Combination of rules or their consequences in fuzzy expert systems

Abstract

Abstract Issues of combination of rules or their consequences in fuzzy expert systems using Compositional Rule of Inference (CRI) are investigated. First, implication operators are classified as the ‘Expansion Type Implication’, the ‘Reduction Type Implication’, and the ‘Other Type Implication’. Further, fuzzy inference processes based on CRI are classified into three categories in terms of their inference results, i.e., the ‘Expansion Type Inference’, the ‘Reduction Type Inference’, and the ‘Other Type Inference’. Finally combination of rules or their consequences is investigated for inference processes based on CRI. It is shown that in order for the basic requirement for fuzzy reasoning to be satisfied, it is necessary that for inference processes using Sup-T composition in the context of CRI, the combination operator be ‘min’ if the implication operator F(a, b) is an ‘Expansion Type’, i.e., F(a, b) = a → b ⩾ b for all aϵ [0, 1], and is a ‘non-increasing’ function of a, i.e., if a1 ⩾ a2, then F(a1, b) ⩽ F(a2, b); and the combination operator be ‘max’ if the implication operator F(a, b) is a ‘Reduction Type’, i.e., F(a, b) = a → b ⩽ b for all aϵ [0,1], and is a ‘non-decreasing’ function of a, i.e., if a1 ⩾ a2, then F(a1, b) ⩾ F(a2, b). The impact of the overlap is studied between membership functions of the antecedent of a rule and an observation on the inference results. In particular, the cases of R-implication for Sup-T composition and min combination are investigated. We also discuss the consistency of our conclusions with respect to the existing conclusions in the current literature. An example is given to illustrate the proposed method for the selection of combination operators.