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Recently, the concept of a fuzzy <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>- neighborhood operator with reflexivity was established and a fuzzy rough set covering based on a fuzzy <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>- neighborhood operator was defined by Zhang <italic>et al.</italic> As a generalization of Zhang <italic>et al.</italic> models, this article aims to introduce the notion of a complementary fuzzy <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-neighborhood operator with reflexivity. Also, three new kinds of a fuzzy rough set covering based on a fuzzy <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>- neighborhood 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.