This paper develops an exponential hesitation fuzzy entropy, and gives a hesitant fuzzy multi-attribute decision-making model based on the entropy weight method. Firstly, an axiomatic definition of exponential hesitant fuzzy element entropy and an exponential hesitation fuzzy element entropy measure formula are proposed. And the exponential hesitation fuzzy element 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, and it is proved that the exponential hesitant fuzzy set entropy measure formula also satisfys the four criteria in the axiomatic definition. 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 illustrate the effectiveness and practicability of the proposed exponential hesitation fuzzy entropy.