**2. Preliminaries**

In this section, some basic concepts related to ordered weighted hesitant fuzzy set (OWHFS) and PA operator are reviewed, which will be useful for later analysis.

**Definition 1.** [58] *Let X be the universe of discourse. An ordered weighted hesitant fuzzy set (OWHFS) on X is defined as:*

$$\,^\omega H = \{<\mathfrak{x}\_\prime\prime^\omega h(\mathfrak{x})>|\mathfrak{x}\in X\}$$

*where, <sup>ω</sup>h*(*x*) = ∪ 1≤*j*≤*Lx* {< *<sup>h</sup><sup>δ</sup>*(*j*)(*x*), *w<sup>δ</sup>*(*j*)(*x*) <sup>&</sup>gt;}*, referred to as the ordered weighted hesitant fuzzy element (OWHFE), is a set of some different values in [0,1]. It denotes all possible membership degrees of the element x* ∈ *X to the set ωH , and w<sup>δ</sup>*(*j*)(*x*) ∈ [0, 1] *is the weight of <sup>h</sup><sup>δ</sup>*(*j*)(*x*) *such that Lx*∑ *j*=1 *w<sup>δ</sup>*(*j*)(*x*) = 1 *for any x* ∈ *X*.

It is worth noting that when *w<sup>δ</sup>*(1)(*x*) = *w<sup>δ</sup>*(2)(*x*) = ... = *w<sup>δ</sup>*(*Lx*)(*x*) = 1/*Lx* for any *x* ∈ *X*, then the OWHFS *ωH* will become a typical HFS. For the convenience of representation, OWHFE can be denoted by *ωh* = *<sup>ω</sup>h*(*x*) = ∪ 1≤*j*≤*Lx*{< *<sup>h</sup><sup>δ</sup>*(*j*), *w<sup>δ</sup>*(*j*) >}

Suppose that the membership degrees provided by *k* experts, of the element *x* in the set *<sup>ω</sup>H*, where *<sup>h</sup><sup>δ</sup>*(*i*)(*x*) is given by *ki* experts, *i* = 1, 2, ... , *L*, *L*∑ *i*=1 *ki* = *k*. It should be noted that every expert cannot persuade other experts to change their opinions. In such a situation, the membership degree of the element *x* in the set *ωH* has *L* possible values *<sup>h</sup><sup>δ</sup>*(1)(*x*), *<sup>h</sup><sup>δ</sup>*(2)(*x*), ... , and *<sup>h</sup><sup>δ</sup>*(*L*)(*x*) associated with weights *w<sup>δ</sup>*(1)(*x*) = *k*1*k* , *w<sup>δ</sup>*(2)(*x*) = *k*2*k* , . . . , and *w<sup>δ</sup>*(*L*)(*x*) = *kLk* respectively.

**Definition 2.** [58] *Let ωh* = ∪1≤*j*≤*L*<sup>&</sup>lt; *<sup>h</sup><sup>δ</sup>*(*j*), *w<sup>δ</sup>*(*j*) >*, <sup>ω</sup>h*1 = ∪1≤*j*≤*L*{<sup>&</sup>lt; *h<sup>δ</sup>*(*j*) 1 , *w<sup>δ</sup>*(*j*) 1 >} *and <sup>ω</sup>h*2 = ∪ 1≤*j*≤*L*{<sup>&</sup>lt; *h<sup>δ</sup>*(*j*) 2 , *w<sup>δ</sup>*(*j*) 2 >} *be three OWHFEs. Then, some operations on the OWHFEs ωh, <sup>ω</sup>h*1 *and <sup>ω</sup>h*2 *are definedasfollows:*

$$(\mathcal{I})^{\quad \quad \omega} h^{\lambda} = \bigcup\_{1 \le j \le L} \left\{ \left< \left( h^{\delta(j)} \right)^{\lambda}, w^{\delta(j)} \right> \right\};$$

*Symmetry* **2019**, *11*, 17

$$\mathcal{L}(\mathcal{Q}) \quad \lambda^{\omega} h = \underset{1 \le j \le L}{\cup} \left\{ \left\langle 1 - \left( 1 - h^{\delta(j)} \right)^{\lambda}, w^{\delta(j)} \right\rangle \right\};$$

$$(3)\quad ^\omega h\_1 \oplus ^\omega h\_2 = \underset{1 \le j \le L}{\cup} \left\{ \left\langle h\_1^{\delta(j)} + h\_2^{\delta(j)} - h\_1^{\delta(j)} h\_2^{\delta(j)}, \overline{(w\_1^{\delta(j)} + w\_2^{\delta(j)})} \right\rangle \right\};$$

$$where \,\lambda > 0 \,\,and \,\,\overline{(w\_1^{\delta(j)} + w\_2^{\delta(j)})} = \frac{w\_1^{\delta(j)} + w\_2^{\delta(j)}}{\sum\_{j=1}^{L} (w\_1^{\delta(j)} + w\_2^{\delta(j)})} (j = 1, 2, \dots, L).$$

**Definition 3.** [59] *Let ωh* = ∪1≤*j*≤*L*<sup>&</sup>lt; *<sup>h</sup><sup>δ</sup>*(*j*), *w<sup>δ</sup>*(*j*) >*, <sup>ω</sup>h*1 = ∪1≤*j*≤*L*<sup>&</sup>lt; *h<sup>δ</sup>*(*j*) 1 , *w<sup>δ</sup>*(*j*) 1 > *and <sup>ω</sup>h*2 = ∪ 1≤*j*≤*L*<sup>&</sup>lt; *h<sup>δ</sup>*(*j*) 2 , *w<sup>δ</sup>*(*j*) 2 > *be three OWHFEs.* Δ(*ωh*) = *L*∑*j*=1 *h<sup>δ</sup>*(*j*)*w<sup>δ</sup>*(*j*) *is called the score function of ωh, and* ∇(*ωh*) = *L* ∑ *j*=1 (Δ(*ωh*) − *<sup>h</sup><sup>δ</sup>*(*j*))<sup>2</sup>*w<sup>δ</sup>*(*j*) *is called the deviation function of <sup>ω</sup>h*.

