setting z11 = −4620.067, z01 = −3944.688, z12 = −2042.5 and z02 = −1503.8 in accordance with Zimmermann’s method (Zimmermann, 1978). Assume that DM1 speciﬁes an initial degree of α-level set at α = 0.9 and the permissible probability levels (θ1 , θ2 ) = (0.9, 0.9), respectively. Then, the maximin problem (5.37) is solved, and the corresponding objective function values and the ratio of the ﬁrst iteration are calculated as shown in Table 5.6. Suppose that DM1 is not satisﬁed with the objective function values and updates the permissible probability levels (θ1 , θ2 ) from (0.9, 0.9) to (0.8, 0.8). For the updated (θ1 , θ2 ), the corresponding maximin problem (5.37) is solved, and the result of the second iteration is shown in Table 5.6. Assume that DM1 is not satisﬁed with the objective function values and prefers to update neither the degree α nor the permissible probability levels θl . Then, suppose that DM1 speciﬁes the minimal satisfactory level at δ = 0.80 and the upper and the lower bounds of the ratio Δ at Δmax = 0.8 and Δmin = 0.7. For the speciﬁed δ = 0.80, the linear programming problem (5.40) is solved, and the result of the third iteration is shown in Table 5.6. A similar procedure continues in this manner until DM1 is satisﬁed with the membership function values. In the ﬁfth iteration, the ratio Δ exists in the interval [0.70, 0.80], and if DM1 is satisﬁed with the membership function values, it follows that a satisfactory solution is derived.

5.1 Cooperative two-level programming

187

Table 5.4 Value of each element of dl , β l and γ l , l = 1, 2. x11

x12

x13

x14

x15

x21

x22

x23

x24

x25

d11 d12 d21 d22

−300 3 −20 1

−270 2 −20 2

−200 3 −20 3

−150 2 −20 2

−100 1 −20 4

−15 2 −130 3

−15 2 −120 3

−15 2 −95 2

−15 2 −75 2

−15 2 −60 3

β1 β2 γ1 γ2

3 2 2.5 1.4

3 1 2.5 1.4

2 1 1.5 1.2

2 1 1 2.3

3 1 3 2.3

1 3 4 2.3

1.5 2 2 2.3

1.3 6 1.5 2.1

1.2 4 1.5 2

2.4 2 1.5 1.5

x22 −40 −80 −80 −150 0 7 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0

x23 −30 −70 −70 −120 0 6 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0

Table 5.5 Value of each element of ai , i = 1, . . . , 26.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26

x11 −100 −50 −250 −100 12 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

x12 −90 −50 −200 −90 11 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

x13 −80 −50 −200 −80 10 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

x14 −70 −40 −150 −70 9 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

x15 −60 −30 −100 −60 8 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

x21 −50 −100 −90 −200 0 8 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

x24 −20 −60 −60 −90 0 5 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0

x25 −10 −50 −50 −50 0 4 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 1

b −2000 −2000 −5500 −4500 220 160 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 8 8 8 8 8 8 8 8 8 8

Table 5.6 Process of interaction. Iteration

1st

2nd

3rd

4th

5th

α θ1 θ2

0.900 0.900 0.900

0.900 0.800 0.800

0.900 0.800 0.800

0.900 0.800 0.700

0.900 0.800 0.700

δ F ∗ μ1 Z1α (x ) F (x∗ ) μ2 Z2α

—

—

0.800

0.700

0.750

0.298 0.298

0.584 0.584

0.800 0.556

0.700 0.579

0.750 0.578

Δ

1.000

1.000

0.695

0.828

0.770

188

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.3 Extensions to integer programming So far, we have discussed stochastic and fuzzy random two-level programming with continuous decision variables, and in this subsection, as we considered in Chapters 3 and 4, we deal with stochastic and fuzzy random two-level programming with not continuous but discrete decision variables. For dealing with large-scale two-level nonlinear integer programming problems to be solved in the interactive fuzzy programming, we employ genetic algorithms to search approximate optimal solutions. In the same composition as 5.1.1, we will give the four models for stochastic and fuzzy random two-level integer programming.

5.1.3.1 Expectation model for stochastic two-level integer programming problems Stochastic two-level integer programming problems are generally formulated as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.42) ¯ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional integer decision variable column vector for DM1, x2 is an n2 dimensional integer decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional random variable row vectors with the mean vectors c¯ l j . The elements of vector b¯ are mutually independent random variables with continuous and increasing probability distribution functions, and νl jl , l = 1, 2, jl = 1, . . . , nl are positive integer values of the upper bounds of the decision variables. In a similar way to the two-level problems with continuous decision variables, by replacing the original stochastic constraints with the chance constrained conditions with satisﬁcing probability levels ηi , i = 1, . . . , m, (5.42) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.43) subject to P(ω | ai1 x1 + ai2 x2 ≤ bi (ω)) ≥ ηi , i = 1, . . . , m ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where ai j is the ith row vector of A j , j = 1, 2, and bi (ω) is a realized value of b¯ i for an elementary event ω. In a similar transformation in 5.1.1, (5.43) can be rewritten as

5.1 Cooperative two-level programming

189

⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎬ for DM1 minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 for DM2 ⎪ ⎭ η), subject to (xT1 , xT2 )T ∈ X int (η

(5.44)

where η) {(xT1 , xT2 )T | A1 x1 + A2 x2 ≤ b(η η), xl jl ∈ {0, 1, . . . , νl jl }, X int (η jl = 1, . . . , nl , l = 1, 2}. Assuming that the DMs are willing to simply minimize the expectation of their own objective functions, by substituting minimization of the expected objective functions for minimization of the original stochastic objective functions in (5.44), we consider the expectation model for the stochastic two-level integer programming problem formulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎬ for DM1 minimize zE2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.45) for DM2 ⎪ ⎭ T T T int η). subject to (x1 , x2 ) ∈ X (η In order to consider the imprecise nature of the DMs’ judgments for the objective functions zEl (x1 , x2 ), l = 1, 2 in (5.45), if we introduce a fuzzy goal such as “zEl (x1 , x2 ) should be substantially less than or equal to a certain value,” (5.45) can be rewritten as the fuzzy two-level integer programming problem ⎫ maximize μ1 (zE1 (x1 , x2 )) ⎪ ⎬ for DM1 maximize μ2 (zE2 (x1 , x2 )) (5.46) ⎪ for DM2 ⎭ η), subject to (xT1 , xT2 )T ∈ X int (η where μl , l = 1, 2 are membership functions to quantify fuzzy goals speciﬁed by DMl, l = 1, 2. To specify the membership functions of the fuzzy goals, the DMs refer to the individual minima and maxima of E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2 obtained by solving the integer programming problems minimize E[¯cl1 ]x1 + E[¯cl2 ]x2 , l = 1, 2, (5.47) η) subject to (xT1 , xT2 )T ∈ X int (η maximize E[¯cl1 ]x1 + E[¯cl2 ]x2 η) subject to (xT1 , xT2 )T ∈ X int (η

, l = 1, 2.

(5.48)

Since (5.47) and (5.48) are deterministic integer programming problems with linear objective functions and constraints, to obtain (approximate) optimal solutions, GADSLPRRSU given in Chapter 2 can be employed. To derive an overall satisfactory solution to (5.46), the ﬁrst step to obtain a satisfactory solution is to ﬁnd a solution maximizing the smaller degree of satisfaction between those of the two DMs by solving the maximin problem

190

5 Stochastic and Fuzzy Random Two-Level Programming

maximize min{μ1 (zE1 (x1 , x2 )), μ2 (zE2 (x1 , x2 ))} η). subject to (xT1 , xT2 )T ∈ X int (η

(5.49)

From the fact that this problem includes a nonlinear objective function and linear constraints, (5.49) can be solved by the revised GADSLPRRSU given in Chapter 2. If DM1 is satisﬁed with the membership function values μl (zEl (x∗1 , x∗2 )), l = 1, 2 of (5.49), the corresponding optimal solution (x∗1 , x∗2 ) can be viewed as a satisfactory solution. However, if DM1 is not satisﬁed with them, assuming that DM1 subjectively speciﬁes the minimal satisfactory level for the DM1’s own membership function value μ1 (zE1 (x1 , x2 )), we consider the fuzzy integer programming problem ⎫ maximize μ2 (zE2 (x1 , x2 )) ⎪ ⎬ (5.50) subject to μ1 (zE1 (x1 , x2 )) ≥ δ ⎪ ⎭ T T T int η). (x1 , x2 ) ∈ X (η From the nonincreasing property of the membership function μl , l = 1, 2, (5.50) can be equivalently transformed into ⎫ minimize zE2 (x1 , x2 ) ⎪ ⎬ E subject to z1 (x1 , x2 ) ≤ μ1 (δ) (5.51) ⎪ η), ⎭ (xT1 , xT2 )T ∈ X int (η where μ1 is a pseudo inverse function of μ1 deﬁned by μ1 (δ) = sup{t | μ1 (t) ≥ δ}. Since both the objective function and the constraints in (5.51) are linear, to solve (5.51), we can directly use GADSLPRRSU. By using the ratio Δ = μ2 (zE2 (x1 , x2 ))/μ1 (zE1 (x1 , x2 )) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the expectation model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the expectation model with integer decision variables Step 1: Ask DM1 to subjectively specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ), by solving (5.47) and (5.48) through GADSLPRRSU. Step 3: Ask each DM to specify the membership function μl (zEl (x1 , x2 )) by considering the individual minima and maxima obtained in step 2. Step 4: Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of the ratio Δ. Step 5: Solve the maximin problem (5.49) through the revised GADSLPRRSU, and calculate the membership function values μl (zEl (x∗1 , x2 )∗ ), l = 1, 2 and the ratio Δ. If DM1 is satisﬁed with the membership function values, then stop. Otherwise, ask DM1 to subjectively specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zE1 (x1 , x2 )).

5.1 Cooperative two-level programming

191

Step 6: For the minimal satisfactory level δ, solve (5.51) through GADSLPRRSU, and calculate the corresponding membership function values μl (zEl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 7: If DM1 is satisﬁed with the current membership function values and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.1.3.2 Variance model for stochastic two-level integer programming problems In the variance model, from the viewpoint of risk-aversion, the variances of the DMs’ objective functions with random variable coefﬁcients are minimized, and the following problem with the constraints for the expectations of the objective functions is formulated: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ DM1 ⎪ ⎪ minimize zV2 (x1 , x2 ) = (xT1 , xT2 )V2 (xT1 , xT2 )T ⎪ ⎪ ⎬ DM2 (5.52) subject to E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ T T T int η), (x1 , x2 ) ∈ X (η where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and γl , l = 1, 2 are permissible expectation levels speciﬁed by the DMs. η, γ ) denote the feasible region of (5.52), For notational convenience, let X int (η namely, η, γ ) {(xT1 , xT2 )T ∈ X int (η η) | E[¯cl1 ]x1 + E[¯cl2 ]x2 ≤ γl , l = 1, 2}. X int (η Considering the imprecise nature of each DM’s judgments for each objective function zVl (x1 , x2 ) in (5.52), if we introduce a fuzzy goal such as “zVl (x1 , x2 ) should be substantially less than or equal to a certain value,” (5.52) can be rewritten as ⎫ maximize μ1 (zV1 (x1 , x2 )) ⎪ ⎬ DM1 V maximize μ2 (z2 (x1 , x2 )) (5.53) DM2 ⎪ ⎭ T T T int η γ subject to (x1 , x2 ) ∈ X (η , ), where μl is a nonincreasing membership function to quantify the fuzzy goal. To help the DM to specify the membership functions μl , it is recommended to calculate the individual minima of zVl (x1 , x2 ) by solving the quadratic integer programming problems minimize zVl (x1 , x2 ) = (xT1 , xT2 )Vl (xT1 , xT2 )T , l = 1, 2. (5.54) η, γ ) subject to (xT1 , xT2 )T ∈ X int (η

192

5 Stochastic and Fuzzy Random Two-Level Programming

To solve (5.54) with the nonlinear objective function and the linear constraints, we can directly apply the revised GADSLPRRSU. In order to obtain an initial candidate for an overall satisfactory solution to the DMs, using the revised GADSLPRRSU, we solve the maximin problem maximize min{μ1 (zV1 (x1 , x2 )), μ2 (zV2 (x1 , x2 ))} (5.55) η, γ ). subject to (xT1 , xT2 )T ∈ X int (η If DM1 is satisﬁed with the membership function values of (5.55), the corresponding optimal solution is regarded as a satisfactory solution. Otherwise, it would be quite natural to assume that DM1 speciﬁes the minimal satisfactory level for the membership function μ1 (zV1 (x1 , x2 )). To calculate a candidate for the satisfactory solution, by introducing the constraint that μ1 (zV1 (x1 , x2 )) is larger than or equal to the speciﬁed minimal satisfactory level δ ∈ (0, 1), we consider the following problem maximizing DM2’s membership function μ2 (zV2 (x1 , x2 )): ⎫ maximize μ2 (zV2 (x1 , x2 )) ⎪ ⎬ subject to μ1 (zV1 (x1 , x2 )) ≥ δ (5.56) ⎪ ⎭ T T T int η, γ ). (x1 , x2 ) ∈ X (η Because, unlike (5.54) and (5.55), (5.56) involves a nonlinear constraint, to solve it, we employ the revised GADSLPRRSU. By using the ratio Δ = μ2 (zV2 (x1 , x2 ))/μ1 (zV1 (x1 , x2 )) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the variance model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the variance model with integer decision variables Step 1: Ask DM1 to specify the satisﬁcing probability levels ηi , i = 1, . . . , m. Step 2: Calculate the individual minima zEl,min and maxima zEl,max of zEl (x1 , x2 ) by solving (5.47) and (5.48) through GADSLPRRSU. Step 3: Ask each DM to specify the permissible expectation level γl by considering the individual minima and maxima obtained in step 2. Step 4: Calculate the individual minima zVl,min of zVl (x1 , x2 ) by solving (5.54) through the revised GADSLPRRSU. Step 5: Ask each DM to specify the membership function μl (zVl (x1 , x2 )) by considering the individual minima calculated in step 4. Step 6 Ask DM1 to specify the upper bound Δmax and the lower bound Δmin of Δ. Step 7: Solve the maximin problem (5.55) through the revised GADSLPRRSU, and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. If DM1 is satisﬁed with the membership function values, then stop. Otherwise, ask DM1 to specify the minimal satisfactory level δ ∈ (0, 1) for the membership function μ1 (zV1 (x1 , x2 )).

5.1 Cooperative two-level programming

193

Step 8: For the minimal satisfactory level δ, solve (5.56) through the revised GADSLPRRSU, and calculate the membership function values μl (zVl (x∗1 , x∗2 )), l = 1, 2 and the ratio Δ. Step 9: If DM1 is satisﬁed with the membership function values and Δ ∈ [Δmin , Δmax ] holds, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 8.

5.1.3.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented interactive fuzzy programming for the variance model with integer decision variables, consider the following numerical example to stochastic two-level integer programming problems: ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 ⎬ for DM2 (5.57) ¯ subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 10 ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , 30}, j1 = 1, . . . , 15 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , 30}, j2 = 1, . . . , 15, where x1 = (x11 , . . . , x115 )T , x2 = (x21 , . . . , x215 )T , and b¯ i , i = 1, . . . , 10 is a Gaussian random variable with the mean si and the variance σ2i given in Table 5.7. The coefﬁcients c¯ l1 and c¯ l2 , l = 1, 2 are Gaussian random variables, and their means are given in Table 5.8. ¯ Table 5.7 Means and variances of b. bi b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 si 78166 150930 231496 293906 369243 423722 497184 588256 667464 731073 σ2i 3502 12002 18002 9802 15002 25202 17502 28002 32002 25002 Table 5.8 Means of coefﬁcients of objective functions. E[¯c11 ] 22 46 −12 38 −41 −2 −50 3 45 22 43 −21 −31 17 38 E[¯c12 ] −5 37 20 45 −8 −19 18 4 8 39 30 47 −46 47 45 13 −13 7 −30 35 −42 18 −36 23 −50 42 −1 27 −2 22 E[¯c21 ] E[¯c22 ] −50 19 −42 24 −11 14 −24 7 −48 46 −6 49 −39 11 −28

Assume that DM1 speciﬁes the satisﬁcing probability levels as η1 = 0.95, η2 = 0.80, η3 = 0.85, η4 = 0.90, η5 = 0.90, η6 = 0.85, η7 = 0.85, η8 = 0.95, η9 = 0.80 and η10 = 0.95, and the individual minima and maxima of zEl (x1 , x2 ) are calculated as zE1,min = −6972, zE2,min = −12006, zE1,max = 15113 and zE2,max = 9745. Taking account of these values, suppose that the DMs subjectively specify the permissible expectation levels at γ1 = −1000 and γ2 = −5000.

