CONTENTS
Preface = xv
1. Introduction = 1
Background = 2
Uncertainty and Imprecision = 4
Statistics and Random Processes = 6
Uncertainty in Information = 9
Fuzzy Sets and Membership = 10
Chance versus Ambiguity = 12
Summary = 14
References = 15
Problems = 15
2. Classical Sets and Fuzzy Sets = 17
Classical Sets = 18
Operations of Classical Sets = 20
Properties of Classical (Crisp) Sets = 21
Mapping of Classical Sets to Functions = 24
Fuzzy Sets = 26
Fuzzy Sets Operations = 27
Properties of Fuzzy Sets = 28
Sets as Points in Hypercubes = 33
Summary = 35
References = 35
Problems = 35
3. Classical Relations and Fuzzy Relations = 46
Cartesian Product = 47
Crisp Relations = 48
Cardinality of Crisp Relations = 49
Operations on Crisp Relations = 50
Properties of Crisp Relations = 50
Composition = 50
Fuzzy Relations = 52
Cardinality of Fuzzy Relations = 52
Operations on Fuzzy Relations = 53
Properties of Fuzzy Relations = 53
Fuzzy Cartesian Product and Composition = 53
Noninteractive Fuzzy Sets = 55
Tolerance and Equivalence Relations = 60
Crisp Equivalence Relations = 61
Crisp Tolerance Relations = 62
Fuzzy Tolerance and Equivalence Relations = 63
Value Assignments = 66
Cosine Amplitude = 67
Max-min Method = 69
Other similarity Methods = 69
Summary = 70
References = 71
Problems = 71
General Relations = 71
Equivalence Relations = 86
4. Membership Functions = 87
Features of the Membership Function = 88
Standard Forms and Boundaries = 90
Fuzzification = 90
Membership Value Assignments = 91
Intuition = 92
Inference = 93
Rank Ordering = 94
Angular Fuzzy Sets = 95
Neural Networks = 98
Genetic Algorithms = 109
Inductive Reasoning = 117
Summary = 122
References = 125
Problems = 126
5. Fuzzy-to-Crisp Conversions = 130
Lambda-Cuts for Fuzzy Sets = 131
Lambda-Cuts for Fuzzy Relations = 133
Defuzzification Methods = 134
Summary = 146
References = 147
Problems = 147
6. Fuzzy Arithmetic, Numbers, Vectors, and the Extension Principle = 151
Extension Principle = 152
Crisp Functions, Mapping, and Relations = 153
Functions of Fuzzy Sets-Extension Principle = 154
Fuzzy Transform (Mapping) = 155
Practical Considerations = 157
Fuzzy Numbers = 162
Interval Analysis in Arithmetic = 164
Approximate Methods of Extension = 166
Vertex Method = 166
DSW Algorithm = 169
Restricted DSW Algorithm = 172
Comparisons = 173
Fuzzy Vectors = 176
Summary = 178
References = 178
Problems = 179
7. Classical Logic and Fuzzy Logic = 183
Classical Predicate Logic = 184
Tautologies = 189
Contradictions = 191
Equivalence = 191
Exclusive Or and Exclusive Nor = 192
Logical Proofs = 192
Deductive Inferences = 194
Fuzzy Logic = 196
Approximate Reasoning = 199
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs = 207
Other Forms of the Implication Operation = 209
Other Forms of the Composition Operation = 210
Summary = 211
References = 211
Problems = 212
8. Fuzzy Rule-Based Systems = 232
Natural Language = 233
Linguistic Hedges = 236
Rule-Based Systems = 239
Canonical Rule Forms = 240
Decomposition of Compound Rules = 241
Likelihood and Truth Qualification = 243
Aggregation of Fuzzy Rules = 245
Graphical Techniques of Inference = 246
Summary = 254
References = 254
Problems = 255
9. Fuzzy Nonlinear Simulation = 266
Fuzzy Relational Equations = 268
Partitioning = 272
Nonlinear Simulation Using Fuzzy Rule-Based Systems = 275
Fuzzy Associative Memories (FAMs) = 278
Summary = 296
References = 298
Problems = 298
10. Fuzzy Decision Making = 313
Fuzzy Synthetic Evaluation = 315
Fuzzy Ordering = 317
Preference and Consensus = 322
Multiobjective Decision Making = 326
Fuzzy Bayesian Decision Method = 331
Decision Making under Fuzzy States and Fuzzy Actions = 341
Summary = 353
References = 353
Problems = 354
Synthetic Evaluation and Rank Ordering = 354
Multiobjective Decision Making = 359
Bayesian Decision Making = 362
Fuzzy Preference and Consensus = 369
11. Fuzzy Classification = 371
Classification by Equivalence Relations = 372
Crisp Relations = 372
Fuzzy Relations = 374
Cluster Analysis = 379
Cluster Validity = 379
c-Means Clustering = 379
Hard c-Means (HCM) = 380
Fuzzy c-Means (FCM) = 388
Classification Metric = 396
Hardening the Fuzzy c-Partition = 398
Similarity Relations from Clustering = 399
Summary = 400
References = 401
Problems = 401
Exercises for Equivalence Classification = 401
Problems for Fuzzy c-Means = 402
12. Fuzzy Pattern Recognition = 411
Feature Analysis = 413
Partitions of the Feature Space = 413
Single Sample Identification = 414
Mulitifeature Pattern Recognition = 419
Image Processing = 430
Syntactic Recognition = 437
Formal Grammar = 440
Fuzzy Grammar and Syntactic Recognition = 441
Summary = 447
References = 447
Problems = 447
Exercises for Syntactic Pattern Recognition = 467
Exercises for Image Processing = 467
13. Fuzzy Control Systems = 469
Review of Control Systems Theory = 470
System Identification Problem = 472
Control System Design Problem = 472
Control (Decision) Surface = 473
Control System Design Stages = 474
Assumptions in a Fuzzy Control System Design = 474
Simple Fuzzy Logic Controllers = 475
General Fuzzy Logic Controllers = 476
Special Forms of Fuzzy Logic Control System Models = 477
Examples of Fuzzy Control System Design = 478
Classical Fuzzy Control Problem : Inverted Pendulum = 483
Aircraft Landing Control Problem = 490
Industrial Applications = 496
Fuzzy Logic Control of Blood Pressure during Anesthesia = 497
Fuzzy Logic Applications to Image-Processing Equipment = 500
Customer Adaptive Fuzzy Control of Home Heating System = 504
Adaptive Fuzzy Systems = 508
Summary = 514
References = 514
Problems = 515
14. Miscellaneous Topics = 524
Fuzzy Optimization = 524
One-Dimensional Optimization = 525
Fuzzy Maximum and Minimum = 530
Inverse Fuzzy Relational Equations = 533
Fuzzy Linear Regression = 539
The Case of Nonfuzzy Data = 541
The Case of Fuzzy Data = 542
Summary = 551
References = 551
Problems = 552
Fuzzy Optimization = 552
Inverse Fuzzy Relational Equations = 553
Regression = 553
15. Fuzzy Measures : Belief, Plausibility, Probability, and Possibility = 556
Fuzzy Measures = 557
Belief and Plausibility = 557
Evidence Theory = 562
Probability Measures = 565
Possibility and Necessity Measures = 567
Possibility Distribution as Fuzzy Sets = 577
Epilogue = 579
References = 584
Problems = 584
Index = 593
