**7. Conclusions**

Combined with the picture fuzzy set and uniformly typical hesitant fuzzy set, this paper develops the picture hesitant fuzzy set, in which the positive, neutral, negative, and refusal membership degrees are expressed by several possible values. Then, the operations and comparison method of picture hesitant fuzzy elements are developed. To solve the multiple criteria decision-making problems under picture hesitant fuzzy environment, the generalized picture hesitant fuzzy weighted averaging and generalized picture hesitant fuzzy weighted geometric operators are put forward to aggregate the picture hesitant elements given by decision maker. Furthermore, considering the

different priorities between the related criteria in practice, the generalized picture hesitant fuzzy prioritized weighted averaging and generalized picture hesitant fuzzy prioritized weighted geometric operators are proposed. Meanwhile, some desirable properties and the reduced operators of them are investigated in detail. Finally, two kinds of multiple criteria decision-making methods combined with the proposed operators are constructed to solve the multiple criteria decision-making problems in different situations. Subsequently, two numerical examples and an application of web service selection are provided to indicate the applications and advantages of the proposed methods.

In future research, we will investigate other operations of picture hesitant fuzzy elements and develop different aggregation operators to aggregate picture hesitant fuzzy elements. In addition, we will propose the consensus model to improve the proposed methods; then, the non-consensus evaluation information of decision makers will be revised to obtain a more accurate ranking result.

**Author Contributions:** R.W. put forward the picture hesitant fuzzy set, and explored the operational laws and comparison method of the picture hesitant fuzzy set. R.W. and Y.L. developed the aggregation operators under picture hesitant fuzzy environment. Then, R.W. wrote the original manuscript, and Y.L. improved the writing.

**Funding:** This study was supported by the National Natural Science Foundation of China (no. 71371156) and the Doctoral Innovation Fund Program of Southwest Jiaotong University (D-CX201727).

**Acknowledgments:** Thank the editor and reviewers for the positive comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
