Optimal Fuzzy Input Set of Fuzzy Relation Equations Using Biogeography-Based Optimization Algorithm

Abstract

This study presents an approach to optimally find a fuzzy input set that can be used with the fuzzy relation to producing a fuzzy output. In this study, the fuzzy relation model is translated into a mathematical optimization model. Although this model can be optimally solved in any derivative-free population-based optimization algorithm, the biogeography-based optimization (BBO) algorithm is implemented just as a practical example and evidence of our success. In this study, it has been observed that the quality of the results depends on two criteria; the resolution of Lambda-cuts and the constraint (whether the estimated fuzzy output is a subset of the actual fuzzy output or not). If this constraint is enabled in the design function, then obtaining the optimal value becomes a hard task, especially if the Lambda-cuts are provided with limited discrete elements of the membership function of the fuzzy input set. Also, it has been found that the optimal value obtained by Sanchez's operator can be met by many alternatives of fuzzy input vectors obtained by BBO.