About the Author xi

Preface to the Fourth Edition xiii

1 Introduction 1

The Case for Imprecision 2

A Historical Perspective 4

The Utility of Fuzzy Systems 7

Limitations of Fuzzy Systems 9

The Illusion: Ignoring Uncertainty and Accuracy 11

Uncertainty and Information 13

Fuzzy Sets and Membership 14

Chance versus Fuzziness 17

Intuition of Uncertainty: Fuzzy versus Probability 19

Sets as Points in Hypercubes 21

Summary 23

References 23

Problems 24

2 Classical Sets and Fuzzy Sets 27

Classical Sets 28

Fuzzy Sets 36

Summary 45

References 46

Problems 46

3 Classical Relations and Fuzzy Relations 51

Cartesian Product 52

Crisp Relations 53

Fuzzy Relations 58

Tolerance and Equivalence Relations 67

Fuzzy Tolerance and Equivalence Relations 70

Value Assignments 72

Other Forms of the Composition Operation 76

Summary 77

References 77

Problems 77

4 Properties of Membership Functions, Fuzzification, and Defuzzification 84

Features of the Membership Function 85

Various Forms 87

Fuzzification 88

Defuzzification to Crisp Sets 90

lambda-Cuts for Fuzzy Relations 92

Defuzzification to Scalars 93

Summary 102

References 103

Problems 104

5 Logic and Fuzzy Systems 107

Part I: Logic 107

Classical Logic 108

Fuzzy Logic 122

Part II: Fuzzy Systems 132

Summary 151

References 153

Problems 154

6 Historical Methods of Developing Membership Functions 163

Membership Value Assignments 164

Intuition 164

Inference 165

Rank Ordering 167

Neural Networks 168

Genetic Algorithms 179

Inductive Reasoning 188

Summary 195

References 196

Problems 197

7 Automated Methods for Fuzzy Systems 201

Definitions 202

Batch Least Squares Algorithm 205

Recursive Least Squares Algorithm 210

Gradient Method 213

Clustering Method 218

Learning from Examples 221

Modified Learning from Examples 224

Summary 233

References 235

Problems 235

8 Fuzzy Systems Simulation 237

Fuzzy Relational Equations 242

Nonlinear Simulation Using Fuzzy Systems 243

Fuzzy Associative Memories (FAMs) 246

Summary 257

References 258

Problems 259

9 Decision Making with Fuzzy Information 265

Fuzzy Synthetic Evaluation 267

Fuzzy Ordering 269

Nontransitive Ranking 272

Preference and Consensus 275

Multiobjective Decision Making 279

Fuzzy Bayesian Decision Method 285

Decision Making under Fuzzy States and Fuzzy Actions 295

Summary 309

References 310

Problems 311

10 Fuzzy Classification and Pattern Recognition 323

Fuzzy Classification 324

Classification by Equivalence Relations 324

Cluster Analysis 332

Cluster Validity 332

c-Means Clustering 333

Hard c-Means (HCM) 333

Fuzzy c-Means (FCM) 343

Classification Metric 351

Hardening the Fuzzy c-Partition 354

Similarity Relations from Clustering 356

Fuzzy Pattern Recognition 357

Single-Sample Identification 357

Multifeature Pattern Recognition 365

Summary 378

References 379

Problems 380

11 Fuzzy Control Systems 388

Control System Design Problem 390

Examples of Fuzzy Control System Design 393

Fuzzy Engineering Process Control 404

Fuzzy Statistical Process Control 417

Industrial Applications 431

Summary 434

References 437

Problems 438

12 Applications of Fuzzy Systems Using Miscellaneous Models 455

Fuzzy Optimization 455

Fuzzy Cognitive Mapping 462

Agent-Based Models 477

Fuzzy Arithmetic and the Extension Principle 481

Fuzzy Algebra 487

Data Fusion 491

Summary 498

References 498

Problems 500

13 Monotone Measures: Belief, Plausibility, Probability, and Possibility 505

Monotone Measures 506

Belief and Plausibility 507

Evidence Theory 512

Probability Measures 515

Possibility and Necessity Measures 517

Possibility Distributions as Fuzzy Sets 525

Possibility Distributions Derived from Empirical Intervals 528

Summary 548

References 549

Problems 550

Index 554