## 4 Experimental Studies
### Comparison with Other PSO Variants  
To compare and analyze the performance of the proposed MLDMS-PSO, nine other state-of-the-art PSO variations were selected and divided into three types.  
There are four PSO variants with different learning strategies, as follows:  
* Traditional global PSO algorithms with inertia weight (GPSO) [17].
* Full information PSO (FIPS) [47]. The FIPS implements the ring topology.
* Comprehensive learning PSO (CLPSO) [21]. All particles update the dimension information of their velocity using _pbest_ and learn from the _pbest_ of different particles.
* Competitive and cooperative (CC) PSO with ISM (CCPSO-ISM) [37]. The CC operator based on the ISM is designed to use the shared information properly and efficiently.  
Two PSOs adopt multiple learning strategies, as follows:  
* A PSO with the multi-exemplar and forgetting ability (XPSO) [48]. XPSO uses two exemplars to update the velocity and configures distinct forgetting abilities to different particles.
* Two-swarm learning PSO (TSLPSO) [20]. In TSLPSO, the dimensional learning strategy and CL strategy are used to guide the local and global search of the particles, respectively.  
Three PSOs divide the whole population into multiple swarms.  
* Dynamic multiple-swarm PSO (DMS-PSO) [11]. DMS-PSO adopts the multiple-swarm strategy and these swarms can be regrouped frequently.
* Multiswarm PSO with a dynamic learning strategy (PSO-DLS) [16]. In PSO-DLS, the particle classification mechanism and the dynamic control mechanism of the strategy promote an information exchange among the subswarms.
* Heterogeneous comprehensive learning and dynamic multiswarm PSO with two mutation operators (HCLDMS-PSO) [49]. In HCLDMS-PSO, a CL strategy and a modified DMS strategy are used to construct the exploitation subpopulation exemplar and the exploration subpopulation exemplar, respectively.  
Figure 5: Parameter investigation result in MLDMS-PSO.  
The parameter settings of the PSO variants above-mentioned and the proposed MLDMS-PSO are listed in Table 2. To make a fair comparison between the proposed MLDMS-PSO and the other state-of-the-art PSO algorithms, the mean error (_mean_), standard deviation (_std_), and rank mean value (_rank_) of 10 PSOs solution were calculated as a basic performance measure to evaluate the accuracy of the algorithm. The experiments were conducted on dimensions of \(D=\) 10, 30, and 50. Each algorithm was run 51 times independently on 10-D, 30-D, and 50-D problems for the 29 benchmark functions, and the termination condition was set to _MaxFEs_ = 10,000 \(\times\)\(D\). The results of 10-D, 30-D, and 50-D are shown in Appendix A, Appendix B, and Appendix C, respectively, where the best experimental result among all of the test functions is shown in bold. Based on this result, all algorithms were sorted according to the average rank (_Ave rank_) of the 29 functions as the final rank (_Final rank_). The _Ave rank_ is the average value of the algorithm's experimental results ranking on each benchmark function. The _Final rank_ is the result of sorting all algorithms by _Ave rank_. To compare the comprehensive performance of MLDMS-PSO and nine other state-of-the-art PSO variations in different dimensions, as shown in Table 3, we calculated the mean of each algorithm's _Ave rank_ in different dimensions as the comprehensive mean rank (_CMean rank_). Based on this _CMean rank_, the comprehensive ranking (_Crank_) of each algorithm was obtained.