Fuzzy Multicriteria Decision-Making E-Book

Witold Pedrycz

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 608

Veröffentlichungsjahr: 2011

Beschreibung

Fuzzy Multicriteria Decision-Making: Models, Algorithms and Applications addresses theoretical and practical gaps in considering uncertainty and multicriteria factors encountered in the design, planning, and control of complex systems. Including all prerequisite knowledge and augmenting some parts with a step-by-step explanation of more advanced concepts, the authors provide a systematic and comprehensive presentation of the concepts, design methodology, and detailed algorithms. These are supported by many numeric illustrations and a number of application scenarios to motivate the reader and make some abstract concepts more tangible. Fuzzy Multicriteria Decision-Making: Models, Algorithms and Applications will appeal to a wide audience of researchers and practitioners in disciplines where decision-making is paramount, including various branches of engineering, operations research, economics and management; it will also be of interest to graduate students and senior undergraduate students in courses such as decision making, management, risk management, operations research, numerical methods, and knowledge-based systems.

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This edition first published 2011

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Library of Congress Cataloguing-in-Publication Data

Pedrycz, Witold, 1953-

Models and algorithms of fuzzy multicriteria decision-making and their applications/Witold Pedrycz, Petr Ekel, Roberta Parreiras. - 1st ed.

p. cm.

Includes index.

ISBN 978-0-470-68225-8 (cloth)

1. Decision-making. 2. Decision-making-Mathematical models. 3. Fuzzy decision-making. I. Ekel, Petr. II. Parreiras, Roberta. III. Title.

T57.95.P42 2011

003′.56-dc22

2010025490

A catalogue record for this book is available from the British Library.

