# A Theoretical Foundation for Context-Aware Cyber-Physical Production Systems
## 4 CPPS Models
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol/Variable** & **Meaning** \\ \hline \(J\) & The total number of different task subnets \\ \(J\) & The set of indices of task subnets, \(J=\{1,2,3,...,J\}\) \\ \(j\) & A type of task subnet, \(j\in J\) \\ \(\Pi\) & The number of periods in the time horizon \\ \(\tau\) & The index of a period, \(\tau\in\{1,2,...,\Pi\}\) \\ \(D_{j}\) & The requested quantity of the products (demand) to be produced by type-\(j\) task subnets, \\ \(\phi\) & The order deadline \\ \(GJ_{j}\) & The model of type-\(j\) task subnet, \(GJ_{j}=(P_{j},T_{j},F_{j},m_{0},\mu_{j})\) \\ \(N_{j}\) & Total operations in \(GJ_{j}\) \\ \(R\) & Total resource types \\ \(R\) & The set of indices of different types of resources, \(R=\{1,2,3,...,R\}\) \\ \(r\) & The index of a resource type, \(r\in R=\{1,2,3,...,R\}\) \\ \(k\) & The \(k\)-th operation performed by a resource \\ \(\|\) & A merging operator to merge two or more Petri nets \\ \(GR_{r}\) & A model of type-\(r\) resource subnet, \(GR_{r}=(P_{r},T_{r},F_{r},m_{r0},\mu_{r})\) \\ \(C_{rt}\) & The nominal capacity of type-\(r\) resources in period \(t\); \(C_{rt}\) is set to \(m_{r0}(r)\) for all \(t\) \\ \(G\) & A CPPS model, \(G=(P,T,F,m_{0},\mu)=G/\|GR\), where \(G=\underset{j\in J}{\|}GJ_{j}\) and \(GR=\underset{r\in R}{\|}GR_{r}\) \\ \(m\) & A marking of \(G\) \\ \(STN_{j}(V_{j},A_{j})\) & The spatial-temporal network of type-\(j\) task subnet, \(j\in J=\{1,2,3,...,J\}\) \\ \(JSTN(V,A)\) & A Joint Spatial–Temporal network \\ \(V(v)\) & The set of nodes in \(V\) directed connected to \(v\) \\ \(V_{j}(s)\) & The set of nodes in \(V_{j}\) directed connected to \(s\) \\ \(V_{j}(e_{j})\) & The set of nodes in \(V_{j}\) directed connected to \(e_{j}\) \\ \(a\) & An arc in \(STN(V,A)\), where \(a\in A\) or an arc in \(STN_{j}(V_{j},A_{j})\), where \(a\in A_{j}\). \\ \(a(v,w)\), \((v,w)\) & An arc from node \(v\) to node \(w\) is represented explicitly by \(a(v,w)\). \\ \(a(v,w)\), \((v,w)\) & The arc \(a(v,w)\) is abbreviated as \((v,w)\) or \(a\) whenever it is clear from the context \\ \(t_{a}\) & The transition associated with arc \(a\) \\ \(f\) & A solution of the Nominal Optimization Problem (NOP) \\ \(f(a(v,w))\) represents the value of the solution \(f\) to NOP on the arc \(a\)\(=\)\(a(v,w)\)\(=(v,w)\) of \\ \(f(a)\), \(f(a(v,w))f(v,w)\) & \(STN(V,A)\). \(f(a(v,w))\) is abbreviated as \(f(v,w)\) or \(f(a)\) with \(a=(v,w)\) whenever it is clear \\  & from the context \\ \(g(\tau)\) & The earliness penalty function for \(\tau\leq\Phi\) \\ \(h(\tau)\) & The lateness penalty function for \(\tau>\Phi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and notations. the initial state and the firing time of a transition \(t\in T\) is \(\mu(t)\delta\), where \(\mu:T\to Z\), \(Z\) is the set of nonnegative integers and \(\delta\) is real._  
To describe the dynamics of a DTPN, we need the following definition about transition firing. **Definition 2**. : _We use \({}^{\bullet}t\) to denote the set of input places of transition \(t\) and use \({}^{\bullet}\) to denote the set of output places of \(t\). If \(m(p)\geq F(p,t)\forall p\in{}^{\bullet}t\), transition \(t\) is enabled under marking \(m\). An enabled transition \(t\) can be fired. A transition \(t\) selected for firing in period \(\tau\) under a control action will be fired at the beginning of period \(\tau\) and the firing will be completed at the end of the designated period \(\tau+\mu(t)-1\). After firing an enabled transition \(t\), the number of tokens in each input place in \({}^{\bullet}t\) will be reduced by one and the number of tokens in each output place in \({}^{\bullet}\) will be increased by one._  
Figure 1a is an example of DTPN, where \({}^{\bullet}t_{1}=\{r_{1}\}\) and \(t_{1}^{\bullet}=\{p_{2}\}\). **Definition 3**.