194

5 Stochastic and Fuzzy Random Two-Level Programming

By using the revised GASDLPRRSU, the individual minima of zVl (x1 , x2 ) are calculated as zV1,min = 260912 and zV2,min = 853492. Assume that based on these values, the DMs subjectively determine the linear membership functions to quantify fuzzy goals as

μl (zVl (x1 , x2 )) =

⎧ 1 ⎪ ⎪ ⎪ ⎨ zV (x1 , x2 ) − zV l l,0 ⎪ ⎪ ⎪ ⎩

zVl,1 − zVl,0

if zVl (x1 , x2 ) ≤ zVl,1 if zVl,1 < zVl (x1 , x2 ) < zVl,0 if zVl (x1 , x2 ) ≥ zVl,0 ,

0

where zVl,1 and zVl,0 , l = 1, 2 are calculated as zV1,1 = 260912, zV1,0 = 2131786, zV2,1 = 853492 and zV2,0 = 2543661 by using Zimmermann’s method (Zimmermann, 1978). For the speciﬁed upper and lower bounds Δmax = 0.85 and Δmin = 0.70 of the ratio Δ, by solving the maximin problem (5.55) through the revised GADSLPRRSU, the membership function values and the ratio are calculated as μ1 (zV1 (x∗1 , x∗2 )) = 0.7573, μ2 (zV2 (x∗1 , x∗2 )) = 0.7663, and Δ = 1.0119. Suppose that DM1 is not satisﬁed with the membership function values and speciﬁes the minimal satisfactory level at δ = 0.80. For the speciﬁed δ = 0.80, by solving (5.56) through the revised GADSLPRRSU, the membership function values and the ratio are calculated as μ1 (zV1 (x∗1 , x∗2 )) = 0.8044, μ2 (zV2 (x∗1 , x∗2 )) = 0.7481 and Δ = 0.9299. Since the ratio Δ is greater than Δmax = 0.85, suppose that DM1 updates the minimal satisfactory level δ from 0.80 to 0.90. A similar procedure continues in this manner until the DM1 is satisﬁed with the membership function values. In the forth interaction, the ratio Δ exists in the interval [0.70, 0.85], and if DM1 is satisﬁed with the membership function values, it follows that the satisfactory solution is derived. Table 5.9 Process of interaction. Iteration 1st δ — μ1 (zV1 (x∗1 , x∗2 )) 0.7573 μ2 (zV2 (x∗1 , x∗2 )) 0.7663 zV1 (x∗1 , x∗2 ) 714946 zV2 (x∗1 , x∗2 ) 1248412 Δ 1.0119

2nd 0.80 0.8044 0.7481 626827 1279327 0.9299

3rd 0.90 0.9077 0.5946 433605 1538687 0.6551

4th 0.85 0.8535 0.6942 535088 1370382 0.8134

5.1.3.4 Possibility-based probability model for fuzzy random two-level integer programming problems As we discussed in 4.1.1, considering that possible realized values of the random parameters involved in two-level integer programming problems are often only am-

5.1 Cooperative two-level programming

195

biguously known to the experts, we deal with two-level linear programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ ˜ ˜ ¯ ¯ ⎬ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 for DM2 (5.58) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, and x2 is an n2 dimensional decision variable column vector for DM2. Each element C˜¯l jk , ˜¯ , l = 1, 2, j = 1, 2 is a fuzzy random variable k = 1, 2, . . . , n j of coefﬁcient vectors C lj whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl jk (ω) − τ ⎪ ⎪ if τ ≤ dl jk (ω) ⎪ ⎨L βl jk μC˜l jk (ω) (τ) = (5.59) ⎪ τ − dl jk (ω) ⎪ ⎪ if τ > dl jk (ω). ⎩R γl jk For notational convenience, let X int denote the feasible region of (5.58) and let ˜¯ ), x = (xT , xT )T . Then, it follows that each objective function C ˜¯ x = ˜C ¯˜ l1 , C ¯ l = (C l2 l 1 2 ˜C ˜ ¯ l1 x1 + C ¯ l2 x2 is a fuzzy random variable whose realized values are fuzzy numbers characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ L if υ ≤ dl (ω)x ⎪ ⎨ βl x (5.60) μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β In order to consider the imprecise nature of each DMs’ judgments for the objective functions zl (x1 , x2 ) in (5.58), we introduce a fuzzy goal such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value.” Let μG˜ l denote the nonincreasing membership function of the fuzzy goal G˜ l with respect to the objective function zl (x1 , x2 ). To elicit the membership function μG˜ l from each DM for the objective function zl (x1 , x2 ), it is recommended to calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving the integer programming problems minimize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2, (5.61) subject to x ∈ X int

196

5 Stochastic and Fuzzy Random Two-Level Programming

maximize E[d¯ l1 ]x1 + E[d¯ l2 ]x2 subject to x ∈ X int

, l = 1, 2.

(5.62)

To solve (5.61) and (5.62) with the linear objective functions and constraints, we employ GADSLPRRSU. Recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. (5.63) y

Along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs prefer to maximize the probabilities with respect to the degrees of possibility, the original fuzzy random two-level integer programming problem (5.58) can be reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ ⎬ for DM1 (5.64) ˜ maximize P ω ΠC˜ 2 (ω)x (G2 ) ≥ h2 ⎪ ⎪ for DM2 ⎪ ⎭ subject to x ∈ X int , where hl , l = 1, 2 are target values for the degrees of possibility speciﬁed by the DMs. As we discussed in 5.1.2.1 for the same model with continuous decision variables, (5.64) can be transformed into #⎫ " β1 − d11 )x + μG˜ (h1 ) ⎪ (h )β (L 1 ⎪ 1 ⎪ ⎪ maximize Z1Π,P (x) T1 ⎪ 2 ⎪ for DM1 d1 x ⎪ ⎬ " # 1 (5.65) β2 − d2 )x + μG˜ (h2 ) ⎪ (L (h )β 2 2 ⎪ maximize Z2Π,P (x) T2 ⎪ ⎪ for DM2 ⎪ d22 x ⎪ ⎪ ⎭ int subject to x ∈ X . In order to obtain an initial candidate for an overall satisfactory solution to (5.65), we solve the maximin problem ⎫ maximize min Z1Π,P (x), Z2Π,P (x) ⎬ (5.66) ⎭ subject to x ∈ X int , which can be solved by using the revised GADSLPRRSU. If DM1 is satisﬁed with the objective function values ZlΠ,P (x∗ ), l = 1, 2, the corresponding optimal solution x∗ to (5.66) is regarded as a satisfactory solution. However, if DM1 is not satisﬁed, by introducing the constraint that Z1Π,P (x) is larger than or equal to the speciﬁed minimal satisfactory level δ ∈ (0, 1), we consider the

5.1 Cooperative two-level programming

following problem maximizing DM2’s objective function: ⎫ ⎪ maximize Z2Π,P (x) ⎬ Π,P subject to Z1 (x) ≥ δ ⎪ ⎭ x ∈ X int .

197

(5.67)

From the assumption that dl2 x > 0, l = 1, 2 together with the nonincreasing property of Tl , (5.67) can be equivalently transformed into ⎫ β2 )x (−d21 + L (h2 )β ⎪ ⎪ maximize ⎪ ⎬ d22 x (5.68) β1 x ≤ μ1 (h1 ) ⎪ subject to T1 (δ)d12 + d11 − L (h1 )β ⎪ ⎪ ⎭ x ∈ X int , which can be solved through the revised GADSLPRRSU. By using the ratio Δ = Z2Π,P (x)/Z1Π,P (x) and its permissible range [Δmin , Δmax ], we present a procedure of interactive fuzzy programming for the possibility-based probability model with integer decision variables in order to derive an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 by solving (5.61) and (5.62) through GADSLPRRSU. Step 2: Ask each DM to specify the membership function μl by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify the target values hl , l = 1, 2. Step 4: For the speciﬁed target values hl , l = 1, 2, solve the maximin problem (5.66) through the revised GADSLPRRSU. Step 5: DM1 is supplied with the objective function values Z1Π,P (x∗ ) and Z2Π,P (x∗ ) for the optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed and prefers updating hl , l = 1, 2, ask DM1 to update hl , and return to step 4. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 6: For the speciﬁed minimal satisfactory δ, solve (5.68) by uisng the revised GADSLPRRSU. Step 7: DM1 is supplied with the objective function values of Z1Π,P (x∗ ), Z2Π,P (x∗ ) and the ratio Δ. If Δ ∈ [Δmin , Δmax ] holds and DM1 is satisﬁed with the objective function values, then stop. If DM1 is not satisﬁed and prefers updating hl , l = 1, 2, ask DM1 to update hl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

198

5 Stochastic and Fuzzy Random Two-Level Programming

5.1.3.5 Level set-based fractile model for fuzzy random two-level programming According to the idea of the level set-based models, suppose that the DMs intend to minimize their objective functions under the condition that all the coefﬁcient vector in the objective functions belong to the α-level set of fuzzy random variable deﬁned by (4.67) in Chapter 4. Then, (5.58) can be interpreted as the following stochastic two-level integer programming problem without fuzzy random variables depending on the the degree α: ⎫ ⎪ ⎪ ⎪ ⎬

minimize z1 (x1 , x2 ) = C¯ 11 x1 + C¯ 12 x2 for DM1 minimize z2 (x1 , x2 ) = C¯ 21 x1 + C¯ 22 x2 for DM2

⎪ subject to x = (xT1 , xT2 )T ∈ X int ⎪ ⎪ ⎭ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ C1 = (C11 , C12 ) ∈ C1α , C2 = (C21 , C22 ) ∈ C2α ,

(5.69)

where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . Considering the imprecise nature of the DMs’ judgments, we assume that each DM has a fuzzy goal for the objective function in (5.69), and the fuzzy goals are determined by the membership functions μl , l = 1, 2 for DM1 and DM2. In the level set-based fractile model, assuming that the DMs prefer to maximize the target values for the satisfaction degrees of the DMs for the objective function values, we consider the following fuzzy random two-level integer programming problem: ⎫ maximize h1 ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ for DM2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.70) ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ X int ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values, and θl , l = 1, 2 are permissible probability levels speciﬁed by the DMs, respectively. As we considered in 5.1.2.2 for the same model with continuous decision variables, letting F βl )x, Zlα (x) (Tl (θl )dl2 + dl1 − L (α)β (5.70) can be transformed into maximize h1 for DM1

maximize h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

for DM2 F subject to μ1 Z1α (x) ≥ h1 ⎪ F ⎪ ⎪ μ2 Z2α (x) ≥ h2 ⎪ ⎪ ⎪ ⎭ x ∈ X int ,

(5.71)

5.1 Cooperative two-level programming

or equivalently

F ⎫ (x) ⎪ maximize μ1 Z1α for DM1 F ⎬ maximize μ2 Z2α (x) ⎪ for DM2 ⎭ subject to x ∈ X int .

199

(5.72)

To derive an overall satisfactory solution to (5.72), it would be useful for DM1 to obtain a solution which maximizes the smaller degree of satisfaction between the two DMs by solving the maximin problem F F (x) , μ2 Z2α (x) maximize min μ1 Z1α (5.73) subject to x ∈ X int , which can be solved by using the revised GADSLPRRSU. F ∗ (x ) , l = 1, 2, If DM1 is satisﬁed with the membership function values μl Zlα the corresponding optimal solution x∗ to (5.73) is regarded as a satisfactory solution. F (x)) is larger than or equal to the Otherwise, introducing the constraint that μ1 (Z1α minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the following F (x)): problem maximizing the membership function μ2 (Z2α ⎫ F maximize μ2 Z2α (x) ⎪ ⎬ F (5.74) (x) ≥ δ subject to μ1 Z1α ⎪ ⎭ int x∈X , which is equivalently expressed as ⎫ β2 )x minimize (T2 (θ2 )d22 + d21 − L (α)β ⎬ β1 )x ≤ μ1 (δ) subject to (T1 (θ1 )d12 + d11 − L (α)β ⎭ x ∈ X int .

(5.75)

It is noted that (5.75) can be solved by GADSLPRRSU. In order to derive the satisfactory solution with well-balanced membership function values between both DMs, we ratio Fintroduce the of the satisfactory degree F of DM2 to that of DM1 Δ = μ2 Z2α (x) /μ1 Z1α (x) , and its permissible range [Δmin , Δmax ]. The, we summarize a procedure of interactive fuzzy programming for the level-based fractile model with integer decision variables in order to derive a satisfactory solution. Interactive fuzzy programming in the level set-based fractile model with integer decision variables Step 1: Calculate the individual minima and maxima of E[d¯ l1 ]x1 + E[d¯ l2 ]x2 , l = 1, 2 by solving (5.61) and (5.62) through GADSLPRRSU. Step 2: Ask DMs to specify membership functions μl , l = 1, 2 by considering the individual minima and maxima obtained in step 1. Step 3: Ask DM1 to specify an initial degree α of the α-level set and the permissible probability levels θl , l = 1, 2.

200

5 Stochastic and Fuzzy Random Two-Level Programming

Step 4: For the speciﬁed values of α and θl , l = 1, 2, solve the maximin problem (5.73) through the revised GADSLPRRSU. F (x∗ ) and μ Z F (x∗ ) for the Step 5: DM1 is supplied with the values of μ1 Z1α 2 2α optimal solution x∗ obtained in step 4. If DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers updating α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , and returnto step 4. Otherwise, ask F (x) and the permisDM1 to specify the minimal satisfactory level δ for μ1 Z1α sible range [Δmin , Δmax ] of the ratio Δ. Step 6: For the minimal satisfactory level δ, solve through (5.75) GADSLPRRSU. F (x∗ ) , μ Z F (x∗ ) and Δ. If Step 7: DM1 is supplied with the values of μ1 Z1α 2 2α Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the membership function values, then stop. If DM1 is not satisﬁed and prefers updating α and/or θl , l = 1, 2, ask DM1 to update α and/or θl , l = 1, 2, and return to step 4. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 6.

5.2 Noncooperative two-level programming In the previous section, from the viewpoint of taking the possibility of coordination or bargaining between two DMs into account in hierarchical decision making situations, it is assumed that there exist communication and a cooperative relationship between the two DMs. However, in the real-world hierarchical managerial or public organizations, observing the decision making situations where the DMs do not make any binding agreement even if there exists such communication, we deal with noncooperative two-level programming problems. As discussed in Chapter 2, the concept of Stackelberg solution can be viewed as one of reasonable solution concepts of noncooperative two-level programming problems, in which the DM at upper level (DM1) makes a decision to optimize the DM1’s own objective function under the assumption that the DM at lower level (DM2) makes a decision to optimize the DM2’s own objective function for a given decision of DM1. Computational methods for obtaining Stackelberg solutions to two-level linear programming problems are classiﬁed roughly into three categories: the vertex enumeration approach (Bialas and Karwan, 1984), the Kuhn-Tucker approach (Bard and Falk, 1982; Bard and Moore, 1990; Bialas and Karwan, 1984; Hansen, Jaumard and Savard, 1992), and the penalty function approach (White and Anandalingam, 1993). The subsequent works on two-level programming problems under noncooperative behavior of the DMs have been appearing (Nishizaki and Sakawa, 1999, 2000; G¨um¨us¸ and Floudas, 2001; Nishizaki, Sakawa and Katagiri, 2003; Colson, Marcotte and Savard, 2005; Faisca et al., 2007) including some applications: aluminum production process (Nicholls, 1996), pollution control policy determination (Amouzegar and Moshirvaziri, 1999), tax credits determination for biofuel producers (Dempe and BardZ, 2001), pricing in competitive electricity markets (Fampa

5.2 Noncooperative two-level programming

201

et al., 2008), supply chain planning (Roghanian, Sadjadi and Aryanezhad, 2007) and so forth. Fuzzy random programming approaches to two-level programming problems were also considered together with the computational methods for obtaining the corresponding Stackelberg solutions (Sakawa and Kato, 2009c; Sakawa and Katagiri, 2010b; Sakawa, Katagiri and Matsui, 2010b) In this section, through the four models as discussed in the previous section, we consider noncooperative two-level programming problems with random variables and fuzzy random variables. Under some assumptions, it is shown that the formulated stochastic and fuzzy random two-level programming problems are transformed into deterministic ones. Computational methods for obtaining Stackelberg solutions for the presented models are developed, and some numerical examples are provided to demonstrate the feasibility and efﬁciency of the computational methods.

