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
?-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
