## 2 Basic Theory  
### Gray-Whitening Weighted Function Clustering  
The gray clustering analysis method uses the whitening weighted function to whiten the 'gray information' and determines the final category of each index by calculating the maximum comprehensive clustering coefficient [12]. The main steps are as follows:  
1. Determining clustering sample, clustering index, and clustering gray number In this study, \(i=1,2,\ldots,m\) are clustering samples, that is, the number of experts participating in the scoring. Let \(j=1,2,\ldots,n\) be the clustering index, that is, the factors affecting the crystallization blockage of the drainage system of the mountain tunnel, \(k=1,2,\ldots,\) and \(p\) is the gray class, that is, the importance of the index. Assuming that \(m\) experts are involved in the scoring of the importance of the indicator, where the \(i\)th experts' rating sample for the \(j\)th indicator is counted as the following matrix \(X\), which can be constructed from the rating data:  
\[X= \begin{bmatrix}x_{11}&x_{12}&\ldots&x_{1n}\\ \vdots&\vdots&\ldots&\vdots\\ x_{r1}&x_{r2}&\ldots&x_{rn}\\ \vdots&\vdots&\ldots&\vdots\\ x_{m1}&x_{m2}&\ldots&x_{mn}\end{bmatrix}  \tag{1}\]
2. Determining the gray-whitening function \(f(x)\) The importance of the factors affecting the crystallization blockage of the tunnel drainage system is divided into three levels: 'high','medium', and 'low'. There are three gray classes, and their scores are quantified as 'high' which corresponds to 5 points,'medium' which corresponds to 3 points, and 'low' which corresponds to 1 point. Then, the corresponding whitening function is determined according to the category of the clustering index. Let \(f_{j}^{k}\big{(}x_{ij}\big{)}\) be the value of the whitening weighted function of the \(j\)th index \(k\) class, \(k=1,2,3\), then the calculation formula of \(f_{j}^{k}\big{(}x_{ij}\big{)}\) is as follows:  
Class I (importance 'high', \(k=1\)):  
The whitening function \(f_{j}^{1}\big{(}x_{ij}\big{)}\) for the \(j\)th index is:  
\[f_{j}^{1}\big{(}x_{ij}\big{)}= \begin{cases}1&x_{ij}\geq 5\\ \frac{x_{ij}-3}{5-3}&3<x_{ij}<5\\ 0&x_{ij}\leq 3\end{cases}  \tag{2}\]  
Class II (importance'medium', \(k=2\)):  
The whitening function \(f_{j}^{2}\big{(}x_{ij}\big{)}\) for the \(j\)th index is:  
\[f_{j}^{2}\big{(}x_{ij}\big{)}= \begin{cases}0&x_{ij}\geq 5\\ \frac{5-x_{ij}}{5-3}&3<x_{ij}<5\\ 1&x_{ij}=3\\ \frac{x_{ij}-1}{3-1}&1<x_{ij}<3\\ 0&x_{ij\leq 1}\end{cases}  \tag{3}\]  
Class III (importance 'low', \(k=3\)):  
The whitening function \(f_{j}^{3}\big{(}x_{ij}\big{)}\) for the \(j\)th index is:  
\[f_{j}^{3}\big{(}x_{ij}\big{)}= \begin{cases}0&x_{ij}\geq 3\\ \frac{3-x_{ij}}{3-1}&1<x_{ij}<3\\ 1&x_{ij}\leq 1\end{cases}  \tag{4}\]3. Gray statistical coefficient for calculating index importance [13]  
From the scoring matrix \(X\) and the whitening weighted function, the gray statistical coefficient of the index \(j\) belonging to the \(k\) category can be calculated, and the calculation formula is as follows:  
\[\eta_{j}^{k}=\sum_{j=1}^{m}f_{j}^{k}\big{(}x_{ij}\big{)} \tag{5}\]  
For the total gray evaluation coefficient \(\eta_{j}\) of index \(j\) belonging to each evaluation gray class, the formula is as follows:  
\[\eta_{j}=\sum_{k=1}^{p}\eta_{j}^{k(A)} \tag{6}\]
4. Calculating gray statistics and weight vector  
Through \(\eta_{j}^{k}\) and another variable, the statistical number \(r_{j}^{k}\) and the corresponding gray statistical weight vector \(r_{j}\) for the index \(j\) belonging to the \(k\)th important degree gray class can be calculated. The formula is as follows:  
\[r_{j}^{k}=\frac{\eta_{j}^{k}}{\eta_{j}} \tag{7}\]  
\[r_{j}=\left(\frac{\eta_{j}^{1}}{\eta_{j}},\ldots,\frac{\eta_{j}^{k}}{\eta_{j}}\right) \tag{8}\]
5.