## 3 The Proposed Method
### The Proposed MDMS Strategy
If \(\textbf{{x}}_{i}\) has the smallest Euclidean distance to \(\textbf{{mast}}_{k}\), it will be assigned to \(\textbf{{subpop}}_{k}\). Different particle numbers are allowed in each subswarm. Figure 2: The multiswarm segmentation scheme with a master–slave structure. The red stars represent master particles and the black dots represent slave particles. In the process of evolution, due to the small scale, the global exploration ability of the subswarm is limited, which makes it easy to lose diversity and fall into the local optimal region. To solve this problem, a multiswarm dynamic adjustment strategy was designed in which the evolutionary ability of each subswarm is evaluated to determine when to update and restructure all the subswarms. When the evolutionary ability of the subswarm is weakened and its advantage cannot be effectively brought into play, the population is reorganized so that the subswarms can exchange information and maintain good diversity. In detail, the evolutionary ability of the population is measured for each interval iteration parameter \(G\). If the subswarm finds a better position after generation \(G\), it means that it still has evolutionary abilities. In contrast, this means that the subswarm has lost its evolution abilities. Finally, the average evolutionary ability of the subswarm (\(SPEA_{ave}\)) is used as a measure of the evolutionary ability of the population and can be calculated by Equations (6) and (7). \[SPEA_{k}=\left\{ \begin{array}{cc}1&\mathit{Lbestval}_{k}^{iter+G}<\mathit{ Lbestval}_{k}^{iter}\\ 0&\mathit{Lbestval}_{k}^{iter+G}\geq\mathit{Lbestval}_{k}^{iter}\end{array} \right. \tag{6}\]  
\[SPEA_{ave}=\frac{1}{sn}{\sum_{k=1}^{sn}{SPEA_{k}}} \tag{7}\]  
where \(SPEA_{k}\) is the evolutionary ability of the \(k\)-th subswarm, \(k=1,2,\ldots,sn\). \(\mathit{Lbestval}_{k}^{iter}\) and \(\mathit{Lbestval}_{k}^{iter+G}\) represent the fitness values of the \(k\)-th subswarm at the local neighbors' best positions in the _iter_ and _iter_ + G generation, respectively. At the same time, a recombination parameter, \(\mathit{Rc}\), is defined. When condition \(SPEA_{ave}<\mathit{Rc}\) is satisfied, the subswarm will be redivided. The values of the interval iteration parameter, \(G\), and recombination parameter, \(\mathit{Rc}\), are investigated in Section 4.1. In summary, the detailed process of the MDMS strategy is shown in Algorithm 1. ```
1: Calculate \(sn\) using Equation (3);
2: [-_,index_f_] = sort(fitness);
3: Index of superior particles: \(ST\_temp=index\_f\) (1:\(M/2\));
4: Index of all potential combinations of master particles \(\mathit{Comb}\) = nchoose(\(ST\_temp\), \(sn\));
5: Calculate all combinations of distance sum Sum_d(\(\mathit{Pos}(\mathit{Comb})\)) using Equations (4) and (5);
6: Index of the combination \(\mathit{Cs}\): [-_,index_m_] = max (\(\mathit{Sum\_d}\));
7: The index of master particles \(\mathit{Master}=\mathit{Comb}(index\_m_,:)\);
8:for\(k=1\): \(sn\)
9:\(\mathit{subx}\)(1,\(:\)) = \(\mathit{Pos}(\mathit{Master}(k),:\)), \(\mathit{subp}\)(1,\(:\)) = \(\mathit{Pbest}(\mathit{Master}(k),:\));
10:endfor
11:for\(l=1\):\(M\)
12:if -ismember(\(l\),\(\mathit{Master}\))
13:\(\mathit{ms\_d}\) = \(\mathbf{pdist2}\)(\(\mathit{Pos}(\mathit{Master},:\)));
14: [-_,index_s_] = min(\(ms\_d\));
15:\(\mathit{subx}\)[\(index\_s\)](end + 1,\(:\)) = \(\mathit{Pos}\)(1,\(:\)), \(\mathit{subp}\)[\(index\_s\)](end + 1,\(:\)) = \(\mathit{Pbest}\)(1,\(:\));
16:endif
17:endfor
```  
**Algorithm 1** MDMS_Phase