# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 4 Numerical Example
### Numerical Example 1  
A five-component series system in \(\boldsymbol{\delta}\) space is considered, for which each component can be defined by a performance function as follows:  
\[\begin{array}{l}G_{1}=-\delta_{1}\cos(-\frac{\pi}{\Delta 0})-\delta_{2}\sin(- \frac{\pi}{\Delta 0})+0.6\\ G_{2}=-\delta_{1}\cos(\frac{\pi}{\Delta 0})-\delta_{2}\sin(\frac{\pi}{\Delta 0})+0.75\\ G_{3}=-\delta_{1}\cos(\frac{\pi}{b})-\delta_{2}\sin(\frac{\pi}{b})+0.6\\ G_{4}=-\delta_{1}\cos(\frac{3\pi}{10})-\delta_{2}\sin(\frac{3\pi}{10})+0.7\\ G_{5}=-\delta_{1}\cos(\frac{3\pi}{10})-\delta_{2}\sin(\frac{3\pi}{10})+0.8\end{array} \tag{19}\]  
Figure 6: Flowchart of the LPNRBM.  
where \(\delta_{1}\) and \(\delta_{2}\) denote the uncertain variables in \(\delta\) space. The two-dimensional ellipsoid model of the uncertainty domain is shown as follows:  
\[\begin{bmatrix}\delta_{1}\\ \delta_{2}\end{bmatrix}^{\mathrm{T}}\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{bmatrix}\delta_{1}\\ \delta_{2}\end{bmatrix}\leq 1 \tag{20}\]  
The system is divided into \(2^{5}=32\) MECE events. Table 2 shows the MECE events and their corresponding design variables.  
The empty events in the subsystem are identified using the method proposed in this paper. First, the empty events of all bicomponent subsystems are identified and marked. Then, based on the bicomponent subsystem, the tricomponent subsystems are found.  
As shown in Figure 7, there are four tricomponent subsystems: \(G_{1}\), \(G_{2}\), \(G_{3}\), \(G_{1}\), \(G_{2}\), \(G_{4}\), \(G_{1}\), \(G_{3}\), \(G_{4}\), and \(G_{2}\), \(G_{3}\), \(G_{4}\)  
Since this example is two-dimensional, a schematic of this five-component series system is shown in Figure 8, from which it can be observed that the proposed identification criteria are in line with the actual situation. In Table 3, it can be seen that a total of 19 zero design variables can be removed. Table 4 shows the non-probabilistic reliability analysis results of the proposed method, NRBM, and exact solution.  
Table 4 shows that the LPNRBM can provide narrower bounds than the NRBM, that is, the former has a greater lower bound and a smaller upper bound than the latter (their maximum relative errors are respectively 0% and 5.6307%). This indicates that even though the NRBM considers all \(5!=120\) possible component-sorting alternatives, it cannot provide the best possible bounds based on the existing information. By contrast, the LPNRBM exhibits markedly higher precision than the NRBM and does not rely on ordering to offer the best possible bounds for the existing information. The proposed method reaches the limit of the bounds method, namely the exact solution in this example. For bounds on a system with a non-probabilistic failure degree, narrower bounds mean a higher accuracy. Thus, for this example, the LPNRBM may be a better choice.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Lower Bounds** & **Upper Bounds** & **Maximum Relative Error** \\ \hline NRBM [26] & \(2.6296\times 10^{-1}\) & \(2.8187\times 10^{-1}\) & 5.6307\% \\ LPNRBM & \(2.7865\times 10^{-1}\) & \(2.7865\times 10^{-1}\) & 0\% \\ Exact solution & \(2.7865\times 10^{-1}\) & — \\ \hline \hline \end{tabular}
\end{table}
Table 4: Reliability analysis results for Example 1.  
Figure 8: Five-component series system for numerical example 1.  
Figure 7: Determining the tricomponent subsystems for numerical example 1.