Rough approximations based on intersection of indiscernibility, similarity and outranking relations

Rough sets theory has proved to be a useful mathematical tool for dealing with the vagueness and granularity in information tables. Classical definitions of lower and upper approximations were originally introduced with reference to an indiscernibility relation. However, indiscernibility relation is still restrictive for many applications. Many real-world problems deal with assignment of some objects to some preference-ordered decision classes. And, the objects are described by a finite set of qualitative attributes and quantitative attributes. In this paper, we construct the indiscernibility relation for the subset of nominal attributes, the outranking relation for the subset of ordinal attributes, and the similarity relation for the subset of quantitative attributes. Then the global binary relation is generated by the intersection of indiscernibility relation, outranking relation and similarity relation. New definitions of lower and upper approximations of the upward and downward unions of decision classes are proposed based on the global relation. We also prove that the lower and upper approximation operations satisfy the properties of rough inclusion, complementarity, identity of boundaries, and monotonicity.