## 4 Evaluation and Discussion
### Speed Reducer  
The speed reducer example is a classic problem of MDO [23], containing two subsystems and seven design variables, as shown in Figure 9. Specifically, design variables \(x_{1}\), \(x_{2}\), \(x_{3}\) and constraints \(g_{1},g_{2},g_{3},g_{4},g_{5}\) belong to gear subsystem; design variables \(x_{4}\), \(x_{5}\), \(x_{6}\), \(x_{7}\) and constraints \(g_{6},g7,g8,g9,g10\), \(g11\) belong to bearing subsystem. This design example includes seven design variables: \(x_{1}\) = face width of the gear teeth, \(x_{2}\) = teeth module, \(x_{3}\) = number of pinion teeth, \(x_{4}\) = shaft-length, \(x_{5}\) = shaft length, \(x_{6}\) = shaft diameter, \(x_{7}\) = shaft diameter. Moreover, this design example is composed of 11 constraints (\(g_{1}\)-\(g_{11}\)) which are related to the bending condition, the compressive stress limitation, the transverse deflection of shafts and the substitute stress conditions. The optimal objective is to minimize the weight of the speed reducer. The constraints of the speed reducer are shown in the Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraints** & **Specification** & **Expression** \\ \hline \(g_{1}\) & Bending stress constraint of gear & \((x_{1}x_{2}^{2}x_{3})/27.0-1.0\geq 0\) \\ \(g_{2}\) & Contact stress constraint of gear & \((x_{1}x_{2}^{2}x_{3}^{2})/397.5-1.0\geq 0\) \\ \(g_{3}\) & Dimensional constraints 1 & \(x_{1}/(50x_{2})-1.0\geq 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The constraints of the speed reducer. Figure 8: Difference of maximum and minimum of the limit-state function. Figure 7: Variation tendency of the limit-state function. Figure 9: The coupling relationship diagram of speed reducer. The functional relationship between two subsystems is shown as below:  
\[\textbf{x}_{1}=\{x_{1},x_{2},x_{3}\}\] \[\textbf{y}_{21}=\{x_{4}:x_{7},g_{6}:g_{11}\} \tag{26}\] \[\textbf{g}_{1}=\{g_{1}:g_{5}\}\]  
Bearing subsystem:  
\[\textbf{x}_{2}=\{x_{4},x_{5},x_{6},x_{7}\}\] \[\textbf{y}_{12}=\{x_{1},x_{2},x_{3},g_{1}:g_{5}\} \tag{27}\] \[\textbf{g}_{2}=\{g_{6}:g_{11}\}\]  
To verify the effectiveness and calculation efficiency of the proposed method, the set of design variables in the single-stage speed reducer example divided into the following two cases:  
Test point 1: Set all the design variable \(\textbf{x}\)-N(\(\mu_{\textbf{x}}\), \(\sigma_{\textbf{x}}\)), \(\sigma_{\textbf{x}}\) = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) and \(\mu_{\textbf{x}}\) = (3.577, 0.700,17.000, 7.300,7.909,3.427,5.363). Test point 2: Set the design variable \(\textbf{x}\) = (\(x_{1}\), \(x_{4}\), \(x_{5}\), \(x_{6}\), \(x_{7}\))-N(\(\mu_{\textbf{x}}\), \(\sigma_{\textbf{x}}\)), \(\sigma_{\textbf{x}}\) = (0.1, 0.1, 0.1, 0.1) and \(\mu_{\textbf{x}}\) = (3.577, 7.300, 7.909, 3.427, 5.363). Specifically, \(x_{2}\) and \(x_{3}\) are interval design variables; the vector \(\overline{\textbf{x}}=[\overline{x_{2}},\overline{x_{3}}]^{T}=[0.700,17.000]^{T}\); the characrisic matrix is that \(\textbf{Wx}\) = [4,0,0,1] with the variation range as \(\textbf{x}\in E=\left\{\textbf{x}\left|(\textbf{x}-\overline{\textbf{x}})^{T }W(\textbf{x}-\overline{\textbf{x}})\leq 0.02^{2}\right.\right\}\). For test point 1, the reliability index of all limit state functions is set as \(g_{\text{t}}\) = 3.0. The sequential multidisciplinary reliability analysis method based on MAMV (SMPRA-MAMV) is used to evaluate the reliability. The values of limit state functions \(g_{1}\) to \(g_{5}\), \(g_{7}\), \(g_{9}\) and \(g_{11}\) (which only contains aleatory uncertain variables) and the MPPs are shown in Figure 8:0 It can be seen that the values of \(g_{3}\) and \(g_{9}\) are less than zero, which does not meet the reliability requirements; the values of \(g_{1}\), \(g_{2}\), \(g_{4}\), \(g_{5}\), \(g_{7}\) and \(g_{11}\) limit state functions are greater than zero, which meets the reliability requirements.