## 14 Applying JADEDO for Solving Engineering Problems
### Pressure Vessel Design Problem  
The pressure vessel design problem involves optimizing the dimensions of a cylindrical pressure vessel with hemispherical ends to minimize the total cost, which includes the material cost, forming cost, and welding cost [49]. Figure 15 illustrates a pressure vessel design consisting of two parts: a hemispherical head and a cylindrical body. The parameters include the radius \(R\), the thickness of the hemispherical head \(T_{h}\), the thickness of the cylindrical shell \(T_{s}\), and the length of the cylindrical shell \(L_{c}\). The design problem focuses on optimizing these parameters to minimize the material used while withstanding the internal pressure. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Optimizer** & **Mean** & **Std** & **Min** & **Max** & **Time** & **Rank** & **BestF** & **x1** & **x2** & **x3** & **x4** \\ \hline GWO & 1.684953 & 0.011193 & 1.677038 & 1.692867 & 0.154987 & 1 & 1.677038 & 0.198901 & 3.374337 & 9.195172 & 0.199005 \\ GOA & 1.719107 & 0.025183 & 1.7013 & 1.736915 & 0.231371 & 2 & 1.7013 & 0.195003 & 3.466748 & 9.093079 & 0.203591 \\ CSA & 1.749686 & 0.09314 & 1.685117 & 1.814255 & 0.122867 & 3 & 1.685117 & 0.196164 & 3.353857 & 9.320198 & 0.19824 \\ CDOA & 1.892749 & 0.035168 & 1.867881 & 1.917616 & 0.155391 & 4 & 1.867881 & 0.196428 & 3.740006 & 10 & 0.200178 \\ AVOA & 1.935251 & 0.011612 & 1.92704 & 1.943462 & 0.11742 & 5 & 1.92704 & 0.10001 & 7.027314 & 9.193464 & 0.198852 \\ SCA & 1.963085 & 0.18394 & 1.83302 & 2.09315 & 0.107374 & 6 & 1.83302 & 0.211406 & 3.377113 & 8.732587 & 0.22836 \\ COOT & 1.975786 & 0.19052 & 1.841068 & 2.110504 & 0.131488 & 7 & 1.841068 & 0.12095 & 5.847412 & 9.189285 & 0.199051 \\ DDO & 2.066372 & 0.178301 & 1.940294 & 2.19245 & 0.1 The objective function to be minimized is given by the following expression, as shown in Equation (53):  
\[ \begin{split} f(x)=& 0.6224\cdot(0.0625\cdot x_{1}) \cdot x_{3}\cdot x_{4}+1.7781\cdot(0.0625\cdot x_{2})\cdot x_{3}^{2}\\ &+3.1661\cdot(0.0625\cdot x_{1})^{2}\cdot x_{4}+19.84\cdot(0.0625 \cdot x_{1})^{2}\cdot x_{3},\end{split}  \tag{53}\]  
where \(x_{1}\) and \(x_{2}\) represent the thickness of the shell and head, respectively, and \(x_{3}\) and \(x_{4}\) represent the inner radius and the length of the cylindrical section of the vessel, excluding the head. The optimization is subject to several constraints, which include both design limitations and practical requirements. These constraints are given by the following inequalities (54)-(61):  
\[c_{1}(x)=-0.0625\cdot x_{1}+0.0193\cdot x_{3}\leq 0, \tag{54}\]  
\[c_{2}(x)=-0.0625\cdot x_{2}+0.00954\cdot x_{3}\leq 0, \tag{55}\]  
\[c_{3}(x)=-\pi\cdot x_{3}^{2}\cdot x_{4}-\frac{4}{3}\cdot\pi\cdot x_{3}^{3}+12 96000\leq 0, \tag{56}\]  
\[c_{4}(x)=x_{4}-240\leq 0, \tag{57}\]  
\[c_{5}(x)=1-x_{1}\leq 0, \tag{58}\]  
\[c_{6}(x)=1-x_{2}\leq 0, \tag{59}\]  
\[c_{7}(x)=10-x_{3}\leq 0, \tag{60}\]  
\[c_{8}(x)=10-x_{4}\leq 0, \tag{61}\]  
The penalty function is incorporated into the objective function to handle constraint violations. The penalty is applied if any of the constraints are violated, as shown in Equation (62):  
\[f(x)=f(x)+10^{5}\cdot\left(\sum_{i=1}^{8}v_{i}+\sum_{i=1}^{8}\max(0,c_{i}(x)) \right), \tag{62}\]  
where \(v_{i}\) is an indicator function that takes the value of 1 if the corresponding constraint \(c_{i}(x)\) is violated and 0 otherwise. This approach ensures that the optimization algorithm focuses on feasible regions of the solution space while minimizing the overall cost of the pressure vessel. In the comparative analysis for the pressure vessel design problem as shown in Table 8, JADEDO achieves the highest overall rank and obtains both the lowest minimum score (5885.334) and a competitive mean value (6067.595). By integrating the dandelion optimizer's explorative phases with JADE's adaptive mutation and crossover, JADEDO demonstrates strong convergence behavior and a remarkable capacity for fine-tuning its solutions.