## 4 Evaluation and Discussion
### Discussion  
From the above analysis of two examples, the proposed methods need few subsystem analysis times while maintaining accuracy and has a evidently higher calculation efficiency than that of other reliability analysis methods. In addition, the MU-DBMRA method achieves a balance between calculation accuracy and efficiency. In short, the proposed method can handle multidisciplinary reliability analysis problems under aleatory uncertainty and epistemic uncertainty at a high calculation efficiency with accuracy. In terms of complexity, both traditional MRA and MU-DBMRA include three modules: probabilistic reliability analysis, convex reliability analysis and multidisciplinary analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Points** & **Method** & **Value of Functons** & **Iteration Times** \\ \hline
1 & SMPRA & \(g_{6}(\textbf{x})=6.341\) & 124 \\ \hline \multirow{4}{*}{2} & MU-DBMRA & \(g_{6}(\textbf{x},\textbf{v})_{\text{min}}=1.1816\) & 163 \\ \cline{2-4}  & \(g_{6}(\textbf{x},\textbf{v})_{\text{max}}=1.3631\) & 163 \\ \cline{1-1} \cline{2-4}  & MCs & \(g_{6}(\textbf{x})=0.416\) & 18,000 \\ \cline{1-1} \cline{2-4}  & MDF + SQP & \(g_{6}(\textbf{x})=1.622\) & 231 \\ \cline{1-1} \cline{2-4}  & IDF + SQP & \(g_{6}(\textbf{x})=1.431\) & 192 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The reliability analysis results of \(g_{6}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test Points** & **Value of Functions** & **Iteration Times** \\ \hline
1 & \(g_{8}(\textbf{x})=0.003\) & 154 \\ \hline \multirow{2}{*}{2} & \(g_{8}(\textbf{x},\textbf{v})_{\text{min}}=-0.055\) & 188 \\ \cline{1-1} \cline{2-4}  & \(g_{8}(\textbf{x},\textbf{v})_{\text{max}}=0.076\) & 188 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The reliability analysis results of \(g_{8}\). The complexity of the algorithm can be analyzed from two aspects, one is the complexity of each module, and the other is the time complexity of the whole strategy. The complexity of each module for MU-DBMRA has been analyzed in Section 3. For probabilistic reliability analysis, the PMA method is used to simplify the model of probabilistic reliability analysis; for convex reliability analysis, KKT conditions is used to replace the expensive extremum analysis. These measures reduce the complexity of the algorithm to some extent. The time complexity of the whole strategy can be express by the total function evaluation times, which is proportional to the number of iteration steps of the algorithm. As analyzed above, traditional MRA under multisource uncertainties is a three-layer iterative cycle. The total iteration times is the number of iterations of the outermost loop that is MPRA loop. The number of function evaluation is shown in Equation (28). \[N_{f}=N_{\text{MPRA}}\cdot N_{\text{MDA}}\cdot\overline{N}_{\text{MCRA}} \tag{28}\]  
where \(N_{\text{MPRA}}\) is the iteration time of the MPRA loop, \(N_{\text{MDA}}\) is the iteration time of the MDA loop, and \(\overline{N}_{\text{MCRA}}\) is the average iteration time of MCRA. The MU-DBMRA method proposed in this paper turns the traditional three-layer nested loop into a serialization process, and its function evaluation times can be expressed by Equation (29). \[N_{f}=N_{\text{Iteration}}\cdot(\overline{N}_{\text{MPRA}}+\overline{N}_{ \text{MCRA}}+2N_{\text{MDA}}) \tag{29}\]  
where \(N_{\text{Iteration}}\) represents the iteration time of the overall large loop, \(\overline{N}_{\text{MPRA}}\) represents the average iteration time of MPRA, and \(\overline{N}_{\text{MCRA}}\) represents the average iterations time of MCRA. Comparing Equations (28) and (29), we can see that function evaluation times of MU-DBMRA is a quadratic polynomial, while the function evaluation times of traditional MRA is a cubic polynomial. It can be seen from the analysis of the above two cases that the iteration time of the outermost cycles of different methods belongs to the same order of magnitude except Monte Carlo.