## 2 Methodology
### Pythagorean Hesitant Fuzzy VIKOR Method  
#### 2.3.1 Overview of Pythagorean Hesitant Fuzzy Sets  
Consider a universe of discourse denoted by \(X\). Within this universe, a set can be characterized as a Pythagorean fuzzy set on \(X\), which is represented as Equation (7):  
\[A=\left\{<x,u_{A}(x),v_{A}(x)>\mid x\in X\right\} \tag{7}\]  
Here, \(u_{A}(x)\) denotes the membership degree of an element \(x\) in set \(A\), and \(v_{A}(x)\) denotes the non-membership degree of \(x\) in set \(A\). These functions must adhere to the condition, as shown in Equation (8):  
\[u_{A}^{2}(x)+v_{A}^{2}(x)\leq 1,\forall x\in X,u_{A}(x),v_{A}(x)\in[0,1] \tag{8}\]  
Extending the concept of Pythagorean fuzzy sets, we introduce a universe of discourse denoted by \(X\). In this context, a set is defined as a Pythagorean hesitant fuzzy set on \(X\), which is given by Equation (9):  
\[A=\left\{<x,h_{A}(x),g_{A}(x)>\mid x\in X\right\} \tag{9}\]  
Here, \(h_{A}(x)\) and \(g_{A}(x)\) are non-empty finite subsets of \([0,1]\), representing the possible membership degree set and the possible non-membership degree set of element \(x\) in set \(A\), satisfying the condition as shown in Equation (10):  
\[u_{A}^{2}(x)+v_{A}^{2}(x)\leq 1,\forall x\in X,u_{A}(x)\in h_{A}(x),v_{A}(x) \in g_{A}(x) \tag{10}\]  
The hesitation degree of \(x\) toward \(A\) is denoted as \(\pi_{A}(x)\), which is defined by Equation (11):  
\[\pi_{A}(x)=\sqrt{1-u_{A}^{2}(x)-v_{A}^{2}(x)} \tag{11}\]  
Such a set as \(\alpha=\left<h_{\alpha},g_{\alpha}\right>\) is referred to as a Pythagorean hesitant fuzzy number, abbreviated as PHFN. For a Pythagorean hesitant fuzzy number \(\alpha=\left<h_{\alpha},g_{\alpha}\right>\), the score function \(S(\alpha)\), which takes values in the interval \([-1,1]\), is defined as Equation (12):  
\[S(\alpha)=\frac{1}{l(h)}\sum\limits_{u\in h_{\alpha}}u^{2}-\frac{1}{l(g)}\sum \limits_{v\in g_{\alpha}}v^{2} \tag{12}\]  
Similarly, the accuracy function \(G(\alpha)\), ranging in \([0,1]\), is given by Equation (13):  
\[G(\alpha)=\frac{1}{l(h)The comparison rules for two Pythagorean hesitant fuzzy numbers, \(\alpha_{i}=\langle h_{i},g_{i}\rangle\), \(i=1,2\) are as follows:  
* If \(S(\alpha_{1})>S(\alpha_{2})\), then \(\alpha_{1}\succ\alpha_{2}\);
* If \(S(\alpha_{1})=S(\alpha_{2})\), then If \(G(\alpha_{1})>G(\alpha_{2})\), then \(\alpha_{1}\succ\alpha_{2}\);
* If \(G(\alpha_{1})=G(\alpha_{2})\), then \(\alpha_{1}\sim\alpha_{2}\). * For the operations of Pythagorean fuzzy numbers:  
Let \(\alpha=\langle h,g\rangle\), \(\alpha_{i}=\langle h_{i},g_{i}\rangle\), \(i=1,2\) and \(\lambda>0\), then  
* \(\alpha_{1}\oplus\alpha_{2}=\left\langle\begin{array}{cc}\cup&\left\{\sqrt{u_ {\alpha_{1}}^{2}+u_{\alpha_{2}}^{2}-u_{\alpha_{1}}^{2}u_{\alpha_{2}}^{2}} \right\},&\cup&\left\{v_{\alpha_{1}}v_{\alpha_{2}}\right\}\right\rangle\); \(u_{\alpha_{2}}\in h_{2}&v_{\alpha_{2}}\in g2\)
* \(\alpha_{1}\otimes\alpha_{2}=\left\langle\begin{array}{cc}\cup&\left\{u_{ \alpha_{1}}u_{\alpha_{2}}\right\},&\cup&\left\{\sqrt{v_{\alpha_{1}}^{2}+v_{ \alpha_{2}}^{2}-v_{\alpha_{1}}^{2}v_{\alpha_{2}}^{2}}\right\}\right\rangle\); \(u_{\alpha_{2}}\in h_{2}&v_{\alpha_{2}}\in g2\)
* \(\lambda\alpha=\left\langle \begin{array}{cc}\cup&\left\{\sqrt{1-\left(1-u_{ \alpha}^{2}\right)^{\lambda}}\right\},&\cup&\left\{v_{\alpha}\right\}\right\} \end{array} \right\rangle\);
* \(\alpha^{\lambda}=\left\langle \begin{array}{cc}\cup&\left\{u_{\alpha_{1}}^{ \lambda}\right\},&\cup&\left\{\sqrt{1-\left(1-\left(v_{\alpha}^{2}\right)^{ \lambda}}\right\}\right\rangle\end{array} \right\rangle\). Suppose \(\alpha_{1}=\langle h_{1},g_{1}\rangle\), \(\alpha_{2}=\langle h_{2},g_{2}\rangle\) are two Pythagorean hesitant fuzzy numbers.