On interval-valued hesitant fuzzy rough approximation operators

Interval-valued hesitant fuzzy set is a generalization of classical interval-valued fuzzy set by returning a family of the interval-valued membership degrees for each object in the universe. By combining interval-valued hesitant fuzzy set and rough set models, the concept of an interval-valued hesitant fuzzy rough set is explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, by employing an interval-valued hesitant fuzzy relation, a pair of lower and upper interval-valued hesitant fuzzy rough approximation operators is first defined. The connections between special interval-valued hesitant fuzzy relations and interval-valued hesitant fuzzy rough approximation operators are further established. In axiomatic approach, an operators-oriented characterization of the interval-valued hesitant fuzzy rough set is presented, that is, interval-valued hesitant fuzzy rough approximation operators are defined by axioms, and then, different axiom sets of lower and upper interval-valued hesitant fuzzy set-theoretic operators guarantee the existence of different types of interval-valued hesitant fuzzy relations producing the same operators. Finally, a practical application is provided to illustrate the validity of the interval-valued hesitant fuzzy rough set model.