## 3 Problem Statement and Formulations
### Mathematical Formulation  
On the basis of the graphical representation that is suggested in Section 3.2, a mixed-integer linear programming model is built to formulate this problem. This paper defines the decision variables listed in Table 3.  
Then, we can mathematically model the problem as the following mixed-integer linear programming model:  
\[\min\rho_{0}\sum_{j\in P\cup D}x_{N_{0j}}+\sum_{i\in P,\cup D\cup N _{0j}}\sum_{j\in P,\cup D\cup N_{end}}(C_{ij}+C_{j})x_{ij} \tag{5}\] \[s.t.\quad\sum_{j\in P\cup D\cup N_{end}}x_{ij}=1,\quad\forall i \in N_{IF}\cup N_{OF}\cup N_{OE}\] (6) \[\sum_{j\in P\cup D\cup N_{end}}x_{ij}\leq 1,\quad\forall i\in N _{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (7) \[\sum_{j\in P\cup D}x_{N_{0j}}\leq K_{truck}\] (8) \[\sum_{j\in P\cup D\cup N_{0}}x_{ji}-\sum_{j\in P\cup D\cup N_{ end}}x_{ij}=0,\quad\forall i\in P\cup D\] (9) \[L_{i}\sum_{j\in P\cup D\cup N_{end}}x_{ij}\leq y_{i}\leq U_{i} \sum_{j\in P\cup D\cup N_{end}}x_{ij},\quad\forall i\in P\cup D\] (10) \[y_{i}+T_{i}+T_{ij}\leq y_{j}+(1-x_{ij})\overline{M},\quad\forall i \in P\cup D\cup N_{0},j\in P\cup D\cup N_{end} \tag{11}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Variables** & \\ \hline \(x_{ij}\) & 1 if a truck travels through arc\((i,j)\), 0 otherwise \\ \(y_{i}\) & the beginning time of traversing node \(i\) \\ \(\omega_{ij}\) & 1 if nodes \(i\) and \(j\) are visited and node \(i\) is visited before node \(j\); 0 otherwise, \\  & \(i,j\in N_{E}^{P}\cup N_{IF}^{P}\cup N_{IF}^{D}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variables. \[y_{i}+T_{i}+T_{iN_{\text{red}}}x_{iN_{\text{red}}}\leq H,\quad\forall i\in P\cup D \tag{12}\] \[\omega_{ij}\leq\sum_{h\in P\cup D\cup N_{\text{red}}}x_{ih},\quad \forall i\in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (13) \[\omega_{ij}\leq\sum_{h\in P\cup D\cup N_{\text{red}}}x_{jh},\quad \forall i\in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (14) \[\omega_{ij}+\omega_{ji}\geq\sum_{h\in P\cup D\cup N_{\text{red}}}x_{ ih}+\sum_{h\in P\cup D\cup N_{\text{red}}}x_{jh}-1,\quad\forall i\in N_{IF}^{P} \cup N_{IF}^{D}\cup N_{E}^{P}\] (15) \[\omega_{ij}+\omega_{ji}\leq 1,\quad\forall i\in N_{IF}^{P}\cup N _{IF}^{D}\cup N_{E}^{P}\] (16) \[y_{j}\leq y_{i}+(2+\omega_{ij}-\sum_{h\in P\cup D\cup N_{\text{ red}}}x_{ih}-\sum_{h\in P\cup D\cup N_{\text{red}}}x_{jh})\overline{M},\quad \forall i\in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (17) \[2-\omega_{ij}-\omega_{jk}\geq 1-\omega_{ik},\quad\forall i,j,k \in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (18) \[0\leq\sum_{i\in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}}\alpha_{ i}\omega_{ij}+\alpha_{j}\sum_{h\in P\cup D\cup N_{\text{red}}}x_{jh}+e\leq Q, \quad\forall j\in N_{IF}^{P}\cup N_{IF}^{D}\cup N_{E}^{P}\] (19) \[y_{i}\geq 0,x_{ij},\omega_{ij}\in\{0,1\},\quad\forall i,j\in P \cup D\cup N_{0}\cup N_{\text{red}} \tag{20}\]  
The goal of Equation (5) is to minimize the total cost. Two components in the objective function correspond to the fixed cost for dispatching trucks and the travel cost, respectively. Constraints (6) stipulate that each inland container transportation request is visited exactly once. Constraints (7) guarantee that each pickup/delivery depot is visited no more than once. Constraints (8) make sure that no more than \(K_{\text{truck}}\) trucks are used to fulfill the requests. Flow conservation is ensured by constraints (9). Constraints (10) restricts the service beginning times of nodes to their respective time windows. Constraints (11) enforce that nodes along a route are properly timed. Constraints (12) imply that all inland container transportation requests should be finished before the end of the planning horizon. Constraints (13)-(18) determine the value of binary variable \(\omega_{ij}\) to indicate whether nodes \(i\) and \(j\) are visited and whether node \(i\) is visited before node \(j\). More specifically, constraints (13) and (14) prohibit \(\omega_{ji}\) from taking a value of \(1\) in cases where either node \(i\) or node \(j\) is not visited.