Generalized Interval-Valued Fuzzy Rough Set and its Application in Decision Making

This paper presents a general study of generalized interval-valued fuzzy rough sets integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. In the constructive approach, by employing an interval-valued fuzzy residual implicator and its dual operator, generalized upper and lower interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy approximation space are first defined. Then properties of generalized interval-valued fuzzy rough approximation operators are discussed. Furthermore, connections between special types of interval-valued fuzzy relations and properties of generalized interval-valued fuzzy approximation operator are also established. In the axiomatic approach, generalized interval-valued fuzzy rough approximation operators are defined by axioms. We prove that different axiom sets can characterize the essential properties of generalized interval-valued fuzzy rough approximation operators. Also the composition of two approximation spaces is explored. Finally, a practical application is provided to illustrate the efficiency of the generalized interval-valued fuzzy rough set model.