Fermatean fuzzy Z-analytic hierarchy process: An application to third party logistics providers

To decrease the impact of ambiguity, fuzzy sets are frequently used to decide if an element belongs to a set. Various fuzzy set extensions differ from others in how uncertainty is expressed in the definition of membership functions. This paper presents a novel extension named Fermatean fuzzy Z-numbers, introducing fuzzy restriction and fuzzy reliability functions with Fermatean fuzzy membership functions. By utilizing fuzzy Z-numbers to delineate membership degrees, considering both restriction and its reliability, in tandem with Fermatean membership functions offering an expanded value spectrum, the assessment of linguistic evaluations using Fermatean fuzzy Z-numbers can yield more pragmatic outcomes in decision evaluations amidst high uncertainty. Moreover, the interval-valued Fermatean fuzzy-Z-analytic hierarchy process is developed to explore the impact of proposed interval-valued Fermatean fuzzy-Z numbers on the multi-criteria decision-making process, demonstrated in the context of third-party logistics provider selection. Adding fuzzy reliability functions to the decision-making process improves how uncertainty is represented by considering variations in confidence levels associated with each decision, thus aiding in the clearer expression of decisions. Additionally, the proposed method's outcomes are compared with the outcomes of the current approaches.