# A Multi-Strategy Improved Honey Badger Algorithm for Engineering Design Problems
## 3 Experiments
### Results of Comparative Experiments
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Criterion**} & \multicolumn{2}{c}{**P/t**} & \multicolumn{1}{c}{**P/t**} & \multicolumn{1}{c}{**P/t**} \\ \cline{3-5}  & & **15/1000** & **30/500** & **60/250** \\ \hline \multirow{4}{*}{\(f_{19}\)} & Mean & \(-3.6990\times 10^{0}\) & \(-3.8626\times 10^{0}\) & \(\mathbf{-3.8628\times 10^{0}}\) \\  & SD & \(2.9147\times 10^{-1}\) & \(1.1146\times 10^{-3}\) & \(\mathbf{1.4827\times 10^{-13}}\) \\  & Rank & \(3\) & \(2\) & \(1\) \\  & Mean & \(\mathbf{-3.2541\times 10^{0}}\) & \(-3.2004\times 10^{0}\) & \(-3.1947\times 10^{0}\) \\ \multirow{2}{*}{\(f_{20}\)} & SD & \(6.5043\times 10^{-2}\) & \(\mathbf{1.4440\times 10^{-2}}\) & \(2.8326\times 10^{-2}\) \\  & Rank & \(1\) & \(2\) & \(3\) \\  & Mean & \(\mathbf{-1.0150\times 10^{1}}\) & \(\mathbf{-1.0153\times 10^{1}}\) & \(-1.0153\times 10^{1}\) \\ \multirow{2}{*}{\(f_{21}\)} & SD & \(1.4456\times 10^{-2}\) & \(\mathbf{3.1420\times 10^{-8}}\) & \(2.7968\times 10^{-7}\) \\  & Rank & \(3\) & \(1\) & \(2\) \\  & Mean & \(\mathbf{-1.0403\times 10^{1}}\) & \(\mathbf{-1.0403\times 10^{1}}\) & \(\mathbf{-1.0403\times 10^{1}}\) \\ \multirow{2}{*}{\(f_{22}\)} & SD & \(3.0841\times 10^{-4}\) & \(8.5220\times 10^{-8}\) & \(\mathbf{4.2074\times 10^{-9}}\) \\  & Rank & \(3\) & \(2\) & \(1\) \\  & Mean & \(\mathbf{-1.0367\times 10^{1}}\) & \(-8.9298\times 10^{0}\) & \(-3.3961\times 10^{0}\) \\ \multirow{2}{*}{\(f_{23}\)} & SD & \(\mathbf{1.1467\times 10^{0}}\) & \(3.2466\times 10^{0}\) & \(2.6635\times 10^{0}\) \\  & Rank & \(1\) & \(2\) & \(3\) \\ \hline \multirow{4}{*}{\(f_{23}\)} & Rank-Count & \(50\) & \(43\) & \(45\) \\  & Ave-Rank & \(2.174\) & \(1.869\) & \(1.957\) \\ \cline{1-1}  & Overall-Rank & \(3\) & \(1\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._In conducting experiments to compare the performance of the HBA with other optimization algorithms, we used a standardized experimental framework to ensure the fairness of the evaluation. In the experiments, the population size was uniformly set to 30 individuals, while the maximum number of iterations of the algorithm was limited to 500 rounds. To ensure the stability and credibility of the results, 50 rounds of independent experiments were executed for each algorithm. In this way, we were able to collect and analyze the mean and standard deviation of each objective function to comprehensively assess the performance and efficiency of different algorithms in solving the optimization problem. Table 4 shows the experimental data of MIHBA with other algorithms after 50 iterations on 23 test functions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **Criterion** & **NO** & **PSO** & **GA** & **DBO** & **HBA** & **MIHBA** \\ \hline \multirow{4}{*}{\(f_{1}\)} & Mean & \(7.3003\times 10^{-16}\) & \(2.4640\times 10^{9}\) & \(9.4741\times 10^{3}\) & \(2.2951\times 10^{-144}\) & \(4.7283\times  
Based on the numerical results obtained in Table 4, we can see that MIHBA exhibits better means and standard deviations than the other compared algorithms on \(f_{1}\)-\(f_{4}\), \(f_{7}\)-\(f_{11}\), and \(f_{17}\). MIHBA has the best average of the six compared algorithms on the \(f_{18}\) and \(f_{21}\)-\(f_{23}\) test functions. And for \(f_{20}\), the standard deviation of MIHBA is optimal compared to other algorithms. For \(f_{15}\) and \(f_{19}\), the improvements to HBA, while effective, are not optimal among all the compared algorithms. On test functions \(f_{5}\), \(f_{6}\),\(f_{12}\)-\(f_{14}\), and \(f_{16}\), the improvements to the HBA do not achieve better results. In terms of rankings, MIHBA performs well in the majority of the tested functions, with an overall ranking of first among the six compared algorithms. Overall, the numerical results of MIHBA on the 23 tested functions show that MIHBA has better stability and stronger ability to jump out of the local optimum.