FUZZY MULTICRITERIA DECISION-MAKING MODELS, METHODS AND APPLICATIONS

Witold Pedrycz University of Alberta, Canada Systems Research Institute, Polish Academy of Sciences, Poland

Petr Ekel and Roberta Parreiras Pontifical Catholic University ofMinas Gerais, Belo Horizonte, Brazil

©WILEY A John Wiley and Sons, Ltd., Publication

Contents

About the Authors xi

Foreword xiii

Preface xvii

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

1.1 Decision-Making and its Support 1 1.2 Optimization and Decision-Making Problems 6 1.3 Multicriteria Decision-Making 9 1.4 Group Decision-Making 11 1.5 Fuzzy Sets and their Role in Decision-Making Processes 14 1.6 Conclusions 17

References 18

2 Notions and Concepts of Fuzzy Sets: An Introduction 21 2.1 Sets and Fuzzy Sets: A Fundamental Departure from the Principle of

Dichotomy 21 2.2 Interpretation of Fuzzy Sets 24 2.3 Membership Functions and Classes of Fuzzy Sets 26 2.4 Fuzzy Numbers and Intervals 28 2.5 Linguistic Variables 29 2.6 A Generic Characterization of Fuzzy Sets: Some Fundamental

Descriptors 31 2.7 Geometric Interpretation of Sets and Fuzzy Sets 37 2.8 Fuzzy Sets and the Family of a-cuts 38 2.9 Operations on Fuzzy Sets 41

2.9.7 Triangular Norms and Triangular Conorms as Models of Operations on Fuzzy Sets 43

2.9.2 Negations 46 2.10 Fuzzy Relations 47

2.10.1 The Concept of Relations 47 2.10.2 Fuzzy Relations 49 2.10.3 Properties of the Fuzzy Relations 50

Contents

2.10.4 Domain and Codomain of Fuzzy Relations 50 2.10.5 Representation of Fuzzy Relations 51 2.10.6 Equality of Fuzzy Relations 51 2.10.7 Inclusion of Fuzzy Relations 51 2.10.8 Operations on Fuzzy Relations 51 2.10.9 Union of Fuzzy Relations 52 2.10.10 Intersection of Fuzzy Relations 52 2.10.11 Complement of Fuzzy Relations 5 2 2.10.12 Transposition of Fuzzy Relations 52 2.10.13 Cartesian Product of Fuzzy Relations 52 2.10.14 Projection of Fuzzy Relations 53 2.10.15 Cylindrical Extension 54 2.10.16 Reconstruction of Fuzzy Relations 55 2.10.17 Binary Fuzzy Relations 55 2.10.18 Transitive Closure 56 2.10.19 Equivalence and Similarity Relations 57 2.10.20 Compatibility and Proximity Relations 58

2.11 Conclusions 59 References 61

3 Selected Design and Processing Aspects of Fuzzy Sets 63 3.1 The Development of Fuzzy Sets: Elicitation of Membership Functions 63

3.1.1 Semantics of Fuzzy Sets: Some General Observations 63 3.1.2 Fuzzy Set as a Descriptor of Feasible Solutions 65 3.1.3 Fuzzy Set as a Descriptor of the Notion of Typicality 66 3.1.4 Vertical and Horizontal Schemes of Membership Function Estimation 68 3.1.5 Saaty 's Priority Approach of Pairwise Membership

Function Estimation 70 3.1.6 Fuzzy Sets as Granular Representatives of Numeric Data -

A Principle of Justifiable Granularity 73 3.1.7 Design of Fuzzy Sets through Fuzzy Clustering: From Data

to their Granular Abstraction 76 3.1.8 Fuzzy Equalization as a Way of Building Fuzzy Sets

Supported by Experimental Evidence 80 3.1.9 Several Design Guidelines for the Formation of Fuzzy Sets 81

3.2 Aggregation Operations 82 3.2.1 Averaging Operations 83

3.3 Transformations of Fuzzy Sets 84 3.3.1 The Extension Principle 84 3.3.2 Fuzzy Numbers and Fuzzy Arithmetic 87 3.3.3 Interval Arithmetic and a-cuts 88 3.3.4 Fuzzy Arithmetic and the Extension Principle 90 3.3.5 Computing with Triangular Fuzzy Numbers 93 3.3.6 Addition 93 3.3.7 Multiplication 95 3.3.8 Division 96

