Problems with the defuzzification method and a new representation using Lukasiewicz logic: Part I

Document Type

Conference Proceeding

Publication Date

12-1-1994

Abstract

We show that the common defuzzification method, which defines the output of a set of fuzzy if-then rules to be y = ∑[μj(x)y j]/∑μj(x), has some serious deficiencies in terms of formal logical properties that must hold for a rule-set. We correct this problem by presenting a new method of representing a functional relationship y = g(x), x = (x1, x2, ..., xn), using Lukasiewicz's logic Lא. The method introduces two sets of normalized fuzzy concepts {Aj: 1 ≤ j ≤ N} and {Cj: 1 ≤ j ≤ N}, a set of rules of the form "if x is Aj then y is Cj" or "if x is ¬Aj then y is C j" where Aj and Cj, have non-decreasing linear membership functions, and an and/or formula φ(r1, r 2, ..., rN) over the rules ri, without involving negation, which describes how to combine the results Rj(x) of inference from the individual rules rj to obtain the final value of y. The result Rj(x) is now a range of values for y instead of the crisp value μj(x)yj as in defuzzification. Our representation uses the true fuzziness of each Cj unlike the defuzzification which uses a fixed crisp representative value yj for Cj. Our fuzzy rules satisfy the formal logical properties indicated above except that some of them may not apply in a given situation in view of φ(r1, r2, ..., rN).

Publication Source (Journal or Book title)

Proceedings of the Joint Conference on Information Sciences

First Page

137

Last Page

140

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