Affiliations: [a] Department of Industrial and Management Systems Engineering, Waseda University, Okubo, Shinjuku, Tokyo, Japan
| [b] Department of Industrial Engineering and Management, Kanagawa University, Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa, Japan
Correspondence:
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Corresponding author. Takashi Hasuike, Department of Industrial and Management Systems Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan. Tel./Fax: +81 3 5286 3294; E-mail: thasuike@waseda.jp.
Abstract: This paper proposes a mathematical programming approach to construct an appropriate membership function extending our previous studies. It is important to set a membership function with both subjectivity and objectivity to obtain a reasonable optimal solution based on decision maker’s feelings in real-world decision making. In order to ensure objectivity of obtained membership function as well as subjectivity, an entropy-based approach based on mathematical programming is integrated into interval estimation considered by the decision maker. As a general entropy with fuzziness, fuzzy Harvda-Charvat entropy is introduced, which is a natural extension of fuzzy Shannon entropy. In addition, qualitative and subjective evaluations based on the pairwise comparison are introduced to represent the differences between two membership values. The main step of our revised approach is 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 approximation of nonlinear functions, and each intermediate value of partial linear function is optimally obtained.
Keywords: Membership function, Harvda-Charvat entropy, scale for measuring, mathematical programming
