## 4 Numerical Examples
### Numerical Example 1  
Consider a series system consisting of six performance functions:  
\[ \begin{array}{l}G_{1}(\delta)=-\cos(\frac{9\pi}{20})\delta_{1}-\sin(\frac{9 \pi}{20})\delta_{2}+0.35(1+\alpha)\\ G_{2}(\delta)=\cos(\frac{9\pi}{20})\delta_{1}-\sin(\frac{9\pi}{20})\delta_{2}+0. 35(1+\alpha)\\ G_{3}(\delta)=\cos(\frac{\pi}{20})\delta_{1}-\sin(\frac{\pi}{20})\delta_{2}+0. 38(1+\alpha)\\ G_{4}(\delta)=\cos(\frac{\pi}{20})\delta_{1}-\sin(\frac{\pi}{20})\delta_{2}+0. 45(1+\alpha)\\ G_{5}(\delta)=-\cos(\frac{\pi}{20})\delta_{1}-\sin(\frac{\pi}{20})\delta_{2}+0. 38(1+\alpha)\\ G_{6}(\delta)=-\cos(\frac{\pi}{20})\delta_{1}-\sin(\frac{\pi}{20})\delta_{2}+0. 45(1+\alpha)\end{array}  \tag{22}\]  
where \(\alpha\) is a control parameter. The uncertainty domain is given as follows:  
\[ \begin{bmatrix}\delta_{1}\\ \delta_{2}\end{bmatrix} ^{\mathrm{T}} \begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{bmatrix}\delta_{1}\\ \delta_{2}\end{bmatrix} \leq 1 \tag{23}\]  
The case of \(\alpha=0.5\) is used as an example to verify the proposed identification criteria. Six failure modes are given in Figure 5, and the identification criteria and redundant failure modes are listed in Table 2. It can be seen from Figure 5 that \(G_{3}\) and \(G_{6}\) are redundant failure modes, which coincide with the results listed in Table 2.  
With the control parameter ranging from 0.1 to 1, \(G_{3}\) and \(G_{6}\) are always redundant failure modes. In this process, the variations in the absolute errors of the proposed method and the NRBM [30] with the control parameter are as given in Figure 6. The absolute error is computed by \(f-f_{\text{exact}}\), where \(f\) represents the results from the proposed method or the NRBM and \(f_{\text{exact}}\) is the exact solution that can be computed analytically. \begin{table}
\begin{tabular}{c c} \hline \hline
**Step 1** & Define the performance function \(g_{j}(\mathbf{X})\). Transform the variables from the original space to the \(\delta\) space, and thereby obtain \(G_{j}(\delta)\). \\ \hline
**Step 2** & Take any two performance functions \(G_{i}(\delta)\) and \(G_{j}(\delta)\) to constitute a subsystem. Identify a redundant failure mode in the above subsystem using Equation (9). The identified redundant failure mode does not participate in the subsequent steps. \\ \hline
**Step 3** & Repeat Step 2 to obtain all the redundant failure modes in subsystems with two failure modes. \\ \hline
**Step 4** & Take any three performance functions \(G_{i}(\delta)\), \(G_{j}(\delta)\), and \(G_{k}(\delta)\) to form a subsystem. Identify a redundant failure mode in the above subsystem using Equation (19) or (20). The identified redundant failure mode does not participate in the subsequent steps. \\ \hline
**Step 5** & Repeat Step 4 to identify all of the redundant failure modes in subsystems with three failure modes. \\ \hline
**Step 6** & Remove all of the redundant failure modes from the system. After renumbering the remaining failure modes, calculate the non-probabilistic failure degree of the system using Equation (21). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Implementation steps of the proposed method. As shown in Figure 6, the proposed method has the same absolute error of the lower bound as the NRBM, which shows a trend of first decreasing and then increasing (the peak occurs at \(\alpha=0.5\)). However, the proposed method has a smaller absolute error of the upper bound than the NRBM. That is, the former has an absolute error that remains constant at zero, while the latter has an absolute error greater than zero, which displays a trend of first increasing and then decreasing (the peak occurs at \(\alpha=0.8\)). Through the above analysis, one can conclude that both methods can provide the reliability bounds containing the exact