Print ISBN: 9780470682258

ePDF ISBN: 9780470974049

oBook ISBN: 9780470974032

Contents

About the Authors

Foreword

Preface

1 Decision-Making in System Project, Planning, Operation, and Control: Motivation, Objectives, and Basic Concepts

1.1 Decision-Making and its Support

1.2 Optimization and Decision-Making Problems

1.3 Multicriteria Decision-Making

1.4 Group Decision-Making

1.5 Fuzzy Sets and their Role in Decision-Making Processes

1.6 Conclusions

References

2 Notions and Concepts of Fuzzy Sets: An Introduction

2.1 Sets and Fuzzy Sets: A Fundamental Departure from the Principle of Dichotomy

2.2 Interpretation of Fuzzy Sets

2.3 Membership Functions and Classes of Fuzzy Sets

2.4 Fuzzy Numbers and Intervals

2.5 Linguistic Variables

2.6 A Generic Characterization of Fuzzy Sets: Some Fundamental Descriptors

2.7 Geometric Interpretation of Sets and Fuzzy Sets

2.8 Fuzzy Sets and the Family of α-cuts

2.9 Operations on Fuzzy Sets

2.10 Fuzzy Relations

2.11 Conclusions

References

3 Selected Design and Processing Aspects of Fuzzy Sets

3.1 The Development of Fuzzy Sets: Elicitation of Membership Functions

3.2 Aggregation Operations

3.3 Transformations of Fuzzy Sets

3.4 Conclusions

References

4 Continuous Models ofMulticriteria Decision-Making and their Analysis

4.1 Continuous Models (X, M Models) of Multicriteria Decision-Making

4.2 Pareto-Optimal Solutions

4.3 Approaches to the Use of DM Information

4.4 Methods of Multiobjective Decision-Making

4.5 Bellman–Zadeh Approach and its Application to Multicriteria Decision-Making

4.6 Multicriteria Resource Allocation

4.7 Adaptive Interactive Decision-Making System for Multicriteria Resource Allocation

4.8 Application of the Bellman–Zadeh Approach to Multicriteria Problems

4.9 Conclusions

References

5 Introduction to Preference Modeling with Binary Fuzzy Relations

5.1 Binary Fuzzy Relations and their Fundamental Properties

5.2 Preference Modeling with Binary Fuzzy Relations

5.3 Preference Structure of Binary Fuzzy Preference Relations

5.4 A Method for Constructing a Fuzzy Preference Structure

5.5 Consistency of Fuzzy Preference Relations

5.6 Conclusions

References

6 Construction of Fuzzy Preference Relations

6.1 Preference Formats

6.2 Ordering of Fuzzy Quantities and the Construction of Fuzzy Preference Relations

6.3 Transformation Functions and their Use for Converting Different Preference Formats into Fuzzy Preference Relations

6.4 A Method for Repairing Inconsistent Judgments

6.5 Conclusions

References

7 Discrete Models of Multicriteria Decision-Making and their Analysis

7.1 Optimization Problems with Fuzzy Coefficients and their Analysis

7.2 Discrete Models (X, R Models) of Multiattribute Decision-Making

7.3 Basic Techniques of Analysis of X, R Models

7.4 Interactive Decision-Making System for Multicriteria Analysis of Alternatives in a Fuzzy Environment

7.5 Multicriteria Analysis of Alternatives with Fuzzy Ordering of Criteria

7.6 Multicriteria Analysis of Alternatives with the Concept of Fuzzy Majority

7.7 Multicriteria Analysis of Alternatives Based on an Outranking Approach (Fuzzy Promethee)

7.8 Application Examples

7.9 Conclusions

References

8 Generalization of a Classic Approach to Dealing with Uncertainty of Information for Multicriteria Decision Problems

8.1 Classic Approach to Dealing with Uncertainty of Information

8.2 Choice Criteria

8.3 Generalization of the Classic Approach

8.4 Modification of the Choice Criteria

8.5 General Scheme of Multicriteria Decision-Making under Uncertainty

8.6 Application Example

8.7 Conclusions

References

9 Group Decision-Making: Fuzzy Models

9.1 Group Decision-Making Problem and its Characteristics

9.2 Strategies for the Analysis of Group Decision-Making Problems: Multiperson and Multiattribute Aggregation Modes

9.3 The Different Levels of Influence of Each Expert in the Construction of the Collective Opinion

9.4 Aggregation Operators for Constructing Collective Opinions on the Basis of Fuzzy Models and their Properties

9.5 Consistency of Pairwise Judgments in Group Decision-Making

9.6 Fuzzy Group Decision-Making Methods

9.7 Conclusions

References

10 Use of Consensus Schemes in Group Decision-Making

10.1 Consensus in Group Decision-Making

10.2 Consensus Schemes: Definition and Motivation

10.3 Fuzzy Concordance and Fuzzy Consensus Measures

10.4 Moderator Interventions

10.5 Optimal Consensus in a Fuzzy Environment

10.6 Consensus Schemes in Fuzzy Environment

10.7 An Application Related to the Balanced Scorecard Methodology

10.8 Conclusions

References

Index

About the Authors

Witold Pedrycz is a Professor and Canada Research Chair (CRC-Computational Intelligence) in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. He is also with the Systems Research Institute of the Polish Academy of Sciences, Warsaw, Poland. He also holds an appointment of special professorship in the School of Computer Science, University of Nottingham, UK. In 2009 Dr. Pedrycz was elected a foreign member of the Polish Academy of Sciences. His main research directions involve Computational Intelligence, fuzzy modeling and Granular Computing, knowledge discovery and data mining, fuzzy control, pattern recognition, knowledge-based neural networks, relational computing, and Software Engineering. He has published numerous papers in this area. He is also an author of 14 research monographs covering various aspects of Computational Intelligence and Software Engineering. Witold Pedrycz has been a member of numerous program committees for IEEE conferences in the area of fuzzy sets and neurocomputing. Dr. Pedrycz is intensively involved in editorial activities; he is an Editor-in-Chief of Information Sciences and Editor-in-Chief of IEEE Transactions on Systems, Man, and Cybernetics-part A. He currently serves as an Associate Editor of IEEE Transactions on Fuzzy Systems and a number of other international journals. He has edited a number of volumes; the most recent one is entitled “Handbook of Granular Computing” (J. Wiley, 2008). In 2007 he received a prestigious Norbert Wiener award from the IEEE Systems, Man, and Cybernetics Council. He is a Fellow of the IEEE and a recipient of the IEEE Canada Computer Engineering Medal 2008. In 2009 he has received a Cajastur Prize for Soft Computing from the European Centre for Soft Computing for “pioneering and multifaceted contributions to Granular Computing”.

