## 5 Example Analysis and Discussion
### Reliability-Based Optimization of a Welded Beam  
A welded beam design problem shown in Figure 9 has four uncertain design variables and four uncertainty constraints [14]. In this problem, the optimization objective is to minimize the welding cost \(C(\mathbf{x})\) and tip deflection \(\delta(\mathbf{x};\mathbf{\xi})\) and tip deflection of the beam. Uncertainty constraints are related to physical characteristics such as bending stress, shear stress, buckling, and displacement of the free end. The input factors are the beam length \(L=355.6\) mm, the Young's Modulus \(E\), shear Modulus \(G=82,740\) MPa, point load \(F=26,688\) N, maximum normal stress \(\sigma\), and maximum shear stress \(\tau\). All uncertain \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Back Plate Thickness (mm)**} & \multicolumn{3}{c}{**Rank-Sum**} \\ \cline{2-10} \multirow{2}{*}{\(Pop\_size=20\)} & \(P_{\text{m}}=0.01\) & 14.9783 & 14.9776 & 14.9781 & 14.9784 & 14.9762 & 14.9782 & 14.9771 & 14.9785 & \multirow{2}{*}{52} \\ \cline{2-10} \cline{3-10}  & \(P_{\text{m}}=0.1\) & 14.9781 & 14.9786 & 14.9793 & 14.9780 & 14.9784 & 14.9783 & 14.9789 & 14.9783 & \multirow{2}{*}{53.5} \\ \cline{2-10} \multirow{2}{*}{\(P_{\text{m}}=0.2\)} & \(Pop\_size=15\) & 14.9780 & 14.9782 & 14.9781 & 14.9774 & 14.9783 & 14.9771 & 14.9781 & 14.9764 & \multirow{2}{*}{52} \\ \cline{2-10}  & \(Pop\_size=30\) & 14.9783 & 14.9772 & 14.9788 & 14.9785 & 14.9780 & 14.9795 & 14.9779 & 14.9782 & \multirow{2}{*}{53.5} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Original sample data of back plate thickness at \(P_{\text{c}}=0.8\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **Optimal Design (mm)** & **F-Evaluations** & **Time (s)** \\ \hline RBDO & RIA & 14.79328 & 169 & 29.4403 \\ URBDO & CEP & 14.97826 & 275 & 73.6282 \\ URBDO & USGA & 14.97828 & 4387 & 6553.0074 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of optimal solutions for brake system problem. variables are independent and characterized by normal or linear uncertainty distributions. The URBDO model of the welded beam design is formulated as follows:  
\[ \begin{split}&\text{find }\mathbf{d}=[d_{1},d_{2},d_{3},d_{4}]^{ \text{T}}\\ &\min\{f_{1},f_{2}\}=\omega_{1}C(\mathbf{x})+\omega_{2}\delta(\mathbf{x}; \mathbf{\xi})\\ &\text{s.t. }\mathcal{M}\Big{\{}g_{j}(\mathbf{x};\mathbf{\xi})\geq 0\Big{\}} \geq R_{\text{U}j},j=1,2,3,4\\ & C(\mathbf{x})\!=\!c_{1}x_{1}^{2}x_{2}+c_{2}x_{3}x_{4}(L+x_{2}), \,\delta(\mathbf{x};\mathbf{\xi})=\frac{4 FL^{3}}{Ex_{3}^{2}x_{4}}\\ & g_{1}(\mathbf{x};\mathbf{\xi})=1-\tau(\mathbf{x};\mathbf{\xi})/\tau,g_{2}(\mathbf{x };\mathbf{\xi})=\ 1-\sigma(\mathbf{x};\mathbf{\xi})/\sigma\\ & g_{3}(\mathbf{x};\mathbf{\xi})=1-\frac{x_{1}}{x_{4}},g_{4}(\mathbf{x};\mathbf{ \xi})=P_{\text{c}}(\mathbf{x};\mathbf{\xi})/F-1\\ &\tau(\mathbf{x};\mathbf{\xi})=\sqrt{L(\mathbf{x};\mathbf{\xi})^{2}+\frac{L(\bm {x};\mathbf{\xi})(\mathbf{x};\mathbf{\xi})}{R(\mathbf{x})}+S(\mathbf{x};\mathbf{\xi})^{2}}+S(\mathbf{x};\mathbf{ \xi})^{2}\\ & L(\mathbf{x};\mathbf{\xi})=\frac{F}{\sqrt{2}x_{1}x_{2}},S(\mathbf{x};\mathbf{ \xi})=\frac{M(\mathbf{x})R(\mathbf{x})}{I(\mathbf{x})}\\ & M(\mathbf{x};\mathbf{\xi})=F(L+0.5x_{2}),R(\mathbf{x})=0.5\sqrt{x_{2}^{2}+ (x_{1}+x_{3})^{2}}\\ & J(\mathbf{x})=\sqrt{2}x_{1}x_{2}\Big{[}\frac{x_{2}^{2}}{12}+\frac{ (x_{1}+x_{3})^{2}}{4}\Big{]},\sigma(\mathbf{x};\mathbf{\xi})=\frac{6 FL}{x_{2}^{2}x_{4} }\\ & P_{\text{c}}(\mathbf{x};\mathbf{\xi})=\frac{4.013x_{3}x_{3}^{3}\sqrt{EG} }{6L^{2}}\Big{(}1-\frac{x_{3}}{4L}\sqrt{\frac{F}{6}}\Big{)}\\ & c_{1}=6.74135\times 10^{-5},\ c_{2}=2.93585\times 10^{-6}\\ & 3.175\leq d_{1}\leq 50.8,\ 0\leq d_{2}\leq 254,\ 0\leq d_{3}\leq 254,\ 0\leq d_{4}\leq 50.8\end{split}  \tag{20}\]  
where \(\mathbf{x}=[x_{1},\,x_{2},\,x_{3},\,x_{4}]^{\text{T}}\) is the vector of uncertain design variables, \(\mathbf{\xi}=[E,\sigma,\tau]^{\text{T}}\) is the vector of uncertain variables, and \(R_{\text{U}j}=0.99\), \(j=1,2,3,4\) is the predetermined reliability.