AN INTRODUCTION TO FUZZY SETS Analysis and Design. Witold Pedrycz and Fernando Gomide
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1 AN INTRODUCTION TO FUZZY SETS Analysis and Design Witold Pedrycz and Fernando Gomide A Bradford Book The MIT Press Cambridge, Massachusetts London, England
2 Foreword - Preface Introduction xiii xxv xxi FUNDAMENTALS OF FUZZY SETS 1 Basic Notions and Concepts of Fuzzy Sets 3.1 Set Membership and Fuzzy Sets 3.2 Basic Definition of a Fuzzy Set 5.3 Types of Membership Functions 8.4 Characteristics of a Fuzzy Set 10.5 Basic Relationships between Fuzzy Sets: Equality and Inclusion Fuzzy Sets and Sets: The Representation Theorem The Extension Principle Membership Function Determination Generalizations of Fuzzy Sets Chapter Summary Problems 29 References 30 Fuzzy Set Operations Set Theory Operations and Their Properties Triangular Norms Aggregation Operations on Fuzzy Sets Sensitivity of Fuzzy Sets Operators Negations Comparison Operations on Fuzzy Sets Chapter Summary Problems 56 References 57 Information-Based Characterization of^fuzzy Sets Entropy Measures of Fuzziness ^ Energy Measures of Fuzziness Specificity of a Fuzzy Set Frames of Cognition Information Encoding and Decoding Using Linguistic Landmarks Decoding Mechanisms for Pointwise Data 72
3 3.7 - Decoding Using Membership Functions of the Linguistic Terms of the Codebook General Possibility-Necessity Decoding Distance between Fuzzy sets Based on Their Internal, Linguistic Representation Chapter Summary Problems 81 References 84 Fuzzy Relations and Their Calculus Relations and Fuzzy Relations Operations on Fuzzy Relations Compositions of Fuzzy Relations Projections and Cylindric Extensions of Fuzzy Relations Binary Fuzzy Relations Some Classes of Fuzzy Relations Fuzzy-Relational Equations Estimation and Inverse Problem in Fuzzy Relational Equations Solving Fuzzy-Relational Equations with the sup-t Composition Solutions to Dual Fuzzy-Relational Equations Adjoint Fuzzy-Relational Equations Generalizations of Fuzzy Relational Equations Approximate Solutions to Fuzzy-Relational Equations Chapter Summary Problems 124 References 126 Fuzzy Numbers Denning Fuzzy Numbers Interval Analysis and Fuzzy Numbers Computing with Fuzzy Numbers Triangular Fuzzy Numbers and Basic Operations General Formulas for LR Fuzzy Numbers Accumulation of Fuzziness in Computing with Fuzzy Numbers Inverse Problem in Computation with Fuzzy Numbers Fuzzy Numbers and Approximate Operations Chapter Summary Problems 148 References 150
4 ix Contents Fuzzy Sets and Probability Introduction Probability and Fuzzy Sets Hybrid Fuzzy-Probabilistic Models of Uncertainty Probability-Possibility Transformations Probabilistic Sets and Fuzzy Random Variables Chapter Summary Problems 162 References 164 Linguistic Variables Introduction Linguistic Variables: Formalization Computing with Linguistic Variables: Hedges, Connectives and Negation Linguistic Approximation Linguistic Quantifiers Applications of Linguistic Variables Chapter Summary Problems 179 References 180 Fuzzy Logic Introduction Propositional Calculus Predicate Logic Many-Valued Logic Fuzzy Logic Computing with Fuzzy Logic Some Remarks about Inference Methods Chapter Summary Problems 203 References 204 Fuzzy Measures and Fuzzy Integrals Fuzzy Measures Fuzzy Integrals Chapter Summary Problems 216 References 217
5 II COMPUTATIONAL MODELS Rule-Based Computations Rules in Knowledge Representation Syntax of Fuzzy Rules Semantics of Fuzzy Rules and Inference Computing with Fuzzy Rules Some Properties of Fuzzy Rule-Based Systems Rule Consistency and Completeness Chapter Summary Problems 261 References Fuzzy Neurocomputation Neural Networks: Basic Notions, Architectures, and Learning Logic-Based Neurons Logic Neurons and Fuzzy Neural Networks with Feedback Referential Logic-Based Neurons Fuzzy Threshold Neurons Classes of Fuzzy Neural Networks Referential Processor Fuzzy Cellular Automata Learning Selected Aspects of Knowledge Representation in Fuzzy Neural Networks Chapter Summary Problems 300 References Fuzzy Evolutionary Computation Introduction Genetic Algorithms Design of Fuzzy Rule-Based Systems with Genetic Algorithms Learning in Fuzzy Neural Networks with Genetic Algorithms Evolution Strategies Hybrid and Cooperating Approaches Chapter Summary 323
6 12.8 Problems 323 References Fuzzy Modeling Fuzzy Models: Beyond Numerical Computations Main Phases of System Modeling Fundamental Design Objectives in System Modeling General Topology of Fuzzy Models Compatibility of Encoding and Decoding Modules Classes of Fuzzy Models Verification and Validation of Fuzzy Models Chapter Summary Problems 354 References 357 III PROBLEM SOLVING WITH FUZZY SETS Methodology Analysis and Design Fuzzy Controllers and Fuzzy Control Mathematical Programming and Fuzzy Optimization Chapter Summary Problems 395 References Case Studies Traffic Intersection Control Distributed Traffic Control Elevator Group Control Induction Motor Control Communication Network Planning Neurocomputation in Fault Diagnosis of Dynamic Systems Multicommodity Transportation Planning in Railways 453 References 460 Index 463
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