# A Hybrid JADE-Sine Cosine Approach for Advanced Metaheuristic Optimization
## 6 Application in Engineering Design
### Pressure Vessel Design Problem  
The pressure vessel design problem is an optimization problem aimed at minimizing the cost of constructing a pressure vessel. This problem includes constraints related to the dimensions and material properties of the vessel to ensure it meets safety and performance standards. The schematic of the pressure vessel is shown in Figure 11. \begin{table}
\begin{tabular}{l l l l l l l} \hline
**Optimizer** & **Min** & **Mean** & **Max** & **Std** & **Time** & **Rank** \\ \hline JADESCA & 1.724852 & 1.839658 & 2.299187 & 0.231405 & 6.924423 & 1 \\ AHA & 1.724853 & 1.724861 & 1.724882 & 9.04E-06 & 3.395266 & 2 \\ MPA & 1.724855 & 1.724864 & 1.724877 & 7.56E-06 & 4.500358 & 3 \\ FDA & 1.724865 & 1.916787 & 2.575409 & 0.288447 & 4.774226 & 4 \\ POA & 1.724876 & 1.72551 & 1.726224 & 0.000493 & 4.629484 & 5 \\ COA & 1.72579 & 1.801442 & 1.995475 & 0.11164 & 4.360163 & 6 \\ MFO & 1.727271 & 1.942539 & 2.512453 & 0.288391 & 2.005437 & 7 \\ GWO & 1.727572 & 1.731249 & 1.738839 & 0.003317 & 2.288584 & 8 \\ GJO & 1.729461 & 1.733087 & 1.738267 & 0.003438 & 3.211563 & 9 \\ MVO & 1.731516 & 1.763911 & 1.844405 & 0.036688 & 3.577161 & 10 \\ MGO & 1.733116 & 1.846738 & 2.293936 & 0.173521 & 15.34788 & 11 \\ HLOA & 1.807531 & 2.654731 & 4.547732 & 0.821542 & 4.3563 & 12 \\ SCA & 1.835108 & 1.923396 & 2.005675 & 0.054927 & 1.769805 & 13 \\ AOA & 2.029756 & 2.392391 & 2.982249 & 0.321545 & 2.825179 & 14 \\ \hline \end{tabular}
\end{table} 
Table 6: Results of the welded beam design problem. \begin{table}
\begin{tabular}{l l l l l l} \hline
**Optimizer** & **Best Score** & **X1** & **X2** & **X3** & **X4** \\ \hline JADESCA & 1.724852 & 0.20573 & 3.470489 & 9.036624 & 0.20573 \\ AHA & 1.724853 & 0.205729 & 3.470499 & 9.036625 & 0.20573 \\ MPA & 1.724855 & 0.205728 & 3.470521 & 9.036621 & 0.20573 \\ FDA & 1.724865 & 0.205725 & 3.470594 & 9.036646 & 0.20573 \\ POA & 1.724876 & 0.205735 & 3.470427 & 9.036516 & 0.205735 \\ COA & 1.72579 & 0.205428 & 3.478037 & 9.036359 & 0.205787 \\ MFO & 1.727271 & 0.20397 & 3.508765 & 9.036624 & 0.20573 \\ GWO & 1.727572 & 0.205872 & 3.471422 & 9.030063 & 0.206191 \\ GJO & 1.729461 & 0.205433 & 3.486698 & 9.051544 & 0.205768 \\ MVO & 1.731516 & 0.202163 & 3.558033 & 9.037382 & 0.205772 \\ MGO & 1.733116 & 0.200261 & 3.589762 & 9.042679 & 0.205699 \\ HLOA & 1.807531 & 0.164289 & 4.655296 & 9.036401 & 0.205741 \\ SCA & 1.835108 & 0.189043 & 3.509404 & 10 & 0.201401 \\ AOA & 2.029756 & 0.202836 & 3.689024 & 10 & 0.218807 \\ \hline \end{tabular}
\end{table} 
Table 7: Optimization algorithms of the welded beam design problem. The objective function to be minimized is expressed as Equation (24):  
\[z =0.6224\times(0.0625\times x_{1})\times x_{3}\times x_{4}+1.7781\times (0.0625\times x_{2})\times x_{3}^{2}\] \[\quad+3.1661\times(0.0625\times x_{1})^{2}\times x_{4}+19.84\times (0.0625\times x_{1})^{2}\times x_{3}, \tag{24}\]  
where \(x_{1},x_{2},x_{3},\) and \(x_{4}\) are the design variables representing the thickness of the shell, the thickness of the head, the inner radius, and the length of the cylindrical section, respectively. The constraints of the problem are defined as follows in Equation (32):  
\[c_{1} =-0.0625\times x_{1}+0.0193\times x_{3}, \tag{25}\] \[c_{2} =-0.0625\times x_{2}+0.00954\times x_{3},\] (26) \[c_{3} =-\pi\times x_{3}^{2}\times x_{4}-\frac{4}{3}\pi\times x_{3}^{3}+ 1296000,\] (27) \[c_{4} =x_{4}-240,\] (28) \[c_{5} =1-x_{1},\] (29) \[c_{6} =1-x_{2},\] (30) \[c_{7} =10-x_{3},\] (31) \[z =0.6224\times(0.0625\times x_{1})\times x_{3}\times x_{4}+1.7781\times (0.0625\times x_{2})\times x_{3}^{2}\]0  
The penalty function used to incorporate constraint violations is defined as Equation (33):  
\[z =0.6224\times(0.0625\times x_{1})\times x_{3}\times x_{4}+1.7781\times (0.0625\times x_{2})\times x_{3}^{2}\]1  
where \(c\) is the vector of constraint violations.