# Phased System Reliability Modeling and Assessment for Construction of Lunar Scientific Base
## 3 Reliability Assessment Method for Three Main Processes
### Reliability Assessment with Different Reliability of Each Fixed-Point Delivery  
Consider the case where each fixed-point delivery reliability is entirely different (their reliability confidence limits are independent). It is inappropriate to apply the conclusion of Section 3.1 directly. In this case, the product of \(R_{\rm Ai,\gamma}(i=1,2,\ldots,m_{1})\) could be taken as an approximate solution to the LCL of \(R_{\rm A}\) with the confidence degree \(\gamma\), and the approximate solution can be expressed as  
\[R_{\rm AL}=\prod_{i=1}^{m_{1}}R_{\rm Ai,\gamma}\hskip 14.226378 pti=1,2, \ldots,m_{1} \tag{9}\]Let \(\gamma^{*}\) denote the actual confidence degree of \(R_{\rm AL}\). It can be calculated by  
\[\gamma^{*}=P\{R_{\rm A}\geq R_{\rm AL}\}=P\Bigg{\{}R_{\rm AI}\geq\,R_{\rm AL} \Bigg{/}\prod_{i=2}^{m_{1}}R_{\rm AI}\Bigg{\}} \tag{10}\]  
Although the true value of \(R_{\rm AI}\) in Formula (10) is unknown, their LCLs at any confidence degree can be determined by Formula (2). Therefore, \(\gamma^{*}\) can be further calculated numerically, that is  
\[\gamma^{*}=P\{R_{\rm A}\geq R_{\rm AL}\}=\frac{1}{M^{m_{1}-1}}\sum_{k_{2}=1}^{M }\sum_{k_{3}=1}^{M}\ldots\sum_{k_{m_{1}}=1}^{M}\widetilde{\gamma}_{k_{2}k_{3} \ldots k_{m_{1}}} \tag{11}\]  
where  
\[\widetilde{\gamma}_{k_{2}k_{3}\ldots k_{m_{1}}}=P\Bigg{\{}R_{\rm A1}\geq\,R_{ \rm AL}\Bigg{/}\prod_{i=2}^{m_{1}}R_{\rm AI_{i}\gamma_{k_{i}}}\Bigg{\}} \tag{12}\]  
\[P\Big{[}R_{\rm AI}\geq R_{\rm AI_{i}\gamma_{k_{i}}}\Big{]}=\gamma_{k_{i}}\ \ \ \ \ i=2,3,\ldots,m_{1} \tag{13}\]  
\[\gamma_{k_{i}}=1-\frac{k_{i}-0.5}{M}\ \ \ k_{i}=1,2,\ldots,M,\ i=2,3,\ldots,m_{1} \tag{14}\]  
A detailed explanation for the numerical algorithm of Formulas (11)-(14) is summarized in Algorithm 1. ```
(1) Select \(M\) according to the requirements of numerical calculation accuracy (e.g., \(10^{4}\), \(10^{5}\), \(10^{6}\)). (2) Calculate the confidence degrees \(\gamma_{k_{i}}\) using Formula (14) for \(i=2,3,\ldots,m_{1}\). (3) Compute the \(R_{\rm AI_{i}\gamma_{i}}(i=2,3,\ldots,m_{1})\) with the corresponding confidence degree \(\gamma_{k_{i}}\) by using Formula (13). (4) Substitute \(R_{\rm AI_{i}\gamma_{k_{i}}}(i=2,3,\ldots,m_{1})\) into Formula (12) to calculate \(\widetilde{\gamma}_{k_{2}k_{3}\ldots k_{m_{1}}}\). (5) The confidence degree \(\gamma^{*}\) is obtained by Formula (11). ```  
**Algorithm 1** Numerical calculation of the confidence degree \(\gamma^{*}\) for \(R_{\rm AL}\)  
It is worth noting that there is a deviation between \(\gamma^{*}\) and the required confidence degree \(\gamma\). In order to obtain the exact LCL for \(R_{\rm A}\) with the confidence degree \(\gamma\), Algorithm 1 should be repeated several times from different start points. Note that Formula (10) is a monotonically increasing function of \(\gamma\). Thus, the value of \(\gamma\) in Formula (9) can be continuously adjusted to \(\gamma^{**}\) by the dichotomy until the value of \(\gamma^{*}\) calculated by Algorithm 1 is equal to the required value of \(\gamma\). In other words, find the value of \(\gamma^{**}\) such that the confidence degree for \(R_{\rm AL}\) in Formula (9) satisfies \(\gamma^{*}=\gamma\). At this time, the LCL of the overall fixed-point delivery reliability \(R_{\rm A}\) with the confidence degree \(\gamma\) is obtained as  
\[R_{\rm A,\gamma}=\prod_{i=1}^{m_{1}}R_{\rm AI_{i}\gamma^{**}}\ \ \ \ \ i=1,2, \ldots,m_{1} \tag{15}\]  
where \(R_{\rm AI_{i}\gamma^{**}}\) is the LCL for each fixed-point delivery reliability \(R_{\rm AI}\) with the confidence degree \(\gamma^{**}\). For all fixed-point deliveries, there is both identical and different reliability. Firstly, the LCL of the reliability of the partial fixed-point delivery consisting of fixed-point deliveries with the same reliability can be calculated by Formula (3). Furthermore, the LCL for the overall fixed-point delivery reliability can be calculated according to Formulas (9)-(15).