# Phased System Reliability Modeling and Assessment for Construction of Lunar Scientific Base
## 5 Example
### Numerical Example of the Proposed Methodology  
Assume that the LCLs for the fixed-point delivery, lunar surface delivery, and docking assembly reliability with the confidence degree \(\gamma=0.9\) are known to be \(R_{\text{A},\gamma}=0.98\), \(R_{\text{B},\gamma}=0.98\) and \(R_{\text{C},\gamma}=0.98\). Moreover, it is known that the fixed-point delivery, lunar surface delivery, and docking assembly tests are all zero-failure data and can be converted into binomial data [33]:  
\[n_{1}=\frac{\ln(1-\gamma)}{\ln R_{\text{A},\gamma}}=114 \tag{24}\]  
\[n_{2}=\frac{\ln(1-\gamma)}{\ln R_{\text{B},\gamma}}=114 \tag{25}\]  
\[n_{3}=\frac{\ln(1-\gamma)}{\ln R_{\text{C},\gamma}}=114 \tag{26}\]  
where \(n_{1}\), \(n_{2}\) and \(n_{3}\) are the pseudo test numbers of the fixed-point delivery, lunar delivery, and docking assembly, respectively.  
In the case of subsystems with zero-failure data, traditional LM and MML methods cannot be applied, as analyzed in Section 5.1. In contrast, the reliability assessment method for the LSB overall construction in Section 4 addresses the above difficulties well. For the cases of fixed-point delivery, lunar surface delivery, and docking assembly with zero-failure data, there are [34]  
\[R_{\text{A},\gamma}=(1-\gamma)^{1/n_{1}} \tag{27}\]  
\[R_{\text{B},\gamma}=(1-\gamma)^{1/n_{2}} \tag{28}\]  
\[R_{\text{C},\gamma}=(1-\gamma)^{1/n_{3}} \tag{29}\]  
Accordingly, using the method in Section 4, it is convenient to obtain the LCL of LSB overall construction reliability \(R\) with the confidence degree \(\gamma=0.9\) as  
\[R_{\text{L},\gamma}=R_{\text{A},\gamma^{\prime\prime}}\cdot R_{\text{B}, \gamma^{\prime\prime}}\cdot R_{\text{C},\gamma^{\prime\prime}}=(1-\gamma^{ \prime\prime})^{\frac{1}{n_{1}}+\frac{1}{n_{2}}+\frac{1}{n_{3}}}\!\!=0.9544 \tag{30}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Date/Results**} & \multirow{2}{*}{**Subsystem 1**} & \multirow{2}{*}{**Subsystem 2**} & \multirow{2}{*}{**Subsystem 3**} & \multicolumn{2}{c}{**System**} \\ \cline{5-6}  & & & & **LM** & **BBA** \\  & & & & **Method** & **Method** \\ \hline Number of samples & 100 & 150 & 200 & 100 & — \\ \hline Number of successes & 100 & 150 & 200 & 100 & — \\ \hline LCL for reliability & 0.977 & 0.984 & 0.989 & 0.977 & 0.962 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Reliability assessment results for the system composed of three zero-failure subsystems using the LM and BBA methods (\(\gamma=0.9\)).  
where \(\gamma^{\prime\prime}=0.8304\), which is calculated from Formulas (19)-(22), and \(\widetilde{\gamma}_{k_{1}k_{2}}\) in Formula (20) is  
\[\widetilde{\gamma}_{k_{1}k_{2}}=1-\frac{R_{\mathrm{L}}^{n_{1}}}{\left(1-\gamma_ {k_{1}}\right)^{n_{1}/n_{2}}\bullet\left(1-\gamma_{k_{2}}\right)^{n_{1}/n_{3}} }=1-\frac{\left(1-\gamma\right)^{1+n_{1}/n_{2}+n_{1}/n_{3}}}{\left(1-\gamma_{ k_{1}}\right)^{n_{1}/n_{2}}\bullet\left(1-\gamma_{k_{2}}\right)^{n_{1}/n_{3}}} \tag{31}\]  
The above results show that the proposed method in this research can fully use the zero-failure information of the three main processes to assess the LSB construction reliability with high confidence.