Fuzzy Set Theory-and Its Applications, Fourth Edition

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Fuzzy Set Theory-and Its Applications, Fourth Edition
Library of Congress Cataloging-in-Publication Data Zimmermann, H.-J(Hans-Jurgen), 1934 Fuzzy set theory--and its applications /H.J. Zimmermann.--4th ed cm Includes bibliographical references and index ISBN978-94-010-3870-6 ISBN978-94-010-0646-0( eBook) DOI10.1007/978-94-010-0646-0 1. Fuzzy sets. 2. Operations research. I. Title QA248Z552001 511.322-dc21 2001038123 Copyright C 2001 by Springer Science+ Business Media New York Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 4th edition 2001 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying recording, or otherwise, without the prior written permission of the publisher, Springer Science+ Business Media, LLC Printed on acid-free paper. Contents List of Figures List of Tables Foreword Preface Preface to the fourth edition Introduction to Fuzzy Sets Crispness, Vagueness, Fuzziness, Uncertainty 12 Fuzzy Set Theory Part I: Fuzzy Mathematics Fuzzy Sets-Basic Definitions 2.1 Basic Definitions 2.2 Basic Set-Theoretic Operations for Fuzzy Sets 16 3 Extensions Types of Fuzzy Sets 23 32 Further Operations on Fuzzy Sets 27 3.2.1 Algebraic Operations 28 3.2.2 Set-Theoretic Operations 29 3. 2. 3 Criteria for Selecting Appropriate Aggregation Operators Fuzzy Measures and Measures of Fuzziness 4.1 Fuzzy Measures 4.2 Measures of Fuzziness The Extension Principle and Applications 5.1 The Extension Principle 3779556 5 52 Operations for Type 2 Fuzzy Sets 53 Algebraic Operations with Fuzzy Numbers 59 53 Special Extended Operations 61 532 Extended Operations for LR-Representation of fuzzy sets 64 CONTENTS v66 Fuzzy Relations and Fuzzy Graphs Fuzzy Relations on Sets and Fuzzy sets 6.1.1 Compositions of Fuzzy Relations 76 6.1.2 Properties of the Min-Max Composition 62 Fuzzy graphs 6.3 Special Fuzzy Relations 86 Fuzzy Analysis Fuzzy Functions on Fuzzy Sets 9 Extrema of Fuzzy Functions 95 7.3 Integration of Fuzzy Functions 7.3.1 Integration of a Fuzzy Function over a Crisp Interval 100 7.3.2 Integration of a(Crisp) Real-valued Function over a Fuzzy Interval 7,4 Fuzzy Differentiation 107 8 Uncertainty Modeling 8.1 Application-oriented Modeling of Uncertainty 111 8.1.1 Causes of Uncertainty 114 8.1.2 Type of Available Information 8.1.3 Uncertainty Methods 118 8.1.4 Uncertainty Theories as Transformers of Information 119 8.1.5 Matching Uncertainty Theory and Uncertain Phenomena 120 8 Possibility Theory 8. 2.1 Fuzzy Sets and Possibility Distributions 8. 2.2 Possibility and Necessity Measures 126 8.3 Probability of Fuzzy Events 129 8.3.1 Probability of a Fuzzy Event as a Scalar 12 8. 3.2 Probability of a Fuzzy Event as a Fuzzy Set 131 Possibility VS. Probability 133 Part I: Applications of Fuzzy Set Theory 139 9 Fuzzy Logic and Approximate Reasoning 4 9.1 Linguistic Variables 141 92 Fuzzy Logic 149 9.2.1 Classical Logics Revisited 149 92.2 Linguistic Truth Tables 153 93 Approximate and Plausible Reasoning 156 94 Fuzzy Languages 160 95 Support Logic Programming and Fril 169 95 Introduction 169 952 Fril rules 170 953 Inference methods in fril 172 9.5. 4 Fril Inference for a Single Rule 175 9.5.5 Multiple Rule Case 176 95.6 Interval and Point Semantic Unification 177 9.5.7 Least Prejudiced Distribution and Learning 179 9.5. 8 Applications of Fril 181 CONTENTS 10 Fuzzy Sets and Expert Systems 185 10.1 Introduction to Expert Systems 185 10.2 Uncertainty Modeling in Expert Systems 193 10.3 Applications 203 Fuzzy Control 223 11.1 Origin and objective Automatic Control 225 11.3 The Fuzzy controller 226 11.4 Types of Fuzzy Controllers 228 11.4.1 the mamdani Controller 228 11.4.2 Defuzzification 232 11.4.3 The Sugeno Controller 239 11.5 Design Parameters 240 11.5.1 Scaling Factors 240 11.5.2 Fuzzy Sets 240 11.53Ru|e 242 11.6 Adaptive fuzzy control 243 Applications 11.7.1 Crane Contro 244 11.7.2 Control of a model car 246 11.7.3 Control of a Diesel Engine 248 11.7. 