Monte Carlo Methods in Fuzzy Optimization

1. 1 Introduction The objective of this book is to introduce Monte Carlo methods to ?nd good approximate solutions to fuzzy optimization problems. Many crisp (nonfuzzy) optimization problems have algorithms to determine solutions. This is not true for fuzzy optimization. There are other things to discuss in fuzzy optimization, which we will do later onin the book, like? and < between fuzzy numbers since there will probably be fuzzy constraints, and how do we evaluate max/minZ for Z the fuzzy value of the objective function. This book is divided into four parts: (1) Part I is the Introduction containing Chapters 1-5; (2) Part II, Chapters 6-16, has the applications of our Monte Carlo method to obtain approximate solutions to fuzzy optimization problems; (3)PartIII,comprisingChapters17-27,outlinesour“un?nishedbusiness”which are fuzzy optimization problems for which we have not yet applied our Monte Carlomethodtoproduceapproximatesolutions;and(4)PartIVisoursummary, conclusions and future research. 1. 1. 1 Part I First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. For a beginning introduction to fuzzy sets and fuzzy logic see [2]. Three other items related to fuzzy sets, needed in this book, are also in Chapter 2: (1) in Section 2. 5 we discuss how we have dealt in the past with determining max/min(Z)for Z a fuzzy set representing the value of anobjective function in a fuzzy optimization problem; (2) in Section 2. cover.jpg......Page 1 front-matter.pdf......Page 2 Part I......Page 14 Part IV......Page 15 Previous Research......Page 16 MATLAB/C++ Programs......Page 17 Fuzzy Sets......Page 19 Fuzzy Numbers......Page 20 Alpha-Cuts......Page 21 Inequalities......Page 22 Extension Principle......Page 23 Interval Arithmetic......Page 24 Fuzzy Arithmetic......Page 25 Extension Principle......Page 26 Differences......Page 27 Min/Max of a Fuzzy Number......Page 28 Buckley's Method......Page 30 Kerre's Method......Page 31 Chen's Method......Page 32 Undominated Fuzzy Vectors......Page 34 Buckley's Method......Page 36 Chen's Method......Page 37 Random Numbers......Page 39 Quasi-random Sequences......Page 40 Random Number Generator......Page 41 Random Vectors: Real Numbers......Page 42 Random Vectors: Non-negative Integers......Page 44 Random Triangular/Trapezoidal Fuzzy Numbers......Page 45 Generated from Implicit Quadratic Functions......Page 46 Generated from Parametric Quadratic Functions, Bézier Fuzzy Numbers......Page 48 Comparison of Random Fuzzy Vectors......Page 50 Random Fuzzy Vectors......Page 51 Run Test......Page 53 Frequency Test......Page 57 Search Space $[a,b]$ for QBGFNs......Page 60 Search Space $\Omega$ for TFNs......Page 62 Other Search Spaces......Page 63 Crisp Linear Program......Page 64 Fuzzy Linear Program......Page 65 Kerre's Method......Page 66 Chen's Method......Page 68 Comparison of Solutions......Page 70 Fully Fuzzified Linear Programming......Page 73 Product Mix Problem......Page 74 Kerre's Method......Page 75 Chen's Method......Page 77 Comparison of Solutions......Page 78 Diet Problem......Page 80 Fuzzy Monte Carlo Method......Page 81 Comparison of Solutions......Page 83 Multiobjective Fully Fuzzified Linear Programming......Page 86 Example Problem......Page 87 Fuzzy Monte Carlo Method......Page 88 Compare Solutions......Page 91 $\overline{A}$ $\overline{X}$+ $\overline{B}$= $\overline{C}$......Page 94 Other Solutions......Page 96 Fuzzy Monte Carlo Method......Page 99 Fuzzy Quadratic Equation......Page 103 Fuzzy Monte Carlo Method......Page 105 Fuzzy Matrix Equation......Page 108 Fuzzy Monte Carlo Method......Page 115 Summary and Conclusions......Page 119 Introduction......Page 121 Random