# Multi-Strategy-Improvement-Based Slime Mould Algorithm
## 4 Experimentation and Application
### Comparison of Algorithm Performance  
To test the performance of the MSSMA, simulation experiments were conducted for the Whale Optimization Algorithm (WOA) [23], K-mean Optimization Algorithm (KO), Beluga Whale Optimization Algorithm (BWO) [24], Ivy Optimization Algorithm (IVY) [25], and SMA algorithm and MSSMA. To ensure the fairness of the test, the number of populations of all algorithms was 50, the maximum number of iterations was 1000, and each algorithm was tested independently for 30 times, and the mean and variance of each method in different benchmarking functions were calculated.  
Table 11 shows the experimental data of the six algorithms on a test set consisting of 50 benchmark test functions, with the best results highlighted in bold.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & **L = 3** & **L = 5** & **L = 10** & **L = 15** & **L = 20** \\ \hline (W \(\mid\) T \(\mid\) L) & (40 \(\mid\) 6 \(\mid\) 4) & (20 \(\mid\) 26 \(\mid\) 4) & (11 \(\mid\) 35 \(\mid\) 4) & (8 \(\mid\) 32 \(\mid\) 10) & (7 \(\mid\) 24 \(\mid\) 19) \\ Ave & 1.56 & 2.88 & 3.54 & 3.40 & 3.62 \\ Rank & 1 & 2 & 3 & 4 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of Friedman test for MSSMA at different values of L.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & **MSSMA** & **SMA** & **MSSMA-1** & **MSSMA-2** & **MSSMA-3** \\ \hline (W \(\mid\) T \(\mid\) L) & (49 \(\mid\) 1 \(\mid\) 10) & (14 \(\mid\) 9 \(\mid\) 27) & (7 \(\mid\) 31 \(\mid\) 12) & (15 \(\mid\) 35 \(\mid\) 0) & (17 \(\mid\) 33 \(\mid\) 0) \\ Ave & 1.68 & 3.20 & 3.54 & 3.40 & 3.14 \\ Rank & 1 & 3 & 5 & 4 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of Friedman test for ablation experiments.  
To further analyze the statistical significance of the selected improvements, the Friedman test and the Wilcoxon signed-rank test with a significant level \(p\)-value of 0.05 were used. The other five algorithms were compared with the MSSMA to obtain the \(p\)-value in the last row of Table 3 (the MSSMA was not compared with itself, so the corresponding cell is \"-\"). If the \(p\)-value was less than 0.05, it meant that there was a significant difference between the two algorithms in terms of performance; on the contrary, it was considered that there was no significant difference between the two algorithms. The results show that all \(p\)-values were less than 0.05, which indicates that the performance of the MSSMA is significantly different from those of the other algorithms and statistically verifies the superiority and reliability of the MSSMA.  
Table 3 shows the results of the six algorithms on the test function. From the (W \(\mid\) T \(\mid\) L) and Friedman average ranking results in the table, it can be seen that the MSSMA significantly outperformed the other five algorithms compared on most of the test functions. The specific analysis is as follows.  
From Table 3, we can see that the MSSMA achieved 14 best results on seventeen single-peak functions, 29 best results on thirty-three multi-peak functions, and only slightly worse test results on seven functions. Specifically, the MSSMA achieved the second-best results on functions \(f7,f9,f26,f40\), and \(f43\) and the third-best results on functions \(f6\) and \(f50\). In addition, the improved algorithm outperformed SMA on all 50 functions tested, reflecting the good results of the improved strategy. The convergence curve of the algorithm is shown in Figure A1.  
The MSSMA performed much better than other algorithms on UN (single-peak non-separable) and US (single-peak separable) functions, which shows that the MSSMA has better exploration capability. It can also be seen that the MSSMA performed better than other algorithms on MS (multimodal separable) and MN (multimodal non-separable) functions, indicating that the MSSMA can better avoid local optima and has better exploration capability. In summary, the solution accuracy and stability of the MSSMA are generally better than those of the other five algorithms.