The differences between the horizontal membership function used in multidimensional fuzzy arithmetic and the inverse membership function used in gradual arithmetic

Abstract

In the last few years, the number of applications of the multidimensional fuzzy arithmetic (MFA) and the multidimensional interval arithmetic is expanding. Authors of new papers about applications of MFA are often faced with comments from other researchers, especially the gradual arithmetic (GA) proponents, that the horizontal membership function (HMF) used in MFA is the same as the inverse membership function (InvMF) used in GA, and that MFA itself adds nothing new to the fuzzy arithmetic. This view leads to unfair evaluations of scientific papers about MFA applications submitted to scientific journals and to unnecessary disagreements between MFA and GA proponents. The purpose of this paper is to carefully analyze the two types of functions (HMF and InvMF) and to demonstrate their important differences. The basic and decisive difference is the dimensionality of both functions, which is illustrated by examples. It should also be added that HMF has proven its usefulness in solving difficult problems such as: systems of fuzzy equations or fuzzy differential equations, which is confirmed by numerous publications. The paper enable the reader to have a deeper understanding of the multidimensional fuzzy arithmetic.