**2. Preliminaries**

### *2.1. Pythagorean Fuzzy Set*

The fundamental definition of PFSs [40,41] are briefly introduced in this section. Then, novel score and accuracy functions of Pythagorean fuzzy numbers (PFNs) are developed. Furthermore, the comparison laws of PFNs are proposed.

**Definition 1** *[40,41]***.** *Let X be a fixed set. A Pythagorean fuzzy set (PFS) is an object which can be denoted as*

$$P = \{ \langle \mathbf{x}, (\alpha p(\mathbf{x}), \beta p(\mathbf{x})) \rangle | \mathbf{x} \in X \} \tag{1}$$

*where the function* α*P* : *X* → [0, 1] *indicates the degree of membership and the function* β*P* : *X* → [0, 1] *indicates the degree of non-membership of the element x* ∈ *X to P, respectively, and, for each x* ∈ *X, it holds that*

$$\left(\alpha\_{\mathcal{P}}(\mathbf{x})\right)^{2} + \left(\beta\_{\mathcal{P}}(\mathbf{x})\right)^{2} \le 1\tag{2}$$

*Mathematics* **2019**, *7*, 344

**Definition 2** *[42]***.** *Assume that p*1 = (<sup>α</sup>1, β1) *, p*2 = (<sup>α</sup>2, β2)*, and p* = (<sup>α</sup>, β) *are three Pythagorean fuzzy numbers (PFNs). Then, some basic operation laws of them can be expressed as:*

$$\begin{aligned} (1) \ p\_1 \oplus p\_2 &= \left(\sqrt{\left(\alpha\_{p\_1}\right)^2 + \left(\alpha\_{p\_2}\right)^2 - \left(\alpha\_{p\_1}\right)^2 \left(\alpha\_{p\_1}\right)^2}, \beta\_{p\_1}\beta\_{p\_2}\right); \\ (2) \ p\_1 \otimes p\_2 &= \left(\alpha\_{p\_1}\alpha\_{p\_2}, \sqrt{\left(\beta\_{p\_1}\right)^2 + \left(\beta\_{p\_2}\right)^2 - \left(\beta\_{p\_1}\right)^2 \left(\beta\_{p\_2}\right)^2}\right); \\ (3) \ \lambda p &= \left(\sqrt{1 - \left(1 - \alpha^2\right)^{\lambda}}, \beta^{\lambda}\right), \lambda > 0; \\ (4) \ (p)^{\lambda} &= \left(\alpha^{\lambda}, \sqrt{1 - \left(1 - \beta^2\right)^{\lambda}}\right), \lambda > 0; \\ (5) \ p^{\varepsilon} &= \left(\beta, \alpha\right). \end{aligned}$$

**Example 1.** *Assume that p*1 = (0.5, 0.7)*, p*2 = (0.3, 0.4)*, and p* = (0.6, 0.3) *are three Pythagorean fuzzy numbers (PFNs). Suppose* λ = 3*. Then, according to the above operation laws, we can obtain:*

$$(1) \; p\_1 \oplus p\_2 = \left(\sqrt{(0.5)^2 + (0.3)^2 - (0.5)^2 \times (0.3)^2}, 0.7 \times 0.4\right) = (0.56, 0.28);$$

$$(2) \; p\_1 \otimes p\_2 = \left(0.5 \times 0.3, \sqrt{\left(0.7\right)^2 + \left(0.4\right)^2 - \left(0.7\right)^2 \times \left(0.4\right)^2}\right) = (0.15, 0.64);$$

$$(3) \; 3 \times p = \left(\sqrt{1 - \left(1 - 0.6^2\right)^3}, 0.3^3\right) = (0.8590, 0.0270);$$

$$(4) \; (p)^3 = \left(0.6^3, \sqrt{1 - \left(1 - 0.3^2\right)^3}\right) = (0.4964, 0.2160);$$

$$(5) \; p^r = (0.3, 0.6).$$

### *2.2. Dual Hesitant Pythagorean Fuzzy Set*

In this section, we shall introduce the basic definition of the dual hesitant Pythagorean fuzzy set (DHPFS), which is the generalization of the PFS [40,41] and the dual hesitant fuzzy set (DHFS) [51,52]. It is obvious that the DHPFSs consist of two parts, namely, the function of membership hesitancy and the function of non-membership hesitancy, which support more exemplary and flexible access to assigning values for each element in the domain, meaning we have to handle two kinds of hesitancy in this situation.

**Definition 3** *[53]***.** *Assume that X is a fixed set. Then, a dual hesitant Pythagorean fuzzy set (DHPFS) on X can be developed as*

$$\overline{P} = \left( \{ \mathbf{x}, h\_{\overline{P}}(\mathbf{x}), g\_{\overline{P}}(\mathbf{x}) \} | \mathbf{x} \in X \right) \tag{3}$$

*in which hP*8(*x*) *and gP*8(*x*) *are two sets of some values in* [0, 1]*, indicating that the function of membership degrees and non-membership degrees of the element x* ∈ *X to the set P, respectively, satisfies the condition* 8

$$a^2 + \beta^2 \le 1$$

*where* α ∈ *hP*8(*x*), β ∈ *gP*8(*x*)*, for all x* ∈ *X. For convenience, the pair* 8*p*(*x*) = *<sup>h</sup>*<sup>8</sup>*p*(*x*), *<sup>g</sup>*<sup>8</sup>*p*(*x*)*is called a dual hesitant Pythagorean fuzzy number (DHPFN) denoted by* 8*p* = (*h*, *g*)*, with the conditions* α ∈ *h*, β ∈ *g ,* 0 ≤ α, β ≤ 1, 0 ≤ α<sup>2</sup> + β2 ≤ 1.

**Definition 4** *[53]***.** *Let* 8*p* = (*h*, *g*) *be a DHPFN. Then, s*(<sup>8</sup>*p*) = 12 1 + 1#*h*α∈*<sup>h</sup>* α<sup>2</sup> − 1#*g*β<sup>∈</sup>*<sup>g</sup>* β2 *is the score function of* 8*p, and <sup>H</sup>*(<sup>8</sup>*p*) = 1#*h*α∈*<sup>h</sup>* α<sup>2</sup> + 1#*g*β<sup>∈</sup>*<sup>g</sup>* β2 *is the accuracy function of* 8*p, where* #*h and* #*g are the numbers of the elements in h and g respectively. Then, let* 8*pi* = (*hi*, *gi*)(*<sup>i</sup>* = 1, 2) *be any two DHPFNs. Subsequently, we have the following comparison laws:*

	- *(1) If p*(<sup>8</sup>*p*1) = *p*(<sup>8</sup>*p*2)*, then* 8*p*1 *is equivalent to* 8*p*<sup>2</sup>*, denoted by* 8*p*1 ∼ 8*p*2 *;*
	- *(2) If p*(<sup>8</sup>*p*1) > *p*(<sup>8</sup>*p*2)*, then* 8*p*1 *is superior to* 8*p*<sup>2</sup>*, denoted by* 8*p*1 8*p*2.

**Definition 5** *[53]***.** *Assume that* 8*p*1 = (*h*1, *g*1) *,* 8*p*2 = (*h*2, *g*2)*, and* 8*p* = (*h*, *g*) *are three DHPFNs. Then, some basic operation laws of these can be expressed as:*

$$\begin{split} (1) \quad & \overline{p}^{\lambda} = \cup\_{\alpha \in h, \theta \in \mathcal{g}} \left\{ \left\{ \alpha^{\lambda} \right\}, \left\{ \sqrt{1 - \left(1 - \beta^{2}\right)^{\lambda}} \right\} \right\}, \lambda > 0; \\ (2) \quad & \lambda \overline{p} = \cup\_{\alpha \in h, \theta \in \mathcal{g}} \left\{ \left\{ \sqrt{1 - \left(1 - \alpha^{2}\right)^{\lambda}} \right\}, \left\{ \beta^{\lambda} \right\} \right\}, \lambda > 0; \\ (3) \quad & \overline{p\_{1}} \otimes \overline{p\_{2}} = \cup\_{\alpha\_{1} \in h\_{1}, \alpha\_{2} \in h\_{2}, \theta\_{1} \in \mathcal{g}\_{1}, \theta\_{2} \in \mathcal{g}\_{2}} \left\{ \left\{ \sqrt{(\alpha\_{1})^{2} + (\alpha\_{2})^{2} - (\alpha\_{1})^{2}(\alpha\_{2})^{2}} \right\}, \left\{ \beta\_{1} \beta\_{2} \right\} \right\}; \\ (4) \quad & \overline{p\_{1}} \otimes \overline{p\_{2}} = \cup\_{\alpha\_{1} \in h\_{1}, \alpha\_{2} \in h\_{2}, \theta\_{1} \in \mathcal{g}\_{1}, \theta\_{2} \in \mathcal{g}\_{2}} \left\{ \left( \alpha\_{1} \alpha\_{2} \right), \left\{ \sqrt{(\beta\_{1})^{2} + (\beta\_{2})^{2} - (\beta\_{1})^{2}(\beta\_{2})^{2}} \right\} \right\}. \end{split}$$

**Example 2.** *Assume that p*1 = {{0.7}, {0.3}}*, p*2 = {{0.1, 0.2}, {0.4}}*, and p* = {{0.5, 0.6}, {0.4}} *are three Pythagorean fuzzy numbers. Suppose* λ = 3*. Then, according to the above operation laws, we can obtain*

