A Subjective and Objective Constructing Approach for Reasonable Membership Function Based on Mathematical Programming

This paper proposes a strict constructing approach for a reasonable membership function as objectively as possible. It is important to set a reasonable membership function to parameters for real-world decision making interactively and objectively. The main contribution of our proposed approach is to integrate a general continuous function derived from mathematical programming under a given probability density function based on real-world data into subjective interval estimation by a heuristic method. The interval estimation is to set intervals that a decision maker confidently judges whether an element is completely or never included in the given set. The main steps of our proposed approach are to solve the mathematical programming problem in terms of objectivity. It is hard to solve the problem with nonlinear function directly and efficiently. In this paper, the given nonlinear membership function is approximately transformed into a piecewise linear membership function, and the appropriate membership values are determined based on the strict optimal solution.