*(1) If* <sup>Δ</sup>(*ωh*1) > <sup>Δ</sup>(*ωh*2)*, then <sup>ω</sup>h*1 >*<sup>ω</sup> h*2 *(2)If*<sup>Δ</sup>(*ωh*1)<<sup>Δ</sup>(*ωh*2)*,then<sup>ω</sup>h*1<*<sup>ω</sup> h*2

$$(3)\quad \text{If } \Delta(^{\omega}h\_1) = \Delta(^{\omega}h\_2), \text{ then} \\
\begin{cases}
\nabla(^{\omega}h\_1) > \nabla(^{\omega}h\_2) \Rightarrow \,\_{\omega}h\_1 < ^{\omega}h\_2 \\
\nabla(^{\omega}h\_1) = \nabla(^{\omega}h\_2) \Rightarrow \,\_{\omega}h\_1 = ^{\omega}h\_2 \\
\nabla(^{\omega}h\_1) < \nabla(^{\omega}h\_2) \Rightarrow \,\_{\omega}h\_1 > ^{\omega}h\_2
\end{cases}$$

**Definition 4.** [59] *Let C* = {*<sup>C</sup>*1, *C*2,..., *Cn*} *be a set of criteria, and there is a prioritization among the criteria expressed by the linear ordering C*1 *C*2 ... *Cn, which indicates that criterion Cj has a higher priority than Ci, if j* < *i. The value Cj*(*x*) *is the performance of any alternative x under criterion Cj, and satisfies Cj*(*x*) ∈ [0, 1]*. If*

$$PA(\mathbb{C}(\mathbf{x})) = \sum\_{j=1}^{n} w\_j \mathbb{C}\_j(\mathbf{x}) \tag{1}$$

*where wj* = *Ti n*∑ *i*=1 *Ti , Tj* = *j*−1 ∏ *l*=1 *Cl*(*x*)(*j* = 1, 2, . . . , *<sup>n</sup>*)*, T*1 = 1*. Then PA is called the prioritized average operator.*

#### **3. GOWHFPWA Operator and Its Properties**

In this section, the GOWHFPWA operator is proposed to aggregate the OWHFEs, and some properties are studied.

The PA operator has been commonly used in situations where the DMs' judgments are the exact values [59]. In this part, we shall extend the PA operator to ordered weighted hesitant fuzzy environments and define the GOWHFPWA operator.

**Definition 5.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ... , *ωhn be a set of OWHFEs, then the GOWHFPWA operator is defined as follows:*

$$\text{GOMHFPWA}(^{\omega}h\_{1}, ^{\omega}h\_{2}, \dots, ^{\omega}h\_{4}) = \left(\frac{\frac{\text{T}\_{1}}{\text{s}}(^{\omega}h\_{1})^{a} \oplus \frac{\text{T}\_{2}}{\text{s}}(^{\omega}h\_{2})^{a} \oplus \dots \oplus \frac{\text{T}\_{n}}{\text{s}}(^{\omega}h\_{4})^{a}\right)^{1/a} = \left(\oplus \frac{\frac{\text{T}\_{1}(^{\omega}h\_{1})^{a}}{\text{s}}}{\frac{\text{T}\_{1}}{\text{s}}\text{t}\_{i}}\right)^{1/a} \tag{2}$$

*where, α* > 0 *is a parameter of GOWHFPWA operator, Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1 *and* Δ(*ωhk*) *is the score function of <sup>ω</sup>hk*.

**Theorem 1.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a set of OWHFEs, then their aggregated value by using the GOWHFPWA operator is also an OWHFE, and*

$$\text{GOWHFPWA}(\prescript{\omega}{}{\mathsf{h}\_{1}}, \prescript{\omega}{}{\mathsf{h}\_{2}}, \dots, \prescript{\omega}{}{\mathsf{h}\_{n}}) = \underset{1 \le j \le L}{\text{U}} \left\{ \left\langle \left(1 - \prod\_{i=1}^{n} \left(1 - \left(\boldsymbol{h}\_{i}^{\delta(j)}\right)^{\mathbf{a}}\right)^{\frac{\sum\_{i}^{T\_{i}}}{\mathbf{n}}}\right)^{1/\mathbf{a}}, \overline{\left(\sum\_{i=1}^{n} \mathbf{w}\_{i}^{\delta(j)}\right)} \right\rangle \right\} \tag{3}$$

*where, Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1*,* Δ(*ωhk*) *is the score function of ωhk, and L is the number of basic units in ωhi*(*i* = 1, 2, ··· , *<sup>n</sup>*).

