# Phased System Reliability Modeling and Assessment for Construction of Lunar Scientific Base
## 4 Reliability Assessment Method for Overall LSB Construction  
The overall construction of the LSB consists of three main processes: lunar surface fixed-point delivery, lunar surface transport, and lunar surface docking assembly. The failure of any one process will lead to the failure of the entire LSB construction. So the overall construction process can be regarded as a series of systems consisting of three independent processes. Figure 6 gives the reliability block diagram of the overall LSB construction. The reliability \(R\) of the overall LSB construction can be expressed as  
\[R=R_{\rm A}\cdot R_{\rm B}\cdot R_{\rm C} \tag{16}\]  
where \(R_{\rm A},~{}R_{\rm B}\) and \(R_{\rm C}\) denote the reliability of fixed-point delivery, lunar surface transport, and docking assembly, respectively. Denote \(R_{\rm A\gamma^{\prime}},~{}R_{\rm B\gamma}\) and \(R_{\rm C,\gamma}\) as the LCLs for the reliability of the three main processes with the confidence degree \(\gamma\), respectively. Taking the product of \(R_{\rm A\gamma,\gamma},~{}R_{\rm B\gamma}\) and \(R_{\rm C,\gamma}\) as an approximate solution of the LCL for overall construction reliability, namely  
\[R_{L}=R_{\rm A\gamma,\gamma}\cdot R_{\rm B\gamma}\cdot R_{\rm C,\gamma} \tag{17}\]  
Figure 5: The flow chart for the fixed-point delivery reliability assessment method. Figure 6: The reliability block diagram of the overall LSB construction. Then the actual confidence degree \(\gamma^{\prime}\) of \(R_{\rm L}\) can be calculated as  
\[\gamma^{\prime}=P\{R\geq R_{\rm L}\}=P\bigg{\{}R_{\rm A}\geq\frac{R_{\rm L}}{R_{ \rm B}\cdot R_{\rm C}}\bigg{\}} \tag{18}\]  
The true values of \(R_{\rm B}\) and \(R_{\rm C}\) in Formula (18) are unknown. However, their confidence limits at any confidence degree can be calculated in Section 3, respectively. Therefore, \(\gamma^{\prime}\) can be further calculated numerically by the following formula:  
\[\gamma^{\prime}=P\{R\geq R_{\rm L}\}=\frac{1}{M^{2}}\sum_{k_{1}=1}^{M}\sum_{k _{2}=1}^{M}\widetilde{\gamma}_{k_{1}k_{2}} \tag{19}\]  
where  
\[\widetilde{\gamma}_{k_{1}k_{2}}=P\bigg{\{}R_{\rm A}\geq\frac{R_{\rm L}}{R_{ \rm B,\gamma_{k_{1}}}\cdot R_{\rm C,\gamma_{k_{2}}}}\bigg{\}} \tag{20}\]  
\[P\Big{[}R_{\rm B}\geq R_{\rm B,\gamma_{k_{1}}}\Big{]}=\gamma_{k_{1}},\ P\Big{[}R_{\rm C}\geq R_{ \rm C,\gamma_{k_{2}}}\Big{]}=\gamma_{k_{2}} \tag{21}\]  
\[\gamma_{k_{i}}=1-\frac{k_{i}-0.5}{M}\ \ \ \ k_{i}=1,2,\ldots,M,\ i=1,2 \tag{22}\]  
A detailed explanation for the numerical algorithm of Formulas (19)-(22) is summarized in Algorithm 2. ```
(1) Select \(M\) according to the requirements of numerical calculation accuracy (e.g., \(10^{4}\), \(10^{5}\), \(10^{6}\)). (2) Calculate the confidence degrees \(\gamma_{k_{1}}\) and \(\gamma_{k_{2}}\) using Formula (22). (3) Compute the \(R_{\rm B,\gamma_{k_{1}}}\) and \(R_{\rm C,\gamma_{k_{2}}}\) with the corresponding confidence degree \(\gamma_{k_{1}}\) and \(\gamma_{k_{2}}\) using Formula (21). (4) Substitute \(R_{\rm B,\gamma_{k_{1}}}\) and \(R_{\rm C,\gamma_{k_{2}}}\) into Formula (20) to calculate \(\widetilde{\gamma}_{k_{1}k_{2}}\). (5) The confidence degree \(\gamma^{\prime}\) is obtained by Formula (19). ```  
**Algorithm 2**Numerical calculation of the \(\gamma^{\prime}\) for \(R_{\rm L}\)  
In order to obtain the exact LCL for overall constructed reliability, it is necessary to adjust the confidence degree \(\gamma\) of \(R_{\rm A,\gamma^{\prime}}\), \(R_{\rm B,\gamma}\) and \(R_{\rm C,\gamma^{\prime}}\) in Formula (17) to \(\gamma^{\prime}\), until \(\gamma^{\prime}\) calculated by Algorithm 2 is equal to the required value of \(\gamma\). In other words, find the value of \(\gamma^{\prime}\) such that the confidence degree for \(R_{\rm L}\) in Formula (17) satisfies \(\gamma^{\prime}=\gamma\).