A comprehensive study of upward fuzzy preference relation based fuzzy rough set models: Properties and applications in treatment of coronavirus disease.

In this paper, we first introduce a new type of rough sets called α -upward fuzzified preference rodownward fuzzy preferenceugh sets using upward fuzy preference relation. Thereafter on the basis of α -upward fuzzified preference rough sets, we propose approximate precision, rough degree, approximate quality and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward β -coverings, fuzzy upward β -neighborhoods and fuzzy upward complement β -neighborhoods and some relavent properties are discussed. Hereby, we formulate a new type of upward lower and upward upper approximations by applying an upward β -neighborhoods. After employing the upward β -neighborhoods based upward rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Subsequently, we presented the idea to combine the fuzzy implicator and t -norm to introduce multigranulation ( ℐ , T ) -fuzzy upward rough set applying fuzzy upward β -covering and some relative properties are discussed. Finally we presented a new technique for the selection of medicine for treatment of coronavirus disease (COVID-19) using multigranulation ( ℐ , T ) -fuzzy upward rough sets.