(1) 8*p*3 = <sup>∪</sup>α∈*h*,β<sup>∈</sup>*g*6-0.53, 0.6<sup>3</sup>, 6 1 − (1 − 0.4<sup>2</sup>)<sup>3</sup>77 = {{0.125, 0.216}, {0.638}}; (2) <sup>3</sup><sup>8</sup>*p* = <sup>∪</sup>α∈*h*,β<sup>∈</sup>*<sup>g</sup>* ⎧⎪⎪⎪⎨⎪⎪⎪⎩⎧⎪⎪⎪⎨⎪⎪⎪⎩ 1 − (1 − 0.5<sup>2</sup>)3, 1 − (1 − 0.6<sup>2</sup>)<sup>3</sup> ⎫⎪⎪⎪⎬⎪⎪⎪⎭,-0.43⎫⎪⎪⎪⎬⎪⎪⎪⎭ = {{0.760, 0.859}, {0.064}}; (3) 8 *p*1 ⊕ 8 *p*2 = <sup>∪</sup><sup>α</sup>1∈*h*1,α2∈*h*2,β1<sup>∈</sup>*g*1,β2<sup>∈</sup>*g*2 ⎧⎪⎨⎪⎩⎧⎪⎨⎪⎩ √0.7<sup>2</sup> + 0.1<sup>2</sup> − 0.7<sup>2</sup> × 0.12, √0.7<sup>2</sup> + 0.2<sup>2</sup> − 0.7<sup>2</sup> × 0.2<sup>2</sup> ⎫⎪⎬⎪⎭, {0.3 × 0.4}⎫⎪⎬⎪⎭ = {{0.704, 0.714}, {0.120}} (4) 8 *p*1 ⊗ 8 *p*2 = <sup>∪</sup><sup>α</sup>1∈*h*1,α2∈*h*2,β1<sup>∈</sup>*g*1,β2<sup>∈</sup>*g*2 -{0.7 × 0.1, 0.7 × 0.2},- √0.3<sup>2</sup> + 0.4<sup>2</sup> − 0.3<sup>2</sup> × 0.4<sup>2</sup> = {{0.070,0.140}, {0.485}}

### *2.3. The Heronian Mean Operator*

**Definition 6** *[65]***.** *Let bi* (*i* = 1, 2, ··· , *n*) *be a group of nonnegative real numbers. Then, the Heronian mean (HM) operator can be defined as:*

$$\text{HM}(b\_1, b\_2, \dots, b\_n) = \frac{2}{n(n+1)} \sum\_{i=1}^{n} \sum\_{j=i}^{n} \left( b\_i b\_j \right)^{\frac{1}{2}} \tag{4}$$

**Definition 7** *[54]***.** *Assume that* ξ, ζ > 0*, and bi* (*i* = 1, 2, ··· , *n*) *are a group of nonnegative real numbers. Then, the GHM operator can be defined as:*

$$\text{GHM}^{\xi,\zeta}(a\_1, a\_2, \dots, a\_n) = \left(\frac{2}{n(n+1)} \sum\_{i=1}^n \sum\_{j=i}^n a\_i^{\xi} a\_j^{\zeta}\right)^{1/(\xi+\zeta)}\tag{5}$$

*When* ξ = ζ = 1/2*, the GHM operator will reduce to the Heronian mean (HM) operator, which indicates that the HM operator is a special case of the GHM operator.*

### **3. Dual Hesitant Pythagorean Fuzzy Heronian Mean Operators**

In the following section Xu et al. [66] proposed the dual hesitant Pythagorean fuzzy generalized Heronian mean (DHPFGHM) operators based on dual hesitant Pythagorean fuzzy numbers (DHPFNs) and GHM operations. In addition, some important properties, such as idempotency, boundedness, and monotonicity are discussed.

### *3.1. The DHPFGHM Aggregation Operator*

**Definition 8** *[66]***.** *Let* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a group of DHPFNs. Then, we can define the DHPFGHM operator as*

$$\text{DHPFRGHM}^{\xi,\zeta}(\widetilde{p\_1}, \widetilde{p\_2}, \dots, \widetilde{p\_n}) = \left(\frac{2}{n(n+1)} \underset{i=1}{\stackrel{n}{\underset{i=1}{\oplus}}} \underset{j=i}{\oplus} \left(\overline{p}\_i^{\xi} \otimes \overline{p}\_j^{\zeta}\right)\right)^{\frac{1}{\xi+\zeta}}\tag{6}$$

*where* " ⊕ " *indicates the addition operation law and* " ⊗ " *indicates the multiplication operation law of the DHPFNs described in Definition 5. Then, according to these operation laws, Xu et al. [66] obtained Theorem 1.*

**Theorem 1** [66]**.** *Let* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a group of dual hesitant Pythagorean fuzzy numbers, meaning their fused results by utilizing the DHPFGHM operator is also a DHPFN, and*

$$\begin{split} & \text{DHPFGHM}^{\mathbb{L},\mathbb{L}}(\overline{p\_{1},\overline{p\_{2}},\ldots,\overline{p\_{n}}}) = \left( \frac{2}{n(n+1)} \sum\_{i=1}^{n} \prod\_{j=i}^{n} \left( \overline{p\_{i}^{\mathcal{L}}} \otimes \overline{p\_{j}^{\mathcal{L}}} \right) \right)^{\frac{1}{1-\zeta}} \\ &= \cup\_{a\_{i}\in\mathbb{h}\_{i},a\_{j}\in\mathbb{h}\_{j},\mathbb{K}\in\mathcal{G},\boldsymbol{\iota}\_{i}\in\mathbb{h}\_{j}} \left\{ \left( \sqrt{1-\prod\_{i=1,\boldsymbol{j}=i}^{n} \left(1-\alpha\_{i}^{2\mathcal{L}}\alpha\_{j}^{\mathcal{L}}\right)^{\frac{2}{n(n+1)}}} \right)^{\frac{1}{1-\zeta}} \right\}, \\ & \left\{ \sqrt{1-\left(1-\prod\_{i=1,\boldsymbol{j}=i}^{n} \left(1-\left(1-\rho\_{i}^{2}\right)^{\mathbb{L}}\left(1-\rho\_{j}^{2}\right)^{\mathbb{L}}\right)^{\frac{2}{n(n+1)}}} \right)^{\frac{1}{1-\zeta}} \right\}, \end{split} \tag{7}$$

**Proof.** Based on Definition 5:

$$\overrightarrow{p\_i^{\xi}} = \cup\_{a\_i \in h\_i \beta\_i \in \mathfrak{g}\_i} \left\{ \left| a\_i^{\xi} \right\rangle, \left\{ \sqrt{1 - \left( 1 - \beta\_i^2 \right)^{\xi}} \right\} \right\} \tag{8}$$

$$\overline{p}\_{\dot{j}}^{\zeta} = \cup\_{\alpha\_{\dot{f}} \in \mathbb{H}\_{\dot{f}}, \theta\_{\dot{f}} \in \mathfrak{F}\_{\dot{f}}} \left\{ \left\{ \alpha\_{\dot{j}}^{\zeta} \right\} \Big/ \sqrt{1 - \left( 1 - \beta\_{\dot{j}}^{2} \right)^{\zeta}} \right\} \tag{9}$$

*Mathematics* **2019**, *7*, 344

Thus,

$$
\overline{p}\_i^{\xi} \otimes \overline{p}\_j^{\zeta} = \cup\_{a\_i \in \mathcal{h}\_i, a\_j \in \mathcal{h}\_j, \beta\_i \in \mathcal{g}\_i, a\_j \in \mathcal{h}\_j} \left\{ \left( a\_i^{\xi} a\_j^{\zeta} \right), \left\{ \sqrt{1 - \left( 1 - \beta\_i^2 \right)^{\zeta} \left( 1 - \beta\_j^2 \right)^{\zeta}} \right\} \right\} \tag{10}
$$

Therefore,

$$\begin{aligned} & \; ^{\textrm{in}}\_{i=1}^{\textrm{in}} \; \Big( \overline{p}\_{i}^{\mathcal{E}} \otimes \overline{p}\_{j}^{\mathcal{E}} \Big) \\ &= \cup\_{a\_{i} \in h\_{i}, a\_{j} \in h\_{j}, \beta\_{i} \in \mathcal{G}, a\_{j} \in h\_{j}} \left\{ \begin{array}{l} \left\{ \sqrt{1 - \prod\_{i=1, j=i}^{n} \left( 1 - \alpha\_{i}^{2\mathcal{E}} \alpha\_{j}^{2\mathcal{E}} \right)} \right\}, \\ \left\{ \sqrt{\prod\_{i=1, j=i}^{n} \left( 1 - \left( 1 - \beta\_{i}^{2} \right)^{\mathbb{C}} \left( 1 - \beta\_{j}^{2} \right)^{\mathbb{C}}} \right)} \right\} \end{array} \right\} \end{aligned} \tag{11}$$

Furthermore,

$$=\begin{pmatrix} \frac{2}{n(n+1)} \stackrel{\text{in}}{\underset{i=1}{n}} \frac{\text{n}}{\sum\limits\_{j=1}^{n} \left\{\overline{p}\_{i}^{\xi} \circledast \overline{p}\_{j}^{\zeta}\right\}} \\\\ =\bigcup\_{\alpha\_{i} \in \mathbb{h}\_{i}, a\_{j} \in \mathbb{h}\_{j}, \theta\_{i} \in \xi\_{i}, a\_{j} \in \mathbb{h}\_{j}} \left\{ \begin{pmatrix} \sqrt{1 - \prod\limits\_{i=1, j=i}^{n} \left(1 - \alpha\_{i}^{2\xi} \alpha\_{j}^{2\zeta}\right)^{\frac{2}{n(n+1)}}} \\\\ \left\{\sqrt{\prod\limits\_{i=1, j=i}^{n} \left(1 - \left(1 - \beta\_{i}^{2}\right)^{\mathbb{L}} \left(1 - \beta\_{j}^{2}\right)^{\mathbb{L}}\right)^{\mathbb{L}}} \right\} \end{pmatrix} \end{pmatrix} \tag{12}$$

Therefore,

$$\begin{split} & \text{DHPFGHM}^{\xi,\overline{\zeta},\overline{\zeta}}(\overline{p\_{1}},\overline{p\_{2}},\ldots,\overline{p\_{n}}) = \left( \frac{2}{n(n+1)} \sum\_{i=1}^{n} \prod\_{j=i}^{n} \left( \overline{p\_{i}^{\xi}} \otimes \overline{p\_{j}^{\zeta}} \right) \right)^{\frac{1}{\xi+\zeta}} \\ &= \cup\_{a\_{i}\in\mathbb{N}\_{i},a\_{j}\in\mathbb{N}\_{j},\theta\_{i}\in\mathsf{g}\_{i},\alpha\_{j}\in\mathsf{h}\_{j}} \left\{ \left\{ \left( \sqrt{1-\prod\_{i=1,j=i}^{n} \left(1-\alpha\_{i}^{2\mathcal{L}}\alpha\_{j}^{2\mathcal{L}}\right)^{\frac{2}{n(n+1)}}}^{2} \right)^{\frac{1}{\xi+\zeta}} \right\}, \\ & \left\{ \sqrt{1-\left(1-\prod\_{i=1,j=i}^{n} \left(1-\left(1-\beta\_{i}^{2}\right)^{\mathcal{L}}\left(1-\rho\_{j}^{2}\right)^{\mathbb{C}}\right)^{\frac{2}{n(n+1)}}}^{2} \right)^{\frac{1}{\xi+\zeta}} \right\}, \end{split} \tag{13}$$

Thus, the proof has been finished.

