## 4 Numerical Optimization Experiments
### Simulation Results Analysis  
In this subsection, comparative numerical optimization experiments were conducted among the WOA [26], MFO [11], HPSO-SSM [48], GWO [10], PSO-GWO [49], LWOA [40], and MSWOA. Table 3 presents the mean and S.D. of the algorithms. Moreover, from Figures 4-13, we can see the convergence curves for these seven metaheuristic swarm optimization algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Function** & **Dim** & **Criteria** & **WOA** & **MPO** & **HPSO-SSM** & **GWO** & **PSO-GWO** & **LWOA** & **MSWOA** \\ \hline \multirow{3}{*}{F1} & \multirow{3}{*}{30} & Mean & \(1.45\times 10^{-80}\) & \(4.00\times 10^{3}\) & \(3.34\times 10^{-3}\) & \(5.67\times 10^{-31}\) & \(3.47\times 10^{-48}\) & \(2.00\times 10^{-9}\) & \(\mathbf{2.53\times 10^{-85}}\) \\  & & S.D. & \(1.36\times 10^{-79}\) & \(1.54\times 10^{4}\) & \(1.12\times 10^{-2}\) & \(1.80\times 10^{-30}\) & \(3.23\times 10^{-4}\) & \(7.64\times 10^{-9}\) & \(\mathbf{2.40\times 10^{-84}}\) \\  & & Rank & 2 & 7 & 6 & 4 & 3 & 5 & **1** \\ \hline \multirow{3}{*}{F2} & \multirow{3}{*}{30} & Mean & \(1.42\times 10^{-33}\) & \(2.73\times 10^{1}\) & 3.25 & \(1.17\times 10^{-18}\) & \(4.22\times 10^{-27}\) & \(5.73\times 10^{-19}\) & \(\mathbf{8.94\times 10^{-36}}\) \\  & & S.D. & \(1.28\times 10^{-52}\) & \(4.48\times 10^{1}\) & 5.23 & \(1.62\times 10^{-18}\) & \(1.65\times 10^{-26}\) & \(2.12\times 10^{-18}\) & \(\mathbf{2.43\times 10^{-55}}\) \\  & & Rank & 2 & 7 & 6 & 5 & 3 & 4 & **1** \\ \hline \multirow{3}{*}{F3} & \multirow{3}{*}{30} & Mean & \(3.36\times 10^{4}\) & \(2.24\times 10^{4}\) & 2.95 & \(4.82\times 10^{-6}\) & \(2.44\times 10^{-33}\) & \(2.31\times 10^{4}\) & \(\mathbf{4.88\times 10^{-41}}\) \\  & & S.D. & \(3.06e+004\) & \(4.00e+004\) & 5.43 & \(3.05\times 10^{-5}\) & \(2.09\times 10^{-32}\) & \(3.41\times 10^{4}\) & \(\mathbf{2.41\times 10^{-40}}\) \\  & & Rank & 7 & 5 & 4 & 3 & 2 & 6 & **1** \\ \hline \multirow{3}{*}{F4} & \multirow{3}{*}{30} & Mean & \(5.58\times 10^{1}\) & \(6.25\times 10^{1}\) & \(9.21\times 10^{-1}\) & \(8.23\times 10^{-8}\) & \(3.91\times 10^{-21}\) & 6.52 & \(\mathbf{1.45\times 10^{-27}}\) \\  & & S.D. & \(8.72\times 10^{1}\) & \(2.32\times 10^{1}\) & 1.83 & \(2.86\times 10^{-7}\) & \(1.87\times 10^{-20}\) & 1.38 & \(\mathbf{6.84\times 10^{-27}}\) \\  & & Rank & 6 & 7 & 4 & 3 & 2 & 5 & **1** \\ \hline \multirow{3}{*}{F5} & \multirow{3}{*}{30} & Mean & \(2.74\times 10^{1}\) & \(1.01\times 10^{4}\) & \(6.15\times 10^{1}\) & \(2.70\times 10^{1}\) & \(2.78\times 10^{1}\) & \(2.64\times 10^{1}\) & \(\mathbf{2.58\times 10^{1}}\) \\  & & S.D. & \(1.20\) & \(8.47\times 10^{1}\) & \(1.66\times 10^{2}\) & 2.22 & 1.23 & \(7.38\times 10^{-1}\) & \(\mathbf{3.45}\) \\  & & Rank & 4 & 7 & 6 & 3 & 5 & 2 & **1** \\ \hline \multirow{3}{*}{F6} & \multirow{3}{*}{30} & Mean & \(1.74\times 10^{-1}\) & \(9.91\times 10^{2}\) & \(1.80\times 10^{-2}\) & \(4.32\times 10^{-1}\) & 3.07 & \(\mathbf{9.73\times 10^{-3}}\) & \(1.67\times 10^{-1}\) \\  & & S.D. & \(3.52\times 10^{-1}\) & \(9.39\times 10^{3}\) & \(6.20\times 10^{-2}\) & 1.02 & \(8.17\times 10^{-1}\) & \(\mathbf{6.48\times 10^{-3}}\) & \(4.53\times 10^{-1}\) \\  & & Rank & 4 & 7 & 2 & 5 & 6 & **1** & 3 \\ \hline \multirow{3}{*}{F7} & \multirow{3}{*}{30} & Mean & \(2.15\times 10^{-1}\) & \(5.36\times 10^{1}\) & \(9.80\times 10^{-1}\) & \(1.24\times 10^{-3}\) & \(\mathbf{8.41\times 10^{-5}}\) & \(2.24\times 10^{-3}\) & \(2.28\times 10^{-4}\) \\  & & S.D. & \(6.37\times 10^{-1}\) & \(2.12\times 10^{1}\) & \(7.07\times 10^{-1}\) & \(1.45\times 10^{-3}\) & \(\mathbf{2.54\times 10^{-4}}\) & \(5.96\times 10^{-3}\) & \(3.08\times 10^{-4}\) \\  & & Rank & 5 & & 6 & **1** & **4** & **1** & **4** \\ \hline \multirow{3}{*}{F8} & \multirow{3}{*}{30} & Mean & \(-1.08\times 10^{4}\) & \(-8.78\times 10^{3}\) & \(-8.74\times 10^{3}\) & \(-6.27\times 10^{3}\) & \(-3.08\times 10^{3}\) & \(\mathbf{-1.11\times 10^{4}}\) & \(-1.01\times 10^{4}\) \\  & & S.D.