## 4 RDO Results and Discussion
\(g_{1}\) and \(g_{2}\) were defined as deterministic constraint functions. In the optimization procedure applying the evolutionary algorithm, MATLAB coding was used. The RDO results according to the reliability analysis methods are represented in Tables 4 and 5, comparing them to the deterministic optimization results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random** & **Part** & \begin{tabular}{c} **Standard** \\ **Deviation** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Distribution** \\ **Type** \\ \end{tabular} &
\begin{tabular}{c} **Design** \\ **Variable** \\ \end{tabular} & **Mean (mm)** \\ \hline \(X_{1}\) & Seat panel & 0.15 & Normal & \(t_{1}\) & 1.6 \\ \(X_{2}\) & Front bracket & 0.20 & Normal & \(t_{2}\) & 2.6 \\ \(X_{3}\) & Side bracket & 0.20 & Normal & \(t_{3}\) & 2.6 \\ \(X_{4}\) & Center bracket & 0.20 & Normal & \(t_{4}\) & 2.6 \\ \(X_{5}\) & Back side plate & 0.15 & Normal & \(t_{5}\) & 1.6 \\ \(X_{6}\) & Pipe & 0.15 & Normal & \(t_{6}\) & 1.6 \\ \(X_{7}\) & Back plate & 0.11 & Normal & \(t_{7}\) & 1.0 \\ \(X_{8}\) & Rear bracket & 0.20 & Normal & \(t_{8}\) & 2.6 \\ \(X_{9}\) & Recliner & 0.17 & Normal & \(t_{9}\) & 2.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of random variables. In order to complete the optimization procedure, each calculation time required by the RDO with MVRM, the RDO with AISM, and the deterministic optimization, was 433.417 h, 1601.483 h, and 20.417 h, respectively, using a server machine which has 6 core Intel Xeon CPU and 128 GB RAM. As represented in Tables 4 and 5, the RDO results of the seat frame design using the MVRM and the AISM all showed conservative design results as compared to the deterministic optimal design results. The weight reduction results using the RDO with the MVRM and the AISM were 3.4% and 1.7%, respectively, based on the initial design weight of 18.38 kg, while the weight reduction of the deterministic optimal design was 17.8%. In the context of the constraint function results for the plastic strain, the RDO with the MVRM and the AISM ensured a safety margin of more than 50% based on the upper limit value of constraint, while the deterministic optimal design was only 6%, which was nearly approaching the failure limit. Regarding the constraint function results for the dynamic deformation, the RDO with the MVRM and the AISM secured a safety margin of about 20% based on the upper limit value of constraint, while the deterministic optimal design was active at the limit value. Reviewing the RDO results according to the reliability analysis method, we see that the RDO results with the AISM were more conservative than the MVRM. The probabilistic optimal design results of \(t_{2}\), \(t_{5}\), \(t_{7}\), and \(t_{8}\) were different according to the reliability analysis method, while there were no differences in the results of the remaining design variables. As described above, it was found that the robustness result for the converged probabilistic optimal design were varied according to the reliability analysis method. The numerical cost was compared based on the number of function evaluations, and the AISM case was the highest because the number of samples increased for the search of the failure region during the reliability analysis. The results of optimization-convergent characteristics were compared for the objective function and the constraint functions, respectively, and are shown in Figures 9 and 10. As shown in Figures 9 and 10, since the reliability analysis was applied to the RDO during the optimization process, the convergence characteristics differ from that of the deterministic optimization. Comparing the results of the MVRM and the AISM, the AISM represented more conservative results than the MVRM in the entire convergence process of both the objective function and the constraint function, and a more conservative convergence characteristic in the constraint function for the plastic strain. In the deterministic optimization, since the uncertainty of the design variable was not included, it was found