## 2 The Establishment of CWM-UM Evaluation Modeling
### Uncertainty Measure Evaluation Model  
This paper uses uncertainty measurement theory to evaluate the construction quality of hydraulic tunnels. The uncertainty measurement theory mainly includes two parts: single-indicator and multiple-indicator uncertain measurements. The evaluation method and corresponding calculation formula are as follows [21; 22]:  
(1) Uncertainty measures for single indicators  
Let the evaluation objects \(T_{1},T_{2},\cdots T_{m}\) form the object space of \(T=\{T_{1},T_{2},\cdots T_{m}\}\). For each object to be evaluated \(T_{j}\in T(j=1,2\cdots,m)\), there exist n indicators \(C_{1},C_{2},\cdots C_{n}\), then \(C=\{C_{1},C_{2},\cdots C_{n}\}\) is called the evaluation space. \(x_{ij}\) is the measured value of the \(i\)th indicator for the \(j\)th object to be evaluated \((i=1,2\cdots,n)\), and the indicator is classified into \(q\) levels \(U_{1},U_{2},\cdots U_{q}\), where \(U_{t}\) is the \(t\)th evaluation level \((t=1,2\cdots,q)\). If the effect of class \(t\) is better than that of class \(t+1\), denoted as \(U_{t}>U_{t+1}\), then \(U=\left\{U_{1},U_{2},\cdots U_{q}\right\}\) is the ordered segmentation class on evaluation space C [23]. \(z_{ijt}=z(x_{ij}\in U_{t})\) is used to indicate that the measured value \(x_{ij}\) belongs to the range of the \(t\)th evaluation grade \(U_{t}\), and the following conditions must be met:  
\[0\leq z\left(x_{ij}\in U_{t}\right)\leq 1 \tag{17}\]  
\[z\left(x_{ij}\in C\right)=1 \tag{18}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Weights** & **Indicator 1** & **Indicator 2** &... & **Indicator n** \\ \hline Subjective weights & \(\alpha_{1}^{\prime}\) & \(\alpha_{2}^{\prime}\) &... & \(\alpha_{n}^{\prime}\) \\ Objective weights & \(\beta_{1}^{\prime}\) & \(\beta_{2}^{\prime}\) &... & \(\beta_{n}^{\prime}\) \\ \hline \end{tabular}
\end{table}
Table 2: Sorting result of indicator weights.  
\[z\left(x_{ij}\in\bigcup_{t=1}^{q}U_{t}\right)=\sum_{t=1}^{q}z\left(x_{ij}\in U_{t}\right) \tag{19}\]  
Equation (17) represents non-negative boundedness, Equation (18) represents normalization, and Equation (19) represents additivity. When z satisfies all three of these formulas at the same time, z is said to be an uncertain measure.  
Construct the uncertainty measurement function \(z\left(x_{ij}\in U_{t}\right)\) for each evaluation index, substitute the measurement value \(x_{ij}\), and calculate the uncertainty measurement value \(z_{ijt}\), which can form the single-indicator measurement evaluation matrix shown in Equation (20).  
\[\left(z_{ijt}\right)_{n\times q}=\begin{bmatrix}z_{j11}&z_{j12}&\ldots&z_{j1q} \\ z_{j21}&z_{j22}&\ldots&z_{j2q}\\ \vdots&\vdots&\vdots&\vdots\\ z_{jn1}&z_{jn2}&\ldots&z_{jnq}\end{bmatrix} \tag{20}\]  
(2) Uncertainty measures for multiple indicators  
If \(z_{jt}=z\left(T_{j}\in U_{t}\right)\) represents the degree to which the \(T_{j}\)th evaluation object belongs to the \(t\)th evaluation level \(U_{t}\), then:  
\[z_{jt}=\sum_{i=1}^{n}\omega_{i}z_{ijt}(j=1,2,\cdots,m;t=1,2,\cdots,q) \tag{21}\]  
where \(z_{jt}\) satisfies \(0\leq z_{jt}\leq 1\), and \(\sum_{t=1}^{q}z_{jt}=1\). Then \(z_{j}=\left[z_{j1},z_{j2},\cdots,z_{jq}\right]\) is the evaluation vector of \(T_{j}\)'s multi-index comprehensive measure [24].  
(3) Confidence identification criteria  
The calculation results are analyzed according to the confidence identification criteria, and the confidence is set as \(\lambda\left(\lambda\geq 0.5\right)\)[25]. If \(\lambda\) satisfies Equation (22), it can be determined that the evaluation object \(T_{j}\) belongs to the \(t_{0}\)th evaluation level \(U_{t0}\).  
\[t_{0}=min\left\{t:\sum_{j=1}^{t}z_{j}\geq\lambda,t=1,2,\cdots,q\right\} \tag{22}\]