Enhancing decision-making with picture fuzzy Einstein interactive operator: An application in multiple attribute group decision-making

Recent advancements in Picture Fuzzy Numbers (PFNs) have significantly contributed to the field of fuzzy systems. However, the conventional operational rules (ORs) for PFNs have limitations in practical decision-making scenarios. In particular, traditional picture fuzzy aggregation methods, such as the picture fuzzy Einstein aggregation operators, while useful in uncertain contexts, do not adequately account for interactions among the degrees of positive, neutral, and negative memberships. This oversight can lead to unsatisfactory outcomes in real-world decision-making. To address these challenges, this paper introduces novel interactive Einstein ORs for PFNs and proposes a new aggregation operator, the Picture Fuzzy Einstein Interactive Weighted Geometric (PFEIWG) operator. Moreover, the proposed operator is applied to the multiple attribute group decision making (MAGDM) problem to evaluate its reliability and demonstrate its superiority through a detailed comparative analysis with established methods.