There are some shortcomings in the existing hesitation fuzzy entropy. Therefore, this paper develops an exponential hesitation fuzzy entropy, then gives a hesitant fuzzy multi-attribute decision-making model based on entropy weight method. Firstly, a new axiomatic definition of exponential hesitant fuzzy entropy and an exponential hesitation fuzzy entropy measure formula are proposed. And the exponential hesitation fuzzy entropy measure formula satisfying the four criteria in the axiomatic definition is proved. Furthermore, the entropy definition and entropy measure formula of hesitant fuzzy sets are derived. The exponential hesitation fuzzy entropy is compared with the existing hesitation fuzzy entropy measure formula. The results show that the proposed exponential hesitation fuzzy entropy is more completely determine the fuzziness of hesitation fuzzy value. Finally, the exponential hesitant fuzzy entropy measure formula is applied to the hesitant fuzzy multi-attribute decision-making model, and an example of supplier selection is given to illustrated the effectiveness and practicability of exponential hesitation fuzzy entropy.