Petr Ekel is a Professor in the Graduate Program in Electrical Engineering, Pontifical Catholic University of Minas Gerais and a Supervisor of Ph.D. Studies in the Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte, Brazil. He is also a President and Principal Consultant in ASOTECH-Advanced System Optimization Technologies, Belo Horizonte, Brazil. He has experience in research, teaching, consulting, and public service. His research activities in the field of system modeling, optimization, and control are concerned with discrete and fuzzy programming, fuzzy multicriteria decision-making, fuzzy preference modeling, fuzzy identification and control, construction of rational mathematical models and their analysis on the basis of experimental design, planning, operation, and control of power systems and subsystems. The results of his investigations have been reflected in more than 260 publications, including pioneering research related to applications of fuzzy set theory in power engineering. More than 45 projects with industry, including projects based on applying techniques of decision-making in a fuzzy environment and fuzzy identification and control, have been realized under his leadership or with his participation. Petr Ekel has been a chair and a member of scientific, advisory, and program committees of more than 80 conferences related to system modeling, optimization, and control, including conferences in the areas of computational intelligence and fuzzy sets. He has been a member of the editorial boards of eight international journals. He is an academician of the Ukrainian Academy of Engineering Sciences.

Roberta Parreiras received the PhD degree in Electrical Engineering from the Federal University of Minas Gerais, Belo Horizonte, Brazil, in 2006. In 2008, she concluded her post-doctoral activities in the Pontifical Catholic University of Minas Gerais, Belo Horizonte, Brazil, where she is currently as an Assistant Professor in the Department of Electrical Engineering. She is also a Consultant in ASOTECH-Advanced System Optimization Technologies, Belo Horizonte, Brazil. Her research has been reported in papers and book chapters in the areas of fuzzy multicriteria decision-making, group decision-making, consensus schemes, multiobjective optimization, evolutionary search algorithms, and robust optimization. Her activities as a consultant in ASOTECH have been mainly associated with planning, including strategic planning, in power engineering.

Foreword

We live in an age which is widely referred to as the age of information. Information has a position of centrality in modern society. Today, it would be hard to live without mobile phones, computers, fax machines, copiers, TV, radio and other artifacts of information. However, in our preoccupation with information we tend to lose sight of the fact that information is not the final destination. What lies beyond information is decision-making. In the final analysis, information is merely a basis for making rational decisions. From the time we wake up in the morning until the time we go to bed, we make a multitude of decisions. Most everyday decisions are made on a subconscious level, but some involve a conscious analysis of consequences of decisions—an analysis aimed at making a decision which in some sense is better than others.

In large measure, the information age began with the pioneering work of Shannon on information theory. Shannon’s first presentation of his theory took place in New York in 1946. As a student at Columbia University, I attended his presentation. I was deeply impressed by Shannon’s ideas. He opened the door to a new world—the world of information and digital information processing.

Two years before the debut of information theory, von Neumann and Morgenstern published a path-breaking book, “Theory of Games and Economic Behavior”. Decision analysis, as we know it today, is the brain-child of two great minds—von Neumann and Morgenstern—and other great minds who followed them. The von Neumann-Morgenstern theory was driven by a quest for a theory which is rigorous, precise and prescriptive. The degree to which von Neumann and Morgenstern have achieved their objective is still a matter of discussion and debate.

