Constructive method for appropriate membership function integrating fuzzy entropy with smoothing function into interval estimation

This paper develops a constructing approach of a membership function to integrate the fuzzy Shannon entropy with a smoothing function into intervals set by the heuristic method based on the human cognitive behavior and subjectivity under a given probability density function. It is important to set a membership function appropriately in decision making. Therefore, the main parts of our proposed approach are to give membership values a decision maker confidently set, and to obtain the others by solving a nonlinear mathematical programming problem objectively. It is difficult to solve the problem efficiently using previous constructing approaches. In this paper, performing deterministic equivalent transformations to the initial problem using nonlinear programming, an efficient algorithm to obtain the optimal condition of each membership value is developed.