## 5 Case Study
### Identification and Calculation of Various Fault Factors  
According to the obtained classification statistics of production system faults, as well as the identification method of the influencing factors of the concrete dam production system fault rate proposed in Section 4.2, combined with the actual engineering production system fault classification statistics, the class A fault number, class B fault number, class C fault number, and class D fault number are selected as reference sequences and denoted as \(y_{1}\), \(y_{2}\), \(y_{3}\), and \(y_{4}\), respectively. Then, the reference sequence matrix is given by:  
\[Y_{i}= \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\end{bmatrix} = \begin{bmatrix}154&66&101\\ 28&75&46\\ 50&116&80\\ 66&30&27\end{bmatrix}  \tag{14}\]  
There are four factors affecting the fault of the CPS studied in this research: human factors, environmental factors, mechanical component service life factors, and other factors. The sequence consisting of the four types of fault factors is taken as the comparison sequence, which is recorded as \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\), and the comparison sequence matrix is given by:  
\[X_{i}= \begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{bmatrix} = \begin{bmatrix}104&97&67\\ 30&57&45\\ 98&75&69\\ 66&58&73\end{bmatrix}  \tag{15}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Category**} & \multirow{2}{*}{**Fault**} & \multicolumn{2}{c}{**Statistical Sample Number S1**} & \multicolumn{2}{c}{**Statistical Sample Number S2**} & \multicolumn{2}{c}{**Statistical Sample Number S3**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Proportion**} & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Proportion**} & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Proportion**} \\ \hline \multirow{6}{*}{C} & C\({}_{1}\) & 16 & & 14 & & & & & & & \\ \cline{2-10}  & C\({}_{2}\) & 8 & & 40 & & & & & & & \\ \cline{2-10}  & C\({}_{3}\) & 14 & & 16.78\% & 31 & & & & & & & & \\ \cline{2-10}  & C\({}_{4}\) & 9 & & 25 & & & & & & & \\ \cline{2-10}  & C\({}_{5}\) & 3 & & 6 & & & & & & & \\ \hline \multirow{6}{*}{D} & D\({}_{1}\) & 15 & & 7 & & & & & & & & \\ \cline{2-10}  & D\({}_{2}\) & 11 & & 5 & & & & & & & & \\ \cline{2-10}  & D\({}_{3}\) & 17 & 22.15\% & 4 & & & & & & & & \\ \cline{2-10}  & D\({}_{4}\) & 14 & & & 11 & & & & & & & \\ \cline{2-10}  & D\({}_{5}\) & 9 & & 3 & & & & & & & \\ \hline \multicolumn{10}{c}{Sum} & 298 & - & 287 & - & & 254 & - & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Quantity**} & \multirow{2}{*}{**Faults**} & \multicolumn{2}{c}{**Number of Faults of Corresponding Types of Faults**} & \multicolumn{2}{c}{**Number of Faults Caused by Various Influencing Factors**} \\ \cline{2-10}  & & **A** & **B** & **C** & **D** & **P** & **E** & **M** & **O** \\ \hline \multirow{2}{*}{\(S_{1}\)} & 154 & 28 & 50 & 66 & 104 & 30 & 98 & 66 \\ \hline \multirow{2}{*}{\(S_{2}\)} & 66 & 75 & 116 & 30 & 97 & 57 & 75 & 58 \\ \hline \multirow{2}{*}{\(S_{3}\)} & 101 & 46 & 80 & 27 & 67 & 45 & 69 & 73 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistics of fault cases for multi-factors of CPS. After the dimensionlessization using Equations (3) and (4), the coefficient matrix is obtained, as follows:  
\[ \begin{bmatrix}y_{1}^{\prime}\\ y_{2}^{\prime}\\ y_{3}^{\prime}\\ y_{4}^{\prime}\\ x_{1}^{\prime}\\ x_{2}^{\prime}\\ x_{4}^{\prime}\\ \end{bmatrix} = \begin{bmatrix}1.4393&0.6168&0.9439\\ 0.5638&1.5101&0.9262\\ 0.6098&1.4146&0.9756\\ 1.6098&0.7317&0.6585\\ 1.1642&1.0858&0.7500\\ 0.6818&1.2955&1.0227\\ 1.2149&0.9298&0.8554\\ 1.0051&0.8832&1.1117\\ \end{bmatrix}  \tag{16}\]  
To determine the degree of the faults of different types of production systems, it is necessary to use the faults in a single category as a reference sequence to calculate the factors and their correlations.