Title Details

About the Author xiii

Preface to the Third Edition xv

1 Introduction 1

The Case for Imprecision 2

A Historical Perspective 3

The Utility of Fuzzy Systems 6

Limitations of Fuzzy Systems 8

The Illusion: Ignoring Uncertainty and Accuracy 10

Uncertainty and Information 13

The Unknown 14

Fuzzy Sets and Membership 14

Chance Versus Fuzziness 16

Sets as Points in Hypercubes 18

Summary 20

References 20

Problems 21

2 Classical Sets and Fuzzy Sets 25

Classical Sets 26

Operations on Classical Sets 28

Properties of Classical (Crisp) Sets 29

Mapping of Classical Sets to Functions 32

Fuzzy Sets 34

Fuzzy Set Operations 35

Properties of Fuzzy Sets 37

Alternative Fuzzy Set Operations 40

Summary 41

References 42

Problems 42

3 Classical Relations and Fuzzy Relations 48

Cartesian Product 49

Crisp Relations 49

Cardinality of Crisp Relations 51

Operations on Crisp Relations 52

Properties of Crisp Relations 52

Composition 53

Fuzzy Relations 54

Cardinality of Fuzzy Relations 55

Operations on Fuzzy Relations 55

Properties of Fuzzy Relations 55

Fuzzy Cartesian Product and Composition 55

Tolerance and Equivalence Relations 62

Crisp Equivalence Relation 63

Crisp Tolerance Relation 64

Fuzzy Tolerance and Equivalence Relations 65

Value Assignments 68

Cosine Amplitude 69

Max–Min Method 71

Other Similarity Methods 71

Other Forms of the Composition Operation 72

Summary 72

References 73

Problems 73

4 Properties of Membership Functions, Fuzzification, and Defuzzification 89

Features of the Membership Function 90

Various Forms 92

Fuzzification 93

Defuzzification to Crisp Sets 95

λ-Cuts for Fuzzy Relations 97

Defuzzification to Scalars 98

Summary 110

References 111

Problems 112

5 Logic and Fuzzy Systems 117

Part I Logic 117

Classical Logic 118

Proof 124

Fuzzy Logic 131

Approximate Reasoning 134

Other Forms of the Implication Operation 138

Part II Fuzzy Systems 139

Natural Language 140

Linguistic Hedges 142

Fuzzy (Rule-Based) Systems 145

Graphical Techniques of Inference 148

Summary 159

References 161

Problems 162

6 Development of Membership Functions 174

Membership Value Assignments 175

Intuition 175

Inference 176

Rank Ordering 178

Neural Networks 179

Genetic Algorithms 189

Inductive Reasoning 199

Summary 206

References 206

Problems 207

7 Automated Methods for Fuzzy Systems 211

Definitions 212

Batch Least Squares Algorithm 215

Recursive Least Squares Algorithm 219

Gradient Method 222

Clustering Method 227

Learning From Examples 229

Modified Learning From Examples 233

Summary 242

References 242

Problems 243

8 Fuzzy Systems Simulation 245

Fuzzy Relational Equations 250

Nonlinear Simulation Using Fuzzy Systems 251

Fuzzy Associative Memories (FAMS) 255

Summary 264

References 265

Problems 266

9 Decision Making with Fuzzy Information 276

Fuzzy Synthetic Evaluation 278

Fuzzy Ordering 280

Nontransitive Ranking 283

Preference and Consensus 285

Multiobjective Decision Making 289

Fuzzy Bayesian Decision Method 294

Decision Making Under Fuzzy States and Fuzzy Actions 304

Summary 317

References 318

Problems 319

10 Fuzzy Classification 332

Classification by Equivalence Relations 333

Crisp Relations 333

Fuzzy Relations 335

Cluster Analysis 339

Cluster Validity 340

c-Means Clustering 340

Hard c-Means (HCM) 341

Fuzzy c-Means (FCM) 349

Fuzzy c-Means Algorithm 352

Classification Metric 357

Hardening the Fuzzy c-Partition 360

Similarity Relations from Clustering 361

Summary 362

References 362

Problems 363

11 Fuzzy Pattern Recognition 369

Feature Analysis 370

Partitions of the Feature Space 371

Single-Sample Identification 371

Multifeature Pattern Recognition 378

Image Processing 390

Summary 398

References 399

Problems 400

12 Fuzzy Arithmetic and the Extension Principle 408

Extension Principle 408

Crisp Functions, Mapping, and Relations 409

Functions of Fuzzy Sets – Extension Principle 411

Fuzzy Transform (Mapping) 411

Practical Considerations 413

Fuzzy Arithmetic 418

Interval Analysis in Arithmetic 420

Approximate Methods of Extension 422

Vertex Method 423

DSW Algorithm 426

Restricted DSW Algorithm 428

Comparisons 429

Summary 432

References 433

Problems 433

13 Fuzzy Control Systems 437

Control System Design Problem 439

Control (Decision) Surface 440

Assumptions in a Fuzzy Control System Design 441

Simple Fuzzy Logic Controllers 441

Examples of Fuzzy Control System Design 442

Aircraft Landing Control Problem 446

Fuzzy Engineering Process Control 453

Classical Feedback Control 453

Fuzzy Control 457

Fuzzy Statistical Process Control 464

Measurement Data – Traditional SPC 466

Attribute Data – Traditional SPC 472

Industrial Applications 478

Summary 479

References 482

Problems 484

14 Miscellaneous Topics 501

Fuzzy Optimization 501

One-Dimensional Optimization 502

Fuzzy Cognitive Mapping 508

Concept Variables and Causal Relations 508

Fuzzy Cognitive Maps 510

Agent-Based Models 520

Summary 524

References 525

Problems 526

15 Monotone Measures: Belief, Plausibility, Probability, and Possibility 530

Monotone Measures 531

Belief and Plausibility 532

Evidence Theory 537

Probability Measures 540

Possibility and Necessity Measures 542

Possibility Distributions as Fuzzy Sets 549

Possibility Distributions Derived from Empirical Intervals 551

Deriving Possibility Distributions from Overlapping Intervals 552

Redistributing Weight from Nonconsonant to Consonant Intervals 554

Comparison of Possibility Theory and Probability Theory 568

Summary 569

References 571

Problems 572

Index 579