## 5 Case Study
### Identification and Calculation of Various Fault Factors
The coefficient matrix is given by:  
\[ \begin{bmatrix}y^{\prime}_{4}\\ x_{1}\\ x^{\prime}_{2}\\ x^{\prime}_{3}\\ x^{\prime}_{4}\end{bmatrix} = \begin{bmatrix}1.6098&0.7317&0.6585\\ 1.1642&1.0858&0.7500\\ 0.6818&1.2955&1.0227\\ 1.2149&0.9298&0.8554\\ 1.0051&0.8832&1.1117\end{bmatrix}  \tag{26}\]  
Substituting the coefficient matrix into Equation (9), the absolute difference matrix is obtained as follows:  
\[ \begin{bmatrix}\Delta_{41}\\ \Delta_{42}\\ \Delta_{43}\\ \Delta_{44}\end{bmatrix} = \begin{bmatrix}0.4456&0.3541&0.0915\\ 0.9279&0.5637&0.3642\\ 0.3949&0.1980&0.1968\\ 0.6047&0.1515&0.4531\end{bmatrix}  \tag{27}\]  
According to Equations (7) and (8), it is found that \(\Delta_{min}=0.0915\) and \(\Delta_{max}=0.9279\). Substituting the above absolute difference matrix into Equation (10), the mean of all absolute differences \(\overline{\Delta}\) can be obtained. \(\overline{\Delta}=\frac{1}{3\times 4}\sum_{j=1}^{4}\sum_{k=1}^{3}\left|y^{ \prime}_{1}(k)-x^{\prime}_{j}(k)\right|=0.3955\), \(E_{\Delta}=\overline{\Delta}/\Delta_{max}=0.4262\), and since \(\Delta_{min}<3\overline{\Delta}=1.1865\), the data sequence is normal. Then, the range of the resolution coefficient can be determined as \(1.5E_{\Delta}\leq\rho\leq 2E_{\Delta}\), that is, \(0.6393\leq\rho\leq 0.8525\). In this study, \(\rho=1.75\) and \(E_{\Delta}=0.7459\). Substituting the above results into Equation (10), the grey correlation coefficient matrix for each influencing factor can be obtained, as follows:  
\[ \begin{bmatrix}\xi_{41}\\ \xi_{42}\\ \xi_{43}\\ \xi_{44}\end{bmatrix} = \begin{bmatrix}0.6888&0.7490&1.0000\\ 0.4837&0.6239&0.7418\\ 0.7209&0.8803&0.8815\\ 0.6043&0.9288&0.6842\end{bmatrix}  \tag{28}\]  
Substituting the above grey correlation coefficient matrix into Equation (5), the correlation coefficient \(r_{4j}\) of the human factor P, environmental factor E, mechanical component life factor M, and other factors O for a class D fault, i.e., a blender malfunction fault, can be obtained. \(j=1,2,3,4\), and \(r_{4}=[r_{41},r_{42},r_{43},r_{44}]=[0.8126,0.6165,0.8275,0.7391]\). It is obvious that \(r_{43}>r_{41}>r_{44}>r_{42}\). Therefore, the relationship between the mechanical component life factor M and class D faults is large. It is indicated that class D faults are greatly affected by the mechanical component life factor. Corresponding measures should be taken to prevent the occurrence of a class D fault in the blender malfunction. Figure 8 shows the comparison results of the grey correlation degrees \(r_{11},r_{12},r_{13}\), \(r_{14},r_{21},r_{22},r_{23},r_{24},r_{31},r_{32},r_{33},r_{34},r_{41},r_{42},r_{ 43},r_{44}\) for the above four reference sequences. The size relationship is as follows:  
\[r_{32}>r_{22}>r_{13}>r_{43}>r_{41}>r_{14}>r_{11}>r_{44}>r_{23}>r_{21}>r_{24}>r_{ 34}>r_{12}>r_{31}>r_{33}>r_{42}.\]  
It can be seen from Figure 8 that the grey correlation degree of the factors affecting concrete dam production system faults for different reference sequences can be divided into three categories: (1) the grey correlation degree is greater than 0.9000. This mainly includes the correlation degrees \(r_{32}\) and \(r_{22}\) of the environment factor E to class C and class B faults. (2) The grey correlation degree is in the range of 0.7000-0.9000. This mainly includes the correlation degrees \(r_{11}\) and \(r_{41}\) of the human factor P to class A and class D faults, the correlation degrees \(r_{13}\), \(r_{23}\), and \(r_{43}\) of the mechanical component life factor M to class A, class B, and class D faults, and the correlation degrees \(r_{14}\) and \(r_{44}\) of the other factors O to class A and class D faults. (3) The grey correlation degree is less than 0.7000.