1 Introduction

Intuitionistic fuzzy set (IFS) theory was first proposed by Atanassov [1] and it is a generalization of Fuzzy set (FS) [2]. Compared with the FS, the IFS have the additional parameters to control the non-membership degree and hesitant degree besides the membership degree function. The non-membership and hesitant parameters make IFS very useful in depicting the fuzziness in the real world and have successfully applied to pattern recognition [3–5], decision making [6–10] and so on.

Torra and Narukawa [11] has pointed out that both FS theory and IFS theory undergo the inherent disadvantages when define the membership degree of a given element to a predefined set. Besides, some other extended FS theory, such as type-2 FS [12, 13] and fuzzy multiset [14, 15] cannot handle this problem [11, 16]. The theory of hesitant fuzzy set (HFS) addresses these limitations [11, 16].

The HFS is based on strong ability on the description of the membership degree. For the membership functions, the HFS is more powerful in descripting the object than the IFS and some other extended FS. One can have some determined number in the description of non-membership degree in the HFS theory. This makes it convenient to describe a group of experts’ opinion, especially when the experts are relatively independent. In this regard, the HFS theory has found many applications such as group decision making (GDM) [17–22].

Based on the extensive research of IFS and HFS, Zhu et al. [23] have combined the idea of IFS theory with HFS theory and introduced the concept of dual hesitant fuzzy set (DHFS). Similar to the IFS theory, DHFS also have membership degree function and non-membership degree function. However, these two functions are expressed by several determined numbers rather a single number and make the descriptions of the fuzziness of the real world more accurately than the other extended FS theory. Specifically, the DHFS is very helpful in GDM problem when the membership degree and non-membership degree functions are hard to be determined. Chen et al. [24] investigated the GDM method under the dual hesitant fuzzy environment. In their contribution, optimization models are adopted to determine the weight vector. Zhu and Xu [25] further extended the DHFS and developed the typical DHFS, some new operations and properties are also investigated. Farhadinia [26] studied the correlation coefficient for DHFS and applied it to medical diagnosis. Wang et al. [27] defined the correlation measures for DHFS and studied the clustering algorithm for DHFS. Ye [28] proposed the concept of weighted correlation coefficient and the predefined idea alternative and applied it investment management. Wang et al. [29] focused on the dual hesitant fuzzy information aggregation methods and proposed a series of aggregation operators, such as dual hesitant fuzzy weighted average (DHFWA) DHFWG, DHFOWA, DHFOWG, DHFHA and DHFHG operators.

It should be noted that the above aggregation operators are based on the assumption that the variables are independent of each other. In this paper, we further research the information fusion methods for dual hesitant fuzzy information which can descript the interrelations between the aggregated arguments quantitatively. To do this, the traditional Heronian mean (HM) [30] is applied to assist in this study. HM is a kind of average mean which can be used to describe the correlations of the aggregated arguments and it have been further researched in our previous works [31, 32]. The structure of HM is very similar to BM [33–37], however, the main difference and the advantage are studied in detail in our previous research results [32]. In this study, we first extended the HM to dual hesitant fuzzy environment and proposed some operators for aggregating dual hesitant fuzzy information, such as dual hesitant fuzzy HM (DHFHM) operator, dual hesitant fuzzy geometric HM (DHFGHM) operator and their weighted forms. Then, we investigate the GDM method based on the above operators. Finally, a real example about supplier selection is shown to illustrate the GDM method.

This paper will be organized as follows: In Sect. 2, we give an overview of the DHFS. We make effort on the extension of HM to DHFS in Sect. 3. In particular, we provide the definition of the DHFHM and DHFGHM aggregation operators and study their desirable properties, a GDM method and its application are presented in Sect. 4. Section 5 concludes this paper.

2 Some basic concepts

FS theory has become an indispensable part in decision making and some other human activities because of the need to express the associated ambiguity. A lot of extended FS theories have been forwarded to meet the growing demand for reality, such as multi-sets [15], vague sets [38], type-2 FSs [39], IFS [1], interval-valued IFS [40], HFS [11, 19] and DHFS [23]. In the past two years, DHFS has attracted the attention of many scholars. In this section, we will introduce the basic concept of DHFS to facilitate the follow-up in following sections.

Definition 1 [23]

Let \(X\) be a given set. A DHFS \(D\) on the set \(X\) is expressed by the following equation.

$$D = \left\{ {\left\langle {x,h(x),g(x)} \right\rangle x \in X} \right\}$$
(1)

In Eq. (1), \(h(x)\) and \(g(x)\) are two HFS, therefore, DHFS is the combination of two HFS. In the research contribution of Zhu et al. [23], the comparison laws of any two dual hesitant fuzzy elements (DHFEs) are also provided.

Definition 2 [23]

Suppose \(d = \left( {h,g} \right)\) be any two DHFEs. The scores function and accuracy function of \(d\) is defined as follows:

$$S(d) = \frac{1}{\# h}\sum\nolimits_{\gamma \in h} {\gamma - } \frac{1}{\# g}\sum\nolimits_{\eta \in g} \eta$$
(2)
$$p(d) = \frac{1}{\# h}\sum\nolimits_{\gamma \in h} {\gamma + } \frac{1}{\# g}\sum\nolimits_{\eta \in g} \eta$$
(3)

In many cases, the comparison can be processed based on Eq. (2). In other words, the larger the score of the DHFE, the larger DHFE.

