## 4 Results and Discussion
### Pressure Vessel Design Problem  
The purpose of this design is to minimize fabrication costs. The vessel has caps at both ends, and its head is hemispherical in shape. This design has four parameters. \(L\), \(R\), \(Th\), and \(Ts\) represent the cylindrical length section without the head, inner radius, head thickness, and shell thickness, respectively. Also, this problem has four limitations, which are presented in Equations (37)-(42). The pressure vessel and the design parameters are shown in Figure 10 [28; 50].  
Consider the following:  
\[\overset{\rightarrow}{z}=[z_{1}\,z_{2}\,z_{3}\,z_{4}]=[T_{s}\,T_{h}\,\,R\,\,L], \tag{37}\]  
Minimize the following:  
\[f(\overset{\rightarrow}{z})=0.6224z_{1}\,z_{3}\,z_{4}+1.7781z_{2}\,z_{3}^{2}+ 3.1661z_{1}^{2}z_{4}+19.84z_{1}^{2}z_{3}, \tag{38}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multicolumn{3}{c}{**Optimum Variables**} & \multirow{2}{*}{**Optimum Cost**} \\  & **d** & **D** & **N** & \\ \hline GWO [25] & 0.05169 & 0.356737 & 11.28885 & 0.012666 \\ \hline MFO [43] & 0.051994457 & 0.36410932 & 10.868421862 & 0.0126669 \\ \hline SSA [27] & 0.051207 & 0.345215 & 12.004032 & 0.0126763 \\ \hline WOA [28] & 0.051207 & 0.345215 & 12.004032 & 0.0126763 \\ \hline HHO [17] & 0.051796393 & 0.359305355 & 11.138859 & 0.012665443 \\ \hline RSRFT [39] & 0.05147146 & 0.3515050 & 11.6013141 & 0.01266617 \\ \hline RFO [39] & 0.052667011 & 0.3806680 & 10.0213925 & 0.0126934 \\ \hline EDBO [40] & 0.0500156 & 0.31777 & 13.7778 & 0.012718751 \\ \hline HHHOWOA1 & 0.0517709162 & 0.3586901539 & 11.174259208 & 0.0126653548 \\ \hline HHHOWOA2 & 0.0516725910 & 0.3563216437 & 11.312225467 & 0.0126652377 \\ \hline HHHOWOA2 PSO & 0.0516901857 & 0.3567431404 & 11.2876518173 & 0.0126654334 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of the results from the literature and the proposed HHHOWOA1, HHHOWOA2, and HHHOWOA2 PSO algorithms for the tension–compression spring design.  
Figure 9: The best optimization convergence curves of the proposed methods and other methods for the tension–compression spring design.  
Variable range as follows:  
\[0\leq z_{1}\leq 99,\ 0\leq z_{2}\leq 99,\ 10\leq z_{3}\leq 200,\ 10\leq z_{4}\leq 200,\]  
Subjected to the following:  
\[g_{1}(\stackrel{{\rightarrow}}{{z}})=-z_{1}+0.0193z_{3}\leq 0, \tag{39}\]  
\[g_{2}(\stackrel{{\rightarrow}}{{z}})=-z_{2}+0.00954z_{3}\leq 0, \tag{40}\]  
\[g_{3}(\stackrel{{\rightarrow}}{{z}})=-\pi z_{3}^{2}z_{4}-\frac{4} {3}\pi z_{3}^{3}+1,296,000\leq 0, \tag{41}\]  
\[g_{4}(\stackrel{{\rightarrow}}{{z}})=z_{4}-240\leq 0, \tag{42}\]  
To compare with the literature, the variables of this problem were analyzed in two ways. First, the variables were examined as continuous. Second, the first two variables (_Ts, Th_) were analyzed as multiples of 0.0625 (material thickness), while the others were examined as continuous variables. Then, the best optimization and its variable values are found. These values are compared to the literature, as given in Table 7. In terms of the best optimization value, the recommended algorithms, HHHOWOA2 PSO and HHHOWOA2, ranked first and second, respectively, for variables considered as continuous. Then, the following algorithms come in order: RSRFT, WOALFVWPSO, the EDBO, CIHHO, HHO, HHHOWOA1, MFO, the WOA, CPSO, and the RFO. Furthermore, Figure 11 illustrates the best optimization convergence curves for the proposed methods and the optimization algorithms they are based on, including the WOA, HHO, and PSO, in the context of the pressure vessel design.  
Figure 11: The best optimization convergence curves of the proposed methods and other methods for the pressure vessel design.  
Figure 10: Pressure vessel design.