**Proof.** For *n* = 1, the result can be obtained easily by Definition 5. In the following, we prove the equation

$$\text{GOWHFPWA}(\prescript{\omega}{}{\mathsf{h}\_{1}}, \prescript{\omega}{}{\mathsf{h}\_{2}}, \dots, \prescript{\omega}{}{\mathsf{h}\_{n}}) = \underset{1 \le j \le L}{\text{U}} \left\{ \left\langle \left(1 - \prod\_{i=1}^{n} (1 - (\mathsf{h}\_{i}^{\delta(j)})^{\frac{\alpha}{\mathsf{h}\_{i}}})^{\frac{\frac{T\_{i}}{\mathsf{h}\_{i}}}{\mathbf{1}\_{i=1}^{T\_{i}}}}\right)^{\frac{T\_{i}}{\mathsf{h}\_{i}}}, \overline{\left(\sum\_{i=1}^{n} w\_{i}^{\delta(j)}\right)} \right\rangle \right\}$$

by using mathematical induction for *n*(*n* ≥ <sup>2</sup>).

> For *n* = 2, since

$$\begin{aligned} \left\{ \begin{array}{l} \frac{T\_1}{2} \omega \mu\_1^{\mathfrak{a}} = \bigcup\_{1 \le j \le L} \left\{ \left< 1 - \left( 1 - \left( h\_1^{\delta(j)} \right)^{\mathfrak{a}} \right)^{\frac{\sum\_{i} T\_i}{i}}, w\_1^{\delta(j)} \right\rangle \right\} \\ \frac{T\_2}{2} \omega \mu\_2^{\mathfrak{a}} = \bigcup\_{1 \le j \le L} \left\{ \left< 1 - \left( 1 - \left( h\_2^{\delta(j)} \right)^{\mathfrak{a}} \right)^{\mathfrak{a}}, w\_2^{\delta(j)} \right\} \right\} \end{array} \end{aligned} $$

then

$$\begin{split} & \frac{\mathbb{V}\_{1}}{\mathbb{V}\_{1}} \, ^{\mathsf{H}}\mathbb{H}\_{1}^{\mathsf{T}} \oplus \frac{\mathbb{V}\_{1}}{\mathbb{V}\_{1}} \, ^{\mathsf{H}}\mathbb{H}\_{2}^{\mathsf{T}} - \\ & \frac{\mathbb{V}\_{1}}{\mathbb{V}\_{1}} \, ^{\mathsf{H}}\mathbb{V}\_{1} \left\{ \left\{ 1 - (\mathbbm{1} - (\mathbbm{h}\_{1}^{\delta(\ell)})^{\mathsf{T}})^{\frac{\mathsf{T}\_{1}}{\mathsf{T}\_{1}}} + (1 - (\mathbbm{h}\_{2}^{\delta(\ell)})^{\mathsf{T}})^{\frac{\mathsf{T}\_{1}}{\mathsf{T}\_{1}} - \mathbb{I}\_{1}} - (1 - (\mathbbm{h}\_{1}^{\delta(\ell)})^{\mathsf{T}})^{\frac{\mathsf{T}\_{1}}{\mathsf{T}\_{1}} - \mathbb{I}\_{1}}) \times (1 - (\mathbbm{1} - (\mathbbm{h}\_{2}^{\delta(\ell)})^{\mathsf{T}})^{\frac{\mathsf{T}\_{1}}{\mathsf{T}\_{1}} - \mathbb{I}\_{1}}), \left( \mathbbm{w}\_{1}^{\delta(\ell)} + \mathfrak{w}\_{2}^{\delta(\ell)} \right) \right\} \\ & - \mathbb{V}\_{1} \left\{ \left\{ 1 - \prod\_{i=1}^{2} \left( 1 - (\mathbbm{h}\_{i}^{\delta(\ell)})^{\mathsf{T}} \right)^{\frac{\mathsf{T}\_{1}}{\mathsf{T}\_{1}} - \mathbb{I}\_{1}}, \left( \mathbbm{w}\_{1}^{\delta(\ell)} + \mathfrak{w}\_{2}^{\delta(\ell)} \right) \right) \right\} \end{split}$$

That is, Equation (7) holds when *n* = 2. Suppose that Equation (3) also holds when for *n* = *l*,

$$\text{GOWHFPWA}(\prescript{\omega}{}{\mathsf{h}\_{1}}, \prescript{\omega}{}{\mathsf{h}\_{2}}, \dots, \prescript{\omega}{}{\mathsf{h}\_{l}}) = \underset{1 \le j \le L}{\text{U}} \left\{ \left\langle \left(1 - \prod\_{i=1}^{l} \left(1 - \left(\boldsymbol{h}\_{i}^{\delta(j)}\right)^{\kappa}\right)^{\frac{\mathsf{T}\_{i}}{\mathsf{h}\_{i}}}\right)^{1/\kappa}, \overline{\left(\sum\_{i=1}^{l} \boldsymbol{w}\_{i}^{\delta(j)}\right)} \right\rangle \right\}$$

when *n* = *l* + 1, the operational laws described in Definition 2 state that

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That is, Equation (3) holds for *n* = *l* + 1.Thus, Equation (3) holds for all *n*. Then,

$$\text{GOWHFPWA}(^{\omega}h\_{1}, ^{\omega}h\_{2}, \cdot, \cdot, ^{\omega}h\_{n}) = \underset{1 \le j \le L}{\text{U}} \left\{ \left\langle \left(1 - \prod\_{i=1}^{n} (1 - \left(h\_{i}^{\delta(j)}\right)^{a})^{\frac{\sum\_{i}^{T\_{i}}}{n}}\right)^{1/a}, \overline{\left(\sum\_{i=1}^{n} w\_{i}^{\delta(j)}\right)} \right\rangle \right\}$$

Now, consider some desirable properties of the GOWHFPWA operator.

