The uncertainty in the data is a hurdle in decision-making problems. Rough set theory and fuzzy set theory are built to handle the uncertainty in data. We introduce the rough bipolar fuzzy sets as a hybridization of rough sets and the bipolar fuzzy sets. After that, we discuss a group decision making problem with the data having fuzziness endowed with bipolarity and iron out this problem by applying the rough bipolar fuzzy sets. We also propose an algorithm for this problem, which yields the best decision, as well as, the worst decision between some objects.

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) theory is a valuable mathematical tool to address incomplete knowledge. In contrast, BS sets (BSSs) can handle the vagueness and the bipolarity of the data in a variety of scenarios. The main goal of the current article is to analyze the concept of rough BS (BS) ideals in semigroup (Sg) by combining RS theory and BSSs. To this end, the ideas of BS left ideals (BS-LIs), BS right ideals (BS-RIs), BS two-sided ideals (BS-2SIs), BS interior ideals (BS-IIs) and BS bi-ideals (BS-BIs) over the Sg ? are developed. Some significant properties associated with these notions are highlighted with several concrete illustrations. Furthermore, it has been noted that the congruence relation (Cr) and complete congruence relation (CCr) are found to be essential for creating rough approximations of BS ideals. As a result, their affiliated features are evaluated through Cr and CCr.