On the monotonicity of functional type SIRMs connected fuzzy reasoning method and T-S reasoning method

Yubazaki et al. have proposed ldquosingle input rule modules connected type fuzzy reasoning methodrdquo (SIRMs method, for short) whose final output is obtained by summarizing the product of the importance degrees and the inference results from single input fuzzy rule module. Moreover, Seki et al. have proposed ldquofunctional type single input rule modules connected fuzzy reasoning methodrdquo (functional type SIRMs method, for short) whose consequent parts are generalized to functions from real numbers. It is expect that inference results of functional type SIRMs method have monotonicity if the antecedent parts and consequent parts of fuzzy rules in the functional type SIRMs rule modules have monotonicity. However, this paper points out that even if fuzzy rules in functional type SIRMs rule modules have monotonicity, the inference results do not necessarily have monotonicity. Moreover, it clarifies the conditions for the monotonicity of inference results by functional type SIRMs method, Takagi-Sugeno reasoning method (T-S reasoning method, for short), and simplified fuzzy reasoning method.