5.2.1 Stochastic two-level linear programming In order to take into account the randomness of the parameters in hierarchical decision making problems under noncooperative environments, we consider the following two-level linear programming problems where coefﬁcients in objective functions and the right-hand side of constraints are represented by random variables: ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.76)

where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors, and b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. By replacing the chance constrained conditions (Charnes and Cooper, 1959) with the original constraints, (5.76) can be rewritten as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(5.77)

202

5 Stochastic and Fuzzy Random Two-Level Programming

where η = (η1 , . . . , ηm )T is a vector of the satisﬁcing probability levels for holding η) = (b1 (η1 ), . . . , bm (ηm ))T is deﬁned by (5.3). the constraints, and b(η

5.2.1.1 Expectation model Assuming that the DMs intend to simply minimize the expectation of their own objective functions, the stochastic two-level linear programming problem can be reformulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ E minimize z2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.78) x2 ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where E denotes the expectation operator. Observing that E[¯cl1 ] and E[¯cl2 ], l = 1, 2 are constant vectors, it is evident that this problem is an ordinary noncooperative two-level linear programming problem. Thus, the computational techniques for obtaining Stackelberg solutions given in Chapter 2 are directly applicable.

5.2.1.2 Variance model In order to take account of each DM’s concern about the ﬂuctuation of the realized objective function values, to stochastic two-level linear programming problems, we consider the following variance minimization model with the constraints for the expectations of the objective functions: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ V T T T T T minimize z2 (x1 , x2 ) = (x1 , x2 )V2 (x1 , x2 ) ⎪ ⎬ x2

η) subject to A1 x1 + A2 x2 ≤ b(η E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 x1 ≥ 0, x2 ≥ 0,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5.79)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of Gaussian random variable vectors (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are the permissible expectation levels speciﬁed by DM1 and DM2, respectively. For a decision xˆ 1 by DM1, a rational response of DM2 is an optimal solution x∗2 of the quadratic programming problem

5.2 Noncooperative two-level programming

⎫ minimize zV2 (ˆx1 , x2 ) = (ˆxT1 , xT2 )V2 (ˆxT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]ˆx1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ E[¯c21 ]ˆx1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ x2 ≥ 0.

203

(5.80)

Observing that the objective function of (5.80) is strictly convex from the assumption that V2 is positive-deﬁnite, it is evident that (5.80) can be solved by a conventional convex programming techniques such as the sequential quadratic programming method. Although a rational response is not always unique generally, the optimal solution x∗2 to (5.80) is unique due to the strictly convexity property of the objective function, and it follows that, for a given x1 , the set of rational responses denoted by R(x1 ) is a singleton. Then, the Stackelberg solution for the variance model is an optimal solution (x1 , x2 ) to the following problem: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎬ subject to A 1 x1 + A 2 x2 ≤ b

(5.81) ⎭ x1 ≥ 0, x2 ∈ R(x1 ), where, for the sake of simplicity, we use coefﬁcient matrices A 1 , A 2 and a coefﬁcient η), γ1 and γ2 in (5.80). vector b instead of E[¯c11 ], E[¯c12 ], E[¯c21 ], E[¯c22 ], b(η Realizing that the constraint x2 ∈ R(x1 ) in (5.81) can be replaced by the KuhnTucker conditions for (5.80), for a DM1’s decision x∗1 , we introduce the Lagrange function deﬁned as T ∗T T T

∗

L(x2 , λ , ζ; x∗1 ) = (x∗T 1 , x2 )V2 (x1 , x2 ) + λ (A1 x1 + A2 x2 − b ) − ζx2 ,

where λ and ζ are Lagrange multiplier vectors. Then, the Kuhn-Tucker conditions are given as ⎫ n1 n1 +n2 ⎪ ∗

⎪ 2 ∑ v2(n1 +i) j x1 j + 2 ∑ v2(n1 +i) j x2, j−n1 + λ A2,·i − ζi = 0, i = 1, . . . , n2 ⎪ ⎪ ⎪ ⎬ j=1 j=n1 +1

∗

A1 x1 + A2 x2 − b ≤ 0 ⎪ ⎪ ⎪ λ (A 1 x∗1 + A 2 x2 − b ) = 0, ζx2 = 0 ⎪ ⎪ ⎭ x2 ≥ 0, λ ≥ 0, ζ ≥ 0, (5.82) where v2i j is the i j-element of V2 , and A 2,·i is an ith column vector of A 2 . Substituting the Kuhn-Tucker conditions (5.82) for the constraint x2 ∈ R(x1 ) in (5.81), (5.81) can be transformed into the following single-level quadratic programming problem with linear complementarity constraints:

204

5 Stochastic and Fuzzy Random Two-Level Programming

minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T n1

n1 +n2

j=1

j=n1 +1

subject to 2 ∑ v2(n1 +i) j x1 j + 2

∑

v2(n1 +i) j x2, j−n1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

λA 2,·i − ζi = 0, i = 1, . . . , n2 ⎪ +λ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0, ζx2 = 0 ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, λ ≥ 0, ζ ≥ 0.

(5.83)

A 1 x1 + A 2 x2 − b ≤ 0 λ (A 1 x1 + A 2 x2 − b ) =

Although (5.83) cannot be solved by a conventional convex programming technique due to the nonconvexity of the constraints, it is signiﬁcant to realize that if the linear complementarity constraints are eliminated, this problem can be viewed as an ordinary quadratic programming problem. With this observation in mind, we solve (5.83) by using the branching technique with respect to the linear complementarity constraints like the Bard-Moore method (Bard and Moore, 1990). Namely, after branching with respect to the linear complementarity constraints, by solving subproblems generated by the branching operation using a conventional convex programming technique, we can obtain a Stackelberg solution for the variance model, i.e., an optimal solution to the deterministic problem (5.83). To improve the readability, in the description of the following algorithm for solving (5.79), let m+n +2 ∑i=1 2 ui gi = 0 denote the linear complementarity constraint in (5.83). Now we are ready to summarize the algorithm for obtaining a Stackelberg solution for the variance model. Algorithm for obtaining a Stackelberg solution for the variance model / Sk− := 0, / Sk0 := W and V c := ∞. Step 1: Set k := 0, Sk+ := 0, + Step 2: Let ui = 0 for Sk and gi = 0 for Sk− . Solve (5.83) without the linear complementarity constraint. If the problem is infeasible, then go to step 6. Otherwise, set k := k + 1 and let (xk1 , xk2 , λ k , ζk ) denote the obtained solution. Step 3: If zV1 (xk1 , xk2 ) ≥ V c , then go to step 6. Step 4: If the linear complementarity constraint is satisﬁed, i.e., ui gi = 0, i = 1, . . . , m + n2 + 2, then go to step 5. Otherwise, ﬁnd i∗ such that the amount of violation of linear complementarity constraint ui∗ gi∗ is the largest, and let Sk+ := Sk+ ∪ i∗ , Sk0 := Sk0 \i∗ , append i∗ to Pk , and return to step 2. Step 5: Set V c := zV1 (xk1 , xk2 ). Step 6: If there exists no unexplored node, then go to step 7. Otherwise, branch to the newest unexplored node, and update Sk+ , Sk− , Sk0 and Pk . Return to step 2. Step 7: Stop the algorithm: if V c = ∞, there is no feasible solution; otherwise, the solution corresponding to the current V c is a Stackelberg solution for the variance model.

5.2.1.3 Numerical example To demonstrate the feasibility and efﬁciency of the presented variance and expectation models, consider the following numerical examples to stochastic two-level

5.2 Noncooperative two-level programming

205

linear programming problems: minimize z1 (x1 , x2 ) = c¯11 x1 + c¯12 x2 x1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

where x2 solves minimize z2 (x1 , x2 ) = c¯21 x1 + c¯22 x2 x2 ⎪ subject to −x1 + 3x2 ≤ b¯ 1 , 10x1 − x2 ≤ b¯ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 3x1 + x2 ≥ b¯ 3 , x1 + 2x2 ≥ b¯ 4 ⎪ ⎭ 3x1 + 2x2 ≥ b¯ 5 , x1 ≥ 0, x2 ≥ 0,

(5.84)

where the means of random variables are shown in Table 5.10, and the variancecovariance matrices of c¯ 1 = (c¯11 , c¯12 ) and c¯ 2 = (c¯21 , c¯22 ) are given as

2 1 1 −1 V1 = , V2 = . 1 3 −1 6 Assume that all the right-hand side constant of the constraints are represented by normal random variables, and the means and the variances of the random variables together with the satisﬁcing probability levels for the stochastic constraints are given in Table 5.11. Table 5.10 Means of random variable coefﬁcients in the objective functions. coefﬁcient mean

c¯11 −2.0

c¯12 −3.0

c¯21 2.0

c¯22 1.0

Table 5.11 Random variable coefﬁcients in the constraints and satisﬁcing probability levels. coefﬁcient mean variance probability

b¯ 1 50.11 9.0 0.85

b¯ 2 113.15 36.0 0.70

b¯ 3 15.16 9.0 0.90

b¯ 4 13.16 4.0 0.70

b¯ 5 25.63 16.0 0.80

Then, the problem formulation in the variance model is given as minimize 2x12 + 2x1 x2 + 3x22 x1

where x2 solves minimize x12 − 2x1 x2 + 6x22 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

subject to −x1 + 3x2 ≤ 47, 10x1 − x2 ≤ 110 ⎪ ⎪ ⎪ −3x1 − x2 ≤ −19, −x1 − 2x2 ≤ −15 ⎪ ⎪ ⎪ ⎪ −3x1 − 2x2 ≤ −29, −2x1 − 3x2 ≤ −31 ⎪ ⎪ ⎭ 2x1 + x2 ≤ 33, x1 ≥ 0, x2 ≥ 0.

(5.85)

Considering the Kuhn-Tucker conditions for the constraint that x2 is a rational response to x1 , (5.85) is transformed into the following single-level quadratic pro-

206

5 Stochastic and Fuzzy Random Two-Level Programming

gramming problem with linear complementarity constraints: ⎫ minimize 2x12 + 2x1 x2 + 3x22 ⎪ ⎪ ⎪ subject to 2x1 + 12x2 + 3λ1 − λ2 − λ3 − 2λ4 − 2λ5 − 3λ6 + λ7 − ζ = 0 ⎪ ⎪ ⎪ ⎪ ⎪ −x1 + 3x2 ≤ 47, 10x1 − x2 ≤ 110 ⎪ ⎪ ⎪ ⎪ −3x1 − x2 ≤ −19, −x1 − 2x2 ≤ −15 ⎪ ⎪ ⎪ ⎪ −3x1 − 2x2 ≤ −29, 2x1 + x2 ≤ 33 ⎪ ⎪ ⎬ −2x1 − 3x2 ≤ −31 (5.86) λ1 (−x1 + 3x2 − 47) = 0, λ2 (10x1 − x2 − 110) = 0 ⎪ ⎪ ⎪ ⎪ λ3 (−3x1 − x2 + 19) = 0, λ4 (−x1 − 2x2 + 15) = 0 ⎪ ⎪ ⎪ ⎪ λ5 (−3x1 − 2x2 + 29) = 0, λ6 (−2x1 − 3x2 + 31) = 0 ⎪ ⎪ ⎪ ⎪ λ7 (2x1 + x2 − 33) = 0, ζx2 = 0 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0, λi ≥ 0, i = 1, . . . , 7 ⎪ ⎪ ⎭ ζ ≥ 0. Fig. 5.1 illustrates the feasible region of (5.85) and the Stackelberg solutions for the variance and expectation models. The Stackelberg solution to (5.85) is denoted by the white circle at the point (5.17, 6.89). In order to illustrate the effect of considering the expectation of the objective function values, by solving (5.85) without the constraint for the expectation, the corresponding Stackelberg solution is denoted by the black circle at the point (7, 4). The black square at the point (1, 16) denotes the Stackelberg solution corresponding to the expectation model. Table 5.12 shows the means and the variances of the objective functions of DM1 and DM2. For simplicity, we denote the expectation and variance models by E-model and V-model, respectively. In particular, we denote the variance model without the constraint for the expectation by V-model without expectation. Table 5.12 Stackelberg solutions of the expectation and variance models. Model

Solution

V-model V-model without expectation E-model

(5.17, 6.89)

Mean of DM1 −31

Mean of DM2 17

Variance of DM1 266.94

Variance of DM2 240.25

(7, 4)

−26

18

202

89

(1, 16)

−50

18

802

1505

From Table 5.12, it is observed that the mean of the DM1’s objective function in the expectation model is smaller than that of the variance model, while the variance of the objective function of each DM in the expectation model is considerably large. In particular, although the mean of the DM2’s objective function in the variance model is almost the same as that in the expectation model, not only the variance of the objective function of DM2 but also that of DM1 are substantially reduced. It is found that the variance model yields the Stackelberg solution in which the variance is reduced adequately with the suitable levels of the mean speciﬁed by DM1.

5.2 Noncooperative two-level programming

207

Fig. 5.1 Feasible region and the Stackelberg solutions.

5.2.2 Fuzzy random two-level linear programming 5.2.2.1 Possibility-based probability model In previous subsections, uncertain parameters or coefﬁcients involved in formulated two-level linear programming problems are assumed to be random variables. However, it would be signiﬁcant to realize that we are faced with decision making situations where possible realized values of the random parameters are often only ambiguously known to the experts. In order to deal with such a hierarchical decision making problem, we formulate the following noncooperative two-level linear programming problem with fuzzy random variable coefﬁcients: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ ˜ ˜ ¯ ¯ (5.87) minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 ⎪ x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2.

208

5 Stochastic and Fuzzy Random Two-Level Programming

˜¯ = (C ˜¯ , C ˜¯ ) For simplicity, let X denote the feasible region of (5.87), and let C l l1 l2 T T T and x = (x1 , x2 ) . As in the possibility-based probability model for cooperative decision making situations described in 5.1.2.1, assume that elements C˜¯l jk , ˜¯ , l = 1, 2, j = 1, 2 are fuzzy random variables k = 1, 2, . . . , n j of coefﬁcient vectors C lj whose realized value for an elementary event ω is characterized by the membership ˜¯ x + C ˜¯ x is ˜¯ x = C function (5.18). Then, it follows that each objective function C l l1 1 l2 2 represented by a single fuzzy random variable whose realized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ if υ ≤ dl (ω)x ⎪ ⎨L βl x μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ R if υ > dl (ω)x, ⎩ γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2, and an where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β element d¯l j of d¯ l is a random variable expressed as d¯l j = dl1j + t¯l dl2j . Considering the imprecise natures of DMs’ judgments, we introduce a fuzzy goal G˜ l such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value” which is assumed to be quantiﬁed by a nonincreasing membership function μG˜ l . Furthermore, recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.88) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. y

It should be noted here that the degree ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of each elementary event ω. With this observation in mind, along the same line as the possibility-based probability model for cooperative decision making situations in 5.1.2.1, assuming that the DMs intend to maximize the probability that ΠC˜ l (ω)x (G˜ l ) is greater than or equal to the speciﬁed target value hl , (5.87) is reformulated as ⎫ maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 ⎪ ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ (5.89) ˜ maximize P ω ΠC˜ 2 (ω)x (G2 ) ≥ h2 ⎪ x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Assuming that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, the objective functions in (5.89) can be equivalently transformed as

5.2 Noncooperative two-level programming

P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl

209

"

βl − dl1 )x + μG˜ (hl ) (L (hl )β l

dl2 x

# ,

where Tl is a probability distribution function of t¯l , and L and μG˜ are pseudo inverse l functions of L and μG˜ l , respectively. Furthermore, since the distribution functions Tl , l = 1, 2 are nondecreasing, (5.89) is equivalently transformed into ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β Π,P ⎪ 1 ⎪ maximize Z1 (x1 , x2 ) = ⎪ ⎪ x1 d12 x ⎪ ⎪ ⎪ ⎪ where x2 solves ⎬ 1 β (5.90) (h )β − d )x + μ (h ) (L 2 2 2 G˜ 2 2 ⎪ ⎪ maximize Z2Π,P (x1 , x2 ) = ⎪ ⎪ x2 d22 x ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. For any feasible decision xˆ 1 of DM1, a rational response of DM2 is given as an optimal solution to the single-level programming problem 1 )x + (L (h )β 1 x + μ (h ) ⎫ β22 − d22 (L (h2 )β 2 2 β 21 − d21 )ˆ 1 G˜ 2 2 ⎪ ⎪ ⎬ 2 2 x2 d22 x2 + d21 xˆ 1 ⎪ subject to A2 x2 ≤ b − A1 xˆ 1 ⎪ ⎭ x2 ≥ 0.

maximize

(5.91)

