Fuzzy Sets and Fuzzy Logic

591 Pages·2011·15.13 MB·English

FUZZY SETS AND FUZZY LOGIC
Theory and Applications
GEORGE J. KLIR AND BO YUAN
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Library of Congress Cataloging-in-Publication Data
Klir, George J., 1932-
Fuzzy sets and fuzzy logic: theory and applications / George J.
Klir, Bo Yuan.
p. cm.
Includes bibliographical references and index.
ISBN 0-13-101171-5
1. Fuzzy sets. 2. Fuzzy logic. I. Yuan, Bo. II. Title.
QA-248.K487 1995
511.3--dc20 94-36398
CIP
Editorial/production supervision: Patti Guerrieri
Manufacturing buyer: Alexis R. Heydt
Acquisitions editor: Paul Becker
Editorial assistant: Maureen Diana
Cover Design: Bo Yuan and Milena Klir
Photographs: Milena Klir
01995 by George J. Klir
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ISBN 0-13-101171-5
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To Lotfi Zadeh,
Who had the courage and
To begin thegrand paradigm shift,
And to the many others,
Whose hard work and healthy thinking
Have contributed to the shifting.

CONTENTS
FOREWORD xi
PREFACE xiii
PART I: THEORY
1
1. FROM CLASSICAL (CRISP) SETS TO FUZZY SETS: A Grand Paradigm Shift
1
1.1 Introduction
1
1.2 Crisp Sets: An Overview 5
1.3 Fuzzy Sets: Basic Types 11
1.4 Fuzzy Sets: Basic Concepts 19
1.5 Characteristics and Significance of the Paradigm Shift 30
Notes 32
Exercises 33
2. FUZZY SETS VERSUS CRISP SETS 35
2.1 Additional Properties of a-Cuts 35
2.2 Representations of Fuzzy Sets 39
2.3 Extension Principle for Fuzzy Sets 44
Notes 48
Exercises 49
3. OPERATIONS ON FUZZY SETS 50
3.1 Types of Operations 50
3.2 Fuzzy Complements 51
3.3 Fuzzy Intersections: t-Norms 61
V
Vi Table of Contents
3.4 Fuzzy Unions: i-Conorms 76
3.5 Combinations of Operations 83
3.6 Aggregation Operations 88
Notes 94
Exercises 95
4. FUZZY ARITHMETIC 97
4.1 Fuzzy Numbers 97
4.2 Linguistic Variables 102
4.3 Arithmetic Operations on Intervals 102
4.4 Arithmetic Operations on Fuzzy Numbers 105
4.5 Lattice of Fuzzy Numbers 109
4.6 Fuzzy Equations 114
Notes 117
Exericses 117
5. FUZZY RELATIONS 119
5.1 Crisp versus Fuzzy Relations 119
5.2 Projections and Cylindric Extensions 122
5.3 Binary Fuzzy Relations 124
5.4 Binary Relations on a Single Set 128
5.5 Fuzzy Equivalence Relations 132
5.6 Fuzzy Compatibility Relations 135
5.7 Fuzzy Ordering Relations 137
5.8 Fuzzy Morphisms 141
5.9 Sup-i Compositions of Fuzzy Relations 144
5.10 Inf- o Compositions of Fuzzy Relations 146
Notes 149
Exercises 149
6. FUZZY RELATION EQUATIONS 153
6.1 General Discussion 153
6.2 Problem Partitioning 154
6.3 Solution Method 156
6.4 Fuzzy Relation Equations Based on Sup-i Compositions 162
6.5 Fuzzy Relation Equations Based on Inf-wi Compositions 164
6.6 Approximate Solutions 166
6.7 The Use of Neural Networks 171
Notes 173
Exercises 175
Table of Contents vii
7. POSSIBILITY THEORY 177
7.1 Fuzzy Measures 177
7.2 Evidence Theory 180
7.3 Possibility Theory 187
7.4 Fuzzy Sets and Possibilty Theory 198
7.5 Possibility Theory versus Probability Theory 200
Notes 208
Exercises 209
8. FUZZYLOGIC 212
8.1 Classical Logic: An Overview 212
