Picture Fuzzy Einstein Interactive Weighted Averaging Aggregation Operator and Its Application in Multiple Attribute Decision-Making

Abstract A Picture Fuzzy Set (PFS) is an effective tool for handling uncertainties and incomplete cognitive information, incorporating membership, hesitancy, non-membership, and refusal degrees to accommodate diverse viewpoints. The Einstein operation has demonstrated good performance in aggregating data within uncertain contexts, garnering attention in the picture fuzzy environment. However, the existing picture fuzzy Einstein aggregation operator has limitations, particularly when a Picture Fuzzy Number (PFN) possesses a non-membership degree of zero. This leads to an aggregated non-membership degree of zero, disregarding the influence of other non-zero non-membership degrees, rendering them independent. To address these issues, a new decision-making approach is proposed, employing Einstein’s operations for PFNs. The Picture Fuzzy Einstein Interactive Weighted Averaging (PFEIWA) operator are proposed and extensively discussed for its desirable properties. Furthermore, a novel approach referring to these operators in Multi-attribute Decision-Making (MADM) problems is presented. The proposed method applied to a sustainable beef supplier selection problem, and its performance is compared with existing methods. The results show consistent optimal rankings of alternatives, indicating superiority of the proposed operator in resolving MADM problems. The suggested operator is more convincing and suitable due to its ability to produce reasonable and reliable results by considering interactions between PFNs’ degrees.