Let R(ˆx1 ) denote a set of rational responses of DM2 for a decision xˆ 1 by DM1 and the inducible region is deﬁned by IR = {(x1 , x2 ) | (x1 , x2 ) ∈ X, x2 ∈ R(x1 )}. Since DM1 should select a solution (x1 , x2 ) to maximize Z1Π,P (x1 , x2 ) from among the inducible region IR, the Stackelberg solution is obtained by solving ⎫ ⎪ maximize Z1Π,P (x1 , x2 ) ⎬ x1 (5.92) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ∈ R(x1 ). In contrast to the variance model discussed in the previous subsection, through the combined use of the variable transformation (Charnes and Cooper, 1962) and the kth best method for two-level linear programming problems (Bialas and Karwan, 1984) we give a computational method for the Stackelberg solution. As the ﬁrst step for obtaining the Stackelberg solution, by removing the objective function of DM2 from (5.90), we consider the single-level programming problem ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β ⎪ 1 ⎬ maximize x1 ,x2 d12 x ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Z1Π,P (x) =

(5.93)

210

5 Stochastic and Fuzzy Random Two-Level Programming

From the fact that the objective function in (5.93) is a linear fractional function with the positive denominator, by employing the variable transformation (Charnes and Cooper, 1962): 1 1 = 2 2 d12 x d11 x1 + d12 x2 y1 = tx1 , y2 = tx2 ,

t=

(5.93) can be equivalently transformed into the linear programming problem ⎫ 1 1 β11 − d11 β12 − d12 )y1 + (L (h1 )β )y2 + μG˜ (h1 )t ⎪ maximize (L (h1 )β ⎪ 1 ⎪ y1 , y2 , t ⎪ ⎬ subject to A1 y1 + A2 y2 − bt ≤ 0 (5.94) 2 y + d2 y = 1 ⎪ d11 ⎪ 1 12 2 ⎪ ⎪ ⎭ y1 ≥ 0, y2 ≥ 0, t ≥ 0. Let (yT1[1] , yT2[1] ,t[1] )T be an optimal solution to (5.94). Then, T (x1[1] , xT2[1] )T of the feasible region of (5.93) is expressed as

an extreme point

(xT1[1] , xT2[1] )T = (yT1[1] /t[1] , yT2[1] /t[1] )T . In order to check whether the extreme point (xT1[i] , xT2[i] )T is in the inducible region IR, i.e., x2[i] is a rational response to x1[i] or not, we solve the linear fractional programming problem ⎫ 1 )x + (L (h )β 1 β22 − d22 (L (h2 )β 2 2 β 21 − d21 )x1[i] + μG˜ (h2 ) ⎪ ⎪ 2 ⎪ ⎬ maximize 2 2 x2 d22 x2 + d21 x1[i] (5.95) ⎪ subject to A2 x2 ≤ b − A1 x1[i] ⎪ ⎪ ⎭ x2 ≥ 0. Using a similar variable transformation u=

1 2 2 d22 x2 + d21 x1[i]

w2 = ux2 , (5.95) can be equivalently written as the linear programming problem ⎫ 1 1 β22 − d22 β21 − d21 maximize (L (h2 )β )w2 +[(L (h2 )β )x1[i] + μG˜ (h2 )]u ⎪ ⎪ 2 ⎪ w2 , u ⎪ ⎬ subject to A2 w2 − (b − A1 x1[i] )u ≤ 0 (5.96) 2 2 ⎪ w2 + d21 x1[i] u = 1 d22 ⎪ ⎪ ⎪ ⎭ w2 ≥ 0, u ≥ 0.

5.2 Noncooperative two-level programming

211

Let (wT2[i] , u[i] )T be an optimal solution of (5.96). If w2[i] /u[i] = x2[i] holds, then the current extreme point (xT1[i] , xT2[i] )T exists in IR, and it follows that (xT1[i] , xT2[i] )T is a Stackelberg solution. Otherwise, we use a set W[i] of feasible extreme points which are adjacent to (xT1[i] , xT2[i] )T and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[i] , x2[i] ). After setting U := U ∪ {(xT1[i] , xT2[i] )T } and W := (W ∪W[i] )\U, let a feasible extreme point

(xT1[i] , xT2[i] )T in W such that Z1Π,P (x1 , x2 ) is the greatest be the next extreme point (xT1[i+1] , xT2[i+1] )T . By repeating this procedure, we can obtain a Stackelberg solution. Now we summarize the algorithm for obtaining a Stackelberg solution for the possibility-based probability model. Algorithm for obtaining a Stackelberg solution for the possibility-based probability model

Step 1: Set i := 1, and solve (5.94). By using the optimal solution (yT1[1] , yT2[1] ,t[1] )T to (5.94), generate the initial extreme point (xT1[1] , xT2[1] )T := (yT1[1] /t[1] , yT2[1] /t[1] )T . / Let W := {(xT1[1] , xT2[1] )T } and U := 0. Step 2: For the current extreme point (xT1[i] , xT2[i] )T , solve the corresponding linear programming problem (5.96). For the obtained optimal solution (wT2[i] , u[i] )T , if w2[i] /u[i] = x2[i] , then stop; then, (xT1[i] , xT2[i] )T is a Stackelberg solution. Otherwise, go to step 3. Step 3: Let W[i] be a set of feasible extreme points which are adjacent to (xT1[i] , xT2[i] )T and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[i] , x2[i] ). Set U := U ∪ {(xT1[i] , xT2[i] )T } and W := (W ∪W[i] )\U. Step 4: Set i := i + 1. Choose an extreme point (xT1[i] , xT2[i] )T such that Z1Π,P (x1[i] , x2[i] ) = max(xT ,xT )T ∈W Z1Π,P (x1 , x2 ), and return to step 2. 1

2

5.2.2.2 Level set-based fractile model Along the same line as the level set-based fractile model for cooperative decision making situations discussed in 5.1.2.2, we consider the level-set based fractile model for noncooperative fuzzy random two-level linear programming problems and present an algorithm for obtaining the Stackelberg solutions. If DM1 supposes that the degree of all of the membership functions of the fuzzy random variables involved in (5.87) should be greater than or equal to some value α, the original fuzzy random two-level linear programming problem (5.87) is reformulated as

212

5 Stochastic and Fuzzy Random Two-Level Programming

minimize C¯ 1 x = C¯ 11 x1 + C¯ 12 x2 x1 , C¯ 1

where x2 solves minimize C¯ 2 x = C¯ 21 x1 + C¯ 22 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

x2 , C¯ 2

⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0 ⎪ ⎪ ⎭ C¯ 1 = (C¯ 11 , C¯ 12 ) ∈ C¯ 1α , C¯ 2 = (C¯ 21 , C¯ 22 ) ∈ C¯ 2α ,

(5.97)

where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . To take into account the imprecise nature of the DMs’ judgments, we assume that DMs have fuzzy goals for their objective functions in the α-stochastic two-level linear programming problem (5.97), and the fuzzy goals are quantiﬁed by the nonincreasing and continuous membership functions μl , l = 1, 2. Then, the satisfaction degrees μl (C¯ l (ω)x), l = 1, 2 may vary randomly due to the stochastic occurrence of an elementary event ω. With this observation in mind, assuming that the DMs intend to maximize the target values for the satisfaction degrees under the probabilistic constraint with permissible probability levels, we consider the following level set-based fractile model for noncooperative two-level fuzzy random linear programming problem: ⎫ maximize h1 ⎪ ⎪ ¯ ⎪ x1 , h1 , C1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ ⎪ x2 , h2 , C¯ 2 ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.98) ⎪ ⎪ P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 ≥ 0, x2 ≥ 0 ⎪ ⎪ ⎭ C¯ 1 ∈ C¯ 1α , C¯ 2 ∈ C¯ 2α , where hl , l = 1, 2 are decision variables representing the target values and θl , l = 1, 2 are the permissible probability levels speciﬁed by DMl, l = 1, 2. From the discussion in 4.1.3.4 of Chapter 4, one ﬁnds that L (ω)x ≤ μl (hl ) ≥ θl , P (ω | μl (Cl (ω)x) ≥ hl ) ≥ θl for all C¯ l ∈ C¯ lα ⇔ P ω | Clα and, from the assumption that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X and the property of the nonincreasing function μl and the nondecreasing function Tl , one ﬁnds that βl − dl1 )x + μl (hl ) (L (α)β L ¯ ≥ θl P ω | Clα x ≤ μl (hl ) ≥ θl ⇔ Tl dl2 x ⇔

βl − dl1 )x + μl (hl ) (L (α)β ≥ Tl (θl ) , dl2 x

5.2 Noncooperative two-level programming

213

where μl and Tl are pseudo inverse functions of μl and Tl , respectively. Furthermore, assuming that the fuzzy goals are quantiﬁed by the linear membership functions deﬁned as ⎧ ⎪ 1 if zl < z1l ⎪ ⎪ ⎨ z − z0 l l (5.99) μl (zl ) = if z1l ≤ zl ≤ z0l 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if zl > z0l , (5.98) can be rewritten as maximize h1 x1 , h1

where x2 solves maximize h2 x2 , h2

subject to

β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β ≥ h1 z01 − z11

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ β2 − d21 − T2 (θ2 )d22 )x + z02 (L (α)β ⎪ ⎪ ≥ h 2⎪ ⎪ 0 1 ⎪ z2 − z2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.100)

Because maximizing h1 is equivalent to maximizing β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β , z01 − z11 (5.100) can be equivalently transformed into ⎫ F β1 − d11 − T1 (θ1 )d12 x ⎪ maximize Z1α (x1 , x2 ) = L (α)β ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎬ F 1 2 β2 − d2 − T2 (θ2 )d2 x maximize Z2α (x1 , x2 ) = L (α)β x2 ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.101)

(5.102)

It should be noted that (5.102) is an ordinary two-level linear programming problem, and therefore it can be solved by the solution technique for obtaining a Stackelberg solution given in Chapter 2.

5.2.2.3 Numerical example In order to demonstrate the feasibility and efﬁciency of the presented computational method for obtaining a Stackelberg solution for the possibility-based probability

214

5 Stochastic and Fuzzy Random Two-Level Programming

model, consider the following numerical example to fuzzy random two-level linear programming problems: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜ ˜ ¯ 21 x1 + C ¯ 22 x2 minimize z2 (x1 , x2 ) = C (5.103) x2 ⎪ ⎪ subject to ai1 x1 + ai2 x2 ≤ bi , i = 1, . . . , 5 ⎪ ⎪ ⎪ ⎪ ⎪ x1 = (x11 , x12 , x13 )T ≥ 0 ⎪ ⎭ T x2 = (x21 , x22 , x23 ) ≥ 0, ˜¯ , l = 1, 2, j = 1, 2 are vectors of fuzzy random variables C¯˜ , k = where C lj l jk ˜¯ is rep1, 2, . . . , n j which are characterized by (5.18). The parameters d¯l jk of C lj resented by d¯l jk = dl1jk + t¯l dl2jk , and t¯l is a Gaussian random variable with the mean 0 and the variance 12 . Values of coefﬁcients in constraints are shown in Table 5.13, and values of the parameters d¯ l1 , d¯ l2 , βl and γl , l = 1, 2 of the fuzzy random variables are shown in Table 5.14. Table 5.13 Values of coefﬁcients in constraints. x11 x12 x13 a1 4 3 1 a2 2 4 5 a3 3 2 3 0 0 0 a4 a5 1 1 1

x21 3 3 3 1 0

x22 2 4 3 1 0

x23 5 2 2 1 0

b 100 115 90 17 22

Table 5.14 Values of dl1 , dl2 , β l and γ l .

d11 d21 d12 d22

β1 β2 γ1 γ2

x11

x12

x13

x21

x22

x23

−1.7 −1.8 0.6 0.7 0.3 0.4 0.3 0.4

−2.4 −1.6 0.1 0.2 0.2 0.2 0.2 0.2

−2.9 −1.2 0.3 0.5 0.6 0.3 0.6 0.3

−1.5 −2.0 0.4 0.1 0.2 0.5 0.2 0.5

−1.0 −2.4 0.7 0.5 0.7 0.2 0.7 0.2

−1.3 −2.8 0.2 0.3 0.4 0.8 0.4 0.8

By solving the linear programming problem (5.94), the initial extreme point (xT1[1] , xT2[1] )T = (0.00, 22.0, 0.00, 0.00, 6.80)T is obtained. / and solve the linear programming problem Let W := {(xT1[1] , xT2[1] )T } and U := 0, (5.96) in order to obtain the rational response for x1[1] . Since the optimal solution w˜ 2[1] /u[1] = (9.00, 0.00, 0.00)T to (5.96) is not equal to x2[1] = (0.00, 0.00, 6.80)T , the current extreme point (xT1[1] , xT2[1] )T is not a Stackelberg solution. Then, we

5.2 Noncooperative two-level programming

215

enumerate feasible extreme points (xT1 , xT2 )T which are adjacent to (xT1[1] , xT2[1] )T

and satisfy Z1Π,P (x1 , x2 ) ≤ Z1Π,P (x1[1] , x2[1] ), and make W[1] . After setting U := U ∪(xT1[1] , xT2[1] )T and W := (W ∪W[1] )\U, we ﬁnd a feasible extreme point (xT1 , xT2 )T in W such that Z1Π,P (x1 , x2 ) is the greatest, and let the point be the next extreme point (xT1[i+1] , xT2[i+1] )T to be examined. By repeating this procedure, we ﬁnally obtain the following Stackelberg solution: (xT1 , xT2 )T = (0.00, 14.56, 7.44, 0.00, 0.00, 9.78)T .

5.2.3 Extensions to integer programming Considering that the real-world decision making problems are often formulated as integer programming problems, as natural extensions of noncooperative stochastic two-level programming problems with continuous variables discussed in the previous section, we present two-level integer programming problems involving random variable coefﬁcients in objective functions and the right-hand side of constraints formulated as ⎫ minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2 (5.104) ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b¯ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, A j , j = 1, 2 are m × n j coefﬁcient matrices, c¯ l j , l = 1, 2, j = 1, 2 are n j dimensional Gaussian random variable row vectors with the mean vectors c¯ l j and the variance-covariance matrices. b¯ is an m dimensional column vector whose elements are independent random variables with continuous and nondecreasing probability distribution function. By introducing the chance constrained conditions (Charnes and Cooper, 1959) instead of the original constraints, (5.104) can be rewritten as minimize z1 (x1 , x2 ) = c¯ 11 x1 + c¯ 12 x2 x1

where x2 solves minimize z2 (x1 , x2 ) = c¯ 21 x1 + c¯ 22 x2 x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ,

(5.105)

216

5 Stochastic and Fuzzy Random Two-Level Programming

where η = (η1 , . . . , ηm )T is a vector of the satisﬁcing probability levels for holding η) = (b1 (η1 ), . . . , bm (ηm ))T is deﬁned by (5.3). the constraints, and b(η

5.2.3.1 Expectation model for stochastic two-level integer programming problems Assuming that the DMs intend to simply minimize the expectation of their own objective functions, the stochastic two-level integer programming problem (5.104) can be reformulated as ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ E minimize z2 (x1 , x2 ) = E[¯c21 ]x1 + E[¯c22 ]x2 (5.106) x2 ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where E denotes the expectation operator. For a decision xˆ 1 by DM1, the rational response of DM2 is obtained by an optimal solution x2 of the following problem: ⎫ ⎪ minimize zE1 (ˆx1 , x2 ) = E[¯c11 ]ˆx1 + E[¯c12 ]x2 ⎬ x2 (5.107) η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . Since (5.107) is a single-objective integer programming problem with the linear objective function and constraints, we employ GADSLPRRSU to solve it. Let R(x1 ) be a set of rational responses of DM2, i.e., the set of optimal solutions x∗2 to (5.107). Then, the Stackelberg solution for the expectation model is an optimal solution (x∗1 , x∗2 ) to the following problem: ⎫ minimize zE1 (x1 , x2 ) = E[¯c11 ]x1 + E[¯c12 ]x2 ⎬ η) subject to A1 x1 + A2 x2 ≤ b(η (5.108) ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ). In order to obtain a Stackelberg solution, x1 must be selected so as to minimize zE1 (x1 , x2 ) under the condition that x2 is selected from among R(x1 ). With this observation in mind, the Stackelberg solution can be obtained by nesting use of the two genetic algorithms: one is a genetic algorithm for ﬁnding a rational response x2 of DM2 for a given decision x1 of DM1, and the other is a genetic algorithm for ﬁnding a decision x1 of DM1 so as to minimize zE1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ).

