# A Hybrid JADE-Sine Cosine Approach for Advanced Metaheuristic Optimization
## 6 Application in Engineering Design
### Welded Beam Design Problem  
The welded beam design problem is a classical optimization problem that aims to minimize the cost of fabricating a welded beam subject to a variety of constraints related to stress, deflection, and geometry. The objective function and constraints are derived from the physical properties and requirements of the beam. The schematic of the welded beam is shown in Figure 10.  
Figure 9: Histogram analysis over CEC202 (F1–F12). The objective function to be minimized is given by Equation (5):  
\[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]  
where \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are the design variables representing the dimensions of the beam. The parameters involved in the problem are as follows:  
\[P =6000\,\mathrm{lb},\] \[L =14\,\mathrm{in},\] \[E =30\times 10^{6}\,\mathrm{psi},\] \[G =12\times 10^{6}\,\mathrm{psi},\] \[\tau_{\mathrm{max}} =13600\,\mathrm{psi},\] \[\sigma_{\mathrm{max}} =30000\,\mathrm{psi},\] \[\delta_{\mathrm{max}} =0.25\,\mathrm{in}.\]  
The moment acting on the weld can be expressed as shown in Equation (6):  
Figure 9:0  
The radius of the critical section is given by Equation (7):  
Figure 9:1  
The polar moment of inertia is defined as shown in Equation (8):  
Figure 9:2  
The stresses in the weld are calculated using Equations (9)-(11):  
Figure 9:3 Figure 9:4 (10) Figure 9:5  
Figure 9:6 Schematic of the welded beam design problem. The normal stress in the beam is expressed by Equation (12):  
Figure 9:7  
The deflection of the beam is given by Equation (13):  
Figure 9:8  
The critical buckling load is defined by Equation (14):  
Figure 9:9  
The constraints of the problem are expressed in Equation (21):  
\[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]0 \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]1 (16) \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]2 (17) \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]3 (18) \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]4 (19) \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]5 (20) \[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]6  
The penalty function to account for constraint violations is defined by Equation (22):  
\[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]7  
where \(c\) is the vector of constraint violations. Finally, the objective function incorporating penalties is as follows:  
\[z=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}), \tag{5}\]8  
where \(v\) is an indicator function that is \(0\) if a constraint is satisfied and \(1\) otherwise. As shown in Equation (23), the overall objective function \(f\) includes a large penalty for any constraint violations, ensuring that the optimizer seeks feasible solutions that also minimize the fabrication cost. As shown in Table 6, the results of the welded beam design problem highlight the performance of various optimizers. Among them, GTO emerged as the best performer, achieving the lowest minimum value (1.724852) with a moderate computation time (6.924 s), demonstrating its effectiveness in finding optimal solutions. AHA and MPA also performed well, ranking second and third, respectively, with slightly higher minimum values but showing superior consistency, as indicated by their low standard deviations (AHA: \(9.04\times 10^{-6}\); MPA: \(7.56\times 10^{-6}\)). In contrast, AOA performed poorly, showing the highest minimum value (2.029756) and standard deviation (0.3215), indicating inefficiency and inconsistency in solution quality. This analysis underscores the importance of choosing the appropriate optimizer based on the specific requirements of the problem, balancing accuracy, consistency, and computational efficiency. As shown in Table 7, the comparison of optimization algorithms applied to the welded beam design problem reveals that JADESCA achieved the best score (1.724852), closely followed by AHA and MPA.