## 2 Materials and Methods
### Ranking Key Indicators  
The fuzzy AHP method is used to determine the weights of factors and rank key indicators. Fuzzy numbers incorporate uncertainty in the decision-making process, making it more realistic. Matrix representation and pairwise computations are conducted using fuzzy numbers. In this analysis, triangular fuzzy numbers \(\widetilde{\widetilde{C}}=(c_{1},c_{2},c_{3})\) with the membership function \(\mu(x)\), shown in Figure 5, were applied. This function maps \((-\infty,\infty)\) to \([0,1]\). Basic operations on triangular fuzzy numbers \(\widetilde{C}=(c_{1},c_{2},c_{3})\) and \(\widetilde{D}=(d_{1},d_{2},d_{3})\) are defined as follows:  
\[\widetilde{C}\oplus\widetilde{D}=(c_{1}+d_{1},c_{2}+d_{2},c_{3}+d_{3}), \tag{5}\]  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]where \(\oplus\) represents the addition of fuzzy numbers, while \(\otimes\) represents the multiplication of fuzzy numbers. Additionally, multiplying by a constant crisp value \(k\) is defined as  
\[k\cdot\widetilde{C}=(k\cdot c_{1},k\cdot c_{2},k\cdot c_{3}), \tag{7}\]  
while the reciprocal of the fuzzy number is determined as  
\[\widetilde{C}^{-1}=\Big{(}c_{3}^{-1},c_{2}^{-1},c_{1}^{-1}\Big{)}. \tag{8}\]  
When making comparisons, experts use the fuzzified scale, as shown in Figure 6. The fuzzy number (1,1,3) represents the equality of alternatives, while (7,9,9) represents absolute dominance. Other basic values of the scale are characterised by the fuzzy distance \(\delta=2\), and can be generally represented as \((c-\delta,c,c+\delta)\). If necessary, during pairwise comparisons, intermediate values can be used, for which \(\delta=1\). The previous figure also shows the fuzzy values for weak (\(W\)), strong (\(S\)), and demonstrated (\(D\)) dominance. Table 3 shows the comparison scale. Experts at each level perform pairwise comparisons of elements. Expert \(k\) describes preferences using a matrix \(F^{(k)}=\left[\widetilde{f_{ij}^{\ (k)}}\right]_{n\ \text{\tiny{$x$}}\ \text{\tiny{$H$}}}\), whose dimensions correspond to the number of elements, with standard characteristics of the pairwise comparison matrix \(\widetilde{f_{ij}^{\ (k)}}=\left(\widetilde{f_{ji}^{\ (k)}}\right)^{-1}\) and \(\widetilde{f_{ii}^{\ (k)}}=1\). Individual preferences are aggregated using the experience coefficient \(\varphi_{k}\)  
\[\widetilde{f_{ij}}=\prod_{k=1}^{\varkappa}\left(\widetilde{f_{ij}^{\ (k)}}\right)^{\phi_{k}}, \tag{9}\]  
where \(\widetilde{f_{ij}}\) is an element of the fuzzy aggregate matrix. Figure 5: Triangular fuzzy number membership function. Figure 6: Fuzzy numbers used in the comparison. In order to fulfil the consistency condition, the \(CCI\) (centric consistency index) is applied, with an acceptable value of less than 0.35 for comparing four elements, and less than 0.37 for comparing more elements [51]. For these values, it can be considered that the experts' comparisons are consistent, and that the aggregate matrix is acceptable. Based on the aggregate matrix, fuzzy weights  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]0  
are determined for \(\varkappa\) different compared elements, where \(j=1,\ldots,\varkappa\). Weights are determined by levels, namely, for aspects  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]1  
where, based on [52],  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]2  
where \(i=1,2,3\); then, for factors in relation to aspects,  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]3  
\[\widetilde{C}\otimes\widetilde{D}=(c_{1}\cdot d_{1},c_{2}\cdot d_{2},c_{3} \cdot d_{3}), \tag{6}\]4  
where \(p_{ij}\) represents the fuzzy assessment of the influence of the \(i\)-th factor on the \(j\)-th aspect.