# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 3 Description and Formulation of BACASP
### Mathematical Formulation  
As described in Section 2.3, Reference [4] proves that their model outperforms other models in terms of both solution speed and solution quality. Therefore, we choose their model as the basis of our study and extend it to construct a bi-objective mathematical model by considering the customer satisfaction of shipowners. In this section, we first define the relevant parameters (as shown in Table 1) and decision variables (as shown in Table 2), and then propose a bi-objective mathematical model to solve BACASP. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Definition** & **Unit** \\ \hline \(T\) & Set of time steps in the planning time horizon & hour (h) \\ \(L\) & Length of the quay & meter (m) \\ \(Q\) & Total number of QCs & unitless \\ V & Set of vessels V \(=\)\(\{1,2,\ldots,|\text{V}|\}\) & unitless \\ a\({}_{\text{i}}\) & Arrival time of vessel i, \(\forall\)i \(\in\) V & h \\ l\({}_{\text{i}}\) & Length of vessel i, \(\forall\)i \(\in\) V & m \\ m\({}_{\text{i}}\) & The QC capacity demand of vessel i given as a & QC.h \\ \(\text{m}_{\text{i}}\) & number of QC hours, \(\forall\)i \(\in\) V & QC.h \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of BACASP. Figure 2: Time–space diagram of a feasible berthing plan. The following mathematical model is proposed:  
\[\text{min}f_{1}\ =\ \sum_{i\in V}\ (\omega_{1}\mu_{1}(g_{i})+\omega_{2}\mu_{2}(y_{i}-a _{i})), \tag{4}\]  
\[\text{min}f_{2}\ =\ \sum_{i\in V}\ (c_{1}r_{i}+c_{2}t_{i}\sum_{q=q_{i}^{\text{max} }}^{q_{i}^{\text{max}}}qh_{iq}+c_{3}(y_{i}+t_{i}-a_{i})). \tag{5}\]  
Subject to:  
\[x_{i}+l_{i}\leq L,\forall i\in V, \tag{6}\]  
\[y_{i}\geq a_{i},\forall i\in V, \tag{7}\]  
\[x_{i}+l_{i}\leq x_{j}+M(1-\sigma_{ij}),\forall i,j\in V,i\
eq j, \tag{8}\]  
\[y_{i}+t_{i}\leq y_{j}+M(1-\delta_{ij}),\forall i,j\in V,i\
eq j, \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Definition** & **Unit** \\ \hline \(\text{q}_{\text{i}}^{\text{min}}\) & Minimum number of cranes that can be assigned & unitless \\  & to vessel i, \(\forall\text{i}\in\text{V}\) & \\ \(\text{q}_{\text{i}}^{\text{max}}\) & Maximum number of cranes that can be assigned & & unitless \\  & to vessel i, \(\forall\text{i}\in\text{V}\) & & \\ \(\text{x}_{\text{i}}^{*}\) & Desired berthing position of vessel i, \(\forall\text{i}\in\text{V}\) & m \\ \(\text{d}_{\text{i}}\) & Desired departure time of vessel i, \(\forall\text{i}\in\text{V}\) & h \\ \(\text{rt}_{\text{i}},\text{rt}_{\text{2i}}\) & Maximum tolerated waiting and delay time of & h \\ \(\text{w}_{1},\text{w}_{2}\) & Coefficients of the value function & unitless \\  & Cost coefficient for vessel deviation from preferred & USD/m (\$/m) \\  & berths & USD/h (\$/m) \\ \(\text{c}_{2}\) & Cost coefficient for QC implementation operations & USD/h (\$/h) \\ \(\text{c}_{3}\) & Cost coefficient for berth occupancy & USD/h (\$/h) \\ \(\text{\alpha}\) & Exponent of interference & unitless \\ \(\text{\beta}\) & Factor of berth deviation & unitless \\ M & A large real number & unitless \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._