Information has many attributes. Among them there are two that stand out in importance—uncertainty and imprecision. Roughly speaking, uncertainty relates to randomness of information while imprecision relates to fuzziness, that is, to unsharpness of class boundaries. Randomness and fuzziness are distinct phenomena, but more often than not what we observe is a mixture. The classical, Aristotelian, bivalent logic is the logic of classes with crisp, (sharp) boundaries. By contrast, fuzzy logic may be viewed as the logic of classes with unsharp boundaries. In fuzzy logic, everything is or is allowed to be a matter of degree.

Like most theories in science, the von Neumann-Morgenstern theory is based on bivalent logic. Based as it is on bivalent logic, the von Neumann-Morgenstern theory addresses the issue of randomness but not that of fuzziness. To deal with randomness, von Neumann and Morgenstern developed the Expected Utility Theory (EUT). The price of being based on bivalent logic is proclivity of EUT to counter-intuitive conclusions. The paradoxes of Allais, Ellsberg and others brought to light serious shortcomings of EUT. In my view, no bivalent-logic-based theory of decision analysis can be paradox-free.

The first attempt to address the issue of fuzziness in decision analysis was made in the 1972 paper by Bellman and I, “Decision-making in a fuzzy environment”. A key idea in this paper involves aggregation of a collection of fuzzy goals (criteria) and fuzzy constraints through conjunction. The issue of uncertainty was not substantively addressed. In a later, 1976 paper, “The linguistic approach and its application to decision analysis”, I introduced the concepts of fuzzy Pareto-optimality and linguistic preference relations. Since then, a number of papers in the literature have addressed the issue of fuzziness in decision analysis.

The book “Fuzzy Multicriteria Decision-Making: Models, Methods and Applications”, or FMD for short, co-authored by Witold Pedrycz, Petr Ekel and Roberta Parreiras, is the first comprehensive treatment of both fuzziness and randomness in decision analysis. As such, it opens the door to construction of far more realistic models of decision problems that can be constructed through the use of bivalent-logic-based theories. In FMD, employment of fuzzy set theory reflects a fact of life: the closer one gets to reality the fuzzier it looks.

What one finds in FMD goes far beyond what has appeared in the literature of decision analysis. Since fuzzy set theory is used extensively in FMD, FMD includes a very thorough and skillfully-organized exposition of those parts of fuzzy set theory which are of relevance to decision analysis. An exposition of applications of fuzzy set theory to decision analysis begins in Chapter 4. In Chapters 4 and 5, principal models of preference relations, among them models based on the ideas of Wald, Laplace, Savage and Hurwicz, are discussed with insight and attention to detail. A realistic example involving resource allocation is analyzed. Chapters 6 and 7 offer a broad panorama of concepts and techniques which have a position of centrality in the fuzzy-set-theory-based approach to decision analysis. In the main, what these chapters offer are important generalizations of existing approaches, but there is much that is new and original. The last two chapters deal with issues which are rarely discussed in textbooks on decision analysis—group decision-making and consensus formation. Based on fuzzy set theory, the authors have succeeded in developing significantly more realistic models of group decision-making and consensus formation that can be found in the literature. What is presented in these chapters is of relevance to policy-making and societal decision processes. What should be underscored is that a concept which has a position of centrality in FMD is that of a fuzzy preference relation. This concept has been an object of discussion in the literature of decision-analysis, but in FMD it plays a far more important role.

A final comment. There is an important issue in decision-making under uncertainty which has not received as much attention in the literature as it deserves. The issue is that of decision-making under second-order uncertainty, that is, uncertainty about uncertainty. Imprecise (uncertain) probabilities fall into this category. The importance of decision-making with imprecise probabilities derives from the fact that most real-world probabilities are not known precisely. In early attempts to deal with imprecise probabilities it was assumed that an imprecise probability distribution is an element of a convex set. Then, a minimax criterion was employed to find an optimal solution. A problem with the minimax criterion is that it is much too conservative. Many variations on the minimax criterion have been suggested, but none has gained wide acceptance. At this juncture, the problem of decision-making under second-order uncertainty is far from solution. I presume that this is the reason why the problem of decision-making under second-order uncertainty is not addressed in Chapter 8.

