The hesitant intuitionistic fuzzy set integrates the advantages of the intuitionistic fuzzy set and the hesitant fuzzy set, and can deal with the uncertainty problems with inconsistent preferences of decision makers perfectly. By considering the hesitancy and fuzziness of information provided by decision makers, the axiomatic definition of the knowledge measure on the hesitant intuitionistic fuzzy set is proposed. Besides, a class of parametric knowledge measure model that comply the axiom is developed, which can not only effectively depict the information in the hesitant intuitionistic fuzzy set, but also indicate the attitude characteristics of decision makers. Next, based on the discussion of the parameters in the knowledge measure, an amount of knowledge measures representing different attitude characteristics of decision makers are obtained. Furthermore, it is verified that the knowledge measure is proportional to the change of the attitude coefficient. Finally, a multiple attribute group decision making method based on the constructed knowledge measure of the hesitant intuitionistic fuzzy set is proposed. Meanwhile, a practical application, i.e., air conditioning installation company selection of an Internet company, demonstrates the effectiveness and practicability of the developed measure.