Approximations de Tchebyshev d’un système incompatible d’équations relationnelles floues de type max?T

In this article, we study the inconsistency of a system of max?T fuzzy relational equations denoted (S) : A?max T x = b where T is a t-norm among the minimum, the product or the one of Lukasiewicz, A is a matrix and b is the column vector used as the second member of the system (S). For each t-norm, we give an analytical formula that allows us to compute the Chebyshev distance ? = infc?C ?b ? c? associated to the second member b of an inconsistent system (S), where C is the set of second members of consistent systems defined with the same matrix A. These formulas are computed from the components of A and b. From ?, we explicitly compute the greatest Chebyshev approximation of b denoted c, i.e ?b ? c? = ? and the system A?max T x = c is consistent. We conclude by proposing some applications of our results.