The reliability of decision makers' judgments together with their degree of hesitancy are often ignored in decision models. The inclusion of judgments with only triangular belonging functions causes a great deal of information loss since non-belongingness functions are ignored in those decision models. In the literature, a method on how to integrate the intuitionistic reliability function into the decision maker's judgment has not been clearly proposed. To handle these issues, we aim at developing a novel intuitionistic Z-AHP (Analytic Hierarchy Process) & Z-TOPSIS (Technique for Order Preference by Similarity to Ideal Solutions) methodology in this paper. Z-numbers can be used to express uncertain quantities with different degrees of precision, which can be very useful in practical applications. Our methodology considers both the restriction function and its reliability function under intuitionistic fuzziness based on α-cut approach and uses non-linear intuitionistic belonging and unbelonging functions. We compare our approach with interval-valued fuzzy TOPSIS and conduct a sensitivity analysis to assess the effects of changes in criteria weights on the rankings of the alternatives. Our results show that the proposed methodology can effectively evaluate and rank the existing alternatives, considering the uncertainties and complexities of the decision environment. This research provides valuable insights for decision-makers and can take part in their decision support systems. The proposed methodology has been applied to a hydrogen storage technology selection problem. According to the results, the most favorable hydrogen storage option in the considered case is chemical storage alternative followed by liquid storage, compressed storage, carbon nanostructure storage, and metal–organic framework storage alternatives. The inference of the comparative analysis with intuitionistic Z-AHP and interval-valued fuzzy TOPSIS method is that the best two alternatives are same in both methodologies whereas the third and the fifth alternatives are replaced since a different linearization approach is applied to the non-linear restriction function multiplied by the reliability function.