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<p>Preface xi</p> <p><b>1 Decision-Making in Problems of System Design, Planning, Operation, and Control: Motivation, Objectives, and Basic Notions 1</b></p> <p>1.1 Decision-Making and Its Support 1</p> <p>1.2 Problems of Optimization and Decision-Making 7</p> <p>1.3 Uncertainty Factor and Its Consideration 11</p> <p>1.4 Multicriteria Decision-Making: Multiobjective and Multiattribute Problems 12</p> <p>1.5 Group Decision-Making: Basic Notions 15</p> <p>1.6 Fuzzy Sets in Problems of Decision-Making 19</p> <p>1.7 Conclusions 23</p> <p>References 24</p> <p><b>2 Notions and Concepts of Fuzzy Sets: An Introduction 29</b></p> <p>2.1 Sets and Fuzzy Sets: A Fundamental Departure from the Principle of Dichotomy 29</p> <p>2.2 Interpretation of Fuzzy Sets 33</p> <p>2.3 Membership Functions and Classes of Fuzzy Sets 35</p> <p>2.4 Information Granules and Granular Computing 37</p> <p>2.4.1 Image Processing 38</p> <p>2.4.2 Processing and Interpretation of Time Series 38</p> <p>2.4.3 Granulation of Time 38</p> <p>2.4.4 Data Summarization 39</p> <p>2.4.5 Design of Software Systems 39</p> <p>2.5 Formal Platforms of Information Granularity 39</p> <p>2.5.1 Symbolic Perspective 41</p> <p>2.5.2 Numeric Perspective 42</p> <p>2.6 Intervals and Calculus of Intervals 42</p> <p>2.6.1 Set-Theoretic Operations 43</p> <p>2.6.2 Algebraic Operations on Intervals 44</p> <p>2.6.3 Distance Between Intervals 45</p> <p>2.7 Fuzzy Numbers and Intervals 45</p> <p>2.8 Linguistic Variables 46</p> <p>2.9 A Generic Characterization of Fuzzy Sets: Some Fundamental Descriptors 48</p> <p>2.10 Coverage of Fuzzy Sets 58</p> <p>2.11 Matching Fuzzy Sets 59</p> <p>2.12 Geometric Interpretation of Sets and Fuzzy Sets 60</p> <p>2.13 Fuzzy Set and Its Family of <i>α</i>-Cuts 61</p> <p>2.14 Fuzzy Sets of Higher Type and Fuzzy Order 64</p> <p>2.14.1 Fuzzy Sets of Type −2 64</p> <p>2.14.2 Fuzzy Sets of Order-2 65</p> <p>2.15 Operations on Fuzzy Sets 65</p> <p>2.16 Triangular Norms and Triangular Conorms as Models of Operations on Fuzzy Sets 68</p> <p>2.17 Negations 70</p> <p>2.18 Fuzzy Relations 71</p> <p>2.19 The Concept of Relations 71</p> <p>2.20 Fuzzy Relations 74</p> <p>2.21 Properties of the Fuzzy Relations 76</p> <p>2.21.1 Domain and Codomain of Fuzzy Relations 76</p> <p>2.21.2 Representation of Fuzzy Relations 76</p> <p>2.21.3 Equality of Fuzzy Relations 76</p> <p>2.21.4 Inclusion of Fuzzy Relations 77</p> <p>2.21.5 Operations on Fuzzy Relations 77</p> <p>2.21.6 Union of Fuzzy Relations 77</p> <p>2.21.7 Intersection of Fuzzy Relations 77</p> <p>2.21.8 Complement of Fuzzy Relations 78</p> <p>2.21.9 Transposition of Fuzzy Relations 78</p> <p>2.21.10 Cartesian Product of Fuzzy Relations 78</p> <p>2.21.11 Projection of Fuzzy Relations 78</p> <p>2.21.12 Cylindrical Extension 80</p> <p>2.21.13 Reconstruction of Fuzzy Relations 81</p> <p>2.21.14 Binary Fuzzy Relations 81</p> <p>2.21.15 Transitive Closure 82</p> <p>2.21.16 Equivalence and Similarity Relations 83</p> <p>2.21.17 Compatibility and Proximity Relations 84</p> <p>2.22 Conclusions 85</p> <p>Exercises 85</p> <p>References 89</p> <p><b>3 Design and Processing Aspects of Fuzzy Sets 91</b></p> <p>3.1 The Development of Fuzzy Sets: Elicitation of Membership Functions 91</p> <p>3.1.1 Semantics of Fuzzy Sets: Some General Observations 92</p> <p>3.1.2 Fuzzy Set as a Descriptor of Feasible Solutions 93</p> <p>3.1.3 Fuzzy Set as a Descriptor of the Notion of Typicality 95</p> <p>3.1.4 Vertical and Horizontal Schemes of Membership Function Estimation 96</p> <p>3.1.5 Saaty’s Priority Approach of Pairwise Membership Function Estimation 99</p> <p>3.1.6 Fuzzy Sets as Granular Representatives of Numeric Data – The Principle of Justifiable Granularity 103</p> <p>3.1.7 From Type-0 to Type-1 Information Granules 107</p> <p>3.2 Weighted Data 108</p> <p>3.3 Inhibitory Data 109</p> <p>3.3.1 Design of Fuzzy Sets Through Fuzzy Clustering: From Data to Their Granular Abstraction 110</p> <p>3.4 Quality of Clustering Results 116</p> <p>3.4.1 Cluster Validity Indexes 117</p> <p>3.4.2 Classification Error 118</p> <p>3.4.3 Reconstruction Error 118</p> <p>3.5 From Numeric Data to Granular Data 119</p> <p>3.5.1 Unlabeled Data 119</p> <p>3.5.2 Labeled Data 119</p> <p>3.5.3 Fuzzy Equalization as a Way of Building Fuzzy Sets Supported by Experimental Evidence 121</p> <p>3.5.4 