Fuzzy Set Theory-and Its Applications, Fourth Edition

1 Fuzzy SetTheory-andItsApplications, Fourth EditionFuzzy Set Theory-and Its Applications Fourth Edition Zimmermann Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data Zimmermann, (Hans-Jiirgen), 1934- Fuzzy set theory--and its applieations / ed. p. em. Includes bibliographieal referenees and index. ISBN 978-94-010-3870-6 ISBN 978-94-010-0646-0 (eBook) DOI 1. Fuzzy sets. 2. Operations researeh. 1. Title. QA248 .Z55 2001 '22--de21 Copyright 2001 by Springer Seience+Business Media New York OriginalIy published by Kluwer Academic Publishers in 2001 Softcover reprint ofthe hardcover 4th Edition 2001 2001038123 AlI rights reserved. No part ofthis 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 pa per.

2 ContentsList of FiguresixList of TablesxiiiForewordxvPrefacexviiPreface to the Fourth Editionxix1 Introduction to Fuzzy , Vagueness, Fuzziness, Set Theory2 Part I: Fuzzy Set-Theoretic Operations for Fuzzy of Fuzzy Operations on Fuzzy for Selecting Appropriate Aggregation Operators434 Fuzzy Measures and Measures of of Fuzziness495 The Extension Principle and Extension for Type 2 Fuzzy Operations with Fuzzy Extended Operations forLR-Representationof Fuzzy Sets64viCONTENTS6 Fuzzy Relations and Fuzzy Relations on Sets and Fuzzy FuzzyRelations867 Fuzzy Fuzzy of a Fuzzy Function over a Crisp a (Crisp) Real-ValuedFunctionover a of asTransformersof Sets Fuzzy a Fuzzy Event as a a Fuzzy Event as a Fuzzy Probability133 Part II: Applications of Fuzzy Set Theory1399 Fuzzy logic and Approximate Logics in Inferencefor a Single Rule and Fril181 CONTENTSvii10 Fuzzy Sets and to Expert Modeling in Expert and Fuzzy of Fuzzy Mamdani Sugeno Fuzzy of a Model of a Diesel Control of a Cement Data Bases and Relational Queries in Crisp Databases26813 Fuzzy Data for Fuzzy Data Net Fuzzy Data of for Analysic Dynamic for Fuzzy Data for FDA of.

3 1 Maintenance Management in Petrochemical Quality Control323viiiCONTENTS14 DecisionMakingin LP with Crisp Objective Dynamic Programming with Crisp Objective Decision Making (MODM) Attributive Decision Making (MADM)35915 Applicationsof Fuzzy Sets inEngineeringand Evaluation and Ranking of Machine Detection Discrete Location Set Models in Approach to LinearProgrammingin Sets with Expert Method to Control Production and Inventory , Instructors, Set Models in Inventory Sets in Banking and onCustomerBehavior43316 EmpiricalResearchin Fuzzy Theories vs. Factual Theories vs. Decision inOperationsResearch Factual Research Research FrequentlyCitedJournals481 Bibliography483 Index507 ListofFiguresFigure 1-1 Concept hierarchy of 2-1 Real numbers close to Fuzzy Fuzzy and intersection of Fuzzy 3-1 Fuzzy sets vs. probabilistic oft-norms, t-conorms,and averaging 5-1 The extension '' Fuzzy number".

4 60 Figure5-3LR-representationof Fuzzy 6-1 Fuzzy that are not 7-1 Maximizing Fuzzy Fuzzy numbers representing a Fuzzy maximum of a Fuzzy bounded 8-1 Uncertainty as situational of a Fuzzy 9-1 Linguistic variable "Age".143 Figure9-2 Linguistic variable "Probability".144 Figure9-3 Linguistic variable "Truth".145 Figure9-4 Terms "True" and "False".146xLIST OF FIGURESF igure10-1 Structure of an expert sets for semantic variables for occurrence and network for damage assessment of existingstructures [Ishizuka et , ].212 Figure10-7 Combination of tree for of "degree of achievement".217 Figure10-10 Aggregation of linguistic variables .218 Figure10-11 Portfolio with linguistic of feedback Mamdani Fuzzy variable "Temperature".229 Figure11-4 Rule consequences in the heating system Value membership membership describing Fuzzy of crane [von Altrock1993].245 Figure11-13 Phases of variables [Sugeno and Nishida1985, ].

5 246 Figure11-15 Trajectories of the Fuzzy controlled model car [Sugenoand Nishida1985, ].247 Figure11-16 Fuzzy model car [von Altrock et , ].248 Figure11-17 Experimental design [von Altrock et , ].249 Figure11-18 FCR vs. fuel injection timing [Murayama et , ].250 Figure11-19 Control algorithm [Murayama et ].251 Figure11-20 Experimental results [Murayama et ].252 Figure11-21 Schematic diagram of rotary cement kiln [Umbers andKing1981, ].252 Figure11-22 ControllerdevelopmentinfuzzyTECH[von Altrock et ].256 Figure11-23 Rule base for model car [von Altrock et ].256 Figure11-24 Simulation screen [von Altrock et ].257 Figure11-25 Fuzzy controller as anonlineartransfer of stability analysis state of data data structure in the of cluster for hierarchical OFFIGURESxiFigure13-5 Fuzzy for graph-theoretic clusters of the 1 of the 2 of the 13-11 Clusters for m= for m= by the FSC. (a) Data set; (b) circles found byFSC; (c) data set; (d) circles found by sets [Krishnapuram and Keller 1993].

