Fuzzy Set Theory - and Its Applications

211 Pages·1991·12.955 MB·English

by  Hans-Jürgen Zimmermann

viii CONTENTS List of Figures
14.3 Empirical Research on Aggregators 355
14.4 Conclusions 367
15 Future Perspectives 369
Bibliography 373
Index 393
Figure1-1 Concepthierarchyofcreditworthiness. 5
Figure2-1 Realnumberscloseto10. 13
Figure2-2a Convexfuzzyset. 15
Figure2-2b Nonconvexfuzzyset. 15
Figure2-3 Unionandintersectionoffuzzysets. 18
Figure3-1 Fuzzysetsvs. probabilisticsets. 26
Figure3-2 MappingofI-norms, I-conorms,andaveragingoperators. 38
Figure5-1 Theextensionprinciple. 54
Figure5-2 Trapezoidal "fuzzynumber." 58
Figure5-3 LR-representationoffuzzynumbers. 63
Figure6-1 Fuzzygraphs. 83
Figure6-2 Fuzzyforests. 84
Figure6-3 Graphsthatarenotforests. 85
Figure7-1 Maximizingset. 94
Figure7-2 Afuzzyfunction. 95
Figure7-3 Triangularfuzzynumbersrepresentingafuzzyfunction. 96
Figure7-4 Themaximumofafuzzyfunction. 97
Figure7-5 Fuzzilyboundedinterval. 102
Figure8-1 Probabilityofafuzzyevent. 122
Figure9-1 Lir7jguisticvariable"Age." 133
Figure9-2 Linguisticvariable"Probability." 134
Figure9-3 Linguisticvariable''Truth." 135
Figure9-4 Terms "True"and "False." 136
Figure9-5 Determiningsupportvaluesonthebasisoffuzzysets. 162
IX

