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

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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