Generalized Concavity in Fuzzy Optimization and Decision Analysis [electronic resource]

299 Pages·2002·36.8 MB·English

by  Ramík, Jaroslav

Generalized Concavity in
Fuzzy Optimization and
Decision Analysis
INTERNATIONAL SERIES IN
OPERATIONS RESEARCH & MANAGEMENT SCIENCE
FrederickS. Hillier, SeriesEditor
StanfordUniversity
Saigal,R. / LINEARPROGRAMMING:AModernIntegratedAnalysis
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Vanderbei, R. / LINEARPROGRAMMING:FoundationsandExtensions
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Tayur, S.,Ganeshan, R. & Magazine,M.IQUANTITATIVEMODELINGFORSUPPLY
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Weyant, J./ENERGYANDENVIRONMENTALPOLICYMODELING
Shanthikumar,J.G. & Sumita,U.lAPPLIEDPROBABILITYANDSTOCHASTICPROCESSES
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Balsamo,S.,Persone,V.,Onvural,R./ANALYSISOFQUEUEINGNETWORKSWITHBLOCKiNG
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Hanne,T./ INTELLIGENTSTRATEGIESFORMETA MULTIPLECRITERIADECISIONMAKiNG
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andApplications
GENERALIZEO CONCAVITY IN
FUZZY OPTIMIZATION ANO
OECISION ANALYSIS
JAROSLAV RAMIK
Silesian University
School of Business Administration
Karvinâ, Czech Republic
MILANVLACH
Japan Advanced Institute of Science and Technology
School of Information Science
Ishikawa, Japan
Charles University
Faculty of Mathematics and Physics
Prague, Czech Republic
....
"
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Ramik, Jaroslav.
Generalized concavity in fuzzy optimization and decision analysis / Jaroslav Ramik,
Milan Vlach.
p. cm. --(International series in operations research & management science; 41)
Inc1udes bibliographical references and index.
ISBN 978-1-4613-5577-9 ISBN 978-1-4615-1485-5 (eBook)
DOI 10.1007/978-1-4615-1485-5
1. Decision making. 2. Matbematical optimization. 3. Fuzzy mathematics. 4. Concave
functions. 1. Vlach, Milan. II. Title. III. Series.
T57.95 .R34 2001
658.4'03--dc21
2001046197
Copyright © 2002 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint ofthe hardcover Ist edition 2002
AII rights reserved. No part ofthis 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 ofthe publisher. Springer Science+Business
Media, LLC.
Printed on acid-free paper.
Contents
Preface xi
Acknowledgments xv
PartI Theory
1. PRELIMINARIES 5
2. GENERALIZED CONVEXSETS 11
2.1 Convex Sets 11
2.2 StarshapedSets 12
2.3 StronglyStarshapedSets 18
2.4 Co-StarshapedSets 20
2.5 Separation ofStarshapedSets 21
2.5.1 Separating Hyperplanes 22
2.5.2 Separation byaFamilyofHyperplanes 24
2.5.3 Separation byaFamily ofLinearFunctionals 26
2.5.4 SeparationbyaCone 27
2.5.5 Separation ofStarshapedSets 31
2.6 Generalizations ofStarshapedSets 34
2.6.1 Path-ConnectedSets 34
2.6.2 Invex and Univex Sets 35
2.6.3 4>-Convex Sets 36
3. GENERALIZED CONCAVE FUNCTIONS 37
3.1 Concave and Quasiconcave Functions 37
3.2 StarshapedFunctions 40
3.3 FurtherGeneralizations ofConcave Functions 46
3.3.1 QuasiconnectedFunctions 46
3.3.2 (4), \lJ)-Concave Functions 50
3.4 DifferentiableFunctions 57
vi GENERALIZED CONCAVITY
3.4.1 DifferentiableQuasiconcaveFunctions 57
3.4.2 PseudoconcaveFunctions 59
3.4.3 Incave, Pseudoincave andPseudounicave Functions 62
3.5 ConstrainedOptimization 66
4. TRIANGULAR NORMS ANDT-QUASICONCAVE FUNCTIONS 73
4.1 TriangularNormsand Conorms 73
4.2 Properties ofTriangularNormsand TriangularConorms 76
4.3 RepresentationsofTriangularNormsandTriangularConorms 79
4.4 Negations and De MorganTriples 82
4.5 DominationofTriangularNorms 84
4.6 T-QuasiconcaveFunctions 85
4.7 (<I>,T)-Concave Functions 95
