## 4 Attribute Weight Calculation Based on Type-2 Hesitant Pythagorean Fuzzy Sets
### Distance Measurement of Type-2 Hesitant Pythagorean Fuzzy Elements  
**Definition 8**. : \(h_{i}=<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(}^{i})}^{i}>\in\mathrm{T2 HPFE}(X)\) _is set, \(i=1,2\), and the distance between two T2 HPFEs is a real-value function \(\mathrm{d}:\mathrm{T2 HPFE}(X)\times\mathrm{T2 HPFE}(X)\rightarrow[0,1]\), which meets the following axiomatization requirements:_  
_(d1) \(0\leq d(h_{1},h_{2})\leq 1\)._  
_(d2) \(d(h_{1},h_{2})=d(h_{2},h_{1})\).__(d3) If_ i \(\
eq j\)_, there is no inclusion relation between_ \((\Gamma_{i}^{1},\Psi_{i}^{1})\) _and_ \((\Gamma_{j}^{1},\Psi_{j}^{1})\) _or between_ \((\Gamma_{i}^{2},\Psi_{i}^{2})\) _and_ \((\Gamma_{j}^{2},\Psi_{j}^{2})\)_, namely,_ \(\Gamma_{i}^{1}\subseteq\Gamma_{j}^{1}\)_,_ \(\Gamma_{i}^{1}\supseteq\Gamma_{j}^{1}\)_,_ \(\Gamma_{i}^{2}\subseteq\Gamma_{j}^{2}\)_, and_ \(\Gamma_{i}^{2}\supseteq\Gamma_{j}^{2}\) _are all not true. When_ \(\Psi_{i}^{1}\subseteq\Psi_{j}^{1},\Psi_{i}^{1}\supseteq\Psi_{j}^{1},\Psi_{i}^{2 }\subseteq\Psi_{j}^{2}\)_, and_ \(\Psi_{i}^{2}\supseteq\Psi_{j}^{2}\) _are not true, either_ \(d(h_{1},h_{2})=0\) _on the premise of_ \(h_{1}=h_{2}\)_._  
_(d4)_ \(d(h_{1},h_{2})=d(h_{1}^{c},h_{2})\)_._  
_(d5)_ \(d(h_{1},h_{2})=1\) _when and only when_ \(h_{1}=<(\{0\},\{1\})>\)_,_ \(h_{2}=<(\{1\},\{0\})>\)_, or_ \(h_{1}=<(\{1\},\{0\})>\) _and_ \(h_{2}=<(\{0\},\{1\})>\)_._  
**Definition 9**. : \(h_{i}=<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(1)}^{i}}^{i}>\) _and \(i=1,2\) are set as two Type-2 hesitant Pythagorean fuzzy elements through supplementing operation, and the Euclidean distance between two T2 HPFEs is solved as follows:_  
\[d_{1}(h_{1},h_{2})=\frac{1}{l_{k}}\sum\limits_{m=1}^{l_{k}}(\frac{1}{2l_{ \Gamma_{m}}}\sum\limits_{i=1}^{l_{\Gamma_{m}}}\mid(\mu_{m^{i}}^{1})^{2}-(\mu_{ m^{i}}^{2})^{2}\mid^{2}+\frac{1}{2l_{\Psi_{m}}}\sum\limits_{i=1}^{l_{\Psi_{m}}} \mid(v_{m^{i}}^{1})^{2}-(v_{m^{i}}^{2})^{2}\mid^{2}) \tag{7}\]  
**Definition 10**. : \(h_{i}=<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(1)}^{i}}^{i}>\) _and \(i=1,2\) are set as two Type-2 hesitant Pythagorean fuzzy elements through supplementing operation, and then the Hausdorff distance between two T2 HPFEs is as follows:_  
\[d_{2}(h_{1},h_{2})=\frac{1}{l_{k}}\sum\limits_{m=1}^{l_{k}}(\frac{1}{2}max \limits_{i}\mid(\mu_{m^{i}}^{1})^{2}-(\mu_{m^{i}}^{2})^{2}\mid^{2}+\frac{1}{2} max\limits_{i}\mid(v_{m^{i}}^{1})^{2}-(v_{m^{i}}^{2})^{2}\mid^{2}) \tag{8}\]  
_The above two distance measurement formulas are commonly used at present. However, the supplementing operation should be performed on two T2 HPFEs first in the use of the above distance measurement formulas. In practical application, the workload is added and the information of T2 HPFE itself is destructed, which has been well solved by the new-type distance formula proposed by [PERSON] et al. [38]. Hereby, this new formula is introduced into Type-2 hesitant Pythagorean fuzzy sets._  
**Definition 11**. : \(h_{i}=<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(1)}^{(1)}}^{i}>\in\mathrm{T2 HPFE}(X)\) _and \(i=1,2\) are set, and then the distance between two T2 HPFEs is as follows:_  
\[d_{3}(h_{1},h_{2})=\frac{1}{2}(\frac{1}{k_{(1)}^{1}}\sum\limits_{i=1}^{k_{(1 )}^{1}}\underset{(\Gamma_{i}^{1},\Psi_{i}^{2})\in h_{2}}{min}(\frac{1}{2\mid \Gamma_{i}^{1}}\mid\sum\limits_{p_{i}^{1}\in\Gamma_{i}^{1}}\underset{\mu^{2} \in\Gamma^{2}}{min}\mid(\mu_{i}^{1})^{2}-(\mu^{2})^{2}\mid+(\frac{1}{2\mid \Psi_{i}^{1}}\mid\sum\limits_{v_{i}^{1}\in\Psi_{i}^{1}}\underset{v^{2}\in \Psi_{i}^{2}}{min}\mid(v_{i}^{1})^{2}-(v^{2})^{2}\mid)\frac{1}{2} \tag{9}\]  
_Compared with the traditional Euclidean distance and Hausdorff distance, for the distance put forward in Definition 11, the supplementing operation is not needed for the two Type-2 hesitant Pythagorean fuzzy elements before the calculation, and the axiomatization requirements for the distance of hesitant fuzzy sets are satisfied._  
Proof. : It can be easily proved that \(d_{3}(h_{1},h_{2})\) meets (d1) and (d2), and it also satisfies (d4) according to Formula (9). Hereby, (d3) and (d5) will be proved. (d3) Sufficiency.