THE MIN-MAX COMPOSITION RULE AND ITS SUPERIORITY OVER THE USUAL MAX-MIN COMPOSITION RULE

A close analysis of the Syllogism inference rule shows that if one use s Zadeh's notion of fuzzy if-then, then the proper way of combining th e membership values of two fuzzy rules r(1): ''if A, then B'' and r(2) : ''if B, then C'' is not by the usual max-min composition rule, but b y the following min-max rule; tau(ij) = min {max(mu(ik), nu(kj)): all j}, where tau(ij) = m(A)(x(i)) -->m(c)(z(j)), mu(ik) = m(A)(x(i)) --> m(B)(y(k)), and v(kj) = m(B)(y(k)) --> m(c)(z(j)). The min-max value g ives an upper bound on tau(ik). The min-max rule results in a new noti on of transitivity and a corresponding notion of a fuzzy equivalence r elation. We demonstrate the superiority of the min-max rule in terms o f the properties of this equivalence relation. In particular, we argue that the new form of transitivity is particularly suitable for studyi ng non-logical (not equal ''<->'') fuzzy equivalence relationships. (C ) 1998 Elsevier Science B.V.