## 5 Type-2 Hesitant Pythagorean Fuzzy Multi-Attribute Decision Method Based on Improved TOPSIS
### Correlation Coefficient of Type-2 Hesitant Pythagorean Fuzzy Elements  
**Definition 14**.: \(h_{i}=\)\(<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(1)}^{i}}^{i}>\in\)\(\text{T2 HPFE}(X)\) _and \(i=1,2\) are set, and then the distance between two T2 HPFEs is a real-value function \(c:\)\(\text{T2 HPFE}(X)\times\text{T2 HPFE}(X)\rightarrow[0,1]\), which meets the following axiomatization requirements:_  
_(c1) \(-1\leq c(h_{1},h_{2})\leq 1\);_  
_(c2) \(c(h_{1},h_{2})=c(h_{2},h_{1})\);_  
_(c3) If \(h_{1}=h_{2}\), \(c(h_{1},h_{2})=1\)._  
**Definition 15**.: \(h_{i}=\)\(<f_{1}^{i},f_{2}^{i},\ldots,f_{k_{(1)}^{i}}^{i}>\in\)\(\text{T2 HPFE}(X)\) _and \(i=1,2\) are set, and then the correlation coefficient between two T2 HPFEs is expressed as follows:_  
\[c(h_{1},h_{2})=\frac{\alpha\cdot d_{3}(h_{1},h_{2}^{c})-(1-\alpha)\cdot d_{3}(h _{1},h_{2})}{\alpha\cdot d_{3}(h_{1},h_{2}^{c})+(1-\alpha)\cdot d_{3}(h_{1},h_ {2})} \tag{12}\]  
_where \(0/0=1\), and \(a\in[0,1]\) is a risk factor whose value depends on the decision-maker's attitude towards the risk. Under a great risk factor, the decision-maker is of risk preference, or otherwise, the decision-maker is of risk aversion. \(c(h_{1},h_{2})\) does not only meet the axiomatization requirements of T2 HPFE but also has the following nature:_  
_(c4) \(c(h_{1},h_{2})=-1\);_  
_(c5) When \(h_{2}=\)\(<(\{1\},\{0\})>\), \(c(h_{1},h_{2})\) is a monotonic decreasing function about \(d_{3}(h_{1},h_{2})\)._  
Proof.: It is easily proven that \(c(h_{1},h_{2})\) satisfies (c1) and (c2), and hereby (c3-c5) will be proved.  
(c3) If \(h_{1}=h_{2}\), \(d_{3}(h_{1},h_{2})=0\) and then \(c(h_{1},h_{2})=\frac{\alpha\cdot d_{3}(h_{1},h_{2}^{c})}{\alpha\cdot d_{3}(h_{1},h_{2}^{c})}=1\).  
(c4) If \(d_{3}(h_{1},(h_{1}^{c})^{c})=0\), then \(c(h_{1},h_{2})=\frac{-(1-\alpha)\cdot d_{3}(h_{1},h_{2})}{(1-\alpha)\cdot d_{3}(h _{1},h_{2})}=-1\).  
\[D_{i}^{+}=\frac{1}{n}{\sum\limits_{j=1}^{n}}\frac{w_{j}}{2^{\frac{1}{4}}}([\frac{1 }{k_{(1)}^{1}}{k_{(1)}^{1}}{\sum\limits_{i=1}^{k_{(1)}^{1}}}(1-\frac{1}{\mid \Gamma_{i}^{1}\mid}\sum\limits_{n_{i}^{l}\in\Gamma_{i}^{1}}{(\mu_{i}^{1})}^{2}+ \frac{1}{\mid\overline{\Psi}_{i}^{1}\mid}\sum\limits_{v_{j}\in\overline{\Psi}_ {i}^{1}}{(v_{i}^{1})}^{2})+\underset{(\Gamma^{1},\Psi^{1})\in h_{1}}{min}(1- \underset{\mu^{1}\in\Gamma^{1}}{\min}(\mu^{1})^{2}+\underset{v^{1}\in\Psi^{1}}{ min}(v^{1})^{2})]\ \ ) \tag{13}\]  
\[D_{i}^{-}=\frac{1}{n}{\sum\limits_{j=1}^{n}}\frac{w_{j}}{2^{\frac{1}{4}}}([ \frac{1}{k_{(1)}^{1}}{\sum\limits_{i=1}^{k_{(1)}^{1}}}(\frac{1}{\mid\Gamma_{i }^{1}\mid}\sum\limits_{n_{i}^{l}\in\Gamma_{i}^{1}}{(\mu_{i}^{1})}^{2}+1-\frac{1 }{\mid\overline{\Psi}_{i}^{1}\mid}\sum\limits_{v_{j}\in\overline{\Psi}_{i}^{1 }}{(v_{i}^{1})}^{2})+\underset{(\Gamma^{1},\Psi^{1})\in h_{1}}{min}(\underset {\mu^{1}\in\Gamma^{1}}{\min}(\mu^{1})^{2}+1-\underset{v^{1}\in\Psi^{1}}{\max}(v ^{1})^{2})]\ \ ) \tag{14}\]  
**Step 4:** After the decision-maker sets the specific value of \(\alpha\) based on his/her risk attitude according to the calculation formula for the correlation coefficient, the closeness of each candidate solution to the ideal solution is calculated:  
\[RC_{i}=\frac{\alpha\cdot D_{i}^{-}-(1-\alpha)\cdot D_{i}^{+}}{\alpha\cdot D_{i }^{-}+(1-\alpha)\cdot D_{i}^{+}} \tag{15}\]  
**Step 5:** The candidate solutions are sorted as per the closeness of each solution to the ideal solution, and the optimal solution is chosen.