Fuzzy soft $$\beta$$-covering based fuzzy rough sets and corresponding decision-making applications

In this article, the concepts of fuzzy soft \(\beta\)-coverings, fuzzy soft \(\beta\)-neighborhoods and fuzzy soft complement \(\beta\)-neighborhoods are firstly proposed and some related properties are studied. Then four distinct types of fuzzy soft \(\beta\)-coverings based fuzzy rough sets are defined. Furthermore, we explore the relationships among the several types of fuzzy soft \(\beta\)-coverings based fuzzy rough sets. In particular, by means of \(\beta\)-level subsets of fuzzy soft \(\beta\)-coverings, some kinds of soft coverings based rough sets are also investigated. Finally, by means of fuzzy soft \(\beta\)-coverings based fuzzy rough sets, we give a detailed description of an algorithmic procedure of decision-making for the new approach.