On the monotonicity of Mamdani-Assilian models with two input variables

In the Mamdani-Assilian models considered in this study, the linguistic values of the two input variables and the output variable are described by trapezial membership functions forming a fuzzy partition, the rule base is monotone and the t-norm T_M, T_P or T_L is applied, combined with either the mean of maxima (MOM) or center of gravity (COG) defuzzification method. For these models, monotonicity cannot be guaranteed for all non-smooth rule bases. If the rule base is smooth, a monotone input-output behaviour is obtained when applying the t-norm T_P combined with the MOM or COG defuzzification method. When applying the t-norm T_M combined with the MOM defuzzification method to a model with a smooth rule base, the lengths of the intervals of non-constant membership degree should be equal for all membership functions used in the consequents of the rules, in order to obtain a monotone input-output behaviour