Generalized multigranulation fuzzy rough sets based on upward additive consistency

Abstract

In this paper, we point out that the transfer function for computing the fuzzy preference degree of Pan et al. (Fuzzy Sets Syst 312:87–108, 2017) for the construction of upward/downward fuzzy relations is not additive consistent. Appropriate counterexample is given. Further their modified versions are presented. Meanwhile, 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 multigranulation optimistic/pessimistic $\left( {\mathcal {I}},{\mathcal {T}}\right)$-fuzzy upward rough set based on fuzzy upward $\beta$-covering and investigate some of their properties. Finally we construct a new approach to multiple attribute decision making problem based on multigranulation optimistic/pessimistic $\left( {\mathcal {I}}, {\mathcal {T}}\right)$-fuzzy upward rough set. The decision making procedure, methodology and the algorithm of the proposed technique are given. The detailed comparison of the present work with other methods to multiple attribute decision making problem illustrates the advantages of the this work and limitations of other studies.