**Theorem 2.** *(Idempotency). Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a set of OWHFs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*), *T*1 = 1 *and* Δ(*ωhl*) *is the score function of ωhl. If <sup>ω</sup>h*1 = *<sup>ω</sup>h*2 = ··· = *ωhn* = *ωh, then*

$$\text{GOWHPWA}(^{\omega}h\_{1\prime} \,^{\omega}h\_{2\prime} \cdot \cdots \cdot \,^{\omega}h\_{\text{fl}}) = \underset{1 \le j \le L}{\cup} \left\{ \left\langle 1 - \prod\_{i=1}^{n} (1 - h\_i^{\delta(j)})^{\frac{\overline{\mathbf{n}}^T}{\mathbf{i} - \mathbf{i}\_i}}, \overline{\left(\sum\_{i=1}^{n} w\_i^{\delta(j)}\right)} \right\rangle \right\} = ^{\omega}h \tag{4}$$

$$\begin{array}{lcl} \textbf{Proof.} & \mathbbm{1}^{\omega}h\_{1} = \ ^{\omega}h\_{2} = \ ^{\omega}h\_{\scriptscriptstyle\rm II} = \ ^{\omega}h\_{\scriptscriptstyle\rm II} = \ ^{\omega}h = \ ^{\cup}\_{1\leq j\leq N} \left\{  \right\}, \ \textbf{then} \ \overline{\begin{subarray}{l} \frac{\pi}{n}w^{\delta(j)}\_{i} \\ i=1 \end{subarray}} \ = \ ^{\cup}\_{1\leq j\leq L} \left\{  \right\}, \ \textbf{then} \ \overline{\begin{subarray}{l} \frac{\pi}{n}w^{\delta(j)}\_{i} \\ i=1 \end{subarray}} \ = \ ^{\omega}\_{1\leq j\leq L} \left\{ \left\{ \left(1-\frac{\pi}{i}\right)\left(1-\left(h^{\delta(j)}\_{i}\right)^{a}\right)^{\frac{\Gamma}{n}}\right\}^{1/a}, \left(\frac{\Gamma}{i=1}w^{\delta(j)}\_{i}\right) \right\} \\ = \ & \bigcup\_{1\leq j\leq L} \left\{ \left\langle \left(1-\frac{\pi}{\prod\_{i=1}^{L}\left(1-\left(h^{\delta(j)}\_{i}\right)^{a}\right)^{\frac{\Gamma}{n}}\right)^{1/a}, w^{\delta(j)}\right\rangle \right\} \ = \ & \bigcup\_{1\leq j\leq L} \left\{ \left\langle 1-(1-h^{\delta(j)}),w^{\delta(j)}\right\rangle \right\} = \ \end{array}$$

**Theorem 3.** *(Boundedness). Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a collection of OWHFEs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1*,* Δ(*ωhl*) *is the score function of ωhl. Let <sup>ω</sup>h*− = {< *h*<sup>−</sup>, 1 >} *ωh*<sup>+</sup> = {< *h*+, 1 <sup>&</sup>gt;}*, h*− = min( min *h<sup>δ</sup>*(*j*) 1 <sup>∈</sup>*<sup>ω</sup>h*1(*h<sup>δ</sup>*(*j*) 1 ), min *h<sup>δ</sup>*(*j*) 2 <sup>∈</sup>*<sup>ω</sup>h*2(*h<sup>δ</sup>*(*j*) 2 ), ··· , min *h<sup>δ</sup>*(*j*) *n* ∈*<sup>ω</sup>hn*(*h<sup>δ</sup>*(*j*) *n* )) *and h*<sup>+</sup> = max( max *h<sup>δ</sup>*(*j*) 1 <sup>∈</sup>*<sup>ω</sup>h*1(*h<sup>δ</sup>*(*j*) 1 ), max *h<sup>δ</sup>*(*j*) 2 <sup>∈</sup>*<sup>ω</sup>h*2(*h<sup>δ</sup>*(*j*) 2 ), ··· , max *h<sup>δ</sup>*(*j*) *n* ∈*<sup>ω</sup>hn*(*h<sup>δ</sup>*(*j*) *n* ))*. Then*

$$^{\omega}h^{-} \le \text{GOWHFPWA}(^{\omega}h\_1 \; ^{\omega}h\_2 \; \cdots \; \; \; \; \; \; \; \; h\_n) \le ^{\omega}h^{+} \tag{5}$$

$$\begin{array}{rclclcl}\textbf{Proof.} & \quad \textbf{Since } f(\textbf{x}) &=& (1-\textbf{x})^{a}(a \in \{0,1\}) \text{ is a decreasing function about } \textbf{x} \in \left[0,1\right], \text{ then,} \\\ h^{-} &=& \begin{pmatrix} \frac{\overline{T\_{i}}}{\textbf{a}} \\ 1-\prod\_{i=1}^{\frac{\overline{T}\_{i}}{\textbf{a}}} (1-\left(h^{-}\right)^{a})^{\frac{\overline{\textbf{a}}-\overline{T}\_{i}}{\textbf{a}}} \end{pmatrix}^{1/a} & \leq & \begin{pmatrix} 1-\prod\_{i=1}^{\frac{\overline{T}\_{i}}{\textbf{a}}} \left(1-\min\_{h\_{i}^{\delta(i)}} \left(h\_{i}^{\delta(i)}\right)^{a}\right)^{\frac{\overline{\textbf{a}}-\overline{T}\_{i}}{\textbf{a}}} \end{pmatrix}^{1/a} & \leq & \begin{pmatrix} \frac{\overline{T}\_{i}}{\textbf{a}} \\ 1-\prod\_{i=1}^{\overline{\textbf{a}}} \left(1-\min\_{h\_{i}^{\delta(i)}} \left(h\_{i}^{\delta(i)}\right)^{a}\right)^{\frac{\overline{\textbf{a}}-\overline{T}\_{i}}{\textbf{a}}} \end{pmatrix}^{1/a} & \leq & \begin{cases} \frac{\overline{T}\_{i}}{\textbf{a}} & \frac{\overline{T}\_{i}}{\textbf{a}} \\ 1-\frac{\overline{T}\_{i}}{\textbf{a}} & \frac{\overline{T}\_{i}}{\textbf{a}} \in \frac{\overline{T}\_{i}}{\textbf{a}} \end{cases} & \leq & \begin{cases} \frac{\overline{T}\_{i}}{\textbf{a}} & \frac{\overline{T}\_{i}}{\textbf{a}} \end{cases} \end{array}$$

