## 2 Methodology
### Pythagorean Hesitant Fuzzy VIKOR Method
#### 2.3.3 The Basic Idea of the Pythagorean Hesitant Fuzzy VIKOR Method  
Consider a set of \(m\) alternative options and a set of \(n\) evaluation criteria, denoted by Equations (26) and (27):  
\[A=\left\{A_{1},A_{2},\cdots A_{m}\right\} \tag{26}\]  
\[F=\left\{F_{1},F_{2},\cdots,F_{n}\right\} \tag{27}\]  
Subject matter experts, leveraging their expertise, assign evaluation values to each criterion for every alternative. The evaluation of the \(i\)-th alternative \(A_{i}\), for the \(j\)-th criterion \(F_{j}\), is encapsulated by a Pythagorean hesitant fuzzy number:  
\[\hat{k}_{ij}=\left\langle h_{ij},g_{ij}\right\rangle\]  
where \(h_{ij}\) and \(g_{ij}\) are the sets of membership degree \(u_{ij}\) and non-membership degree \(v_{ij}\) of the \(i\)-th alternative \(A_{i}\) for the \(j\)-th evaluation criterion \(F_{j}\), respectively. We can obtain a Pythagorean hesitant fuzzy matrix about various alternative options and evaluation criteria. The computational process for the Pythagorean hesitant fuzzy VIKOR method is as follows:  
1. Construct the Expert Decision Fuzzy Matrix \[K=\left(\hat{k}_{ij}\right)_{m\times n}= \begin{bmatrix}\hat{k}_{11}&\hat{k}_{1 2}&\cdots&\hat{k}_{1n}\\ \hat{k}_{21}&\hat{k}_{22}&\cdots&\hat{k}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \hat{k}_{m1}&\hat{k}_{m2}&\cdots&\hat{k}_{mn}\end{bmatrix} \] (28)  
In Equation (28), \(m\) is the number of alternative solutions, \(n\) is the number of evaluation criteria, and \(\hat{k}_{ij}\) is the fuzzy evaluation result of the \(i\)-th alternative under the \(j\)-th criterion made by experts. 2. Determine the Weights of Each Evaluation Criterion  
Employing a method for weight determination, we identify the relative weights of the evaluation criteria. These weights are denoted by the set \(W\), which is defined as Equation (29):  
\[W=\left\{w_{1},w_{2},\cdots,w_{n}\right\} \tag{29}\]  
Here, \(w_{i}\) represents the weight of the \(i\)-th evaluation criterion. Determine the Pythagorean Hesitant Fuzzy Ideal Solution \(K^{+}\) and Anti-Ideal Solution \(K^{-}\) The calculation is based on the method shown in Equation (30): \[ \begin{array}{l}K^{+}=\left(k_{1}^{+},k_{2}^{+},\cdots,k_{n}^{+}\right)\\ K^{-}=\left(k_{1}^{-},k_{2}^{-},\cdots,k_{n}^{-}\right)\end{array} \] (30) Here, \(k_{j}^{+}=\max_{i}(\hat{k}_{ij})\), \(k_{j}^{-}=\min_{i}(\hat{k}_{ij})\). 4. Determine the Individual Regret Value \(R_{i}\) and Group Utility Value \(S_{i}\) for Each Alternative with Respect to Each Safety Evaluation Standard Level The calculation methods are shown in Equations (31) and (32): \[S_{i}=\sum_{j=1}^{n}\frac{w_{j}\times d\left(\hat{k}_{j}^{+},\hat{k}_{ij} \right)}{d\left(\hat{k}_{j}^{+},\hat{k}_{j}^{-}\right)},i=1,2,\cdots,m\] (31) \[R_{i}=\max_{j}\sum_{j=1}^{n}\frac{w_{j}\times d\left(\hat{k}_{j}^{+},\hat{k}_{ ij}\right)}{d\left(\hat{k}_{j}^{+},\hat{k}_{j}^{-}\right)},i=1,2,\cdots,m\] (32)
5. Calculate the Compromise Value \(Q_{i}\) After obtaining the group utility value \(S_{i}\) and individual regret value \(R_{i}\) for each alternative, use these to calculate the compromise value for each alternative. The specific calculation method is shown in Equation (33): \[Q_{i}=\alpha S_{i}+(1-\alpha)R_{i},i=1,2,\cdots,m\] (33) where \(Q_{i}\) is the compromise value for the \(i\)-th alternative, and \(\alpha\) is the decision-making coefficient. 6. Ranking of Alternatives Order the options according to their compromise values \(Q_{i}\) from lowest to highest; a smaller value indicates a more favorable alternative. 7. Acquiring the Optimal Solution The alternative with the smallest compromise value \(Q_{i}\) denoted as \(X^{(1)}\), is considered the optimal solution if it satisfies the following two conditions: Condition 1: \(Q(X^{(2)})-Q(X^{(1)})\geq DQ\), where \(X^{(2)}\) represents the alternative with the second smallest compromise value \(Q_{i}\) when sorted in ascending order.