Fuzzy Set Theory - and Its Applications

Cover\r......Page 1
Title Page\r......Page 3
Publication Data......Page 4
Contents......Page 5
List of Figures......Page 9
List of Tables......Page 13
Foreword......Page 15
Preface......Page 17
Preface to the Third Edition......Page 19
1.1 Crispness, Vagueness, Fuzziness, Uncertainty......Page 21
1.2 Fuzzy Set Theory......Page 24
PART I. FUZZY MATHEMATICS......Page 28
2.1 Basic Definitions......Page 29
2.2 Basic Set-Theoretic Operations for Fuzzy Sets......Page 34
3.1 Types of Fuzzy Sets......Page 40
3.2 Further Operations on Fuzzy Sets......Page 44
3.2.1 Algebraic Operations......Page 45
3.2.2 Set-Theoretic Operations......Page 46
3.2.3 Criteria for Selecting Appropriate Aggregation Operators......Page 55
Exercises......Page 59
4.1 Fuzzy Measures......Page 61
4.2 Measures of Fuzziness......Page 63
Exercises......Page 67
5.1 The Extension Principle......Page 68
5.2 Operations for Type 2 Fuzzy Sets......Page 69
5.3 Algebraic Operations with Fuzzy Numbers......Page 72
Exercises......Page 81
6.1 Fuzzy Relations on Sets and Fuzzy Sets......Page 83
6.1.1 Compositions of Fuzzy Relations......Page 88
6.1.2 Properties of the Min-Max Composition......Page 91
6.2 Fuzzy Graphs......Page 95
6.3 Special Fuzzy Relations......Page 98
Exercises......Page 102
7.1 Fuzzy Functions on Fuzzy Sets......Page 104
7.2 Extrema of Fuzzy Functions......Page 106
7.3 Integration of Fuzzy Functions......Page 110
7.3.1 Integration of a Fuzzy Function over a Crisp Interval......Page 111
7.3.2 Integration of a (Crisp) Real-Valued Function over aFuzzy Interval......Page 114
7.4 Fuzzy Differentiation......Page 118
8 Possibility Theory, Probability Theory, and Fuzzy Set Theory......Page 121
8.1.1 Fuzzy Sets and Possibility Distributions......Page 122
8.1.2 Possibility and Necessity Measures......Page 126
8.2.1 Probability of a Fuzzy Event as a Scalar......Page 129
8.2.2 Probability of a Fuzzy Event as a Fuzzy Set......Page 130
8.3 Possibility vs. Probability......Page 134
Exercises......Page 137
Part II. APPLICATIONS OF FUZZY SET THEORY......Page 139
9.1 Linguistic Variables......Page 141
9.2.1 Classical Logics Revisited......Page 149
9.2.2 Linguistic Truth Tables......Page 153
9.3 Approximate and Plausible Reasoning......Page 156
9.4 Fuzzy Languages......Page 160
9.5.1 Introduction......Page 169
9.5.2 Fril Rules......Page 170
9.5.3 Inference Methods in Fril......Page 172
9.5.4 Fril Inference for a Single Rule......Page 175
9.5.5 Multiple Rule Case......Page 176
9.5.6 Interval and Point Semantic Unification......Page 177
9.5.7 Least Prejudiced Distribution and Learning......Page 180
9.5.8 Applications of Fril......Page 181
Exercises......Page 183
10.1 Introduction to Expert Systems......Page 185
10.2 Uncertainty Modeling in Expert Systems......Page 192
10.3 Applications......Page 195
Exercises......Page 212
11.1 Origin and Objective......Page 214
11.2 Automatic Control......Page 216
11.3 The Fuzzy Controller......Page 217
11.4.1 The Mamdani Controller......Page 219
11.4.2 The Sugeno Controller......Page 226
11.5.2 Fuzzy Sets......Page 227
11.5.3 Rules......Page 229
11.6 Adaptive Fuzzy Control......Page 230
11.7. 1 Crane Control......Page 231
11.7.2 Control of a Model Car......Page 233
11.7.3 Control of a Diesel Engine......Page 235
11.7.4 Fuzzy Control of a Cement Kiln......Page 236
11.8 Tools......Page 242
11.9 Stability......Page 244
11.10 Extensions......Page 249
Exercises......Page 250
12.1 Introduction......Page 252
12.2 Methods for Fuzzy Data Analysis......Page 254
12.2.1.1.1 Clustering Methods......Page 256
12.2.1.1.2 Cluster Validity......Page 271
12.2.2 Knowledge-Based Approaches......Page 278
12.2.3 Neural Net Approaches......Page 279
12.3.1 Requirements for FDA Tools......Page 280
12.3.2 DataEngine......Page 282
12.4.1 Maintenance Management in Petrochemical Plants......Page 284
12.4.2 Acoustic Quality Control......Page 288
Exercises......Page 291
13.1 Fuzzy Decisions......Page 292
13.2 Fuzzy Linear Programming......Page 298
13.2.1 Symmetric Fuzzy LP......Page 300
13.2.2 Fuzzy LP with Crisp Objective Function......Page 304
13.3.1 Fuzzy Dynamic Programming with Crisp State Transformation Function......Page 311
13.4 Fuzzy Multicriteria Analysis......Page 314
13.4.1 Multi Objective Decision Making (MODM)......Page 315
13.4.2 Multi Attributive Decision Making (MADM)......Page 321
Exercises......Page 329
14.1 Introduction......Page 332
14.2.1 Fuzzy Approach to the Transportation Problem [Chanas et al. 1984]......Page 334
14.2.2 Fuzzy Linear Programming in Logistics......Page 340
14.3.1 A Fuzzy Set Decision Model as Optimization Criterion [von Altrock 1990]......Page 342
14.3.2 Job-Shop Scheduling with Expert Systems [Bensana et al. 1988]......Page 344
14.3.3 A Method to Control Flexible Manufacturing Systems [Hintz and Zimmermann 1989]......Page 347
14.3.4 Aggregate Production and Inventory Planning [Rinks 1982a, b]......Page 354
14.3.5 Fuzzy Mathematical Programming for Maintenance Scheduling......Page 360
14.3.6 Scheduling Courses, Instructors, and Classrooms......Page 362
14.4 Fuzzy Set Models in Inventory Control......Page 367
14.5 A Discrete Location Model [Darzentas 1987]......Page 373
Exercises......Page 376
15.1 Formal Theories vs. Factual Theories vs. Decision Technologies......Page 379
15.1.1 Models in Operations Research and Management Science......Page 383
15.1.2 Testing Factual Models......Page 385
15.2 Empirical Research on Membership Functions......Page 389
15.2.1 Type A-Membership Model......Page 390
15.2.2 Type B-Membership Model......Page 391
15.3 Empirical Research on Aggregators......Page 400
15.4 Conclusions......Page 410
16 Future Perspective......Page 412
Abbreviations of Frequently Cited Journals......Page 415
Bibliography......Page 416
Index......Page 436