## 5 Case Study
### Identification and Calculation of Various Fault Factors
Therefore, the key influencing factors of each category fault can be identified. (1) The influence degrees of the four kinds of influencing factors on the class A faults are studied. For material conveying system faults, the reference sequence is \(y_{1}^{\prime}\) and the comparison sequences are \(x_{1}^{\prime}\), \(x_{2}^{\prime}\), \(x_{3}^{\prime}\), and \(x_{4}^{\prime}\). The coefficient matrix is given by:  
\[ \begin{bmatrix}y_{1}^{\prime}\\ x_{1}^{\prime}\\ x_{2}^{\prime}\\ x_{3}^{\prime}\\ x_{4}^{\prime}\\ \end{bmatrix} = \begin{bmatrix}1.4393&0.6168&0.9439\\ 1.1642&1.0858&0.7500\\ 0.6818&1.2955&1.0227\\ 1.2149&0.9298&0.8554\\ 1.0051&0.8832&1.1117\\ \end{bmatrix}  \tag{17}\]  
Substituting the coefficient matrix into Equation (9), the absolute difference matrix is obtained as follows:  
\[ \begin{bmatrix}\Delta_{11}\\ \Delta_{12}\\ \Delta_{13}\\ \Delta_{14}\\ \end{bmatrix} = \begin{bmatrix}0.2751&0.4690&0.1939\\ 0.7574&0.6786&0.0788\\ 0.2244&0.3129&0.0886\\ 0.4342&0.2664&0.1677\\ \end{bmatrix}  \tag{18}\]  
According to Equations (7) and (8), it is found that \(\Delta_{min}\) = 0.0788 and \(\Delta_{max}\) = 0.7574. Substituting the above absolute difference matrix into Equation (10), the mean of all absolute differences \(\overline{\Delta}\) can be obtained. \(\overline{\Delta}=\frac{1}{3\times 4}\sum_{j=1}^{4}\sum_{k=1}^{3}\left|y_{1 }^{\prime}(k)-x_{j}^{\prime}(k)\right|=0.3289\), \(E_{\Delta}=\overline{\Delta}/\Delta_{max}\) = 0.4343, and since \(\Delta_{min}<3\overline{\Delta}\) = 0.9868, 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, 0.6514 \(\leq\rho\leq 0.8685\). In this research, \(\rho\) = 1.75 and \(E_{\Delta}\) = 0.7600. Substituting the above results into Equation (10), the grey correlation coefficient matrix for each influencing factor can be obtained as follows:  
\[ \begin{bmatrix}\xi_{11}\\ \xi_{12}\\ \xi_{13}\\ \xi_{14}\\ \end{bmatrix} = \begin{bmatrix}0.7693&0.6265&0.8504\\ 0.4909&0.5218&1.0000\\ 0.8180&0.7365&0.9853\\ 0.6481&0.7772&0.8803\\ \end{bmatrix}  \tag{19}\]  
Substituting the above grey correlation coefficient matrix into Equation (5), the correlation coefficient \(r_{1j}\) of the human factor P, environmental factor E, mechanical component life factor M, and other factors O for a class A fault, i.e., a material conveying system fault, can be obtained. \(j=1,2,3,4\), \(r_{1}=[r_{11},r_{12},r_{13},r_{14}]=[0.7487,0.6709,0.8466,0.7685]\). It is obvious that \(r_{13}>r_{14}>r_{11}>r_{12}\). Therefore, the relationship between the mechanical component life factor M and class A faults is large. It is indicated that class A faults are greatly affected by the mechanical component life factor. Corresponding measures should be taken to prevent the occurrence of a class A fault in the material conveying system. (2) The influence degrees of the four kinds of influencing factors on a class B fault, i.e., a batch metering system fault, are studied. The reference sequence is \(y^{\prime}_{2}\) and the comparison sequences are \(x^{\prime}_{1}\), \(x^{\prime}_{2}\), \(x^{\prime}_{3}\), and \(x^{\prime}_{4}\). The coefficient matrix is given by:  
\[ \begin{bmatrix}y^{\prime}_{2}\\ x^{\prime}_{1}\\ x^{\prime}_{2}\\ x^{\prime}_{3}\\ x^{\prime}_{4}\end{bmatrix} = \begin{bmatrix}0.5638&1.5101&0.9262\\ 1.1642&1.0858&0.7500\\ 0.6818&1.2955&1.0227\\ 1.2149&0.9298&0.8554\\ 1.0051&0.8832&1.1117\end{bmatrix}  \tag{20}\]  
Substituting the coefficient matrix into Equation (9), the absolute difference matrix is obtained as follows:  
\[ \begin{bmatrix}\Delta_{21}\\ \Delta_{22}\\ \Delta_{23}\\ \Delta_{24}\end{bmatrix} = \begin{bmatrix}0.6004&0.4242&0.1762\\ 0.1181&0.2146&0.0966\\ 0.6511&0.5803&0.0708\\ 0.4413&0.6268&0.1855\end{bmatrix}  \tag{21}\]  
According to Equations (7) and (8), it is found that \(\Delta_{min}=0.0708\) and \(\Delta_{max}=0.6511\). Substituting the above absolute difference matrix into Equation (10), the mean of all absolute differences \(\overline{\Delta}\) can be obtained.