A hybrid decision-making model in a complex Fermatean picture fuzzy environment with application to emergency management

This study presents innovative concepts and practical theories to improve the precise representation of ambiguous and complex information in real-world scenarios. To achieve this objective, develop the ideas of a complex Fermatean picture fuzzy set, which integrates the Fermatean fuzzy set and complex picture fuzzy set. This method expands applicability by membership, neutral, and non-membership degrees with complex values, facilitating a more comprehensive representation of uncertainty. To handle uncertain information, Einstein operators are crucial in efficiently aggregating data through various structures. This study extends the application of Einstein operators to the complex Fermatean picture fuzzy environment. Several new operators are presented in the complex Fermatean picture fuzzy context, such as the Einstein weighted average, Einstein ordered weighted average, Einstein weighted geometric, and Einstein ordered weighted geometric, each with distinct characteristics appropriate for specific decision-making environments. These operators act as tools in a multi-attribute group decision-making model to identify and evaluate the most effective emergency management solutions from a variety of options. The findings suggest that the mobile emergency response unit is the most effective alternative, with the highest assessment score, while the advanced emergency alert system is the least successful, with the lowest evaluation score. Finally, comparative analyses and validation against existing techniques highlight the proposed operators’ advantages, limitations, and reliability.