New insights into rough approximations of a fuzzy set inspired by soft relations with decision making applications

In the fields of mathematics and information sciences, binary relations are vital. Fuzzy sets (FSs), rough sets (RSs), and soft sets (SSs) are mathematical strategies that effectively handle ambiguous and imprecise data in practical situations. This work presents various properties of the roughness of fuzzy sets regarding foresets (F-sets) and aftersets (A-sets) using soft binary relations (SBRs). Initially, two pairs of fuzzy soft sets (FSSs) are obtained by approximating a fuzzy subset using an SBR, and their distinctive axiomatic systems are explored. Additionally, two types of fuzzy topologies that result from soft reflexive relations (SRRs) are examined. Numerous similarity relations allied with SBRs are also investigated. In addition, we present the accuracy measure and roughness measure for a fuzzy subset based on the mass assignment of the fuzzy subset through soft relations. Next, we outline a decision-making (DM) approach within the context of the proposed method. In addition, we provide two algorithms and decision phases. Ultimately, an applied example is used to evaluate the reliability of the decision processes. An extensive comparison study confirms the proposed method's feasibility and superiority over current DM methods.