Monotone Mamdani–Assilian models under mean of maxima defuzzification

In the Mamdani–Assilian models considered in this study, the linguistic values of the input variables as well as the output variable are described by trapezoidal membership functions forming a fuzzy partition, the rule base is monotone and the crisp output is obtained by the Mean of Maxima (MOM) defuzzification method. It is verified for each of the three basic t-norms TM, TP and TL whether a monotone input–output behaviour is obtained for any monotone rule base, or at least for any monotone smooth rule base. For models applying TL a monotone input–output behaviour can only be guaranteed if the model has a single input variable. The monotonicity of models with two or more input variables applying TM cannot be guaranteed for any monotone rule base, whereas when applying TP monotonicity is only guaranteed for any monotone rule base when using a fuzzy output partition belonging to a restricted class of nine types of fuzzy partitions which contain at most five linguistic values. Finally, it is proved that a monotone input–output behaviour is always obtained for models with a monotone smooth rule base applying TM when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degree of equal length—a restriction that can easily be bypassed by an auxiliary interpolation procedure—and for models with a monotone smooth rule base applying TP for any fuzzy output partition.