## 4 The Proposed ECPSO-MTP
### The Common Framework of ECPSO-MTP  
The membrane system of the proposed ECPSO-MTP is defined and described as a tuple, which is completely depicted in the following:  
\[\Pi=\Big{(}\Gamma,\varepsilon,\mu,\omega_{1},\cdots,\omega_{m},R,R^{{}^{ \prime}},\sigma_{in},\sigma_{out}\Big{)},\]  
where  
1. \(\Gamma\) is a non-empty, finite alphabet for objects;
2. \(\varepsilon\subseteq\Gamma\) is a finite alphabet set for objects that are initially placed in the input membrane;
3. \(\mu\) is the structure of the ECPSO-MTP, which contains \(m+4\) elementary membranes;
4. \(\omega_{1},\cdots,\omega_{m}\) are finite multisets for objects that are initially placed in \(m\) elementary membranes, where \(\omega_{i}\subseteq\Gamma\), for \(1\leq i\leq m\);
5. \(R\) is the finite set of evolution rules for objects in the proposed ECPSO-MTP, where \(R=\{R_{1},R_{2},\cdots,R_{m},R_{m+2}\}\). \(R_{i}\) is a finite subset of \(R\) associated with membrane \(i\), for \(1\leq i\leq m\) or \(i=m+2\), and is of the form: \(R_{i}=\Big{[}u\to u^{{}^{\prime}}\Big{]}_{i}\), for \(u,u^{{}^{\prime}}\in\Gamma^{+}\). When such an evolution rule \(R_{i}\) in \(i\) is applied, objects \(u\) in \(i\) evolve into \(u^{{}^{\prime}}\);
6. \(R^{{}^{\prime}}\) is the finite set of communication rules for objects in the proposed ECPSO-MTP, where \(R^{{}^{\prime}}=\Big{\{}R^{{}^{\prime}}_{1},R^{{}^{\prime}}_{2},\cdots,R^{{}^{ \prime}}_{m+2}\Big{\}}\). \(R^{{}^{\prime}}_{i}\) is a finite subset of \(R^{{}^{\prime}}\) associated with \(i\) using the evolutional support with promoters of MTP. Specifically, \(R^{{}^{\prime}}_{ij}\subseteq R^{{}^{\prime}}_{i}\) and is of the form: \(R^{{}^{\prime}}_{i,j}:\ \big{[}u\big{|}_{p}\big{]}_{i}[\lambda]_{j}\to\big{[} \lambda]_{i}\Big{[}u^{{}^{\prime}}\Big{]}_{j}\), or \(R^{{}^{\prime}}_{i,j}:\ \big{[}\lambda\big{|}_{p}\big{]}_{i}[u]_{j}\to\Big{[}u^{{}^{ \prime}}\Big{]}_{i}[\lambda]_{j}\), where \(1\leq i\
eq j\leq m+2\), \(u\in\Gamma^{+}\), \(u^{\prime}\in\Gamma^{*}\), \(p\in\Gamma\), \(|u|>0\);
7. \(\sigma_{in}\) is the input membrane of the proposed ECPSO-MTP;(8) \(\sigma_{out}\) is the output membrane of the proposed ECPSO-MTP. When the computation of this extended P system is completed, objects in the output membrane \(\sigma_{out}\) will be sent to the environment \(\sigma_{0}\), which is regarded as the final computational result of the system. The membrane structure of the proposed ECPSO-MTP is graphically depicted in Figure 1.  
The membrane structure of ECPSO-MTP can be abstracted as the hierarchical structure in mathematics, which is depicted in Figure 1. This extended P system contains \(m+4\) membranes, including an input membrane \(\sigma_{in}\) and an output membrane \(\sigma_{out}\), while the others are labeled from \(1\) to \(m+2\). Specifically, if there are no membranes contained within a membrane, it is called an elementary membrane. In this case, membranes \(\sigma_{1}\) to \(\sigma_{m+2}\) are elementary membranes. Additionally, \(\sigma_{m+2}\) is also called a comparison membrane, and the best object selected from \(\sigma_{1}\) to \(\sigma_{m+1}\) is stored as the best object of \(\sigma_{m+2}\).