ISBN: 3790814563 TITLE: Fuzzy Mathematics in Economics and Engineering AUTHOR: Buckley TOC: 1 Introduction 1 Bibliography 3 2 Fuzzy Sets 5 2.1 Fuzzy Sets 5 2.1.1 Fuzzy Numbers 5 2.1.2 Alpha-Cuts 7 2.1.3 Inequalities 9 2.1.4 Discrete Fuzzy Sets 9 2.2 Fuzzy Arithmetic 9 2.2.1 Extension Principle 9 2.2.2 Interval Arithmetic 10 2.2.3 Fuzzy Arithmetic 11 2.3 Fuzzy Functions 12 2.3.1 Extension Principle 13 2.3.2 Alpha-Cuts and Interval Arithmetic 13 2.3.3 Differences 14 2.4 Possibility Theory 15 Bibliography 17 3 Solving Fuzzy Equations 19 3.1 A^_X^_+B^_=C^_ 19 3.2 New Solutions 22 3.3 Systems of Fuzzy Linear Equations 24 3.4 Applications 33 3.4.1 Fuzzy Linear Equation 33 3.4.2 Fuzzy Quadratic Equation 34 3.4.3 System of Linear Equations 35 3.5 Fuzzy Input-Output Analysis 39 3.5.1 The Open Model 39 3.5.2 Fuzzy Model 41 3.6 Summary and Conclusions 43 Bibliography 45 4 Fuzzy Mathematics in Finance 47 4.1 Future Value 48 4.2 Present Value 50 4.3 Annuities 53 4.3.1 Future Value 53 4.3.2 Present Value 54 4.4 Portfolio Analysis 55 4.4.1 NPV Method 55 4.4.2 IRR Method 59 4.5 Summary and Conclusions 62 Bibliography 65 5 Fuzzy Non-Linear Regression 69 5.1 Univariate Non-Linear Fuzzy Regression 70 5.1.1 Testing the EA 71 5.1.2 Application 72 5.2 Multivariate Non-Linear Fuzzy Regression 74 5.2.1 Testing 75 5.2.2 Application 75 5.3 Conclusions and Results 76 Bibliography 77 6 Operations Research 81 6.1 Fuzzy Linear Programming 81 6.1.1 Maximize Z^_ 82 6.1.2 Fuzzy Inequality 85 6.1.3 Evolutionary Algorithm 88 6.1.4 Applications 89 6.1.5 Summary and Conclusions 97 6.2 Fuzzy PERT 98 6.2.1 Job Times Fuzzy Numbers 102 6.2.2 Job Times Discrete Fuzzy Sets 104 6.2.3 Summary 104 6.3 Fuzzy Inventory Control 105 6.3.1 Demand Not Fuzzy 109 6.3.2 Fuzzy Demand 111 6.3.3 Backorders 117 6.3.4 Evolutionary Algorithm 118 6.4 Fuzzy Queuing Theory 118 6.4.1 Service 120 6.4.2 Arrivals 121 6.4.3 Finite or Infinite System Capacity 122 6.4.4 Machine Servicing Problem 124 6.4.5 Fuzzy Queuing Decision Problem 126 6.4.6 Summary and Conclusions 129 6.5 Fuzzy Network Armlysis 129 6.5.1 Fuzzy Shortest Route 130 6.5.2 Fuzzy Min-Cost Capacitated Network 132 6.5.3 Evolutionary Algorithm 136 6.5.4 Summary and Conclusions 137 6.6 Summary and Conclusions 137 Bibliography 139 7 Fuzzy Differential Equations 145 7.1 Fuzzy Initial Conditions 146 7.1.1 Electrical Circuit 150 7.1.2 Vibrating Mass 153 7.1.3 Dynamic Supply and Demand 155 7.2 Other Fuzzy Parameters 158 7.3 Summary and Conclusions 161 Bibliography 163 8 Fuzzy Difference Equations 165 8.1 Difference Equations 166 8.2 Fuzzy Initial Conditions 167 8.2.1 Classical Solution 167 8.2.2 Extension Principle Solution 169 8.2.3 Interval Arithmetic Solution 172 8.2.4 Summary 174 8.3 Recursive Solutions 175 8.4 Applications 176 8.4.1 National Income 176 8.4.2 Transmission of Information 178 8.4.3 Fuzzy Fibonacci Numbers 179 8.5 Summary and Conclusions 180 Bibiiography 183 9 Fuzzy Partial Differential Equations 185 9.1 Elementary Partial Differential Equations 185 9.2 Classical Solution 187 9.3 Extension Principle Solution 190 9.4 Summary and Conclusions 194 Bibliography 197 10 Fuzzy Eigenvalues 199 10.1 Fuzzy Eigenvalue Problem 199 10.1.1 Algorithm 203 10.2 Fuzzy Input-Output Analysis 206 10.3 Fuzzy Hierarchical Analysis 209 10.3.1 The lambda_max-Method 210 10.3.2 Fuzzy lambda_max-Method 212 10.3.3 Fuzzy Geometric Row Mean Method 222 10.4 Summary and Conclusions 224 Bibliography 227 11 Fuzzy Integral Equations 229 11.1 Resolvent Kerne1 Method 230 11.1.1 Classical Solution 231 11.1.2 Second Solution Method 235 11.2 Symmetric Kerne1 Method 237 11.2.1 Classical Solution 237 11.2.2 Second Solution Method 239 11.3 Summary and Conclusions 240 Bibliography 241 12 Summary and Conclusions 243 12.1 Summary 243 12.1.1 Chapter 3: Solving Fuzzy Equations 243 12.1.2 Chapter 4: The Fuzzy Mathematics in Finance 244 12.1.3 Chapter 5: Fuzzy Non-Linear Regression 244 12.1.4 Chapter 6: Operations Research 245 12.1.5 Chapter 7: Fuzzy Differential Equations 247 12.1.6 Chapter 8: The Fuzzy Difference Equations 248 12.1.7 Chapter 9: Fuzzy Partial Differential Equations 248 12.1.8 Chapter 10: Fuzzy Eigenvalues 249 12.1.9 Chapter 11: Fuzzy Integral Equations 249 12.2 Research Agenda 250 12.2.1 Chapter 3: Solving Fuzzy Equations 250 12.2.2 Chapter 4: The Fuzzy Mathematics of Finance 250 12.2.3 Chapter 5: Fuzzy Non-Linear Regression 250 12.2.4 Chapter 6: Operations Research 250 12.2.5 Chapter 7: Fuzzy Differential Equations 251 12.2.6 Chapter 8: Fuzzy Difference Equations 252 12.2.7 Chapter 9: Fuzzy Partial Differential Equations 252 12.2.8 Chapter 10: Fuzzy Eigenvalues 252 12.2.9 Chapter 11: Fuzzy Integral Equations 252 12.3 Conclusions 252 13 Evolutionary Algorithms 253 13.1 Introduction 253 13.2 General Purpose Algorithm 253 Bibliography 257 Index 259 List of Figures 267 List of Tables 271 END