Subsethood Measures on a Bounded Lattice of Continuous Fuzzy Numbers with an Application in Approximate Reasoning

Analogical reasoning schemes have found widespread use ever since the inception of the Mamdani fuzzy rule-based system which can be seen both as an interpolative and a relational fuzzy inference system (FIS). Note that in this case Zadeh’s compositional rule of inference (CRI) is used as an inference mechanism. As an alternative characterization in terms of analogical reasoning, we have that an input fuzzy set is compared with the antecedents of the rules in terms of Zadeh’s consistency index. Recently, there have been numerous proposals to replace Zadeh’s consistency index by similarity measures and overlap indices. Cornelis and Kerre previously presented a promising approach toward analogical reasoning that is based on the fuzzification of set inclusion and analyzed its properties such as monotonicity, compatibility with modus ponens, coherence and consistency. Other fuzzy extensions of set inclusion are given by Kaburlasos’ et al.’s inclusion measure approaches. The latter are related to subsethood measures that can be defined on arbitrary bounded lattices such as the class of closed subintervals of $$[0, 1]^n$$ . In this paper, we introduce a bounded lattice consisting of the empty set and the fuzzy sets on a closed interval whose $$\alpha $$ -cuts are closed and whose membership functions (MFs) are continuous. Then we define specific types of subsethood measures on this bounded lattice. These subsethood measures are then used in analogical reasoning schemes for fuzzy rule-based systems. Finally, we provide an application in mechanical condition monitoring.