$$\left(1-\prod\_{i=1}^{\frac{n}{t}}\left(1-\left(h\_{i}^{\delta(j)}\right)^{a}\right)^{\frac{\frac{T\_{j}}{n}}{\frac{n}{t}T\_{i}}}\right)^{1/a} \leq \qquad\left(1-\prod\_{i=1}^{\frac{n}{t}}\left(1-\max\_{h\_{i}^{\delta(j)}\in\mathcal{C}^{\omega}h\_{i}}\left(h\_{i}^{\delta(j)}\right)^{a}\right)^{\frac{T\_{j}}{n}}\right)^{1/a} \leq \frac{1}{n}$$

$$\left(1-\prod\_{i=1}^{\frac{T\_i}{n}}(1-(h^+)^{\kappa})^{\frac{\kappa}{1+\kappa}}\right)^{1/\kappa} = \quad h^+, \text{ thus } \Delta(^{\omega}h^-) \le \quad \Delta(^{\omega}h\_i) \le \quad \Delta(^{\omega}h^+) \text{ and } \; ^{\omega}h^- \le \quad \text{as } \mathbb{R}^N \text{ is a measurable function.}$$

**Theorem 4.** *(Monotonicity). Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn and ωh* 1, *ωh* 2, ··· , *ωh n be two sets of OWHFEs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T i* = *i*−1 ∏ *l*=1 Δ(*ωh l*)(*<sup>i</sup>* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = *T* 1 = 1*,* Δ(*ωhl*) *is the score function of ωhl and* Δ(*ωh l*) *is the score function of ωh l, if h<sup>δ</sup>*(*j*) *i* ≤ *h i<sup>δ</sup>*(*j*)(*<sup>i</sup>* = 1, 2, ··· , *n*, *j* = 1, 2, ··· , *L*) *and w δ*(*j*) *i* = *w i<sup>δ</sup>*(*j*)(*<sup>i</sup>* = 1, 2, ··· , *n*, *j* = 1, 2, ··· , *<sup>L</sup>*)*, then*

$$\text{GOWHPWA}(^{\omega}h\_{1\prime} \,^{\omega}h\_{2\prime} \cdots \,^{\omega}h\_n) \le \text{GOWHPWA}(^{\omega}h'\_{1\prime} \,^{\omega}h'\_{2\prime} \cdots \,^{\omega}h'\_n) \tag{6}$$

**Proof.** According to the proof of Theorem 3, it is easy to prove that the GOWHFPWA operator satisfies the above monotonicity, thus the proof process is omitted. -

**Theorem 5.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a set of OWHFEs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1 *and* Δ(*ωhl*) *is the score function of ωhl. If <sup>ω</sup>g is an OWHFE. Then*

$$\text{GOM}\text{HFPWA}(^{\omega}\text{h}\_1 \oplus ^{\omega}\text{g}\_2 \,^{\omega}\text{h}\_2 \oplus ^{\omega}\text{g}, \dots, ^{\omega}\text{h}\_n \oplus ^{\omega}\text{g}) = \text{GOM}\text{HFPWA}(^{\omega}\text{h}\_1 \,^{\omega}\text{h}\_2 \,^{\omega}\text{h}\_2, \dots, ^{\omega}\text{h}\_n) \oplus ^{\omega}\text{g} \tag{7}$$

**Theorem 6.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a set of OWHFEs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1 *and* Δ(*ωhl*) *is the score function of ωhl. Then*

$$\text{GONHFPWA}(r^{\omega}h\_1, r^{\omega}h\_2, \dots, r^{\omega}h\_n) = r \text{GONHFPWA}(^{\omega}h\_1, ^{\omega}h\_2, \dots, ^{\omega}h\_n) \tag{8}$$

*where r is an arbitrary number greater than 0.*

**Theorem 7.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn be a set of OWHFEs, where Ti* = *i*−1 ∏ *l*=1 Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T* = 1 *and* Δ(*ωhl*) *and is the score function of ωhl. If <sup>ω</sup>g is an OWHFE. Then*

$$\text{GOMHFPWA}(r^{\omega}h\_1, r^{\omega}h\_{2\prime}, \dots, r^{\omega}h\_n) \oplus {}^{\omega}\mathbf{g} = r \text{GOMHFPWA}(^{\omega}h\_1, ^{\omega}h\_{2\prime}, \dots, ^{\omega}h\_n) \oplus {}^{\omega}\mathbf{g} \tag{9}$$