5.2 Noncooperative two-level programming

217

5.2.3.2 Variance model for stochastic two-level integer programming problems In order to consider each DM’s concern about the ﬂuctuation of the realized objective function values, from the viewpoint of leveling of the minimized objective function values, by introducing the constraints that the expectation of each objective function must be less than or equal to a certain value, we consider the variance minimization model for noncooperative stochastic two-level integer programming problems formulated as minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T x1

where x2 solves minimize zV2 (x1 , x2 ) = (xT1 , xT2 )V2 (xT1 , xT2 )T x2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

η) subject to A1 x1 + A2 x2 ≤ b(η ⎪ ⎪ E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ,

(5.109)

where V1 and V2 are positive-deﬁnite variance-covariance matrices of (¯c11 , c¯ 12 ) and (¯c21 , c¯ 22 ), and γ1 and γ2 are the permissible expectation levels speciﬁed by DM1 and DM2, respectively. For a decision xˆ 1 by DM1, the rational response of DM2 is obtained by an optimal solution x2 of the following problem: ⎫ minimize zV2 (ˆx1 , x2 ) = (ˆxT1 , xT2 )V2 (ˆxT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 xˆ 1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]ˆx1 + E[¯c12 ]x2 ≤ γ1 (5.110) ⎪ ⎪ E[¯c21 ]ˆx1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . To solve (5.110) with the quadratic objective function and linear constraints, we use the revised GADSLPRRSU. Let R(x1 ) be a set of rational responses of DM2, i.e., the set of optimal solutions x∗2 to (5.110). Then, the Stackelberg solution for the variance model is an optimal solution (x∗1 , x∗2 ) to the following problem: ⎫ minimize zV1 (x1 , x2 ) = (xT1 , xT2 )V1 (xT1 , xT2 )T ⎪ ⎪ ⎪ ⎪ η) subject to A1 x1 + A2 x2 ≤ b(η ⎬ E[¯c11 ]x1 + E[¯c12 ]x2 ≤ γ1 ⎪ ⎪ ⎪ E[¯c21 ]x1 + E[¯c22 ]x2 ≤ γ2 ⎪ ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ).

(5.111)

Similarly to solving (5.108) in the expectation model, to ﬁnd a Stackelberg solution, we nest two genetic algorithms: one searches a rational response x2 of DM2 for

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5 Stochastic and Fuzzy Random Two-Level Programming

a given decision x1 of DM1, and the other ﬁnds a decision x1 of DM1 minimizing zV1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ). 5.2.3.3 Possibility-based probability model for fuzzy random two-level integer programming problems In previous subsections, it is assumed that uncertain parameters or coefﬁcients involved in the formulated two-level programming problems are given as random variables. However, it would be signiﬁcant to realize that the possible realized values of these random parameters are often only ambiguously known to the experts. For handling such a hierarchical decision making problem, we deal with two-level integer programming problems involving fuzzy random variable coefﬁcients in objective functions formulated as ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜¯ x + C ˜¯ x minimize z2 (x1 , x2 ) = C 21 1 22 2 (5.112) x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 , where x1 is an n1 dimensional decision variable column vector for DM1, x2 is an n2 dimensional decision variable column vector for DM2, z1 (x1 , x2 ) is the objective function of DM1 and z2 (x1 , x2 ) is the objective function of DM2. ˜¯ = For simplicity, let X int denote the feasible region of (5.112), and let C l ˜¯ ˜¯ , C T T T (C l1 l2 ) and x = (x1 , x2 ) . As we examined in the possibility-based probability model for two-level programming problems with continuous decision variables in 5.1.2.1, assume that el˜¯ , l = 1, 2, j = 1, 2 are fuzzy ements C˜¯l jk , k = 1, 2, . . . , n j of coefﬁcient vectors C lj random variables whose realized value for each elementary event ω is characterized by the membership function (5.18). Then, it follows that each objective function ˜¯ x = C ˜¯ x + C ˜¯ x is represented by a single fuzzy random variable whose realC l l1 1 l2 2 ized value for an elementary event ω is an L-R fuzzy number characterized by the membership function ⎧ dl (ω)x − υ ⎪ ⎪ L if υ ≤ dl (ω)x ⎪ ⎨ βl x μC˜ l (ω)x (υ) = ⎪ υ − dl (ω)x ⎪ ⎪ if υ > dl (ω)x, ⎩R γl x βl1 , β l2 ) and γ l = (γγl1 , γ l2 ), l = 1, 2. where dl (ω) = (dl1 (ω), dl2 (ω)), β l = (β

5.2 Noncooperative two-level programming

219

Considering the imprecise nature of DMs’ judgments, we introduce a fuzzy goal G˜ l such as “zl (x1 , x2 ) should be substantially less than or equal to a certain value” which is quantiﬁed by a membership function μG˜ l . Furthermore, recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the objective function value C˜ l (ω)x attains the fuzzy goal G˜ l for an elementary event ω is expressed as (5.113) ΠC˜ l (ω)x (G˜ l ) = sup min μC˜ l (ω)x (y), μG˜ l (y) , l = 1, 2. y

It should be noted here that the degree of possibility ΠC˜ l (ω)x (G˜ l ) varies randomly due to the stochastic occurrence of an elementary event ω. With this observation in mind, along the same line as the possibility-based probability model discussed in Chapter 4, assuming that the DMs are willing to maximize the probability that ΠC˜ l (ω)x (G˜ l ) is greater than or equal to target value hl , (5.112) is reformulated as maximize P ω ΠC˜ 1 (ω)x (G˜ 1 ) ≥ h1 x1

where x2 solves maximize P ω ΠC˜ 2 (ω)x (G˜ 2 ) ≥ h2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.114)

Assuming that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, where X denotes the feasible region of (5.114), the objective functions in (5.114) can be equivalently transformed as # " βl − dl1 )x + μG˜ (hl ) (L (hl )β l . P ω ΠC˜ l (ω)x (G˜ l ) ≥ hl = Tl dl2 x Since the distribution functions Tl , l = 1, 2 are nondecreasing functions, (5.114) is equivalently transformed into ⎫ β1 − d11 )x + μG˜ (h1 ) ⎪ (L (h1 )β Π,P ⎪ 1 ⎪ maximize Z1 (x1 , x2 ) ⎪ ⎪ x1 d12 x ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ 1 β (h )β − d )x + μ (h ) (L 2 2 ˜ 2 2 G Π,P 2 (5.115) maximize Z2 (x1 , x2 ) ⎪ ⎪ x2 d22 x ⎪ ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 . For any feasible decision xˆ 1 of DM1, a rational response is given as an optimal solution to the single-level integer programming problem

220

5 Stochastic and Fuzzy Random Two-Level Programming 1 )x + (L (h )β 1 x + μ (h ) ⎫ β22 − d22 (L (h2 )β 2 2 β 21 − d21 )ˆ 1 G˜ 2 2 ⎪ ⎪ ⎬ maximize 2 2 x2 d22 x2 + d21 xˆ 1 ⎪ subject to A2 x2 ≤ b − A1 xˆ 1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.116)

Let R(ˆx1 ) denote a set of rational responses of DM2 for a decision xˆ 1 by DM1 and IR = {(x1 , x2 ) | (x1 , x2 ) ∈ X, x2 ∈ R(x1 )} the inducible region. Then, since DM1 selects a solution (x1 , x2 ) maximizing Z1Π,P (x1 , x2 ) from among the inducible region IR, the Stackelberg solution can be obtained by solving ⎫ ⎪ maximize Z1Π,P (x1 , x2 ) ⎬ x1 (5.117) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 , x2 ∈ R(x1 ). To solve (5.117), we employ two genetic algorithms which are nested: one searches a rational response x2 of DM2 for a given decision x1 of DM1, and the other seeks a decision x1 of DM1 maximizing zV1 (x1 , x2 ) under the condition of x2 ∈ R(x1 ). 5.2.3.4 Level set-based fractile model for fuzzy random two-level integer programming problems Along the same line as the level set-based optimization models discussed in Chapter 4, assuming that DM1 decides that the degree of all of the membership functions of the fuzzy random variables involved in (5.112) should be greater than or equal to some value α, the original fuzzy random two-level programming problem (5.112) is reformulated as ⎫ minimize C¯ 1 x = C¯ 11 x1 + C¯ 12 x2 ⎪ ⎪ x1 , C¯ 1 ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ¯ ¯ ¯ ⎪ minimize C2 x = C21 x1 + C22 x2 ⎬ ¯ x2 , C2 (5.118) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎪ ⎪ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ⎪ ⎪ ⎭ C¯ 1 = (C¯ 11 , C¯ 12 ) ∈ C¯ 1α , C¯ 2 = (C¯ 21 , C¯ 22 ) ∈ C¯ 2α , where C¯ lα = (C¯ l1α , C¯ l2α ), l = 1, 2 are α-level sets deﬁned as the Cartesian product of α-level sets C˜¯l jkα of fuzzy random variables C˜¯l jk , j = 1, 2, k = 1, 2, . . . , n j . In order to consider the imprecise nature of the DMs’ judgments, assuming that each DM has a fuzzy goal for the objective function in the α-stochastic two-level integer programming problem (5.118), we introduce the nonincreasing and continuous membership functions μ1 and μ2 to quantify the fuzzy goals of DM1 and DM2 for their own objective function values, respectively. Then, observing that the satisfaction degrees μl (Cl (ω)x), l = 1, 2 vary randomly due to the stochastic occurrence of

5.2 Noncooperative two-level programming

221

an elementary event ω, it is evident that computational techniques for conventional two-level programming problems cannot be applied. With this observation in mind, assuming that the DMs intend to maximize the target values for the satisfaction degrees under the probabilistic constraint with permissible probability levels, we consider the level set-based fractile model for noncooperative two-level fuzzy random integer programming problem formulated as ⎫ maximize h1 ⎪ ⎪ x1 , h1 , C¯ 1 ⎪ ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎪ ⎪ maximize h2 ⎪ ⎪ ¯ ⎪ x2 , h2 , C2 ⎪ ⎬ subject to P (ω | μ1 (C1 (ω)x) ≥ h1 ) ≥ θ1 (5.119) P (ω | μ2 (C2 (ω)x) ≥ h2 ) ≥ θ2 ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎪ ⎪ ⎪ ⎪ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 ⎪ ⎪ ⎭ ¯ ¯ ¯ ¯ C1 ∈ C1α , C2 ∈ C2α , where hl and θl , l = 1, 2 are decision variables representing the target values and the permissible probability levels speciﬁed by DMl, l = 1, 2, respectively. If we assume that dl2 x > 0, l = 1, 2 for any feasible solution x ∈ X, where X denotes the feasible region of (5.119), and the property of the nonincreasing function μl and the nondecreasing function Tl , one ﬁnds that together with βl − dl1 )x + μl (hl ) (L (α)β L ≥ Tl (θl ) , P ω | μl (C¯ lα x) ≥ hl ≥ θl ⇔ dl2 x where μl and Tl are pseudo inverse functions of μl and Tl , respectively. Furthermore, assuming that the fuzzy goals are quantiﬁed by linear membership functions deﬁned as ⎧ ⎪ 1 if zl < z1l ⎪ ⎪ ⎨ z − z0 l l (5.120) if z1l ≤ zl ≤ z0l μl (zl ) = 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if zl > z0l , (5.119) can be rewritten as

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5 Stochastic and Fuzzy Random Two-Level Programming

maximize h1 x1 , h1

where x2 solves maximize h2 x2 , h2

subject to

β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β ≥ h1 z01 − z11

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ β2 − d21 − T2 (θ2 )d22 )x + z02 (L (α)β ⎪ ⎪ ≥ h 2⎪ ⎪ 0 1 ⎪ z2 − z2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(5.121)

Because maximizing h1 is equivalent to maximizing β1 − d11 − T1 (θ1 )d12 )x + z01 (L (α)β , z01 − z11 (5.121) can be equivalently transformed as ⎫ F (x , x ) = L (α)β β1 − d11 − T1 (θ1 )d12 x ⎪ maximize Z1α 1 2 ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ F 1 2 β2 − d2 − T2 (θ2 )d2 x maximize Z2α (x1 , x2 ) = L (α)β x2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , ν1 j1 }, j1 = 1, . . . , n1 ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , ν2 j2 }, j2 = 1, . . . , n2 .

(5.122)

Observing that (5.122) is the same type of problem as (5.106), a Stackelberg solution of (5.122) can be obtained by using a computational method for (5.106).

5.2.3.5 Numerical example In order to demonstrate the feasibility and efﬁciency of the proposed computational methods for obtaining a Stackelberg solution for the possibility-based probability model, consider the following numerical example to fuzzy random two-level integer programming problems: ⎫ ˜¯ x + C ˜¯ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ x1 ⎪ ⎪ ⎪ where x2 solves ⎪ ⎪ ⎬ ˜ ˜ ¯ ¯ minimize z2 (x1 , x2 ) = C21 x1 + C22 x2 (5.123) x2 ⎪ ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎪ x1 j1 ∈ {0, 1, . . . , 30}, j1 = 1, . . . , 15 ⎪ ⎪ ⎭ x2 j2 ∈ {0, 1, . . . , 30}, j2 = 1, . . . , 15,

5.2 Noncooperative two-level programming

223

˜¯ , l = 1, 2, j = 1, 2 are vectors whose elements C¯˜ , k = 1, 2, . . . , n are where C j lj l jk fuzzy random variables, and the random parameter t¯l of C˜¯l jk is a uniform random variable distributed at the closed interval [−10, 10]. Elements of dl1 , dl2 , β l and γ l , l = 1, 2 are randomly chosen from the closed intervals [−500, 499], [0, 50], [0, 25] and [0, 25], respectively. The values of bi , i = 1, . . . , 26 are determined by ⎞ ⎛ bi =

∑− 30ai j + 5 ⎝ ∑+ ai j − ∑− ai j ⎠ ,

j∈Jai

j∈Jai

j∈Jai

where Ja+i { j | ai j > 0, 1 ≤ j ≤ 30} and Ja−i { j | ai j < 0, 1 ≤ j ≤ 30}. Assume that the DM1 speciﬁes θ1 = θ2 = 0.7 and α = 0.75. Then, by the computational method using the genetic algorithm, the (approximate) Stackelberg solution x∗ for the level set-based fractile model is obtained, and the correspondF (x∗ ) = 0.838054 and ing optimal objective function values are calculated as Z1α F ∗ Z2α (x ) = 0.497792.

Chapter 6

Future Research Directions

This chapter outlines some future research directions. In contrast to the fuzzy random variables treating the ambiguity of realized values of the random parameters discussed in Chapter 4, by considering the ambiguity of means and/or variances characterizing random variables, the concept of random fuzzy variables is introduced to express the coefﬁcients involved in mathematical programming problems under fuzzy stochastic environments. In particular, attention is focused on the case where the mean of each of random variables is represented by a fuzzy number. From the viewpoint of simultaneous maximization of the possibility and the probability under some speciﬁed target value for the objective function, the linear programming problem involving random fuzzy variable coefﬁcients is transformed into a deterministic nonlinear programming problem. Furthermore, assuming that the decision maker intends to optimize the target variable under the permissible possibility level, it is shown that the original random fuzzy linear programming problem is equivalently transformed into a convex programming problem. The presented decision making models are extended to deal with multiobjective and two-level programming problems with random fuzzy variables.

6.1 Random fuzzy variable In the framework of stochastic programming, as discussed in Chapter 3, it is implicitly assumed that the uncertain parameter which well represents the stochastic factor of real systems can be deﬁnitely expressed as a single random variable. However, from the expert’s experimental point of view, the experts may think of a collection of random variables to be appropriate to express stochastic factors rather than only a single random variables. In this case, reﬂecting the expert’s conviction degree that each of random variables properly represents the stochastic factor, it would be quite reasonable to assign the different degrees of possibility to each of random variables. For handling such an uncertain parameter, a random fuzzy variable was deﬁned by Liu (2002) as a function from a possibility space to a collection of random variables, M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9_6, © Springer Science+Business Media, LLC 2011

225

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6 Future Research Directions

which is considered to be an extended concept of fuzzy variable (Nahmias, 1978). It should be noted here that the fuzzy variables can be viewed as another way of dealing with the imprecision which was originally represented by fuzzy sets. Although we can employ Liu’s deﬁnition, considering that we have consistently discussed various concepts in relation to the fuzzy sets in the previous chapters, we deﬁne the random fuzzy variables by extending not the fuzzy variables but the fuzzy sets. Deﬁnition 6.1 (Random fuzzy variable). Let Γ be a collection of random variables. Then, a random fuzzy variable C¯˜ is deﬁned by its membership function μC¯˜ : Γ → [0, 1].