**Example 3.** *Assume that* 8*p*1 = {{0.7, 0.8}, {0.4}}*,* 8*p*2 = {{0.3}, {0.6, 0.7}}*,* 8*p*3 = {{0.1, 0.3}, {0.4, 0.6}} *and* 8 *p*4 = {{0.5}, {0.5}} *are four DHPFNs, and suppose that* ξ = 2, ζ = 3*. Then according to the DHPFGHM operator, we can obtain the fused results as follows. For the membership degree function* α*, the fused results are shown as:*

$$\begin{split} &a\_{1} = \mathrm{DHPFGHM}^{2,3}(0.7,0.3,0.1,0.5) = \left(\sqrt{1-\prod\_{i=1,j=1}^{n}\left(1-\alpha\_{i}^{2\mathcal{L}}\alpha\_{j}^{\mathcal{L}}\right)^{2}}\right)^{\frac{1}{1+\mathcal{L}}} \\ &= \left\{ \sqrt{1-\left(\frac{1-0.7^{2\times2}\times0.7^{2\times3}\right)\times\left(1-0.7^{2\times2}\times0.3^{2\times3}\right)\times\left(1-0.7^{2\times2}\times0.1^{2\times3}\right)}} \times 10^{\frac{1}{1+\mathcal{L}}} \\ &= \left\{ \sqrt{1-\left(\frac{1-0.7^{2\times2}\times0.5^{2\times3}}{\times\left(1-0.3^{2\times2}\times0.5^{2\times3}\right)\times\left(1-0.1^{2\times2}\times0.1^{2\times3}\right)\times\left(1-0.1^{2\times2}\times0.5^{2\times3}\right)}\right)^{\frac{1}{1+\mathcal{L}}}} \\ &\quad \times \left\{ 1-0.3^{2\times2}\times0.5^{2\times3}\right\}\times\left(1-0.1^{2\times2}\times0.1^{2\times3}\right)\times\left(1-0.1^{2\times2}\times0.5^{2\times3}\right) \Bigg)^{\frac{1}{1+\mathcal{L}}} \\ &= 0.5658 \\ &= 0.5658 \end{split}$$

Similarly, we can obtain

> α2 = DHPFGHM2,3(0.7, 0.3, 0.3, 0.5) = 0.5664 α3 = DHPFGHM2,3(0.8, 0.3, 0.1, 0.5) = 0.6492 α4 = DHPFGHM2,3(0.8, 0.3, 0.3, 0.5) = 0.6432

Hence, we can ge<sup>t</sup> α = {0.5658, 0.5664, 0.6429, 0.6432}. For the non-membership degree function β, the fused results are shown as:

β1 = DHPFGHM2,3(0.4, 0.6, 0.4, 0.5) = 34451 − ⎛⎜⎜⎜⎜⎜⎝1 − 2*n <sup>i</sup>*=1,*j*=*<sup>i</sup>*&<sup>1</sup> − 1 − β2*i* ξ1 − β2*j*ζ' 2 *n*(*n*+<sup>1</sup>) ⎞⎟⎟⎟⎟⎟⎠ 1 ξ+ζ = 3444444444444444444444444444444444444444444444444444445⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 − ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup> × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.6<sup>2</sup><sup>3</sup> ×1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup> × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> ×1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.6<sup>2</sup><sup>3</sup> × 1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup> ×1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup> ×1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> × 1 − 1 − 0.5<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 110 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 1 2+3

= 0.4698

Similarly, we can obtain

 $\beta\_2 = \text{DHPFGHM}^{2,3}(0.4, 0.6, 0.6, 0.5) = 0.5211$ 
 $\beta\_3 = \text{DHPFGHM}^{2,3}(0.4, 0.7, 0.4, 0.5) = 0.4856$ 
 $\beta\_4 = \text{DHPFGHM}^{2,3}(0.4, 0.7, 0.6, 0.5) = 0.5383$ 

Hence, we can ge<sup>t</sup> β = {0.4698, 0.5211, 0.4856, 0.5383}.Therefore,

$$\text{IDHPFGHM}(\overline{p}\_1, \overline{p}\_2, \overline{p}\_3, \overline{p}\_4) = \left\{ \begin{array}{c} \{0.5658, 0.5664, 0.6429, 0.6432\}, \\ \{0.4698, 0.5211, 0.4856, 0.5383\} \end{array} \right\}.$$

It can be easily proven that the DHPFGHM operator satisfies the following properties.

**Property 1.** *(Idempotency) If all* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are equal, i.e.,* 8*pj* = 8*p for all j, then*

$$\text{DHPFGHM}^{\xi,\overline{\zeta}}(\overline{p}\_1, \overline{p}\_2, \dots, \overline{p}\_n) = \overline{p} \tag{14}$$

**Property 2.** *(Boundedness) Let* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a collection of DHPFNs, and let*

$$
\widehat{p}^+ = \cup\_{a\_{\widehat{\jmath}} \in \mathfrak{h}\_{\widehat{\jmath}} \theta\_{\widehat{\jmath}} \in \mathfrak{g}\_{\widehat{\jmath}}} \{ \langle \mathsf{max}\_{i}(a\_{i}) \rangle\_{\prime} \{ \mathsf{min}\_{i}(\beta\_{i}) \} \rangle\_{\prime} \widehat{p}^- \\
= \cup\_{a\_{\widehat{\jmath}} \in \mathfrak{h}\_{\widehat{\jmath}} \theta\_{\widehat{\jmath}} \in \mathfrak{g}\_{\widehat{\jmath}}} \{ \langle \mathsf{min}\_{i}(a\_{i}) \rangle\_{\prime} \{ \mathsf{max}\_{i}(\beta\_{i}) \} \}
$$

*Then*

$$
\widetilde{p}^- \le \text{DPPFGHM}^{\xi,\zeta}(\widetilde{p}\_1, \widetilde{p}\_2, \dots, \widetilde{p}\_n) \le \widetilde{p}^+ \tag{15}
$$

**Property 3.** *(Monotonicity) Let* 8*pj* = *hj*, *gj and* 8*p j* = *h j*, *g j*, *j* = 1, 2, ··· , *n*, *be two sets of DHPFNs. If* 8*pj* ≤ 8*p j, for all j, then*

$$\text{DHPFGHM}^{\mathbb{E},\mathbb{L}}(\widetilde{p}\_1, \widetilde{p}\_2, \dots, \widetilde{p}\_n) \le \text{DHPFGHM}^{\mathbb{E},\mathbb{L}}(\widetilde{p}\_1', \widetilde{p}\_2', \dots, \widetilde{p}\_n') \tag{16}$$

### *3.2. The DHPFGWHM Aggregation Operator*

Using Definition 8, we can conclude that the DHPFGHM operator didn't take the importance of arguments being fused into account. However, in many practical MADM problems, we should consider the weights of attributes. To overcome this limitation of the DHPFGHM operator, we propose a novel DHPFGWHM operator as follows.

**Definition 9.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a group of dual hesitant Pythagorean fuzzy numbers (DHPFNs). Then, we define the DHPFGWHM operator as follows:*

$$\text{DHPFGWHM}\_{w}^{\xi,\zeta}(\overline{p}\_{1}, \overline{p}\_{2}, \dots, \overline{p}\_{n}) = \left(\stackrel{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\Gamma}}}}}}}{\oplus} \stackrel{\text{\tiny{\tiny{\tiny{\text{\tiny{\Gamma}}}}}}}{\oplus} \left(w\_{l}w\_{\langle\rangle}(\overline{p}\_{i}^{\xi}\overline{p}\_{j}^{\zeta})\right)\right)^{\frac{1}{\xi+\zeta}}\tag{17}$$

*According to the operation laws of the DHPFNs described in Definition 5, we can obtain Theorem 2.*

