Picture fuzzy Choquet integral based Einstein operations and its application in selection of the best mobile apps for online education.

This article focuses on the development of novel aggregation operators for multi-attribute decision-making (MADM) problems under picture fuzzy set (PFS) theory, which extends intuitionistic fuzzy sets by incorporating degrees of membership, non-membership, and neutrality. Specifically, it proposes the Choquet integral picture fuzzy Einstein geometric aggregation (CIPFEGA) operator and the Choquet integral picture fuzzy Einstein geometric ordered aggregation (CIPFEOGA) operator, designed to capture correlative relationships among attributes using fuzzy measures (FM). The operators' mathematical properties—such as idempotency, boundedness, and monotonicity—are established, and their effectiveness is demonstrated through illustrative examples, including the selection of the best mobile applications for online education. Comparative analysis shows that these operators consider attribute interdependencies more comprehensively than existing methods, offering a robust and adaptable approach for decision-making problems involving uncertain and correlated information.