As a test of a theory to deal with first-order and second-order uncertainties, I should like to suggest a quartet of problems.

Assume that I am given two boxes, each containing twenty black and white balls.

Problem l. A ball is picked at random from box 1. If I pick a white ball, I get a1 dollars. If I pick a black ball, I lose b1 dollars. If I pick a ball at random from box 2, I get a2 dollars if the ball is white and I lose b2 dollars if the ball is black. I can count the number of white balls and black balls in each box. Which box should I choose?

In Problem 2, I am shown the boxes for a few seconds, not long enough to count the balls. I form a perception of the number of white and black balls in each box. These perceptions lead to perception-based (fuzzy) imprecise probabilities. The question is the same: Which box should I choose?

In Problem 3, I am given enough time to be able to count the number of white and black balls, but it is the gains and losses that are perception-based (fuzzy). The question remains the same.

In Problem 4, probabilities, gains and losses are perception-based (fuzzy). The question remains the same.

These four problems are representative of problems which decision-makers encounter in the real world.

The work of Pedrycz, Ekel and Parreiras is a role model of exposition. Carefully worked out examples are crafted to facilitate understanding. Definitions are carefully formulated and make reference to earlier work. Exercises at the end of chapters make the book very useful as a textbook. FMD is highly informative and highly reader-friendly.

In sum, the importance of the work of Pedrycz, Ekel and Parreiras is hard to exaggerate. However, their work is not intended to be read during lunch hour. FMD requires careful study. FMD’s wealth of new, original and applicable results makes it a must reading for all who are concerned with decision analysis. The authors and the publisher, John Wiley, deserve a loud applause.

Lotfi A. Zadeh

August 23, 2010

Berkeley, CA

Preface

This book presents a comprehensive, constructive, well-balanced, fuzzy set modeling framework for a timely, challenging, and important area of multicriteria decision-making. It focuses on ways of representing and handling diverse manifestations of uncertainty and the remarkably multicriteria nature of problems encountered in system projects, planning, operation, and control. The focus of the book is on multiobjective and multiattribute individual and group decision-making. We stress the hands-on nature of the exposition of the overall material and the book comes with a wealth of detailed appealing examples and carefully selected real-world case studies.

We stress the existence of alternative methods for the solution of the most complicated decision-making problems. Especially, diverse techniques for multicriteria analysis of alternatives on the basis of fuzzy preference modeling are presented. The choice of a specific technique is a prerogative of a decision-maker or of a group of decision-makers; it is based on the specificity of the problem and possible sources of available information and its uncertainty.

There have been a number of comprehensive publications in the area of fuzzy decisionmaking, each of them adhering to some pedagogy and highlighting a certain perspective on the decision-making process. The key features of this book, which determine its focus, can be highlighted as follows:

• It describes a complete set of models and methods based on the direct application of fuzzy sets or their combination with other approaches to uncertainty representation and handling for multicriteria decision-making, including multiobjective, multiattribute, and group decisionmaking. We aim at providing constructive answers to the fundamental decision questions “what should we do?” and “how should we do it?” which emerge in the planning, design, operation, and control of complex systems.

• Taking into account that different experts involved in a decision-making process as well as different criteria taken into consideration can demand the use of different ways to represent preferences, the book includes the description of several preference formats, which cover a majority of real situations encountered when preparing information for decision-making. The book presents transformation functions for converting different preference formats into fuzzy preference relations. It bridges an acute gap between decision-making in a fuzzy environment and classical, widely applied decision-making technologies, such as utility theory and an analytic hierarchy process (AHP) approach.

• It describes different aggregation strategies and procedures for constructing collective opinions in group decision-making. The main differences between these strategies are associated with: the points in the process of the multicriteria analysis in which aggregation of the opinion of experts is carried out; the way the experts are considered (mutually dependent or independent); and the character of estimates being aggregated (fuzzy or linguistic estimates, fuzzy preference relations, or fuzzy nondominance degrees).

