Some Notes on the Addition of Interactive Fuzzy Numbers

Author(s)

Esmi, Estevao ; de Barros, Laecio Carvalho ; Wasques, Vinicius Francisco ; Kearfott, RB ; Batyrshin, I ; Reformat, M ; Ceberio, M ; Kreinovich, V

• Document type: Journal article
• Source: FUZZY TECHNIQUES: THEORY AND APPLICATIONS; v. 1000, p. 12-pg., 2019-01-01.

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+B-J (or A-B-J), 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 + B-J = C? (2) Given fuzzy numbers A, B, and C, is there a joint possibility distribution J of A and B such that A+B-J = C? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of (A + B-J) and B such that (A + B-J) - B-N = 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. (AU)

• FAPESP's process: 16/26040-7 - Differential and integral calculus based on arithmetic of interactive fuzzy numbers
• Grantee: Estevão Esmi Laureano
• Support Opportunities: Regular Research Grants