## 5 Case Studies
### Case One
#### 5.1.2 Extension Theory Calculation of Activation Grade  
(1) Extension evaluation of stability at coal mine goat site  
The matter-element to be evaluated in this case is:  
\[R_{01}=\begin{bmatrix}p_{1}&\text{stability grade of coal mine goat}&0.64 \end{bmatrix}\]  
Through the calculation of Equations (18) and (19), the correlation matrix \(K_{1}\) is obtained.  
\[K_{1}=\begin{bmatrix}-0.471&-0.413&-0.212&0.369\end{bmatrix}\]  
The normalization of the correlation matrix \(K^{{}^{\prime}}_{1}\) is conducted based on Equation (20).  
\[K^{{}^{\prime}}_{1}=\begin{bmatrix}-1&-0.876&-0.451&0.783\end{bmatrix}\]  
Equation (21) is utilized to compute the comprehensive correlation degree, resulting in the acquisition of the matrix \(K_{p1}\). As this evaluation layer solely encompasses a single index, the weight coefficient is set to 1.  
\[K_{p1}=A_{1}\bullet K^{{}^{\prime}}_{1}=\begin{bmatrix}-1&-0.876&-0.451&0.783 \end{bmatrix}\]  
(2) Extension evaluation of train influence  
Based on the overall analysis of the project, it can be seen that the matter-element to be evaluated for the Taijiao high-speed railway is:  
\[R_{1}=\begin{bmatrix}p_{1}&\text{train axle}&18\\ &\text{train speed}&250\\ &\text{subgrade height}&4.15\end{bmatrix}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** & **C7** & **C8** & **C9** & **C10** & **C11** & **C12** \\ \hline Taijiao & Actual & & & & & & & & & & & & \\ high-speed & parameters & 110 & 14 & 1980 & 3 & 4 & 28 & 3 & 4 & 4 & 3 & 3 & 3 \\ railway & Index after & & & & & & & & & & & & \\ normalization & 0.432 & 0.140 & 0.823 & 0.333 & 0.000 & 0.28 & 0.333 & 0.000 & 0.000 & 0.333 & 0.333 & 0.333 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Engineering parameters and normalized results of the Taijiao high-speed railway.  
Through the calculation of Equations (18) and (19), the correlation matrix \(K_{1}\) of the Taijiao high-speed railway is obtained.  
\[K_{1}=\begin{bmatrix}0&0&-0.333&-0.667\\ -0.5&-0.333&1&-0.333\\ -0.448&-0.333&1&-0.304\end{bmatrix}\]  
The correlation matrix \(K^{\prime}_{1}\) of the Taijiao high-speed railway is obtained via normalization according to Equation (20).  
\[K^{\prime}_{1}=\begin{bmatrix}0&0&-0.5&-1\\ -0.5&-0.333&1&-0.333\\ -0.448&-0.333&1&-0.304\end{bmatrix}\]  
The weight coefficient obtained from Section 4.2 is:  
\[A_{1}=\begin{bmatrix}0.15&0.14&0.71\end{bmatrix}\]  
Calculate the comprehensive correlation degree using Equation (17) to obtain the Taijiao high-speed railway matrix \(K_{p1}\):  
\[K_{p1}=A_{1}\bullet K^{\prime}_{1}=\begin{bmatrix}-0.388&-0.283&0.775&-0.413 \end{bmatrix}\]  
(3) Extension pre-evaluation of \"activation\" grade in coal mine goat ground  
The current weighting coefficient \(A=\begin{bmatrix}0.8&0.2\end{bmatrix}\) is obtained from Section 4.2, and the first-grade comprehensive evaluation of Taijiao high-speed railway is obtained according to Equation (17) as follows:  
\[K=A\bullet K_{p}=\begin{pmatrix}0.8&0.2\end{bmatrix}\begin{bmatrix}-1&-0.876& -0.451&0.783\\ -0.388&-0.283&0.775&-0.413\end{bmatrix}\]  
The final pre-evaluation result is calculated as follows:  
\[K=\begin{bmatrix}-0.878&-0.757&-0.206&0.545\end{bmatrix}\]  
The final evaluation result is \"activation\". In the actual project, the coal mine goat was grouted during the construction of the regional section, and the corresponding anti-settlement design of the subgrade was carried out. The evaluation results align with the engineering evaluation report, demonstrating that this method possesses a particular point of reference and engineering significance in the construction of high-speed railway coal mine goat site.