Example 1

Let \(d_{1} = \left\{ {\left\{ {0.6,0.7} \right\},\left\{ {0.1,0.3} \right\}} \right\}, d_{2} = \left\{ {\left\{ {0.2,0.3,0.4} \right\},\left\{ {0.6} \right\}} \right\}, d_{3} = \left\{ {\left\{ {0.6,0.7,0.8} \right\},\left\{ {0.1} \right\}} \right\} ,\) be three DHFEs. According to the Definition 2, the following score functions can be obtained.

$$S(d_{1} ) = \frac{1}{2}\left( {0.6 + 0.7} \right) - \frac{1}{2}\left( {0.1 + 0.3} \right) = 0.45$$
$$S(d_{2} ) = \frac{1}{3}\left( {0.2 + 0.3 + 0.4} \right) - \frac{1}{1}\left( {0.6} \right) = - 0.3$$
$$S(d_{3} ) = \frac{1}{3}\left( {0.6 + 0.7 + 0.8} \right) - \frac{1}{1}\left( {0.1} \right) = 0.6$$

Since

$$S(d_{3} ) > S(d_{1} ) > S(d_{2} )$$

We have

$$d_{3} \succ d_{1} \succ d_{2}$$

Definition 3

Let \(d ,\) \(d_{1}\) and \(d_{2}\) be three DHFEs, Then

  1. (1)

    \(d_{1} \oplus d_{2} = \cup_{{\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} ,\eta_{1} \in g_{1} ,\eta_{2} \in g_{2} }} \left\{ {\left\{ {\gamma_{1} + \gamma_{2} - \gamma_{1} \gamma_{2} } \right\},\left\{ {\eta_{1} \eta_{2} } \right\}} \right\}\)

  2. (2)

    \(d_{1} \otimes d_{2} = \cup_{{\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} ,\eta_{1} \in g_{1} ,\eta_{2} \in g_{2} }} \left\{ {\left\{ {\gamma_{1} \gamma_{2} } \right\},\left\{ {\eta_{1} + \eta_{2} - \eta_{1} \eta_{2} } \right\}} \right\}\)

  3. (3)

    \(nd\) = \(\cup_{\gamma \in h,\eta \in g} \left\{ {\left\{ {1 - (1 - \gamma )^{n} } \right\},\left\{ {\eta^{n} } \right\}} \right\}\)

  4. (4)

    \(d^{n}\) = \(\cup_{\gamma \in h,\eta \in g} \left\{ {\left\{ {\gamma^{n} } \right\},\left\{ {1 - (1 - \eta )^{n} } \right\}} \right\}\)

3 Dual hesitant fuzzy information aggregation with HM techniques

Based on the above operations given in Definition 3 [23], some aggregation operators for DHFE have been proposed, such as DHFWA operator, DHFWG operator, DHFOWA operator, DHFOWG operator, DHFHA operator and DHFHG operator [29]. However, the above aggregation operators can only deal with the situation that the aggregated arguments are independent. As everything in the real world is related to and interacted with each other. Investigate the correlated dual hesitant fuzzy information fusion method has important meaning to the development of DHFS theory and information fusion theory. HM is an important technique to deal with this situation and the definition is given as follows [30–32].

Definition 4

Suppose there are two parameters \(p,\text{ }q > 0 ,\) and \(a_{i}\) \((i = 1,2,{ \ldots },n)\) be a group of non-negative real number. If

$${\text{HM}}^{p,q} (a_{1} ,a_{2} ,{ \ldots },a_{n} ) = \left( {\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {a_{i}^{p} a_{j}^{q}}}}\right)^{1/p+q}$$
(4)

then \({\text{HM}}\) is called the Heronian mean.

Example 2

Let \(a_{1} = 2, a_{2} = 3, a_{3} = 4\) be three non-negative real number, based on the \({\text{HM}}\) operator given above, we can have the following aggregated result (take \(p = 3,q = 5\) for example).

$${\text{HM}}(2,3,4)$$
$$= \left( {\frac{2}{3(3 + 1)}(2^{3} \times 2^{5} + 2^{3} \times 3^{5} + 2^{3} \times 4^{5} + 3^{3} \times 3^{5} + 3^{3} \times 4^{5} + 4^{3} \times 4^{5} )} \right)^{\frac{1}{8}} = 3.41$$

Existing \({\text{HM}}\) operator can be used to aggregate the real number [30], IFS [31] and Interval-valued IFS [32]. In this paper, we mainly focus on the \({\text{HM}}\) operator under DHFS environment.

Definition 5

Let \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a group of DHFEs. If

$${\text{DHFHM}}\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right) = \left( {\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {d_{i}^{p} \otimes d_{j}^{q} } } } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {p + q}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${p + q}$}}}}$$
(5)

then \({\text{DHFHM}}\) is called the dual hesitant fuzzy Heronian mean operator (\({\text{DHFHM}}\)).

