## 4 Experimental Studies
### Results Analysis and Discussion  
#### 4.3.1 Solutions Accuracy  
The comparison of the experimental results for the 10-D problems is shown in Appendix A. The proposed MLDMS-PSO achieved the best results on f6, f9, f11, f13, f14, f16, f18, f20, f23, and f29. The second-best solution was obtained on f12, f27, and f30. The result ranked third on f1, f4, f5, f8, f15, f17, f19, f21, f22, f24, f26, and f28, and the result ranked fourth on f7 and f10, which was better than most comparison algorithms. On f3, although the mean value of its solution ranked lower, the accuracy of the solution was satisfactory, which was 6.96 \(\times\)\(10^{-12}\). In addition, MLDMS-PSO achieved the minimum average rank. Therefore, MLDMS-PSO outperformed the other comparison algorithms on 10-D problems. The comparison of the experimental results for the 30-D problems is shown in Appendix B. TSLPSO and XPSO had the best performance on unimodal functions f1 and f3, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters Setting** & **Year** \\ \hline GPSO & \(w=0.9\sim 0.4\), \(c_{1}=c_{2}=2\) & 1998 \\ FIPS & \(\chi=0.729\), \(\sum c_{i}=4.1\) & 2004 \\ CLPSO & \(w=0.9\sim 0.4\), \(c=1.49445\) & 2006 \\ CCPSO-ISM & \(P=0.05\), \(\sigma=5\), \(w=0.6\), \(c=2\) & 2015 \\ TSLPSO & \(w=0.9\sim 0.4\), \(c_{1}=c_{2}=1.49445\), \(c_{3}=0.5\sim 2.5\) & 2019 \\ XPSO & \(\eta=0.2\), \(Stag_{\max}=5\), \(p=0.5\) & 2020 \\ DMS-PSO & \(w=0.9\sim 0.2\), \(c_{1}=c_{2}=1.49445\) & 2005 \\ PSO-DLS & \(w=0.9\sim 0.4\), \(c_{1}=c_{2}=1.49445\) & 2017 \\ HCLDMS-PSO & \(w=0.99\sim 0.29\), \(c_{1}=2.5\sim 0.5\), \(c_{2}=0.5\sim 2.5\), \(P_{m}=0.1\) & 2020 \\ MLDMS-PSO & \(w=0.9\sim 0.4\), \(c_{1}=c_{2}=1.49445\), \(G=12\), \(Rc=0.1\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings of the ten PSOs. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Dim** & **Index** & **GPSO** & **FIPS** & **CLPSO** & **CCPSO-ISM** & **XPSO** & **TSLPSO** & **DMS-PSO** & **PSO-DLS** & **HCLDMS-PSO** & **MLDMS-PSO** \\ \hline
10-D & _Ave_ & 8.62 & 9.93 & 4.28 & 4.14 & 5.69 & 6.72 & 3.07 & 5.12 & 4.41 & 2.41 \\
30-D & _Ave_ & 8.97 & 10 & 5.03 & 4.72 & 6.52 & 4.55 & 4.93 & 3.97 & 3.28 & 3 \\
50-D & _Ave_ & 8.97 & 10 & 4.9 & 5.1 & 5.9 & 5.79 & 4.86 & 3.41 & 3.07 & 3 \\ _CMean rank_ & 8.85 & 9.98 & 4.74 & 4.65 & 6.04 & 5.69 & 4.29 & 4.17 & 3.59 & 2.80 \\ _Crank_ & 9 & 10 & 6 & 5 & 8 & 7 & 4 & 3 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comprehensive rank of MLDMS-PSO and other PSO variations for 10-D, 30-D, and 50-D problems. The proposed MLDMS-PSO ranked third and fifth on f1 and f3, respectively. In the simple multimodal functions, hybrid function, and composition function, the proposed MLDMS-PSO achieved the best solution on f6, f9, f11, f12, f17, f18, f20, f22, f28, and f29 and the second-best solution on f16, f25, and f30. CCPSO-ISM obtained the best performance on f6, f13, f15, f19, and f26. PSO-DLS obtained the best solution on f5, f7, f8, f21, f23, and f24. DMS-PSO found the smallest mean value of all solutions on f4, f14, f16, f25, f27, and f30. HCLDMS-PSO showed the best results on f10. In total, depending on the final rank, the MLDMS-PSO algorithm was the first. Hence, MLDMS-PSO consistently performed well and achieved the best overall performance on the 30-D problems. As shown in Appendix C, for the 50-D problems, it can be seen that the proposed MLDMS-PSO had the highest performance on the whole test suite. Compared with other comparison algorithms, it achieved the best results on f6, f11, f13, f14, f16, f17, f20, and f29. Depending on the final rank, the MLDMS-PSO achieved the optimal performance, ranking first on eight problems, second on two problems, and third on eleven problems out of the twenty-nine benchmark problems.