(6.1)

In Deﬁnition 6.1, the membership function μC¯˜ assigns each random variable γ¯ ∈ Γ to a real number μC¯˜ (¯γ). It should be noted here that if Γ is deﬁned as R, then (6.1) becomes equivalent to the membership function of an ordinary fuzzy set discussed in (2.1). In this sense, a random fuzzy variable can be regarded as an extended concept of fuzzy sets. On the other hand, if Γ is deﬁned as a singleton Γ = {¯γ} and μC¯˜ (¯γ) = 1, then the corresponding random fuzzy variable C¯˜ can be viewed as an ordinary random variable. When taking account of the imprecise nature of the realized values of random variables, as discussed in Chapter 4, it would be appropriate to employ the concept of fuzzy random variables. However, it should be emphasized here that if mean and/or variance of random variables are speciﬁed by the expert as a set of real values or fuzzy sets, such uncertain parameters can be represented by not fuzzy random variables but random fuzzy variables. As a simple example of random fuzzy variables, we consider a Gaussian random variable whose mean value is not deﬁnitely speciﬁed as a constant. For example, when some random parameter γ¯ is represented by the Gaussian random variable N(si , 102 ) where the expert identiﬁes a set {s1 , s2 , s3 } of possible mean values as (s1 , s2 , s3 ) = (90, 100, 110), if the membership function μC¯˜ is deﬁned by ⎧ 0.5 ⎪ ⎪ ⎪ ⎨0.7 μC¯˜ (¯γ) = ⎪ 0.3 ⎪ ⎪ ⎩ 0

if γ¯ ∼ N(90, 102 ) if γ¯ ∼ N(100, 102 ) if γ¯ ∼ N(110, 102 ) otherwise,

then C¯˜ is a random fuzzy variable. More generally, when the mean values are expressed as fuzzy sets or fuzzy numbers, the corresponding random variable with the fuzzy mean is represented by a random fuzzy variable, as we will discuss in the following sections.

6.2 Random fuzzy linear programming

227

6.2 Random fuzzy linear programming In Chapter 4, considering the imprecise nature of human judgments for the realized values of random variable coefﬁcients, we have discussed mathematical programming problems with fuzzy random variable coefﬁcients. However, if the mean values of random variables are expressed as fuzzy numbers, it is not hard to imagine that the methodology of fuzzy random programming is not applicable. With this observation in mind, for handling such random variable coefﬁcients with fuzzy mean values involved in objective functions, we consider the random fuzzy version of linear programming problems formulated as ¯˜ minimize Cx (6.2) subject to Ax ≤ b, x ≥ 0, where x is an n-dimensional decision variable column vector, A is an m × n matrix, and b is an m-dimensional column constant vector. Considering that the real data with uncertainty are often distributed normally, from the practical point of view, we assume that each C¯˜ j of the coefﬁcient vector ¯˜ = (C¯˜ , . . . , C¯˜ ) is a Gaussian random variable whose mean value is represented C 1 n by a fuzzy number. To be more explicit, C¯˜ j , j = 1, . . . , n are mutually independent Gaussian random variable N(M˜ j , σ2j ) where the mean value M˜ j is expressed as a fuzzy set characterized by the membership function ⎧ mj −t ⎪ ⎪ if m j ≥ t ⎪ ⎨L αj μM˜ j (t) = (6.3) ⎪ t −mj ⎪ ⎪ ⎩R if m j < t, βj where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, the membership function of C¯˜ j is given as μC¯˜ (¯γ j ) = {μM˜ j (s j ) | γ¯ j ∼ N(s j , σ2j )}, ∀¯γ j ∈ Γ. j

(6.4)

¯˜ can be exThrough the Zadeh’s extension principle, the objective function Cx pressed as a random fuzzy variable with the membership function n (6.5) μCx ¯ = sup min μC¯˜ (¯γ j ) u¯ = ∑ γ¯ j x j , ∀u¯ ∈ Γ, ¯˜ (u) j 1≤ j≤n γ¯ j=1 where γ¯ = (¯γ1 , . . . , γ¯ n ). It should be noted here that if each γ¯ j is a Gaussian random variable and x j , j = 1, . . . , n are ﬁxed as real values, then u¯ = ∑nj=1 γ¯ j x j is also a Gaussian random variable due to its reproductive property.

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6 Future Research Directions

By substituting (6.4) into (6.5), it follows that n 2 μCx ¯ = sup min μM˜ j (s j ) γ¯ j ∼ N(s j , σ j ), u¯ = ∑ γ¯ j x j ¯˜ (u) 1≤ j≤n s j=1 # " n n 2 2 , = sup min μM˜ j (s j ) u¯ ∼ N ∑ s j x j , ∑ σ j x j 1≤ j≤n s j=1 j=1

(6.6)

where s = (s1 , . . . , sn ). ¯˜ is represented by a random fuzzy variable with the membership Realizing that Cx function μCx ¯˜ deﬁned as (6.6), it is evident that conventional fuzzy and stochastic programming approaches, even the fuzzy random programming approach discussed in Chapter 4, cannot be directly applicable.

6.2.1 Possibility-based probability model Assuming that the decision maker (DM) concerns about the probability that the objective function is smaller than or equal to a certain target value, we consider the ˜ probability P ω C(ω)x ≤ f , where P is a probability measure, and f is a target ¯˜ value for the objective function Cx. Through the Zadeh’s extension principle, the ˜ probability P ω C(ω)x ≤ f is expressed by a fuzzy set P˜ with the membership function ¯ | p = P (ω | u(ω) ≤ f ) . μP˜ (p) = sup μCx ¯˜ (u)

(6.7)

u¯

By substituting (6.6) into (6.7), the membership function μP˜ is rewritten as n n 2 2 μP˜ (p) = sup min μM˜ j (s j ) p = P (ω | u(ω) ≤ f ) , u¯ ∼ N ∑ s j x j , ∑ σ j x j . s 1≤ j≤n j=1 j=1 Considering the imprecise nature of the DM’s judgments for the probability with respect to the random fuzzy objective function value, we introduce a fuzzy goal such as “P˜ should be greater than or equal to a certain value,” which is assumed to be quantiﬁed by a nondecreasing continuous membership function μG˜ . Having determined the fuzzy goal of the DM, if we regard μP˜ (p) as a possibility distribution on the basis of the concept of possibility measure, the degree of possibility that the fuzzy goal G˜ is fulﬁlled under the possibility distribution μP˜ (p) is given as ˜ = sup min{μP˜ (p), μG˜ (p)}. (6.8) ΠP˜ (G) p

Then, the original random fuzzy linear programming problem (6.2) is interpreted as the possibility-based probability maximization problem

6.2 Random fuzzy linear programming

229

˜ maximize ΠP˜ (G) subject to Ax ≤ b, x ≥ 0 or equivalently

(6.9)

⎫ maximize h ⎬ ˜ ≥h subject to ΠP˜ (G) ⎭ Ax ≤ b, x ≥ 0.

(6.10)

˜ ≥ h in (6.10) is equivalently replaced by the From (6.8), the constraint ΠP˜ (G) condition that there exists a p such that μP˜ (p) ≥ h and μG˜ (p) ≥ h, namely, n n 2 2 ≥h sup min μM˜ j (s j ) p = P (ω | u(ω) ≤ f ) , u¯ ∼ N ∑ s j x j , ∑ σ j x j s 1≤ j≤n j=1 j=1 (6.11) and p ≥ μG˜ (h), where s = (s1 , . . . , sn ) and μG˜ (h) is a pseudo inverse function deﬁned ¯ such as μG˜ (h) = sup{p | μG˜ (p) ≥ h}. This implies that there exists a vector (p, s, u) that # " min μM˜ j (s j ) ≥ h, u¯ ∼ N

1≤ j≤n

n

n

j=1

j=1

∑ s j x j , ∑ σ2j x2j

, p = P (ω | u(ω) ≤ f ) , p ≥ μG˜ (h),

which can be equivalently transformed into the condition that there exists a vector (s, u) ¯ such that # " μM˜ j (s j ) ≥ h, j = 1, . . . , n, u¯ ∼ N

n

n

j=1

j=1

∑ s j x j , ∑ σ2j x2j

, P (ω | u(ω) ≤ f ) ≥ μG˜ (h). (6.12)

In view of (6.3), it follows that μM˜ j (s j ) ≥ h ⇔ s j ∈ [m j − L (h)α j , m j + R (h)β j ], where L (h) and R (h) are pseudo inverse functions deﬁned as L (h) = sup{t | L(t) ≥ h} and R (h) = sup{t | R(t) ≥ h}. Hence, (6.12) is rewritten as the equivalent condition that there exists a u¯ such that " # P (ω | u(ω) ≤ f ) ≥ μG˜ (h), u¯ ∼ N

n

n

j=1

j=1

∑ {m j − L (h)α j }x j , ∑ σ2j x2j

.

Since P (ω | u(ω) ≤ f ) is transformed into ⎛ n n f − ∑ {m j − L (h)α j }x j ⎜ u(ω) − ∑ {m j − L (h)α j }x j ⎜ j=1 j=1 0 0 ≤ P⎜ ⎜ω n n ⎝ 2 2 σjxj σ2j x2j ∑ ∑ j=1

j=1

(6.13)

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

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6 Future Research Directions

in consideration of n

u¯ − ∑ {m j − L (h)α j }x j j=1

0

∼ N(0, 1),

n

∑ σ2j x2j

j=1

(6.13) is equivalently transformed as ⎛ n

⎞

⎜ f − ∑ {m j − L (h)α j }x j ⎟ ⎟ ⎜ j=1 ⎟ ≥ μ˜ (h), 0 Φ⎜ ⎟ ⎜ G n ⎠ ⎝ 2 2 ∑ σjxj

(6.14)

j=1

where Φ is a probability distribution function of the standard Gaussian random variable N(0, 1). From the monotone increasingness of Φ, (6.14) is rewritten as 0 n

n

j=1

j=1

∑ {m j − L(h)α j }x j + Φ−1 (μG˜ (h)) ∑ σ2j x2j ≤ f ,

(6.15)

where Φ−1 is the inverse function of Φ. From (6.11)−(6.15), it holds that ˜ ≥h⇔ ΠP˜ (G)

0

n

∑ {m j − L (h)α j }x j + Φ

−1

j=1

n

∑ σ2j x2j ≤ f .

(μG˜ (h))

(6.16)

j=1

Consequently, (6.10) is equivalently transformed into maximize h n

subject to

0

∑ {m j − L (h)α j }x j + Φ

j=1

Ax ≤ b, x ≥ 0.

−1

(μG˜ (h))

n

⎫ ⎪ ⎪ ⎪ ⎬

∑ σ2j x2j ≤ f ⎪

j=1

⎪ ⎪ ⎭

(6.17)

Noting that this problem is a nonconvex programming problem, an (approximate) optimal solution of (6.17) is obtained through some nonlinear programming techniques or meta heuristic approaches such as genetic algorithms.

6.2 Random fuzzy linear programming

231

6.2.2 Possibility-based fractile model The possibility-based probability for the DM who prefers model is recommended ˜ to maximize the probability P ω C(ω)x ≤ f for the speciﬁed target value f . On ¯˜ under the other hand, if the DM would like to minimize the objective function Cx the condition that the degree of possibility with respect to the attained probability ¯˜ formulated as ˜ P ω C(ω)x ≤ f , we consider the problem of minimizing f for Cx ⎫ minimize f ⎬ ˜ ≥h subject to ΠP˜ (G) (6.18) ⎭ Ax ≤ b, x ≥ 0, where h is a permissible possibility level speciﬁed by the DM. Recalling the equivalent transformation (6.16), (6.18) is transformed into minimize f

0

n

∑ {m j − L (h)α j }x j + Φ

subject to

−1

j=1

(μG˜ (h))

n

∑ σ2j x2j ≤ f

j=1

Ax ≤ b, x ≥ 0 or equivalently n

minimize

0

∑ {m j − L (h)α j }x j + Φ

j=1

subject to Ax ≤ b, x ≥ 0.

−1

(μG˜ (h))

n

∑

j=1

σ2j x2j

⎫ ⎪ ⎬ ⎪ ⎭

(6.19)

From the viewpoint of reliability assurance, considering that the DM often feels that the attained probability p should be greater than or equal to at least 1/2, it would be natural to assume that μG˜ (p) ≥ 0 for any p ≥ 1/2, which implies that Φ−1 (μG˜ (h)) ≥ 0 holds for any h. Then, (6.19) is a convex programming problem, and it can be solved by convex programming techniques such as the sequential quadratic programming method. So far, we have discussed linear programming problems in which the coefﬁcients of the objective function are represented by random fuzzy variables. In the near future, we will extend the proposed models to more general cases where not only the objective function but also the constraints involve random fuzzy variables coefﬁcients. Furthermore, extensions to integer problems as well as developments of expectation and variance models will be considered elsewhere.

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6.3 Multiobjective random fuzzy programming In the previous section, we have dealt with random fuzzy linear programming problems with a single objective function. However, most of the real-world decision making problems usually involve multiple, noncommensurable, and conﬂicting objectives which should be considered simultaneously. In this section, as a multiobjective version of random fuzzy linear programming problems, we consider the multiobjective random fuzzy programming problems formulated as ⎫ ¯˜ x minimize C ⎪ 1 ⎪ ⎬ ··· (6.20) ¯˜ x ⎪ minimize C k ⎪ ⎭ subject to Ax ≤ b, x ≥ 0 , where x is an n-dimensional decision variable column vector, A is an m × n matrix, ¯˜ = (C¯˜ , . . . , C¯˜ ) is a random fuzzy b is an m × 1 constant column vector, and C l l1 ln variable coefﬁcient vector. Here, assume that C¯˜l j is a Gaussian random variable whose mean value is a fuzzy number M˜ l j characterized by the membership function ⎧ ml j − t ⎪ ⎪ if ml j ≥ t ⎪ ⎨L αl j (6.21) μM˜ l j (t) = ⎪ t − ml j ⎪ ⎪ ⎩R if ml j < t, βl j where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, the membership function of C¯˜l j is expressed as μC¯˜ (¯γl j ) = {μM˜ l j (sl j )|¯γl j ∼ N(sl j , σ2l j )}, ∀¯γl j ∈ Γ. lj

(6.22)

¯˜ x is expressed Using the Zadeh’s extension principle, each objective function C l as a random fuzzy variable characterized by the membership function n (6.23) μC¯˜ l x (u¯l ) sup min μC¯˜ (¯γl j ) u¯l = ∑ γ¯ l j x j , ∀u¯l ∈ Γ, 1≤ j≤n l j γ¯l j=1 where γ¯ l = (¯γ1 , . . . , γ¯ n ). By substituting (6.22) into (6.23), the membership function of a random fuzzy ¯˜ x in (6.20) is rewritten as variable corresponding to the objective function C l n n 2 2 μC¯˜ l x (u¯l ) = sup min μM˜ l j (sl j ) u¯l ∼ N ∑ sl j x j , ∑ σl j x j , (6.24) 1≤ j≤n sl j=1 j=1

6.3 Multiobjective random fuzzy programming

233

where sl = (sl1 , . . . , sln ). ¯˜ x is expressed as a random fuzzy variable with the membership Observing C l function μCx ¯˜ deﬁned by (6.24), it is signiﬁcant to realize that the fuzzy random programming models discussed in Chapter 4 cannot be applied.

6.3.1 Possibility-based probability model Assuming that the DM concerns about the probability that each of the objective ¯˜ x is smaller than or equal to a certain target values f , we introduce function values C l l the probability P ω C˜ l (ω)x ≤ fl which is expressed as a fuzzy set P˜l with the membership function (6.25) μP˜l (pl ) = sup μC¯˜ l x (u¯l ) pl = P (ω | ul (ω) ≤ f l ) , u¯l

where fl , l = 1, . . . , k are target values speciﬁed by the DM as constants. In order to consider the imprecise nature of the DM’s judgments for the probability with respect to the lth objective function, we introduce the fuzzy goal G˜ l with the nondecreasing continuous membership function μG˜ l . Recalling that the membership function μP˜l (pl ) can be viewed as a possibility distribution on the basis of the concept of possibility measure, the degree of possibility that the fuzzy goal G˜ l for the probability P˜l is satisﬁed is expressed by ΠP˜l (G˜ l ) sup min{μP˜l (pl ), μG˜ l (pl )}. pl

Then, we consider the possibility-based probability model for multiobjective random fuzzy programming problems formulated as ⎫ maximize ΠP˜1 (G˜ 1 ) ⎪ ⎪ ⎬ ······ (6.26) maximize ΠP˜k (G˜ k ) ⎪ ⎪ ⎭ subject to Ax ≤ b, x ≥ 0 or equivalently maximize h1 .. .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

maximize hk ⎪ ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , 0 ≤ hl ≤ 1, l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0.

