Bipolarity highlights the positive and negative facets of a certain dilemma. This script aims to propose a novel multi-criteria decision-making (MCDM) approach based on the bipolar fuzzy preference delta-covering based bipolar fuzzy rough set (BFPδC-BFRS) model by combining the VIKOR (VIseKriterijumska Optimizacija I Kompromisno Rasenje) scheme. The VIKOR scheme is viewed as a beneficial MCDM strategy, particularly in situations where an expert is incapable of making the right decision at the beginning of system design.The VIKOR method works well for problems with competing attributes because it operates under the presumptions that compromise is acceptable in conflict resolution, the expert seeks a solution that is extremely close to the best, and all developed attributes are taken into consideration when processing the various alternatives. In this study, firstly, we propose an integrated MCDM based on BFPδC-BFRSs using the VIKOR methodology. Moreover, we solve a real-world illustration to show the effectiveness of the expanded VIKOR approach. Finally, we demonstrate a detailed comparative analysis of the proposed methodology with some prevalent decision-making approaches to substantiate the accountability of the recommended scheme.

Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-Rs and complete cng-Rs.

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. [8] 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.

Preference analysis is a significant component in decision-making (DM) when selecting an optimal alternative. By comparing any two alternatives pairwise, preference relations (PRs) effectively depict the preference degrees of decision-makers (DMrs). The rough set theory (RST) has been effectively applied to cope with preference analysis by swapping the equivalence relation (Er) with the dominance relation (DR). In this study, we propose new transfer functions to construct alternatives’ upward/downward fuzzy preference degree (FPD) for evaluating upward and downward fuzzy PRs (FPRs). Based on these newly proposed transfer functions, we present a novel method for fuzzifying RSs called the upward $\alpha $ -fuzzified preference rough sets ( $\alpha ^{\uparrow }$ -FPRSs). The basic properties of the proposed $\alpha ^{\uparrow }$ -FPRSs are thoroughly studied. Moreover, several uncertainty measures related to $\alpha ^{\uparrow }$ -FPRSs are presented. Meanwhile, we offered the notion of upward fuzzy $\beta $ -covering (UF $\beta $ C) and upward fuzzy $\beta $ -neighborhood (UF $\beta $ -nghd), upward $\beta $ -neighborhood (U $\beta $ -nghd), and several related properties are explored. Based on UF $\beta $ -nghd and U $\beta $ -nghd, we construct two new models of UF $\beta $ C rough sets (UF $\beta $ -CRSs) along with their properties. We formulate a novel technique of multi-attribute DM (MADM). To legitimise the practicality of our proposed model, we provide a real-life example of selecting an appropriate medication to treat a specific disease. Finally, we look into the efficacy of the launched scheme through a comparison study.