Several Design Guidelines for the Formation of Fuzzy Sets 122</p> <p>3.6 Aggregation Operations 123</p> <p>3.6.1 Averaging Operations 124</p> <p>3.7 Transformations of Fuzzy Sets 125</p> <p>3.7.1 The Extension Principle 125</p> <p>3.7.2 Fuzzy Numbers and Fuzzy Arithmetic 128</p> <p>3.7.3 Interval Arithmetic and α-Cuts 130</p> <p>3.7.4 Fuzzy Arithmetic and the Extension Principle 131</p> <p>3.7.5 Computing with Triangular Fuzzy Numbers 136</p> <p>3.7.6 Addition 136</p> <p>3.7.7 Multiplication 138</p> <p>3.7.8 Division 139</p> <p>3.8 Conclusions 140</p> <p>Exercises 140</p> <p>References 144</p> <p><b>4 <<i>X, F</i>> Models of Multicriteria Decision-Making and Their Analysis 147</b></p> <p>4.1 Models of Multiobjective Decision-Making 147</p> <p>4.2 Pareto Optimal Solutions 148</p> <p>4.3 Approaches to Incorporating Decision-Maker Information 150</p> <p>4.4 Methods of Multiobjective Decision-Making 152</p> <p>4.4.1 Normalization of Objective Functions 152</p> <p>4.4.2 Choice of the Principle of Optimality 152</p> <p>4.4.3 Consideration of Priorities of Objective Functions 152</p> <p>4.5 Bellman–Zadeh Approach to Decision-Making in a Fuzzy Environment and Its Application to Multicriteria Decision-Making 159</p> <p>4.6 OWA Operator Applied to Multiobjective Decision-Making 162</p> <p>4.7 Multiobjective Allocation of Resources and Their Shortages 166</p> <p>4.7.1 Model 1: Allocation of Available Resources 170</p> <p>4.7.2 Model 2: Allocation of Resource Shortages with Unlimited Cuts 170</p> <p>4.7.3 Model 3: Allocation of Resource Shortages with Limited Cuts 171</p> <p>4.8 Practical Examples of Analyzing Multiobjective Problems 178</p> <p>4.9 Conclusions 189</p> <p>Exercises 190</p> <p>References 192</p> <p><b>5 <<i>X, R</i>> Models of Multicriteria Decision-Making and Their Analysis 199</b></p> <p>5.1 Introduction to Preference Modeling with Binary Fuzzy Relations 200</p> <p>5.2 Construction of Fuzzy Preference Relations 205</p> <p>5.3 Preference Formats 215</p> <p>5.3.1 Ordering of Alternatives 216</p> <p>5.3.2 Utility Values 216</p> <p>5.3.3 Fuzzy Estimates 219</p> <p>5.3.4 Multiplicative Preference Relations 220</p> <p>5.4 Transformation Functions and Their Application to Unifying Different Preference Formats 222</p> <p>5.4.1 Transformation of the Ordered Array into the Additive Reciprocal Fuzzy Preference Relation 223</p> <p>5.4.2 Transformation of the Utility Values into the Additive Reciprocal Fuzzy Preference Relation 224</p> <p>5.4.3 Transformation of the Multiplicative Preference Relation into the Additive Reciprocal Fuzzy Preference Relation 225</p> <p>5.4.4 Transformation of the Nonreciprocal Fuzzy Preference Relation into the Additive Reciprocal Fuzzy Preference Relation 226</p> <p>5.4.5 Transformation of the Additive Reciprocal Fuzzy Preference Relation into the Nonreciprocal Fuzzy Preference Relation 228</p> <p>5.4.6 Transformation of the Ordered Array into the Nonreciprocal Fuzzy Preference Relation 229</p> <p>5.4.7 Transformation of the Utility Values into the Nonreciprocal Fuzzy Preference Relation 230</p> <p>5.4.8 Transformation of the Multiplicative Preference Relation into the Nonreciprocal Fuzzy Preference Relation 232</p> <p>5.4.9 Transformation of the Quantitative Information into the Fuzzy Preference Relation 232</p> <p>5.5 Optimization Problems with Fuzzy Coefficients and Their Analysis 233</p> <p>5.6 <<i>X, R</i>> Models of Multicriteria Decision-Making 241</p> <p>5.7 Techniques for Analyzing <<i>X, R</i>> Models 242</p> <p>5.8 Practical Examples of Analyzing <<i>X, R</i>> Models 251</p> <p>5.9 Conclusions 264</p> <p>Exercises 265</p> <p>References 268</p> <p><b>6 Dealing with Uncertainty of Information: A Classic Approach 275</b></p> <p>6.1 Characterization of the Classic Approach to Dealing with Uncertainty of Information 275</p> <p>6.2 Payoff Matrices and Characteristic Estimates 276</p> <p>6.3 Choice Criteria and Their Application 281</p> <p>6.4 Elements of Constructing Representative Combinations of Initial Data, States of Nature, or Scenarios 283</p> <p>6.5 Application Example 285</p> <p>6.6 Conclusions 288</p> <p>Exercises 288</p> <p>References 290</p> <p><b>7 Generalization of the Classic Approach to Dealing with Uncertainty of Information and General Scheme of Multicriteria Decision-Making under Conditions of Uncertainty 291</b></p> <p>7.1 Generalization of the Classic