Double-Framed Bipolar Fuzzy Soft Sets and Algorithmic Approaches with Symmetry for Multi-Criteria Decision-Making Under Uncertainty

The bipolar fuzzy set and bipolar soft set have inspired the development of a new framework called double-framed bipolar fuzzy soft sets (DFBFSSs). This structure represents positive and negative membership information through ordered pairs, enabling a balanced treatment of uncertainty, imprecision, and bi-directional information in complex decision-making scenarios. The fundamental concepts and operations of DFBFSSs are rigorously defined and analyzed. The double-framed formulation is symmetric: exchanging the frames preserves the structure of DFBFSSs. This symmetry enables balanced handling of opposing or complementary information. The key properties of the proposed set show improved handling of uncertainty over existing fuzzy and soft set models. In addition, a decision-making algorithm based on DFBFSSs is developed and applied to a real-world problem to validate the framework’s feasibility. Comparative analysis confirms the method’s robustness and advantages in uncertain, dual-information settings.