Double-quantitative decision-theoretic approach to multigranulation approximate space

The decision-theoretic rough set, as a special case of probabilistic rough set, mainly utilizes conditional probability to express relative quantitative information, while the graded rough set is characterized by absolute quantitative information between the partitions and basic concept. Thus, the double-quantification integrating relative and absolute quantitative information has become a fundamental topic for model construction, especially for developing the decision-theoretic rough set. In this study, we propose a basic framework of double-quantitative decision-theoretic rough set based on Bayesian decision and graded rough set approach in multigranulation approximate space. Three pairs of double-quantitative multigranulation decision-theoretic rough set models are established, which consist of a dual of optimistic double-quantitative multigranulation decision-theoretic rough sets, pessimistic double-quantitative multigranulation decision-theoretic rough sets and mean double-quantitative multigranulation decision-theoretic rough sets. These models essentially indicate the relative and absolute information quantification. Furthermore, some essential properties of these models are addressed and the decision rules which incorporate the relative and absolute quantitative information are investigated. Finally, an illustrative case about medical diagnosis is conducted to interpret and evaluate the double-quantitative decision-theoretic approach.