Theorem 1

Let \(p,q > 0,\) and \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a group of DHFEs, based on the \({\text{DHFHM}}\) operator, the aggregated DHFE was given as follows.

$${\text{DHFHM}}\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right)$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\}$$
(6)

Proof

According to the operations defined in Definition 2, we know,

$$d_{i}^{p} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i}^{p} } \right\},\left\{ {1 - (1 - \eta_{i} )^{p} } \right\}} \right\},\quad d_{j}^{q} = \cup_{{\xi_{j} \in h_{j} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\xi_{j}^{q} } \right\},\left\{ {1 - (1 - \eta_{j} )^{q} } \right\}} \right\},$$
(7)

and

$$d_{i}^{p} \otimes d_{j}^{q} = \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\xi_{i}^{p} \xi_{j}^{p} } \right\},\left\{ {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right\}} \right\}$$
(8)

Then

$$\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {d_{i}^{p} \otimes d_{j}^{q} } }$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)} } \right\},\left\{ {\prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\}} \right\}$$
(9)

and

$$\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {h_{i}^{p} \otimes h_{j}^{q} } }$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\},\left\{ {\prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\}} \right\}$$
(10)

Therefore, we have

$${\text{DHFHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} )$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\}$$
(11)

which completes the proof of Theorem 1.The \({\text{DHFHM}}\) operator has excellent properties such as idempotency, monotonicity, permutation and boundary.□

Property 1 (Idempotency)

If all DHFEs \(d_{i} (i = 1,2,{ \ldots },n)\) are equal, i.e., \(d_{i} = d =\) \(\cup_{\xi \in h,\eta \in g} \left\{ {\left\{ \xi \right\},\left\{ \eta \right\}} \right\}\) for all \(i ,\) \(p > 0,\text{ }q > 0 .\) Then

$${\text{DHFHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} ) = {\text{DHFHM}}(d,d,{ \ldots },d) = d$$
(12)

Proof

Based on known information, we have

$${\text{DHFHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} )= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\} = \cup_{\xi \in h,\eta \in g} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{{}}^{p} \xi_{{}}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta )^{p} (1 - \eta )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\} = \cup_{\xi \in h,\eta \in g} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{{}}^{p + q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {1 - \left( {(1 - \eta )^{p + q} } \right)^{{\frac{1}{p + q}}} } \right\}} \right\} = \cup_{\xi \in h,\eta \in g} \left\{ {\left\{ \xi \right\},\left\{ \eta \right\}} \right\} = d$$

which completes the proof.□

Property 2 (Monotonicity)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) and \(k_{i} = \cup_{{\alpha_{i} \in s_{i} ,\beta_{i} \in t_{i} }} \left\{ {\left\{ {\alpha_{i} } \right\},\left\{ {\beta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be two sets of DHFEs, if \(\xi_{i} \le \alpha_{i}\) and \(\eta_{i} \ge \beta_{i}\) for all \(i,\) then

$${\text{DHFHM}}(h_{1} ,h_{2} ,{ \ldots },h_{n} ) \le {\text{DHFHM}}(k_{1} ,k_{2} ,{ \ldots },k_{n} )$$
(13)

Proof

On the one hand, Since \(\xi_{i} \le \alpha_{i}\) for all \(i ,\) then

$$\xi_{i}^{p} \xi_{j}^{q} \le \alpha_{i}^{p} \alpha_{j}^{q}$$
$$\Rightarrow \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } \ge \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \alpha_{i}^{p} \alpha_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} }$$
$$\Rightarrow 1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } \le 1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \alpha_{i}^{p} \alpha_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} }$$
$$\Rightarrow \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} \le \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \alpha_{i}^{p} \alpha_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}}$$
$$\Rightarrow \cup_{{\xi_{i} \in d_{i} ,\xi_{j} \in d_{j} }} \left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \xi_{i}^{p} \xi_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\} \le \cup_{{\alpha_{i} \in k_{i} ,\alpha_{j} \in k_{j} }} \left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \alpha_{i}^{p} \alpha_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}$$
(14)

On the other hand, put the same thought into the proof of non-membership degree, we can easily get.

$$\cup_{{\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i} )^{p} (1 - \eta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\} \ge \cup_{{\beta_{i} \in t_{i} ,\beta_{j} \in t_{j} }} \left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \beta_{i} )^{p} (1 - \beta_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}$$
(15)

United Eqs. (14) with (15), we have

$${\text{DHFHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} ) \le {\text{DHFHM}}(k_{1} ,k_{2} ,{ \ldots },k_{n} )$$
(16)

which completes the proof.□

Property 3 (Permutation)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be a set of DHFEs. Then

$${\text{DHFHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} ) = {\text{DHFHM}}(\dot{d}_{1} ,\dot{d}_{2} ,{ \ldots },\dot{d}_{n} )$$
(17)

where \((\dot{d}_{1} ,\dot{d}_{2} ,{ \ldots },\dot{d}_{n} )\) is any permutation of \((d_{1} ,d_{2} ,{ \ldots },d_{n} ) .\)

Proof

According to the operations of DHFE, we can get DHFHM

$$\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right)\,=\,\left( {\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {d_{i}^{p} \otimes d_{j}^{q} } } } \right)^{1/p + q} = \left( {\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {\dot{d}_{i}^{p} \otimes \dot{d}_{j}^{q} } } } \right)^{1/p + q} = {\text{DHFHM}}(\dot{d}_{1} ,\dot{d}_{2} ,{ \ldots },\dot{d}_{n} )$$

which completes the proof.□

Property 4 (Boundary)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be a set of DHFEs, and

$$d^{ - } = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\hbox{min} \left( {\xi_{i} } \right)} \right\},\left\{ {\hbox{max} \left( {\eta_{i} } \right)} \right\}} \right\}$$
$$d^{ + } = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\hbox{max} \left( {\xi_{i} } \right)} \right\},\left\{ {\hbox{min} \left( {\eta_{i} } \right)} \right\}} \right\}$$
(18)

then

$$d^{ - } \le {\text{DHFHM}}(h_{1} ,h_{2} ,{ \ldots },h_{n} ) \le d^{ + }$$
(19)

Proof

According to the proof of properties 1–3, the property 4 can be proved easily. We do not repeat here.