**Theorem 2.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a collection of DHPFNs with the weighting vector w* = (*<sup>w</sup>*1, *w*2, ... , *wn*)*T, which satisfies wj* > 0*, i* = 1, 2, ... , *n and nj*=1 *wj* = 1*. Then, their fused result obtained by utilizing the DHPFGWHM operator is also a DHPFN, and*

$$\begin{split} \text{DHPFGWM}\_{\text{w}}^{\xi,\zeta}(\widetilde{p\_{1}},\widetilde{p\_{2}},\ldots,\widetilde{p\_{n}}) &= \left( \prod\_{i=1}^{n} \underset{j=i}{\oplus} \left( w\_{i} w\_{j} \middle| \left< \widetilde{p\_{i}}^{\xi} \widetilde{p\_{j}}^{\zeta} \right> \right) \right)^{\frac{1}{1-\zeta}} \\ &= \cup\_{a\_{i} \in \mathbb{N}\_{\boldsymbol{\nu}}, a\_{j} \in \mathbb{N}\_{\boldsymbol{\nu}}, \boldsymbol{\xi} \in \mathcal{S}\_{\boldsymbol{\nu}}, a\_{j} \in \mathbb{N}\_{\boldsymbol{\nu}}} \left\{ \left( \sqrt{1-\prod\_{i=1, j=i}^{n} \left( 1-\alpha\_{i}^{2\zeta} \alpha\_{j}^{2\zeta} \right)^{w\_{i}w\_{j}}} \right)^{\frac{1}{1-\zeta}} \right\}, \\ &= \cup\_{a\_{i} \in \mathbb{N}\_{\boldsymbol{\nu}}, a\_{j} \in \mathbb{N}\_{\boldsymbol{\nu}}, \boldsymbol{\xi} \in \mathcal{S}\_{\boldsymbol{\nu}}, a\_{j} \in \mathbb{N}\_{\boldsymbol{\nu}}. \end{split} \tag{18}$$

**Proof.** Based on Definition 5, we can obtain:

$$\overline{p}\_i^{\xi} = \cup\_{\alpha\_i \in \mathbb{N}\_i, \beta\_i \in \xi\_i} \left\{ \left| \alpha\_i^{\xi} \right|, \left\{ \sqrt{1 - \left( 1 - \beta\_i^2 \right)^{\xi}} \right\} \right\} \tag{19}$$

$$\overleftarrow{p}\_{j}^{\zeta} = \cup\_{\alpha\_{j} \in \mathbb{H}\_{j}, \beta\_{j} \in \mathbb{g}\_{j}} \left\{ \left\{ \alpha\_{j}^{\zeta} \right\}, \left\{ \sqrt{1 - \left( 1 - \beta\_{j}^{2} \right)^{\zeta}} \right\} \right\} \tag{20}$$

Thus,

$$\overrightarrow{p\_i^{\xi}}\overrightarrow{p\_j^{\xi}} = \cup\_{a\_i \in \mathsf{h}\_i, a\_j \in \mathsf{h}\_j, \theta\_i \in \mathsf{g}\_i, a\_j \in \mathsf{h}\_j} \left\{ \left< a\_i^{\xi} a\_j^{\zeta} \right> , \left< \sqrt{1 - \left(1 - \beta\_i^2\right)^{\xi} \left(1 - \beta\_j^2\right)^{\zeta}} \right> \right\} \tag{21}$$

*Mathematics* **2019**, *7*, 344

Therefore,

$$\begin{aligned} \forall w\_{i} w\_{j} \Big( \overline{p}\_{i}^{\varepsilon} \overline{p}\_{j}^{\varepsilon} \Big) \\ = \cup\_{\alpha\_{i} \in h\_{i}, a\_{j} \in h\_{j}, \theta\_{i} \in \mathcal{G}\_{i}, a\_{j} \in h\_{j}} \left\{ \begin{array}{l} \left\{ \sqrt{1 - \left( 1 - \alpha\_{i}^{2\mathcal{L}} \alpha\_{j}^{2\mathcal{L}} \right)^{w\_{i} w\_{j}}} \right\}\_{\prime} \\ \left\{ \left( \sqrt{1 - \left( 1 - \beta\_{i}^{2} \right)^{\mathcal{L}} \left( 1 - \beta\_{j}^{2} \right)^{\mathcal{L}}} \right)^{w\_{i} w\_{j}} \right\}\_{\prime} \end{array} \tag{22}$$

Thereafter,

$$\begin{aligned} \stackrel{\text{in}}{\underset{i=1}{n}} & \stackrel{\text{in}}{\underset{j=1}{\text{in}}} \left( w\_{i} w\_{j} \Big\langle \overline{p\_{i}^{\xi}} \overline{p\_{j}^{\zeta}} \Big\rangle \right) \\ &= \bigcup\_{a\_{i} \in \mathbb{H}\_{i}, a\_{j} \in \mathbb{H}\_{j}, \beta\_{i} \in \operatorname{\mathcal{G}}\_{i}, a\_{j} \in \operatorname{\mathcal{H}}\_{j}} \left\{ \begin{Bmatrix} \left\{ \sqrt{1 - \prod\_{i=1, j=i}^{n} \left( 1 - \alpha\_{i}^{2\zeta} \alpha\_{j}^{2\zeta} \right)^{w\_{i} w\_{j}}} \right\}, \\ \left\{ \left( \sqrt{\prod\_{i=1, j=i}^{n} \left( 1 - \left( 1 - \beta\_{i}^{2} \right)^{\zeta} \left( 1 - \beta\_{j}^{2} \right)^{\zeta}} \right) \right)^{w\_{i} w\_{j}} \right\} \end{Bmatrix} \end{aligned} \tag{23}$$

Therefore,

$$\begin{split} \text{DHPFGWHM}\_{w}^{\xi,\mathsf{f}}(\widetilde{p\_{1}},\widetilde{p\_{2}},\ldots,\widetilde{p\_{n}}) &= \left(\prod\_{i=1}^{n}\underset{j=i}{\underset{i=1}{\underset{i=1}{\underset{i=1}}}}\langle w\_{i}w\_{j}\middle|\widetilde{p\_{i}}\widetilde{p\_{j}}\right)\right)^{\frac{1}{1+\zeta}} \\ &= \cup\_{a\_{i}\in\mathsf{h}\_{i},a\_{j}\in\mathsf{h}\_{j},\theta\_{i}\in\mathsf{g},a\_{j}\in\mathsf{h}\_{j}} \left\{ \left\{ \left(\sqrt{1-\prod\_{i=1,j=i}^{n}\left(1-\alpha\_{i}^{2\mathcal{L}}\alpha\_{j}^{2}\right)^{w\_{i}w\_{j}}}\right)^{\frac{1}{1+\zeta}} \right\}, \\ &= \cup\_{a\_{i}\in\mathsf{h}\_{i},a\_{j}\in\mathsf{h}\_{j},\theta\_{i}\in\mathsf{g},a\_{j}\in\mathsf{h}\_{j}} \left\{ \left\{ \sqrt{1-\left(1-\prod\_{i=1,j=i}^{n}\left(1-\left(1-\beta\_{i}^{2}\right)^{\mathcal{L}}\left(1-\beta\_{j}^{2}\right)^{\mathbb{C}}\right)^{w\_{i}w\_{j}}\right)^{\frac{1}{1+\zeta}}} \right\} \end{split} \tag{24}$$

Thus, we have finished the proof.

**Example 4.** *Assume that* 8*p*1 = {{0.7, 0.8}, {0.4}}*,* 8*p*2 = {{0.3}, {0.6, 0.7}}*,* 8*p*3 = {{0.1, 0.3}, {0.4, 0.6}} *and* 8 *p*4 = {{0.5}, {0.5}} *are four DHPFNs, and suppose that* ξ = 2, ζ = 3 *and wj* = (0.3, 0.2, 0.1, 0.4)*. Then, according to the DHPFGWHM operator, we can obtain the fused results as follows. For the membership degree function* α*, the fused results are shown as:*

$$\begin{split} &a\_{1} = \text{DHPPGWM}^{2,3}(0.7,0.3,0.1,0.5) = \left(\sqrt{1-\int\_{1}^{\infty}\prod\_{i=1,j=1}^{n}\left(1-\alpha\_{i}^{2\mathcal{E}}\alpha\_{j}^{\mathcal{K}}\right)^{\mathbb{1}\times\mathbb{Q}}}\right)^{\frac{1}{1+\zeta}} \\ &= \left\{ \left[\begin{array}{cc} 1-\left(1-0.7^{2\times2}\times0.7^{2\times3}\right)^{0.3\times0.3}\times\left(1-0.7^{2\times2}\times0.3^{2\times3}\right)^{0.3\times0.2}\times\left(1-0.7^{2\times2}\times0.1^{2\times3}\right)^{0.3\times0.1} \\ \times\left(1-0.7^{2\times2}\times0.5^{2\times3}\right)^{0.3\times0.4}\times\left(1-0.3^{2\times2}\times0.3^{2\times3}\right)^{0.2\times0.2}\times\left(1-0.3^{2\times2}\times0.1^{2\times3}\right)^{0.2\times0.1} \\ \times\left(1-0.3^{2\times2}\times0.5^{2\times3}\right)^{0.2\times0.4}\times\left(1-0.1^{2\times2}\times0.1^{2\times3}\right)^{0.1\times0.1}\times\left(1-0.1^{2\times2}\times0.5^{2\times3}\right)^{0.1\times0.4} \\ \end{array}\right.\right\}^{\frac{1}{1+\zeta}} \\ &= 0.5630 \\ \pm 0.5630 \end{split}$$

Similarly, we can obtain

$$\begin{aligned} \alpha\_2 &= \text{DHPFGWHM}^{2,3}(0.7, 0.3, 0.3, 0.5) = 0.5632\\ \alpha\_3 &= \text{DHPFGWHM}^{2,3}(0.8, 0.3, 0.1, 0.5) = 0.6376\\ \alpha\_4 &= \text{DHPFGWHM}^{2,3}(0.8, 0.3, 0.3, 0.5) = 0.6377 \end{aligned}$$

Hence, we can ge<sup>t</sup> α = {0.5630, 0.5632, 0.6376, 0.6377}. For the non-membership degree function β, the fused results are shown as:

