Multi-Granulation Fuzzy Rough Sets Based on Fuzzy Preference Relations and Their Applications

In this paper, we pointed out that the transfer function for computing the fuzzy preference degree for the construction of upward/downward fuzzy relations are not additive consistent. Appropriate counterexample is given. Further their modified versions are presented. Similarly, we construct upward consistency matrices of experts which satisfy the upward additive consistency and the upward order consistency simultaneously. After that, by introducing some new fuzzy upward $\beta $ -coverings, fuzzy upward $\beta $ -neighborhoods and fuzzy upward complement $\beta $ -neighborhoods are proposed and related properties are studied. Furthermore, we propose multi-granulation optimistic/pessimistic fuzzy upward rough set based on fuzzy upward $\beta $ -covering and investigate some of their properties. Finally we developed a new technique to multiple attribute decision making problem based on multi-granulation optimistic/pessimistic fuzzy upward rough set. The decision making procedure and the methodology as well as the algorithm of the proposed technique are given. The detailed comparison of the present work with other methods to multiple attribute decision making problem illustrate the advantages of the this work and limitations of other studies.