New categories of coverings in terms of rough fuzzy sets

In this article, we develop innovative models of multigranulation rough fuzzy sets, utilizing fuzzy \(\beta \)-covering and incorporating a range of techniques such as fuzzy \(\beta \)-neighborhood, fuzzy complementary \(\beta \)-neighborhood, fuzzy \(\beta \)-minimal description, and fuzzy \(\beta \)-maximal description. The axiomatic characteristics of fuzzy \(\beta \)-neighborhoods within the context of fuzzy \(\beta \)-covering based multigranulation rough fuzzy sets (F\(\beta \)CMGRFS) are analyzed. Thus, we introduce seven new classes of F\(\beta \)CMGRFS and investigate their relevant properties. Furthermore, the connections and associations between these techniques are established. Thus, we provide a set of observations and propositions that highlight the value and dissimilarities of our proposed models compared to others. A test example is produced to verify the applicability of the presented strategies and treat MCGDM issues. Finally, we compare the outcomes of our approaches and the existing studies to check the reliability and the validity of our work.