• It presents different consensus schemes which allow different ways of organizing the meetings among the experts involved in a decision-making process. We show how a level of consensus among the experts and a level of concordance among pairs of opinions can be assessed and monitored.

• It describes ways of evaluating the consequences of decision-making, including the quantification of particular risks or regrets (monocriteria estimates) and aggregated risks or regrets (multicriteria estimates), which are based on a generalization of the classic approach to dealing with uncertainty in decision-making problems.

Due to the coverage of the material, the book will appeal to those active in various areas in which decision-making becomes of paramount relevance: operational research, systems analysis, engineering, management, and economics. Given the way in which the material is structured, the book can serve as a useful reference source for graduate and senior undergraduate students in courses related to the areas indicated above, as well as for courses on decisionmaking, risk management, numerical methods, and knowledge-based systems. The book will be of interest to system analysts and researchers in areas where decision-making technologies are paramount.

The book is organized into 10 chapters. In Chapter 1, which is of an introductory nature, we offer the reader a broad perspective on the fundamentals of decision-making problems and discuss generic notions of decision-making problems such as criteria, objectives, and attributes. Diverse manifestations of the uncertainty factor, its relevance, and visibility in decisionmaking problems are stressed. We also discuss fundamental differences between optimization and decision-making problems. The main objectives, concepts, and characteristics of group decision-making are presented. The role of fuzzy sets is stressed in the general framework of decision-making processes along with their advantages in application to individual and group decision-making problems. The chapter also presents all the required notation and terminology used throughout the book.

The basic concepts of fuzzy sets are introduced in Chapter 2. The fundamental idea of partial memberships, which are conveniently quantified through membership functions and individual membership degrees, is discussed. We present the underlying rationale behind fuzzy sets regarded as information granules and then move on to a detailed description of fuzzy sets by considering the most commonly encountered classes of membership functions and directly relating these classes to the semantics of fuzzy sets. The basic operations on fuzzy sets are further elaborated. The fundamental concepts of fuzzy relations and their main properties, which are of direct relevance to decision-making problems, are discussed. In Chapter 3, which is an immediate continuation of Chapter 2, we present the development aspects of fuzzy set ideas by focusing on the main issues related to the design of fuzzy sets, logic operations, and aggregation of fuzzy sets, and their transformations (mappings).

In Chapter 4, the questions of the construction, analysis, and application of continuous multicriteria decision-making models (multiobjective or X, M models) are considered. The basic definitions related to multicriteria decision-making as well as the commonly utilized approaches to multiobjective decision making are discussed. Particular attention is given to the classic and well-established Bellman-Zadeh approach to decision-making in a fuzzy environment and its application to multicriteria problems. We show that this approach is a convincing means to develop harmonious solutions to multiobjective problems. We illustrate its direct use by solving problems on the multicriteria allocation of resources (or their shortages) as well as some important power engineering problems.

Chapter 5 provides an introduction to preference modeling realized in terms of binary fuzzy relations and addresses certain difficulties that arise in the extension of the classical or Boolean preference structures of binary relations to the fuzzy environment. To alleviate these difficulties, we recall some concepts related to binary fuzzy relations and specific t-norms, t-conorms, and negation operators. We introduce fuzzy preference structures of binary fuzzy relations as well as develop a method for constructing these fuzzy structures, without losing important characteristics that are present in the classical preference structures of binary relations.

Chapter 6 is dedicated to an important problem of forming fuzzy preference relations to analyze multiattribute decision-making models (X, R models). Techniques based on the direct and indirect construction of preference relations are considered. Experts involved in an individual or group decision-making process may present their preferences in heterogeneous forms. Different criteria can also demand the use of different preference forms. Taking this into account, the chapter considers five preference formats which cover a significant part of real situations and which arise in preparing preference information. Considering this as well as the rationality of utilizing fuzzy preference relations for a uniform preference representation, the chapter studies diverse transformation functions required to convert different preference formats into fuzzy preference relations. Some aspects of eliminating inconsistencies in the judgments provided by experts are also tackled here.

