Generalized multigranulation double-quantitative decision-theoretic rough set

The principle of the minority subordinate to the majority is the most feasible and credible when people make decisions in real world. So generalized multigranulation rough set theory is a desirable fusion method, in which upper and lower approximations are approximated by granular structures satisfying a certain level of information. However, the relationship between a equivalence class and a concept under each granular structure is very strict. Therefore, more attention are paid to fault tolerance capabilities of double-quantitative rough set theory and the feasibility of majority principle. By considering relative and absolute quantitative information between the class and concept, we propose two kinds of generalized multigranulation double-quantitative decision-theoretic rough sets(GMDq-DTRS). Firstly, we define upper and lower approximations of generalized multigranulation double-quantitative rough sets by introducing upper and lower support characteristic functions. Then, important properties of two kinds of GMDq-DTRS models are explored and corresponding decision rules are given. Moreover, internal relations between the two models under certain constraints and GMDq-DTRS and other models are explored. The definition of the approximation accuracy in GMDq-DTRS is proposed to show the advantage of GMDq-DTRS. Finally, an illustrative case is shown to elaborate the theories advantage of GMDq-DTRS which are valuable to deal with practical problems. Generalized multigranulation double-quantitative decision-theoretic rough set theory will be more feasible when making decisions in real life.