# Self-Adaptive Differential Evolution with Gauss Distribution for Optimal Mechanism Design
## 5 Numerical Simulation
### Modified Differential Evolution for Real-World Problem
#### 5.4.2 Design Problem of Air-Storage Tank  
In this section, the dimension of the air-storage tank is optimized, high-compression pressure of 3000 psi is applied to the structure [24]. The design domain is a cylindrical structure with half of a sphere at both heads as Figure 24. The target is to minimize the manufacturing cost by evaluating design variables, including end thickness \(x_{1}(Ts)\), head thickness \(x_{2}(Th)\), inner radius \(x_{3}(R)\), and cylindrical length \(x_{4}(L)\). This optimization problem is described by Equations (28)-(32) as below.  
Minimize:  
\[f(x)=0.6224x_{1}x_{3}x_{4}+1.7781x_{2}x_{3}^{2}+3.1661x_{1}^{2}x_{4}+19.84x_{1} ^{2}x_{3} \tag{28}\]  
Subject to  
\[g_{1}\ =\ -x_{1}+0.0193x_{3}\leq 0; \tag{29}\]  
\[g_{2}\ =\ -x_{2}+0.00954x_{3}\leq 0; \tag{30}\]  
\[g_{3}\ =\ -\pi x_{3}^{2}x_{4}-\frac{4}{3}\pi x_{3}^{3}+1,296,000.0\leq 0; \tag{31}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **Best** & **Mean** & **Worst** & **Standard Deviation** \\ \hline CDE & 1.733461 & 1.768158 & 1.824105 & 0.022 \\ CPSO & 1.728024 & 1.748831 & 1.782143 & 0.013 \\ CGA & 1.748309 & 1.771973 & 1.785835 & 0.011 \\ EOBIDE & 1.724852 & 1.724856 & 1.725001 & \(2.131\times 10^{-5}\) \\ G-ISADE & 1.713571 & 1.713571 & 1.713571 & **0.00** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical results for welded beam problem.  
Figure 24: Design problem of air-storage tank.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Design Variables** & **CDE** & **CPSO** & **CGA** & **EOBLDE** & **G-ISADE** \\ \hline \(x_{1}\) & 0.203137 & 0.202369 & 0.208800 & 0.205729 & 0.207407 \\ \(x_{2}\) & 3.542998 & 3.544214 & 3.420500 & 3.470488 & 3.448728 \\ \(x_{3}\) & 9.033498 & 9.048210 & 8.99750 & 9.036623 & 9.00000 \\ \(x_{4}\) & 0.206179 & 0.205723 & 0.210000 & 0.205729 & 0.207407 \\ \(g_{1}\) & \(-44.57856\) & \(-12.83979\) & \(-0.337812\) & \(-1.4790\times 10^{-7}\) & \(-1.8\times 10^{-12}\) \\ \(g_{2}\) & \(-44.66353\) & \(-1.247467\) & \(-353.9026\) & \(-1.8863\times 10^{-7}\) & \(-7.3\times 10^{-12}\) \\ \(g_{3}\) & \(-0.003042\) & \(-0.001498\) & \(-0.001200\) & \(-9.7584\times 10^{-11}\) & 0.00000 \\ \(g_{4}\) & \(-3.423726\) & \(-3.429347\) & \(-3.411865\) & \(-3.432983\) & \(-3.428506\) \\ \(g_{5}\) & \(-0.078137\) & \(-0.079381\) & \(-0.083800\) & \(-0.080729\) & \(-0.082407\) \\ \(g_{6}\) & \(-0.235557\) & \(-0.235536\) & \(-0.235540\) & \(-0.235540\) & \(-0.235481\) \\ \(g_{7}\) & \(-38.02826\) & \(-11.68135\) & \(-363.2323\) & \(-4.4691\times 10^{-8}\) & \(-131.577828\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Best solution for welded beam problem.  
\[g_{4}\ =\ x_{4}-240.0\leq 0; \tag{32}\]  
where \(1\times 0.0625\leq x_{1},x_{2}\leq 99\times 0.0625\), \(10.0\leq x_{3}\), and \(x_{4}\leq 200.0\).  
The four methods, such as CDE, CPSO, CGA, and EOBLDE, were continued compared with our solution for this problem. The comparison results were given in Tables 5 and 6. The best solution of G-ISADE is better than that of the other algorithms. As can be seen in Table 6, the search ability of G-ISADE is also better than the remains since the worst solution found by G-ISADE is better than those of the others.