Example 1

Let \(d_{1} = \left\{ {\left\{ {0.3,0.4,0.5} \right\},\left\{ {0.2,0.3} \right\}} \right\}, h_{2} = \left\{ {\left\{ {0.7,0.8} \right\},\left\{ {0.2} \right\}} \right\}, h_{3} = \left\{ {\left\{ {0.6,0.7} \right\},} \right. \left. {\left\{ {0.1,0.2,0.3} \right\}} \right\}\) be three DHFEs, then by Definitions of DHFHM operator and Theorem 1 described above, when the values of the parameters be set the determined numbers, the final aggregated DHFEs can be obtained, for example,

When \(p = 0.5,q = 0.5\)

$${\text{DHFHM}}(d_{1} ,d_{2} ,d_{3}) = \left\{ {\left\{ { 0.5421 , 0.5793 , 0.5 8 5 8 , 0 . 6 2 1 9 , 0 . 5 7 2 7 , 0 . 6 0 8 7 , 0 . 6 1 4 8 , 0 . 6 4 9 5 , 0 . 6 0 3 3 , 0 . 6 3 8 0 , 0. 6 4 3 8 , 0 . 6 7 7 1} \right\}} , {\left\{ { 0. 1 6 2 4 , 0 . 2 0 0 0 , 0 . 2 3 1 0 , 0 . 1 9 0 1 , 0 . 2 3 1 0 , 0 . 2 6 4 4} \right\}} \right\}$$

the score of the aggregated DHFE is 0.3982.

When \(p = 1,q = 10 ,\)

$${\text{DHFHM}}(d_{1} ,d_{2} ,d_{3}) = \left\{ {\left\{ { 0. 6 3 1 3 , 0 . 6 7 2 6 , 0 . 7 0 6 5 , 0 . 7 2 3 8 , 0 . 6 3 6 3 , 0 . 6 7 7 0 , 0 . 7 1 2 0 , 0 . 7 2 8 9 , 0 . 6 4 1 5 , 0 . 6 8 1 4 , 0 . 7 1 7 2 , 0 . 7 3 3 8} \right\}} , {\left\{ { 0. 1 3 8 2 , 0 . 2 0 0 0 , 0 . 2 3 2 6 , 0 . 1 4 5 5 , 0 . 2 1 3 1 , 0 . 2 5 2 1} \right\}} \right\}$$

the score of the aggregated DHFE is 0.4916.

When \(p = 10,q = 10,\)

$${\text{DHFHM}}(d_{1} ,d_{2} ,d_{3} ) = \left\{ {\left\{ { 0.6475, 0.6762 , 0.7337,\,0.7421, 0.6476, 0.6762, 0.7338, 0.7422, 0.6486, 0.6769, 0.7341, 0.7425} \right\}}, {\left\{ {0.1494, 0.2000, 0.2202, 0.1585, 0.2202, 0.2481} \right\}} \right\}$$

the score of the aggregated DHFE is 0.5007.□

From the definition of the DHFHM operator, we can find out that there are two parameters (\(p,q\)) to control the aggregated DHFE simultaneously. When one of these two parameters is set, the scores trends can be shown.

Case 1

Take \(p = 1(q \in \left( {0,10} \right])\) or \(q = 1(p \in \left( {0,10} \right])\) for example, scores trends were shown in Fig. 1.

Fig. 1
figure 1

Scores trends \(p = 1\) and \(q = 1\)

Full size image

Case 2

Take \(p = 0.1,1,5,10(q \in \left( {0,10} \right])\) for example, scores trends was shown in Fig. 2.

Fig. 2
figure 2

Scores trends \(p = 0.01,1,5,10\)

Full size image

Case 3

Take \(q = 0.1,1,5,10(p \in \left( {0,10} \right])\) for example, scores trends was shown in Fig. 3.

Fig. 3
figure 3

Scores trends \(q = 0.01,1,5,10\)

Full size image

The effects of simultaneous varieties of \(p\) and \(q\) on score functions were analyzed on Fig. 4.

Fig. 4
figure 4

Scores for DHFEs obtained by the DHFHM operator (\(p \in \left( {0,10} \right],\,q \in \left( {0,10} \right]\))

Full size image

The DHFHM operator is one kind of average mean operator and it is the extension of some existing operators such as HFWA operator, HFOWA operator [18], DHFWA operator and DHFOWA operator [29]. Another kind of mean operator is geometric mean and it has been extended to fuzzy environment [41], IFS environment [42–45], interval-valued IFS environment [46, 47]. In the following, we study the geometric mean under dual hesitant fuzzy environment and introduce the dual hesitant fuzzy geometric Heronian mean (DHFGHM) operator.

Definition 6

Let \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a collection of DHFEs. If

$${\text{DHFGHM}}\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right) = \frac{1}{p + q}\left( {\mathop \otimes \limits_{i = 1,j = i}^{n} (pd_{i} \oplus qd_{j} )^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {n(n + 1)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n(n + 1)}$}}}} } \right)$$
(20)

then \({\text{DHFGHM}}\) is called the dual hesitant fuzzy geometric HM operator (\({\text{DHFGHM}}\)).