In Chapter 7, the essence and key features of problems of multicriteria evaluation, comparison, choice, prioritization, and/or ordering of alternatives in a fuzzy environment, based on the analysis of X, R models, are discussed. There exist two types of situations which generate these models. The first type is associated with a direct statement of multiattribute decisionmaking problems when the consequences of the problems’ solution cannot be estimated with a single criterion. The second type, illustrated in the chapter by analyzing continuous as well as discrete optimization models with fuzzy coefficients, is related to problems that may be solved on the basis of a single criterion; however, if the uncertainty of information does not permit one to obtain unique solutions, it is possible to reduce these problems to multiattribute decisionmaking by applying additional criteria. Diverse techniques of the multicriteria analysis of alternatives in a fuzzy environment developed on the basis of fuzzy preference modeling are considered. These techniques are directly aimed at individual decision-making. However, they can be and are applied to decision-making in a group environment. We stress that although the presented techniques can lead to different solutions, this situation is quite natural and should not be treated as an impediment of the underlying methods. On the contrary, given several methods, the most adequate technique can be selected by taking into account the essence of the problem, possible sources of information, and associated uncertainty.

In Chapter 8, the generalization of the classic approach to dealing with uncertainty of information (based on constructing and analyzing payoff matrices) in monocriteria decisionmaking for multicriteria problems is discussed. The ways of constructing aggregated payoff matrices, modifying the choice criteria, and evaluating particular (monocriteria) and aggregated (multicriteria) risks or regrets in decision-making are studied. We propose a general scheme of multicriteria decision-making, based on a unified application of the generalization of the classic approach and the use of the analysis of X, M and X, R models. The special feature of this scheme is the utilization of all available quantitative information to the highest extent in order to reduce the decision uncertainty regions; if a resolving capacity of the processing of formal information does not lead to unique solutions, the scheme resorts to the application of qualitative information based on the knowledge, experience, and intuition of experts involved in a decision-making process.

The last two chapters are dedicated to different approaches for solving decision-making problems in a group environment. In particular, Chapter 9 is concerned with a certain approach which consists of using aggregation procedures regarded as the exclusive arbitration scheme to arrive at an evaluation, comparison, choice, prioritization, and/or ordering of alternatives for the group. This type of dictatorial arbitration scheme does not require achieving a consensus within a group of decision-makers. Three strategies, based on different aggregation mechanisms, are considered. In each of them, the experts involved in the decision process are seen in a different way: either as mutually dependent individuals who act synergistically in the process of decision-making; or as independent individuals who are capable of solving the decisionmaking problem independently of the other members. We include some examples to illustrate how these strategies are utilized to solve group decision problems by means of different techniques for multiattribute decision-making.

Finally, Chapter 10 presents a suite of procedures for achieving a consensus in the analysis of discrete multicriteria decision-making problems, which involves the evaluation, comparison, choice, prioritization, and/or ordering of alternatives, in a group environment. The chapter presents two different approaches for constructing collective opinions under a rubric of satisfactory consensus: the consensus schemes and the procedures for constructing an optimized consensus. Whereas the former approach requires the experts to review and update their respective opinions in an iterative discussion process, the latter approach represents an attempt to automate the process of constructing and improving the collective opinion, in such a way that the level of consensus in the group is elevated. Each approach has its own advantages and drawbacks. The selection of the most suitable method for a specific application depends mostly on the available time and on the cost of facilitating meetings among the members of the group.

As has been noted, the book can be used in a variety of senior undergraduate and graduate courses. While, in general, one can adhere to the linear flow of coverage of the main topics presented in the consecutive chapters, depending upon the prerequisites, some chapters can be briefly reviewed. For instance, assuming familiarity with the concepts of fuzzy sets, Chapters 2 and 3 could be briefly reviewed with more focus on the design of fuzzy sets and their operational framework.

