## 3 Study Methods and Evaluation Process
### Weighting of Evaluation Indicators
After the hierarchical structure is established, a judgment matrix is constructed by comparing the importance of the two indicators within each level to each other. If the previous level element \(\mathrm{B_{k}}\) is used as a criterion for the next level element \(\mathrm{C_{1}}\), \(\mathrm{C_{2}}\),..., \(\mathrm{C_{n}}\) has a dominant relationship, for these n elements, a two-by-two judgment matrix \(\mathrm{A=(a_{ij})n}\)\(\times\) n is obtained, where \(\mathrm{a_{ij}}\) indicates the importance values of factor \(\mathrm{C_{i}}\) and factor \(\mathrm{C_{j}}\) relative to the target \(\mathrm{B_{k}}\). It is clear that the judgment matrix \(\mathrm{A}\) is a positive and negative matrix with the following properties:  
1. \(\mathrm{a_{ij}>0}\)
2. \(\mathrm{a_{ij}=1/a_{ij}(i\
eq j)}\)
3. \(\mathrm{a_{ii}=1(i,j=1,2,3,\ldots,n)}\)  
For the positive and negative matrix \(\mathrm{A}\), if for any i, j, k there is \(\mathrm{a_{ij}}-\mathrm{a_{jk}=a_{ik}}\), then the judgment matrix \(\mathrm{A}\) is said to be consistent. In the hierarchical analysis method, the importance level is often expressed according to the numerical value, and the importance level between two indicators is scored according to the experts' experience and theoretical knowledge for quantification, and then the importance level is formed into a judgment matrix. The following Table 2 shows a commonly used 1-9 scale method. 4. Consistency check of the judgment matrix To ensure that the importance scores between the indicators are reasonable, the consistency of the constructed judgment matrix needs to be checked. Assume that the characteristic roots of the discriminant matrix \(\mathrm{A}\) are \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\); if satisfied by \(\mathrm{Ax}=\lambda\mathrm{x}\), there will be  
\[\sum_{i=1}^{n}\lambda_{i}=n \tag{1}\]  
When matrix \(\mathrm{A}\) has perfect consistency, the largest eigenroot is \(\lambda_{\mathrm{max}}=n\), and the rest of the eigenroots are zero; When matrix \(\mathrm{A}\) is not perfectly consistent, then we have \(\lambda_{\mathrm{max}}>n\). The remaining characteristic roots \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\) are related as follows:  
\[\sum_{i=2}^{n}\lambda_{i}=n-\lambda_{\mathrm{max}} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Importance Rating** & **Assignment \(\mathrm{a_{ij}}\)** \\ \hline
1 & elements I and j are equally important & 1 \\
2 & element i is slightly more important than element j & 3 \\
3 & element i is significantly more important than element j & 5 \\
4 & element i is strongly more important than element j & 7 \\
5 & element i is extremely more important than element j & 9 \\
6 & element i is slightly less important than element j & 1/3 \\
7 & element i is significantly less important than element j & 1/5 \\
8 & It is not important that element i is stronger than element j & 1/7 \\
9 & element i is less important than element j in the extreme & 1/9 \\ \hline \hline \end{tabular} Note: \(\mathrm{a_{ij}=(2,4,6,8,1/2,1/4,1/6,1/8)}\) indicates a level of importance between \(\mathrm{a_{ij}=(1,3,5,7,9,1/3,1/5,1/7,1/9)}\).  
\end{table} 
Table 2: 1–9 scale method. From the above conclusions, it is clear that when the judgment matrix is not guaranteed to have complete consistency, the corresponding characteristic roots of the judgment matrix will change so that the changes in the characteristic roots of the judgment matrix can be used to check the degree of consistency of the judgment.