## 3 Methodology
### Fuzzy Hierarchy Analysis
\begin{array}{ll}1&\text{if }m_{j}\geq m_{i}\\ 0&\text{if }l_{i}\geq u_{j}\\ \frac{l_{i}-u_{j}}{\big{(}m_{j}-u_{j}\big{)}-(m_{i}-l_{i})}&\text{otherwise} \end{array} \right. \tag{10}\]  
when \(m_{j}<l_{i}<u_{j}<m_{i}\), d is the value of the transverse coordinate of the highest intersection point between \(S_{j}\) and \(S_{i}\). Meanwhile, we need the value of \(N\big{(}S_{j}\geq S_{i}\big{)}\) and \(N\big{(}S_{i}\geq S_{j}\big{)}\) to compare \(S_{i}\) and \(S_{j}\). The degree of probability that a convex fuzzy number is greater than \(k\) convex fuzzy numbers \(S_{i}\)\((i=1,\;2,...,k)\) can be defined as:  
\[N(S\geq S_{1},S_{2},\ldots,S_{k})=N\big{[}(S\geq S_{1})\text{and}(S\geq S_{2}) \text{ and} \tag{11}\] \[\ldots\text{and}(S\geq S_{k})]=\text{min }N(S\geq S_{i}),i=1,2,3,\ldots,k\]  
Assume that:  
\[d^{\prime}(A_{i})=minN(S_{i}\geq S_{k}) \tag{12}\]  
The weight values are as follows:  
\[W^{\prime}=\big{(}d^{\prime}(A_{1}),d^{\prime}(A_{2}),\ldots,d^{\prime}(A_{n} )\big{)}^{\text{T}} \tag{13}\]  
where \(A_{i}\) (\(i=1,2,\ldots,n\)) are \(n\) elements. By normalization, the normalized weight vector is:  
\[W=(d(A_{1}),d(A_{2}),\ldots,d(A_{n}))^{\text{T}} \tag{14}\]  
In this study, we applied the triangular fuzzy transformation scale of Table 1 as the evaluation model. We define a problem with \(k\) indicator factors, where the relative importance of factor \(i\) to \(j\) is denoted by the triangular fuzzy number as \(\widetilde{a}_{ij}=\big{(}l_{ij},m_{ij},u_{ij}\big{)}\). For example, if \(j\) is thought to be Weakly more important, it could be represented as \(\widetilde{a}_{ji}=(1,3/2,2)\). Then, the comparison matrix \(\widetilde{A}\) can be constructed as:  
\[\widetilde{A}= \begin{bmatrix}1&\widetilde{a}_{12}&\ldots&\widetilde{a}_{1n} \\ \widetilde{a}_{21}&1&\ldots&\widetilde{a}_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ \widetilde{a}_{n1}&\widetilde{a}_{n2}&\ldots&1\end{bmatrix} = \begin{bmatrix}1 &\widetilde{a}_{12}&\ldots&\widetilde{a}_{1n}\\ 1/\widetilde{a}_{12}&1&\ldots&\widetilde{a}_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ 1/\widetilde{a}_{1n}&1/\widetilde{a}_{2n}&\ldots&1\end{bmatrix}  \tag{15}\]  
Consistency index (CI) and consistency ratio (CR) \begin{table}
\begin{tabular}{c c c} \hline
**Linguistic Scale** & **Triangular Fuzzy Scale** & **Triangular Fuzzy Reciprocal Scale** \\ \hline Just equal & (1, 1, 1) & (1, 1, 1) \\ Equally important & (1/2, 1, 3/2) & (2/3, 1, 2) \\ Weakly more important & (1, 3/2, 2) & (1/2, 2/3, 1) \\ Strongly more important & (3/2, 2, 5/2) & (2/5, 1/2, 2/3) \\ Very strongly more important & (2, 5/2, 3) & (1/3, 2/5, 1/2) \\ Absolutely more important & (5/2, 3, 7/2) & (2/7, 1/3, 2/5) \\ \hline \end{tabular}
\end{table} 
Table 1: Linguistic scales and fuzzy scales for importance. Since the trait roots continuously depend on \(\widetilde{a}_{ij}\), the larger \(y_{max}\) is compared to \(n\), the more serious the degree of non-consistency of the data in matrix \(\widetilde{A}\) is, and the standardized eigenvectors corresponding to \(y_{max}\) do not truly reflect the weights of individual factors. Therefore, it is necessary to do a CI test on the judgment matrix provided by the decision-maker to decide whether to accept it or not. For the decisions to reach a good level of quality, the consistency of the assessment must be analyzed. As \(CR\leq 0.1\), it can proceed to the next step; otherwise, the judgment matrix should be appropriately corrected [53]. The triangular fuzzy number \(\widetilde{a}_{ij}=\left(l_{ij},m_{ij},u_{ij}\right)\) can be defuzzied into a clear value in the following way:  
\[\left(a_{ij}^{\alpha}\right)^{x}=\left[x\cdot l_{ij}^{\alpha}+\left(1-x \right)u_{ij}^{\alpha}\right],\quad 0\leq x\leq 1,0\leq\alpha\leq 1 \tag{16}\]  
\[l_{ij}^{\alpha}=\left(m_{ij}-l_{ij}\right)\times\alpha+l_{ij} \tag{17}\]  
\[u_{ij}^{\alpha}=u_{ij}-\left(u_{ij}-m_{ij}\right)\times\alpha \tag{18}\]  
where \(l_{ij}^{\alpha}\) and \(u_{ij}^{\alpha}\) denotes the left-end and the right-end value of \(\alpha\)-cut for \(a_{ij}\).