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