## 2 Methodology
### Pythagorean Hesitant Fuzzy VIKOR Method
#### 2.3.3 The Basic Idea of the Pythagorean Hesitant Fuzzy VIKOR Method
For this study, the ideal solution corresponds to the highest value in every column of the Pythagorean hesitant fuzzy evaluation matrix, and the separation between each option and the ideal solution can be determined by employing the distance equation for Pythagorean hesitant fuzzy numbers outlined in Section 2.3. Calculate the individual regret value \(R_{i}\) and the group utility value \(S_{i}\) for the safety risk level boundaries and the crane under evaluation using the Pythagorean hesitant fuzzy VIKOR method, where \(R_{i}\) represents the worst-case scenario of a particular evaluation criterion, and \(S_{i}\) represents the overall state after considering all evaluation criteria. In gantry crane safety risk assessments, there are two scenarios: one where a particular evaluation criterion is excessively poor, and the other where no individual criterion reaches an extremely poor condition, but the coupling of various indicators leads to poor overall equipment condition. Therefore, both \(R_{i}\) and \(S_{i}\) should be considered, and appropriate weights should be selected for them to calculate the compromise value of the Pythagorean hesitant fuzzy VIKOR. Sorting these values will yield the safety risk level of the crane under evaluation. Given that the objective is solely to ascertain the safety risk level of the crane being assessed, there is no need for an optimal solution judgment. The evaluation process is as follows:  
Step 1: Experts deliberate to represent the boundaries of each safety level using Pythagorean hesitant fuzzy numbers \(\hat{k}_{ij}=\left<h_{ij},g_{ij}\right>\), where \(h_{ij}\) and \(g_{ij}\) are the sets of membership \(u_{ij}\) and non-membership degrees \(v_{ij}\), respectively, for the \(i\)-th safety level \(A_{i}\) boundary concerning the \(j\)-th evaluation criterion \(F_{j}\). We can obtain a Pythagorean hesitant fuzzy matrix for various safety level boundaries and evaluation criteria, as shown in Equation (36):  
\[K_{L}= \begin{bmatrix}\hat{k}_{11}&\hat{k}_{12}&\cdots&\hat{k}_{1n}\\ \hat{k}_{21}&\hat{k}_{22}&\cdots&\hat{k}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \hat{k}_{61}&\hat{k}_{62}&\cdots&\hat{k}_{6n}\end{bmatrix}  \tag{36}\]  
Step 2: Experts deliberate and determine the Pythagorean hesitant fuzzy numbers \(\hat{k}_{xj}=\left<h_{xj},g_{xj}\right>\) for each evaluation criterion of the quasugde container crane equipment under evaluation, where \(h_{xj}\) and \(g_{xj}\) are the sets of membership degree \(u_{xj}\) and non-membership degree \(v_{xj}\) for the \(j\)-th evaluation criterion \(F_{j}\) of the equipment under evaluation. These are then expanded to \(K_{L}\), resulting in the expert decision Pythagorean hesitant fuzzy evaluation matrix, as shown in Equation (37):  
\[K= \begin{bmatrix}\hat{k}_{11}&\hat{k}_{12}&\cdots&\hat{k}_{1n}\\ \hat{k}_{21}&\hat{k}_{22}&\cdots&\hat{k}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \hat{k}_{61}&\hat{k}_{62}&\cdots&\hat{k}_{6n}\\ \hat{k}_{x1}&\hat{k}_{x2}&\cdots&\hat{k}_{xn}\end{bmatrix}  \tag{37}\]  
Step 3: Experts deliberate to determine the relative importance of each evaluation criterion and use the BWM, as described in Section 2.2, to determine the relative weights of each evaluation criterion, as shown in Equation (38):  
\[W=(w_{1},w_{2},\cdots,w_{n}) \tag{38}\]  
where \(w_{i}\in[0,1]\) for \(i=1,2,\cdots,n\), and the weights satisfy Equation (39):  
\[\sum_{i=1}^{n}w_{i}=1 \tag{39}\]  
Step 4: Determine the Pythagorean hesitant fuzzy ideal solution \(K^{+}\) and anti-ideal solution \(K^{-}\) using the calculation method shown in Equation (30). Step 5: Determine the individual regret value \(R_{i}\) and the group utility value \(S_{i}\) for each alternative and safety evaluation standard level using the Pythagorean hesitant fuzzy VIKOR method.