Algebraic Product is the only T-norm for Which Optimization Under Fuzzy Constraints is Scale-Invariant

Abstract

Abstract. In many practical situations, we need to optimize under fuzzy constraints. There is a known Bellman-Zadeh approach for solving such problems, but the resulting solution, in general, depends on the choice of a not well-defined constant M . We show that this dependence disappears if we use an algebraic t-norm (and-operation) f & (a, b) = a · b, and we also prove that the algebraic product is the only t-norm for which the corresponding solution is independent on M .