*5.1. Numerical Example*

In this section we present a numerical example for supplier selection in supply chain managemen<sup>t</sup> with dual hesitant Pythagorean fuzzy information in order to demonstrate the method proposed in this paper. Suppose there is a problem to do with the supplier selection in supply chain managemen<sup>t</sup> which is a classical MADM problem. There are five prospective suppliers η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) for four attributes <sup>δ</sup>*j*(*j* = 1, 2, 3, <sup>4</sup>). The four attributes include product quality (<sup>δ</sup>1), service (<sup>δ</sup>2), delivery, (<sup>δ</sup>3) and price (<sup>δ</sup>4), respectively. In order to avoid influencing each other, the decision makers are required to evaluate the five suppliers η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) under the above four attributes in anonymity. The decision matrix *P* 8 = 8*pij*<sup>5</sup>×<sup>4</sup> is presented in Table 1, where 8*pij*(*<sup>i</sup>* = 1, 2, 3, 4, 5, *j* = 1, 2, 3, 4) are in the form of DHPFNs. (Suppose the weighting vector is *wj* = (0.25, 0.34, 0.27, 0.14))


**Table 1.** Dual hesitant Pythagorean fuzzy decision matrix.

In what follows, we can utilize our developed methods to deal with the supplier selection in supply chain managemen<sup>t</sup> with dual hesitant Pythagorean fuzzy information.

**Step 1.** We aggregate the dual hesitant Pythagorean fuzzy information given in the matrix by utilizing the DHPFGWHM operator to obtain the overall preference values 8*pi* of the supplier in supply chain managemen<sup>t</sup> η*i*(*<sup>i</sup>* = 1, 2, 3, 4, <sup>5</sup>). Taking the alternative η1 as an example (here, we take ξ = ζ = 2), we have

8 *p*1 = DHPFGWHMξ,<sup>ζ</sup> *w* (<sup>8</sup>*p*11,<sup>8</sup>*p*12,<sup>8</sup>*p*13,<sup>8</sup>*p*14) = & 4⊕*k*=1 4⊕*j*=*k wikwij*8*p*ξ*ik*<sup>8</sup>*p*ζ*ij*' 1 ξ+ζ = <sup>∪</sup><sup>α</sup>*ik*<sup>∈</sup>*hik*,<sup>α</sup>*ij*<sup>∈</sup>*hij*,β*ij*<sup>∈</sup>*gij*,<sup>α</sup>*ij*<sup>∈</sup>*hij*, ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎧⎪⎪⎪⎨⎪⎪⎪⎩⎛⎜⎜⎜⎜⎜⎝ 91 − 24 *<sup>k</sup>*=1,*j*=*<sup>k</sup>*<sup>1</sup> − α2ξ*ik* α2ζ*ij wiwj*⎞⎟⎟⎟⎟⎟⎠ 1 ξ+ζ ⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ 3451 − ⎛⎜⎜⎜⎜⎝1 − 24 *<sup>k</sup>*=1,*j*=*<sup>k</sup>*&<sup>1</sup> − 1 − <sup>β</sup><sup>2</sup>*ik*<sup>ξ</sup><sup>1</sup> − <sup>β</sup><sup>2</sup>*ij*<sup>ζ</sup>'*wiwj*⎞⎟⎟⎟⎟⎠ 1 ξ+ζ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = {{0.4, 0.5}, {0.7}}, {{0.5, 0.6}, {0.4, 0.5)}, {{0.3, 0.4}, {0.8)}, {{0.5, 0.6}, {0.6}} = {{0.4234, 0.4461, 0.4335, 0.4547, 0.4824, 0.4964, 0.4887, 0.5023, 0.4448, 0.4642,

0.4536, 0.4719, 0.4957, 0.5087, 0.5016, 0.5143}, {0.6319, 0.6725}}

**Step 2.** Compute the scores results *s*(<sup>8</sup>*pi*) (*i* = 1, 2, 3, 4, 5) of the overall dual hesitant Pythagorean fuzzy preference values 8*pi* (*i* = 1, 2, 3, 4, <sup>5</sup>):

$$s(\overline{p}\_1) = 0.3998, s(\overline{p}\_2) = 0.4536, s(\overline{p}\_3) = 0.4669 \\ s(\overline{p}\_4) = 0.6255, s(\overline{p}\_5) = 0.4674$$

**Step 3.** Determine the ordering of all the suppliers η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) based on the scores values *s*(<sup>8</sup>*pi*) (*i* = 1, 2, 3, 4, <sup>5</sup>): η4 η5 η3 η2 η1, and it is clear that the most desirable supplier is η4.

Similarly, if we utilize the DHPFGGWHM operator to solve this MADM, the decision making steps can be described as follows.

