Three-way group decisions based on multigranulation hesitant fuzzy decision-theoretic rough set over two universes

Three-way decisions, deduced from decision-theoretic rough set (DTRS), provide useful approaches for uncertain decision-making problems from a new perspective. Considering situations where decision-makers hesitate among several evaluation values, the hesitant fuzzy set, an innovative extension of fuzzy set, is introduced into multigranulation DTRS under multi-criteria group decision-making (MCGDM) environment. More specifically, we incorporate DTRS with hesitant fuzzy information into MCGDM in this paper, and explore the related decision-making mechanism. Firstly, the variable precision multigranulation hesitant fuzzy DTRS over two universes is defined by utilizing hesitant relation and conditional probability of a hesitant fuzzy event, and the associated decision rules and properties are derived and carefully investigated. Accordingly, as two special types, optimistic and pessimistic multigranulation hesitant fuzzy DTRS over two universes are constructed similarly at the same time. Then, different from loss functions with fixed values in most of the existing DTRS models, the loss functions in this paper are not given directly but calculated from evaluation values expressed by hesitant fuzzy elements. With the aid of the distance measure of hesitant fuzzy elements, the calculation of loss functions is presented. Furthermore, the three-way group decision-making model based on multigranulation hesitant fuzzy DTRS over two universes is established to address MCGDM problems, and key steps of this model are designed. Finally, we elaborate the application of the model by an example of green supplier selection problem.