*Article* **Picture Hesitant Fuzzy Set and Its Application to Multiple Criteria Decision-Making**

#### **Rui Wang 1 and Yanlai Li 1,2,\***


Received: 21 June 2018; Accepted: 18 July 2018; Published: 20 July 2018

**Abstract:** To address the complex multiple criteria decision-making (MCDM) problems in practice, this article proposes the picture hesitant fuzzy set (PHFS) theory based on the picture fuzzy set and the hesitant fuzzy set. First, the concept of PHFS is put forward, and its operations are presented, simultaneously. Second, the generalized picture hesitant fuzzy weighted aggregation operators are developed, and some theorems and reduced operators of them are discussed. Third, the generalized picture hesitant fuzzy prioritized weighted aggregation operators are put forward to solve the MCDM problems that the related criteria are at different priorities. Fourth, two novel MCDM methods combined with the proposed operators are constructed to determine the best alternative in real life. Finally, two numerical examples and an application of web service selection are investigated to illustrate the effectiveness of the proposed methods. The sensitivity analysis shows that the different values of the parameter *λ* affect the ranking of alternatives, and the proposed operators are compared with several existing MCDM methods to illustrate their advantages.

**Keywords:** multiple criteria decision-making; picture hesitant fuzzy set; generalized picture hesitant fuzzy weighted aggregation operator; generalized picture hesitant fuzzy prioritized weighted aggregation operator
