Fuzzy Decision Procedures with Binary Relations: Towards A Unified Theory

271 Pages·1993·11.95 MB·English

by  Leonid Kitainik (auth.)

FUZZY DECISION PROCEDURES
WITH BINARY RELATIONS
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
Series A: Philosophy and Methodology of the Social Sciences
Series B: Mathematical and Statistical Methods
Series C: Game Theory, Mathematical Programming and Operations Research
Series D: System Theory, Knowledge Engineering and Problem Solving
SERIES D: SYSTEM THEORY, KNOWLEDGE ENGINEERING
AND PROBLEM SOLVING
VOLUME 13
Editor: R. Lowen (Antwerp); Editorial Board: G. Feichtinger (Vienna), G. J. Klir (New
York) O. Opitz (Augsburg), H. J. Skala (Paderbom), M. Sugeno (Yokohama), H. J.
Zimmermann (Aachen).
Scope: Design, study and development of structures, organizations and systems aimed at
formal applications mainly in the social and human sciences but also relevant to the
information sciences. Within these bounds three types of study are of particular interest.
First, formal definition and development of fundamental theory and/or methodology, second,
computational and/or algorithmic implementations and third, comprehensive empirical
studies, observation or case studies. Although submissions of edited collections will appear
occasionally, primarily monographs will be considered for publication in the series. To
emphasize the changing nature of the fields of interest we refrain from giving a clear
delineation and exhaustive list of topics. However, certainly included are: artificial
intelligence (including machine learning, expert and knowledge based systems approaches),
information systems (particularly decision support systems), approximate reasoning
(including fuzzy approaches and reasoning under uncertainty), knowledge acquisition and
representation, modeling, diagnosis, and control.
The titles published ill this series are listed at the elld of this volwne.
FUZZY DECISION PROCEDURES
WITH BINARY RELATIONS
Towards A Unified Theory
by
Leonid Kitainik
Computing Center of the
Russian Academy of Science
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Puhlication Data
Kitainik, Leonid.
Fuzzy decision procedures with binary relations : towards a unified
theory / by Leonid Kitainik.
p. cm. -- (Theory and decision library. Series D)
Includes bibliographical references and index.
ISBN 978-94-010-4866-8 ISBN 978-94-011-1960-3 (eBook)
DOI 10.1007/978-94-011-1960-3
1. Decision-making. 2. Fuzzy sets. 3. Ranking and selection
(Statistics) 1. Title II. Series: Theory and decision library.
Series D, System theory, knowledge engineering, and problem solving.
QA279.4.K58 1993
003' .56--dc20 93-23851
CIP
Copyright @ 1993 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1s t edition 1993
AII 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,
photo-copying, recording, or otherwise, without the prior written permission of
the publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
To
Dennis and Irina
vii
Table of Contents
FOREWORD by Didier Dubois and Henry Prade . . . . . . . .. ix
PREFACE • • • • • • • • xi
ACKNOWLEDGMENTS • • • • • xxiii
CHAPl'ER 1. INTRODUCl'ION 1
CHAPl'ER 2. cx::t-mN NOTATIONS • 7
CHAPl'ER 3. SYSTEMATIZATION OF CHOICE RULES WITH BINARY RELATIONS 11
3. 1. RATIONALITY CONCEPI'. MULTIFOLD CHOICE • • • • • • • • • •• 11
3.2. BASIC DICHO'1UfiES: INVARIANT DESCRIPrION 15
