## 3 Materials and Methods
### Dynamic Supply Chain Operation  
Once the supply chain has been designed, the operations plan is generated. Nodes represent processes selected in the dynamic supply chain design phase. We also introduce a dummy node, \(I_{d}\), so that finished products manufactured in several factories are delivered to the customer. The supply chain operations model, using the notations in Table 2, is as follows.  
\[min\sum_{i\in I}\sum_{j=I_{d}}\sum_{t\in T}p^{t}a_{ij}x_{ij}^{t} \tag{11}\]  
\[\text{s.t.}\qquad\qquad y_{i}^{t}\geq c_{i}^{t}\qquad\forall\;i,t \tag{12}\]  
\[II_{i}^{t}=II_{i}^{t-1}+\sum_{j\in I}a_{ji}x_{ji}^{t-LT_{ji}}-y_{i}^{t}\quad \forall\;i,t \tag{13}\]  
\[IO_{i}^{t}=IO_{i}^{t-1}+y_{i}^{t}+\sum_{j\in I}a_{ij}x_{ij}^{t}\quad\forall\;i,t \tag{14}\]  
\[\sum_{i\in I}\sum_{t\in T}a_{iI_{d}}x_{iI_{d}}^{t}\geq d \tag{15}\]  
\[x_{ij}^{t}=0\quad\forall\;(i,j)\in V,t\in T\backslash T_{d} \tag{16}\]  
\[x_{ij}^{t},y_{i}^{t},II_{i}^{t},IO_{i}^{t}\geq 0\quad\forall\;i,j,t \tag{17}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Sets**} \\ \hline I & Set of processes or nodes (\(i,j\in\) I) \\ T & Time periods (t \(\in\) T) \\ T\({}_{d}\) & Time periods when transportation model is available. (\(T_{d}\subset T\)) \\ V & A pair of nodes connecting two consecutive processes in different factories \\ \hline \hline \multicolumn{1}{c}{**Parameters**} \\ \(a_{ij}\) & Adjacency matrix resenting the relationship between processes \\ \(p^{l}\) & Penalty cost having an incremental function for time \\ \(c_{i}^{t}\) & Available capacity of node \(i\) at time t \\ \(LT_{ij}\) & Transportation lead time from node \(i\) to node \(j\) \\ d & Demand for the product \\ \hline \hline \multicolumn{1}{c}{**Decision Variables**} \\ \(x_{ij}^{t}\) & Quantity of product sent from node \(i\) to node \(j\) at time t \\ \(y_{i}^{t}\) & Quantity of product processed in node \(i\) at time t \\ \(II_{i}^{t}\) & Raw material inventory level of node \(i\) at time t \\ \(IO_{i}^{t}\) & Manufactured product inventory level of node \(i\) at time t \\ \hline \hline \end{tabular}
\end{table}
Table 2: Notations for dynamic supply chain operation.  
The objective function, Equation (11), aims to ensure that the production is finished as soon as possible by employing penalty cost, which increases with time. The objective function can be modified according to the decision maker's goal. Equation (12) limits the production quantity depending on the capacity of the process at each time unit. Equations (13) and (14) represent inventory balance equations for raw material and manufactured product at each process, respectively. \(LT_{ij}\) represents the transportation lead time between processes \(i\) and \(j\). Equation (15) is a constraint expressing the demand of the product. Equation (16) shows that inter-factory movement is only possible when the transportation mode is available. The manufactured product in a factory can be sent to other factories when \(t\) belongs to \(T_{d}\). For example, all factories operate ten hours a day, and inter-factory transportation takes place at night after work hours. Equation (17) defines the decision variables.