## 3 Materials and Methods
### Dynamic Supply Chain Design  
The purpose of the dynamic supply chain design phase is to select the optimal combination of nodes to be used. We developed an integer programming model by modifying a set-covering problem. The concept is that all the required processes must be covered with at the minimum total cost. We assume that there are enough raw materials and no work-in-progress (WIP) in the process. The notations used in the optimization model are summarized in Table 1.  
\[min\sum_{k\in K}FC_{k}x_{k}+\sum_{i\in I}\sum_{k\in K}\left(SC_{ik}y_{ik}+PC_{ ij}q_{ik}\right)+\sum_{i\in I}\sum_{k\in K}\frac{PT_{ik}q_{ik}}{\text{CAP}_{ik}} TC_{jk}v_{ikl} \tag{1}\]  
s.t. \[\sum_{k\in K}q_{ik}\geq d\quad\forall\;i\] (2)  
\[\sum_{k\in K}a_{ik}x_{k}\geq 1\qquad\forall\;i \tag{3}\]  
\[y_{ik}\leq a_{ik}x_{k}\qquad\forall\;i,k \tag{4}\]  
\[p_{ik}q_{ik}\leq u_{ik}CAP_{ik}y_{ik}\qquad\forall\;i,k \tag{5}\]  
\[v_{ikl}\leq y_{ik}\qquad\forall\;i,k,l \tag{6}\]  
\[v_{ikl}\leq y_{i+1l}\qquad\forall\;i,k,l \tag{7}\]  
\[v_{ikl}\geq y_{il}+y_{i+1l}-1\qquad\forall\;i,k,l \tag{8}\]  
\[x_{k},\;y_{ik}\in\{0,1\}\quad\forall\;i,k \tag{9}\]  
\[q_{ik}\geq 0,\;v_{ikl}\geq 0\quad\forall\;i,k,l \tag{10}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sets** & \\ \hline I & Set of processes or nodes (\(i\in\mathbb{I}\)) \\ K & Set of factories (\(k,l\in\mathbb{K}\)) \\ Parameters & \\ D & Demand for the product \\ \(CAP_{ik}\) & Average time available per day for process \(i\) in factory \(k\) in the planning horizon \\ \(u_{ik}\) & Utilization of process \(i\) in factory \(k\) \\ \(FC_{k}\) & Fixed cost for selecting factory \(k\) \\ \(SC_{k}\) & Setup cost for process \(i\) in factory \(k\) \\ \(PC_{ik}\) & Process cost per unit for process \(i\) in factory \(k\) \\ \(PT_{ik}\) & Processing time per unit for process \(i\) in factory \(k\) \\ \(TC_{il}\) & Transportation cost from factory \(k\) to factory \(l\) \\ \(a_{ik}\) & Set-covering matrix representing the relationship between process \(i\) and factory \(k\) \\ \hline Decision Variables & \\ \(x_{k}\) & Factory selection, 1 if factory \(k\) is selected; 0, otherwise \\ \(y_{ik}\) & Process selection; 1 if process \(i\) in factory \(k\) is selected; 0, otherwise \\ \(v_{il}\) & Transportation selection; 1 if product from process \(i\) in factory \(k\) is sent to factory \(l\) \\ \(q_{ik}\) & Production quantity of process \(i\) in factory \(k\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations for dynamic supply chain design.  
The objective function is expressed by Equation (1): it minimizes the total costs composed of fixed, production, and transportation costs. Fixed costs are incurred when a factory is selected, and the production costs are expressed as the sum of the setup cost and the production cost per unit multiplied by the production quantity. Finally, transportation costs arise when consecutive processes are carried out in different factories. Equation (2) represents the demand for a product, which requires a supply chain design. Equation (3) requires that all processes be covered, and Equation (4) indicates that the available process at a factory can be selected only if the factory is selected. Equation (5) represents a capacity constraint, and capacity is calculated in terms of available time. Equations (6) and (8) determine whether transportation between the factories is necessary or not. We note that the decision variable \(v_{ikl}\) is not necessarily defined as a binary variable within the constraints. Equations (9) and (10) define the decision variables.