*5.3. Comparative Analysis*

The prominent characteristic of the DHPFGWHM and DHPFGGWHM operators is that they can consider the interrelationship among the DHFNs. Next, we shall compare our developed methods with the dual hesitant Pythagorean fuzzy weighted average (DHPFWA) and dual hesitant Pythagorean fuzzy weighted geometric (DHPFWG) operators [53], with the comparative analysis results listed as follows.

According to Table 1 and attribute weights, the fused values obtained by the DHPFGWA operator are:

$$\begin{aligned} \widetilde{p}\_1 &= \text{DHPFWA}(\widetilde{p}\_{11}, \widetilde{p}\_{12}, \widetilde{p}\_{13}, \widetilde{p}\_{14}) = \underset{j=1}{\text{€}} w\_j \widetilde{p}\_{1j} \\ &= \left\{ \left\{ \begin{array}{l} 0.4325, 0.4527, 0.4522, 0.4711, 0.4793, 0.4967, 0.4962, 0.5126, \\ 0.4580, 0.4766, 0.4761, 0.4936, 0.5013, 0.5174, 0.5170, 0.5323 \end{array} \right\} \Big/ \begin{array}{l} 0.5872, \\ 0.6334 \end{array} \right\}; \end{aligned}$$

0.4288, 0.4393, 0.4535, 0.4247, 0.4353,

8 *p*2 = DHPFWA(<sup>8</sup>*p*21,<sup>8</sup>*p*22,<sup>8</sup>*p*23,<sup>8</sup>*p*24) = 4⊕*<sup>j</sup>*=<sup>1</sup>*wj*<sup>8</sup>*p*2*<sup>j</sup>* = {{0.5005, 0.5461, 0.5788}, {0.5476, 0.5617, 0.5677, 0.5823, 0.5860, 0.6011}}; 8*p*3 = DHPFWA(<sup>8</sup>*p*31,<sup>8</sup>*p*32,<sup>8</sup>*p*33,<sup>8</sup>*p*34) = 4⊕*<sup>j</sup>*=<sup>1</sup>*wj*<sup>8</sup>*p*3*<sup>j</sup>* =66 0.4547, 0.4812, 0.5178, 0.4775, 0.5022, 0.5364, 0.5071, 0.5296, 0.5610, 0.5595, 0.5784, 0.6051, 0.5757, 0.5936, 0.6190, 0.5973, 0.6139, 0.6376 7, 6 0.5480, 0.5704 77;8 *p*4 = DHPFWA(<sup>8</sup>*p*41,<sup>8</sup>*p*42,<sup>8</sup>*p*43,<sup>8</sup>*p*44) = 4⊕*<sup>j</sup>*=<sup>1</sup>*wj*<sup>8</sup>*p*4*<sup>j</sup>* = {{0.6278, 0.6347, 0.6796, 0.6852, 0.7529, 0.7570}, {0.3923, 0.3997, 0.4063}}; 8 *p*5 = DHPFWA(<sup>8</sup>*p*51,<sup>8</sup>*p*52,<sup>8</sup>*p*53,<sup>8</sup>*p*54) = 4⊕*<sup>j</sup>*=<sup>1</sup>*wj*<sup>8</sup>*p*5*<sup>j</sup>* = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩0.3434, 0.3574, 0.3761, 0.3936, 0.4053, 0.4213, 0.3763, 0.3888, 0.4056, 0.4214, 0.4321, 0.4467, 0.4172, 0.4281, 0.4428, 0.4568, 0.4663, 0.4794, 0.3531,0.3666,0.3848,0.4017,0.41320.4287,0.3849,0.3971,0.4134,⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭, {0.4184}⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭.

 Then, based on the score function of the dual hesitant Pythagorean fuzzy elements (DHPFEs), we can obtain the score results of 8*pi* as:

0.4497, 0.4634, 0.4727, 0.4855

$$s(\overline{p}\_1) = 0.4317, s(\overline{p}\_2) = 0.4822, s(\overline{p}\_3) = 0.4977\\s(\overline{p}\_4) = 0.6592, s(\overline{p}\_5) = 0.5005$$

Then, we rank all the suppliers in supply chain managemen<sup>t</sup> η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) in accordance with the scores *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, ··· , <sup>5</sup>): η4 η5 η3 η2 η1 and thus the most desirable supplier in supply chain managemen<sup>t</sup> is obtained, which is η4.

