Integration of hesitant fuzzy set and interval type-2 fuzzy set in analytic hierarchy process

In the broad application of the analytic hierarchy process (AHP) to address decision problems pertaining to multi-criteria decision-making, crisp values are often used to represent the linguistic judgement made by experts or decision makers. The numerous fuzzy approaches proposed in prior studies, such as interval type-2 fuzzy set (IT2FS) and hesitant fuzzy set (HFS), may serve as alternative models to tackle both vagueness and uncertainty during the decision process. As such, this study offers a new AHP framework characterised by the integration of IT2FS and HFS for linguistic variables, called Interval Type-2 Fuzzy Hesitant Number (IT2FHN). Unlike AHP, which uses crisp numbers in a direct manner, the method introduces IT2FS and HFS approaches to improve judgement within the fuzzy decision-making setting. This approach incorporates several linguistic variables into IT2FHN, while the technique of rank value normalises both the lower and upper memberships of IT2FHN. The integrated method proposed in this study is presented based on three numerical instances outlined by past research. The comparative findings demonstrate the feasibility of the decision model. The model captures nuanced differences in judgments, ensuring that each contributes meaningfully to the final outcome.