Fuzzy Set Theory-and Its Applications, Second, Revised Edition Fuzzy Set Theory and Its Applications Second, Revised Edition H.-J. Zimmermann . ., ~ Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data Zimmermann, H.-J. (Hans·Jiirgen), 1934- Fuzzy set theory and its applicationslH.-J. Zimmermann.-2nd ed. p. cm. Includes bibliographical relerences and index. ISBN 978-94-015-7951-3 ISBN 978-94-015-7949-0 (eBook) DOI 10.1007/978-94-015-7949-0 1. Fuzzy sets. 2. Operations research. 1. Title. QA24B.Z55 1990 511.3'22-dc20 90-38077 CIP Copyright © 1991 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1991 Softcover reprint of the hardcover 2nd edition 1991 AII righls reserved. No pari 01 this publicat ion may be reproduced, stored in a retrieval system or transmitted in any lorm 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 ix List of Tables xiii Foreword xv Preface xvii Preface for the Revised Edition xix 1 Introduction to Fuzzy Sets 1 1.1 Crispness, Vagueness, Fuzziness, Uncertainty 1 1.2 Fuzzy Set Theory 5 Part One: Fuzzy Mathematics 9 2 Fuzzy Sets-Basic Definitions 11 2.1 Basic Definitions 11 2.2 Basic Set-Theoretic Operations for Fuzzy Sets 16 3 Extensions 23 3.1 Types of Fuzzy Sets 23 3.2 Further Operations on Fuzzy Sets 28 3.2.1 Aigebraic Operations 28 3.2.2 Set -Theoretic Operations 29 3.2.3 Criteria for Selecting Appropriate Aggregation Operators 39 4 Fuzzy Measures and Measures of Fuzziness 45 4.1 Fuzzy Measures 45 4.2 Measures of Fuzziness 47 5 The Extension Principle and Applications 53 5.1 The Extension Principle 53 5.2 Operations for Type 2 Fuzzy Sets 55 5.3 Aigebraic Operations with Fuzzy Numbers 57 v vi CONTENTS 5.3.1 Special Extended Operations 59 5.3.2 Extended Operations for LR-Representation of Fuzzy Sets 62 6 Fuzzy Relations and Fuzzy Graphs 69 6.1 Fuzzy Relations on Sets and Fuzzy Sets 69 6.1.1 Compositions of Fuzzy Relations 74 6.1.2 Properties of the Min-Max Composition 77 6.2 Fuzzy Graphs 82 6.3 Special Fuzzy Relations 85 7 Fuzzy Analysis 91 7.1 Fuzzy Functions on Fuzzy Sets 91 7.2 Extrema of Fuzzy Functions 93 7.3 Integration of Fuzzy Functions 98 7.3.1 Integration of a Fuzzy Function over a Crisp Interval 98 7.3.2 Integration of a (Crisp) Real-Valued Function over a Fuzzy 101 Interval 7.4 Fuzzy Differentiation 105 8 Possibility Theory, Probability Theory, and Fuzzy Set 109 Theory 8,1 Possibility Theory 110 8.1.1 Fuzzy Sets and Possibility Distributions 110 8.1.2 Possibility and Necessity Measures 114 8.2 Probability of Fuzzy Events 117 8.2.1 Probability of a Fuzzy Event as a Scalar 118 8.2.2 Probability of a Fuzzy Event as a Fuzzy Set 119 8.3 Possibility vs. Probability 121 Part Two: Applications of Fuzzy Set Theory 129 9 Fuzzy logic and Approximate Reasoning 131 9.1 linguistic Variables 131 9.2 Fuzzy Logic 139 9.2.1 Classical Logics Revisited 139 9.2.2 Truth Tables and linguistic Approximation 143 9.3 Approximate Reasoning 146 9.4 Fuzzy Languages 148 9.5 Support Logic Programming 157 9.5.1 Support Horn Clause Logic Representation 158 9.5.2 Approximate Reasoning in SLOP 162 10 Expert Systems and Fuzzy Control 171 10.1 Fuzzy Sets and Expert Systems 171 10.1.1 Introduction to Expert Systems 171 10.1.2 Uncertainty Modeling in Expert Systems 179 10.1.3 Applications 182 10.2 Fuzzy Control 201 CONTENTS vii 10.2.1 Introduction to Fuzzy Control 201 10.2.2 Process of Fuzzy