## 3 Problem Formulation
### Model Formulation
\[\sum\limits_{s\in S}n_{sr}=N_{r}\cdot x_{r} \forall r\in R\] (1-3) \[\sum\limits_{r\in R}n_{sr}\leq A_{s}+n_{s}^{\mathrm{in}}-n_{s}^{ \mathrm{out}} \forall s\in S\] (1-4) \[n_{s}^{\mathrm{out}}\leq A_{s} \forall s\in S\] (1-5) \[\sum\limits_{r\in R_{\mathit{wd}}}z_{odr}\leq D_{\mathit{wd}} \forall(o,d)\in W\] (1-6) \[\sum\limits_{(o,d)\in W_{\mathit{br}}}z_{odr}\leq\sum\limits_{s\in S }\frac{Q_{s}\cdot n_{sr}}{N_{r}} \forall r\in R,I_{r}\in L_{r},\] (1-7) \[x_{r}\in\{0,1\} \forall r\in R\] (1-8) \[n_{sr}\in\mathbb{Z}_{\geq 0} \forall s\in S,\forall r\in R\] (1-9) \[\max \sum\limits_{(o,d)\in W}\sum\limits_{r\in R_{\mathit{wd}}}p_{od} \cdot z_{odr}-\sum\limits_{r\in R}\sum\limits_{s\in S}\frac{n_{sr}}{N_{r}}(F_{ sr}+B_{sr})-\sum\limits_{r\in R}\sum\limits_{s\in S}\frac{n_{sr}}{N_{r}}\mathrm{ Bextra}-\sum\limits_{s\in S}\mathrm{c_{s}^{in}}\cdot n_{s}^{in}+\sum \limits_{s\in S}\mathrm{c_{s}^{out}}\cdot n_{s}^{out}\]0 (1-10) \[\max \sum\limits_{(o,d)\in W}\sum\limits_{r\in R_{\mathit{wd}}}p_{od} \cdot z_{odr}-\sum\limits_{r\in R}\sum\limits_{s\in S}\frac{n_{sr}}{N_{r}}(F_{ sr}+B_{sr})-\sum\limits_{r\in R}\sum\limits_{s\in S}\frac{n_{sr}}{N_{r}}\mathrm{ Bextra}-\sum\limits_{s\in S}\mathrm{c_{s}^{in}}\cdot n_{s}^{in}+\sum \limits_{s\in S}\mathrm{c_{s}^{out}}\cdot n_{s}^{out}\]1 (1-11)  
The objective function (1-1) maximizes the total weekly profit, which consists of four major components: the total revenue generated from accepted OD transport demand, the transportation cost (including fuel, berthing, and extra penalty costs) associated with route operation and ship assignment, the chartering cost for ships chartered in, and the revenue earned from chartering out surplus owned ships. The objective function (1-2) maximizes the total weekly transport volume across all OD pairs. Constraints (1-3) ensure that the quantity of ships assigned to each operated route equals the total number of ships required to operate that route. Constraints (1-4) ensure that the total number of ships of each type assigned to all routes does not exceed the available fleet after chartering in and chartering out. Constraints (1-5) require that the number of ships chartered out for each ship type does not exceed the company's owned fleet. Constraints (1-6) ensure that, for each OD pair, the total accepted transport volume allocated to all routes does not exceed the total market demand for that OD pair. Constraints (1-7) denote that, for each leg on each operated route, the total transport volume assigned across all relevant OD pairs does not exceed the transport capacity of that route. Constraints (1-8)-(1-11) are the domains of the decision variables. Figure 2 illustrates the decision-making flow of our MIP model. \begin{table}
\begin{tabular}{l l} \hline \multicolumn{2}{c}{**Transport-Related Parameters**} \\ \hline \(F_{sr}\) & The fuel cost of one ship of type \(s\) completing one full trip on route \(r\) \\ \(B_{sr}\) & The berthing cost for a ship of type \(s\) for a full round trip on route \(r\) \\ \(B_{sr}^{\mathrm{extra}}\) & The extra penalty cost for a ship of type \(s\) for a full round trip on route \(r\) \\ \hline \multicolumn{2}{c}{**Decision Variables**} \\ \hline \(x_{r}\) & Binary variable, which equals 1 if the candidate route \(r\) is operated, and 0 otherwise \\ \(n_{sr}\) & Integer variable, indicating the number of ships of type \(s\) assigned to route \(r\) \\ \(n_{s}^{\mathrm{in}}\) & Integer variable, indicating the number of ships of type \(s\) chartered in \\ \(n_{s}^{\mathrm{out}}\) & Integer variable, indicating the number of ships of type \(s\) chartered out \\ \(z_{odr}\) & Continuous variable, indicating the number of accepted TEUs per week for \((o,d)\) \\  & on route \(r\) \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._