*where r is an arbitrary number greater than 0.*

**Theorem 8.** *Let <sup>ω</sup>h*1, *<sup>ω</sup>h*2, ··· , *ωhn and <sup>ω</sup>g*1, *<sup>ω</sup>g*2, ··· , *<sup>ω</sup>gn be two set of OWHFEs, where Ti* = *i*−1 ∏ *l*=1Δ(*ωhl*)(*i* = 1, 2, ··· , *<sup>n</sup>*)*, T*1 = 1 *and* Δ(*ωhl*) *is the score function of ωhl. Then*

$$\text{GMOHFPWA}(\,^{\omega}\text{h}\_{1}\,^{\omega}\text{h}\_{2},\dots,\,^{\omega}\text{h}\_{n}) \oplus \text{GONHFPWA}(\,^{\omega}\text{g}\_{1}\,^{\omega}\text{g}\_{2},\dots,\,^{\omega}\text{g}\_{n}) \\ - \text{GMOHFPWA}(\,^{\omega}\text{h}\_{1}\,\cap\,^{\omega}\text{g}\_{2},\,^{\omega}\text{h}\_{2}\,\cap\,^{\omega}\text{g}\_{3},\,\dots,\,^{\omega}\text{h}\_{n}\,\cap\,^{\omega}\text{g}\_{n}) \\ \tag{10}$$

**Proof.** According to Definition 2, it is easy to prove that the GOWHFPWA operator satisfies Theorem 5, 6, 7, and 8, so the proof process is omitted. -

#### **4. The MCGDM Approach with Order Weighted Hesitant Fuzzy Information**

In this section, we present a novel MCGDM method based on ordered weighted hesitant fuzzy information, which utilizes the above GOWHFPWA operator to rank the alternatives of GSS. Consider a MCGDM for GSS problem, let *X* = {*<sup>x</sup>*1, *x*2,... *xm*} be a set of suppliers, *C* = {*<sup>c</sup>*1, *c*2,... *cn*} be a set of criteria, and *E* = {*<sup>e</sup>*1,*e*2,...*ek*} be a set of DMs. In practice, there is a priority relationship among the GSS evaluation criteria. For example, if DMs believe that environmental protection is the most important criterion, they should take precedence over price, quality, and other criteria. Secondly, if price is more important than quality and other criteria, the priority of price is higher than quality, and so on. Such a prioritization among the criteria can be expressed by the ordering *c*1 *c*2 ... *cn*, in which criterion *cj* has a higher priority than *ci* if *j* < *i*.

For an alternative under a criterion, all the DMs provide their evaluated values anonymously. The evaluation values of alternative *xp* under criteria *cq* are provided by DM *eu*(*u* = 1, 2, ... , *k*), which can be represented by an OWHFE *<sup>ω</sup>hpq*. The ordered weighted hesitant fuzzy group decision matrix *M* = (*ωhpq*)*m*×*n*is constructed from all of these OWHFEs.

In view of the above analysis, the procedure of the proposed approach is described under the following steps:

**Step 1**. Calculate the values of *Tpq*(*p* = 1, 2, . . . *m*; *q* = 1, 2, . . . , *n*) based on Equation (11).

$$T\_{pq} = \prod\_{q=1}^{n-1} \Delta(^{\omega}h\_{pq}) (p = 1, 2, \dots, m, q = 1, 2, \dots, n) \tag{11}$$

where *Tp*1 = 1. **Step 2**. Aggregate the OWHFEs *ωhpq* for each supplier *xp*(*p* = 1, 2, ... , *m*) by the GOWHFPWA operator, then we can ge<sup>t</sup> the overall OWHFE *<sup>ω</sup>hp*(*p* = 1, 2, ... , *m*) for the supplier *xp*(*p* = 1, 2, ... , *m*) as follows:

$$\mathbb{E}^{\omega\_{\mathsf{H}}}h\_{\mathsf{P}} - \text{GOWHPPA}\{\boldsymbol{\omega}^{\omega}\boldsymbol{h}\_{\mathsf{P}L}, \boldsymbol{\omega}^{\omega}\boldsymbol{h}\_{\mathsf{P}L}, \dots, \boldsymbol{\omega}^{\omega}\boldsymbol{h}\_{\mathsf{P}R}\} - \underset{1 \leq \boldsymbol{j} \leq L\_{\mathsf{T}}}{\displaystyle}{\displaystyle} \left\{ \left\langle \left(1 - \prod\_{q=1}^{n} \left(1 - \left(\boldsymbol{h}\_{\mathsf{P}q}^{\boldsymbol{\ell}(\boldsymbol{\ell})}\right)^{\boldsymbol{\ell}}\right)^{\boldsymbol{\ell} + \boldsymbol{1}}\right)^{1/\boldsymbol{\ell}}, \left(\sum\_{q=1}^{n} \boldsymbol{w}\_{q\boldsymbol{\ell}}^{\boldsymbol{\ell}(\boldsymbol{\ell})}\right)^{\boldsymbol{\ell}}\right\rangle \right\}} - \underset{1 \leq \boldsymbol{j} \leq L\_{\mathsf{T}}}{\displaystyle}{\displaystyle} \left\{ \left. \left\langle \boldsymbol{h}\_{\mathsf{P}}^{\boldsymbol{\ell}(\boldsymbol{\ell})},\boldsymbol{w}\_{p}^{\boldsymbol{\ell}(\boldsymbol{\ell})}\right\rangle \right\}. \tag{12}$$

**Step 3**. Calculate the score functions <sup>Δ</sup>(*ωhp*)(*p* = 1, 2, ... , *m*) of the OWHFE *<sup>ω</sup>hp*(*p* = 1, 2, ... , *m*) for the supplier *xp*(*p* = 1, 2, . . . , *<sup>m</sup>*), that is,

$$\Delta(^{\omega}h\_p) = \sum\_{j=1}^{L\_p} h\_p^{\delta(j)} w\_p^{\delta(j)} \tag{13}$$

**Step 4**. Rank the score functions Δ(*ωhp*) in ascending order. Then, the supplier with the highest priority is the most desirable green supplier.