Approach to Dealing with Uncertainty of Information in Multicriteria Decision Problems 292</p> <p>7.2 Consideration of Choice Criteria of the Classic Approach to Dealing with Uncertainty of Information as Objective Functions within the Framework of <<i>X, F</i>> Models 299</p> <p>7.3 Construction of Objectives and Elaboration of Representative Combination of Initial Data, States of Nature, or Scenarios using Qualitative Information 309</p> <p>7.3.1 Elicitation of Preferences 310</p> <p>7.3.2 Representation of Preferences Within Multiplicative Preference Relations 312</p> <p>7.3.3 Definition of Preference Vectors on the Basis of Applying the AHP 314</p> <p>7.3.4 Aggregation of Preferences and Generation of Representative Combinations of Initial Data, States of Nature, or Scenarios 314</p> <p>7.4 General Scheme of Multicriteria Decision-Making under Conditions of Uncertainty 315</p> <p>7.5 Application Studies 317</p> <p>7.6 Conclusions 333</p> <p>Exercises 333</p> <p>References 335</p> <p>Index 339</p>

<p><b>PETR EKEL,</b> DSc (habil.), PhD, is a Full Professor in the Graduate Program of Electrical Engineering, Pontifical Catholic University of Minas Gerais, Belo Horizonte, Brazil. <p><b>WITOLD PEDRYCZ,</b> DSc (habil.), PhD, is a Full Professor and Canada Research Chair (CRC) in Computational Intelligence in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. <p><b>JOEL PEREIRA, J<small>R</small>.,</b> PhD, is a Researcher in ASOTECH – Advanced System Optimization Technologies Ltda., Belo Horizonte, Brazil.

<p><b>A GUIDE TO THE VARIOUS MODELS AND METHODS TO MULTICRITERIA DECISION-MAKING IN CONDITIONS OF UNCERTAINTY PRESENTED IN A SYSTEMATIC APPROACH</b> <p><i>Multicriteria Decision-Making Under Conditions of Uncertainty</i> presents approaches that help to answer the fundamental questions at the center of all decision-making problems: "What to do?" and "How to do it?" The book explores methods of representing and handling diverse manifestations of the uncertainty factor and a multicriteria nature of problems that can arise in system design, planning, operation, and control. The authors—noted experts on the topic—and their book covers essential questions, including notions and fundamental concepts of fuzzy sets, models and methods of multiobjective as well as multiattribute decision-making, the classical approach to dealing with uncertainty of information and its generalization for analyzing multicriteria problems in condition of uncertainty, and more. <p>This comprehensive book contains information on "harmonious solutions" in multiobjective problem-solving (analyzing <<i>X, F></i> models), construction and analysis of <<i>X</i>, <i>R</i>> models, results aimed at generating robust solutions in analyzing multicriteria problems under uncertainty, and more. In addition, the book includes illustrative examples of various applications, including real-world case studies related to the authors' various industrial projects. This important resource: <ul> <li>Explains the design and processing aspect of fuzzy sets, including construction of membership functions, fuzzy numbers, fuzzy relations, aggregation operations, and fuzzy sets transformations</li> <li>Describes models of multiobjective decision-making (<<i>X</i>. <i>M</i>> models), their analysis on the basis of using the Bellman-Zadeh approach to decision-making in a fuzzy environment, and their diverse applications, including multicriteria allocation of resources</li> <li>Investigates models of multiattribute decision-making (<<i>X</i>, <i>R</i>> models) and their analysis on the basis of the construction and processing of fuzzy preference relations as well as demonstrating their applications to solve diverse classes of multiattribute problems</li> <li>Explores notions of payoff matrices and fuzzy-set-based generalization and modification of the classic approach to decision-making under conditions of uncertainty to generate robust solutions in analyzing multicriteria problems</li> </ul> <p>Written for students, researchers and practitioners in disciplines in which decision-making is of paramount relevance, <i>Multicriteria Decision-Making Under Conditions of Uncertainty</i> presents a systematic and current approach that encompasses a range of models and methods as well as new applications.

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