Aggregation operators based on interval-valued Fermatean fuzzy linguistic sets for medical waste disposal planning

This study introduces a novel decision-making framework based on interval-valued Fermatean fuzzy linguistic sets, a new theoretical model designed to capture linguistic uncertainty, interval-valued hesitation, and Fermatean membership characteristics simultaneously. The primary objective is to construct the mathematical foundations of interval-valued Fermatean fuzzy linguistic sets, develop corresponding aggregation operators, propose a new score function, and formulate a complete multi-criteria decision-making algorithm. The research begins by defining interval-valued Fermatean fuzzy linguistic sets and exploring their fundamental algebraic and ordering properties. Several aggregation operators (Weighted Aggregation and Geometric Weighted; Ordered Weighted Average and Ordered Geometric Weighted; Generalized Weighted Averaging and Generalized Weighted Geometric; Generalized Ordered Weighted Averaging and Generalized Ordered Weighted Geometric; and Linguistic Hybrid) tailored to the interval-valued Fermatean fuzzy linguistic structure are proposed, and their properties are rigorously proven. A novel score function is introduced to provide a meaningful ranking mechanism for interval-valued Fermatean fuzzy linguistic evaluations. Building on these components, a new multi-criteria decision-making algorithm is developed. The algorithm is applied to selecting medical waste disposal techniques, a domain characterized by high uncertainty and conflicting criteria. The robustness of the results is examined using the interval-valued Fermatean fuzzy-COPRAS method, and comparative analyses are conducted with previously established fuzzy multi-criteria decision-making approaches. A comprehensive sensitivity analysis is also performed to validate the stability of the proposed framework. The empirical results indicate that the proposed interval-valued Fermatean fuzzy linguistic-based multi-criteria decision-making framework effectively handles linguistic imprecision and interval-valued hesitation, producing reliable and interpretable rankings. For the medical waste disposal problem, the final ranking is obtained as follows: [Formula: see text]. Robustness and sensitivity analyses confirm the stability of the ranking across different parameter settings and weighting schemes. Comparative results show that the proposed method provides more consistent and resilient outcomes than traditional fuzzy and classical decision-making models. The study enriches the fuzzy multi-criteria decision-making literature by establishing a robust framework for managing complex linguistic uncertainty in real-world decision-making scenarios.