Contents Vll

3.4 Conclusions 97 References 101

4 Continuous Models of Multicriteria Decision-Making and their Analysis 103 4.1 Continuous Models ((X, M) Models) of Multicriteria Decision-Making 103 4.2 Pareto-Optimal Solutions 104 4.3 Approaches to the Use of DM Information 105 4.4 Methods of Multiobjective Decision-Making 107 4.5 Bellman-Zadeh Approach and its Application to Multicriteria

Decision-Making 113 4.6 Multicriteria Resource Allocation 115 4.7 Adaptive Interactive Decision-Making System for Multicriteria Resource

Allocation 118 4.8 Application of the Bellman-Zadeh Approach to Multicriteria Problems 120 4.9 Conclusions 132

References 134

5 Introduction to Preference Modeling with Binary Fuzzy Relations 137

5.1 Binary Fuzzy Relations and their Fundamental Properties 137 5.2 Preference Modeling with Binary Fuzzy Relations 141 5.3 Preference Structure of Binary Fuzzy Preference Relations 145 5.4 A Method for Constructing a Fuzzy Preference Structure 146 5.5 Consistency of Fuzzy Preference Relations 148 5.6 Conclusions 151

References 152

6 Construction of Fuzzy Preference Relations 155 6.1 Preference Formats 155

6.1.1 Ordering of Alternatives 156 6.7.2 Utility Values 157 6.7.5 Fuzzy Estimates 160 6.7.4 Multiplicative Preference Relations 161 6.7.5 Fuzzy Preference Relations 163

6.2 Ordering of Fuzzy Quantities and the Construction of Fuzzy Preference Relations 166

6.3 Transformation Functions and their Use for Converting Different Preference Formats into Fuzzy Preference Relations 172 6.5.7 Transformation Functions for ARFPR 172 6.3.2 Transformation Functions for NRFPRs 180

6.4 A Method for Repairing Inconsistent Judgments 185 6.4.1 Method for Repairing an Inconsistent ARFPR to Satisfy

Weak Transitivity 186 6.5 Conclusions 188

References 189

V11I Contents

7 Discrete Models of Multicriteria Decision-Making and their Analysis 193 7.1 Optimization Problems with Fuzzy Coefficients and their Analysis 193 7.2 Discrete Models ((X, R) Models) of Multiattribute Decision-Making 202 7.3 Basic Techniques of Analysis of (X, R) Models 203 7.4 Interactive Decision-Making System for Multicriteria Analysis of

Alternatives in a Fuzzy Environment 210 7.5 Multicriteria Analysis of Alternatives with Fuzzy Ordering of Criteria 219 7.6 Multicriteria Analysis of Alternatives with the Concept of Fuzzy Majority 221 7.7 Multicriteria Analysis of Alternatives Based on an Outranking Approach

(Fuzzy Promethee) 226 7.8 Application Examples 232 7.9 Conclusions 243

References 245

8 Generalization of a Classic Approach to Dealing with Uncertainty of Information for Multicriteria Decision Problems 247

8.1 Classic Approach to Dealing with Uncertainty of Information 247 8.2 Choice Criteria 250 8.3 Generalization of the Classic Approach 251 8.4 Modification of the Choice Criteria 252 8.5 General Scheme of Multicriteria Decision-Making under Uncertainty 253 8.6 Application Example 254 8.7 Conclusions 260

References 261

9 Group Decision-Making: Fuzzy Models 263 9.1 Group Decision-Making Problem and its Characteristics 263 9.2 Strategies for the Analysis of Group Decision-Making Problems:

Multiperson and Multiattribute Aggregation Modes 265 9.3 The Different Levels of Influence of Each Expert in the Construction of

the Collective Opinion 268 9.4 Aggregation Operators for Constructing Collective Opinions on the Basis

of Fuzzy Models and their Properties 269 9.5 Consistency of Pairwise Judgments in Group Decision-Making 275 9.6 Fuzzy Group Decision-Making Methods 278 9.7 Conclusions 288

References 291

10 Use of Consensus Schemes in Group Decision-Making 293 10.1 Consensus in Group Decision-Making 293 10.2 Consensus Schemes: Definition and Motivation 295 10.3 Fuzzy Concordance and Fuzzy Consensus Measures 295 10.4 Moderator Interventions 304 10.5 Optimal Consensus in a Fuzzy Environment 305

Contents

10.6 Consensus Schemes in Fuzzy Environment 309 10.6.1 Guidance Procedure of CS FE 311 10.6.2 Guidance Procedure of CSFPR 312 10.6.3 Guidance Procedure of CSRA 318

10.7 An Application Related to the Balanced Scorecard Methodology 324 10.8 Conclusions 330

References 332

Index 335