## 4 Experimental Studies
### Parameter Tuning  
The value of some parameters may greatly affect the performance of the algorithm. In the proposed MLDMSPSO, two important parameters, the interval iteration parameter \(G\) and the recombination parameter \(Rc\), need to be calibrated. The parameter tuning used the same population size of 40, maximum number of fitness evaluations (_MaxFEs_) of 1.00 \(\times\) 10\({}^{5}\), run time of 51, and 10 dimensions. A smaller value for parameter \(G\) and a larger value for \(Rc\) may lead to frequent reorganization of the subswarms in the iterative process, which makes the subpopulation unable to make full use of the internal information to improve itself and fully exploit the local area where it is located. In contrast, for a larger parameter \(G\) and a smaller \(Rc\), the population may have the risk of a stagnant evolution and a waste of the evaluation times. Therefore, to obtain an insightful view of how the interval iteration parameter \(G\) and recombination parameter \(Rc\) affect the performance of MLDMS-PSO, this paper tested 48 different combinations of \(G\) values, from 5 to 20, when \(Rc\) was 0.1, 0.2, and 0.3. The experimental results were ranked by the average rank with the mean of 29 benchmark functions as the measurement standards. When the average rank value corresponding to a combination of \(G\) and \(Rc\) is smaller, it indicates that the combination performs better in terms of overall performance and can exhibit better performance on more functions; on in contrast, the larger the average rank value, the weaker its comprehensiveness. The average rank value of the 29 benchmark functions was counted when MLDMS-PSO, using different \(G\) and \(Rc\), plotted these, as shown in Figure 5. It can be clearly seen \begin{table}
\begin{tabular}{c l c c c} \hline \hline
**No.** & **Function** & **Range** & **Category** & \(f_{\text{opt}}\) \\ \hline f1 & Shifted and Rotated Bent Cigar Function & \([-100,100]\) & UN & 100 \\ f3 & Shifted and Rotated Zakharov Function & \([-100,100]\) & UN & 300 \\ f4 & Shifted and Rotated Rosenbrock’s Function & \([-100,100]\) & MN & 400 \\ f5 & Shifted and Rotated Rastrigin’s Function & \([-100,100]\) & MN & 500 \\ f6 & Shifted and Rotated Expanded Scaffer’s F6 Function & \([-100,100]\) & MN & 600 \\ f7 & Shifted and Rotated Lunacek Bi-Rastrigin Function & \([-100,100]\) & MN & 700 \\ f8 & Shifted and Rotated Non-Continuous & \([-100,100]\) & MN & 800 \\ f9 & Shifted and Rotated Levy Function & \([-100,100]\) & MN & 900 \\ f10 & Shifted and Rotated Schwefel’s Function & \([-100,100]\) & MN & 1000 \\ f11 & Hybrid Function 1 (N = 3) & \([-100,100]\) & H & 1100 \\ f12 & Hybrid Function 2 (N = 3) & \([-100,100]\) & H & 1200 \\ f13 & Hybrid Function 3 (N = 3) & \([-100,100]\) & H & 1300 \\ f14 & Hybrid Function 4 (N = 4) & \([-100,100]\) & H & 1400 \\ f15 & Hybrid Function 5 (N = 4) & \([-100,100]\) & H & 1500 \\ f16 & Hybrid Function 6 (N = 4) & \([-100,100]\) & H & 1600 \\ f17 & Hybrid Function 6 (N = 5) & \([-100,100]\) & H & 1700 \\ f18 & Hybrid Function 6 (N = 5) & \([-100,100]\) & H & 1800 \\ f19 & Hybrid Function 6 (N = 5) & \([-100,100]\) & H & 1900 \\ f20 & Hybrid Function 6 (N = 6) & \([-100,100]\) & H & 2000 \\ f21 & Composition Function 1 (N = 3) & \([-100,100]\) & C & 2100 \\ f22 & Composition Function 2 (N = 3) & \([-100,100]\) & C & 2200 \\ f23 & Composition Function 3 (N = 4) & \([-100,100]\) & C & 2300 \\ f24 & Composition Function 4 (N = 4) & \([-100,100]\) & C & 2400 \\ f25 & Composition Function 5 (N = 5) & \([-100,100]\) & C & 2500 \\ f26 & Composition Function 6 (N = 5) & \([-100,100]\) & C & 2600 \\ f27 & Composition Function 7 (N = 6) & \([-100,100]\) & C & 2700 \\ f28 & Composition Function 8 (N = 6) & \([-100,100]\) & C & 2800 \\ f29 & Composition Function 9 (N = 3) & \([-100,100]\) & C & 2900 \\ f30 & Composition Function 10 (N = 3) & \([-100,100]\) & C & 3000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The information of the benchmark function used in this paper.