(6.27)

To generate a candidate for a satisﬁcing solution of the DM for the reference levels hˆ l , l = 1, . . . , k speciﬁed by the DM, called reference possibility levels, the corresponding optimal solution is obtained by solving the augmented minimax problem

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6 Future Research Directions

minimize max

l=1,...,k

⎫ ⎪ ⎪ hˆ l − hl + ρ ∑ (hˆ l − hl ) ⎪ ⎬ k

l=1

subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k Ax ≤ b, x ≥ 0,

⎪ ⎪ ⎪ ⎭

(6.28)

where ρ is a sufﬁciently small positive number. Quite similar to the equivalent transformation (6.16), it holds that 0 n n −1 ˜ ΠP˜l (Gl ) ≥ hl ⇔ ∑ {ml j − L (hl )αl j }x j + Φ (μG˜ (hl )) ∑ σ2l j x2j ≤ fl , (6.29) j=1

l

j=1

where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (hl ) = sup{pl | μG˜ l (pl ) ≥ hl } l and L (hl ) = sup{t | L(t) ≥ hl } are pseudo inverse functions. Consequently, (6.28) is rewritten as ⎫ k ⎪ ⎪ ⎪ minimize max hˆ l − hl + ρ ∑ (hˆ l − hl ) ⎪ ⎪ ⎪ l=1,...,k ⎬ l=1 0 subject to

n

n

j=1

j=1

⎪ ∑ {ml j − L (hl )αl j }x j + Φ−1 (μG˜ l (hl )) ∑ σ2l j x2j ≤ fl , l = 1, . . . , k ⎪ ⎪ ⎪

Ax ≤ b, x ≥ 0.

⎪ ⎪ ⎭

(6.30) Observing that (6.30) is a nonconvex nonlinear programming problem, we can employ genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001) in order to obtain an approximate solution. Now we are ready to construct an interactive algorithm for deriving the satisﬁcing solution for the DM from among the Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based probability model Step 1: Ask the DM to specify the membership functions μG˜ l and the target values fl , l = 1, . . . , k. Step 2: Set the initial reference possibility levels at 1s, which can be viewed as the ideal values, i.e., hˆ l = 1, l = 1, . . . , k. Step 3: For the current reference possibility levels hˆ l , l = 1, . . . , k, solve the augmented minimax problem (6.30). Step 4: The DM is supplied with the corresponding Pareto optimal solution x∗ . If the DM is satisﬁed with the current objective function values h∗l , l = 1, . . . , k, then stop. Otherwise, ask the DM to update the reference possibility levels hˆ l , l = 1, . . . , k by considering the current values of objective functions hl , l = 1, . . . , k, and return to step 3.

6.3 Multiobjective random fuzzy programming

235

It should be noted for the DM that any improvement of one objective function value can be achieved only at the expense of at least one of the other objective function values for the ﬁxed target values fl , l = 1, . . . , k.

6.3.2 Possibility-based fractile model As discussed in the when the DM is willing to maximize the previous subsection, probability P ω C˜ l (ω)x ≤ fl for the speciﬁed target value fl , the possibilitybased probability model is appropriate. However, if we assume that the DM would ¯˜ x, l = 1, . . . , k under the condition that like to minimize the objective functions C l the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, we consider the multiobjective programming problem ⎫ minimize f1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎬ . (6.31) minimize fk ⎪ ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0, where hl , l = 1, . . . , k are permissible possibility levels speciﬁed by the DM. To consider the imprecise nature of human judgments for the target variables fl , l = 1, . . . , k, by introducing the fuzzy goals characterized by the nonincreasing concave and continuous membership functions μl , l = 1, . . . , k, we consider the fuzzy multiobjective programming problem ⎫ maximize μ1 ( f1 ) ⎪ ⎪ ⎪ ⎪ ··· ⎬ maximize μk ( fk ) (6.32) ⎪ subject to ΠP˜l (G˜ l ) ≥ hl , l = 1, . . . , k ⎪ ⎪ ⎪ ⎭ Ax ≤ b, x ≥ 0. Recalling the equivalent transformation (6.16), it holds that 0 n n −1 ΠP˜l (G˜ l ) ≥ hl ⇔ ∑ {ml j − L (hl )αl j }x j + Φ (μG˜ (hl )) ∑ σ2l j x2j ≤ fl , (6.33) j=1

l

j=1

where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (hl ) = sup{pl | μG˜ l (pl ) ≥ hl } l and L (hl ) = sup{t | L(t) ≥ hl } are pseudo inverse functions. From (6.33), (6.32) is transformed into

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6 Future Research Directions

maximize μ1 ( f1 ) ··· maximize μk ( fk )

0

n

subject to

∑ {ml j − L (hl )αl j }x j + Φ

−1

j=1

(μG˜ (hl )) l

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ n

⎪ ⎪ ⎪ ⎭

⎪ ∑ σ2l j x2j ≤ fl , l = 1, . . . , k ⎪ ⎪ ⎪

j=1

Ax ≤ b, x ≥ 0

(6.34) or equivalently

⎫ maximize μ1 Z1Π,F (x) ⎪ ⎪ ⎪ ⎪ ⎬ ·· · · · · · · · maximize μk ZkΠ,F (x) ⎪ ⎪ ⎪ ⎪ ⎭ subject to Ax ≤ b, x ≥ 0,

(6.35)

where ZlΠ,F (x) =

n

0

∑ {ml j − L (hl )αl j }x j + Φ

−1

j=1

(μG˜ (hl )) l

n

∑ σ2l j x2j .

j=1

As mentioned in the previous section, assuming μG˜ l (pl ) ≥ 0 for any pl ≥ 1/2,

l = 1, . . . , k, each of the objective functions ZlΠ,F (x), l = 1, . . . , k is convex due to the property of Φ−1 (μG˜ (hk )) ≥ 0. k In order to ﬁnd a candidate for the satisﬁcing solution, through the introduction of the reference membership levels μˆ l , l = 1, . . . , k speciﬁed by the DM, a Pareto optimal solution is obtained by solving the augmented minimax problem minimize

max

l=1,...,k

k μˆ l − μl ZlΠ,F (x) + ρ ∑ μˆ l − μl ZlΠ,F (x)

subject to Ax ≤ b, x ≥ 0

⎫ ⎪ ⎬

l=1

⎪ ⎭

(6.36)

or equivalently ⎫ ⎪ ⎪ ⎪ k ⎪ ⎪ Π,F Π,F ⎪ subject to μˆ 1 − μ1 Z1 (x) + ρ ∑ μˆ l − μl Zl (x) ≤ v ⎪ ⎪ ⎪ ⎬ l=1 ··············· ⎪ ⎪ k ⎪ ⎪ μˆ k − μk ZkΠ,F (x) + ρ ∑ μˆ l − μl ZlΠ,F (x) ≤ v ⎪ ⎪ ⎪ ⎪ ⎪ l=1 ⎭ Ax ≤ b, x ≥ 0, minimize v

(6.37)

where ρ is a sufﬁciently small positive number. When μl is concave, similar to the convex property shown in (3.22), it is easily shown that (6.37) is a convex programming problem. Hence, an optimal solution of

6.4 Random fuzzy two-level programming

237

(6.37) can be obtained by using some convex programming techniques such as the sequential quadratic programming method. We now summarize an interactive algorithm in order to derive the satisﬁcing solution for the DM from among the Pareto optimal solution set. Interactive satisﬁcing method for the possibility-based fractile model Step 1: Ask the DM to specify the membership functions μG˜ l and the permissible possibility levels hl , l = 1, . . . , k. Step 2: Set the initial reference membership levels at 1s, which can be viewed as the ideal values, i.e., μˆ l = 1, l = 1, . . . , k. Step 3: For the current reference fractile levels μˆ l , l = 1, . . . , k, solve the corresponding minimax problem (6.36). Step 4: The DM is supplied with the corresponding Pareto optimal solution x∗ . If Π,F ∗ the DM is satisﬁed with the current membership function values μl Zl (x ) , l = 1, . . . , k, then stop. Otherwise, ask the DM to update the reference membership levels μˆ l , l = 1, . . . , k, and return to step 3. It should be stressed for the DM that any improvement of one membership function value can be achieved only at the expense of at least one of the other membership function values for the ﬁxed permissible possibility levels hl , l = 1, . . . , k. In the future, along the same line as in Chapter 4, we will consider random fuzzy versions of possibility-based expectation and variance models. Extensions to more general problems where random fuzzy variables coefﬁcients are involved not only in the objective function but also in the constraints will be worth considering. Furthermore, extensions of the proposed models to integer problems will be discussed elsewhere.

6.4 Random fuzzy two-level programming Although we have restricted ourselves to the problems where decisions are made by a single DM in the previous section, it should be noted here that decision making problems in hierarchical managerial or public organizations are often formulated as two-level programming problems, where there exist two DMs. In contrast to the fuzzy random two-level programming discussed in Chapter 5, we consider the random fuzzy version of two-level linear programming problems formulated as ⎫ ¯˜ x + C ¯˜ x ⎪ minimize z1 (x1 , x2 ) = C 11 1 12 2 ⎪ ⎪ for DM1 ⎪ ¯˜ x + C ¯˜ x ⎬ minimize z2 (x1 , x2 ) = C 21 1 22 2 (6.38) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where x1 is an n1 dimensional decision variable column vector for the DM at the upper level (DM1), x2 is an n2 dimensional decision variable column vector for the

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6 Future Research Directions

DM at the lower level (DM2), and zl (x1 , x2 ), l = 1, 2 are the objective functions ¯˜ , for DMl, l = 1, 2, respectively. Suppose that each of C¯˜l jk , k = 1, 2, . . . , n j of C lj l = 1, 2, j = 1, 2 is the Gaussian random variable with fuzzy mean value M˜ l jk which is represented by the fuzzy set with the membership function ⎧ ml jk − τ ⎪ ⎪ if ml jk ≥ τ ⎪ ⎨L αl jk (6.39) μM˜ l jk (τ) = ⎪ τ − ml jk ⎪ ⎪ ⎩R if ml jk < τ, βl jk where L and R are nonincreasing continuous functions from [0, ∞) to [0, 1]. Let Γ be a collection of all possible Gaussian random variables N(s, σ2 ) where s ∈ (−∞, ∞) and σ2 ∈ (0, ∞). Then, C¯˜l jk is expressed as a random fuzzy variable with the membership function μC¯˜ (¯γl jk ) = {μM˜ l jk (sl jk ) | γ¯ l jk ∼ N(sl jk , σ2l jk )}, ∀¯γl jk ∈ Γ. l jk

(6.40)

Through the extension principle, in view of (6.40), the membership function of a random fuzzy variable corresponding to each of objective functions zl (x1 , x2 ), l = 1, 2 is given as 2 nj μ ¯˜ (¯γl jk ) u¯l = ∑ ∑ γ¯ l jk x jk μC¯˜ l x (u¯l ) = sup min 1≤k≤n j , j=1,2 Cl jk γ¯ l j=1 k=1 " # 2 nj 2 nj 2 2 μM˜ l jk (sl jk ) u¯l ∼ N ∑ ∑ sl jk x jk , ∑ ∑ σl jk x jk = sup min , 1≤k≤n j , j=1,2 sl j=1 k=1 j=1 k=1 (6.41) where γ¯ l = (¯γl11 , . . . , γ¯ l1n1 , γ¯ l21 , . . . , γ¯ l2n2 ) and sl = (sl11 , . . . , sl1n1 , s121 , . . . , sl2n2 ).

6.4.1 Possibility-based probability model Assuming that the DMs concerns about the probabilities that their own objective ¯˜ x are smaller than or equal to certain target values f , l = 1, 2, we function values C l l introduce the probabilities P ω C˜ l (ω)x ≤ fl which are expressed as fuzzy sets P˜l with the membership functions μP˜l (pl ) = sup μC¯˜ l x (u¯l ) | pl = P (ω | ul (ω) ≤ fl ) , (6.42) u¯l

where fl , l = 1, 2 are target values speciﬁed by the DMs as constants. In order to consider the imprecise nature of the human judgments, we introduce the fuzzy goals G˜ l of the DMs for the probabilities P˜l , l = 1, 2, which is characterized by a nondecreasing membership function μG˜ l . Recalling that the membership

6.4 Random fuzzy two-level programming

239

function is regarded as a possibility distribution, the degree of possibility that the probability P˜l attains the fuzzy goal G˜ l is expressed as ΠP˜l (G˜ l ) = sup min{μP˜l (pl ), μG˜ l (pl )}. pl

Then, assuming that the DMs are willing to maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probability model for random fuzzy two-level programming problems formulated as ⎫ maximize Z1Π,P (x1 , x2 ) = ΠP˜1 (G1 ) ⎪ ⎪ ⎪ for DM1 ⎪ ⎬ maximize Z2Π,P (x1 , x2 ) = ΠP˜2 (G2 ) (6.43) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. As an initial candidate for an overall satisfactory solution to (6.43), it would be useful for DM1 to ﬁnd a solution which maximize the smaller degree of satisfaction between the two DMs by solving the maximin problem ⎫ maximize min Z1Π,P (x1 , x2 ), Z2Π,P (x1 , x2 ) ⎪ ⎬ (6.44) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently maximize v subject to Z1Π,P (x1 , x2 ) ≥ v

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

Z2Π,P (x2 , x2 ) ≥ v ⎪ ⎪ A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.45)

Similar to the equivalent transformation (6.16), in view of ZlΠ,P (x1 , x2 ) = ΠP˜l (Gl ), it holds that 1 2 2 nj 2 nj 2 Π,P −1 Zl (x1 , x2 ) ≥ v ⇔ ∑ ∑ {ml jk −L (v)αl jk }x jk +Φ (μG˜ (v))3 ∑ ∑ σ2l jk x2jk ≤ fl , j=1 k=1

l

j=1 k=1

(6.46) where Φ−1 is the inverse function of the probability distribution function Φ of the standard Gaussian random variable N(0, 1), and μG˜ (v) = sup{pl | μG˜ l (pl ) ≥ v} and l L (v) = sup{t | L(t) ≥ v} are pseudo inverse functions. Consequently, (6.45) is equivalently transformed as

240

6 Future Research Directions

maximize v 2

subject to

∑

1

j=1 k=1 2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ 2 2 ⎪ σ x ≤ f ⎪ ∑ 1 jk jk 1 ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎬

1 2 2 nj 2 −1 3 {m − L (v)α }x + Φ (μ (v)) 1 jk jk ∑ 1 jk ∑ G˜ 1 2 2 2 −1 3 {m − L (v)α }x + Φ (μ (v)) 2 jk jk ∑ 2 jk ∑ G˜ nj

2

j=1 k=1

nj

⎪

⎪ ⎪ ∑ σ22 jk x2jk ≤ f2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

(6.47) Although (6.47) is nonconvex, an approximate solution can be obtained by genetic algorithms such as GENOCOPIII (Michalewicz and Nazhiyath, 1995) or the revised GENOCOPIII (Sakawa, 2001). If DM1 is satisﬁed with the degrees ZlΠ,P (x∗1 , x∗2 ), l = 1, 2, the corresponding optimal solution (x∗1 , x∗2 ) to (6.47) is regarded as the satisfactory solution. However, if DM1 is not satisﬁed, by introducing the constraint that Z1Π,P (x1 , x2 ) is larger than or equal to the minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the maximization problem formulated as ⎫ ⎪ maximize Z2Π,P (x1 , x2 ) ⎪ ⎪ ⎬ Π,P subject to Z1 (x1 , x2 ) ≥ δ (6.48) A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently

⎫ maximize h ⎪ ⎪ ⎪ Π,P ⎪ subject to Z2 (x1 , x2 ) ≥ h ⎪ ⎬ Π,P

(6.49)

Z1 (x1 , x2 ) ≥ δ ⎪ ⎪ A1 x1 + A 2 x2 ≤ b ⎪ ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. Similar to the equivalent transformation (6.16), it follows that Z2Π,P (x1 , x2 ) ≥ h ⇔

2

∑

j=1 k=1

2

Z1Π,P (x1 , x2 ) ≥ δ ⇔

2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ ⎪ 2 2 ⎪ σ x ≤ f , ⎪ 2 ∑ 2 jk jk ⎪ ⎬ j=1 k=1

1 2 2 nj 2 −1 ∑ {m2 jk − L (h)α2 jk }x jk + Φ (μG˜ (h))3 ∑ 1 2 2 2 −1 3∑ {m − L (δ)α }x + Φ (μ (δ)) 1 jk jk ∑ 1 jk G˜ nj

j=1 k=1

1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ σ x ≤ f . ⎪ ∑ 1 jk jk 1 ⎪ ⎭ nj

j=1 k=1

(6.50) Consequently, (6.49) is rewritten as

6.4 Random fuzzy two-level programming

maximize h 2

subject to

∑

241

2

j=1 k=1 2

∑

⎫ ⎪ ⎪ ⎪ ⎪ nj ⎪ ⎪ 2 2 ⎪ σ x ≤ f ⎪ ∑ 2 jk jk 2 ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎬

1 2 2 nj 2 −1 3 {m − L (h)α }x + Φ (μ (h)) 2 jk jk ∑ 2 jk ∑ G˜ 1 2 2 2 −1 3 {m − L (δ)α }x + Φ (μ (δ)) 1 jk jk ∑ 1 jk ∑ G˜ nj

1

j=1 k=1

nj

⎪

⎪ ⎪ ∑ σ21 jk x2jk ≤ f1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

(6.51) In view of the nonconvexity of (6.51), quite similar to (6.47), genetic algorithms can be applied for solving this problem. Realizing that the objective functions of DM1 and DM2 often conﬂict with each other, it should be noted here that the larger the minimal satisfactory level δ for the DM1’s objective function value is speciﬁed, the smaller the DM2’s objective function value becomes, which may lead to the unbalanced satisfactory degrees of DM1 and DM2 due to the large difference between the objective function values of both DMs. In order to derive the satisfactory solution which has well-balanced objective function values of DM1 and DM2, by introducing the ratio Δ between the satisfactory degrees of both DMs expressed as Δ=

Z2Π,P (x1 , x2 )

Z1Π,P (x1 , x2 )

,

(6.52)

we assume that DM1 speciﬁes the lower bound Δmin and the upper bound Δmax of Δ, which are used for evaluating the appropriateness of the ratio Δ. To be more speciﬁc, if it holds that Δ ∈ [Δmin , Δmax ], then DM1 regards the corresponding solution as a promising candidate for the satisfactory solution with well-balanced membership function values. Now we construct a procedure of interactive fuzzy programming for the possibilitybased probability model in order to derive an overall satisfactory solution. Interactive fuzzy programming in the possibility-based probability model Step 1: Ask each DM to specify the membership function μG˜ l and the target values fl , l = 1, 2. Step 2: For the current target values fl , l = 1, 2, solve the maximin problem (6.44). Step 3: DM1 is supplied with the objective function values Z1Π,P (x∗1 , x∗2 ) and Z2Π,P (x∗1 , x∗2 ) for the optimal solution (x∗1 , x∗2 ) obtained in step 2. If DM1 is satisﬁed with the current membership function values, then stop. If DM1 is not satisﬁed and prefers to update fl , l = 1, 2, ask DM1 to update fl , and return to step 2. Otherwise, ask DM1 to specify the minimal satisfactory level δ and the permissible range [Δmin , Δmax ] of Δ. Step 4: For the current minimal satisfactory δ, solve (6.48).

