Alpha Level Sets Research Articles

An interactive fuzzy satisficing method for multiobjective nonconvex programming problems with fuzzy numbers through coevolutionary genetic algorithms

In this paper, by considering the experts' fuzzy understanding of the nature of the parameters in the problem-formulation process, multiobjective nonconvex nonlinear programming problems with fuzzy numbers are formulated and an interactive fuzzy satisficing method through coevolutionary genetic algorithms is presented. Using the alpha-level sets of fuzzy numbers, the corresponding nonfuzzy alpha-programming problem is introduced. After determining the fuzzy goals of the decision maker, if the decision maker specifies the degree alpha and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the augmented minimax problems for which the coevolutionary genetic algorithm, called GENOCOP III, is applicable. In order to overcome the drawbacks of GENOCOP III, the revised GENOCOP III is proposed by introducing a method for generating an initial feasible point and a bisection method for generating a new feasible point efficiently. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented together with an illustrative numerical example.

A neural fuzzy system with fuzzy supervised learning

A neural fuzzy system learning with fuzzy training data (fuzzy if-then rules) is proposed in this paper. This system is able to process and learn numerical information as well as linguistic information. At first, we propose a five-layered neural network for the connectionist realization of a fuzzy inference system. The connectionist structure can house fuzzy logic rules and membership functions for fuzzy inference. We use alpha-level sets of fuzzy numbers to represent linguistic information. The inputs, outputs, and weights of the proposed network can be fuzzy numbers of any shape. Furthermore, they can be hybrid of fuzzy numbers and numerical numbers through the use of fuzzy singletons. Based on interval arithmetics, a fuzzy supervised learning algorithm is developed for the proposed system. It extends the normal supervised learning techniques to the learning problems where only linguistic teaching signals are available. The fuzzy supervised learning scheme can train the proposed system with desired fuzzy input-output pairs which are fuzzy numbers instead of the normal numerical values. With fuzzy supervised learning, the proposed system can be used for rule base concentration to reduce the number of rules in a fuzzy rule base. Simulation results are presented to illustrate the performance and applicability of the proposed system.