Discernibility-Matrix Method Based On Dominance Relations

In some practical problems, particularly in multi-criteria decision analysis, there are some attributes with ordered values. In this case, the preference-orders of attribute values should not be ignored. The concept of equivalence relation cannot reflect this kind of order,the framework of dominance relation-based rough set theory is developed, in which the original equivalence relation is substituted by dominance relation. According to diverse purposes, different types of reductions based on dominance relations are then proposed. Based on the research situation above, here is the main contribution in this paper:Firstly, by incorporating dominance principle in inconsistent decision systems based on dominance relations, two new types of distribution reductions are proposed, i.e., generalized distribution reduction and generalized maximum distribution reduction, and their properties and relationship are also discussed. The corresponding generalized distribution discernibility matrix is then defined to provide a convenient computation method to obtain the generalized distribution reductions. The validation of this method is showed by both theoretical proofs and illustrative examples. Next, we also provide discernibility-matrix method to obtain the positive domain reduction. The Discernibility-Matrix (DM) method of reduct generation is compared with the attribute significance method. Last, for some information systems whose attribute set is composed of both regular attributes and criteria, equivalence-dominance relation is then defined. Equivalence relation should be defined on the regular attributes and dominance relation on the criteria. The reduction based on equivalence-dominance relation is then given and Discernibility-Matrix approach of reduction generation is also provided.