## **5. Numerical Example**

In light of the above discussion, we will further illustrate the procedure of the proposed method by an example of GSS. The GSCM of manufacturing enterprises is affected by its green suppliers' performance, and GSCM is considered as a strategic decision for manufacturing enterprises to maintain a competitive advantage in the international market. Inspired by the advantages of GSCM, there is a bus manufacturing enterprise who wants to choose the most appropriate green supplier for purchasing the key components of its new bus equipment. After initial screening, five potential suppliers *xi* (*i* = 1, 2, 3, 4, 5) have been determined for further assessment. In order to choose the most suitable supplier,

the company established a team of six DMs *eu*(*u* = 1, 2, ... , 6) from the department of purchasing, quality, and production who have abundant knowledge and experience in GSCM. Finally, four criteria are chosen from the Table 1 criteria list by experts to evaluate possible green suppliers. The four selected criteria are quality (*c*1), technology(*c2*), environment(*c3*), cost(*c4*), and the priority relationship among the criteria is *c*1 *c*2 *c*3 *c*4 in the evaluation process. For a supplier under a criterion, six DMs need to give their evaluation values. As an instance, for the supplier *x*1 under the criterion *c*1, the evaluation values 0.3, 0.5, and 0.8 are provided by two, one and three DMs, respectively, and then an OWHFE *<sup>ω</sup>h*11 can be represented by {<0.3,2/6>,<0.5,1/6>,<0.8,3/6>}.

In the same manner, all of OWHFEs *<sup>ω</sup>hpq*(*p* = 1, 2, ... , 5, *q* = 1, 2, 3, 4) can be obtained, as shown in Table 2.


**Table 2.** Ordered weighted hesitant fuzzy decision matrix.

**Step 1**. According to Equation (11), *Tpq*(*p* = 1, 2, . . . , 5, *q* = 1, 2, 3, 4) are calculated as follows:


**Step 2**. Aggregate *<sup>ω</sup>hpq*(*p* = 1, 2, ... , 5, *q* = 1, 2, 3, 4) by using a GOWHFPWA (*α* = 1) operator to derive the overall OWHFEs *<sup>ω</sup>hp*(*p* = 1, 2, . . . , 5) for the supplier *xp*(*p* = 1, 2, . . . , <sup>5</sup>).

> *<sup>ω</sup>h*1 = {< 0.3091, 2/6 >, < 0.5462, 1/6 >, < 0.7470, 3/6 >} *<sup>ω</sup>h*2 = {< 0.1305, 2/6 >, < 0.3755, 1/6 >, < 0.4973, 3/6 >} *<sup>ω</sup>h*3 = {< 0.1000, 2/6 >, < 0.2000, 1/6 >, < 0.3190, 3/6 >} *<sup>ω</sup>h*4 = {< 0.2514, 2/6 >, < 0.3866, 1/6 >, < 0.6654, 3/6 >} *<sup>ω</sup>h*5 = {< 0.5703, 2/6 >, < 0.7163, 1/6 >, < 0.8233, 3/6 >}

**Step 3**. Calculate the score functions <sup>Δ</sup>(*ωhp*)(*p* = 1, 2, ... , 5) of the OWHFEs *<sup>ω</sup>hp*(*p* = 1, 2, ... , 5) for the supplier *xp*(*p* = 1, 2, . . . , <sup>5</sup>), that is,

$$
\Delta(\,^\omega h\_2) = 0.5676,\\
\Delta(\,^\omega h\_2) = 0.3547,\\
\Delta(\,^\omega h\_3) = 0.2262,\\
\Delta(\,^\omega h\_4) = 0.4809,\\
\Delta(\,^\omega h\_5) = 0.7211
$$

**Step 4**. Rank all the suppliers *xp*(*p* = 1, 2, ... , 5) in accordance with the score functions <sup>Δ</sup>(*ωhp*)(*p* = 1, 2, . . . , 5) and the priority relationship of five suppliers can be obtained, that is,

$$\mathbf{x\_5 \succ x\_1 \succ x\_4 \succ x\_2 \succ x\_3}$$

Thus, the most desirable green supplier is *x*5.

#### **6. Performance Analysis and Comparation Analysis**

In this section, performance analysis is provided based on the numerical example above to prove the validation and verification of the proposed method, including sensitivity analysis and effectiveness analysis. Additionally, the proposed GOWHFPWA operator is further compared with the hesitant fuzzy prioritized weighted average (HFPWA) operator suggested by Wei [61].

The sensitivity analysis is used to identify and determine the robustness of the proposed method. In Equation (2), the parameter α may affect the final ranking result, so the sensitivity analysis can be carried out by taking different α. The score functions Δ(*ωhp*) with different α can be calculated, and all of the results are presented in Table 3 and Figure 1.

**Table 3.** The results of the generalized ordered weighted hesitant fuzzy prioritized weighted average operator (GOWHFPWA) operator with different α.


**Figure 1.** The curve of the score function with different *α.*

It can be seen from Table 3 that as parameter *α* takes different values, the priority relationships of five suppliers are unchanged and the most desirable supplier is still *x*5. Therefore, the parameter *α* is insensitive to the proposed method and the obtained result ranking is robustness.

