**6. Conclusions**

Covering rough set models are important research topics, which investigate data mining in a more general manner. Huang et al. [33] presented an IF rough set model and an optimistic multi-granulation IF rough set model. By investigation, we have found that no one has applied multi-granulation IF rough set models to MCGDM problems. In this paper, by showing some new notions and properties of IF *β*-covering approximation spaces, we mainly study Huang et al.'s models and propose a novel approach to MCGDM problems. The main conclusions in this paper and the further work are listed as follows.


**Author Contributions:** All authors have contributed equally to this paper. The individual responsibilities and contribution of all authors can be described as follows: the idea of this whole thesis was put forward by X.Z.; he also completed the preparatory work of the paper; J.W. analyzed the existing work of rough sets and IF sets, and wrote the paper.

**Funding:** This work is supported by the National Natural Science Foundation of China under Grant Nos. 61573240 and 61473239.

**Conflicts of Interest:** The authors declare no conflict of interest.
