Optimal Fuzzy Model Construction with Statistical Information using Genetic Algorithm

Fuzzy rule based models have a capability to approximate any continuous function to any degree of accuracy on a compact domain. The majority of FLC design process relies on heuristic knowledge of experience operators. In order to make the design process automatic we present a genetic approach to learn fuzzy rules as well as membership function parameters. Moreover, several statistical information criteria such as the Akaike information criterion (AIC), the Bhansali-Downham information criterion (BDIC), and the Schwarz-Rissanen information criterion (SRIC) are used to construct optimal fuzzy models by reducing fuzzy rules. A genetic scheme is used to design Takagi-Sugeno-Kang (TSK) model for identification of the antecedent rule parameters and the identification of the consequent parameters. Computer simulations are presented confirming the performance of the constructed fuzzy logic controller

Fuzzy Bilevel Optimization

In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1 1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2 1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 11 2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 3 Optimization problem with fuzzy objective 19 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25 4 Linear optimization with fuzzy objective 27 4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 5 Optimality conditions 47 5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48 5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49 5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51 5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52 5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Fuzzy linear optimization problem over fuzzy polytope 59 6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65 6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Bilevel optimization with fuzzy objectives 73 7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78 7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Conclusions 95 Bibliography 9

Maximin and maximal solutions for linear programming problems with possibilistic uncertainty

We consider linear programming problems with uncertain constraint coefficients described by intervals or, more generally, possi-bility distributions. The uncertainty is given a behavioral interpretation using coherent lower previsions from the theory of imprecise probabilities. We give a meaning to the linear programming problems by reformulating them as decision problems under such imprecise-probabilistic uncer-tainty. We provide expressions for and illustrations of the maximin and maximal solutions of these decision problems and present computational approaches for dealing with them

Uncertain Multi-Criteria Optimization Problems

Most real-world search and optimization problems naturally involve multiple criteria as objectives. Generally, symmetry, asymmetry, and anti-symmetry are basic characteristics of binary relationships used when modeling optimization problems. Moreover, the notion of symmetry has appeared in many articles about uncertainty theories that are employed in multi-criteria problems. Different solutions may produce trade-offs (conflicting scenarios) among different objectives. A better solution with respect to one objective may compromise other objectives. There are various factors that need to be considered to address the problems in multidisciplinary research, which is critical for the overall sustainability of human development and activity. In this regard, in recent decades, decision-making theory has been the subject of intense research activities due to its wide applications in different areas. The decision-making theory approach has become an important means to provide real-time solutions to uncertainty problems. Theories such as probability theory, fuzzy set theory, type-2 fuzzy set theory, rough set, and uncertainty theory, available in the existing literature, deal with such uncertainties. Nevertheless, the uncertain multi-criteria characteristics in such problems have not yet been explored in depth, and there is much left to be achieved in this direction. Hence, different mathematical models of real-life multi-criteria optimization problems can be developed in various uncertain frameworks with special emphasis on optimization problems

Structural optimization of large structural systems by optimality criteria methods

The fundamental concepts of the optimality criteria method of structural optimization are presented. The effect of the separability properties of the objective and constraint functions on the optimality criteria expressions is emphasized. The single constraint case is treated first, followed by the multiple constraint case with a more complex evaluation of the Lagrange multipliers. Examples illustrate the efficiency of the method