Theorem 2

Let \(p,q > 0,\) and \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a group of DHFEs, based on the \({\text{DHFGHM}}\) operator, the aggregated DHFE was given as follows.

$${\text{DHFGHM}}\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right)$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \xi_{i} )^{p} (1 - \xi_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \eta_{i}^{p} \eta_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\}$$
(21)

Proof

According to the operations defined in Definition 2, we know,

$$pd_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {1 - (1 - \xi_{i} )^{p} } \right\},\left\{ {\eta_{i}^{p} } \right\}} \right\}$$
(22)
$$qd_{j} = \cup_{{\xi_{j} \in h_{j} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - (1 - \xi_{j} )^{q} } \right\},\left\{ {\eta_{j}^{q} } \right\}} \right\}$$
(23)

and

$$pd_{i}^{{}} \oplus qd_{j} = \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - (1 - \xi_{i} )^{p} (1 - \xi_{j} )^{q} } \right\},\left\{ {\eta_{i}^{p} \eta_{j}^{p} } \right\}} \right\}$$
(24)

Then

$$\mathop \otimes \limits_{i = 1,j = i}^{n} \left( {pd_{i} \oplus qd_{j} } \right)$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \xi_{i} )^{p} (1 - \xi_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\},\left\{ {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \eta_{i}^{p} \eta_{j}^{q} } \right)} } \right\}} \right\}$$
(25)

and

$$\mathop \otimes \limits_{i = 1,j = i}^{n} \left( {pd_{i} \oplus qd_{j} } \right)^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {n(n + 1)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n(n + 1)}$}}}}$$
$$= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \xi_{i} )^{p} (1 - \xi_{j} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\},\left\{ {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - \eta_{i}^{p} \eta_{j}^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right\}} \right\}$$
(26)

Therefore, we have

$${\text{DHFGHM}}(d_{1} ,d_{2} , \ldots ,d_{n} ) = \cup _{{\xi _{i} \in h_{i} ,\xi _{j} \in h_{j} ,\eta _{i} \in g_{i} ,\eta _{j} \in g_{j} }} \left\{ \begin{gathered} \left\{ {1 - \left( {1 - \prod\limits_{{i = 1,j = i}}^{n} {\left( {1 - (1 - \xi _{i} )^{p} (1 - \xi _{j} )^{q} } \right)^{{\frac{2}{{n(n + 1)}}}} } } \right)^{{\frac{1}{{p + q}}}} } \right\}, \hfill \\ \left\{ {\left( {1 - \prod\limits_{{i = 1,j = i}}^{n} {\left( {1 - \eta _{i}^{p} \eta _{j}^{q} } \right)^{{\frac{2}{{n(n + 1)}}}} } } \right)^{{\frac{1}{{p + q}}}} } \right\} \hfill \\ \end{gathered} \right\}$$
(27)

which completes the proof of Theorem 2.

Similar to the DHFHM operator, the DHFGHM operator also has some excellent properties such as idempotency, monotonicity, permutation and boundary.□

Property 5 (Idempotency)

If all DHFEs \(d_{i} (i = 1,2,{ \ldots },n)\) are equal, i.e., \(d_{i} = d = \cup_{\xi \in h,\eta \in g} \left\{ {\left\{ \xi \right\},\left\{ \eta \right\}} \right\},\) for all \(i ,\) \(p > 0,\text{ }q > 0 .\) Then

$${\text{DHFGHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} ) = {\text{DHFGHM}}(d,d,{ \ldots },d) = d$$
(28)

Property 6 (Monotonicity)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) and \(k_{i} = \cup_{{\alpha_{i} \in s_{i} ,\beta_{i} \in t_{i} }} \left\{ {\left\{ {\alpha_{i} } \right\},\left\{ {\beta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be two sets of DHFEs, if \(\xi_{i} \le \alpha_{i}\) and \(\eta_{i} \le \beta_{i}\) for all \(i,\) then

$${\text{DHFGHM}}(h_{1} ,h_{2} ,{ \ldots },h_{n} ) \le {\text{DHFGHM}}(k_{1} ,k_{2} ,{ \ldots },k_{n} )$$
(29)

Property 7 (Permutation)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be a set of DHFEs. Then

$${\text{DHFGHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} ) = {\text{DHFGHM}}(\dot{d}_{1} ,\dot{d}_{2} ,{ \ldots },\dot{d}_{n} )$$
(30)

where \((\dot{d}_{1} ,\dot{d}_{2} ,{ \ldots },\dot{d}_{n} )\) is any permutation of \((d_{1} ,d_{2} ,{ \ldots },d_{n} ).\)

Property 8 (Boundary)

If \(d_{i} = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\xi_{i} } \right\},\left\{ {\eta_{i} } \right\}} \right\}(i = 1,2,{ \ldots },n)\) be a set of DHFEs, and

$$d^{ - } = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\hbox{min} \left( {\xi_{i} } \right)} \right\},\left\{ {\hbox{max} \left( {\eta_{i} } \right)} \right\}} \right\}, d^{ + } = \cup_{{\xi_{i} \in h_{i} ,\eta_{i} \in g_{i} }} \left\{ {\left\{ {\hbox{max} \left( {\xi_{i} } \right)} \right\},\left\{ {\hbox{min} \left( {\eta_{i} } \right)} \right\}} \right\}$$

then

$$d^{ - } \le {\text{DHFGHM}}(h_{1} ,h_{2} ,{ \ldots },h_{n} ) \le d^{ + }$$
(31)

Example 2

Let \(d_{1} = \left\{ {\left\{ {0.12,0.34} \right\},\left\{ {0.51,0.65} \right\}} \right\},h_{2} = \left\{ {\left\{ {0.56,0.61,0.74} \right\},\left\{ {0.21} \right\}} \right\},h_{3} = \left\{ {\left\{ {0.37} \right\},\left\{ {0.46,0.57,0.62} \right\}} \right\}\) be three DHFEs, then by Definitions of DHFGHM operator and Theorem 2 described above, when the parameters be set the determined numbers, the final aggregated DHFEs can be obtained, for example,When \(p = 0.5,q = 0.5 {\text{DHFGHM}}(d_{1} ,d_{2} ,d_{3} ) {\text{DHFHM}}(d_{1} ,d_{2} ,d_{3} )\) the score of the aggregated DHFE is −0.0571.When \(p = 1,q = 10, {\text{DHFHM}}(d_{1} ,d_{2} ,d_{3} ) = \left\{ {\left\{ {0.2400, 0.2401, 0.2404, 0.3867, 0.3881, 0.3910} \right\}} , {\left\{ {0.4 577, 0.5275, 0.5678, 0.5556, 0.5762, 0.5986} \right\}} \right\}\) the score of the aggregated DHFE is −0.2329.

