## 4 Results and Discussion
### Three-Bar Truss Design Problem  
The civil engineering optimization problem wants to find the minimal weight based on stress, deflection, and buckling limitations. This is accomplished by using two parameters. The different components of this problem are given in Figure 12. There are 4 nodes, numbered from 1 to 4, and 3 bars in the three-bar truss design, as shown in Figure 12. These nodes represent the connection points where the bars meet, forming the structure of the truss. Equations (43)-(47) express this problem [43].  
Consider the following:  
\[\stackrel{{\rightarrow}}{{z}}=[z_{1}\,z_{2}]=[A_{1}\,A_{2}], \tag{43}\]  
Minimize the following:  
\[f(\stackrel{{\rightarrow}}{{z}})=(2\sqrt{2}z_{1}+z_{2})*l, \tag{44}\]  
Subject to the following:  
\[g_{1}(\stackrel{{\rightarrow}}{{z}})=\frac{\sqrt{2}z_{1}+z_{2}}{ \sqrt{2}z_{1}^{2}+2z_{1}z_{2}}P-\sigma\leq 0, \tag{45}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & \multicolumn{4}{c}{**Optimum Variables**} \\  & \(\mathbf{T_{s}}\) & \(\mathbf{T_{k}}\) & \(\mathbf{R}\) & \(\mathbf{L}\) & **Optimum Cost** \\ \hline WOALFVWPSO [5] & 0.7831596 & 0.3944979 & 40.35439 & 200 & 5955.7996 \\ \hline CPSO [50] & 0.8125 & 0.4375 & 42.091266 & 176.7465 & 6061.0777 \\ \hline MFO [43] & 0.8125 & 0.4375 & 42.098445 & 176.636596 & 6059.7143 \\ \hline WOA [28] & 0.8125 & 0.4375 & 42.0982699 & 176.638998 & 6059.741 \\ \hline HHO [17] & 0.81758383 & 0.4072927 & 42.09174576 & 176.7196352 & 6000.46259 \\ \hline RSBFT [39] & 0.81612257 & 0.403409949 & 42.2861349 & 174.325078 & 5953.4364 \\ \hline RFO [39] & 0.81425 & 0.44521 & 42.20231 & 176.62145 & 6113.3195 \\ \hline EDBO [40] & 0.7827496 & 0.3943 & 40.38594 & 200 & 5957.489796 \\ \hline CIHHO [41] & 1.07055 & 0.52863 & 55.38854 & 60.615468 & 5962.00814 \\ \hline HHHOWOA1 & 0.8049634 & 0.3890709 & 40.88383 & 192.6467 & 6023.1709 \\ \hline HHHOWOA2 & 0.8125 & 0.4375 & 42.0974671 & 176.64872135 & 6059.8335 \\ \hline HHHOWOA2 & 0.7818555 & 0.3920745 & 40.31962 & 200 & 5933.5439 \\ \hline  & 0.8125 & 0.4375 & 42.095787915 & 176.66953154 & 6060.0380 \\ \hline HHHOWOA2 PSO & 0.7869923 & 0.3888867 & 40.77421 & 193.7898 & 5901.0625 \\ \hline  & 0.8125 & 0.4375 & 42.098444394 & 176.63768237 & 6059.7395 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of the results from the literature and the proposed HHHOWOA1, HHHOWOA2, and HHHOWOA2 PSO algorithms for the pressure vessel design.  
Figure 12: Three-bar truss design.  
\[g_{2}(\vec{z})=\frac{z_{2}}{\sqrt{2}z_{1}^{2}+2z_{1}z_{2}}P-\sigma\leq 0, \tag{46}\]  
\[g_{3}(\vec{z})=\frac{1}{\sqrt{2}z_{2}+z_{1}}P-\sigma\leq 0, \tag{47}\]  
Variable range as follows:  
\[0\leq z_{1},\;z_{2}\leq l,\]  
where  
\[l=100\;\text{cm},P=2\;\text{KN}/\text{cm}^{2},\sigma=2\;\text{KN}/\text{cm}^{2}\]  
The best optimization and its variable values are found for this problem. These values are compared to the literature, as shown in Table 8. RSRFT, CIHHO, PSO-DE, HHHOWOA1, HHHOWOA2, and HHO are the most cost-effective. Then, the following algorithms come in order: HHHOWOA2 PSO, MFO, MBA, the EDBO, CS, and the RFO. Furthermore, Figure 13 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 three-bar truss design.