## 3 Methodology
### Fuzzy VIKOR Method  
The VIKOR method was firstly proposed by [PERSON] and [PERSON] for MCDM with conflicting criteria [61]. The VIKOR technique is capable of ranking alternatives and selecting the optimal solution among different potential alternatives. The VIKOR method determines the most compromising solution by measuring the \"closeness\" of the individual alternative to the ideal solution. The compromise solution is often regarded as the solution with the biggest advantage [62; 63; 64]. For many multi-criteria problems with various conflicting criteria, decision makers often need to deal with imprecise and uncertain information. We used the fuzzy VIKOR method to evaluate the overall performance of various potential alternatives and prioritize the alternatives. The detailed phases of the developed model are described as follows. Step 1: identify the linguistic variables for alternative assessment. The linguistic variables are used to assess the performance of all alternatives on each criterion and rate the alternatives through fuzzy rating. In this research, linguistic variables are divided into 5 levels, which are shown in Table 3. Each linguistic variable can be transformed into a corresponding fuzzy number, as shown in Figure 4. The membership function can be defined through Equation (5). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Linguistic Variables** & **Abbreviations** & **Triangular Fuzzy Scale** \\ \hline Very low & VL & (0.00, 0.00, 0.25) \\ Low & L & (0.00, 0.25, 0.50) \\ Fair & F & (0.25, 0.50, 0.75) \\ High & H & (0.50, 0.75, 1.00) \\ Very high & VH & (0.75, 1.00, 1.00) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The linguistic scales for PPE performance evaluation in the fuzzy VIKOR method. Figure 3: The fuzzy triangular number membership function. Step 2: establish the fuzzy decision matrix for the PPE performance assessment with respect to each criterion. \[\widetilde{D}= \begin{bmatrix}p_{11}&p_{12}&p_{13}&\cdots&p_{1n}\\ p_{21}&p_{22}&p_{23}&\cdots&p_{2n}\\ \vdots&\vdots&\vdots&&\ddots&\vdots\\ p_{m1}&p_{m2}&p_{m3}&\cdots&p_{mm}\end{bmatrix}  \tag{6}\]  
where \(p_{ij}\) is the assessment value of the \(j\)th criterion of the \(i\)th alternative in the initial matrix \(\widetilde{D}\), \(i=1,\ldots,m,j=1,\ldots,n\). \(m\) is the number of PPE alternatives, and \(n\) is the total number of evaluation criteria. Step 3: generate the normalized fuzzy decision matrix. For each criterion, determine the worst value \(f_{j}^{-}\) and the best value \(f_{j}^{*}\), where \(j=1,\,2,\,\ldots,\,n\). Calculate the normalized value (\(v_{ij}\)) for alternative \(i\) with respect to criterion \(j\) based on Equation (7). Then, the normalized fuzzy decision matrix is determined. \[v_{ij}=\frac{w_{j}\Big{(}f_{j}^{*}-f_{ij}\Big{)}}{\Big{(}f_{j}^{*}-f_{j}^{-} \Big{)}},\,\,i=1,\ldots,m,\,\,j=1,\ldots,n \tag{7}\]  
where \(w_{j}\) represents the weight value of criterion \(j,f_{j}^{-}\) represents the worst value among the assessment values of criterion \(j,f_{j}^{*}\) represents the best value among the assessment values of criterion \(j\), and \(f_{ij}\) refers to the value score for alternative \(i\) with respect to criterion \(j,i=1,\,\ldots,m,j=1,\,\ldots,\,n\). Step 4: calculate the Utility, Regret, and VIKOR index values. The Utility (\(S_{i}\)), Regret (\(G_{i}\)), and VIKOR index (\(Q_{i}\)) values are computed using Equations (8)-(10). \[S_{i}=\sum_{j=1}^{n}\frac{w_{j}\Big{(}f_{j}^{*}-f_{ij}\Big{)}}{\Big{(}f_{j}^{ *}-f_{j}^{-}\Big{)}},\,\,i=1,\ldots,m \tag{8}\]  
Figure 4: Fuzzy numbers and the corresponding linguistic variables. \[G_{i}=max_{k}\left[\frac{w_{j}\left(f_{j}^{*}-f_{ij}\right)}{\left(f_{j}^{*}-f_{j} ^{-}\right)}\right] \tag{9}\]  
\[Q_{i}=v\frac{\left(S_{i}-S^{*}\right)}{\left(S^{-}-S^{*}\right)}+\left(1-v \right)\frac{\left(G_{i}-G^{*}\right)}{\left(G^{-}-G^{*}\right)} \tag{10}\]  
where \(i=1,\ldots,\)\(m;\)\(w_{j}\) represents the weight of criterion \(j;\)\(v\) is the group utility weight; and \(\left(1-v\right)\) represents the weight of individual regret.