β1 = DHPFGWHM2,3(0.4, 0.6, 0.4, 0.5) = 3451 − ⎛⎜⎜⎜⎜⎝1 − 2*n <sup>i</sup>*=1,*j*=*<sup>i</sup>*&<sup>1</sup> − 1 − β2*i* ξ1 − <sup>β</sup><sup>2</sup>*j*<sup>ζ</sup>'*wiwj*⎞⎟⎟⎟⎟⎠ 1 ξ+ζ = 3444444444444444444444444444444444444444444444444444444445⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 − 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup>0.3×0.3 × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.6<sup>2</sup><sup>3</sup>0.3×0.2 ×1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup>0.3×0.1 × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.3×0.4 ×1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.6<sup>2</sup><sup>3</sup>0.2×0.2 × 1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup>0.2×0.1 ×1 − 1 − 0.6<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.2×0.4 × 1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.4<sup>2</sup><sup>3</sup>0.1×0.1 ×1 − 1 − 0.4<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.1×0.4 × 1 − 1 − 0.5<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.4×0.4 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 1 2+3

= 0.5333

Similarly, we can obtain

 $\beta\_2 = \text{DHPFGWHM}^{2,3}(0.4, 0.6, 0.6, 0.5) = 0.5480$ 
 $\beta\_3 = \text{DHPFGWHM}^{2,3}(0.4, 0.7, 0.4, 0.5) = 0.5438$ 
 $\beta\_4 = \text{DHPFGWHM}^{2,3}(0.4, 0.7, 0.6, 0.5) = 0.5586$ 

Hence, we can ge<sup>t</sup> β = {0.5333, 0.5480, 0.5438, 0.5586}. Therefore,

$$\text{DHPFGWM}(\overleftarrow{p}\_1, \overleftarrow{p}\_2, \overleftarrow{p}\_3, \overleftarrow{p}\_4) = \left\{ \begin{array}{l} \langle 0.5630, 0.5632, 0.6376, 0.6377 \rangle, \\ \langle 0.5333, 0.5480, 0.5438, 0.5586 \rangle \end{array} \right\}.$$

It can be easily proven that the DHPFGWHM operator satisfies the following properties.

**Property 4.** *(Idempotency) If all* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are equal, i.e.,* 8*pj* = 8*p for all j, then*

$$\text{DHPFGWHM}\_{w}^{\xi,\zeta}(\overline{p}\_{1}, \overline{p}\_{2}, \dots, \overline{p}\_{n}) = \overline{p} \tag{25}$$

**Property 5.** *(Boundedness) Let* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a collection of DHPFNs, and let*

$$
\widehat{p}^+ = \cup\_{a\_j \in h\_j \theta\_j \in \mathfrak{g}\_j} \{ \langle \max\_i(a\_i) \rangle, \{ \min\_i(\beta\_i) \} \rangle, \widehat{p}^- = \cup\_{a\_j \in h\_j \theta\_j \in \mathfrak{g}\_j} \{ \langle \min\_i(a\_i) \rangle, \{ \max\_i(\beta\_i) \} \}
$$

*Then*

$$
\overleftarrow{p}^{-} \le \text{DHPFGWM}\_{w}^{\mathbb{E},\mathbb{L}}(\overleftarrow{p}\_{1}, \overleftarrow{p}\_{2}, \dots, \overleftarrow{p}\_{n}) \le \overleftarrow{p}^{+} \tag{26}
$$

**Property 6.** *(Monotonicity) Let* 8*pj* = *hj*, *gj and* 8*p j* = *h j*, *g j*, *j* = 1, 2, ··· , *n*, *be two sets of DHPFNs. If* 8 *pj* ≤ 8 *p j, for all j, then*

$$\text{DHPFGWMM}\_{w}^{\xi,\zeta}(\overline{p}\_{1}, \overline{p}\_{2}, \dots, \overline{p}\_{n}) \leq \text{DHPFGWMM}\_{w}^{\xi,\zeta}(\overline{p}\_{1}^{\prime}, \overline{p}\_{2^{\prime}}^{\prime}, \dots, \overline{p}\_{n}^{\prime}) \tag{27}$$

### *3.3. The DHPFGGHM Aggregation Operator*

In the following, based on the geometric mean (GM) operator, Yu [54] extended the GHM operator to a GGHM operator which can be depicted as follows.

**Definition 10** *[54]***.** *Assume that* ξ, ζ > 0 *and bi*(*i* = 1, 2, ··· , *n*) *are a group of non-negative real numbers. Then, the generalizeGGHM) operator can be expressed as:*

$$\text{GHM}^{\xi,\zeta}(a\_1, a\_2, \dots, a\_n) = \frac{1}{\xi + \zeta} \left( \prod\_{i=1, j=i}^n \left( \xi a\_i + \zeta a\_j \right) \right)^{\frac{2}{n(n+1)}} \tag{28}$$

In this section, we introduced the GGHM operator with dual hesitant Pythagorean fuzzy information. According to Definition 5, Xu et al. [66] gave the definition of the DHPFGGHM operator as follows.

**Definition 11** *[66]***.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a collection of DHPFNs. Then, the DHPFGGHM operator can be defined as:*

$$\text{DHPFGGHM}^{\xi,\zeta}(\overline{p}\_1, \overline{p}\_2, \dots, \overline{p}\_n) = \frac{1}{\xi + \zeta} \left( \bigotimes\_{i=1}^n \bigotimes\_{j=1}^n \left( \xi \overline{p}\_i \oplus \zeta \overline{p}\_j \right) \right)^{\frac{2}{n(n+1)}} \tag{29}$$

*According to the operation laws of the DHPFNs described in Definition 5, Xu et al. [66] obtained Theorem 3.*

**Theorem 3** [66]**.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a group of DHPFNs. Then, their fused results obtained by utilizing the DHPFGGHM operator is also a DHPFN, and*

$$\begin{split} & \mathbb{E} \| \text{DHPGGHM}^{\xi,\zeta} (\overline{p}\_1, \overline{p}\_2, \dots, \overline{p}\_n) = \frac{1}{\xi + \zeta} \left( \sum\_{i=1}^n \frac{n}{\zeta} \left( \zeta \overline{p}\_i \oplus \zeta \overline{p\_j} \right) \right)^{\frac{2}{\overline{n}(n+1)}} \\ & = \mathbb{U}\_{a\_i \in \mathbb{H}\_i, a\_j \in \mathbb{H}\_j, \beta\_i \in \mathbb{g}, a\_j \in \mathbb{H}\_j} \left\{ \left( \sqrt{1 - \left( 1 - \prod\_{i=1, j=i}^n \left( 1 - \left( 1 - \alpha\_i^2 \right)^{\mathbb{E}} \left( 1 - \alpha\_j^2 \right) \right)^{\frac{2}{\overline{n}(n+1)}} \right)^{\frac{2}{\overline{n}(n+1)}}} \right)^{\frac{1}{\overline{\xi} + \zeta}} \right\}, \\ & \left\{ \left( \sqrt{1 - \prod\_{i=1, j=i}^n \left( 1 - \beta\_i^{2\mathcal{L}} \beta\_j^{2\prime} \right)^{\frac{2}{\overline{n}(n+1)}}} \right)^{\frac{1}{\overline{\xi} + \zeta}} \right\} \end{split} \tag{30}$$

**Proof.** Based on Definition 5:

$$\xi \overleftarrow{p\_i} = \cup\_{a\_i \in h\_i, \theta\_i \in \mathfrak{g}\_i} \left\{ \left\{ \sqrt{1 - \left( 1 - \alpha\_i^2 \right)^\xi} \right\} \left\{ \beta\_i^\xi \right\} \right\} \tag{31}$$

$$\zeta \overleftarrow{p\_j} = \cup\_{a\_j \in h\_j, \theta\_j \in \mathfrak{g}\_j} \left\{ \left\{ \sqrt{1 - \left( 1 - a\_j^2 \right)^\zeta} \right\} \left\{ \beta\_j^\zeta \right\} \right\} \tag{32}$$

*Mathematics* **2019**, *7*, 344

Thus,

$$\zeta \widetilde{p\_i} \oplus \zeta \widetilde{p\_j} = \cup\_{a\_i \in \mathbb{h}\_i, a\_j \in \mathbb{h}\_j, \beta\_i \in \mathbb{g}\_i, a\_j \in \mathbb{h}\_j} \left\{ \left( \sqrt{1 - \left( 1 - \alpha\_i^2 \right)^{\mathbb{\zeta}} \left( 1 - \alpha\_j^2 \right)^{\mathbb{\zeta}}} \right), \left\{ \beta\_i^{\mathbb{\zeta}} \beta\_j^{\mathbb{\zeta}} \right\} \right\} \tag{33}$$

Therefore,

$$\begin{aligned} & \left\{ \begin{array}{l} \stackrel{\text{a}}{\otimes} \stackrel{\text{b}}{\otimes} \left( \xi \overleftarrow{p\_{i}} \oplus \zeta \overleftarrow{p\_{j}} \right) \\\\ = & \cup\_{a\_{i} \in h\_{i}, a\_{j} \in h\_{j}, \beta\_{i} \in \xi\_{i}, a\_{j} \in h\_{j} \end{array} \right\} \left\{ \left\{ \sqrt{\prod\_{i=1, j=i}^{n} \left( 1 - \left( 1 - \alpha\_{i}^{2} \right)^{\mathbb{E}} \left( 1 - \alpha\_{j}^{2} \right)^{\mathbb{E}} \right)} \right\} \end{aligned} \tag{34}$$

Furthermore,

$$\begin{split} \mathbb{E}\_{\begin{subarray}{c}i=1 \ \overset{n}{\geqslant} \end{subarray}} & \left(\xi\_{i}^{\tau}\overline{p\_{i}}\oplus\zeta\overline{p\_{j}}\right)^{\frac{2}{n(n+1)}} \\ &=\mathbb{U}\_{a\_{i}\in h\_{i},a\_{j}\in h\_{j}\notin\xi\_{i},a\_{j}\in h\_{j}} \left\{ \left\{ \sqrt{\overbrace{\prod\limits\_{i=1,i=i}^{n}\Big(1-\left(1-\alpha\_{i}^{2}\right)^{\mathbb{C}}\Big(1-\alpha\_{j}^{2}\Big)^{\mathbb{C}}\Big)}^{2}}^{n}}^{2} \right\} \\ & \left\{ \sqrt{1-\prod\limits\_{i=1,i=i}^{n}\Big(1-\beta\_{i}^{2,\xi}\beta\_{j}^{2,\zeta}\Big)^{\frac{2}{n(n+1)}}} \right\} \end{split} \tag{35}$$