6 301 Figure13-15 Knowledge-based variables "Depth of Cut" and "Feed".304 Figure13-17 Knowledge structure of the knowledge-based (a) States of objects at a point of time; (b) projections oftrajectories over time into the feature and pointwise 13-21 Fictitious developments of share characteristic patterns of time signals for (a) anintact engine; (b) an engine with some (a) The Fuzzy set "approximately zero" (ll(y)), the functionf(t) and the resulting pointwise similarityll(f(t)); (b)projection of pointwise similarity into the plane (t,ll(f(t))).311 Figure13-24 Transformation of a feature vector containing trajectoriesinto trajectories into a usual feature and output of the functional Fuzzy of shot of classification of continuous 13-31 Application of DataEngine for acoustic quality 14-1 A classical decision under Fuzzy dividend as maximizing regions forllR(X)=0 andllR(X)= structure of a dynamic programming vector-maximum LP with sets representing weights and ratings of 14-11 Preferability of alternative 2 over all 15-1 Linguistic values for variable "rigidity".

7 376 Figure15-2 Linguistic values for variable "elements' rigidity".377 Figure15-3 Linguistic values for variable "significance".379 Figure15-4 Linguistic evaluation values of lathes B, C, D, OF FIGURESF igure15-5 Membershipfunctions resulting from incrementalclassifierdesign andclassificationof data obtained till for timewindow(230, 330).385 Figure15-7 Membershipfunctions for timewindow(240, 340).386 Figure15-8 Membershipfunctions for timewindow(250, 350).386 Figure15-9 Proportional difference between class centers1and2(withrespect to thecenterof class2)in time window(250, 350).387 Figure15-10 Membershipfunctions for timewindow(3014, 3114).388 Figure15-11 Membershipfunctions for time window(3064,3200).388 Figure15-12 Road trapezoidal form of a fuzzynumberiii=(qJ,aJ,a7,al).394 Figure15-15 The membership function of the Fuzzy solution of the numerical example .399 Figure15-17 Structure of sets for the ratio in the"if' part of the of an FMS [Hartley1984, ].

8 405 Figure15-20 Criteria hierarchies. (a) Release scheduling; (b) for several linguistic work force of a of :currentend-of-monthbalance for "Y".438 Figure15-28 Feature1:current end-of-month balance for "N".439 Figure16-1 Calibration of the interval ,"Old Man".460 Figure16-3 Subject58,"Very Old Man".461 Figure16-4 Subject5,"Very Young Man".461 Figure16-5 Subject15,"Very Young Man".462 Figure16-6 Subject17,"Young Man".462 Figure16-7 Subject32,"Young Man".463 Figure16-8 Empiricalmembershipfunctions "Very Young Man","Young Man", "Old Man", "Very Old Man".464 Figure16-9 Empirical unimodelmembershipfunctions "Very YoungMan", "Young Man".464 Figure16-10 Min-operator: Observed vs. expected grades : Observed vs. expected grades vs. observed data: vs. observed data: vs. observed data:Geometricmean vs. observed data: hierarchy of creditworthiness together with individualweights d and g-values for each level 3-1 Classification ofcompensatoryand of aggregation betweenparameterizedoperators and 6-1 Properties of Fuzzy 8-1 Rough taxonomy of uncertainty 's vs.

9 Kolmogoroff's between Boolean algebra, probabilities, 9-1 Formal quality of implication 10-1 Expert crisp data extended data possibilistic data 11-1 Rule of 14-1 Ratings and weights of alternative 15-1 Selected applications inmanagementand quality parameters (output data).376 Table15-4 Boundary values of the linguistic variable"significance". OF TABLEST able15-5bTable15-6 Table15-7 Table15-8 Table15-9 Table15-10 Table 15-11 Table15-12 Table15-13 Table15-14 Table15-15 Table15-16 Table15-17 Table15-18 Table15-19 Table15-20 Table 15-21 Table15-22 Table15-23 Table15-24 Table15-25 Table15-26 Table15-27 Table15-28 Table15-29 Table15-30 Table 16-1 Table16-2 Table16-3 Distances between of the Fuzzy set of the parametric transportation grades for slack time and waiting grades for conditional parts of the grades for the of linguistic variables [Rinks 1982].Membership of of instruction of of weeks for week's final centers of nine optimal features describing bank statistics of each feature of the data group "Y".

10 Main statistics of each feature of data group "N".Scope of the analysis of bank and stray customers for "Y" and stray customers for "N" change of assignment of customers in group"Y" to change of assignment of customers in group"N" to of scale determined grades of vs. predicted grades of its name implies, the theory of Fuzzy sets is, basically, a theory of graded con-cepts-atheory in which everything is a matter of degree or, to put it figuratively,everything has the two decades since its inception, the theory has matured into a wide-rangingcollectionof concepts and techniques for dealing with complex phe-nomena that do not lend themselves to analysis by classical methods based onprobabilitytheory and bivalent ,a question that is frequentlyraised by the skeptics is: Are there, in fact, anysignificantproblem-areasin whichthe use of the theory of Fuzzy sets leads to results that could not be obtained byclassical methods?