Meanwhile, it can be observed from Figure 1 that the values of the score function for each alternative will increase as *α* increases. From this point of view, the parameter *α* can be regarded as a DM's risk attitude. As the DMs can select different *α* in accordance with their own risk preferences, the proposed GOWHFPWA operator can offer more choice opportunities for the DMs in the actual GSS problems.

Additionally, since we proposed the GOWHFPWA operator based on the HFPWA operator [61], a comparative analysis was conducted in order to illustrate the effectiveness of the proposed GOWHFPWA operator. For convenience of comparison, we apply the HFPWA operator to the above numerical example in this paper. The hesitant fuzzy decision matrix is shown in Table 4.


**Table 4.** Hesitant fuzzy decision matrix.

Then *tpq*(*p* = 1, 2, . . . , 5, *q* = 1, 2, 3, 4) are calculated as follows:


We aggregate all hesitant fuzzy elements *hpq*(*p* = 1, 2, ... , 5, *q* = 1, 2, 3, 4) by using the HFPWA operator to derive the overall hesitant fuzzy elements *hp*(*p* = 1, 2, ... , 5) of the suppliers *xp*(*p* = 1, 2, ... , <sup>5</sup>). Taking supplier *x*1 as an example, we have *h*1 = *HFPWA*(*h*11, *h*12, *h*13, *h*14) = {0.3083,0.3179,0.3295,0.3620,0.3709,0.3816,0.3879,0.3965,0.4067,0.4055,0.4138,0.4238,0.4517,0.4593,0.4685, 0.4740,0.4813,0.4902,0.4501,0.4578,0.4670,0.4928,0.4998,0.5084,0.5134,0.5202,0.5284,0.4169,0.4250,0.4348, 0.4622,0.4697,0.4787,0.4841,0.4912,0.4999,0.4989,0.5059,0.5143,0.5378,0.5442,0.5520,0.5566,0.5628,0.5702, 0.5365,0.5429,0.5507,0.5724,0.5784,0.5856,0.5898,0.5956,0.6024,0.6338,0.6389,0.6451,0.6623,0.6670,0.6727, 0.6760,0.6805,0.6860,0.6853,0.6897,0.6950,0.7098,0.7138,0.7187,0.7216,0.7255,0.7301,0.7089,0.7130,0.7179, 0.7315,0.7353,0.7398,0.7424,0.7460,0.7504}.

The scores *s*(*hp*)(*p* = 1, 2, ... , 5) of the suppliers *xp*(*p* = 1, 2, ... , 5) are obtained as the following: *<sup>s</sup>*(*h*1) = 0.5539, *<sup>s</sup>*(*h*2) = 0.3408, *<sup>s</sup>*(*h*3) = 0.2080, *<sup>s</sup>*(*h*4) = 0.4524, *<sup>s</sup>*(*h*5) = 0.7174. Finally, ranking all the suppliers *xp*(*p* = 1, 2, ... , 5) according to the scores *s*(*hp*)(*p* = 1, 2, ... , <sup>5</sup>), we can ge<sup>t</sup> the priority relationship of six suppliers, that is,

$$\mathbf{x\_5 \succ x\_1 \succ x\_4 \succ x\_2 \succ x\_3}$$

Thus, the most desirable supplier by using the HFPWA operator proposed by Wei [61] is also *x*5. The comparative results can be shown in Table 5.


**Table 5.** The result of different approaches.

From Table 5, despite the evaluation result obtained by using the HFPWA operator being the same as that of the GOWHFPWA operator, the proposed method has some advantages over the previous method. Firstly, the proposed method in this paper extends a prioritized weighted average operator from HFS to OWHFS which can solve the problem of the importance of the experts' evaluation results that the previous method cannot solve. Secondly, the computational complexity of the proposed approach is much lower than that of the previous method. Therefore, the introduced model for GSS in practice is more objective and reasonable than that obtained by using the HFPWA operator proposed by Wei [61].

#### **7. Conclusions and Further Directions**

In this paper, in order to overcome the limitation of MCGDM problems with GSS in practice, we have focused on a novel MCGDM approach with a priority relationship under the ordered weighted hesitant fuzzy environment to evaluate green suppliers, which can present the importance of each DM's judgment. Firstly, based on the ideal of the PA operator and HFPWA operator, the OWHFPWA operator was introduced and the prominent characteristics of the propose operator were studied. Secondly, we have utilized the OWHFPWA operator to develop MCGDM approaches to solve the GSS problem. Finally, a practical example of GSS in bus manufacturing enterprise was given to verify the practicality of the proposed method, meanwhile, its feasibility and effectiveness in dealing with MCGDM problems was carried out by the performance analysis and comparative analysis.

In future research, we will develop another hesitant fuzzy prioritized aggregation operator to solve the ordered weighted hesitant fuzzy MCGDM for GSS problems, namely, the generalized ordered weighted hesitant fuzzy prioritized weighted geometric (GOWHFPWG) operator. Moreover, we will combine the expanded hesitant fuzzy set (EHFS) [67] with the PA operator to deal with the MCDGM for GSS problems for future research, which take into account that a single DM gives several hesitant fuzzy elements in MCDGM problems.

**Author Contributions:** Authors Y.L. and L.J. conceived and designed the model for research, pre-processed and analyzed the data and the obtained inference. Anthors F.Z. and L.J. processed the data collected and wrote the paper. The final manuscript has been read and approved by all authors.

**Funding:** The work is supported by the National Natural Science Foundation of China (NSFC) under Projects 71672182, 71711540309, U1604262 and 71272207.

**Acknowledgments:** The authors are grateful to the editors and the anonymous reviewers for providing us with insightful comments and suggestions throughout the revision process.

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