This paper proposes several distance measures about hesitant fuzzy sets based on the Lance distance at first. Then, aiming at the unequal number of membership numbers in two hesitant fuzzy elements, a new dimension reduction scheme of hesitant fuzzy elements is introduced. The scheme does not need to add the maximum or minimum membership value to the hesitant fuzzy element, which not only preserves the original information of the data well, but also reduces the computation of the calculated distance. In the case that the information of attribute weights are completely unknown, the hesitant fuzzy exponential entropy is constructed by using the actual data information. This paper uses the principle of entropy minimization to calculate attribute weights. Finally, a medical diagnosis example is provided to illustrate the weighted hesitant fuzzy Lance distance measure of dimension reduction based on the exponential entropy. The results show that, the proposed distance measure not only achieves the consistent diagnosis results when $\lambda$ takes different values, but also reduces the computation and improves the diagnostic efficiency, which has certain application value for real-time and effective medical diagnosis.