Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets

Decision-theoretic rough sets (DTRS) are a representative rough set model. The loss function is a pivotal ingredient of DTRS, which is associated with the decision maker's evaluation. Considering the value of loss function with the imprecise evaluation, interval-valued DTRS (IVDTRS) and its mechanism in this paper are explored. First, we construct a basic model of IVDTRS. The comparison between DTRS and IVDTRS is discussed. In the frame of IVDTRS, we then focus on deriving three-way decisions with the aid of two conventional methods, i.e., a certain ranking method and a degree of possibility ranking method, respectively. The certain ranking method converts an interval value into single and derives decision rules under a certain risk attitude of decision maker; the degree of possibility ranking method assumes the flexibility of interval and utilizes the preference between interval values. All the combinations and their prerequisites are summarized, in which we obtain two types of decision rules. Based on the above analysis, we further propose an optimization method for three-way decisions with IVDTRS, which is designed to minimize the overall uncertainty based on the Shannon entropy. We also compare these methods based on standard data sets. Finally, the criteria for choosing a suitable method to three-way decisions with IVDTRS are generated. These results can support decision making in the uncertain environment.