## 3 Methodology  
### Fuzzy Hierarchy Analysis  
In Multiple Criteria Decision-Making (MCDM), FAHP can reconcile the fuzziness and uncertainty of a problem well and, therefore, is a practical and popular method with a wide range of application [44]. Specifically, the incorporation of fuzzy mathematics into quantum evaluation allows for a more nuanced and flexible approach to measurement and analysis. By leveraging hierarchical structures, researchers can more effectively capture the complex relationships and interactions between quantum variables, leading to more accurate and comprehensive evaluations. [PERSON] and [PERSON] [45] extended [PERSON]'s [46] AHP method directly with the use of a triangular fuzzy number (TFN), which compared fuzzy ratios described by triangular membership functions. Numerous scholars have since exploited this method in their studies [47; 48; 49; 50]. However, [PERSON] [51; 52] proposed a pairwise comparison scale using the TFN to conduct pairwise comparisons with integrated range values, which can make the transformation of fuzzy and uncertain problems clearer and measurable. Moreover, [PERSON]'s method of FAHP stands out for its simple steps and calculations compared to others. Therefore, this study employs [PERSON]'s approach to compute the weights of obtained indicators and perform pairwise comparisons. The outline for FAHP is described in the following. Let \(X=\{x_{1},x_{2},x_{3},\ldots,x_{n}\}\) be an object set, and \(U=\{u_{1},u_{2},u_{3},\ldots,u_{m}\}\) is a goal set. According to [PERSON]'s extent analysis, each object is taken, and extent analysis for each goal, \(g_{i}\) is performed, respectively. Therefore, \(m\) extent analysis values for each object can be obtained with the following equations:  
\[M_{g_{i}}^{1},M_{g_{i}}^{2},\ldots,M_{g_{i}}^{m},i=1,2,\ldots,n \tag{1}\]  
where all the \(M_{g_{i}}^{j}\) (\(\mathbf{j}=1,2,\ldots,m\)) are TFNs whose parameters are \(l\), \(m\), and \(u\). They are the lowest possible value, the highest possible value, and the largest possible value, respectively. The triplet (\(l\), \(m\), \(u\)) can represent a TFN. A triplet array can represent a fuzzy set. The value of the fuzzy synthesis range of the \(i\)th object is defined as follows:  
\[S_{i}=\sum_{j=1}^{m}M_{g_{i}}^{j}\otimes\left[\sum_{i=1}^{n}\sum_{j=1}^{m}M_{g_ {i}}^{j}\right]^{-1} \tag{2}\]  
and:  
\[\sum_{j=1}^{m}M_{ij}=\left(\sum_{j=1}^{m}l_{ij},\sum_{j=1}^{m}m_{ij},\sum_{j=1 }^{m}u_{ij}\right),i=1,2,\ldots,n \tag{3}\]  
\[\sum_{i=1}^{n}\sum_{j=1}^{m}M_{ij}=\left(\sum_{i=1}^{n}\sum_{j=1}^{m}l_{ij}, \sum_{i=1}^{n}\sum_{j=1}^{m}m_{ij},\sum_{i=1}^{n}\sum_{j=1}^{m}u_{ij}\right) \tag{4}\]  
\[\left[\sum_{i=1}^{N}\sum_{j=1}^{m}M_{ij}\right]^{-1}=\left(\frac{1}{\sum_{i= 1}^{n}\sum_{j=1}^{m}u_{ij}},\frac{1}{\sum_{i=1}^{n}\sum_{j=1}^{m}m_{ij}},\frac {1}{\sum_{i=1}^{n}\sum_{j=1}^{m}l_{ij}}\right) \tag{5}\]  
* Fuzzy Arithmetic: Each triangular fuzzy value is evaluated by experts based on Table 1. The evaluation results of multiple experts can be averaged as a triangular fuzzy number by applying operations of triangular fuzzy numbers. The algorithm of two TFNs \(A(l_{1},m_{1},u_{1})\) and \(B(l_{2},m_{2},u_{2})\) is as follows. \[\text{A}_{1}\oplus\text{B}_{2}=(l_{1}+l_{2},m_{1}+m_{2},u_{1}+u_{2})\] (6)  
\[\text{A}_{1}\otimes\text{B}_{2}\approx(l_{1}l_{2},m_{1}m_{2},u_{1}u_{2}), \text{for}l_{i}>0,m_{i}>0,u_{i}>0,i=1,2 \tag{7}\]  
\[\text{A}_{1}/\text{B}_{2}=\left(\frac{l_{1}}{u_{2}},\frac{m_{1}}{m_{2}},\frac {u_{1}}{l_{2}}\right),\text{for}l_{i}>0,m_{i}>0,u_{i}>0,i=1,2 \tag{8}\]  
\[\text{A}_{1}^{-1}\approx\left(\frac{1}{u_{1}},\frac{1}{m_{1}},\frac{1}{l_{1}} \right),\text{for}l_{1}>0,m_{1}>0,u_{1}>0 \tag{9}\]  
* Compare the values of fuzzy synthesis range \(S_{i}\)It is first necessary to calculate the degree of probability that \(S_{j}=\big{(}l_{j},m_{j},u_{j}\big{)}\geq S_{i}=(l_{i},m_{i},u_{i})\). As follows:  
\[M\big{(}S_{j}\geq S_{i}\big{)}=\text{height}\big{(}S_{i}\cap S_{j}\big{)}=\left\{