## 2 Methodology
### Pythagorean Hesitant Fuzzy VIKOR Method
#### 2.3.2 The Basic Principle of the VIKOR Method  
The VIKOR method is a multi-attribute decision-making approach that falls under the category of optimized compromise solutions in multi-attribute decision making. It utilizes an LP metric aggregation function and adopts a compromise philosophy. This method initially analyzes the ideal and anti-ideal solutions and then calculates various alternative solutions. The options are ordered and assessed in accordance with the ideal solution, achieving a compromise through mutual concessions among the attributes of each solution [39]. The strength of VIKOR lies in its capability to offer compromise solutions amidst a series of conflicting evaluation criteria, thereby facilitating problem resolution. Compared to the TOPSIS method, VIKOR is more effective in avoiding inversions, and its outcomes are generally more acceptable to decision makers. Considering a decision-making scenario with \(m\) alternative solutions, we define the set of alternatives as Equation (15):  
\[X=\{X_{1},X_{2},\cdots,X_{m}\} \tag{15}\]  
Furthermore, each alternative is assessed based on \(n\) evaluation criteria, denoted by Equation (16):  
\[Y=\{Y_{1},Y_{2},\cdots,Y_{n}\} \tag{16}\]  
Here, \(h_{ij}\) represents the evaluation value of the \(j\)-th criterion for the \(i\)-th alternative. The weights assigned to each evaluation criterion are represented by the weight vector, as shown in Equation (17):  
\[W=\{w_{1},w_{2},\cdots,w_{n}\} \tag{17}\]The VIKOR method, derived from the LP metric function, is founded on Equation (18):  
\[L_{p,i}=\left\{\sum\limits_{j=1}^{n}\left[w_{j}\left(h_{j}^{+}-h_{ij}\right)/ \left(h_{j}^{+}-h_{j}^{-}\right)\right]^{p}\right\}^{1/p},1\leq p\leq\infty;i=1, 2,\cdots,m \tag{18}\]  
The calculation process for the VIKOR method is as follows:  
1. Data Normalization Normalize the data using the method shown in Equation (19): \[h_{ij}=\frac{x_{ij}}{\sqrt{\sum\limits_{i=1}^{m}x_{ij}^{2}}},i=1,2,\cdots,m;j=1,2,\cdots,n\] (19)
2. Determining the Ideal and Anti-ideal Solutions Calculate the ideal and anti-ideal solutions using the method shown in Equation (20): \[h_{j}^{+}=\text{max}h_{ij},h_{j}^{-}=\text{min}h_{ij}\] (20)
3. Calculating the Distance Ratios to the Ideal and Anti -Ideal Solutions This includes calculating the group utility value \(S_{i}\) and individual regret value \(R_{i}\) for each alternative. The calculation methods are shown in Equations (21) and (22): \[S_{i}=\sum\limits_{j=1}^{n}w_{j}\left(h_{j}^{+}-h_{ij}\right)/\left(h_{j}^{+}- h_{j}^{-}\right)\] (21)  
\[R_{i}=\text{max}\left[w_{j}\left(h_{j}^{+}-h_{ij}\right)/\left(h_{j}^{+}-h_{j }^{-}\right)\right] \tag{22}\]  
Here, \(S_{i}\) is the group utility value, where a smaller value indicates a greater group benefit. \(R_{i}\) is the individual regret value, where a smaller value indicates less individual regret. 4. Calculating the Compromise Value After calculating 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 (23):  
\[Q_{i}=\alpha S_{i}+(1-\alpha)R_{i},i=1,2,\cdots,m \tag{23}\]  
Here, \(Q_{i}\) is the compromise value for the \(i\)-th alternative, and \(\alpha\) is the decision-making coefficient. The coefficient \(\alpha\) reflects the weight of the group benefit value, while \(1-\alpha\) reflects the weight of the individual regret value. When \(\alpha\) is greater than 0.5, the group utility is primarily considered; when \(\alpha\) is less than 0.5, the individual regret mechanism is primarily considered for decision making; when \(\alpha\) equals 0.5, it represents a balanced consideration of both group utility and individual regret. The most commonly used decision-making method is when \(\alpha\) is set to 0.5. 5. Ranking of Alternatives Once the compromise values \(Q_{i}\) for all alternatives are calculated, arrange them in ascending order. The smaller the compromise value of an alternative, the better it is considered. 6.