## 4 The Proposed ECPSO-MTP
### Evolution Rules  
In ECPSO-MTP, two types of evolution rules are defined and described for objects in different elementary membranes, including elementary membrane \(\sigma_{o}\) (\(1\leq o\leq m\)) and elementary membrane \(\sigma_{m+1}\), respectively. The evolution rules are used to evolve objects. And the evolutionary mechanism of modified PSO with environmental factors and crossover operators is adopted to accomplish this evolution in different membranes. Specifically, an object \(u_{i}\) is composed of two essential components, including velocity \(V_{i}\) and position \(X_{i}\), where \(u_{i}=\{V_{i},X_{i}\}\).  
Two types of evolution rules are introduced to modify the velocity of objects in elementary membranes, based on the velocity updating model of classic PSO and modified PSO with environmental factors. Thus, \(V_{i}(t+1)\) of \(u_{i}\) in \(\sigma_{o}\) at \(t+1\) is defined by (6) in the following:  
\[V_{i}(t+1)=\omega*V_{i}(t)+c_{1}*r_{1}*\left(X_{i}^{\textit{lbest}}(t)-X_{i}(t )\right)+c_{2}*r_{2}*\left(X_{o}^{\textit{best}}(t)-X_{i}(t)\right), \tag{6}\]  
Another evolution rule is defined by (7) in the following:  
\[V_{i}(t+1)=\omega*V_{i}(t)+c_{1}*r_{1}*\left(X_{i}^{\textit{lbest}}(t)-X_{i}(t )\right)+c_{2}*r_{2}*\left(X_{i}^{\textit{best}}(t)-X_{i}(t)\right), \tag{7}\]  
And \(\omega\) is determined by (8) in the following:  
\[\omega(t)=\omega_{\text{min}}+(\omega_{\text{max}}-\omega_{\text{min}})*(t/t_ {\text{max}}), \tag{8}\]  
Figure 1: The membrane structure of the proposed ECPSO-MTP.  
where \(\omega_{\min}\) and \(\omega_{\max}\) represent the minimum and maximum values of \(\omega\). The maximum number of iterations is denoted by \(t_{\max}\). \(c_{1}\) and \(c_{2}\) are typically set to 2. \(X_{i}^{\textit{Ibest}}(t)\) represents the local best of \(u_{i}\), and is also indicated as \(u_{i}^{\textit{Ibest}}(t)\). \(X_{o}^{\textit{gbest}}(t)\) represents the global best of \(u_{i}\) in \(\sigma_{o}\), and is also denoted by \(u_{o}^{\textit{gbest}}(t)\). \(X_{i}^{\textit{best}}(t)\) is the environmental factor of \(u_{i}\) and is determined by Equation (2).  
The position updating model of the classic PSO is employed to modify the position of objects, and \(X_{i}(t+1)\) of \(u_{i}\) in \(\sigma_{o}\) is defined by (9) in the following:  
\[X_{i}(t+1)=X_{i}(t)+V_{i}(t+1), \tag{9}\]  
Then, \(u_{i}^{\textit{Ibest}}(t+1)\) of \(u_{i}\) at \(t+1\) is described by (10) in the following:  
\[u_{i}^{\textit{Ibest}}(t+1)=\left\{\begin{array}{ll}X_{i}(t+1),\ \text{if}\ f \big{(}X_{i}(t+1)\big{)}<f\Big{(}X_{i}^{\textit{Ibest}}(t)\Big{)},\\ X_{i}^{\textit{Ibest}}(t),\ \text{otherwise}\end{array}\right. \tag{10}\]  
where \(f()\) represents the fitness of the fitness function. And \(u_{o}^{\textit{gbest}}(t+1)\) of \(\sigma_{o}\) at \(t+1\) is described by (11) in the following:  
\[u_{o}^{\textit{gbest}}(t+1)=\left\{\begin{array}{ll}X_{i}^{\textit{Ibest}}( t+1),\ \text{if}\ f\Big{(}X_{i}^{\textit{Ibest}}(t+1)\Big{)}<f\Big{(}X_{o}^{\textit{gbest }}(t)\Big{)}\\ X_{o}^{\textit{gbest}}(t),\ \text{otherwise}\end{array}\right. \tag{11}\]  
In addition, the position updating model based on the crossover operator is adopted to modify the position of objects in \(\sigma_{m+1}\) using the Equations (3)-(5). And the global best \(u_{m+1}^{\textit{gbest}}(t+1)\) of \(\sigma_{m+1}\) at \(t+1\) is the best position in \(\sigma_{m+1}\).