Dominance-based rough fuzzy set approach and its application to rule induction

Theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness, respectively. Marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information. In this paper, our attention is paid to ordered fuzzy decision systems, where condition criteria are preference-ordered and decision classes are not only ordered but also fuzzy. First, the dominance-based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems. Second, lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system. Then, two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy, respectively. Also, decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers. Finally, a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods.