Dominance-based Rough Interval-valued Fuzzy Set in Incomplete Fuzzy Information System

Abstract

The fuzzy rough set is a fuzzy generalization of the classical rough set. In the traditional fuzzy rough model, the set to be approximated is a fuzzy set. This paper deals with an incomplete fuzzy information system with interval-valued decision by means of generalizing the rough approximation of a fuzzy set to the rough approximation of an interval-valued fuzzy set. Since all condition attributes are considered as criteria in such incomplete fuzzy information system, the interval-valued fuzzy set is approximated by using the information granules, which are constructed on the basis of a dominance relation. By the proposed rough approximation, the ``at least'' and ``at most'' decision rules can be generated from the incomplete fuzzy information system with interval-valued decision. To obtain the optimal ``at least'' and ``at most'' decision rules, the concepts of the lower and upper approximate reducts, the relative lower and upper approximate reducts for an object are proposed in the incomplete fuzzy information system with interval-valued decision. The judgement theorems and discernibility matrixes associated with these reducts are also obtained. Some numerical examples are employed to substantiate the conceptual arguments.