Extended rough set-based attribute reduction in inconsistent incomplete decision systems

A systematic study of attribute reduction in inconsistent incomplete decision systems (IIDSs) has not yet been performed, and no complete methodology of attribute reduction has been developed for IIDSs to date. In an IIDS, there are various ways to handle missing values. In this paper, a missing attribute value may be replaced with any known value of a corresponding attribute (such a missing attribute value is called a “do not care” condition). In this way, this paper establishes reduction concepts specifically for IIDSs, mainly by extending related reduction concepts from other types of decision systems into IIDSs, and then derives their relationships and properties. With these derived properties, the extended reducts are divided into two distinct types: heritable reducts and nonheritable reducts, and algorithms for computing them are presented. Using the relationships derived here, the eight types of extended reducts established for IIDSs can be converted to five equivalent types. Then five discernibility function-based approaches are proposed, each for a particular kind of reduct. Each approach can find all reducts of its associated type. The theoretical analysis of the proposed approaches is described in detail. Finally, numerical experiments have shown that the proposed approaches are effective and suitable for handling both numerical and categorical attributes, but that they have different application conditions. The proposed approaches can provide a solution to the reduction problem for IIDSs.