Fuzzy Mathematical Programming And Fuzzy Matrix Games

Free ebooks ==> www.Ebook777.com +:*MK\WZ;+PIVLZI ]bba5I\PMUI\QKIT8ZWOZIUUQVOIVL.]bba5I\ZQ`/IUM[ www.Ebook777.com ;\]LQM[QV.]bbQVM[[IVL;WN\+WUX]\QVO>WT]UM! -LQ\WZQVKPQMN 8ZWN2IV][b3IKXZbaS ;a[\MU[:M[MIZKP1V[\Q\]\M 8WTQ[P)KILMUaWN;KQMVKM[ ]T6M_MT[SI ?IZ[I_ 8WTIVL -UIQT"SIKXZbaS(QJ[XIV_I_XT ]Z\PMZ^WT]UM[WN\PQ[[MZQM[ KIVJMNW]VLWVW]ZPWUMXIOM" [XZQVOMZWVTQVMKWU >WT66MLRIP4LM5IKMLW5W]ZMTTM -L[ -^WT^IJTM5IKPQVM[ 1;*6! >WT2.]TKPMZ4+2IQV-L[ )XXTQML1V\MTTQOMV\;a[\MU[ 1;*6 >WT61KPITSIZIVRM:3PW[TI 4+2IQV ,M[QOVWN1V\MTTQOMV\5]T\Q)OMV\;a[\MU[  1;*6! >WT*4Q] =VKMZ\IQV\aWT/:M[KWVQ242IQV 1V\MTTQOMV\)OMV\[ 1;*6 >WT:WT56QSZI^M[P4)BILMP 23IKXZbaS-L[ ;WN\+WUX]\QVONWZ1VNWZUI\QWV8ZWLM[[QVO IVL)VITa[Q[ 1;*6! >WT:))TQM^..IbTWTTIPQ::)TQM^ ;WN\+WUX]\QVOIVLQ\[)XXTQKI\QWV[QV *][QVM[[IVL-KWVWUQK[ 1;*6  >WT).:WKPI-5I[[IL )8MZMQZI2Z MZTIO*MZTQV0MQLMTJMZO 8ZQV\MLQV/MZUIVa www.Ebook777.com Preface VII After presenting some basic facts on fuzzy sets and fuzzy arithmetic, the main topics namely fuzzy linear and quadratic programming, fuzzy matrix games, fuzzy bi-matrix games and modality constrained programming are discussed in Chapters to 10 Our presentation is certainly not exhaustive and some topics e.g fuzzy multi-objective programming and fuzzy multi-objective games have been left deliberately to remain focussed and to keep the book to a reasonable size Nevertheless these topics are important and therefore appropriate references are provided whenever desirable This book is primarily addressed to senior undergraduate students, graduate students and researchers in the area of fuzzy optimization and related topics in the department of Mathematics, Statistics, Operational Research, Industrial Engineering, Electrical Engineering, Computer Science and Management Sciences Although every care has been taken to make the presentation error free, some errors may still remain and we hold ourselves responsible for that and request that the error if any, be intimated by e-mailing at chandra@maths.iitd.ernet.ac.in (e-mail address of S.Chandra) In the long process of writing this book we have been encouraged and helped by many individuals We would first and foremost like to thank Professor Janusz Kacprzyk for accepting our proposal and encouraging us to write this book We are highly grateful to Professors I Nishizaki, M Inuiguchi, J Ramik, D Li, T Maeda and H-C Wu for sending their reprints / preprints and answering to our queries at the earliest Their research has certainly been a source of inspiration for us We would also like to thank the editors and publishers of the journals “Fuzzy Sets and Systems”, “Fuzzy Optimization and Decision Making” and “Omega” for publishing our papers in the area of fuzzy linear programming and fuzzy matrix games which constitute the core of this book We also appreciate our students Ms Vidyottama Vijay and Ms Reshma Khemchandani for their tremendous help during the preparation of the manuscript in LATEX and also reading the manuscript from a student point of view We also acknowledge the book grant provided by IIT Delhi and thank Prof P.C Sinha for all help in this regard Our special thanks are due to Dr J.L.Gray, Dean, Faculty of Management, University of Manitoba for his encouragement and interest in this work Last but not the least, we are obliged to Dr Thomas Ditzinger and Ms Heather King of International Engineering Department, and Mr Nils Schleusner of Production Department, Springer-Verlag for all www.Ebook777.com VIII Preface their help, cooperation and understanding in the publication of this book (Winnipeg), (New Delhi), C.R.Bector S.Chandra Free ebooks ==> www.Ebook777.com Contents Crisp matrix and bi-matrix games: some basic results 1.1 Introduction 1.2 Duality in linear programming 1.3 Two person zero-sum matrix games 1.4 Linear programming and matrix game equivalence 1.5 Two person non-zero sum (bi-matrix) games 1.6 Quadratic programming and bi-matrix game 1.7 Constrained matrix games 1.8 Conclusions 1 11 13 17 20 Fuzzy sets 2.1 Introduction 2.2 Basic definitions and set theoretic operations 2.3 α-Cuts and their properties 2.4 Convex fuzzy sets 2.5 Zadeh’s extension principle 2.6 Fuzzy relations 2.7 Triangular norms (t-norms) and triangular conorms (t-conorms) 2.8 Conclusions 21 21 21 24 26 29 30 Fuzzy numbers and fuzzy arithmetic 3.1 Introduction 3.2 Interval arithmetic 3.3 Fuzzy numbers and their representation 3.4 Arithmetic of fuzzy numbers 3.5 Special types of fuzzy numbers and their arithmetic 39 39 39 42 44 46 www.Ebook777.com 33 38 X Contents 3.6 Ranking of fuzzy numbers 53 3.7 Conclusions 56 Linear and quadratic programming under fuzzy environment 4.1 Introduction 4.2 Decision making under fuzzy environment and fuzzy linear programming 4.3 LPPs with fuzzy inequalities and crisp objective function 4.4 LPPs with crisp inequalities and fuzzy objective functions 4.5 LPPs with fuzzy inequalities and objective function 4.6 Quadratic programming under fuzzy environment 4.7 A two phase approach for solving fuzzy linear programming problems 4.8 Linear goal programming under fuzzy environment 4.9 Conclusions 57 57 58 61 64 67 72 78 83 94 Duality in linear and quadratic programming under fuzzy environment 95 5.1 Introduction 95 5.2 Duality in LP under fuzzy environment: Ră odderZimmermanns model 95 5.3 A modified linear programming duality under fuzzy environment 101 5.4 Verdegay’s dual for fuzzy linear programming 108 5.5 Duality for quadratic programming under fuzzy environment 112 5.6 Conclusions 116 Matrix games with fuzzy goals 117 6.1 Introduction 117 6.2 Matrix game with fuzzy goals: a generalized model 118 6.3 Matrix game with fuzzy goals: Nishizaki and Sakawa model 122 6.4 Special cases 126 6.5 Conclusions 131 Matrix games with fuzzy pay-offs 133 7.1 Introduction 133 7.2 Definitions and preliminaries 134 ... basic facts on fuzzy sets and fuzzy arithmetic, the main topics namely fuzzy linear and quadratic programming, fuzzy matrix games, fuzzy bi -matrix games and modality constrained programming are... linear programming, two-person zero-sum matrix games, linear programming and matrix game equivalence, two person non-zero sum (bi -matrix) games, quadratic programming and bi -matrix games, and constrained... editors and publishers of the journals ? ?Fuzzy Sets and Systems”, ? ?Fuzzy Optimization and Decision Making” and “Omega” for publishing our papers in the area of fuzzy linear programming and fuzzy matrix