Therefore,

$$\begin{split} & \text{DHPFGGGHM}^{\xi,\zeta}(\overline{p}\_{1}, \overline{p}\_{2}, \dots, \overline{p}\_{n}) = \frac{1}{\xi + \zeta} \Bigg( \prod\_{i=1}^{n} \frac{\eta}{\zeta} \left( \zeta \overline{p\_{i}} \otimes \zeta \overline{p\_{j}} \right) \Bigg)^{\frac{2}{n(n+1)}} \\ & = \big( \omega\_{\text{a};\text{f}\boldsymbol{a},\boldsymbol{a}\_{j} \in \text{hi}\_{j};\text{f}\boldsymbol{\xi} \in \mathbf{y},\boldsymbol{a}\_{j} \neq \text{hi}\_{j} \right) \Bigg)^{\frac{1}{n(n+1)}} \Bigg( \Bigg( \sqrt{1 - \left( 1 - \prod\_{i=1, i \neq j}^{n} \left( 1 - \left( 1 - \alpha\_{i}^{2} \right)^{\zeta} \left( 1 - \alpha\_{j}^{2} \right)^{\zeta} \right)^{\frac{2}{n(n+1)}}}{\left( \left( \sqrt{1 - \prod\_{i=1, i \neq j}^{n} \left( 1 - \rho\_{i}^{2\zeta} \rho\_{j}^{2\zeta} \right)^{\frac{2}{n(n+1)}}} \right)^{\frac{1}{\zeta + \zeta}} \right)} \Bigg) \Bigg) \Bigg( \Bigg( \sqrt{1 - \prod\_{i=1, i \neq j}^{n} \left( 1 - \rho\_{i}^{2\zeta} \rho\_{j}^{2\zeta} \right)^{\frac{1}{n(n+1)}}}^{\frac{1}{\zeta + \zeta}} \Bigg) \end{split} \tag{36}$$

Thus, the proof have been finished.

**Example 5.** *Assume that* 8*p*1 = {{0.7, 0.8}, {0.4}}*,* 8*p*2 = {{0.3}, {0.6, 0.7}}*,* 8*p*3 = {{0.1, 0.3}, {0.4, 0.6}} *and* 8 *p*4 = {{0.5}, {0.5}} *are four DHPFNs, and suppose that* ξ = 2, ζ = 3*. Then, according to the DHPFGGHM operator, we can obtain the fused results as follows. For the membership degree function* α*, the fused results are shown as:*

α1 = DHPFGGHM2,3(0.7, 0.3, 0.1, 0.5) = 34451 − ⎛⎜⎜⎜⎜⎜⎝1 − 2*n <sup>i</sup>*=1,*j*=*<sup>i</sup>*&<sup>1</sup> − 1 − α2*i* ξ1 − α2*j*ζ' 2 *n*(*n*+<sup>1</sup>) ⎞⎟⎟⎟⎟⎟⎠ 1 ξ+ζ = 3444444444444444444444444444444444444444444444444444445⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 − ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.7<sup>2</sup><sup>3</sup> × 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.3<sup>2</sup><sup>3</sup> ×1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup> × 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> ×1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.3<sup>2</sup><sup>3</sup> × 1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup> ×1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> × 1 − 1 − 0.1<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup> ×1 − 1 − 0.1<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> × 1 − 1 − 0.5<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 110 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 1 2+3 = 0.3461

Similarly, we can obtain

$$\begin{aligned} \alpha\_2 &= \text{DHPFGGHM}^{2,3}(0.7, 0.3, 0.3, 0.5) = 0.4236\\ \alpha\_3 &= \text{DHPFGGHM}^{2,3}(0.8, 0.3, 0.1, 0.5) = 0.3545\\ \alpha\_4 &= \text{DHPFGGHM}^{2,3}(0.8, 0.3, 0.3, 0.5) = 0.4343 \end{aligned}$$

Hence, we can ge<sup>t</sup> α = {0.3461, 0.4236, 0.3545, 0.4343}.

For the non-membership degree function β, the fused results are shown as:

$$\beta\_{1} = \text{DHIPGGHM}^{2,3}(0.4, 0.6, 0.4, 0.5) = \left(\sqrt{1 - \prod\_{i=1, i=1}^{n} \left(1 - \beta\_{i}^{2\times 2} \beta\_{j}^{2\times 3}\right)^{\frac{2}{n(n+1)}}}\right)^{\frac{1}{1+\zeta}}$$

$$= \left(\sqrt{1 - \left(\frac{\left(1 - 0.4^{2\times 2} \times 0.4^{2\times 3}\right) \times \left(1 - 0.4^{2\times 2} \times 0.6^{2\times 3}\right) \times \left(1 - 0.4^{2\times 2} \times 0.4^{2\times 3}\right)}\right)^{\frac{1}{1+\zeta}}}{\times \left(1 - 0.4^{2\times 2} \times 0.5^{2\times 3}\right) \times \left(1 - 0.6^{2\times 2} \times 0.4^{2\times 3}\right) \times \left(1 - 0.6^{2\times 2} \times 0.4^{2\times 3}\right)}\right)^{\frac{1}{1+\zeta}}\right)^{\frac{1}{1+\zeta}}$$

$$\left(\sqrt{1 - 0.6^{2\times 2} \times 0.5^{2\times 3}}\right) \times \left(1 - 0.4^{2\times 2} \times 0.4^{2\times 3}\right) \times \left(1 - 0.4^{2\times 2} \times 0.4^{2\times 3}\right)$$

$$= 0.5100$$

 Similarly, we can obtain

 $\beta\_2 = \text{DHPFGGHM}^{2.5}(0.4, 0.6, 0.6, 0.5) = 0.5516$   $\beta\_3 = \text{DHPFGGHM}^{2.3}(0.4, 0.7, 0.4, 0.5) = 0.5734$   $\beta\_4 = \text{DHPFGGHM}^{2.3}(0.4, 0.7, 0.6, 0.5) = 0.5968$ 

Hence, we can ge<sup>t</sup> β = {0.5100, 0.5516, 0.5734, 0.5968}.Therefore,

$$\text{DHPFGGHM}(\overleftarrow{p}\_1, \overleftarrow{p}\_2, \overleftarrow{p}\_3, \overleftarrow{p}\_4) = \left\{ \begin{array}{c} \langle 0.3461, 0.4236, 0.3545, 0.4343 \rangle, \\ \langle 0.5100, 0.5516, 0.5734, 0.5968 \rangle \end{array} \right\}.$$

It can be easily proven that the DHPFGGHM operator satisfies the following properties.

**Property 7.** *(Idempotency) If all* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are equal, i.e.,* 8*pj* = 8*p for all j, then*

$$\text{'DHPFGGHM}^{\mathbb{E},\mathbb{E}}(\overline{p}\_1, \overline{p}\_2, \dots, \overline{p}\_n) = \overline{p} \tag{37}$$

**Property 8.** *(Boundedness) Let* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a collection of DHPFNs, and let*

$$
\widetilde{p}^+ = \cup\_{a\_j \in h\_j, \beta\_j \in \mathcal{g}\_j} \{ \langle \max\_i(a\_i) \rangle, \{ \min\_i(\beta\_i) \} \},
\widetilde{p}^- = \cup\_{a\_j \in h\_j, \beta\_j \in \mathcal{g}\_j} \{ \langle \min\_i(a\_i) \rangle, \{ \max\_i(\beta\_i) \} \}
$$

*Then*

$$
\widetilde{p}^- \le \text{DHPFGGHM}^{\mathbb{E}, \mathbb{Q}}(\widetilde{p}\_1, \widetilde{p}\_2, \dots, \widetilde{p}\_n) \le \widetilde{p}^+ \tag{38}
$$

**Property 9.** *(Monotonicity) Let* 8*pj* = *hj*, *gj and* 8*p j* = *h j*, *g j*, *j* = 1, 2, ··· , *n*, *be two set of DHPFNs. If* 8*pj* ≤ 8*p j, for all j, then*

$$\text{DHPFGGHM}^{\mathbb{E},\mathbb{L}}(\overline{p}\_1, \overline{p}\_2, \dots, \overline{p}\_n) \le \text{DHPFGGHM}^{\mathbb{E},\mathbb{L}}(\overline{p}\_1', \overline{p}\_2', \dots, \overline{p}\_n') \tag{39}$$

### *3.4. The DHPFGGWHM Aggregation Operator*

Using Definition 11, we can conclude that the DHPFGGHM operator didn't take the importance of arguments being fused into account. However, in many practical MADM problems, we should consider the weights of attributes. To overcome the limitations of the DHPFGGHM operator, we propose a novel DHPFGGWHM operator as follows.

