Covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and applications

As an extension of the classical Pawlak rough sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough set models have been researched, in which objects approximated are crisp sets or accurate concepts of the universe of discourse, and there are only two states, which are disjoint and opposite each other for an accurate concept of the universe of discourse for a decision-making problem. However, the objects of many decision-making problems can have more than two states in practice. Moreover, the states of the decision object are not necessarily disjoint and opposite each other, but these decision states are fuzzy descriptions of the states of the object on the universe. In order to deal with decision-making problems in which decision states are fuzzy sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models are proposed in this paper. In addition, the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets are investigated and characterized. Furthermore, the relationships between covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models are discussed. Finally, an example is presented to illustrate the practicality and generality of the proposed models.