4.8 Propertiesof(<I>,T)-ConcaveFunctions 97
5. AGGREGATIONOPERATORS 101
5.1 Introduction 101
5.2 Definitionand Basic Properties 102
5.3 ContinuityProperties 104
5.4 Averaging AggregationOperators 106
5.4.1 CompensativeAggregationOperators 106
5.4.2 Order-StatisticAggregationOperators 108
5.4.3 OrderWeightedAveraging Operators 108
5.5 Sugeno andChoquetIntegrals 110
5.6 OtherAggregationOperators 113
5.7 AggregationofFunctions 114
6. FUZZYSETS 121
6.1 Introduction 121
6.2 DefinitionandBasic Properties 122
6.3 OperationswithFuzzy Sets 126
6.4 ExtensionPrinciple 127
6.5 Binary andValuedRelations 129
6.6 FuzzyRelations 131
6.7 Fuzzy ExtensionsofValuedRelations 133
6.8 Fuzzy Quantitiesand Fuzzy Numbers 137
6.9 FuzzyExtensionsofReal-Valued Functions 140
6.10 HigherDimensionalFuzzy Quantities 145
6.11 FuzzyExtensionsofValued Relations 150
Contents vii
PartII Applications
7. FUZZY MULTI-CRITERIA DECISION MAKING 163
7.1 Introduction 163
7.2 FuzzyCriteria 164
7.3 Pareto-OptimalDecisions 166
7.4 CompromiseDecisions 170
7.5 Generalized CompromiseDecisions 173
7.6 AggregationofFuzzy Criteria 177
7.7 ExtremalProperties 178
7.8 ApplicationtoLocationProblem 179
7.9 ApplicationinEngineeringDesign 186
8. FUZZYMATHEMATICALPROGRAMMING 193
8.1 Introduction 193
8.2 ModellingRealitybyFuzzy MathematicalProgramming 195
8.3 MathematicalProgrammingProblemswithParameters 195
8.4 FormulationofFuzzy MathematicalProgrammingProblems 197
8.5 FeasibleSolutionsofFMPProblems 199
8.6 Properties ofFeasibleSolution 200
8.7 OptimalSolutionsofthe FMPProblem 208
9. FUZZYLINEAR PROGRAMMING 217
9.1 Introduction 217
9.2 FormulationofFLPproblem 217
9.3 Properties ofFeasible Solution 220
9.4 Properties ofOptimalSolutions 223
9.5 ExtendedAdditioninFLP 227
9.6 Duality 231
9.7 Special Models ofFLP 235
9.7.1 IntervalLinearProgramming 235
9.7.2 Flexible LinearProgramming 238
9.7.3 FLPProblems withInteractiveFuzzyParameters 240
9.7.4 FLP Problems withCenteredParameters 242
9.8 IllustrativeExamples 244
10. FUZZY SEQUENCING AND SCHEDULING 253
10.1 Introduction 253
10.2 DeterministicModels 254
10.3 StochasticModels 259
10.4 Fuzzy Models 265
10.4.1 FuzzyDueDates 266
Vlll GENERALIZEDCONCAVITY
10.4.2 Fuzzy ProcessingTimes 268
10.4.3 Fuzzy Precedence 276
10.4.4 ConcludingRemarks 281
List ofSymbols
Symbol Description
Rn n-dimensional(Euclidean)realvectorspace
I:X-,Y mappingorfunctionI thatmapsasetX intoasetY
Ran(f) rangeofI
1(-1) pseudo-inversefunctionto I
(x,y) innerproductofx andy
Ilxll normofx
d(x,y) distancebetweenx andy
B(x,8) openballwithcenterx andradius8
[0,1] unitintervalinR
C(S) complementofS
Ker(X) kernelofX
Ker*(X) strongkernelofX
Keroo(X) co-kernelofX
Ker;'.,(X) strongco-kernelofX
Int(S) interiorofS
Rlint(S) relativeinteriorofS
CI(S) closureofS
Bd(S) boundaryofS
U(J,a) upperlevelsetofI ata
L(J,a) lowerlevelsetofI ata
H(J,a) levelsetofI ata
Epi(J) epigraphofI
Hyp(J) hypographofI
I(x,y) linesegmentjoiningx andy
L(x,y) linegoingthroughx andy
H(x,y) halflineemanatingfromx throughy
Conv(S) convexhullofS
dimeS) dimensionofS
Card(S) cardinalityofS,numberofelementsofS
Ext(S) setofallextremepointsofS
Core(J..L) coreofJ..L
Supp(J..L) supportofJ..L
\11(x) gradientvectorofI atx
\12/(x) HessianmatrixofI atx
TM,SM minimumt-norm,maximumt-conorm
Tp,Sp productt-norm,probabilisticsumt-conorm
TL,SL Lukasiewiczt-norm,boundedsumt-conorm
Tv,Sv drasticproductt-norm,drasticsumt-conorm
OSk k-orderstatisticaggregationoperator
OWA'I\r orderweightedaveragingoperatorofdimensionn
[A]", a-cutofafuzzysetA
.r(X) setofallfuzzysubsetsofX
CNA complementoffuzzysetAwithrespecttonegationN
J..LilT(A,B) T-fuzzy extensionofrelationRoffuzzysetsAandB