## 2 Modeling and Problem Formulation
### A Multi-Agent System Architecture and Models of Agents in CPSs
\\ \hline \(x\) & The vector of the decision variables \(x_{aj}\), where \(a\in\{1,2,3,\ldots,|A|\}\) and \(j\in\{1,2,3,\ldots,J_{a}\}\) \\ \hline \(\Gamma(x,B)\) & A function that maps a solution for \(x_{aj}\) and \(B_{aj}\) to the total processing time of a process, where \(a\in\{1,2,3,\ldots,|A|\}\) and \(j\in\{1,2,3,\ldots,J_{a}\}\) \\ \hline \(WA\) & The set of process agents, where \(WA=\{1,2,\ldots,N\}\) \\ \hline \(n\) & A process agent, where \(n\in WA\) \\ \hline \(\Omega_{n}^{k}\) & The Cyber World model for the \(k\)-th operation of process agent \(n\in WA\) \\ \hline \(\Omega_{n}\) & The Cyber World model for process agent \(n\) \\ \hline \(R_{n}^{k}\) & The set of resource agents involved in performing the \(k\)-th operation in \(\Omega_{n}\) \\ \hline \(A_{a}^{k}\) & The Cyber World model of the activity that represents the \(k\)-th operation is performed by resource agent \(a\), where \(a\in RA\) \\ \hline \(\Psi_{n}\) & The Cyber World model of a configuration for process agent \(n\); \(\overline{\
u}_{n}=(P_{n},T_{n},F_{n},m_{n0},\mu_{n})=\Omega_{n}\|_{k\in\{1,2, \ldots,K_{a}\  
**Definition 2**. : _Given two DTPNs, \(PN_{1}=\left(P_{1}\), \(T_{1}\), \(F_{1}\), \(m_{10}\), \(\mu_{1}\right)\) and \(PN_{2}=\left(P_{2}\), \(T_{2}\), \(F_{2}\), \(m_{20}\), \(\mu_{2}\right)\), \(PN_{1}\|PN_{2}\ =\left(P\), \(T\), \(F\), \(m_{0}\), \(\mu\right)\), where \(P=P_{1}\cup P_{2}\), \(T=T_{1}\cup T_{2}\), \(F(p,t)=\left\{ \begin{array}{ll}F_{1}(p,t)\ if\ p\in P_{1}\text{ and }t\in T_{1}\\ F_{2}(p,t)\ if\ p\in P_{2}\text{ and }t\in T_{2}\end{array} \right.,\ F(t,p)=\left\{ \begin{array}{ll}F_{1}(t,p)\ if\ p\in P_{1}\text{ and }t\in T_{1}\\ F_{2}(t,p)\ if\ p\in P_{2}\text{ and }t\in T_{2}\end{array} \right.\text{ and }\ m_{0}(p)=\left\{ \begin{array}{ll}m_{10}(p)\ if\ p\in P_{1}\\ m_{20}(p)\ if\ p\in P_{2}\end{array} \right.\text{. }\]  
Let \(K_{n}\) denote the maximum number of operations involved in the workflow of process agent \(n\). We define the Cyber World model of process agent \(n\) as follows. **Definition 3**. : _The Cyber World model of process agent \(n\), where \(n\in WA\) is an acyclic DTPN \(\Omega_{n}=\|_{k\in\left\{K\in\left\{1,2,\ldots,K_{n}\right\}|d_{nk}=1\right\} \Omega_{n}^{k}\)._  
Figure 2 shows the DTPN model, \(\Omega_{n}\), for process agent \(n\), where  
\[\Omega_{n}=\Omega_{n}^{1}\|\Omega_{n}^{2}\|\Omega_{n}^{3}\|\Omega_{n}^{4}\| \Omega_{n}^{5}.\]  
Operations in a CPS are performed by the resource agents. In this paper, the action that an agent performs an operation is called an activity. An activity can be described by a Cyber World model. Let \(A_{nn}^{k}\) denote the Cyber World model described by a DTPN of the activity performed by resource agent \(a\) for \(k\)-th operation of process agent \(n\), defined as follows. **Definition 4**. : _We use DTPN \(A_{na}^{k}=(P_{na}^{k},T_{na}^{k},F_{na}^{k},m_{n0^{\prime}}^{k},\mu_{na}^{k})\) as the Cyber World model to represent that the \(k\)-th operation of process agent \(n\) is performed by resource agent \(a\), where \(a\in RA\), the initial marking \(m_{n0}^{k}(r_{a})\) is the number of available resources and \(r_{a}\) is the idle state place. There is no common transition between \(A_{na}^{k}\) and \(A_{na}^{k^{\prime}}\) for \(k\
eq k^{\prime}\)._  
For the motivating example, to represent the capability of a certain type of resource agents, we will construct the resource activity model for each type of resource agent. We use the resource activity model \(A_{na}^{k}\) to denote the activity that the \(k\)-th operation of process agent \(n\) can be performed by resource agent \(a\).