## 3 Materials and Methods
### AI-Based Optimization
Table 2 provides a comprehensive comparison of the RBDD results with other RBDD results, along with the comparison of results obtained in this study by RBDD and other studies that used regular optimization methods for the 10-bar truss structure. It is important to note that the first two studies [8; 21] employed real-world discrete cross-sectional areas as design variables, while the last [57] and current study used continuous sectional properties. However, there is a clear difference in reliability between the two methods. The proposed method in this study successfully achieved a better objective function without violating the reliability constraint (3) compared with the [57]. According to the results, there is a 51.1% chance of failure of the optimized structure by [21] while the present study optimized the structure with a probability of failure of less than 0.15%. It also provided a comparable objective result while significantly improving reliability compared to traditional methods. This demonstrates that RBDD \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Structural Optimization** & \multicolumn{2}{c}{**RBDD**} \\ \hline  & [21] & [8] & [57] & **This Study** \\ \hline Objective (kg) & 2549.22 & 2763.06 & 2817.40 & 2787.55 \\ \(\beta^{MCS}\) & \(-\)0.027 & 0.29 & 3.01 & 3.0022 \\ Employed MA & GA & GA & GA & DE \\ Population Size & 20 & 20 & 100 & 50 \\ Generation Number & 20 & 20 & 500 & 100 \\ Separate Runs & - & 10 & 50 & 2 \(\times\) GA, \(\times\) DE \\ A1 (cm\({}^{2}\)) & 216.128600 & 181.289960 & 221.625363 & 223.489362 \\ A2 (cm\({}^{2}\)) & 10.451592 & 23.161244 & 0.645160 & 0.783208 \\ A3 (cm\({}^{2}\)) & 141.935200 & 165.160960 & 191.502843 & 196.122331 \\ A4 (cm\({}^{2}\)) & 99.999800 & 128.386840 & 169.515790 & 125.815885 \\ A5 (cm\({}^{2}\)) & 10.451592 & 10.322560 & 0.645160 & 0.749852 \\ A6 (cm\({}^{2}\)) & 11.612880 & 27.225752 & 0.645160 & 0.679507 \\ A7 (cm\({}^{2}\)) & 91.612720 & 115.483640 & 21.528989 & 26.795538 \\ A8 (cm\({}^{2}\)) & 128.386840 & 125.161040 & 182.928666 & 189.808710 \\ A9 (cm\({}^{2}\)) & 128.386840 & 115.483640 & 168.631921 & 174.270183 \\ A10 (cm\({}^{2}\)) & 16.903192 & 22.516084 & 0.645160 & 0.652246 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of 10-bar truss results. mass and enhance safety under uncertain conditions. Figure 8 presents the convergence history of the DE algorithm, which provided a better result compared with the GA for this example, with Adaptive Epsilon Constraint Handling for the best run cycle. This constraint handling technique allows relaxing the constraint, epsilon (\(\varepsilon\)), using a dynamic threshold value which adapts overtime during the optimization process instead of strictly enforcing constraints. Over time, this \(\varepsilon\) is adjusted to become stricter, allowing infeasible solutions early on. However, as the epsilon value decreases, only solutions that closely satisfy the constraints are accepted, guiding the algorithm toward a feasible solution space as shown in Figure 8. The workflow took approximately 190 min to finalize 10,000 FORM evaluations along with FEM evaluations in 200 generations which is acceptable by considering the authors previously studied structural optimization methods with the RSA [8; 17]. To adapt the proposed method to frame structures, a three-bay, four-story frame structure is optimized on the cross-sectional areas using RBDO. Sectional areas, as mentioned above, consist of discrete and real-world cross-sectional areas. Table 3 presents the comparison of results with open literature and optimized sections, along with other important outcomes. The results in Table 3 indicate that RBDO provides more reliable structures under uncertainty. Overall, DE with the adaptive epsilon constraint handling method yielded a better objective with a smaller generation population size and required fewer runs compared to the results reported in the literature.