3.3. CCMPOSITION LAWS • • • • • • • • • • • • • • • • • • • •• 19
3.4. SYNTHESIS OF RATIONALITY CONCEPI'S 23
CHAPl'ER 4. FUZZY DECISION PROCEDURES 31
4.1. FUZZY RATIONALITY CONCEPI' • 32
4.2. MULTIFOLD FUZZY CHOICE • • • • • • • • • • • • • • • • •• 33
4.3. FAMILIES OF FUZZY DICHO'l'CM:XJS DECISION PROCEDURES • • • •• 34
CHAPl'ER 5. OONTENSIVENESS CRITERIA • • • • • • • • • • • • • • •• 37
5. 1. IDI'IVATIONS AND POSTULATES FOR MULTIFOLD FUZZY CHOICE 37
5.2. DICH<Yl'(M)USNESS AND o-CJONTENSIVENESS OF MULTIFOLD
FUZZY CHOICE, PROCEDURES, AND RELATIONS • • • • • • • •• 38
5.3. RANKING ALTERNATIVES USING MULTIFOLD FUZZY CHOICE • • • •• 41
CHAPl'ER 6. FUZZY INCLUSIONS • • • • • • • • • • • • • • • • 47
6.1. IDI'IVATIONS. FUZZV INCLUSION AND FUZZY IMPLICATION 49
6.2. AXIOMATICS • • • • • • • • • • • 57
6. 3. REPRESENTATION THIlX>RFM • • • • • • • • • 58
6.4. PROPERTIES OF FUZZV INCLUSIONS. • • • • 62
6.5. BINARY OPERATIONS WITH FUZZV INCWSIONS 84
6.6. CHARACTERISTIC FUZZY INCLUSIONS
(POLYN~AL AND PIECEWISE-POL~AL MJDELS) 86
6.7. exMPARATIVE STUDY OF FUZZV INCWSIONS • • • 97
CHAPl'ER 7. CONTENSIVENESS OF FUZZV DICHO'J.'CM)US DECISION PROCEDURES
IN UNIVERSAL ENVIRONMENT • • • • • • • • • • • • • • •• 103
viii
wrm
CHAPl'ER 8. CHOICE FUZZY RELATlOOS • • • • • • • • • • • • •• 109
8. L BASIC TECHNIQUE. ELFMENTS OF MULTlFOLD FUZZY CHOICE. •• 110
8.2. (X-curs, AND MULTlFOLD FUZZY CHOICE wrTII BASIC DICHO'J.'a1IES. 116
8.3. THE ClCRE IS UNFIT • • • • • • • • • • • • • • • • • • •• 119
8.4. FUZZY von NmJMANN - ~ SOUJI'ION. FUZZY STABLE ClCRE 122
8.5. POOCEI>URBS BASED ON THE DUAL <n1POSITION LAW • • • • • 133
CHAPl'ER 9. RANKING AND C-SPECI'RAL moPERTIES OF FUZZY RELATlOOS
(FUZZY von NmJMANN - ~ - ZADEH SOUJI'IOOS) 137
9. L BASIC CHARAm'ERISTICS. x-MAPPING • • • • • • • 138
9.2. BOUNDS OF MULTIFOLD FUZZY CHOICE. • • • • • • 140
9.3. OONNECTED SPECTRUM, AND SPECI'RAL moPERTIES OF
A FUZZY RELATION •••••••••••• 150
9.4. CLASSIFICATION OF MULTlFOLD FUZZY CHOICES • • • 153
9.5. FUZZY L.ZADEH' STABLE ClCRE • • • • • • • • • • • • 160
9.6. INCONTENSIVE mx::EDURES BASED ON L.ZADEH' INCLUSION • •• 163
CHAPl'ER 10. INVARIANT, ANTIINVARIANT AND EIGEN FUZZY SUBSFl'S.
MAINSPRINGS OF CUT TECHNIQUE IN FUZZY RELATIONAL SYSTl!MS 165
CHAPl'ER 11. OONTENSIVENESS OF FUZZY DECISION POOCEDURES IN
RESTRICTED ~ • • • • • • • • • 183
CHAPl'ER 12. EFFICIENCY OF FUZZY DECISION POOCEDURES • • • • 199
CHAPl'ER 13. DECISION-MAKING wrTII SPECIAL CLASSES OF FUZZY BINARY
RELATlOOS • • • • • 207
13.1. FUZZY PREORDERINGS • 207
13.2. RECTImCAL RELATlOOS 211
CHAPl'ER 14. APPLICATlOOS TO CRISP CHOICE RULES 217
14.1. ADJUSTING CRISP CHOICE • • • • • • • • • • •• 218
14.2. PRODUCING NEW CHOICE RULES (FNM'Z8 AND DIPOLE DEmfPOSITION) 221
CHAPl'ER 15. APPLICATIONS TO DECISION SUPPORT SYSTl!MS
AND TO MULTIPURPOSE DECISION-MAKING • • • • • • • 225
15.1. GENERAL APPLICATIONS TO DECISION SUPPORT SYSTl!MS 225
15.2. APPLICATIOOS TO MULTIPURPOSE DECISION-MAKING 227
15.3. EXPERT ASSISTANT FICCKAS
(in collaboration with S.Orlovski) 237
LlTER.A.n.JRB • . • • • • • • • • • • • • • • • • 240
INDEl{ • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 250
ix
Foreword
In decision theory there are basically two appr~hes to the modeling of
individual choice: one is based on an absolute representation of
preferences leading to a ntDnerical expression of preference intensity.
