An Interactive Algorithm to Construct an Appropriate Nonlinear Membership Function Using Information Theory and Statistical Method

This paper develops a constructing algorithm for an appropriate membership function as objectively as possible. It is important to set an appropriate membership function for real-world decision making. The main academic contribution of our proposed algorithm is to integrate a general continuous and nonlinear function with fuzzy Shannon entropy into subjective interval estimation by a heuristic method under a given probability density function based on real-world data. Two main steps of our proposed approach are to set membership values a decision maker confidently judges whether an element is included in the given set or not and to obtain other values objectively by solving a mathematical programming problem with fuzzy Shannon entropy. It is difficult to solve the problem efficiently using previous constructing approaches due to nonlinear function. In this paper, the given nonlinear membership function is approximately transformed into a piecewise linear membership function, and the appropriate values are determined. Furthermore, by introducing natural assumptions in the real-world and interactively adjusting the membership values, an algorithm to obtain the optimal condition of each appropriate membership value is developed.