## 14 Applying JADEDO for Solving Engineering Problems  
### Welded Beam Design Problem  
The welded beam design problem is an engineering optimization problem that seeks to minimize the cost of fabricating a welded beam while satisfying constraints related to stress, deflection, and geometry [48]. Figure 14 represents a welded beam design problem, where the objective is to minimize the cost of welding while meeting structural constraints. The design variables include the weld thickness \(x_{1}\), the beam width \(x_{2}\), the beam height \(x_{3}\), and the beam thickness \(x_{4}\). The force \(P\) applied at the free end of the beam and the length \(L\) are also considered for the design constraints. The objective function to be minimized is given as shown in Equation (34):  
\[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{34}\] \begin{table}
\begin{tabular}{l c c c c c c} \hline
**Optimizer** & **NumFeasible** & **BestCompisiteObj** & **MeanCompositeObj** & **StdCompositeObj** & **MaxEffect** & **MeanEffect** & **MeanCPUTime** \\ \hline JADEDO & 30 & \(-52.4180\) & \(-36.8937\) & 6.0205 & 52.4180 & 36.8937 & 0.6995 \\ FLO & 30 & \(-48.8887\) & \(-36.3929\) & 6.6850 & 45.8857 & 36.3929 & 0.0845 \\ GWO & 30 & \(-51.9165\) & \(-36.4190\) & 6.0185 & 51.9165 & 36.4190 & 0.0783 \\ SSOA & 30 & \(-43.6779\) & \(-33.0937\) & 4.9245 & 43.6779 & 33.0937 & 0.5819 \\ Chimp & 30 & \(-47.9557\) & \(-35.9650\) & 7.4236 & **47**\({}^{\prime}\)9557 & 35.9650 & 0.6059 \\ \hline \end{tabular}
\end{table} 
Table 6: Results for 30 runs per method, each with potentially different threat factor \(\alpha\). A more negative Composite Objective (i.e., lower in the table) is better. FeasRatio = 1 indicates all runs satisfied the cost, risk, and time constraints. Figure 14: Welded beam design problem. where \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are the design variables. The moment acting on the weld can be expressed as shown in Equation (35):  
\[M=P\Big{(}L+\frac{x_{2}}{2}\Big{)}, \tag{35}\]  
and the radius of the critical section is given in Equation (36):  
\[R=\sqrt{\frac{x_{2}^{2}}{4}+\left(\frac{x_{1}+x_{3}}{2}\right)^{2}}, \tag{36}\]  
while the polar moment of inertia is defined as shown in Equation (37):  
\[J=2\Bigg{(}\sqrt{2}x_{1}x_{2}\Bigg{(}\frac{x_{2}^{2}}{12}+\left(\frac{x_{1}+x_ {3}}{2}\right)^{2}\Bigg{)}\Bigg{)}, \tag{37}\]  
and the stresses in the weld are calculated using Equations (38)-(40):  
\[\tau_{1} =\frac{P}{\sqrt{2}x_{1}x_{2}}, \tag{38}\] \[\tau_{2} =\frac{MR}{J},\] (39) \[\tau =\sqrt{\tau_{1}^{2}+2\tau_{1}\tau_{2}\Big{(}\frac{x_{2}}{2R} \Big{)}+\tau_{2}^{2}}. \tag{40}\]  
The normal stress in the beam is expressed by Equation (41):  
\[\sigma=\frac{6 PL}{x_{4}x_{3}^{2}}, \tag{41}\]  
and the deflection of the beam is given in Equation (42):  
\[\delta=\frac{4 PL^{3}}{Ex_{4}x_{3}^{3}}. \tag{42}\]  
The critical buckling load is defined in Equation (43):  
Figure 14:0  
while the constraints of the problem are expressed in Equation (50):  
Figure 14:1 Figure 14:2 (45) Figure 14:3 (46) Figure 14:4 (47) Figure 14:5 (48) Figure 14:6 (49) Figure 14:7  
The penalty function to account for constraint violations is defined in Equation (51):  
Figure 14:8where \(c\) is the vector of constraint violations. Finally, the objective function incorporating penalties is shown in Equation (52):  
Figure 14:9  
where \(v\) is an indicator function that is 0 if a constraint is satisfied and 1 otherwise, as shown in Equation (52). Discussion of JADEDO performance: In comparing JADEDO with benchmarked methods on the Welded Beam design problem as shown in Table 7, JADEDO demonstrates a balance of exploration and exploitation that often enables it to achieve solutions close to or surpassing the best-performing optimizers such as GWO and GOA.