The objective of this book is to present an uncertainty modeling approach using a new type of fuzzy system model via "Fuzzy Functions". Since most researchers on fuzzy systems are more familiar with the standard fuzzy rule bases and their inference system structures, many standard tools of fuzzy system modeling approaches are reviewed to demonstrate the novelty of the structurally different fuzzy functions, before we introduced the new methodologies. To make the discussions more accessible, no special fuzzy logic and system modeling knowledge is assumed. Therefore, the book itself may be a reference for some related methodologies to most researchers on fuzzy systems analyses. For those readers, who have knowledge of essential fuzzy theories, Chapter 1, 2 should be treated as a review material. Advanced readers ought to be able to read chapters 3, 4 and 5 directly, where proposed methods are presented. Chapter 6 demonstrates experiments conducted on various datasets.
Author(s): Asli Celikyilmaz, I. Burhan Türksen
Series: Studies in Fuzziness and Soft Computing 240
Publisher: Springer
Year: 2009
Language: English
Pages: 383
Cover......Page 1
Optimal Models and Methods with Fuzzy Quantities (Springer, 2010)......Page 4
ISBN 978-3-642-10710-8......Page 5
Preface......Page 7
Contents......Page 9
Fuzzy Sets......Page 12
Operations in Fuzzy Sets......Page 16
α–Cut Set......Page 20
Convex Fuzzy Sets......Page 21
Fuzzy Relations......Page 23
The Operation Properties of the Fuzzy Relation......Page 26
Special Fuzzy Operators......Page 28
Fuzzy Function from Universe $X$ to Another One $Y$......Page 29
Fuzzy Functions from Fuzzy Set \~A to Another One \~B......Page 30
Fuzzy Constrained Function......Page 31
Decomposition Theorem......Page 32
Extension Principle......Page 34
Representation Theorem......Page 36
Interval and Fuzzy Numbers......Page 38
Type $(·, c), T, L − R$ and Flat Fuzzy Numbers......Page 40
Introduction......Page 44
Definitions and Concepts of Fuzzy Parameters......Page 45
Establishment of Linear Regression Model......Page 46
Introduction......Page 50
Model......Page 51
Conclusion......Page 54
Exponential Model with Fuzzy Coefficients......Page 55
Practical Example......Page 59
Conclusion......Page 60
Linear Regression Model with Flat Fuzzy Parameters......Page 61
Self-regression Forecasting Model with Flat Fuzzy Parameters......Page 66
Preliminary......Page 68
Fuzzy Linear Regression......Page 69
Comparison of Two-Kind Distance Formula......Page 72
Basic Property......Page 74
Regression Model with $T$ -Fuzzy Variables......Page 77
Regression Model with $T$ -Fuzzy Data......Page 79
Self-regression Model with $T$-Fuzzy Variables......Page 82
Determination of the Modal with $(·, c)$ Fuzzy Variables......Page 87
Obtaining $(·, c)$ Fuzzy Data......Page 88
Self-regression with $(·, c)$ Fuzzy Variables......Page 89
Linear Model......Page 90
Non-linear Model......Page 92
Prepare Theorem and Property......Page 96
Two Kinds of Non-T -Fuzzily Approach and Its Equivalence......Page 97
Weight of Linearized Nonlinear Regression with T -Fuzzy Variables......Page 99
Numeric Example......Page 101
Determination of the Model with Flat Fuzzy Variables......Page 102
Conclusion......Page 105
Introduction......Page 106
Models......Page 107
Introduction......Page 109
Models and Its Properties......Page 110
Numerical Example......Page 116
Definitions and Properties......Page 119
Model......Page 121
Numerical Example......Page 123
Fuzzy Cluster......Page 127
Cluster Analysis Model with T -Fuzzy Data......Page 129
Fuzzy Recognition......Page 137
Common Method in Pattern Recognition......Page 138
Pattern Recognition Model with T -Fuzzy Data......Page 139
Application of Recognition Model with $T$ -Fuzzy Data......Page 144
Fuzzy Linear Programming and Its Algorithm......Page 148
Replacement Solution Method in Fuzzy Linear Programming......Page 150
Zimmermann Algorithm to Fuzzy Linear Programming......Page 152
Relevant Theorems of Parameter Linear Programming $(LPα)$......Page 155
Optimal Solution and Algorithm for Fuzzy Linear Programming......Page 158
Example......Page 162
Introduction......Page 163
Analysis of Fuzzy Linear Programming......Page 164
Algorithm to Fuzzy Linear Programming......Page 166
