## 2 Methods
### Determining the Weight of the Impact Factor  
1. Weight-based least squares method to determine the objective weight  
In 1979, [PERSON] proposed the theory of the weighted least squares method [27], which is a new ranking method in the analytic hierarchy process. The traditional analytic hierarchy process greatly interferes with human factors and lacks the necessary theoretical basis. The weighted least square method can solve this problem. It calculates the minimum solution \(\omega_{\theta}\) of the judgment matrix function relationship under the limited constraints and solves the ranking weight vector. The detailed method is described as follows. 1. Construction of judgment matrix \(A\)  
Judgment matrix \(A\) is constructed considering the influence of multiple parameters on rock mass quality evaluations. \[A= \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{21}&\cdots&a_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ a_{n1}&a_{n1}&\cdots&a_{nn}\end{bmatrix}  \tag{8}\]Each element in the judgment matrix \(A\) is assigned a value, referring to the assignment table in Table 1.  
(b) Solving Function Minimization \(\omega_{\theta}\). The function relation is \(J=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\left(\omega_{i}-a_{ij}\omega_{j} \right)^{2}\). The constraint condition is \(\sum\limits_{i=1}^{n}\omega_{i}=1\), minimization \(\omega_{\theta}=(\omega_{1}\), \(\omega_{2}\), \(\ldots\), \(\omega_{n})^{T}\). According to [PERSON]'s [28] solution of the above relation, the formula of \(\omega_{\theta}\) is given in Equation (9). \[\omega_{\theta}=\frac{C^{-1}e}{e^{T}C^{-1}e} \tag{9}\]  
where \(C\) and \(e\) are  
\[C= \begin{bmatrix}\sum\limits_{i=1}^{n}a_{i1}^{2}+n-2&-(a_{12}+a_{21})&\cdots& -(a_{1n}+a_{n1})\\ -(a_{21}+a_{12})&\sum\limits_{i=1}^{n}a_{i2}^{2}+n-2&\cdots&-(a_{2n}+a_{n2})\\ \cdots&\cdots&\cdots&\cdots\\ -(a_{n1}+a_{n1})&-(a_{n2}+a_{2n})&\cdots&\sum\limits_{i=1}^{n}a_{in}^{2}+n-2 \end{bmatrix}  \tag{10}\]  
\[e=(1,\,1,\ldots,\,1)^{T} \tag{11}\]  
(2) Information entropy determines the objective weight of the evaluation index. As a means of objective weighting, the information entropy is used to determine the weight of influencing factors. Its evaluation process relies entirely on the laws of objective data, making the weight absolutely objective, and thus avoiding the interference of human factors to a large extent. The specific operation steps are given as follows. There are \(m\) evaluation objects and \(n\) evaluation indexes in the system to be evaluated, then the entropy value \(H_{i}\)[29; 30] of the \(i\)th evaluation index is given in Equation (12). \[H_{i}=-k\sum\limits_{j=1}^{m}P_{ij}\ln P_{ij} \tag{12}\]  
where \(H_{i}>0\); \(P_{ij}=\,u_{ij}/\sum\limits_{j=1}^{n}u_{ij};k=1/1\)nm; when \(\mu_{ij}=0\), \(\mu_{ij}\)n\(\mu_{ij}=0\), (\(i=1,\,2,\ldots,\,n\)). \[\theta_{i}=\frac{d_{i}}{\sum\limits_{i=1}^{n}d_{i}}=\frac{1-H_{i}}{n-\sum \limits_{i=1}^{n}H_{i}} \tag{13}\]  
where \(d_{i}=1-H_{i}\). \begin{table}
\begin{tabular}{c c} \hline \hline _The Value_ & **Description of the Importance of Indicator \(i\) Relative to Indicator \(j\)** \\ _of a\({}_{ij}\)_ & **Description of the Importance of Indicator \(i\) Relative to Indicator \(j\)** \\ \hline
1 & equal importance \\
3 & a little important \\
5 & obviously important \\
7 & strongly important \\
9 & top importance \\
2, 4, 6, 8 & the intermediate values of 1-3, 3–5, 5–7, 7–9 are denoted by 2, 4, 6, 8, respectively \\ reciprocal & If the importance of index \(i\) relative to index \(j\) is \(a_{ij}\). \\ The importance of indicator \(j\) relative to indicator \(i\) is \(a_{ji}=1/a_{ij}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantitative assignment table of index importance. 3. Combined weighting determined by game theory For a weight vector \(\delta_{i}=\{\delta_{1},\,\delta_{2},\ldots\delta_{n}\}\), the linear combination is \[\delta=\sum_{k=1}^{n}\alpha_{k}\delta_{k}^{T}(\alpha_{k}>0)\] (14) where \(\alpha_{k}\) is the weight coefficient.