Control 202 10.2.3 Applications of Fuzzy Control 209 11 Pattern Recognition 217 11.1 Models for Pattern Recognition 217 11.1.1 The Data 219 11.1.2 Structure or Pattern Space 219 11.1.3 Feature Space and Feature Selection 219 11.1.4 Classification and Classification Space 220 11.2 Fuzzy Clustering 220 11.2.1 Clustering Methods 220 11.2.2 Cluster Validity 236 12 Decision Making in Fuzzy Environments 241 12.1 Fuzzy Decisions 241 12.2 Fuzzy Linear Programming 248 12.2.1 Symmetric Fuzzy LP 250 12.2.2 Fuzzy LP with Crisp Objective Function 254 12.3 Fuzzy Dynamic Programming 261 12.4 Fuzzy Multi Criteria Analysis 265 12.4.1 Multi Objective Decision Making 265 12.4.2 Multi Attributive Decision Making 272 13 Fuzzy Set Models in Operations Research 283 13.1 Introduction 283 13.2 Fuzzy Set Models in Logistics 285 13.2.1 Fuzzy Approach to the Transportation Problem 286 13.2.2 Fuzzy Linear Programming in Logistics 290 13.3 Fuzzy Set Models in Production Control and Scheduling 293 13.3.1 A Fuzzy Set Decision Model as Optimization Criterion 294 13.3.2 Job-Shop Scheduling with Expert Systems 296 13.3.3 A Method to Control Flexible Manufacturing Systems 299 13.3.4 Aggregate Production and Inventory Planning 307 13.3.5 Fuzzy Mathematical Programming for Maintenance 313 Scheduling 13.3.6 Scheduling Courses, Instructors, and Classrooms 314 13.4 Fuzzy Set Models in Inventory Control 321 13.5 A Discrete Location Model 327 14 Empirical Research in Fuzzy Set Theory 333 14.1 Formal Theories vs. Factual Theories vs. Decision 333 Technologies 14.1.1 Models in Operations Research and Management Science 337 14.1.2 Testing Factual Models 339 14.2 Empirical Research on Membership Functions 344 14.2.1 Type A-Membership Model 345 14.2.2 Type B-Membership Model 346 viii CONTENTS 14.3 Empirical Research on Aggregators 355 14.4 Conclusions 367 15 Future Perspectives 369 Bibliography 373 Index 393 List of Figures Figure 1-1 Concept hierarchy of creditworthiness. 5 Figure 2-1 Real numbers close to 10. 13 Figure2-2a 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. 54 Figure 5-2 Trapezoidal "fuzzy number." 58 Figure5-3 LR-representation of fuzzy numbers. 63 Figure 6-1 Fuzzy graphs. 83 Figure 6-2 Fuzzy forests. 84 Figure 6-3 Graphs that are not forests. 85 Figure 7-1 Maximizing set. 94 Figure 7-2 A fuzzy function. 95 Figure 7-3 Triangular fuzzy numbers representing a fuzzy function. 96 Figure 7-4 The maximum of a fuzzy function. 97 Figure 7-5 Fuzzily bounded interval. 102 Figure 8-1 Probability of a fuzzy event. 122 Figure 9-1 Linguistic variable "Age." 133 Figure 9-2 Linguistic variable "Probability." 134 Figure9-3 Linguistic variable "Truth." 135 Figure 9-4 Terms "True" and "False." 136 Figure 9-5 Determining support values on the basis of fuzzy sets. 162 ix x LIST OF FIGURES Figure 10-1 Structure of an expert system. 174 Figure 10-2 Semantic net. 177 Figure 10-3 The max-min inference method. 181 Figure 10-4 The max-dot inference method. 182 Figure 10-5 Linguistic descriptors. 184 Figure 10-6 Label sets for semantic representation. 185 Figure 10-7 Linguistic variables for occurrence and confirmability. 188 Figure 10-8 Inference network for damage assessment of existing 191 structures. [Ishizuka et al. 1982, p. 263]. Figure 10-9 Combination of two two-dimensional portfolios. 194 Figure 10-10 Criteria tree for technology attractiveness. 