## 3 Example Analyses
### Determination of Weight and Results of Cluster Analysis  
#### 3.3.1 Determination of Weight (w\({}_{1}\))  
(1) Weight-based least squares method to determine the objective weight  
In this paper, five factors of saturated compressive strength (\(R_{c}\)), spacing (\(D\)), wave velocity (\(v\)), rock quality designation (RQD) and integrity coefficient (\(Kv\)) are considered to evaluate the quality of rock mass. According to the importance of each factor, they are compared in turn to establish judgment matrix A, as follows:  
\[A=\begin{bmatrix}1&3&5&3&4\\ 1/3&1&3&1&2\\ 1/5&1/3&1&1/3&1/2\\ 1/3&1&3&1&1\\ 1/4&1/2&2&1/2&1\end{bmatrix}\]  
The intermediate matrix C is solved by Formula (10), as follows:  
\[C=\begin{bmatrix}4.32&-3.33&-5.2&-3.33&-4.25\\ -3.33&14.36&-3.33&-2&-2.5\\ -5.2&-3.33&51&-3.33&-2.5\\ -3.33&-2&-3.33&14.36&-2.5\\ -4.25&-2.5&-2.5&28.25\end{bmatrix}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Zone**} & \multicolumn{2}{c}{\(C_{1}\)**/Saturated**} & \multirow{2}{*}{\(C_{2}\)**/Spacing**} & \multirow{2}{*}{\(C_{3}\)**/Wave Velocity \(v\)**} & \multirow{2}{*}{\(C_{4}\)**/RQD**} & \multirow{2}{*}{\(C_{5}\)**/Integrity Coefficient**} \\ \cline{3-5}  & & & & & & \\ \cline{3-5}  & & & & & \\ \hline \(P_{8}\) & 103 & 0.8 & 4789 & 79 & 68 \\ \(P_{9}\) & 110 & 0.83 & 5890 & 88 & 82 \\ \(P_{10}\) & 98 & 0.69 & 5342 & 78 & 78 \\ \(P_{11}\) & 116 & 0.97 & 5203 & 90 & 75 \\ \(P_{12}\) & 120 & 1.1 & 5578 & 93 & 79 \\ \hline \(P_{13}\) & 134 & 0.94 & 5275 & 88 & 76 \\ \(P_{14}\) & 145 & 1.5 & 6085 & 95 & 88 \\ \(P_{15}\) & 143 & 1.43 & 5765 & 94 & 80 \\ \hline \hline \end{tabular}
\end{table}
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Zone**} & \multicolumn{2}{c}{\(C_{1}\)**/Saturated**} & \multirow{2}{*}{\(C_{2}\)**/Spacing**} & \multirow{2}{*}{\(C_{3}\)**/Wave Velocity \(v\)**} & \multirow{2}{*}{\(C_{4}\)**/RQD**} & \multirow{2}{*}{\(C_{5}\)**/Integrity Coefficient**} \\ \cline{3-5}  & & & & & \\ \cline{3-5}  & & & & & \\ \hline \(P_{1}\) & 0.04 & 0.08 & 0.23 & 0.22 & 0.12 \\ \(P_{2}\) & 0.13 & 0.13 & 0.36 & 0.24 & 0.16 \\ \(P_{3}\) & 0.26 & 0.21 & 0.43 & 0.50 & 0.28 \\ \(P_{4}\) & 0.29 &  
The weight values of the five parameters can be calculated by Formula (9), \(w_{1}\), as shown in Table 6.  
(2) Information entropy determines the objective weight of the evaluation index (\(w_{2}\)).  
The rock mass quality data of 15 tunnel sections were transformed into standard data, and the objective weight \(w_{2}\) of rock mass quality evaluation index was 0.305, 0.113, 0.074, 0.103, 0.124, using the entropy method combined with Formulas (12) and (13).  
(3) Combined weighting (w) determined by game theory  
In this paper, game theory was used to determine the comprehensive weight of sample rock mass quality. According to the least square and entropy methods, the subjective weight value and objective weight value of five evaluation indexes of rock mass quality evaluation were obtained. Game theory was used to establish the optimal combination coefficient equations:  
\[\begin{cases}aw_{1}w_{1}^{T}+bw_{1}w_{2}^{T}=w_{1}w_{1}^{T}\\ aw_{2}w_{1}^{T}+bw_{2}w_{2}^{T}=w_{2}w_{2}^{T}\end{cases}\]  
According to Formula (17), the comprehensive weight (\(w\)) of the evaluation factor was calculated to be 0.142, 0.179, 0.035, 0.116, and 0.108.