## 3 Model Formulation
\begin{table}
\begin{tabular}{c l l} \hline \hline \multicolumn{2}{c}{**Model Component**} & \multicolumn{1}{c}{**Description**} \\ \hline
**Type** & **Nomenclature** & \\ \hline \multirow{3}{*}{Sets} & \(I=\{1,\dots,m\}\) & set of arriving vessels (vessels) \\  & \(J=\{1,\dots,n\}\) & set of available berthing positions (berthing positions) \\  & \(K=\{1,\dots,s\}\) & set of vessel service orders (service orders) \\ \hline \multirow{3}{*}{Decision Variables} & \(\mathbf{x}_{ijk}\), \(i\in I\), \(j\in J\), & =1 if vessel \(i\) is assigned to berthing position \(j\) in service order \\  & \(k\in K\) & \(k\) (=0 otherwise) \\ \hline \multirow{3}{*}{Auxiliary Variables} & \(T_{ijk}^{ID}\), \(i\in I\), \(j\in J\), & idle time of berthing position \(j\) between the start service time \\  & \(k\in K\) & of vessel \(i\) and a preceding vessel served as \(k-1\) vessel (h) \\ \cline{1-1}  & \(\mathbf{T}_{i}^{ST}\), \(i\in I\) & start service time of vessel \(i\) (h) \\ \cline{1-1}  & \(\mathbf{T}_{i}^{FT}\), \(i\in I\) & finish service time of vessel \(i\) (h) \\ \cline{1-1}  & \(\mathbf{T}_{i}^{WT}\), \(i\in I\) & waiting time of vessel \(i\) (h) \\ \cline{1-1}  & \(\mathbf{T}_{i}^{LD}\), \(i\in I\) & late departure time of vessel \(i\) (h) \\ \hline \multirow{3}{*}{Parameters} & \(m\) & number of arriving vessels (vessels) \\  & \(n\) & number of available berthing positions (berthing positions) \\ \cline{1-1}  & \(s\) & number of vessel service orders (service orders) \\ \cline{1-1}  & \(T_{j}^{B}\), \(j\in J\) & time when berthing position \(j\) becomes available in the \\ \cline{1-1}  & \(T_{i}^{A}\), \(i\in I\) & planning horizon for the first time (h) \\ \cline{1-1}  & \(T_{ij}^{H}\), \(i\in I\) & arrival time of vessel \(i\) at the MCT (h) \\ \cline{1-1}  & \(T_{ij}^{HT}\), \(i\in I\), \(j\in J\) & handling time of vessel \(i\) at berthing position \(j\) (h) \\ \cline{1-1}  & \(T_{j}^{D}\), \(i\in I\) & negotiated departure time of vessel \(i\) (h) \\ \cline{1-1}  & \(L_{i}^{V}\), \(i\in I\) & length of vessel \(i\) (ft.) \\ \cline{1-1}  & \(L_{i}^{B}\), \(j\in J\) & length of berthing position \(j\) (ft.) \\ \cline{1-1}  & \(D_{i}^{H}\), \(i\in I\) & horizontal clearance requirement for vessel \(i\) (ft.) \\ \cline{1-1}  & \(H_{i}^{V}\), \(i\in I\) & draft of vessel \(i\) (ft.) \\ \cline{1-1}  & \(H_{i}^{B}\), \(j\in J\) & depth of berthing position \(j\) (ft.) \\ \cline{1-1}  & \(D_{i}^{V}\), \(i\in I\) & vertical clearance requirement for vessel \(i\) (ft.) \\ \cline{1-1}  & \(W_{i}\), \(i\in I\) & weight of vessel \(i\) (value ranging from 0.10 to 1.00) \\ \cline{1-1}  & \(\Gamma\) & large positive number \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the mathematical model components. Constraint set (6) guarantees that service of each vessel will start after its arrival at the MCT. \[\sum_{i^{\prime}\in I:I:\
eq i}\sum_{k^{\prime}\in K:K^{\prime}<k}(T^{HT}_{i^{ \prime}j}\mathbf{x}_{i^{\prime}jk^{\prime}}+\mathbf{T}^{ID}_{i^{\prime}jk^{\prime}})+\bm {T}^{ID}_{ijk}-(T^{A}_{i}-T^{B}_{j})\mathbf{x}_{ijk}\geq 0\ \forall i\in I,\ j\in J,\ k\in K \tag{6}\]  
Constraint set (7) calculates the start service time for each vessel calling the MCT. \[\mathbf{T}^{ST}_{i}\geq\sum_{i^{\prime}\in I:i^{\prime}\
eq i}\sum_{k^{\prime}\in K :K^{\prime}<k}(T^{HT}_{i^{\prime}j}\mathbf{x}_{i^{\prime}jk^{\prime}}+\mathbf{T}^{ID}_ {i^{\prime}jk^{\prime}})+\mathbf{T}^{ID}_{ijk}+T^{B}_{j}-\Gamma(1-\mathbf{x}_{ijk})\ \forall i\in I,\ j\in J,\ k\in K \tag{7}\]  
Constraint set (8) computes the waiting time of each vessel before its service start at the MCT. \[\mathbf{T}^{WT}_{i}\geq\mathbf{T}^{ST}_{i}-T^{A}_{i}\ \forall i\in I \tag{8}\]  
Constraint set (9) computes the finish service time for each vessel calling the MCT. \[\mathbf{T}^{FT}_{i}=\mathbf{T}^{ST}_{i}+\sum_{j\in I}\sum_{k\in K}(T^{HT}_{ij}\mathbf{x}_{ ijk})\ \forall i\in I \tag{9}\]  
Constraint set (10) estimates the late departure hours for each vessel calling the MCT.