8.2 Multivalued Logics 217
8.3 Fuzzy Propositions 220
8.4 Fuzzy Quantifiers 225
8.5 Linguistic Hedges 229
8.6 Inference from Conditional Fuzzy Propositions 231
8.7 Inference from Conditional and Qualified Propositions 236
8.8 Inference from Quantified Propositions 239
Notes 242
Exercises 242
9. UNCERTAINTY-BASED INFORMATION 245
9.1 Information and Uncertainty 245
9.2 Nonspecificity of Crisp Sets 247
9.3 Nonspecificity of Fuzzy Sets 250
9.4 Fuzziness of Fuzzy Sets 254
9.5 Uncertainty in Evidence Theory 258
9.6 Summary of Uncertainty Measures 267
9.7 Principles of Uncertainty 269
Notes 277
Exercises 278
PART II. APPLICATIONS 280
10. CONSTRUCTING FUZZY SETS AND OPERATIONS ON FUZZY SETS 280
10.1 General Discussion 280
10.2 Methods of Construction: An Overview 281
10.3 Direct Methods with One Expert 282
10.4 Direct Methods with Multiple Experts 283
10.5 Indirect Methods with One Expert 287
10.6 Indirect Methods with Multiple Experts 288
10.7 Constructions from Sample Data 290
Notes 300
Exercises 301
viii Table of Contents
11. APPROXIMATE REASONING 302
11.1 Fuzzy Expert Systems: An Overview 302
11.2 Fuzy Implications 304
11.3 Selection of Fuzzy Implications 312
11.4 Multiconditional Approximate Reasoning 317
11.5 The Role of Fuzzy Relation Equations 321
11.6 Interval-Valued Approximate Reasoning 323
Notes 325
Exercises 325
12. FUZZY SYSTEMS 327
12.1 General Discussion 327
12.2 Fuzzy Controllers: An Overview 330
12.3 Fuzzy Controllers: An Example 339
12.4 Fuzzy Systems and Neural Networks 344
12.5 Fuzzy Neural Networks 347
12.6 Fuzzy Automata 349
12.7 Fuzzy Dynamic Systems 353
Notes 354
Exercises 356
13. PATTERN RECOGNITION . 357
13.1 Introduction 357
13.2 Fuzzy Clustering 358
13.3 Fuzzy Pattern Recognition 365
13.4 Fuzzy Image Processing 374
Notes 377
Exercises 378
14. FUZZY DATABASES AND INFORMATION RETRIEVAL SYSTEMS 379
14.1 General Discussion 379
14.2 Fuzzy Databases 381
14.3 Fuzzy Information Retrieval 385
Notes 388
Exercises 388
15. FUZZY DECISION MAKING 390
15.1 General Discussion 390
15.2 Individual Decision Making 391
15.3 Multiperson Decision Making 395
15.4 Multicriteria Decision Making 399
15.5 Multistage Decision Making 401
15.6 Fuzzy Ranking Methods 405
Table of Contents ix
15.7 Fuzzy Linear Programming 408
Notes 415
Exercises 416
16. ENGINEERING APPLICATIONS 418
16.1 Introduction 418
16.2 Civil Engineering 419
16.3 Mechanical Engineering 426
16.4 Industrial Engineering 432
16.5 Computer Engineering 436
16.6 Reliability Theory 439
16.7 Robotics 440
Notes 440
Exercises 441
17. MISCELLANEOUS APPLICATIONS 443
17.1 Introduction 443
17.2 Medicine 443
17.3 Economics 450
17.4 Fuzzy Systems and Genetic Algorithms 452
17.5 Fuzzy Regression 454
17.6 Interpersonal Communication 459
17.7 Other Applications 463
Notes 465
Exercises 466
APPENDIX A. NEURAL NETWORKS: An Overview 467
APPENDIX B. GENETIC ALGORITHMS: An Overview 476
APPENDIX C. ROUGH SETS VERSUS FUZZY SETS 481
APPENDIX D. PROOFS OF SOME MATHEMATICAL THEOREMS 484
APPENDIX E. GLOSSARY OF KEY CONCEPTS 487
APPENDIX F GLOSSARY OF SYMBOLS 490
BIBLIOGRAPHY 494
BIBLIOGRAPHICAL INDEX 548
NAME INDEX 552
SUBJECT INDEX 563