# Multi-Strategy Hybrid Whale Optimization Algorithm Improvement
## 4 Simulation Experiments and Comparative Analysis
### [PERSON]'s Rank-Sum Test Analysis
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **MSWOA** & **MWOA** & **eWOA** \\ \hline F1 & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) \\ F2 & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) \\ F3 & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) & \(2.64\times 10^{-7}\) \\ F4 & \(3.15\times 10^{-11}\) & \(3.15\times 10^{-11}\) & \(3.15\times 10^{-11}\) \\ F5 & \(3.15\times 10^{-11}\) & \(3.15\times 10^{-11}\) & \(4.62\times 10^{-5}\) \\ F6 & \(2.52\times 10^{-7}\) & \(2.52\times 10^{-7}\) & \(2.49\times 10^{-6}\) \\ F7 & \(2.65\times 10^{-5}\) & \(2.65\times 10^{-5}\) & \(2.73\times 10^{-5}\) \\ F8 & \(2.52\times 10^{-7}\) & \(2.52\times 10^{-7}\) & \(2.51\times 10^{-7}\) \\ F9 & \(2.52\times 10^{-7}\) & \(2.52\times 10^{-7}\) & \(2.52\times 10^{-7}\) \\ F10 & \(2.52\times 10^{-7}\) & \(2.52\times 10^{-7}\) & \(3.77\times 10^{-5}\) \\ F11 & \(4.17\times 10^{-10}\) & \(4.17\times 10^{-10}\) & \(4.35\times 10^{-8}\) \\ F12 & \(4.17\times 10^{-10}\) & \(4.17\times 10^{-10}\) & \(4.35\times 10^{-8}\) \\ F13 & \(4.17\times 10^{-10}\) & \(4.05\times 10^{-10}\) & \(4.35\times 10^{-8}\) \\ F14 & \(4.61\times 10^{-9}\) & \(4.61\times 10^{-9}\) & \(4.61\times 10^{-9}\) \\ F15 & \(4.61\times 10^{-9}\) & \(6.63\times 10^{-1}\) & \(4.61\times 10^{-9}\) \\ F16 & \(4.90\times 10^{-5}\) & \(3.02\times 10^{-11}\) & \(5.57\times 10^{-3}\) \\ F17 & \(4.15\times 10^{-11}\) & \(4.15\times 10^{-11}\) & \(3.01\times 10^{-4}\) \\ F18 & \(6.38\times 10^{-8}\) & \(2.69\times 10^{-2}\) & \(7.73\times 10^{-1}\) \\ F19 & \(3.15\times 10^{-11}\) & \(3.15\times 10^{-11}\) & \(6.01\times 10^{-11}\) \\ F20 & \(2.69\times 10^{-2}\) & \(3.02\times 10^{-11}\) & \(1.06\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculating the resulting \(p\)-value based on [PERSON]’s rank-sum test for MHWOA and WOA variant. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **No.** & **Functions** & \(\boldsymbol{F}_{i}^{*}\) \\ \hline Unimodal Function & 1 & \begin{tabular}{c} Shifted and full Rotated \\ Zakharov Function \\ \end{tabular} & 300 \\ \hline \multirow{6}{*}{Basic Functions} & 2 & \begin{tabular}{c} Shifted and full Rotated \\ Rosenbrock’s Function \\ Shifted and full Rotated \\ Expanded Schaffer’s F6 Function \\ Shifted and full Rotated \\ Non-Continuous Rastrigin’s \\ Function \\ Shifted and full Rotated Levy \\ Function \\ \end{tabular} & 400 \\ \cline{1-1}  & 3 & \begin{tabular}{c} Shifted and full Rotated \\ Expanded Schaffer’s F6 Function \\ Shifted and full Rotated \\ Non-Continuous Rastrigin’s \\ Function \\ Shifted and full Rotated Levy \\ Function \\ \end{tabular} & 600 \\ \cline{1-1}  & 4 & \
\begin{tabular}{c} Non-Continuous Rastrigin’s \\ Function \\ Shifted and full Rotated Levy \\ Function \\ \end{tabular} & 800 \\ \cline{1-1}  & 5 &
\begin{tabular}{c} Shifted and full Rotated Levy \\ Function \\ \end{tabular} & 900 \\ \hline \multirow{3}{*}{Hybrid Function} & 6 & Hybrid Function 1 (\(N=3\)) & 1800 \\  & 7 & Hybrid Function 2 (\(N=6\)) & 2000 \\  & 8 & Hybrid Function 3 (\(N=5\)) & 2200 \\ \hline \multirow{3}{*}{Compositions} & 9 & Composition Function 1 (\(N=5\)) & 2300 \\  & 10 & Composition Function 2 (\(N=4\)) & 2400 \\ \cline{1-1}  & 11 & Composition Function 3 (\(N=5\)) & 2600 \\ \cline{1-1}  & 12 & Composition Function 4 (\(N=6\)) & 2700 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: CEC2022 test functions. Table 8 shows the mean, standard deviation, and comprehensive ranking of each algorithm obtained from 30 runs with dimension 10 where the bold part is the optimal value in each piece of data. In the comparison test, MHWOA performs better on F1 to F5, F7 to F10, and F12 test functions, which is reflected in that the average value obtained is closer to the optimal solution than other algorithms, which means MHWOA has higher accuracy in solving these problems.