## 4 Evaluation and Discussion
### Speed Reducer
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraints** & **Specification** & **Expression** \\ \hline \(g_{4}\) & Dimensional constraints 2 & \(12.0x_{2}/x_{1}-1.0\geq 0\) \\ \(g_{5}\) & Dimensional constraints 3 & \(40.0/(x_{2}x_{3})-1.0\geq 0\) \\ \(g_{6}\) & Small shaft lateral displacement constraint & \((x_{2}x_{3}x_{4}^{b})/1.925x_{4}^{b}-1.0\geq 0\) \\ \(g_{7}\) & Large shaft lateral displacement constraint & \((x_{2}x_{3}x_{7}^{d})/1.925x_{3}^{d}-1.0\geq 0\) \\ \(g_{8}\) & Minor shaft stress constraint & \(110x_{6}^{2}/\sqrt{(\frac{2458x_{3}}{32x_{3}})^{2}+1.691\times 10^{7}-1.0\geq 0}\) \\ \(g^{9}\) & Major shaft stress constraint & \(85x_{7}^{2}/\sqrt{(\frac{2458x_{3}}{32x_{3}})^{2}+1.575\times 10^{8}-1.0\geq 0}\) \\ \(g_{10}\) & Dimensional constraints 4 & \(x_{4}/(1.5x_{6}+1.9)-1.0\geq 0\) \\ \(g_{11}\) & Dimensional constraints 5 & \(x_{5}/(1.1x_{7}+1.9)-1.0\geq 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**MPP**} \\ \cline{2-6}
**Limit State Function Value** & \(\textbf{x}_{1}\) & \(\textbf{x}_{2}\) & \(\textbf{x}_{3}\) & \(\textbf{x}_{5}\) & \(\textbf{x}_{7}\) \\ \hline \(g_{1}\) = 0.4915 & 2.9916 & 0.6755 & 22.9989 & 7.8000 & 5.2500 \\ \hline \(g_{2}\) = 0.4414 & 2.9916 & 0.6755 & 22.9978 & 7.8000 & 5.2500 \\  
For test point 2, the reliability index of all limit state function is set as \(\beta_{\text{t}}=3\). Obtained MPPs, limit-state function values and iteration times of \(g_{7}\) and \(g_{8}\) through respective multi-disciplinary reliability analysis are shown in Tables 4 and 5. We chose the method proposed in reference [14] to compare the results for \(g_{6}\). The solver of PMA in the contrasting method is sequential quadratic programming (SQP). From Table 4, it can ve discovered that the values of limit-state function under single aleatory uncertainties and under aleatory and interval uncertainties are all positive, indicating that the constraint condition \(g_{6}\) satisfies the given reliability requirement. Moreover, the value of limit state function under single aleatory uncertainties belongs to the value range of limit state function under random and interval uncertainties, which verifies the correctness of the proposed MU-DBMRA method. In terms of computational efficiency, the proposed MU-DBMRA method needs the least number of iterations when considering the aleatory and interval uncertainties, and its efficiency is 41.7% and 17.8% higher than MDF + SQP and IDF + SQP, respectively, which verifies the correctness and efficiency of the proposed method. From Table 5, the values of limit-state function under single aleatory uncertainty are positive. In that circumstance, that x = {3.577, 7.300, 7.9000999, 3.427, 5.363} is regarded as a design point for the constraint condition to satisfy the reliability requirment, while for the limit-stage function under aleatory uncertainties and interval uncertainties, its minimum is negative. This is because interval uncertainty changes limit-stage function from a single value to an interval. Only when the interval minimum is positive can the reliability requirement be satisfied. In this circumstance, that x = {3.577, 7.300, 7.9000999, 3.427, 5.363} cannot be viewed as a design point where the constraint condition satisfies the reliability requirement. Instead, new design point should be chosen in the proces of RBMDO.