4 Fuzzy Control of a Cement Kiln 249 118 Tools 255 Stability 257 11.10 Extensions 262 Fuzzy Data Bases and Queries 265 12.1 Introduction 265 12.2 Fuzzy Relational Databases 266 123 Fuzzy Queries in Crisp Databases 268 13 Fuzzy Data Analysis 277 13.1 Introduction 277 132 Methods for Fuzzy Data Analysis 279 13.2.1 Algorithmic Approaches 281 13.2.2 Knowledge-Based Approaches 302 13.2.3 Neural Net Approaches 304 133 Dynamic Fuzzy Data Analysis 306 13.3. Problem Description 306 13.3.2 Similarity of Functions 307 13.3.3 Approaches for Analysic Dynamic Systems 313 134 Tools for Fuzzy Data Analysis 317 13.4.1 Requirements for FDa tools 317 13.4.2 Data Engine 318 Applications of FDA 322 13.5. 1 Maintenance Management in Petrochemical Plants 322 13.5.2 Acoustic Quality Contro 323 CONTENTS 14 Decision Making in Fuzzy Environments 329 14.1 Fuzzy Decisions 14.2 Fuzzy Linear Programming 14.2.1 Symmetric Fuzzy LP 337 14.2.2 Fuzzy LP with Crisp Objective Function 342 14.3 Fuzzy Dynamic Programming 348 14.3.1 Fuzzy Dynamic Programming with Crisp State Transformation Function 349 14.4 Fuzzy Multicriteria Analysis 352 14.4.1 Multi Objective Decision Making(MODM) 353 14.4.2 Multi Attributive Decision Making(MADM 359 Applications of fuzzy sets in Engineering and management 371 15.1 Introduction 371 152 Engineering Applications 373 152 Linguistic Evaluation and Ranking of Machine Tools 375 15.2.2 Fault Detection in Gearboxes 38 153 Applications in Management 389 15.3.1 A Discrete Location Mode 390 15.3.2 Fuzzy Set Models in Logistics 393 15.3.2. 1 Fuzzy Approach to the Transportation Problem 393 15.3.2.2 Fuzzy Linear Programming in Logistics 398 15.3. 3 Fuzzy Sets in Scheduling 40 15.3.3.1 Job-Shop Scheduling with Expert Systems 401 15.3.3.2 A Method to Control Flexible Manufacturing Systems 405 15.3.3.3 Aggregate Production and Inventory Planning 411 15.3. 3.4 Fuzzy Mathematical Programming for Maintenance Scheduling 418 15.3. 3.5 Scheduling Courses, Instructors, and Classrooms 419 15.3.4 Fuzzy Set Models in Inventory Control 426 15. 3.5 Fuzzy Sets in Marketing 432 15.3.5. 1 Customer Segmentation in Banking and Finance 432 15.3.5.2 Bank Customer Segmentation based on Customer Behavior 433 16 Empirical Research in Fuzzy Set Theory 443 16.1 Formal Theories vs Factual Theories vs. Decision Technologies 443 16.1.1 Models in Operations Research and Management Science 16.1.2 Testing Factual models 449 162 Empirical Research on Membership Functions 453 16. 2.1 Type A-Membership Model 454 16. 2.2 Type B-Membership Model 456 16.3 Empirical Research on Aggregators 63 164 Conclusions 474 Future Perspectives 477 Abbreviations of Frequently Cited Journals 481 Bibliography 483 Index 507 List of Figures Figure 1-1 Concept hierarchy of creditworthiness 5 Figure 2-1 Real numbers close to 10. Figure 2-2a Convex fuzzy set 15 Figure 2-2b Nonconvex fuzzy set 15 Figure 2-3 Union and intersection of fuzzy sets 18 Figure 3-1 Fuzzy sets Vs probabilistic sets 26 Figure 3-2 Mapping of t-norms, t-conorms, and averaging operators. 38 Figure 5-1 The extension principle 57 Figure5-2 Trapezoidal“ fuzzy number” 60 Figure 5-3 LR-representation of fuzzy numbers 65 Figure 6-1 Fuzzy graphs 84 Figure 6-2 Fuzzy forests 86 Figure 6-3 Graphs that are not forests 86 Figure 7-1 Maximizing set 96 Figure 7-2 a fuzzy function 97 Figure 7-3 Triangular fuzzy numbers representing a fuzzy function 98 Figure 7-4 The maximum of a fuzzy function 99 Figure 7-5 Fuzzily bounded interval 1044 Figure 8-1 Uncertainty as situational property 113 Figure 8-2 Probability of a fuzzy event 134 Figure 9-1 Linguistic variable"Age 143 Figure 9-3 Linguistic variable"Truth"lyn Figure 9-2 Linguistic variable "Probabi 144 145 Figure94 Terms“ True" and" False”. 146 LIST OF FIGURES Figure 10-1 Structure of an expert system 189 Figure 10-2 Semantic net 19 Figure 10-3 Linguistic descriptors 205 Figure 10-4 Label sets for semantic representation 205 Figure 10-5 Linguistic variables for occurrence and confirmability 209 Figure 10-6 Inference network for damage assessment of existing structures [Ishizuka et al. 1982, p. 263 212 Figure 10-7 Combination of two two-dimensional portfolios 215 Figure 10-8 Criteria tree for technology attractiveness 216 Figure 10-9 Terms of"degree of achievement. 