Complex interval-valued Fermatean fuzzy sets with Einstein aggregation operators: Application to the selection of the best physical education instructor

The quality assessment of physical education (PE) programs in higher education plays a vital role in promoting student development and ensuring institutional excellence. However, evaluating PE quality is inherently complex due to various ambiguous and uncertain factors that influence the assessment process. To address these challenges, this research presents complex interval-valued Fermatean fuzzy sets (CIVFFS), a robust model for effectively managing uncertainty in evaluating PE quality. CIVFFSs represent a novel mathematical approach that integrates the strengths of interval-valued Fermatean fuzzy sets (IVFFSs) and complex fuzzy sets (CFSs), enhancing decision-making (DM) under complex and imprecise conditions. IVFFSs handle uncertainty using intervals for both membership and non-membership degrees, while the addition of complexity from CFSs allows these values to exist in a complex plane. It gives a more powerful way to deal with imprecise and uncertain information. We introduce the Einstein operations for CIVFFSs, expanding the computational framework for handling uncertainty in complex DM environments. Using these novel operations, we present a series of Einstein aggregation operators (EAOs) specifically designed for CIVFFSs. These include the complex interval-valued Fermatean fuzzy Einstein weighted averaging (CIVFFEWA), complex interval-valued Fermatean fuzzy Einstein weighted geometric (CIVFFEWG), complex interval-valued Fermatean fuzzy Einstein ordered weighted averaging (CIVFFEOWA), complex interval-valued Fermatean fuzzy Einstein ordered weighted geometric (CIVFFEOWG), complex interval-valued Fermatean fuzzy Einstein hybrid averaging (CIVFFEHA), and complex interval-valued Fermatean fuzzy Einstein hybrid geometric (CIVFFEHG) operators. We also develop some structure properties of these operators namely idempotency, boundedness, and monotonicity to ensure their theoretical robustness. The multi-attribute group decision-making (MAGDM) problem serves as a powerful tool for addressing complex real-life application challenges. To demonstrate the compatibility of proposed model, a case study in PE is considered using numerical examples and innovative mathematical approaches. A comparative analysis is also discussed to evaluate the performance of the proposed approaches against existing techniques. This demonstrates the effectiveness and practical relevance of the proposed methods.