## 4 RDO Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{\(t_{1}\)} & \multirow{2}{*}{\(t_{2}\)} & \multirow{2}{*}{\(t_{3}\)} & \multirow{2}{*}{\(t_{4}\)} & \multirow{2}{*}{\(t_{5}\)} & \multirow{2}{*}{\(t_{6}\)} & \multirow{2}{*}{\(t_{7}\)} & \multirow{2}{*}{\(t_{8}\)} & \multirow{2}{*}{\(t_{9}\)} \\  & (\(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{ \text{\text{\text{ \texttexttexttext{  }}}}}}}}}}}}}}}\) & & & & & & \\ \hline RDO with MVRM & 1.5 & 2.9 & 2.5 & 3.0 & 1.4 & 1.4 & 0.9 & 2.9 & 2.5 \\  & (3.1) & (3.0) & (4.1) & (3.0) & (4.6) & (3.0) & (3.0) & (3.0) & (3.6) \\ RDO with AISM & 1.5 & 3.0 & 2.5 & 3.0 & 1.5 & 1.4 & 1.0 & 3.0 & 2.5 \\  & (3.1) & (3.6) & (4.2) & (3.0) & (4.3) & (3.0) & (3.6) & (3.3) & (3.6) \\ Deterministic optimization & 1.4 & 2.8 & 2.3 & 2.9 & 1.2 & 1.2 & 0.8 & 2.8 & 2.3 \\ \hline \hline \end{tabular} \({}^{1}\) Resultant sigma level robustness (SLR).  
\end{table} 
Table 4: Comparison of design optimum (unit: mm). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Constraints**} \\ \cline{2-5}  & **Plastic Strain (\%)** & **Dynamic Deformation (mm)** & **Obj. (kg)** & **\# of FE \({}^{1}\)** \\ \hline RDO with MVRM & 9.40 & 59.95 & 17.75 & 3715 \\ RDO with AISM & 8.85 & 58.73 & 18.07 & 13,727 \\ Deterministic optimization & 18.77 & 74.00 & 15.11 & 175 \\ \hline \hline \end{tabular} \({}^{1}\) FE: function evaluation.  
\end{table} 
Table 5: Comparison of objective and constraint functions. that the constraint function for the dynamic deformation at the optimal design point was active at the failure limit. Figure 10: _Cont._  
Figure 9: Convergence results of objective function. Reliability analyses using the MCS were carried out to review the difference between the results of the probabilistic optimal design of the RDO using the MVRM and those using the AISM, in terms of the reliability probability in the constraint functions. In order to review the difference between the probabilistic optimal design results of the RDO using the MVRM and the AISM, reliability analyses using the MCS were carried out in terms of the reliability probability of the constraint functions. In the MCS-based reliability analyses, the number of samples was 10,000, and the reliability probabilities of the constraint functions for the optimal design results summarized in Table 4 were evaluated. The results of the probability of failure are summarized in Table 6, and the PDF results are shown in Figures 11 and 12 by comparing the constraint functions for the plastic strain and the dynamic deformation. As shown in Table 6 and Figures 11 and 12, the probability of failure for the probabilistic optimal design of the RDO using the MVRM and the AISM all presented results of three sigma or higher. However, since the probabilities of failure in the deterministic optimal design were 10.3% for the constraint function of the plastic strain and 26.3% for the constraint function of the dynamic deformation, with the deterministic optimization it was difficult to guarantee adequate structural safety in cases wherein the design uncertainty was included. In the case of the RDO with the AISM, the probability of failure was evaluated as 0.0% for all the constraint functions, so the most reliable structure safety was guaranteed. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Probability of Failure (\%)**} \\ \cline{2-3}  & **Plastic Strain** & **Dynamic Deformation** \\ \hline RDO with MVRM & 0.0001 & 0.001 \\ RDO with AISM & 0.000001 & 0.0000001 \\ Deterministic optimization & 10.3 & 26.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of probability of failure. Figure 10: Convergence results of constraint functions. Figure 11: Comparison of PDF results for constraint function—plastic strain. Figure 12: Comparison of PDF results for constraint function—dynamic deformation.