**Definition 12.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a collection of DHPFNs. Then, the DHPFGGWHM operator can be defined as:*

$$\text{DHPFGGWH}\_{w}^{\xi,\zeta}(\widetilde{p\_1}, \widetilde{p\_2}, \dots, \widetilde{p\_n}) = \frac{1}{\zeta + \zeta} \left( \mathop{\otimes}\_{i=1}^{n} \mathop{\otimes}\_{j=i}^{n} \left( \zeta \widetilde{p\_i} \oplus \zeta \widetilde{p\_j} \right)^{w\_i w\_j} \right) \tag{40}$$

*According to the operation laws of the DHPFNs described in Definition 5, we can obtain Theorem 4.*

**Theorem 4.** *Assume that* ξ, ζ > 0 *and* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are a collection of DHPFNs with the weighting vector w* = (*<sup>w</sup>*1, *w*2, ... , *wn*)*<sup>T</sup> which satisfies wj* > 0*, i* = 1, 2, ... , *n and nj*=1 *wj* = 1*. Then, their fused result obtained by utilizing the DHPFGGWHM operator is also a DHPFN, and*

$$\begin{aligned} \text{DHPFGGWHM}\_{w}^{\boldsymbol{\zeta},\boldsymbol{\zeta}}(\overline{p}\_{i},\overline{p}\_{2},...,\overline{p}\_{n}) &= \frac{1}{\zeta+\zeta} \left( \underset{i=1}{\overset{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\boldsymbol{\forall{\boldsymbol{\boldsymbol{\alpha}}}}}}}}}}}}}}}}\right)}} \right)} \left\{\overset{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\rm{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\boldsymbol{\alpha}}}}}}}}}}}}\right\}}{\boldsymbol{\zeta}}\}}{\left\{\boldsymbol{\zeta}-{\boldsymbol{\zeta}}\right\}} \left\{\sqrt{\sqrt{1-\left(1-\frac{1}{i}\prod\_{i=1}^{n}\left(1-\left(1-\alpha\_{i}^{2}\right)^{\boldsymbol{\zeta}}\left(1-\alpha\_{i}^{2}\right)\right)^{\boldsymbol{\zeta}}}^{\boldsymbol{\omega},w\_{j}}\right)^{\frac{1}{1+\boldsymbol{\zeta}}}}\right\}, \tag{41}$$

*Mathematics* **2019**, *7*, 344

**Proof.** Based on Definition 5, we can obtain:

$$\xi \overleftarrow{p\_i} = \cup\_{a\_i \in h\_i, \theta\_i \in \mathfrak{g}\_i} \left\{ \left( \sqrt{1 - \left( 1 - \alpha\_i^2 \right)^{\xi}} \right) \cdot \left| \beta\_i^{\xi} \right| \right\} \tag{42}$$

$$\widetilde{\zeta\_j p\_j} = \cup\_{a\_j \in h\_j, \theta\_j \in \mathfrak{g}\_j} \left\{ \left( \sqrt{1 - \left( 1 - \alpha\_j^2 \right)^{\zeta}} \right), \left\{ \beta\_j^{\zeta} \right\} \right\} \tag{43}$$

Thus,

$$\zeta \widetilde{p\_i} \oplus \zeta \widetilde{p\_j} = \cup\_{a \in \mathbb{N}\_i, a\_j \in \mathbb{N}\_j, \beta\_i \in \mathbb{g}, a\_j \in \mathbb{h}\_j} \left\{ \left( \sqrt{1 - \left( 1 - \alpha\_i^2 \right)^\zeta \left( 1 - \alpha\_j^2 \right)^\zeta} \right) \cdot \left( \beta\_i^\zeta \beta\_j^\zeta \right) \right\} \tag{44}$$

Therefore,

$$\mathbf{w}\_{i} = \cup\_{a\_{i} \in \mathbb{H}\_{i}, a\_{j} \in \mathbb{H}\_{j}, \theta\_{i} \in \mathcal{G}\_{i}, a\_{j} \in \mathbb{H}\_{j}} \left\{ \begin{array}{l} \left\{ \left( \sqrt{1 - \left( 1 - \alpha\_{i}^{2} \right)^{\mathbb{L}} \left( 1 - \alpha\_{j}^{2} \right)^{\mathbb{L}}} \right)^{w\_{i} w\_{j}} \right\}, \\\\ \left\{ \sqrt{1 - \left( 1 - \beta\_{i}^{2} \beta\_{j}^{2} \right)^{w\_{i} w\_{j}}} \right\} \end{array} \tag{45}$$

Thereafter,

$$\begin{aligned} &\left\{ \begin{array}{l} \stackrel{\text{\tiny\tiny\text{\tiny\raisebox{0.5pt}{ $\Gamma\_{i}$ }}}{\longrightarrow} \Big( \xi \not\!\!/ \mkern-1.5pt \end{array} \Big\}\_{i}^{w\_{i}w\_{j}} \\ &= \cup\_{a\_{i}\in\text{hi}\_{i},a\_{j}\in\text{hi}\_{j},\emptyset\in\xi\_{i},a\_{j}\in\text{hi}\_{j}} \Bigg\{ \left\{ \left( \sqrt{\prod\limits\_{i=1,j=i}^{n} \Big( 1-\left(1-\alpha\_{i}^{2}\right)^{\xi} \binom{1}{i} - \alpha\_{j}^{2}\Big)} \right)^{w\_{i}w\_{j}} \right\}\_{i} \end{aligned} \tag{46}$$

Therefore,

$$\begin{aligned} \text{DHPFGG}(\text{WHHM}^{\xi,\zeta}\_{\text{uv}}(\overline{p}\_{1},\overline{p}\_{2},\ldots,\overline{p}\_{n}) &= \frac{1}{\zeta+\zeta} \Bigg\{ \sum\_{i=1}^{n} \int\_{i=1}^{\overline{p}\_{i}} \Bigg\{ \left( \xi \overline{p}\_{i} \oplus \zeta \overline{p}\_{j}^{\text{uv}} \right)^{w\_{i}w\_{j}} \Bigg\} \\ &= \cup\_{\text{in}\in\text{bi}\_{i},a\_{i}\in\text{b}\_{i},\overline{p}\_{i}\in\text{g}\_{i},\alpha\_{i}\in\text{h}\_{j}, \\ &\left( \begin{cases} \sqrt{1-\left(1-\prod\_{i=1,j=i}^{n}\left(1-\left(1-\alpha\_{i}^{2}\right)^{\mathbb{L}}\left(1-\alpha\_{j}^{2}\right)\right)^{\mathbb{L}}\right)^{w\_{i}w\_{j}}}^{1-\text{e}\_{i}} \Bigg\}, \\ &\left\{ \left(\sqrt{1-\prod\_{i=1,j=i}^{n}\left(1-\rho\_{i}^{2\mathcal{L}}\rho\_{j}^{2\mathcal{L}}\right)^{w\_{i}w\_{j}}}\right)^{\frac{1}{\mathbb{L}-\zeta}} \right\} \end{aligned} \tag{47}$$

Thus, we have finished the proof.

**Example 6.** *Assume that* 8*p*1 = {{0.7, 0.8}, {0.4}}*,* 8*p*2 = {{0.3}, {0.6, 0.7}}*,* 8*p*3 = {{0.1, 0.3}, {0.4, 0.6}} *and* 8 *p*4 = {{0.5}, {0.5}} *are four DHPFNs, and suppose that* ξ = 2, ζ = 3 *and wj* = (0.3, 0.2, 0.1, 0.4)*. Then, according to the DHPFGGWHM operator, we can obtain the fused results as follows. For the membership degree function* α*, the fused results are shown as:*

α1 = DHPFGGWHM2,3(0.7, 0.3, 0.1, 0.5) = 3451 − ⎛⎜⎜⎜⎜⎝1 − 2*n <sup>i</sup>*=1,*j*=*<sup>i</sup>*&<sup>1</sup> − 1 − α2*i* ξ1 − <sup>α</sup>2*j*<sup>ζ</sup>'*wiwj*⎞⎟⎟⎟⎟⎠ 1 ξ+ζ = 3444444444444444444444444444444444444444444444444444444445⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 − 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.7<sup>2</sup><sup>3</sup>0.3×0.3 × 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.3<sup>2</sup><sup>3</sup>0.3×0.2 ×1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup>0.3×0.1 × 1 − 1 − 0.7<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.3×0.4 ×1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.3<sup>2</sup><sup>3</sup>0.2×0.2 × 1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup>0.2×0.1 ×1 − 1 − 0.3<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.2×0.4 × 1 − 1 − 0.1<sup>2</sup><sup>2</sup> × 1 − 0.1<sup>2</sup><sup>3</sup>0.1×0.1 ×1 − 1 − 0.1<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.1×0.4 × 1 − 1 − 0.5<sup>2</sup><sup>2</sup> × 1 − 0.5<sup>2</sup><sup>3</sup>0.4×0.4 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 1 2+3

= 0.5156

Similarly, we can obtain

$$\begin{aligned} a\_2 &= \text{DHPFGGWHHM}^{2,3}(0.7, 0.3, 0.3, 0.5) = 0.5378\\ a\_3 &= \text{DHPFGGWHHM}^{2,3}(0.8, 0.3, 0.1, 0.5) = 0.5273\\ a\_4 &= \text{DHPFGGGWHM}^{2,3}(0.8, 0.3, 0.3, 0.5) = 0.5503 \end{aligned}$$

Hence, we can ge<sup>t</sup> α = {0.5156, 0.5378, 0.5273, 0.5503}.

