q-Fractional Fuzzy Frank Aggregation Operators and Their Application in Decision-Making

Multi-criteria decision-making (MCDM) involves evaluating alternatives under uncertain, vague, and conflicting criteria. While fuzzy set theories, such as intuitionistic, pythagorean, fermatean, and q-rung orthopair fuzzy sets have advanced uncertainty modeling, they remain limited to capturing extreme judgments where membership reaches a value of one alongside significant non-membership. The recently introduced q-fractional fuzzy set (q-FrFS) addresses these shortcomings via a flexible constraint, making it suitable for extreme contexts. However, existing q-FrFS methodologies lack robust aggregation mechanisms capable of balancing trade-offs and modulating compensation during information fusion. To overcome this, this study proposes a novel class of Frank-based aggregation operators tailored specifically to q-FrFS environments. Leveraging the parameterized structure of Frank t-norms and t-conorms, we develop two operators: q-FrFFWA (Frank weighted averaging) and q-FrFFWG (Frank weighted geometric) alongside their essential algebraic properties. These operators enhance the representation and fusion of complex and uncertain data. Furthermore, we present a comprehensive MCDM framework utilizing the proposed operators and demonstrate its applicability by selecting optimal vehicle routing software for last-mile delivery. Sensitivity and comparative analyses affirm the stability and credibility of the proposed methodology. This research contributes to the evolving landscape of fuzzy decision-making by integrating the expressive power of q-FrFS with the adaptive flexibility of Frank aggregation, offering a potent tool for modeling and analyzing multidimensional uncertainties in complex decision environments.