Intuitionistic fuzzy extensions are the most used type of fuzzy extensions in the literature since they better represent decision makers strength of commitment on the considered subject in an effective way including membership and non-membership functions. On the other hand, decision makers may assign more than one intuitionistic fuzzy number in order to capture their hesitancies when they are hesitant in assigning a membership degree and a non-membership degree. Hesitant fuzzy sets, another extension of ordinary fuzzy sets, help decision makers to assign different values on the same element aiming at reflexing decision makers’ hesitation. Utilizing these two types of fuzzy sets capture both uncertainty and ambiguity of the considered problem and helps to eliminate the weaknesses of each fuzzy extension. In this study, hesitant intuitionistic fuzzy linguistic sets (HIFLSs) are used to extend the Analytical Hierarchy Process (AHP). The developed method is applied to investment prioritization problem, based on relevant risk factors. Comparative analyses with intuitionistic fuzzy AHP and hesitant fuzzy AHP methods are realized in order to validate the proposed method. A sensitivity analysis is also conducted for the stability of the results of the hesitant intuitionistic fuzzy linguistic AHP method.