Monotonicity of implicative fuzzy models

Frequent practical problems from decision-making as well as automatic control lead to intuitively monotone fuzzy rule bases. let us assume an appropriate ordering of fuzzy sets is defined. Then by the monotone fuzzy rule base we mean a rule base consisting of such fuzzy rules expressing the monotone dependence of consequent fuzzy sets on antecedent fuzzy sets. In other words the “bigger” antecedent fuzzy is present in a fuzzy rule the “bigger” consequent fuzzy set appears on the right hand side of the same fuzzy rule. Very often real-world applications require some defuzzification to be employed at the end of the inference process. The problem is that after the defuzzification we obtain a crisp input-output function which is not necessarily monotone anymore. Obviously, such behavior is not only counterintuitive but also dangerous. Most of the attention has been paid to the Mamdani-Assilian conjunctive kind of models of fuzzy rule bases built with help of particular t-norms. This paper focuses on the implicative approach for arbitrary residual implication.