List ofTables
Table3-1 Classificationofcompensatoryandnoncompensatory 39
operators.
Table3-2 Classificationofaggregationoperators. 40
Table3-3 Relationshipbetweenparametrizedoperatorsandtheir 41
parameters.
Table6-1 Propertiesoffuzzyrelations. 88
Table8-1 Possibilityfunctions. 117
Table8-2 Koopman'svs. Kolmogoroff'sprobabilities. 124
Table8-3 RelationshipbetweenBooleanalgebra,probabilities,and 125
possibilities.
Table9-1 Supportpairs. 163
Table9-2 Rowandcolumnconstraints. 167
Table10-1 ExpertSystems. 178
Table10-2 FuzzyControlSystems. 209
Table12-1 Ratingsandweightsofalternativegoals. 277
Table13-1 Applicationsoffuzzysettheoryinoperationsresearch. 285
Table13-2 Tableoftheparametrictransportationproblem. 289
Table13-3 Solutiontotransportationproblem. 290
Table13-4 Membershipgradesforslacktimeandwaitingtime. 305
Table13-5 Membershipgradesforconditionalpartsoftherules. 305
Table13-6 Membershipgradesoftherules. 306
Table13-7 Results. 306
Table13-8 Definitionoflinguisticvariables. 309
Table13-9 MembershipFunctions. 310
Table13-10 Costresults. 311
Table 13-11 Comparisonofperformances. 312
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xiv LISTOFTABLES Foreword
Table13-12 Structureofinstructionprogram. 318
Table13-13 Availabilityofinstructors. 320
Table13-14 PERToutput. 320
Table13-15 Availabilityofweeksforcourses. 321
Table13-16 Firstweek'sfinalschedule. 321
Table13-17a Populations. 329
Table13-17b Distancesbetweenvillages. 329
Table13-18 Determinationofthefuzzysetdecision. 330
Table14-1 Hierarchyofscalelevels. 342
Table14-2 Empiricallydeterminedgradesofmembership. 346 I'
Table14-3 Empiricalvs. predictedgradesofmembership. 358
As its name implies, the theory of fuzzy sets is, basically, a theory of
graded concepts-a theory in which everything is a matterofdegree or, to
put it figuratively, everything has elasticity.
In the two decades since its inception, the theory has matured into a
wide-ranging collection of concepts and techniques for dealing with
complex phenomena which do not lend themselves to analysis by classical
methods based on probability theory and bivalent logic. Nevertheless, a
question that is frequently raised by the skeptics is: Are there, in fact, any
significant problem-areas in which the use ofthe theory of fuzzy sets leads
to results that could not be obtained by classical methods?
Professor Zimmermann's treatise provides an affirmative answer to this
question. His comprehensive exposition of both the theory and its
applications explains in clear terms the basic concepts that underlie the
theory and how they relate to their classical counterparts. He shows
through a wealth ofexamples the ways in which the theory can be applied
to the solution of realistic problems, particularly in the realm of decision
analysis, and motivates the theory by applications in which fuzzy sets play
an essential role.
An important issue in the theory of fuzzy sets which does not have a
counterpart in the theory of crisp sets relates to the combination of fuzzy
sets through disjunction and conjunction or, equivalently, union and
intersection. Professor Zimmermann and his associates at the Technical
University of Aachen have made many important contributions to this
problem and were the first to introduce the concept ofa parametric family
ofconnectives which can be chosen to fit a particular application. In recent
years, this issue has given rise to an extensive literature dealing with
{-norms and related conceptswhich link someaspectsofthe theory offuzzy
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xviii PREFACE Preface for the Revised Edition
been selected for more detailed consideration. The information has been
divided into two volumes. The first volume contains the basic theory of
fuzzy sets and some areasofapplication. Itis intended to provide extensive
coverage of the theoretical and application'll approaches to fuzzy sets.
Sophisticated formalisms have not been included. I have tried to present
the basic theory and its extensions as detailed as necessary to be
comprehended by those who have not been exposed to fuzzy set theory.
Examples and exercises serve to illustrate the concepts even more clearly.
For the interested or more advanced reader, numerous references to
recent literature are included that should facilitate studies ofspecific areas
in more detail and on a more advanced level.
The second volume is dedicated to the application offuzzy set theory to
the area ofhuman decision making. Itis self-contained in the sense that all
concepts used are properly introduced and defined. Obviously this cannot
be done in the same breadth as in the first volume. Also the coverage of Since this book was first published in 1985, Fuzzy Set Theory has had an
fuzzy concepts in the second volume is restricted to those that are directly unexpected growth. It was further developed theoretically and it was
used in the models of decision making. applied to new areas. A number of very good books have appeared,
It is advantageous but not absolutely necessary to go through the first primarily dedicated to special areas such as PossibilityTheory [Dubois and
volume before studying the second. The material in both volumes has Prade 1988a], Fuzzy Control [Sugeno 1985a; Pedrycz 1989], Behavioral
served as texts in teaching classes in fuzzy set theory and decision mak and Social Sciences [Smithson 1987], and others have been published.
ing in the United States and in Germany. Each time the material was Many newedited volumes, eitherdedicated to special areas orwith a much
used, refinements were made, but the author welcomes suggestions for wider scope, have been added to the existing ones. Thousands of articles
further improvements. have been published on fuzzy sets in various journals. Successful real
The target groups were students in business administration, manage applications of fuzzy set theory have also increased in number and in
ment science, operations research, engineering, and computer science. quality. In particular, applications of fuzzy control, fuzzy computers,
Even though no specific mathematical background is necessary to under expert system shells with capabilities to process fuzzy information, and
stand the books, it is assumed that the students have some background fuzzy decision support systems have become known and have partly
in calculus, set theory, operations research, and decision theory. already proven their superiority over more traditional tools.
I would like to acknowledge the help and encouragement of all the One thing, however, does not seem to have changed since 1985: access
students, particularly those at the Naval Postgraduate School in Monterey to the area has not become easier for newcomcrs. I do not know of any
and at the Institute ofTechnology in Aachen (F.R.G.), who improved the introductory yet comprehensive book or textbook that will facilitate
manuscripts before they became textbooks. I also thank Mr. Hintz who entering into the area of fuzzy sets or that can be used in classwork.
helped to modify the different versions of the book, worked out the Iam, therefore, very grateful to KluwerAcademic Publishers for having
examples, and helped to make the text as understandable as possible. Ms. agreed to publish a revised edition of the book, which four timcs has
Grefen typed the manuscript several times without losing her patience. I already been printed without improvcment. In this revised edition all
am also indebted to Kluwer Academic Publishers for making the typing and other errors have bcen eliminated. All chapters have been
publication of this book possible. updated. The chapters on possibility theory (8), on fuzzy logic and
approximate reasoning (9), on cxpert systems and fuzzy control (10), on
H.-.1. Zimmermann decision making (12), and on fuzzy set models in operations research (13)
have been restructured and rewritten. Exerciseshave been added to almost
all chapters and a teacher's manual is available on request.
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