For the non-membership degree function β, the fused results are shown as:

$$\beta\_{1} = \text{DHPFGGWHM}^{2,3}(0.4, 0.6, 0.4, 0.5) = \left(\sqrt{1 - \prod\_{i=1, i \neq 1}^{n} \left(1 - \beta\_{i}^{2\varepsilon} \beta\_{j}^{2\varepsilon}\right)^{\otimes 2\varepsilon \mu\_{j}}}\right)^{\frac{1}{\varepsilon \cdot \varepsilon}}$$

$$= \left(\sqrt{1 - \left(1 - 0.4^{2\varepsilon \times 2} \times 0.4^{2\times 3}\right)^{0.3 \times 0.3} \times \left(1 - 0.4^{2\times 2} \times 0.6^{2\times 3}\right)^{0.3 \times 0.2}} \times \left(1 - 0.4^{2\times 2} \times 0.4^{2\times 3}\right)^{0.3 \times 0.1}}\right)^{0.3 \times 0.1}$$

$$= \left(\sqrt{1 - 0.4^{2\times 2} \times 0.5^{2\times 3}}\right)^{0.3 \times 0.4} \times \left(1 - 0.6^{2\times 2} \times 0.4^{2\times 3}\right)^{0.2 \times 0.2} \times \left(1 - 0.6^{2\times 2} \times 0.4^{2\times 3}\right)^{0.1 \times 0.1} \times \left(1 - 0.4^{2\times 2} \times 0.5^{2\times 3}\right)^{0.1 \times 0.4}}\right)^{0.1 \times 0.4}$$

$$\sqrt{1 - 0.5^{2\times 2} \times 0.5^{2\times 3}}^{0.4 \times 0.4}$$

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1

5

= 0.4850

Similarly, we can obtain

$$\beta\_2 = \text{DHIPGGWHM}^{2,3}(0.4, 0.6, 0.6, 0.5) = 0.50068\\\beta\_3 = \text{DHIPGGWHM}^{2,3}(0.4, 0.7, 0.4, 0.5) = 0.5338\\\beta\_4 = \text{DHIPGGWHM}^{2,3}(0.4, 0.7, 0.6, 0.5) = 0.5433$$

Hence, we can ge<sup>t</sup> β = {0.4850, 0.5006, 0.5338, 0.5433}. Therefore,

$$\text{IDHPFGGWHM}(\overleftarrow{p}\_1, \overleftarrow{p}\_2, \overleftarrow{p}\_3, \overleftarrow{p}\_4) = \left\{ \begin{array}{c} \langle 0.5156, 0.5378, 0.5273, 0.5503 \rangle, \\ \langle 0.4850, 0.5006, 0.5338, 0.5433 \rangle \end{array} \right\}.$$

It can be easily proven that the DHPFGGWHM operator satisfies the following properties.

**Property 10.** *(Idempotency) If all* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *are equal, i.e.,* 8*pj* = 8*p for all j, then*

$$\text{DHPFGGWHM}\_{w}^{\xi,\zeta}(\widetilde{p}\_1, \widetilde{p}\_2, \dots, \widetilde{p}\_n) = \widetilde{p} \tag{48}$$

**Property 11.** *(Boundedness) Let* 8*pj* = *hj*, *gj*(*<sup>j</sup>* = 1, 2, ··· , *n*) *be a collection of DHPFNs, and let*

$$
\widetilde{p}^+ = \cup\_{a\_{\widetilde{\boldsymbol{\eta}}} \in \mathfrak{h}\_{\widetilde{\boldsymbol{\eta}}} \underline{\boldsymbol{\eta}}\_{\widetilde{\boldsymbol{\eta}}} \{ \{ \max\_{i} (\boldsymbol{\alpha}\_{i}) \} , \{ \min\_{i} (\boldsymbol{\beta}\_{i}) \} \} \prime \\
\widetilde{p}^- = \cup\_{a\_{\widetilde{\boldsymbol{\eta}}} \in \mathfrak{h}\_{\widetilde{\boldsymbol{\eta}}} \underline{\boldsymbol{\eta}}\_{\widetilde{\boldsymbol{\eta}}} \{ \{ \min\_{i} (\boldsymbol{\alpha}\_{i}) \} , \{ \max\_{i} (\boldsymbol{\beta}\_{i}) \} \} \prime
$$

*Then*

$$
\overleftarrow{p}^{\sim} \in \text{DHPFGGWM}\_{w}^{\xi,\zeta}(\overleftarrow{p}\_{1}, \overleftarrow{p}\_{2}, \dots, \overleftarrow{p}\_{n}) \le \overleftarrow{p}^{+} \tag{49}
$$

**Property 12.** *(Monotonicity) Let* 8*pj* = *hj*, *gj and* 8*p j* = *h j*, *g j*, *j* = 1, 2, ··· , *n*, *be two sets of DHPFNs. If* 8 *pj* ≤ 8 *p j, for all j, then*

$$\text{DHPFGCWHM}\_{w}^{\mathbb{C}\mathbb{Z}}(\widetilde{p}\_{1}, \widetilde{p}\_{2}, \dots, \widetilde{p}\_{n}) \leq \text{DHPFGCWHM}\_{w}^{\mathbb{C}\mathbb{Z}}(\widetilde{p}\_{1}^{\prime}, \widetilde{p}\_{2}^{\prime}, \dots, \widetilde{p}\_{n}^{\prime}) \tag{50}$$

### **4. An Approach to MADM with DHPFNs Information**

In this section, we shall use the DHPFGWHM and DHPFGGWHM operators to deal with MADM problems with dual hesitant Pythagorean fuzzy information. Suppose that there are *m* alternatives η = η1, η2, ··· , η*m*, and each alternative is characterized by *n* attributes δ = {δ1, δ2, ··· , δ*n*} with the weighting vector being *wj* = {*<sup>w</sup>*1, *w*2, ··· , *wn*}. Then, the dual hesitant Pythagorean fuzzy matrix can be constructed as *P* 8 = 8*pijm*×*n*, with each element 8*pij* = *hij*, *gij*(*<sup>i</sup>* = 1, 2, ··· , *m*, *j* = 1, 2, ··· , *n*) indicating a dual hesitant Pythagorean fuzzy number, where *hij* means the membership degree set with several values in [0, 1], and *gij* means the no-membership degree set with several values in [0, 1].

In what follows, we apply the DHPFGWHM or DHPFGGWHM operator to MADM problems for supplier selection in supply chain managemen<sup>t</sup> with dual hesitant Pythagorean fuzzy information.

**Step 1.** In order to derive the fused results of each alternative, for alternatives η = η1, η2, ··· , η*m*, based on the weighting vector *wj* = {*<sup>w</sup>*1, *w*2, ··· , *wn*} and dual hesitant Pythagorean fuzzy information 8 *pij* = *hij*, *gij*(*<sup>i</sup>* = 1, 2, ··· , *m*, *j* = 1, 2, ··· , *n*) given in matrix *P*8 = 8*pijm*×*n*, we can aggregate all the DHPFNs by the DHPFGWHM operator

$$\begin{aligned} \widetilde{p\_{i}} &= \text{DHPFGWM}\_{w}^{\mathbb{E},\mathbb{E}}(\widetilde{p\_{i}},\widetilde{p\_{i}},\ldots,\widetilde{p\_{in}}) = \left(\prod\_{k=1}^{n}\underset{j=k}{\mathbb{E}}\left(w\_{ik}w\_{ij}\left(\widetilde{p\_{ik}^{\mathbb{E}}}\widetilde{p\_{ij}^{\mathbb{E}}}\right)\right)\right)^{\frac{1}{\zeta-\zeta}} \\ &= \text{U}\_{a\_{k}\in\mathbb{I}[a\_{i},a\_{j}\in\mathbb{K}\_{ij},\mathbb{K}\_{ij}\in\mathbb{K}\_{ij}]}\left\{\left\{\left(\sqrt{1-\prod\_{k=1,j=k}^{n}\left(1-\alpha\_{ik}^{2\zeta}\alpha\_{ij}^{2\zeta}\right)^{w\_{j}w\_{j}}}\right)^{\frac{1}{\zeta-\zeta}}\right\}, \end{aligned} \tag{51}$$

### or the DHPFGGWHM operator

$$\begin{split} \widetilde{p\_{i}} &= \text{DHPFGGWHM}\_{w}^{\xi,\zeta}(\widetilde{p\_{i1}}, \widetilde{p\_{i2}}, \dots, \widetilde{p\_{in}}) = \frac{1}{\zeta + \zeta} \Bigg\{ \sum\_{k=1}^{n} \sum\_{j=k}^{n} \left( \xi \widetilde{p\_{k}} \otimes \zeta \widetilde{p\_{j}} \right)^{w\_{k} w\_{j}} \Bigg\} \\ &= \text{U}\_{a\_{\overline{k}} \in \text{h}\_{k}, a\_{ij} \in \text{h}\_{ij}, \widehat{p\_{ij}} \in \text{g}\_{ij}, \widehat{\sigma}\_{ij} \Bigg\} \Bigg\{ \left\{ \left( \sqrt{1 - \left( 1 - \prod\_{k=1, j=k}^{n} \left( 1 - \left( 1 - \alpha\_{ik}^{2} \right)^{\zeta} \left( 1 - \alpha\_{ij}^{2} \right)^{\zeta} \right)^{w\_{i} w\_{j}} \right)^{1 + \zeta}} \right) \Bigg\} \Bigg\} \Bigg\} \end{split} \tag{52} \end{split}$$

to obtain the overall fused results 8*pi*(*<sup>i</sup>* = 1, 2, ··· , *<sup>m</sup>*).

**Step 2.** To obtain the rank of all the alternatives, we need to adapt the score function and accuracy function described in Definition 4. Firstly, based on the score function equation, we can compute the score values *<sup>S</sup>*(<sup>8</sup>*pi*) (*i* = 1, 2, ··· , *m*) of 8*pi*(*<sup>i</sup>* = 1, 2, ··· , *<sup>m</sup>*). If all the score values of 8*pi*(*<sup>i</sup>* = 1, 2, ··· , *m*) are different, we can easily obtain the ordering of alternatives. Then, if there is no difference between any two scores *<sup>S</sup>*(<sup>8</sup>*pi*) and *<sup>S</sup>*8*pj*, we need to compute the accuracy values *<sup>H</sup>*(<sup>8</sup>*pi*) and *<sup>H</sup>*8*pj* of 8*pi* and 8*pj*, respectively, and then determine the ordering of all the alternatives η*i* and η*j* based on the accuracy results *<sup>H</sup>*(<sup>8</sup>*pi*) and *<sup>H</sup>*8*pj*.

**Step 3.** Determine the ordering of all the alternatives η*i*(*<sup>i</sup>* = 1, 2, ··· , *m*) and select the best one(s) according to the scores values *<sup>S</sup>*(<sup>8</sup>*pi*)(*<sup>i</sup>* = 1, 2, ··· , *m*) and accuracy results *<sup>H</sup>*(<sup>8</sup>*pi*). Thus, we have finished the decision making process by using the DHPFGWHM operator or the DHPFGGWHM operator.

### **5. Numerical Example and Comparative Analysis**
