Inverse Operations for Fuzzy Numbers

Abstract

A fuzzy number is defined as a convex normalized fuzzy set of the real line with an upper semi-continuous membership function. The addition of fuzzy numbers denoted ⊕, and defined via a sup- t-norm convolution is now well-understood from both theoretical and computational points of view, at least for the t-norm ’min’. Moreover, there exists an “inverse”, -A, of a fuzzy number A, i.e. -A restricts the possible values of the variable -u iff A restricts the possible values of u. However, the identify A ⊕ [— A] = 0 does not hold but when A is a genuine real number. Thus, the solution X of the equation of fuzzy numbers : X ⊕ A = B, when it exists, is NOT given by X = B ⊕ [-А].An operation) + (is introduced, which enables to express the solution, when it exists, under the form X = B ) + ([-A] ; for instance A) + ([-A] = 0. This operation) + (is shown to be related to Godel-Brouwer implication, when a “sup-min” convolution is used in the definition of ⊕. On crisp intervals,) + (reduces to an operation sometimes known as “MINKOWSKI subtraction” : if A and B are crisp intervals and the length of A is greater than that of B, then A) + (B is an interval whose mean value (in the sense of arithmetic mean) is the sum of the mean values of A and B, and whose length is equal to the difference of the respective lengths of A and B. In terms of error analysis,) + (corresponds to the maximal compensation of errors (i.e. the optimistic case) while ⊕ corresponds to the cumulation of errors (i.e. the pessimistic case).) + (is also defined when a t-norm other than “min” is used in the convolution defining ⊕. For instance, if ⊕ employs the product, then) + (is based on an implication earlier considered by Goguen and Gaines.The same approach can be applied to the multiplication.