Case with Fuzzy Coefficients......Page 168
Case with Fuzzy Variables......Page 170
Reason for Antinomy Emergence......Page 174
Example......Page 179
Distance......Page 180
Ranking Fuzzy Numbers......Page 183
Linear Programming in Constraint with Fuzzy Coefficients......Page 184
Numerical Example......Page 185
Introduction......Page 186
Linear Programming in Constraints with $L-R$ Coefficients......Page 187
Linear Programming in Object with $L-R$ Coefficient......Page 189
Introduction......Page 191
Linear Programming with $T$ -Fuzzy Variables......Page 192
Dual Problem......Page 193
Numerical Example......Page 194
Conclusion......Page 195
Non Fuzzification of Model......Page 196
Finding Solution......Page 199
Conclusion......Page 200
Fuzzy Posynomial Geometric Programming......Page 201
Extension in Fuzzy Geometric Programming......Page 207
Fuzzy Reversed Posynomial Geometric Programming Model......Page 209
Fuzzy Lagrange Problem and Algorithm......Page 211
Antinomy in Fuzzy Posynomial Geometric Programming......Page 214
Example of Antinomy......Page 217
Extension......Page 221
Geometric Programming with Fuzzy Coefficients......Page 222
Objective Function with Fuzzy Coefficients......Page 223
Mixed with Fuzzy Coefficients in Objective and Constraints......Page 224
Nonfuzzification Model......Page 226
Algorithm and Numerical Example......Page 229
Conclusion......Page 231
Properties......Page 232
Fuzzy Model......Page 233
Numerical Example......Page 236
Change of Fuzzy Objective Function......Page 237
Determination of Fuzzy Constraints......Page 238
Equivalent Form......Page 240
Primal Geometric Programming with $T$-Fuzzy Variables......Page 243
Primal Geometric programming with Trapezoidal Fuzzy Variables......Page 245
Introduction......Page 248
Dual Geometric Programming with $T$−Fuzzy Variables......Page 249
Dual Geometric Programming with Trapezoidal Fuzzy Variables......Page 253
Disposal of Nonfuzzification in Fuzzy Number......Page 254
Conclusions......Page 255
Modeling......Page 256
Fuzzy Dual Problem......Page 259
Numerical Example......Page 260
$(\vee, \wedge)$ Fuzzy Relative Equation......Page 262
Solubility of $(\vee, ·)$ Fuzzy Relative Equations and Theorem for Greatest Solution......Page 268
Application in Business Management......Page 273
Comparison in Algorithm......Page 278
Introduction......Page 280
Characteristic of Optimal Solution......Page 281
Method to Optimal Solution......Page 282
Numerical Example......Page 285
Structure of Solution Set on Model......Page 287
Solution on Model......Page 290
Model Algorithm......Page 291
Examples......Page 292
Introduction......Page 293
Structure of Solution Set on Equation......Page 294
Solving Solution on Model......Page 296
Algorithm to Model......Page 298
Conclusion......Page 299
Interval Ordinary Differential Equations......Page 300
Fuzzy-Valued Ordinary Differential Equations......Page 306
Ordinary Differential Equations with Fuzzy Variables......Page 313
Building Model......Page 316
Solution and Its Properties in Fuzzy Duoma Debt Model......Page 317
Building of Fuzzy Solow Economic Growth Model......Page 322
Solution and Its Properties of Model......Page 324
Application of Fuzzy Duoma Debt Model......Page 327
Application of Fuzzy Solow Economic Growth Model......Page 330
Interval Functional and Its Variation......Page 334
Variation of Fuzzy-Value Functional at Ordinary Point......Page 339
Variation of Ordinary or Fuzzy-Valued Functional at Fuzzy Points......Page 344
Convex Interval and Fuzzy Function and Functional......Page 345
Convex Interval Function with Functional......Page 346
Convex Function with Functional at Fuzzy Points......Page 348
Conclusion......Page 351
Convex Fuzzy-Valued Function and Functional at Ordinary Points......Page 352
Convex Fuzzy-Valued Function and Functional at Fuzzy Points......Page 355
Variation of Condition Extremum in Interval Functional......Page 357
Variation on Fuzzy-Valued Functional Condition Extremum at Ordinary Points......Page 359
Numerical Example......Page 362
Condition Extremum Variation of Functional with Fuzzy Function......Page 363
Variation of Fuzzy-Valued Functional Condition Extremum with Fuzzy Function......Page 367
Conclusion......Page 368
References......Page 369
Index......Page 378