We would like to take this opportunity to acknowledge support from the National Council for Scientific and Technological Development of Brazil (CNPq) - the research presented in this book was partially supported under CNPq grants 307406/2008-3 and 307574/2008-9. Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada Research Chair (CRC) Program is highly acknowledged.

We would like to express our thanks to colleagues and friends, namely R. C. Berredo, A. F. Bondarenko, E. A. Galperin, I. V. Kokshenev, O. Machado Jr. J. S. C. Martini, C. A. P. S. Martins, R. M. Palhares, V. A. Popov, A. V. Prakhovnik, J. C. B. Queiroz, F. H. Schuffner Neto, G. L. Soares, R. Schinzinger (in memoriam), L. D. B. Terra, J. A. Vasconcelos, and V. V. Zorin, for their encouragement and support. We would also like to thank our graduate students W. J. Araujo, M. F. D. Junges, B. Mendonça Neta, J. G. Pereira Jr. (software development), M. R. Silva (software development), and V. V. Tkachenko for their dedication and hard work.

We are very grateful to the editorial team at John Wiley & Sons, Ltd, especially Debbie Cox and Nicky Skinner, for providing truly professional assistance, expert advice, and continuous encouragement during the realization of this project.

Witold Pedrycz

Petr Ekel

Roberta Parreiras

Edmonton and Belo Horizonte, April 2010

1

Decision-Making in System Project, Planning, Operation, and Control: Motivation, Objectives, and Basic Concepts

The intent of this introductory chapter is to offer the reader a broad perspective on the fundamentals of decision-making problems, provide their general taxonomy in terms of criteria, objectives, and attributes involved, stress the relevance and omnipresence of the uncertainty factor, and highlight the aspects of rationality of decision-making processes. We also highlight the fundamental differences between optimization and decision-making problems. The main objectives, concepts, and characteristics of group decision-making are presented. The role of fuzzy sets is stressed in the general framework of decision-making processes. The main advantages of their application to individual and group decision-making processes are briefly discussed. The chapter also clarifies necessary notations and terminology (such as X, M models and X, R models) used throughout the book.

1.1 Decision-Making and its Support

The life of each person is filled with alternatives. From the moment of conscious thought to a venerable age, from morning awakening to nightly sleeping, a person meets the need to make a decision of some sort. This necessity is associated with the fact that any situation may have two or more mutually exclusive alternatives and it is necessary to choose one among them. The process of decision-making, in the majority of cases, consists of the evaluation of alternatives and the choice of the most preferable from them.

Making the “correct” decision means choosing such an alternative from a possible set of alternatives, in which, by considering all the diversified factors and contradictory requirements, an overall value will be optimized (Pospelov and Pushkin, 1972); that is, it will be favorable to achieving the goal sought to the maximal degree possible.

If the diverse alternatives, met by a person, are considered as some set, then this set usually includes at least three intersecting subsets of alternatives related to personal life, social life, and professional life. The examples include, for instance, deciding where to study, where to work, how to spend time on a weekend, who to elect, and so on.

At the same time, if we speak about any organization, it encounters a number of goals and achieves these goals through the use of diverse types of resources (material, energy, financial, human, etc.) and the performance of managerial functions such as organizing, planning, operating, controlling, and so on (Lu et al., 2007). To carry out these functions, managers engage in a continuous decision-making process. Since each decision implies a reasonable and justifiable choice made among diverse alternatives, the manager can be called a decisionmaker (DM). DMs can be managers at various levels, from a technological process manager to a chief executive officer of a large company, and their decision problems can vary in nature. Furthermore, decisions can be made by individuals or groups (individual decisions are usually made at lower managerial levels and in small organizations, and group decisions are usually made at high managerial levels and in large organizations). The examples include, for instance, deciding what to buy, when to buy, when to visit a place, who to employ, and so on. These problems can concern logistics management, customer relationship management, marketing, and production planning.