A Novel Evidence-Based Framework for Picture Fuzzy Sets: Theory and Applications of Belief and Plausibility

Picture Fuzzy Sets (PiFSs) have appeared as an effective tool to tackle ambiguity in decision-making and offer greater flexibility than traditional extensions of Fuzzy Sets (FSs). Under the framework of evidence theory (ET), the concepts of belief and plausibility significantly boost the representative capacity of PiFSs, which enables the management of uncertain and ambiguous data. We constructed both distance and similarity measures specifically for Belief and Plausible Picture Fuzzy Sets (BP-PiFSs). The constructed measures detect the differences and connections between BP-PiFSs and addressed the key shortcomings in current methodologies. They are mathematically validated and applied to real-world scenarios, such as fault detection in complex systems and antenna design optimization, where managing uncertainty is critical. A modified decision-making method, Belief and Plausible SMART (BP-SMART), extends the classical SMART approach to more effectively handle multi-criteria decision-making (MCDM) in uncertain environments. Numerical evaluations across pattern recognition, clustering, fault detection, and MCDM demonstrates the effectiveness and robustness of the suggested framework, contributing significantly to both the theoretical and practical development of fuzzy set theory.