Novel rough set models based on hesitant fuzzy information

Granular computing, an attractive branch of artificial intelligence, focuses on constructing, processing and communicating information granules. Although various useful structures involving fuzzy sets, rough sets and their extensions have been discussed in relation to this literature, there is still a research gap regarding the connections between hesitant fuzzy sets and rough sets. In response to this, we lay a preliminary foundation for two kinds of new rough structures induced by hesitant fuzzy sets. Using hesitant fuzzy information, we put forward one lower approximation, and two related upper approximations, for every crisp subset of the universe of discourse. We investigate the fundamental properties of the generalized rough structures produced from these approximations. We also reveal their connections with granular structures defined from binary relations: under very mild cardinality restrictions, both Pawlak's classical model and its extension to preordered-based approximations become particular cases of the new structures. In this framework, we introduce and investigate the notion of equivalent behavior toward granularity that can occur in various forms (when different hesitant fuzzy approximation spaces produce either the same upper or lower approximation, or the same collection of definable subsets). We describe an elegant mechanism which can be utilized to generate equivalent hesitant fuzzy approximation spaces in every possible way. Finally, these theoretical achievements are supported by a real-world application example dealing with life cycle assessment.