242

6 Future Research Directions

Step 5: DM1 is supplied with the current values of Z1Π,P (x∗1 , x∗2 ), Z2Π,P (x∗1 , x∗2 ) and Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the current objective function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 4.

6.4.2 Possibility-based fractile model When the DMs would like to maximize the probability for the speciﬁed target value, the possibility-based probability model is recommended for the DM1. On the other hand, if the DMs would like to minimize their own objective function values under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, we consider the two-level programming problem ⎫ maximize μ1 ( f1 ) ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize μ2 ( f2 ) ⎪ ⎪ for DM2 ⎬ subject to ΠP˜1 (G1 ) ≥ h1 (6.53) ⎪ ΠP˜2 (G2 ) ≥ h2 ⎪ ⎪ ⎪ ⎪ A1 x 1 + A 2 x 2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0, where h1 and h2 are permissible possibility levels speciﬁed by the DMs, and μ1 and μ2 are the membership functions of the fuzzy goals for the target variables f 1 and f 2 , respectively. Recalling the equivalent transformation (6.50), (6.53) is transformed into ⎫ maximize μ1 ( f1 ) ⎪ ⎪ ⎪ for DM1 ⎪ ⎪ ⎪ maximize μ2 ( f2 ) ⎪ ⎪ for DM2 ⎪ 1 ⎪ 2 ⎪ n ⎪ j 2 nj 2 2 ⎪ ⎪ −1 2 2 3 subject to ∑ ∑ {m1 jk − L (hl )α1 jk }x jk + Φ (μG˜ (h1 )) ∑ ∑ σ1 jk x jk ≤ f1 ⎪ ⎪ ⎬ 1 j=1 k=1 2

∑

j=1 k=1

1 2 2 nj 2 −1 ∑ {m2 jk − L (h2 )α2 jk }x jk + Φ (μG˜ (h2 ))3 ∑

j=1 k=1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0

2

⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∑ σ2 jk x jk ≤ f2 ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ nj

(6.54) or equivalently

6.4 Random fuzzy two-level programming

243

⎫ maximize μ1 Z1Π,F (x1 , x2 ) ⎪ ⎪ for DM1 ⎪ ⎪ ⎬ maximize μ2 Z2Π,F (x1 , x2 ) for DM2 ⎪ ⎪ subject to A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0,

(6.55)

where ZlΠ,F (x1 , x2 ) =

2

∑

1 2 2 2 −1 3∑ {m − L (h )α }x + Φ (μ (h )) l l jk jk l ∑ l jk G˜ nj

j=1 k=1

l

nj

∑ σ2l jk x2jk ,

j=1 k=1

l = 1, 2. (6.56) are convex functions of (x1 , x2 ). As the ﬁrst step toward deriving an overall satisfactory solution to (6.55), in order to ﬁnd a solution which maximize the smaller degree of satisfaction between the two DMs, we solve the maximin problem ⎫ maximize min μ1 Z1Π,F (x1 , x2 ) , μ2 Z2Π,F (x1 , x2 ) ⎪ ⎬ (6.57) subject to A1 x1 + A2 x2 ≤ b ⎪ ⎭ x1 ≥ 0, x2 ≥ 0. By introducing an auxiliary variable v, this problem is rewritten as ⎫ maximize v ⎪ ⎪ ⎪ ⎪ subject to μ1 Z1Π,F (x1 , x2 ) ≥ v ⎪ ⎪ ⎪ ⎬ Π,F μ2 Z2 (x1 , x2 ) ≥ v ⎪ ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.58)

Although the membership function does not always need to be linear, for the sake of simplicity, we adopt a linear membership function which characterizes the fuzzy goal of each decision maker. The linear membership functions μl , l = 1, 2 are deﬁned as ⎧ ⎪ 1 if ZlΠ,F (x1 , x2 ) ≤ z1l ⎪ ⎪ ⎨ Π,F Zl (x1 , x2 ) − z0l μl ZlΠ,F (x1 , x2 ) = if z1l < ZlΠ,F (x1 , x2 ) < z0l 1 − z0 ⎪ z ⎪ l l ⎪ ⎩ 0 if ZlΠ,F (x1 , x2 ) ≥ z0l . Then, (6.58) is equivalently transformed into the convex programming problem

244

6 Future Research Directions

⎫ ⎪ ⎪ ⎪ ⎪ 2 nj ⎪ ⎪ ⎪ subject to ∑ ∑ {m1 jk − L (h1 )α1 jk }x jk ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ 1 ⎪ ⎪ 2 2 nj ⎪ ⎪ 2 2 2 −1 1 0 0⎪ 3 +Φ (μG˜ (h1 )) ∑ ∑ σ1 jk x jk ≤ (z1 − z1 )v + z1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ j=1 k=1 ⎬

maximize v

nj

2

∑ ∑ {m2 jk − L (h2 )α2 jk }x jk

⎪ ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ 1 ⎪ 2 2 nj ⎪ ⎪ 2 ⎪ ⎪ 2 2 −1 1 0 0 ⎪ +Φ (μG˜ (h2 ))3 ∑ ∑ σ2 jk x jk ≤ (z2 − z2 )v + z2 ⎪ ⎪ 2 ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0.

(6.59)

If DM1 is satisﬁed with the membership function values μl ZlΠ,F (x∗1 , x∗2 ) , l = 1, 2, the corresponding optimal solution x∗ to (6.58) is regarded as the satisfactory solution. Otherwise, by introducing the constraint that μ1 (Z1Π,F (x)) is larger than or equal to the minimal satisfactory level δ ∈ (0, 1) speciﬁed by DM1, we consider the Π,F problem of maximizing the membership function μ2 (Z2 (x1 , x2 )) formulated as ⎫ Π,F ⎪ maximize μ2 Z2 (x1 , x2 ) ⎪ ⎪ ⎪ ⎪ ⎬ Π,F subject to μ1 Z1 (x1 , x2 ) ≥ δ (6.60) ⎪ ⎪ ⎪ A1 x1 + A2 x2 ≤ b ⎪ ⎪ ⎭ x1 ≥ 0, x2 ≥ 0 or equivalently, we obtain the convex programming problem 2

minimize

∑

1 2 2 2 −1 ∑ {m2 jk − L (h2)α2 jk }x jk + Φ (μG˜ (h2 ))3 ∑ nj

2

j=1 k=1 2

subject to

nj

∑ ∑ {m1 jk − L (h1)α1 jk }x jk

j=1 k=1

1 2 2 2 −1 +Φ (μG˜ (h1 ))3 ∑ 1

A1 x1 + A2 x2 ≤ b x1 ≥ 0, x2 ≥ 0.

⎫ ⎪ ⎪ ⎪ ⎪ ∑ σ22 jk x2jk ⎪ ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ nj

nj

∑

j=1 k=1

σ21 jk x2jk

≤

(z11 − z01 )δ + z01

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(6.61) In general, when the objective functions of DM1 and DM2 conﬂict with each other, it should be noted here that the larger the minimal satisfactory level δ for μ1 is speciﬁed by DM1, the smaller the satisfactory degree for μ2 becomes, which may

6.4 Random fuzzy two-level programming

245

lead to the improper satisfactory balance between DM1 and DM2 due to the large difference between the membership function values of both DMs. In order to derive the satisfactory solution which has well-balanced membership function values between both DMs, by introducing the ratio Δ expressed as μ2 Z2Π,F (x1 , x2 ) , (6.62) Δ= μ1 Z1Π,F (x1 , x2 ) the lower bound Δmin and the upper bound Δmax of Δ, speciﬁed by DM1, are introduced to determine whether or not the ratio Δ is appropriate. To be more explicit, if it holds that Δ ∈ [Δmin , Δmax ], then DM1 regards the corresponding solution as a preferable candidate for the satisfactory solution with well-balanced membership function values. Now we summarize a procedure of interactive fuzzy programming for the possibilitybased fractile model in order to derive a satisfactory solution. Interactive fuzzy programming in the possibility-based fractile model Step 1: Ask DMs to specify the membership functions μl , l = 1, 2. Step 2: Ask DM1 to specify the permissible possibility levels hl , l = 1, 2. Step 3: For the current hl , l = 1, 2, solve the maximin problem (6.57). Step 4: DM1 is supplied with the current values of the membership functions μ1 and μ2 for the optimal solution obtained in step 3. If DM1 is satisﬁed with the current membership function values, then stop. If DM1 is not satisﬁed and prefers ask to update hl , l = 1, 2, ask DM1 to update hl , and return to step 3. Otherwise, Π,F DM1 to specify the minimal satisfactory level δ for μ1 Z1 (x1 , x2 ) and the permissible range [Δmin , Δmax ] of the ratio Δ. Step 5: For the current minimal satisfactory level δ, solve the convex programming problem (6.61). Step 6: DM1 is supplied with the current values of the membership function μ1 , μ2 and the ratio Δ. If Δ ∈ [Δmin , Δmax ] and DM1 is satisﬁed with the current membership function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory level δ, and return to step 5. In this section, from the viewpoint of taking the possibility of coordination or bargaining between the DMs into account, it is assumed that there exist communication and a cooperative relationship between the DMs. However, as discussed in Chapter 5, we are faced with the decision making situations where the DMs do not make any binding agreement even if there exists some communication. For dealing with such situations, extensions to noncooperative cases as well as the developments of the computational methods for obtaining Stackelberg solutions will be discussed elsewhere. Furthermore, an integer generalization along the same line as in Chapter 5 will be reported elsewhere.

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Index

L-R fuzzy number, 13, 119, 132, 141, 178, 195, 208, 218 L∞ norm, 27 α-level optimal parameter, 135, 138, 141 α-level set, 12, 102, 133, 136, 138, 141, 160 α-linear programming problem, 19 σ-ﬁeld, 20, 102 kth best method, 37, 209 algorithm of interactive fuzzy programming, 36 augmented minimax problem, 57, 87, 136, 233, 236 bisection method, 31, 54, 66, 110, 124, 143, 181, 185 Borel σ-ﬁeld, 102 Borel set, 21 Cartesian product, 198, 212, 220 chance constrained condition, 51, 171, 188, 215 chance constraint programming, 24 continuous fuzzy random variable, 119 crossover, 41 discrete fuzzy random variable, 102 distribution function, 21 double strings, 44 dual problem, 39 duality gap, 39 E-model, 25 E-P-Pareto optimal solution, 108, 151 E-P-Pareto optimality test, 113 extended Dinkelbach-type algorithm, 116, 129 extension principle, 105, 132, 227, 232, 238

F-P-Pareto optimal solution, 127, 147 feasible extreme point, 211 feasible region, 20, 27, 31, 37, 57, 104, 171 ﬁtness function, 42 ﬁtness scaling, 42 ﬁtness value, 41 fractile criterion model, 25 fuzzy decision, 15, 29 fuzzy goal, 52, 57, 105, 173, 228 fuzzy multiobjective programming, 28 fuzzy number, 13, 182 fuzzy programming for two-level linear programming, 33 fuzzy random variable, 101, 178, 182, 195, 198, 207, 211, 218, 220 fuzzy set, 12 fuzzy variable, 226 GADSLPRRSU, 44, 86, 87, 91, 95, 151, 162 Gaussian random variable, 50, 72, 84, 91, 127, 146, 157, 226, 227 genetic algorithm, 40 GENOCOPIII, 86, 136, 140, 234, 240 inducible region, 210, 220 integer programming problem, 44 interactive fuzzy multiobjective stochastic integer programming, 83 interactive fuzzy satisﬁcing method, 53, 55, 58, 69, 75, 92, 95 interactive satisﬁcing method, 112, 151 Kuhn-Tucker condition, 38, 203 Kuhn-Tucker necessity theorem, 68 Lagrange function, 67, 112, 125, 203 Lagrange multiplier, 203

M. Sakawa et al., Fuzzy Stochastic Multiobjective Programming, International Series in Operations Research & Management Science, DOI 10.1007/978-1-4419-8402-9, © Springer Science+Business Media, LLC 2011

263

264 level set-based model, 18 linear fractional programming problem, 18, 66, 110, 182 linear membership function, 15, 28, 106, 128, 146, 150, 186, 213, 221, 243 linear programming problem involving fuzzy parameters, 16 linear programming problem with a fuzzy goal and fuzzy constraints, 14 linear programming relaxation, 44 M-Pareto optimal solution, 30, 53, 87 M-Pareto optimality, 53 Manhattan distance, 27 maximin problem, 173, 180, 186, 189, 192, 196, 199, 239 minimal satisfactory level, 34, 185, 186, 192, 196, 199, 240 minimax problem, 109, 124, 142, 147, 151 minimum operator, 53 mixed zero-one programming problem, 39 multiobjective stochastic integer programming problem, 84 multiobjective stochastic linear programming problem, 81 multiobjective stochastic programming problem, 51 mutation, 41 nonlinear integer programming problem, 91 NP-hard problem, 37 P-M-α Pareto optimality test, 144 P-model, 25 P-P-Pareto optimal solution, 123, 142 P-P-Pareto optimality test, 125 Pareto optimal solution, 27, 236 Pareto optimality in two-level programming, 33 permissible expectation level, 57, 85, 113 permissible possibility level, 231 permissible probability level, 75, 127 phase one of the two-phase simplex method, 54, 110, 124, 143 possibilistic programming, 107, 118 possibility distribution, 107, 179, 228, 233 power set, 102 probability density function, 23 probability measure, 21, 102 probability measure space, 21

Index probability space, 21, 102 product operator, 53 pseudo inverse function, 123, 143, 229 quadratic integer programming problem, 86 random variable, 21 ratio of the satisfactory degree, 34, 199 rational response, 37, 202, 209, 216, 217, 219 realized value, 102 reference membership level, 53 reference point method, 27 reference solution updating, 44 reproduction, 41 revised GADSLPRRSU, 86, 152, 153, 163 revised GENOCOPIII, 136, 140, 234, 240 roulette wheel selection, 42 sample space, 20, 102 satisfactory solution, 33, 172, 173, 180, 184, 189, 192, 196, 199, 239 satisﬁcing probability level, 51, 171 satisﬁcing solution, 85, 86, 109, 151, 236 sequential quadratic programming method, 24, 147, 148, 174, 203, 231 shape function, 13 simple recourse problem, 22, 78 simplex multiplier, 68, 112 Stackelberg equilibrium solution, 36 Stackelberg solution, 36, 202, 203, 209, 213, 216, 217 standard Gaussian random variable, 72, 230 target value, 62, 90, 122, 228 target variable, 72, 235 Tchebyshev norm, 27 trade-off rate, 55, 112 triangular fuzzy number, 105, 120 two-phase simplex method, 31, 54, 66 two-stage model, 77 two-stage programming, 22 V-model, 25 V-P-Pareto optimal solution, 114, 153 variable transformation, 66 variance-covariance matrix, 25, 50, 173 weak Pareto optimality, 27 Zimmermann method, 29, 61, 177

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