This is utility theory. Another approach is based on binary relations that
encode pairwise preference. While the former has mainly blossomed in the
Anglo-Saxon academic world, the latter is mostly advocated in continental
Europe, including Russia. The advantage of the utility theory approach is
that it integrates uncertainty about the state of nature, that may affect
the consequences of decision. Then, the problems of choice and ranking
from the knowledge of preferences become trivial once the utility function
is known. In the case of the relational approach, the model does not
explicitly accounts for uncertainty, hence it looks less sophisticated. On
the other hand it is more descriptive than normative in the first stand
because it takes the pairwise preference pattern expressed by the
decision-maker as it is and tries to make the best out of it. Especially
the preference relation is not supposed to have any property. The main
problem with the utility theory approach is the gap between what
decision-makers are and can express, and what the theory would like them
to be and to be capable of expressing. With the relational approach this
gap does not exist, but the main difficulty is now to build up convincing
choice rules and ranking rules that may help the decision process.
This book is devoted to a systematic investigation of choice and
ranking rules for both fuzzy and non-fuzzy binary preference relations, a
topic to which Russian scholars have contributed a lot. At the
representation level, the main benefit of the use of fuzzy relations is to
x
account for shades of preference rather than uncertainty about states of
nature. Most of this book may look like a paradox to utility theory
tenants because the algebraic framework of binary preference relations,
fully explored here, is cast in a continuous setting. Yet, the "tour de
force" of the author is to show that the study of fuzzy relations can shed
new light on classical choice rules in the non-fuzzy case by laying bare
reasons why some choice rules are better than others, and by suggesting
that the fuzzy setting can enrich the potential rankings of alternatives
stemming from standard binary relations. The study of multiple-valued
decision rules also gives the opportunity for a thorough and original
study of basic issues in fuzzy set theory such as fuzzy set inclusions and
eigen-fuzzy sets on which many results are presented in the following
pages. This book will certainly be considered as a worthwhile contribution
to the mathematics of decision science and. of fuzzy sets. It is also a
j,indow that opens on the research world of Eastern Europe in these fields.
May it help starting again the dialogue between Eastern and Western
mathematics of decision, a dialogue that had been cut off for so-many
years.
Didier DUBOIS Henri PRADE
xi
Preface
This book is aimed at building new bridges between Fuzzy Set Theory,
and Crisp Set Theory. It often occurs in fuzzy studies that generalized
concepts, originated with crisp prototypes, keep aloof from crisp problems
and never return to their crisp "motherland". In our opinion, the backward
influence of a generalized theory upon its source should (and can) be much
more fertilizing. Dealing with vast material and possessing highly
developed technique, fuzzy concepts can be used to achieve a more profound
understanding of the nature of crisp problems. In many cases, they can
suggest new solutions of crisp problems, though sometimes these solutions
may seem somewhat paradoxical. Furthermore, fuzzy ideas can be
instrumental in building new crisp constructions.
Therefore, the interaction between fuzzy and crisp theories can be
conceived of as a two-sided road
generalization
CRISP FUZZY
explanation, new constructions
The author thanks in advance all the readers who will make efforts to
follow this general idea everywhere in the book, irrespectively of a
specific subject, as well as of technical details.
In the present research, the above "super-task" is developed within a
conventional branch of decision theory, namely, in decision-making with
binary relations.
Aggregation of preferences, choice and ranking of alt ernativ es with
respect to binary relations represent an acknowledged tool both in theory