195 Figure 10-11 Terms of "degree of achievement." 196 Figure 10-12 Aggregation of linguistic variables. 197 Figure 10-13 Portfolio with linguistic input. 199 Figure 10-14 Structure of ESP. 200 Figure 10-15 DDC-control system. 203 Figure 10-16 Fuzzy logic control system. 203 Figure 10-17 Functions of FLC. 204 Figure 10-18 Schematic diagram of rotary cement kiln. [Umbers and 210 King 1981, p. 371] Figure 11-1 Pattern recognition. 218 Figure 11-2 Possible data structures in the plane. 221 Figure 11-3 Performance of cluster criteria. 222 Figure 11-4 Dendogram for hierarchical clusters. 223 Figure 11-5 Fuzzy graph. 224 Figure 11-6 Dendogram for graph-theoretic clusters. 225 Figure 11-7 The butterfly. 226 Figure 11-8 Crisp clusters of the butterfly. 226 Figure 11-9 Cluster 1 of the butterfly. 227 Figure 11-10 Cluster 2 of the butterfly. 228 Figure 11-11 Clusters for m = 1 .25. 235 Figure 11-12 Clusters for m = 2. 235 Figure 12-1 A classical decision under certainty. 242 Figure 12-2 A fuzzy decision. 244 Figure 12-3 Optimal dividend as maximizing decision. 245 Figure 12-4 Feasible regions for IlR(x) = 0 and IlR(x) = 1. 257 Figure 12-5 Fuzzy decision. 259 Figure 12-6 Basic structure of a dynamic programming model. 262 Figure 12-7 The vector-maximum problem. 268 Figure 12-8 Fuzzy LP with min-operator. 270 Figure 12-9 Fuzzy sets representing weights and ratings. 278 Figure 12-10 Final ratings of alternatives. 280 Figure 12-11 Preferability of alternative 2 over all others. 281 Figure 13-1 The trapezoid al form of a fuzzy number 286 Figure 13-2 The membership function of the fuzzy goal ~. 287 Figure 13-3 The solution of the numerical example. 291 LIST OF FIGURES xi Figure 13-4 Membership function of CDs in the set of fully satisfying 295 CDs. Figure 13-5 Structure of OPAL. 297 Figure 13-6 Fuzzy sets for the ratio in the "if" part of the rules. 298 Figure 13-7 Example for an FMS. 300 Figure 13-8 Criteria hierarchies. 302 Figure 13-9 Principle of approximate reasoning. 304 Figure 13-10 Membership functions for severallinguistic terms. 308 Figure 13-11 Comparison of work force algorithms. 311 Figure 13-12 Flowtime of a course. 316 Figure 13-13 The scheduling process. 317 Figure 13-14 Courses of one instruction program. 319 Figure 13-15 Road network. 328 Figure 13-16 Feasible covers. 329 Figure 14-1 Calibration of the interval for measurement. 349 Figure 14-2 Subject 34, "Old Man." 351 Figure 14-3 Subject 58, "Very Old Man." 352 Figure 14-4 Subject 5, "Very Young Man." 352 Figure 14-5 Subject 15, "Very Young Man." 353 Figure 14-6 Subject 17, "Young Man." 353 Figure 14-7 Subject 32, "Young Man." 354 Figure 14-8 Empirical membership functions "Very Young Man," 354 "Young Man," "Old Man," "Very Old Man." Figure 14-9 Empirical unimodal membership functions "Very Young 355 Man," "Young Man." Figure 14-10 Min-operator: Observed vs. expected grades of 359 membership. Figure 14-11 Product-operator: Observed vs. expected grades of 360 membership. Figure 14-12 Predicted vs. Observed data: Min-operator. 364 Figure 14-13 Predicted vs. Observed data: Max-operator. 364 Figure 14-14 Predicted vs. Observed data: Geometric Mean Operator. 365 Figure 14-15 Predicted vs. Observed data: y-operator. 365 Figure 14-16 Concept hierarchy of creditworthiness together with 366 individual weights ö and y-values for each level of aggregation.