Implication-based models of monotone fuzzy rule bases

In many modelling problems, there is some inherent monotone relationship between one or more of the input variables and the output variable. We consider the prototypical case of an increasing relationship between each of the input variables and the output variable. When using fuzzy rule-based models, this desired monotonicity is reflected in the rule base, given an appropriate ordering on the fuzzy sets involved in the respective input and output domains. More specifically, the larger the antecedent fuzzy sets, the larger the consequent fuzzy set. However, fuzzy rule-based modelling involves a final defuzzification step, possibly resulting in a function that is no longer monotone. In the context of Mamdani–Assilian conjunctive fuzzy models, ample attention has been paid to this problem, both for the centre-of-gravity defuzzification and mean-of-maxima defuzzification methods. In this paper, we show that for implicative fuzzy models, the non-monotonicity problem can be circumvented by making explicit the semantics of the fuzzy rules by subjecting the antecedent and consequent fuzzy sets to the at-least and/or at-most modifiers.