On the Monotonicity of Fuzzy-Inference Methods Related to T-S Inference Method

Yubazaki et al. have proposed a "single-input rule modules connected-type fuzzy-inference method" (SIRMs method) whose final output is obtained by combining the products of the importance degrees and the inference results from single-input fuzzy-rule modules. Moreover, Seki et al. have proposed a "functional-type SIRMs method" whose consequent parts are generalized to functions from real numbers. It is expected that inference results from the functional-type SIRMs method are monotone, if the antecedent parts and the consequent parts of fuzzy rules in the functional-type SIRMs rule modules are monotone. However, this paper points out that even if consequent parts in the functional-type SIRMs rule modules are monotone, the inference results are not necessarily monotone when the antecedent parts are noncomparable fuzzy sets, and it clarifies the conditions for the monotonicity of inference results from the functional-type SIRMs method. Moreover, for the Takagi-Sugeno (T-S) inference method, the monotonicity condition is clarified in the case of two inputs by using the equivalence relation of fuzzy inference.