217 Figure 10-10 Aggregation of linguistic variables 218 Figure 10-11 Portfolio with linguistic input 220 Figure 10-12 Structure of ESP 221 Figure 11-1 Automatic feedback control Figure 11-2 Generic Mamdani fuzzy controller 223 Figure 11-3 Linguistic variableTemperature 229 Figure 11 Rule consequences in the heating system example 232 Figure 11-5 Extreme Value Strategies 234 Figure 11-6 CoA Defuzzification 235 Figure 11-7 Neighboring membership functions 236 Figure 11-8 Separate membership functions 236 Figure 11-9 Parameters describing fuzzy sets 241 Figure 11-10 Influence of symmetry 242 Figure 11-11 Condition width 242 Figure 11-12 Container crane [von Altrock 1993] 245 Figure 11-13 Phases of motion 245 Figure 11-14 Input variables [Sugeno and Nishida 1985, p. 106 246 Figure 11-15 Trajectories of the fuzzy controlled model car [Sugeno and Nishida 1985, p. 112 247 Figure 11-16 Fuzzy model car [von Altrock et aL. 1992, p 42 248 Figure 11-17 Experimental design [von Altrock et aL. 1992, p. 48 249 Figure 11-18 FCR VS fuel injection timing [Murayama et al. 1985, p. 64]. 250 Figure 11-19 Control algorithm [Murayama et al. 1985 251 Figure 11-20 Experimental results [Murayama et aL. 1985 252 Figure 11-21 Schematic diagram of rotary cement kiln [Umbers and King1981,p.371] 252 Figure 11-22 Controller development in fuzzyTECH [von Altrock et al 1992] 256 Figure 11-23 Rule base for model car [von Altrock et al. 1992 Figure 11-24 Simulation screen [von Altrock et al. 1992 257 Figure 11-25 Fuzzy controller as a nonlinear transfer element 258 Figure 11-26 Classification of stability analysis approaches 259 Figure 11-27 Linguistic state space 260 Figure 11-28 Linguistic trajectory Figure 13-1 Scope of data analysis. Figure 13-2 Possible data structure in the plane 22 Figure 13-3 Performance of cluster criteria 283 Figure 13-4 Dendogram for hierarchical clusters LIST OF FIGURES X Figure 13-5 Fuzzy graph 285 Figure 13-6 Dendogram for graph-theoretic clusters 285 Figure 13-7 The butterfly 286 Figure 13-8 Crisp clusters of the butterfly 287 Figure 13-9 Cluster 1 of the butterfly 287 Figure 13-10 Cluster 2 of the butterfly 288 Figure 13-11 Clusters for m= 1.25 295 Figure 13-12 Clusters for m= 2. 295 Figure 13-13 Clusters by the FSC(a)Data set; (b)circles found by FSC;(c)data set; (d) circles found by FSC 300 Figure 13-14 Data sets [Krishnapuram and Keller 1993 301 Figure 13-15 Knowledge-based classification 303 Figure 13-16 Linguistic variables "Depth of Cut"and"Feed 304 Figure 13-17 Knowledge base 304 Figure 13-18 Basic structure of the knowledge-based system 305 Figure 13-19 (a) States of objects at a point of time; (b) projections of rajectories over time into the feature space 307 Figure 13-20 Structural and pointwise similarity 308 Figure 13-21 Fictitious developments of share prices 309 Figure 13-22 Idealized characteristic patterns of time signals for(a)an intact engine; (b)an engine with some defect. 309 Figure 13-23(a) The fuzzy set"approximately zero"(uly)), the function f(t)and the resulting pointwise similarity u(f (t));(b) projection of pointwise similarity into the plane(t,u(f(t))) 311 Figure 13-24 Transformation of a feature vector containing trajectories into trajectories into a usual feature vector 314 Figure 13-25 Input and output of the functional fuzzy C-means 315 Figure 13-26 Structure of DataEngine 318 Figure 13-27 Screen shot of DataEngine 320 Figure 13-28 Cracking furnace 324 Figure 13-29 Furnace temperature Figure 13-30 Fuzzy classification of continuous process 325 Figure 13-31 Application of DataEngine for acoustic quality control 327 Figure 14-1 A classical decision under certainty 330 Figure 14-2 A fuzzy decision 332 Figure 14-3 Optimal dividend as maximizing decision Figure144 Feasible regions forμ成(x)=0and成x= 344 Figure 14-5 Fuzzy decision 34 Figure 14-6 Basic structure of a dynamic programming model 349 Figure 14-7 The vector-maximum problem Figure 14-8 Fuzzy LP with min-operator 357 Figure 14-9 Fuzzy sets representing weights and ratings 366 Figure 14-10 Final ratings of alternatives 368 Figure 14-11 Preferability of alternative 2 over all others Figure 15-1 Linguistic values for variable "rigidity 376 Figure 15-2 Linguistic values for variable"elements'rigidity' 377 Figure 15-3 Linguistic values for variable"significance 379 Figure 15 Linguistic evaluation values of lathes B, C, D,E 380 【实例截图】
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