Rough Fuzzy Set in Incomplete Fuzzy Information System Based on Similarity Dominance Relation

The purpose of this paper is to introduce the concept of dominance-based rough set into the incomplete fuzzy decision system. In such information system, all unknown values are considered as lost, the similarity dominance relation is then used to construct information granules. The lower and upper rough fuzzy approximations in terms of the similarity dominance relation are presented, from which one can derive all “at least” and “at most” decision rules from the incomplete fuzzy decision system. Moreover, to obtain the optimal decision rules, we propose two types of knowledge reductions, relative lower and upper approximate reducts. These two reducts are minimal subsets of the condition attributes, which preserve the lower and upper approximate memberships for an object respectively. Some numerical examples are employed to substantiate the conceptual arguments and related patents are also reviewed in the paper.