## 4 Reliability-Based Optimum Design of Preform
### Optimum Design Outcomes  
The reliability of the target performance was assessed for each design derived from the initial design, DDO, and RBDO. While Monte Carlo simulation is commonly used for reliability assessment, it requires significant computational resources to ensure accurate predictions. Therefore, this study employed the enhanced dimension reduction method (eDR), a technique provided by PIAnO 2024. eDR is an approximation integration method that converts a multi-integral problem into a multi-variable first-order integral using the additive decomposition method. This method is recognized as being more precise and efficient than traditional reliability calculation methods [21]. For eDR, reliability analysis can be conducted using only 2n + 1, 4n + 1, and 6n + 1 experimental points, significantly reducing the computational burden. In this paper, 4n + 1 sampling was used to ensure accuracy, resulting in a total of 37 experimental points for the reliability assessment. Figure 13 displays the results of the reliability analysis, including evaluations for the initial design, DDO, and RBDO in terms of the effective strain (a), defect occurrence (b), and forming load (c). The results for the effective strain are shown in Figure 13a. The initial design exhibited the lowest average effective strain at 4.60, while DDO and RBDO recorded values of 4.70 and 4.71, respectively. This difference can be attributed to the tolerance variability present in the initial design, whereas the optimization outcomes of DDO and RBDO reflected a relatively increased effective strain due to reduced variability. The green areas in Figure 13b,c represent the objective function ranges where no defects are observed and the minimum load requirement is satisfied. The probability that the minimum forming load will not be met for the initial design is 69.59% and 17.15%, respectively, with the probability of defect occurrence exceeding 50%. It is anticipated that the optimal design will improve the defective elements of the initial design, thereby reducing the number of defects and ensuring the minimum forming load is achieved when defects are not present. However, as the material's void space is filled, the effective strain will increase. Furthermore, when comparing DDO with RBDO, the average effective strain for DDO is 0.01 lower, indicating that the two approaches are quite similar. In terms of reliability, RBDO demonstrates a 17.2% lower probability of defect occurrence and a 2.05% lower probability of failing to meet the minimum forming load compared to DDO, resulting in a failure probability of less than 1% and ensuring high reliability. This signifies a level of reliability that nearly eliminates the likelihood of product defects during the actual process, achieving the objectives of design optimization. Additionally, in the case of Figure 13b, the DDO and RBDO results also represent the probability of underfill with negative values, which reflect the likelihood of material overfilling during the preform stage. Table 6 quantitatively illustrates the results of the reliability-based optimization design, comparing the effective strain, defect occurrence, and forming load values for each design, with RBDO exhibiting the highest reliability. Furthermore, validation analysis based on the optimal design variables identified through reliability-based optimization confirmed that the analysis results were within the average and error range of the objective function values. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Approximation**} & \multicolumn{3}{c}{**Predicted R\({}^{2}\)**} \\ \cline{2-4}  & **Amount of Underfill** & **Effective Strain** & **Load Force** \\ \hline KRG & 0.999 & 0.234 & 0.818 \\ \hline RBF & 0.999 & 0.136 & 0.759 \\ \hline PR & 0.999 & 0.197 & 0.798 \\ \hline EDT & 0.955 & 0.150 & 0.702 \\ \hline MLP & 0.996 & 0.092 & 0.710 \\ \hline \end{tabular}
\end{table} 
Table 5: Predicted R\({}^{2}\) of metamodels on last iteration.