When \(p = 10,q = 10, {\text{DHFHM}}(d_{1} ,d_{2} ,d_{3} ) = \left\{ {\left\{ { 0. 1 9 2 6 , 0 . 1 9 2 7 , 0 . 1 9 2 7 , 0 . 3 7 4 5 , 0 . 3 7 4 8 , 0 . 3 7 4 9} \right\}} , {\left\{ { 0. 4 7 5 6 , 0 . 5 3 0 7 , 0 . 5 7 1 1 , 0 . 5 9 5 3 , 0 . 6 0 3 1 , 0 . 6 1 5 4} \right\}} \right\}\) the score of the aggregated DHFE is −0.2815.

The definition of DHFGHM operator shows that there are two parameters (\(p,q\)) control the aggregated DHFE simultaneously. When one of these two parameters is set, the scores trends can be described in detail.

Case 1

Take \(p = 1(q \in \left( {0,10} \right])\) or \(q = 1(p \in \left( {0,10} \right])\) for example, scores trends were shown in Fig. 5.

Fig. 5
figure 5

Scores trends \(p = 1\) and \(q = 1\)

Full size image

Case 2

Take \(p = 0.01,1,5,10(q \in \left( {0,10} \right])\) for example, scores trends was shown in Fig. 6.

Fig. 6
figure 6

Scores trends \(p = 0.01,1,5,10\)

Full size image

Case 3

Take \(q = 0.01,1,5,10(p \in \left( {0,10} \right])\) for example, scores trends was shown in Fig. 7.

Fig. 7
figure 7

Scores trends \(q = 0.01,1,5,10\)

Full size image

Influences of simultaneous varieties of \(p\) and \(q\) on score functions were analyzed on Fig. 8.

Fig. 8
figure 8

Scores for DHFEs obtained by the DHFGHM operator (\(p \in \left( {0,10} \right],\) \(q \in \left( {0,10} \right]\))

Full size image

4 GDM under dual hesitant fuzzy environment

Fuzziness, uncertainty and insufficient information are almost inevitable in all GDM and classification problems [48–50]. The GDM problem under dual hesitant fuzzy environment is chiefly researched in this section. In Sect. 4.1, we propose the weighted DHFHM and weighted DHFGHM operators. Algorithm for GDM using dual hesitant fuzzy information is investigated in Sect. 4.2. In Sect. 4.3, a real example of supplier selection presented to illustrate the proposed method.

4.1 Weighted DHFHM and weighted DHFGHM operators

The effects of the importance of aggregated arguments are not taken into account by DHFHM and DHFGHM operators. In the following, we introduced the weighted DHFHM and weighted DHFGHM operators.

Definition 7

Let \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a collection of DHFEs whose weight vector is \(w = (w_{1} ,w_{2} ,{ \ldots },w_{n} )^{T} ,\) which satisfies \(w_{i} > 0,\) \(i = 1,2,{ \ldots },n\) and \(\sum\limits_{i = 1}^{n} {w_{i} } = 1.\) If

$${\text{DHFWHM}}\left( {h_{1} ,h_{2} ,{ \ldots },h_{n} } \right) = \left( {\frac{2}{n(n + 1)}\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {\left( {w_{i} d_{i} } \right)^{p} \otimes \left( {w_{j} d_{j} } \right)^{p} } } } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {p + q}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${p + q}$}}}}$$
(32)
$${\text{DHFWGHM}}(h_{1} ,h_{2} ,{ \ldots },h_{n} ) = \frac{1}{p + q}\left( {\mathop \otimes \limits_{i = 1,j = i}^{n} \left( {\left( {pd_{i} } \right)^{{w_{i} }} \oplus \left( {qd_{j} } \right)^{{w_{j} }} } \right)^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {n(n + 1)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n(n + 1)}$}}}} } \right)$$
(33)

Then DHFWHM is called the dual hesitant fuzzy weighted Heronian mean operator (DHFWHM) and DHFWGHM is called the dual hesitant fuzzy weighted geometric Heronian mean operator (DHFWGHM).

