A Study on Soft Multi-Granulation Rough Sets and Their Applications

Rough set (RS) and soft set (SS) theories are two successful mathematical approaches to dealing with uncertainty in data analysis. The classical soft rough set (SRS) theory proposed by Feng et al. (2011) offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is essential to note that the SRS theory cannot be applied in the context of multi-granulation in the real world. To address this issue, in this paper, we introduce the idea of soft multi-granulation RS (SMGRS) model based on two soft binary relations (S-BRs). Axiomatic operations, lower soft rough approximation space (lower SRA-space) and upper soft rough approximation space (upper SRA-space), are defined through after sets of soft relations. After that, the concept of SMGRSs is applied to a significant part of commutative algebra, group theory. In this respect, the primitive notions of SRA-spaces are defined with the help of two normal soft groups (NSGs). In groups, several important structural properties related to SMGRS are investigated in detail with illustrative examples. It is shown that SMGRS in groups may be influential in decision-making (DM) by some numerical examples. To demonstrate the flexibility, superiority, and effectiveness of the suggested technique, some comparative examples are given with some existing methods.