Arithmetic operations on triangular fuzzy numbers via credibility measures: An inverse distribution approach

Article type: Research Article

Authors: Xie, Xuehui^a | Liu, Yuanyuan^b | Gu, Yujie^a | Zhou, Jian^{a; *}

Affiliations: [a] School of Management, Shanghai University, Shanghai, China | [b] School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, China

Correspondence: [*] Corresponding author. Jian Zhou, School of Management, Shanghai University, Shanghai 200444, China. Tel.: +86 21 6613 4414 805. E-mail: zhou_jian@shu.edu.cn.

Abstract: With extensive applications of fuzzy numbers, many methods for fuzzy arithmetic especially the basic operations have been developed based on Zadeh’s extension principle. Among these methods, the interval arithmetic approach and the standard approximation method are the most important and commonly used exact and approximate methods, respectively. In this paper, regarding the continuous and strictly monotone functions of triangular fuzzy numbers, we propose an inverse distribution approach to deriving the exact or well approximate membership functions for arithmetic results by embedding the credibility measure of fuzzy sets into fuzzy arithmetic. Besides, some non-complicated and complicated examples are given to illustrate the performance of the new approach, together with a detailed comparison with the interval arithmetic approach and the standard approximate method. Furthermore, the inverse distribution approach is also applied to the fuzzy system reliability calculation based on fault tree compared with several current related researches.

Keywords: Triangular fuzzy number, fuzzy arithmetic, membership function, credibility measure, inverse distribution, reliability analysis

DOI: 10.3233/JIFS-172186

Journal: Journal of Intelligent & Fuzzy Systems, vol. 35, no. 3, pp. 3359-3374, 2018