According to Table 1 and attribute weights, the fused values by the DHPFWG operator are:

$$\begin{aligned} \widetilde{p\_1} &= \text{DHPFWG}(\widetilde{p\_{11}}, \widetilde{p\_{12}}, \widetilde{p\_{13}}, \widetilde{p\_{14}}) = \mathop{\otimes}\_{j=1}^{4} \{ \widetilde{p\_{1j}} \}^{w\_j} \\ &= \left\{ \left\{ \begin{array}{l} 0.4119, 0.4226, 0.4452, 0.4567, 0.4383, 0.4496, 0.4737, 0.4859, \\ 0.4356, 0.4468, 0.4708, 0.4829, 0.4634, 0.4754, 0.5009, 0.5138 \end{array} \right\} \Big| \begin{array}{l} 0.6574, \\ 0.6735 \end{array} \right\}; \end{aligned}$$

$$\begin{aligned} \widetilde{p}\_2 &= \text{DHPFWG}(\widetilde{p}\_{21}, \widetilde{p}\_{22}, \widetilde{p}\_{23}, \widetilde{p}\_{24}) = \underset{j=1}{\text{\textquotedbl{}}} \{ \widetilde{p}\_{2j} \}^{w\_j} \\ &= \{ \langle 0.4086, 0.4861, 0.5171 \rangle, \langle 0.5694, 0.5822, 0.6203, 0.6311, 0.6945, 0.7026 \rangle \} \end{aligned}$$

8 *p*3 = DHPFWG(<sup>8</sup>*p*31,<sup>8</sup>*p*32,<sup>8</sup>*p*33,<sup>8</sup>*p*34) = 4⊗*<sup>j</sup>*=<sup>1</sup>8*<sup>p</sup>*3*<sup>j</sup>wj* = 66 0.3931, 0.4017, 0.4093, 0.4335, 0.4430, 0.4513, 0.4677, 0.4779, 0.4869, 0.4224,0.4316,0.4398,0.4658,0.4760, 0.4850,0.5026,0.5135,0.52327, 6 0.6622, 0.7404

$$
\widetilde{p}\_4 = \text{DHPFWG}(\widetilde{p}\_{41}, \widetilde{p}\_{42}, \widetilde{p}\_{43}, \widetilde{p}\_{44}) = \mathop{\otimes}\_{j=1}^4 (\widetilde{p}\_{4j})^{w\_j}
$$

= {{0.6278, 0.6347, 0.6796, 0.6852, 0.7529, 0.7570}, {0.3923, 0.3997, 0.4063}};

 77;

$$
\begin{split}
\widetilde{p}\_{5} &= \mathrm{DHPFWG}(\widetilde{p}\_{51}, \widetilde{p}\_{52}, \widetilde{p}\_{53}, \widetilde{p}\_{54}) = \bigwedge\_{j=1}^{4} \left( \widetilde{p}\_{5j} \right)^{w\_{j}} \\ &= \left\{ \begin{pmatrix} 0.2617, 0.2724, 0.2811, 0.2749, 0.2862, 0.2952, 0.2885, 0.3004, 0.3099 \\ 0.3031, 0.3156, 0.3256, 0.3113, 0.3241, 0.3344, 0.3270, 0.3404, 0.3512, \\ 0.3112, 0.3240, 0.3342, 0.3269, 0.3403, 0.3511, 0.3431, 0.3572, 0.3686 \\ 0.3605, 0.3753, 0.3872, 0.3702, 0.3976, 0.3889, 0.4049, 0.4177 \end{pmatrix}, \begin{pmatrix} 0.4811 \\ 0.4811 \end{pmatrix} \right\}. \end{split}
$$

Then, based on the score function of the DHPFEs, we can obtain the score results of 8*pi* as:

$$s(\overline{p}\_1) = 0.3851, s(\overline{p}\_2) = 0.4099, s(\overline{p}\_3) = 0.3584, 
 s(\overline{p}\_4) = 0.4939, s(\overline{p}\_5) = 0.4410$$

Then, we rank all the suppliers in supply chain managemen<sup>t</sup> η*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) in accordance with the scores *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, ··· , <sup>5</sup>): η4 η5 η2 η1 η3 and thus the most desirable supplier in supply chain managemen<sup>t</sup> is obtained, which is η4.

According to Table 4, we can easily conclude that the ordering is slightly di fferent and that these are some of the best alternatives. However, our defined operators are mainly characteristic of the advantages that can consider the interrelationship between the arguments being fused into consideration and consider the human hesitance in practical MADM problems. Obviously, the DHPFWA and DHPFWG operators defined by Wei and Lu [53] cannot consider the interrelationship between the arguments being fused. In addition, in a complicated decision-making environment, the decision maker's risk attitude is an important factor to think about, and our methods can do this by altering the parameters ξ and ζ, whereas the DHPFWA and DHPFWG operators presented by Wei and Lu [53] do not have the ability to dynamically adjust to the parameters according to the decision maker's risk attitude, meaning it is di fficult to solve risk multiple attribute decision making in real practice.

**Table 4.** Ordering of the suppliers by the DHPFGGWHM operators.

