# Multi-Strategy Hybrid Whale Optimization Algorithm Improvement
## 4 Simulation Experiments and Comparative Analysis
### An Introduction to Test Functions  
The comprehensive test set consists of 20 test functions as shown in Table 2. It includes five high-dimensional unimodal functions (F1\(\sim\)F3, F6\(\sim\)F7), five high-dimensional multimodal functions (F4\(\sim\)F5, F8\(\sim\)F10), seven high-dimensional hybrid functions (F11\(\sim\)F17), and three composition functions (F18\(\sim\)F20). The high-dimensional unimodal and multimodal functions combine classical test functions with offset and rotation functions to comprehensively evaluate the algorithm's local convergence capability, calculation speed, and accuracy. They also assess the algorithm's ability to perform global optimal search and avoid local optima. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Basic Parameter** \\ \hline MHWOA & \(a=1-\frac{t}{t}\) \\ BWO [40] & \(Wf=0.1-0.5\times\frac{t}{t}\) \\ GJO [41] & \(E=1.5\times E_{0}\times(1-\frac{t}{t})\) \\ RSO [42] & \(A=R\times(1-\frac{t}{t})\) & \(C\in(0,2)\) \\ SO [43] & \(C_{1}=0.5\) & \(C_{2}=0.05\) & \(C_{3}=2\) \\ WOA [44] & \(a=2\times(1-\frac{t}{t})\) \\ MSWOA [45] & \(a=\frac{m}{e^{\frac{t}{n\max}}}\omega\) \\ MWOA [46] & \(a=2\times(1-\frac{t}{t})\) \\ eWOA [47] & \(a=2\times(1-\frac{t}{t})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of standard swarm intelligence algorithm and improved swarm intelligence algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Function** & **Dim** & **Range** &
\begin{tabular}{c} **Optimum** \\ **Value** \\ \end{tabular} \\ \hline \multirow{8}{*}{**Unimodal**} & \(F_{1}(x)=\sum_{i=1}^{n}\left(\sum_{j=1}^{i}x_{i}\right)^{2}\) & 30 & \([-100,100]\) & 0 \\  & \(F_{2}(x)=\) & 30 & \([-100,100]\) & 0 \\  & \(F_{3}(x)=\) & 30 & \([-100,100]\) & 0 \\  & \(\sum_{i=1}^{n}x_{i}^{2}+random[0,1]\) & & & \\  & \(F_{4}=\) Shifted and Rotated & 30 & \([-100,100]^{D}\) & 100 \\  & Bent Cigar Function & \(F_{5}=\) Shifted and Rotated & 30 & \([-100,100]^{D}\) & 300 \\  & Zakharov Function & & & \\ \hline \multirow{8}{*}{**Multimodal**} & \(\frac{E_{5}(x)}{\sum_{i=1}^{n}\left[x_{i}^{2}-100 cs(2 mx_{i})+10\right]}\) & 30 & \([-5.12,5.12]\) & 0 \\  & \(\frac{1}{100}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}cos(\frac{x_{i}}{t^{3}})+1\) & 30 & \([-600,600]\) & 0 \\  & \(F_{6}=\) Shifted and Rotated & 30 & \([-100,100]^{D}\) & 400 \\  & Rosenbrock’s Function & & & \\  & \(F_{n}=\) Shifted and Rotated & 30 & \([-100,100]^{D}\) & 900 \\  & Levy Function & \(F_{10}=\) Shifted and Rotated & 30 & \([-100,100]^{D}\) & 1000 \\  & Schwef's Function & & & & \\ \hline \multirow{8}{*}{**Hybrid Functions**} & \(F_{11}=\) Hybrid Function 1 & & & 1100 \\  & \(F_{12}=\) Hybrid Function 2 & & & 1200 \\  & \(F_{13}=\) Hybrid Function 3 & & & 1300 \\ \cline{1-1} \cline{2-4}  & \(F_{14}=\) Hybrid Function 4 & 30 & \([-100,100]^{D}\) & 1400 \\  & \(F_{15}=\) Hybrid Function 5 & & & 1500 \\  & \(F_{16}=\) Hybrid Function 8 & & & 1800 \\  & \(F_{17}=\) Hybrid Function 9 & & & 1900 \\ \hline \multirow{8}{*}{**Composition**} & \(F_{8}=\) Composition & & & 2500 \\  & Function 5 & & & \\ \cline{1-1} \cline{2-4}  & \(F_{19}=\) Composition & \(F_{18}=\) Composition & 30 & \([-100,100]^{D}\) & 2800 \\ \cline{1-1} \cline{2-4}  & \(F_{20}=\) Composition & & & 3000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Introduction of about 20 test functions. and it is 6.22% slower compared with SO. When dealing with composition functions, MHWOA achieves a speedup of 3.65 times compared with BWO, 2.37 times compared with GJO, 5.6% compared with RSO, 46.68% compared with WOA, and it is 1.96% slower compared with SO. Overall, in most functions, MHWOA's computation time is only second to SO, but it shows a significant improvement compared with WOA. In handling increasingly difficult problems across the four categories, the time difference between MHWOA and SO gradually decreases, and MHWOA still demonstrates a significant advantage over other algorithms in tackling complex problems. This further highlights the superiority of MHWOA in handling challenging problems.