Fuzzy Vectors......Page 122 First Choice of Intervals......Page 123 Second Choice of Intervals......Page 124 Comparison of Solutions......Page 126 Summary and Conclusions......Page 127 Introduction......Page 130 Univariate Fuzzy Nonlinear Regression......Page 131 Fuzzy Monte Carlo Method......Page 132 First Choice of Intervals......Page 133 Second Choice of Intervals......Page 135 First Choice of Intervals......Page 136 Second Choice of Intervals......Page 137 Comparison of Solutions......Page 138 Summary and Conclusions......Page 139 Universal Approximator......Page 141 Evolutionary Algorithm......Page 142 Fuzzy Monte Carlo Method......Page 143 First Application......Page 144 Second Application......Page 147 Third Application......Page 149 Fourth Application......Page 151 Summary and Conclusions......Page 155 Introduction......Page 157 Error Measures......Page 158 Example Problem......Page 159 First Choice of Intervals......Page 160 Second Choice of Intervals......Page 161 Comparison of Solutions......Page 162 Summary and Conclusions......Page 163 MATLAB Program......Page 164 Two-Person Zero-Sum Games......Page 167 Fuzzy Two-Person Zero-Sum Games......Page 168 Fuzzy Monte Carlo......Page 171 Max/Min of Fuzzy Numbers......Page 172 Fuzzy Monte Carlo Solution Method......Page 173 Conclusions and Future Research......Page 174 Queuing Model......Page 176 Fuzzy Queuing Model......Page 178 Maximum of Fuzzy Profit......Page 181 Fuzzy Monte Carlo Solution......Page 182 Summary and Conclusions......Page 184 Min-Cost Capacitated Network......Page 186 Fuzzy Monte Carlo Method......Page 189 Fuzzy Shortest Path Problem......Page 190 Monte Carlo Method......Page 191 Max-Flow Problem......Page 193 Fuzzy Max-Flow Problem......Page 194 Fuzzy Monte Carlo Solution......Page 195 Inventory Problem......Page 197 Monte Carlo Method......Page 200 Monte Carlo Solution......Page 201 Inventory Model......Page 203 Monte Carlo Solution......Page 209 Inventory Model......Page 211 Monte Carlo Method......Page 212 Transportation Problem......Page 215 Fuzzy Transportation Problem......Page 216 Fuzzy Integers......Page 220 A Fuzzy Integer Programming Problem......Page 221 Fuzzy Monte Carlo Solution......Page 222 A Dynamic Programming Problem......Page 224 A Fuzzy Dynamic Programming Problem......Page 225 Fuzzy Monte Carlo Solution......Page 226 Introduction......Page 228 Job Times Fuzzy Numbers......Page 229 Fuzzy Monte Carlo Method......Page 231 Max/Min $f(\overline{X})$......Page 234 Max/Min $f(\overline{X},\overline{Y})$......Page 235 Summary......Page 237 Future Research......Page 238 Conclusions......Page 239 back-matter.pdf......Page 240 Monte Carlo Methods in Fuzzy Optimization is a clear and didactic book about Monte Carlo methods using random fuzzy numbers to obtain approximate solutions to fuzzy optimization problems. The book includes various solved problems such as fuzzy linear programming, fuzzy regression, fuzzy inventory control, fuzzy game theory, and fuzzy queuing theory. The book will appeal to engineers, researchers, and students in Fuzziness and applied mathematics. This book is a concise and readable introduction to Monte Carlo methods to find good approximate solutions to fuzzy optimization problems. Various basic applications and illustrative examples are presented in an understandable way. The aim of the book is to convince the reader that Monte Carlo methods can be useful in generating approximate solutions to fuzzy optimization problems "The objective of this book is to introduce Monte Carlo methods to find good approximate solutions to fuzzy optimization problems. Various basic applications and illustrative examples are presented in an understandable way."--Jacket