Some Notes on the Addition of Interactive Fuzzy Numbers

Abstract

This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say A and B, is described by a joint possibility distribution J. One can define a fuzzy number \(A +_J B\) (or \(A -_J B\)), called J-interactive sum (or difference) of A and B, in terms of the sup-J extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers B and C, is there a fuzzy number X and a joint possibility distribution J of X and B such that \(X +_J B = C\)? (2) Given fuzzy numbers A, B, and C, is there a joint possibility distribution J of A and B such that \(A +_J B = C\)? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of \((A +_J B)\) and B such that \((A +_J B) -_N B = A\)? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers A, B and C are real numbers, since the fuzzy arithmetic \(+_J\) and \(-_N\) are extension of the classical arithmetic for real numbers.