A Comprehensive study on $(\alpha ,\beta )$-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present $(\alpha ,\beta )$-optimistic multi-granulation bipolar fuzzified preference rough sets ( $(\alpha ,\beta {)}^o$-MG-BFPRSs) and $(\alpha ,\beta )$-pessimistic multi-granulation bipolar fuzzified preference rough sets ( $(\alpha ,\beta {)}^p$-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both $(\alpha ,\beta {)}^o$-MG-BFPRSs and $(\alpha ,\beta {)}^p$-MG-BFPRSs are investigated in detail. At the same time, a relationship among the $(\alpha ,\beta )$-BFPRSs, $(\alpha ,\beta {)}^o$-MG-BFPRSs and $(\alpha ,\beta {)}^p$-MG-BFPRSs is given.