As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzy β -neighborhoods and thus three kinds of covering-based multigranulation ( ℐ , T )-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ( ℐ , T )-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.

Recently, the concept of a fuzzy αneighborhood operator with reflexivity was established and a fuzzy rough set covering based on a fuzzy αneighborhood operator was defined by Zhang et al. As a generalization of Zhang et al. models, this paper aims to introduce the notion of a complementary fuzzy α-neighborhood operator with reflexivity. Also, three new kinds of a fuzzy rough set covering based on a fuzzy αneighborhood operator are constructed and some of their properties are discussed. Further, the relationships between these models are studied. Several new kinds of a fuzzy rough set covering based variable precision are put forward and the relevant properties are constructed. Finally, an application to MADM to solve realistic problems is illustrated.INDEX TERMS Complementary fuzzy α-neighborhood operator, fuzzy rough set covering, fuzzy rough set covering based variable precision, Multi-attribute decision making.