There have always been problems associated with managing vague or imprecise information, and therefore, various tools have recently been investigated to handle this uncertainty. In this study, hesitant intuitionistic fuzzy sets are introduced to resolve situations where decision-makers hesitate in selecting between several intuitionistic fuzzy values when assessing alternatives. Furthermore, new operations, based on the Archimedean t-norms and t-conorms, are developed and corresponding properties and the ranking method of HIFNs are also investigated. Additionally, the hesitant intuitionistic fuzzy weighted averaging operator and the hesitant intuitionistic fuzzy power weighted averaging operator based on the Archimedean t-norms and t-conorms are proposed to aggregate decision-makers’ information in multi-criteria decision-making (MCDM) problems, and an approach to MCDM problems that employ hesitant intuitionistic fuzzy information is constructed based on the proposed aggregation operators. Lastly, an example of selecting project is investigated to verify the applicability and validity of the proposed approach and the study is supported by comparative analyses.