Three-way decisions based on decision-theoretic rough sets with dual hesitant fuzzy information.

Decision-theoretic rough sets (DTRSs) as a classic model of three-way decisions have been widely applied in the area of risk decision-making. When we confront the complicated and uncertain environment, one of challenges is to estimate the loss function of DTRSs. As a new generalization of fuzzy sets, dual hesitant fuzzy sets (DHFSs) can handle uncertain information more flexibly in the process of decision making and give a new measure for the determination of loss functions of DTRSs. To have more interesting results in the context of three-way decisions, we introduce the new hesitant format of DHFSs into DTRSs and explore a new three-way decision model. Firstly, we take into account the loss functions of DTRSs with dual hesitant fuzzy elements (DHFEs) and propose a dual hesitant fuzzy DTRS model. In order to satisfy the preconditions of three-way decisions, we analyze the normalized principle of loss functions under the dual hesitant fuzzy environment. Meanwhile, some properties of the expected losses are carefully investigated. Then, we further design two approaches for deriving three-way decisions with the new DTRS model, i.e., Method 1 and Method 2, which mainly relies on the comparisons among the expected losses. Method 1 is a general method based on the scores and the accuracies of DHFEs. Method 2 is a ranking method of possibility degrees with a stochastic strategy and enriches the comparisons among the expected losses. Finally, the assessment of emergency blood transshipment is used to illustrate and compare these proposed methods.