Dominance-based rough set approach and knowledge reductions in incomplete ordered information system

Many methods based on the rough set to deal with incomplete information systems have been proposed in recent years. However, they are only suitable for the incomplete systems with regular attributes whose domains are not preference-ordered. This paper thus attempts to present research focusing on a complex incomplete information system—the incomplete ordered information system. In such incomplete information systems, all attributes are considered as criterions. A criterion indicates an attribute with preference-ordered domain. To conduct classification analysis in the incomplete ordered information system, the concept of similarity dominance relation is first proposed. Two types of knowledge reductions are then formed for preserving two different notions of similarity dominance relations. With introduction of the approximate distribution reduct into the incomplete ordered decision system, the judgment theorems and discernibility matrixes associated with four novel approximate distribution reducts are obtained. A numerical example is employed to substantiate the conceptual arguments.