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JACIII Vol.20 No.4 pp. 535-542
doi: 10.20965/jaciii.2016.p0535
(2016)

Paper:

An Objective Approach for Constructing a Membership Function Based on Fuzzy Harvda-Charvat Entropy and Mathematical Programming

Takashi Hasuike* and Hideki Katagiri**

*Faculty of Science and Technology, Waseda University
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

**Faculty of Engineering, Kanagawa University
3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa 221-8686, Japan

Received:
December 25, 2015
Accepted:
March 26, 2016
Published:
July 19, 2016
Keywords:
membership function, constructing approach with subjectivity and objectivity, fuzzy entropy, mathematical programming
Abstract

This paper proposes an objective approach to the construction of an appropriate membership function that extends to our previous studies. It is important to set a membership function with subjectivity and objectivity to obtain a reasonable optimal solution that complies with the decision maker’s feelings in real-world decision making. To ensure objectivity and subjectivity of the obtained membership function, an entropy-based approach based on mathematical programming is integrated into the interval estimation considered by the decision maker. Fuzzy Harvda-Charvat entropy, which is a natural extension of fuzzy Shannon entropy, is introduced as general entropy with fuzziness. The main steps of our proposed approach are to set intervals with membership values 0 and 1 to enable a decision maker to judge confidently, and to solve the proposed mathematical programming problem strictly using nonlinear programming. In this paper, the given membership function is assumed to be a piecewise linear membership function as an approximation of nonlinear functions, and each intermediate value of partial linear function is optimally obtained.

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