Granular representation and granular computing with fuzzy sets

In this study, we introduce a concept of a granular representation of numeric membership functions of fuzzy sets, which offers a synthetic and qualitative view at fuzzy sets and their ensuing processing. The notion of consistency of the granular representation is formed, which helps regard the problem as a certain optimization task. More specifically, the consistency is referred to a certain operation ϕ, which gives rise to the concept of ϕ-consistency. Likewise introduced is a concept of granular consistency with regard to a collection of several operations, Given the essential role played by logic operators in computing with fuzzy sets, detailed investigations include and- and or-consistency as well as (and, or)-consistency of granular representations of membership functions with the logic operators implemented in the form of various t-norms and t-conorms. The optimization framework supporting the realization of the ϕ-consistent optimization process is provided through particle swarm optimization. Further conceptual and representation issues impacted processing fuzzy sets are discussed as well.