## 1 Introduction  
In our daily life, there are many problems where uncertainty is involved in the data associated with them. These problems are very common in engineering, economics, medicine and social sciences. There always occurs difficulties in addressing such problems by applying classical mathematical techniques because such techniques are structured for clear situations. One of the best known theories to deal with uncertainty is a fuzzy set theory. Following the pioneering work of [PERSON] [1], the fuzzy set theory has expanded in a numerous directions, such as intuitionistic fuzzy set [2], hesitant fuzzy set [3], hesitant probabilistic fuzzy set [4], expanded hesitant fuzzy set [5], and necessary and possible hesitant fuzzy set [6]. [PERSON] [2] generalized the notion of fuzzy set to intuitionistic fuzzy set (IFS) by adding the concept of a non-membership function to a fuzzy set. IFS is a powerful tool used to handle the problems of decision-making that a fuzzy set could not address properly. [PERSON] [7] suggested the concept of Pythagorean fuzzy subsets which are the most effective ones that relate to the representation of the membership degrees of the underlying fuzzy set. [PERSON] [3] introduced a new generalized form of fuzzy set called hesitant fuzzy set (HFS), which opens a new dimension of further research in decision theory. HFS has a lot of merits over the traditional fuzzy set and its extensions, especially in group decision-making with anonymity. HFS has attracted several researchers' attention. [PERSON] and [PERSON] [8; 9] initially provided the mathematical expression of HFS and investigated the distance, similarity and correlation measure for HFSs. [PERSON] also set up the relation between HFS and IFS, according to which [PERSON] and [PERSON] [8] defined some basic operational laws for HFSs. In addition, many extensions of HFS [10; 11; 12; 13] were proposed to model the fuzzy hesitant problems from different prospectives. The HFS is a widely used fuzzy set in decision-making, especially, in those situations in which DMs hesitate to give their preferences. Anyhow, it has the following two major deficiencies:  
1. HFS provides only the membership information of DMs' opinions and pay no attention to the non-membership information. In order to get accurate results, DMs should be allowed to provide the membership as well as the non-membership information. 2. HFS ignores the importance of multiple times occurring opinions, which may lead to unfruitful decision ranking results. To handle real life scenarios more accurately, [PERSON] et al. [13] proposed a dual hesitant fuzzy set (DHFS) that permits both the membership and non-membership grades of an element to a given set having different values. The DHFS is a more flexible tool for DMs to assign their views for each element in the domain. [PERSON] and [PERSON] [14] extended the TOPSIS method for DHFS and describe the application of this new type of set in the framework of decision making. [PERSON] et al. [15] investigated the correlation measure of DHFS and develop a direct transfer algorithm regarding the problem of complex operation of matrix synthesis. [PERSON] and [PERSON] [16] extended the DHFS to typical dual hesitant fuzzy set (TDHFS) and construct a number of operations and properties to generalize the operations of this new type of set. After that, the distance and similarity measure [17; 18], the entropy measure [19] and the evaluation model [20] regarding DHFS have been developed to handle the MAGDM problems. For the purpose of comparison of DHFSs, [PERSON] et al. [21] studied the score function and distance measure. In addition to this, they also extended the VIKOR method for DHFSs. Anyhow, DHFSs, as well as its extensions, cover the deficiency 1 of HFS, but they are not suitable for the situation where DMs could not reach agreement. For example, assume that a decision is made by a decision-making group that contains no less than two DMs.