## 3 Problem Description and Formulation
### Mathematical Model  
Let us introduce the following parameters:  
\(I:\) Set of senders/receivers. \(K:\) Set of candidates sites for intermodal terminals. \(l:\) Fixed number of links to be established. \(q_{ij}:\) The amount of goods to be shipped from the customer \(i\) to customer \(j\). \(c_{ij}^{km}:\) The intermodal unit cost for transportation from sender \(i\) to receiver \(j\) using the two rail terminals \(k\) and \(m\). \(c_{ij}:\) The unimodal unit cost for transporting goods by truck from sender \(i\) to receiver \(j\). \(f_{k}:\) The investment cost for location of a terminal on site \(k\). \(C_{k}:\) Fixed capacity of a located rail terminal \(k\). Let us consider the following decision variables:  
\(l_{km}:\) A binary variable equal 1 if the rail link between \(k\) and \(m\) is established, 0 otherwise. Accordingly, \(l_{kk}\) equal 1 if the node \(k\) is a terminal, 0 otherwise. \(r_{ij}:\) The amount of goods routed from the sender \(i\) to receiver \(j\). \(s_{ij}^{km}:\) The amount of goods routed from the sender \(i\) to receiver \(j\) through the two rail terminals \(k\) and \(m\). The incomplete location problem of intermodal terminals may be formulated as follows:  
\[\text{Min }C_{1}=\sum_{i,j\in I}\sum_{k,m\in K}c_{ij}^{km}s_{ij}^{km}+\sum_{i,j \in I}c_{ij}r_{ij}+\sum_{k\in K}f_{k}l_{kk} \tag{1}\]  
The linear objective function (1) minimizes the total cost for routing goods and the total cost for locating rail terminals. The first term expresses the cost for routing goods through the intermodal chain: from the initial sender \(i\) to the first rail terminal \(k\) using trucks, then from the terminal \(k\) to the second terminal \(m\) by rail mode, and later from the terminal \(m\) to the final customer \(j\) using road mode. The second term is the cost for routing goods directly using trucks from \(i\) to \(j\) while the third term is the overall investment cost for locating rail terminals. \[\sum_{k,m\in K}s_{ij}^{km}+r_{ij}=q_{ij},\forall i,j\in I \tag{2}\]  
Figure 2: Small incomplete network. The constraint (2) states that the sum of goods transported via the intermodal chain through the terminals \(k\) and \(m\) and that routed directly from \(i\) to \(j\) equal exactly the total goods to be transported from \(i\) to \(j\). \[\sum_{i,j\in\mathbb{I}}\sum_{m\in K}s_{ij}^{bm}+\sum_{i,j\in\mathbb{I}}\sum_{m \in K}s_{ij}^{mk}\leq C_{k}l_{kk},\forall k\in K \tag{3}\]  
The constraint (3) imposes to respect the terminals capacities. \[l_{km}\leq l_{kk},\forall k,m\in K \tag{4}\]  
\[l_{km}\leq l_{mm},\forall k,m\in K \tag{5}\]  
The two inequalities (4) and (5) guarantee that a rail link is established if the two adjacent terminals are located. In fact, we have the following implications:  
\[\text{if }l_{kk}=0\text{ or }l_{mm}=0\implies l_{km}=0\]  
\[l_{km}=l_{mk},\forall k,m\in K \tag{6}\]  
Equation (6) guarantees that a rail link is opened in the two directions. \[\sum_{k,m\in K}l_{km}=l \tag{7}\]  
Equation (7) guarantees the establishment of exactly the given number of rail links. The decision variable \(l_{km}\) controls the rail links to be opened. Thus, if due to economic, geographic or environmental constraints a railway link is impossible between two terminals, such scenario is forbidden by \(l_{kl}=0\). \[s_{ij}^{km}\leq l_{km},\forall k,m\in K,\forall i,j\in I \tag{8}\]  
Equation (8) forbids the rail flows between terminals if they are not located. The developed model is a Mixed binary Integer Program (MIP). If we denote by \(p\) the number of potential sites (the cardinal of \(K\)) and by \(n\) the numbers of senders/receivers (the cardinal of \(I\)), then the program has \(n^{2}p^{2}+3p^{2}+n^{2}+p+1\) constraints and \(n^{2}p^{2}+n^{2}+p^{2}\) variables.