Theorem 3

Let \(p,q > 0,\) and \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a collection of DHFEs whose weight vector is \(w = (w_{1} ,w_{2} ,{ \ldots },w_{n} )^{T},\) which satisfies \(w_{i} > 0,\) \(i = 1,2,{ \ldots },n\) and \(\sum\limits_{i = 1}^{n} {w_{i} } = 1 .\) then the aggregated value based on the DHFWHM or DHFWGHM is

$$\begin{aligned} &{\text{DHFWHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} )\\ &= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - (1 - \xi_{i} )^{{w_{i} }} )^{p} (1 - (1 - \xi_{j} )^{{w_{j} }} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\},} \right.\\& \left. {\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \eta_{i}^{{w_{i} }} )^{p} (1 - \eta_{j}^{{w_{j} }} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\} \end{aligned}$$
(34)
$$\begin{aligned} {\text{DHFWHM}}(d_{1} ,d_{2} ,{ \ldots },d_{n} )\,= \cup_{{\xi_{i} \in h_{i} ,\xi_{j} \in h_{j} ,\eta_{i} \in g_{i} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - \left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - \xi_{i}^{{w_{i} }} )^{p} (1 - \xi_{j}^{{w_{j} }} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right.,\\ \left. {\left\{ {\left( {1 - \prod\limits_{i = 1,j = i}^{n} {\left( {1 - (1 - (1 - \eta_{i} )^{{w_{i} }} )^{p} (1 - (1 - \eta_{j} )^{{w_{j} }} )^{q} } \right)^{{\frac{2}{n(n + 1)}}} } } \right)^{{\frac{1}{p + q}}} } \right\}} \right\} \end{aligned}$$
(35)

The proof of Theorem 3 is very similar to Theorems 1 and 2 and not to repeat them.

4.2 Dual hesitant fuzzy GDM method

Consider a multi-criteria GDM problem with a group of experts \(\left( {e_{1} ,e_{2} ,{ \ldots },e_{p} } \right),\) a set of alternatives \(\left( {a_{1} ,a_{2} ,{ \ldots },a_{m} } \right)\) and a collection of criteria \(\left( {c_{1} ,c_{2} ,{ \ldots },c_{m} } \right).\) The experts \(\left( {e_{1} ,e_{2} ,{ \ldots },e_{p} } \right)\) are responsible for the evaluation of each alternative against each criterion. The evaluation results of alternative \(a_{i}\) with the criteria \(c_{j}\) is presented by a DHFE \(d_{ij} = \left\{ {\left\{ {h_{ij} } \right\},\left\{ {g_{ij} } \right\}} \right\}.\) The decision matrix is illustrated as follows.

The decision making method consists of the following steps.

Step 1. Access to decision-making information based on the experts’ evaluation and expressed by DHFE.

Step 2. Aggregate all the dual hesitant fuzzy values \(d_{ij} (j = 1,2,{ \ldots },n)\) with respect of alternative \(a_{i} \left( {i = 1,2,{ \ldots },m} \right) ,\) and obtain the comprehensive evaluation result DHFE \(d_{i}\) based on the DHFWHM or DHFWGHM operators.

Step 3. Rank the DHFEs \(d_{i}\) \((i = 1,2,{ \ldots },m)\) and choice the most appropriate one.

4.3 Supplier selection

Supplier selection is an indispensable part of supply chain and it is the activity to identify the most suitable supplier. In addition, supplier selection problem is difficult due to the following two reasons, (1) the evaluation criteria are correlated and conflicted; (2) the performance of the evaluation criteria is hard to be characterized quantitatively. However, the GDM method proposed in Sect. 4.2 can solve the above two difficulties effectively. The main reasons are (1) the aggregation operators such as DHFWHM and DHFWGHM are characterized by the ability to cope with the decision making problem which the attribute are correlated; (2) DHFS is a powerful technique to describe the vagueness of the performance of the evaluation criteria (Table 1).

Table 1 The decision matrix
Full size table

Based on the existing research contributions [51–53], the following four criteria are selected to assess the performance of the alternatives and their weight are supposed as \(\left( {0.17,0.32,0.38,0.13} \right)^{T} .\) (1) Relationship closeness \({\text{c}}_{1} ;\) (2) Product quality \({\text{c}}_{2} ;\) (3) Price competitiveness \({\text{c}}_{3};\) (4) Delivery performance \({\text{c}}_{4}.\) Suppose there are four alternatives \(\left( {a_{1} ,a_{2} ,a_{3} ,a_{4} } \right)\) need to be evaluated and three experts \(\left( {e_{1} ,e_{2} ,e_{3} } \right)\) responsible for this evaluations. In order to access to decision-making information, the review table (Table 2) is designed and sent to three experts. There experts are requested to fill in Table 2 with the numbers between 0 and 1. The decision making information can be obtained after the three experts expressed their preferences.

Table 2 Review table for experts
Full size table

Take the performance of alternative \(a_{1}\) against criterion \(c_{1}\) for example, The DHFE {{0.3, 0.4},{0.6}} means (1) two of the three experts believe that the degree of alternative \(a_{1}\) meet criteria \(c_{1}\) is 0.3, and the remaining one expert think the degree is 0.4; (2) three experts all account the degree of alternative \(a_{1}\) does not meet criteria \(c_{1}\) is 0.6. According to the above analysis, the final evaluation result of the alternative \(a_{1}\) against criterion \(c_{1}\) can be expressed as a DHFE {{0.3, 0.4}, {0.6}}. Similarly, other DHFEs in Table 3 indicated the corresponding evaluation information.

Table 3 Dual hesitant fuzzy decision matrix
Full size table

Based on the decision making information described in Table 3, we first use the DHFWHM operator to obtain the comprehensive evaluation dual hesitant fuzzy information for every alternatives. Due to the relatively large numbers, we just list the score functions of the DHFEs.

From the Table 4, we find that different values of the parameters could lead to different ranking of the alternatives. For example, when \(p = 0.01,q = 10,\) the best alternative is \(a_{3}\) and this result would be changed when parameters \(p\) and \(q\) be set the different values. When the value of parameter \(p\) fixed, the trends of the scores of four alternatives with different values of parameter \(q\) are presented in Fig. 9 \(\left( {p = 0.01} \right)\) and Fig. 10 \(\left( {p = 10} \right)\) respectively.

