## 4 The Proposed ECPSO-MTP
### Computation of Proposed ECPSO-MTP  
1. Initialization \(<\)1\(>\) Parameters initialized In the proposed ECPSO-MTP, the values of adjusting parameters are preset by adding the following values to the existing ones, including \(N\) and \(n_{o}\), where \(N\) is the totality of objects in the system, \(n_{o}\) is the totality of objects in \(\sigma_{o}\) (\(1\leq o\leq m\)). The static membrane structure of the proposed ECPSO-MTP is shown in Figure 1 in more details. All objects involved in the computational procedure of the system are located in \(\sigma_{in}\) during initialization; \(<\)2\(>\) Velocity and position initialized To initialize the velocity and position for all objects in \(\sigma_{in}\), a random generation strategy in search space is employed. Then these objects are sent to the elementary membrane \(\sigma_{1}\) to \(\sigma_{m}\) respectively, where \(n_{o}=N/m\); \(<\)3\(>\) Local and global best updated The local best \(u_{i}^{\mathit{lbest}}\) (\(1\leq i\leq n_{o}\)) and global best \(u_{o}^{\mathit{gbest}}\) in \(\sigma_{o}\) are generated by the Equations (10) and (11); 2. Evolution mechanism for \(\sigma_{1}\) to \(\sigma_{m}\) \(<\)1\(>\) Velocity and position updated The evolution rules for the velocity updating of all objects in \(\sigma_{o}\), based on classic PSO and modified PSO with environmental factors, are randomly employed according to Equations (6)-(8), and the position updating is determined by Equation (9); \(<\)2\(>\) Local and global best updated The local best \(u_{i}^{\mathit{lbest}}\) (\(1\leq i\leq n_{o}\)) and global best \(u_{o}^{\mathit{gbest}}\) in \(\sigma_{o}\) are generated by the Equations (10) and (11); 3. Forward communication mechanism from \(\sigma_{o}\) to \(\sigma_{m+1}\) Under the presence of both \(p\) and \(u_{o}^{\mathit{gbest}}\) in \(\sigma_{o}\), the first type of forward communication rules are employed to transmit \(u_{o}^{\mathit{gbest}}\) from \(\sigma_{o}\) to \(\sigma_{m+1}\) and evolve it into \(u_{o}\) in \(\sigma_{m+1}\). Specifically, the promoter objects \(p\) can be described as logical judgements, where \(p=\{t<t_{\max}\}\); 4. Evolution mechanism for \(\sigma_{m+1}\) \(<\)1\(>\) Position updated The evolution rules in \(\sigma_{m+1}\) based on the crossover operator are employed to update objects in \(\sigma_{m+1}\) according to Equations (3)-(5); \(<\)2\(>\) Local best updated Update \(u_{o}^{\mathit{lbest}}\) for all objects in \(\sigma_{m+1}\) according to Equation (10); 5. Forward communication mechanism from \(\sigma_{m+1}\) to \(\sigma_{m+2}\) Under the presence of both \(p\) and \(u_{o}^{\mathit{lbest}}\) in \(\sigma_{m+1}\), the second type of forward communication rules is employed to transmit \(u_{o}^{\mathit{lbest}}\) from \(\sigma_{m+1}\) to \(\sigma_{m+2}\) and evolve it into \(u_{o}\) in \(\sigma_{m+2}\). The best object from \(\sigma_{m+1}\) is regarded as the global best \(u_{m+2}^{\mathit{gbest}}\) of \(\sigma_{m+2}\). Then \(u_{m+2}^{\mathit{gbest}}\) is sent to \(\sigma_{out}\) as the computed result of the system at the moment;(6)  
Backward communication mechanism from \(\sigma_{o}\) to \(\sigma_{m+2}\)  
Under the presence of \(q\) in \(\sigma_{o}\) and \(u_{m+2}^{gbest}\) in \(\sigma_{m+2}\), the backward communication rules from \(\sigma_{o}\) to \(\sigma_{m+2}\) are utilized to transmit \(u_{m+2}^{gbest}\) from \(\sigma_{m+2}\) to \(\sigma_{o}\) and evolve it into \(u_{o}^{gbest}\) in \(\sigma_{o}\). Specifically, the promoter objects \(q\) can be described as logical judgements, where \(q=\left\{f\Big{(}u_{m+2}^{gbest}\Big{)}<f\Big{(}u_{o}^{gbest}\Big{)}\right\}\). The communication relationships within the system are graphically depicted in Figure 2, with black arrows indicating the direction of communication;  
(7)  
Termination and output  
The evolution-communication mechanism in ECPSO-MTP is executed repeatedly through iteration until the maximum number of iterations is attained.