**Step 1**-**.** Aggregate all dual hesitant Pythagorean fuzzy values 8*pij*(*j* = 1, 2, 3, 4) by using the DHPFGGWHM operator to derive the overall dual hesitant Pythagorean fuzzy values 8*pi*(*<sup>i</sup>* = 1, 2, ··· , 5) of the supplier η*i*. Taking supplier η1 for an example (here, we take ξ = ζ = 2), we have

$$
\begin{split}
\widetilde{p\_{1}} &= \mathrm{DHPFGG}(\mathrm{WHM}\_{\widetilde{w}}^{\xi,\zeta}(\widetilde{p\_{1}},\widetilde{p\_{1}}\_{\widetilde{r}1\*},\widetilde{p\_{1}}\_{\widetilde{r}1\*},\widetilde{p\_{1}}\_{\widetilde{r}1\*}) = \frac{1}{\zeta+\zeta} \Biggl( \mathop{\rm d}\_{k=1}^{4} \oint\_{\begin{subarray}{c} 4 \\ k=1} \frac{4}{j-k} \left( \left( \widetilde{c\_{\widetilde{r}k}} \oplus \zeta \widetilde{p\_{\widetilde{r}}} \right)^{w\_{\widetilde{k}}w\_{\widetilde{r}}} \right)^{w\_{\widetilde{k}}w\_{\widetilde{r}}} \Biggr) \\ & \quad \cdot \Bigg( \mathop{\rm d}\_{\widetilde{a}\_{\widetilde{x}}\in h\_{\widetilde{a}},\widetilde{a}\_{\widetilde{y}}\in \widetilde{g\_{\widetilde{x}}},\widetilde{a}\_{\widetilde{y}}\in \widetilde{a}\_{\widetilde{y}}} \Bigg) \Bigg\{ \left\{ \left( \sqrt{1-\left(1-\prod\_{k=1,j\to k}^{4} \left(1-\left(1-\alpha\_{\widetilde{a}k}\right)^{\xi}\left(1-\alpha\_{ij}^{2}\right)\right)^{\xi}\right)^{w\_{\widetilde{r}}w\_{j}}} \right)^{\frac{1}{\xi+\zeta}} \right\} \Biggr{)} \Bigg{]} \end{split}
$$

= {{0.4, 0.5}, {0.7}}, {{0.5, 0.6}, {0.4, 0.5)}, {{0.3, 0.4}, {0.8)}, {{0.5, 0.6}, {0.6}}

= {{0.4929, 0.5022, 0.5190, 0.5287, 0.5186, 0.5283, 0.5460, 0.5560, 0.5140, 0.5236, 0.5406, 0.5506, 0.5406, 0.5505, 0.5686, 0.5790}, {0.6421, 0.6493}}

**Step 2**-**.** Compute the scores results *s*(<sup>8</sup>*pi*) (*i* = 1, 2, 3, 4, 5) of the overall dual hesitant Pythagorean fuzzy values 8*pi*(*<sup>i</sup>* = 1, 2, 3, 4, 5) of the supplier 8*pi*:

$$s(\overline{p}\_1) = 0.4349, s(\overline{p}\_2) = 0.4549, s(\overline{p}\_3) = 0.3976, s(\overline{p}\_4) = 0.5240, s(\overline{p}\_5) = 0.4780$$

**Step 3**- **.** Determine the ordering of all the suppliers η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) based on the score results *s*(<sup>8</sup>*pi*) (*i* = 1, 2, 3, 4, 5) of 8*pi*(*<sup>i</sup>* = 1, 2, ··· , <sup>5</sup>): η4 η5 η2 η1 η3 and it is clear that the most desirable supplier in supply chain managemen<sup>t</sup> is η4.

According to the above analysis, we can easily find that although the overall rating values of the alternatives are slightly di fferent by using two operators respectively, the most desirable supplier in supply chain managemen<sup>t</sup> is η4.

### *5.2. Influence of Parameters on the Final Result*

The parameters ξ and ζ play an important role in the final ranking of alternatives. We may obtain di fferent ordering results by assigning di fferent values to ξ and ζ. By altering the values of ξ and ζ, di fferent ranking results are obtained, as shown in Tables 2 and 3. Therefore, the DHPFGWHM and DHPFGGWHM operators are shown to be considerably flexible by using a parameter vector. Tables 2 and 3 show that the ranking results increase and become steady with the increase of values in the parameter vector. That is, the final results become increasingly objective by considering the interrelationship among the attribute values. These features of the DHPFGWHM and DHPFGGWHM operators are crucial in real MADM problems.

**Table 2.** Ordering by the DHPFGWHM operators.