Table 4 Scores obtained by the DHFWHM operator and the rankings of alternatives
Full size table
Fig. 9
figure 9

Change trends of the scores of four alternatives \(\left( {q \in \left( {0,10} \right],\,\,p = 0.01} \right)\)

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Fig. 10
figure 10

Change trends of the scores of four alternatives \(\left( {q \in \left( {0,10} \right],\,\,p = 10} \right)\)

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From Table 4 and Figs. 9 and 10, we find that different score functions can be obtained when \(p\) and \(q\) are set different values. Figures 11, 12, 13, 14 show the scores for the four alternatives when the values of parameters \(p\) and \(q\) change simultaneously.

Fig. 11
figure 11

Scores for alternative \(a_{1}\) obtained by the DHFWHM operator

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Fig. 12
figure 12

Scores for alternative \(a_{2}\) obtained by the DHFWHM operator

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Fig. 13
figure 13

Scores for alternative \(a_{3}\) obtained by the DHFWHM operator

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Fig. 14
figure 14

Scores for alternative \(a_{4}\) obtained by the DHFWHM operator

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4.3.1 Comparative study

The above ranking results and the analysis about the scores for every alternative are based on the DHFWHM operator. In order to show the advantages and rationality of the method proposed in the above Section, we compare it with DHFWA operator proposed by Wang et al. [29]. The DHFWA operator is defined by Wang et al. as follows. Let \(d_{j} = \left( {h_{j} ,g_{j} } \right)(j = 1,2,{ \ldots },n)\) be a collection of DHFEs. A dual hesitant fuzzy weighted averaging (DHFWA) operator is a mapping \(D^{n} \to D\) such that

$${\text{DHFWA}}\left( {d_{1} ,d_{2} ,{ \ldots },d_{n} } \right)\mathop { = \oplus }\limits_{j = 1}^{n} \left( {\omega_{j} d_{j} } \right) = \omega_{1} d_{1} \oplus \omega_{2} d_{2} \oplus \cdots \oplus \omega_{n} d_{n}$$
$${ = } \cup_{{\gamma_{j} \in h_{j} ,\eta_{j} \in g_{j} }} \left\{ {\left\{ {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \gamma_{j} } \right)^{{\omega_{j} }} } } \right\},\left\{ {\prod\limits_{j = 1}^{n} {\eta_{j}^{{\omega_{j} }} } } \right\}} \right\}$$
(36)

Based on DHFWA operator, we can get the overall preference values for each alternative (\(a_{i}\)). \(d(a_{1} )\) = {{0.5182, 0.5319, 0.6610, 0.6707, 0.5306, 0.5440, 0.6697, 0.6792}, {0.1931, 0.2253}}. The score of \(d(a_{1} )\) is \(s\left( {d(a_{1} )} \right)\) = 0.3915. Similarly, \(d(a_{2} )\) = {{0.5966, 0.6542, 0.6321, 0.6846, 0.6056, 0.6619, 0.6403, 0.6917}, {0.2136, 0.2337, 0.2464}}. The score of \(d(a_{2} )\) is \(s\left( {d(a_{2} )} \right)\) = 0.4147. \(d(a_{3} )\) = {{0.5380, 0.6040, 0.5574, 0.6206, 0.5787, 0.6388}, {0.2184, 0.2303, 0.2340, 0.2467}}. The score of \(d(a_{3} )\) is \(s\left( {d(a_{3} )} \right)\) = 0.3573. \(d(a_{4} )\) = {{0.4398, 0.4686, 0.4717, 0.4988, 0.4666, 0.4940, 0.4969, 0.5227}, {0.3462, 0.3789}}. The score of \(d(a_{4} )\) is \(s\left( {d(a_{4} )} \right)\) = 0.1198.

Since \(s\left( {d(a_{2} )} \right)\) > \(s\left( {d(a_{1} )} \right)\) > \(s\left( {d(a_{3} )} \right)\) > \(s\left( {d(a_{4} )} \right),\) then the second alternative should be selected. This result is quite different from the result based on DHFWHM operator which was shown in Table 4. The main reason is that the DHFWA operator [29] does not take into account the relationship between various attributes but considered the aggregated arguments independent. Therefore, the dual hesitant fuzzy aggregation operators proposed in this paper are good complement to existing research.

5 Concluding remarks

This paper proposes some dual hesitant fuzzy information aggregation operators by introducing two parameters to reflect the decision makers’ preference. Based on which, a multi-criteria GDM method under dual hesitant fuzzy environment is introduced and applied to evaluate suppliers, where criteria are described quantitatively with the DHFE. The final evaluation DHFE of each supplier can be achieved by DHFWHM operator. Decision can be made depends on the comprehensive DHFE for every alternative, the bigger the score function of the comprehensive DHFE the better the performance. A numerical example is presented to illustrate the effectiveness of the proposed method. The proposed multi-criteria GDM method can also be used to other evaluation problem under dual hesitant fuzzy environment. The following points should be taken into consideration when the method is adopted.

(1) Different values of the parameters represent different attitude of the decision makers which produce different comprehensive values and further cause different ranking results. Attention should be paid to the selection of appropriate values for parameter according the decision makers’ preference.

(2) Access to decision-making information based on the experts’ evaluation and expressed by DHFE correctly is a critical step of the proposed method. It should be noted that there are two kinds of information must be collected, namely membership degree and non-membership degree.

(3) The GDM method proposed in this paper should be compared with some other traditional methods to observe whether the final decision result changes or not. Further study about real case in real situations worth studied which would enlarge the scope of the proposed GDM approach.