## 2 The Establishment of CWM-UM Evaluation Modeling  
### Determination of Index Weight  
#### 2.1.1 IAHP Method to Determine the Subjective Weight  
In this paper, the IAHP method [15] is used to determine the subjective weights of the construction quality evaluation indexes of hydraulic tunnels. Hierarchical analysis has a strong advantage in addressing multi-factor problems, as it can split different factors, determine the corresponding level, and then construct a hierarchical relationship model between the factors to calculate their subjective weights. In this paper, the judgment matrix is established based on the theory of four scales, as shown in Equation (1), and the scale of the matrix is shown in Table 1.  
\[A=\left(a_{ij}\right)_{n\times n}=\begin{bmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ a_{n1}&a_{n1}&\ldots&a_{nn}\end{bmatrix} \tag{1}\]  
where, \(i=1,\ 2,\ \cdots,\ n;\ j=1,\ 2,\ \cdots,n;\ a_{ij}\) is the result of the importance of comparing the two evaluation indicators \(i\) and \(j\) in guideline A, \(a_{ji}=\frac{1}{a_{ij}}\).  
Construct the antisymmetric matrix \(B\) of the judgment matrix \(A\). The values of the matrix \(B\) and its element \(b_{ij}\) can be calculated by Equation (2).  
\[B=\lg A;b_{ij}=-b_{ji} \tag{2}\]  
Construct the optimal transfer matrix \(C\) of the antisymmetric matrix \(B\). The values of the elements \(c_{ij}\) in the matrix \(C\) can be calculated by Equation (2).  
\[c_{ij}=\frac{1}{n}\sum_{g=1}^{n}\left(b_{ig}-b_{ig}\right) \tag{3}\]  
Construct the proposed good agreement matrix \(A^{\prime}\) of the judgment matrix \(A\). The values of the element \(a^{\prime}_{ij}\) in the matrix \(A^{\prime}\) can be calculated by Equation (4).  
\[a^{\prime}_{ij}=10^{c_{ij}} \tag{4}\]  
Normalize the proposed good agreement matrix \(A^{\prime}\) by columns. The values of the elements \(\overline{a}^{\prime}_{ij}\) in the normalized matrix can be calculated by Equation (5).  
\[\overline{a}^{\prime}_{ij}=\frac{a^{\prime}_{ij}}{\frac{\sum\limits_{i}a^{ \prime}_{ij}}{i}} \tag{5}\]  
The sum vector \(a^{\prime}_{i}\) is calculated by adding rows, as shown in Equation (6):  
\[a^{\prime}_{i}=\sum_{i=1}^{n}\overline{a}^{\prime}_{ij} \tag{6}\]  
The weight vector \(\alpha_{i}\) can be calculated by normalizing the sum vector through Equation (7).  
\[\alpha_{i}=\frac{\alpha^{\prime}_{i}}{\sum\limits_{i=1}^{n}\alpha^{\prime}_{i}} \tag{7}\]  
The analytic hierarchy process (AHP) [16] is used to determine the weights of evaluation indicators, consistency tests are needed to ensure that the indicators are harmonized.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Elemental** & **Numerical Value** & **Degree** \\ \hline \multirow{3}{*}{\(a_{ij}\)} & 1 & Indicator \(i\) is as important as indicator \(j\) \\  & 2 & Indicator \(i\) is slightly more important than indicator \(j\) \\  & 3 & Indicator \(i\) is more important than indicator \(j\) \\  & 4 & Indicator \(i\) is very important compared to indicator \(j\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numerical values and significance relationships.  
This paper optimizes the hierarchical analysis method through the optimal matrix method, which enables the calculation results to meet the consistency requirement by itself.