## 5 Numerical Example
### Outputs  
After running the simulation tool, Table 1 shows the total solution to the BAP, QCAP, and QCSP. For ship 7, it is assigned to berth 1 and QC 1 is assigned to handle tasks 1 and 5; QC 2 handles the task 7; and QC 3 handles tasks 11 and 13. For each task \(k\), a QC \(q\) is assigned to work it during the time period [\(T1_{i\!qk}\), \(T2_{i\!qk}\)]. For instance, the QC 1 is assigned to handle task 1 of ship 7 during the time period [0,174]. Table 2 shows the schedules for all QCs. The solution to the BAP is denoted as  
\[\{\bigcup_{i=1}^{m}(i,j):\text{X}_{ij}^{t}=1\}=\{(1,3),(2,2),(3,2),(4,1),(5,3),(6,2),(7,1),(8,1),(9,3)\}\]  
Furthermore, based on Equation (10), the \(BT_{1}\) to \(BT_{9}\) of the calling ships are derived as follows.  
\[BT_{1} =[1002,1757]\] \[BT_{2} =[474,1044]\] \[BT_{3} =[1044,1524]\] \[BT_{4} =[1140,1590]\] \[BT_{5} =[0,570]\] \[BT_{6} =[0,432]\] \[BT_{7} =[0,570]\] \[BT_{8} =[570,1140]\] \[BT_{9} =[570,1002]\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(j\)} & \multicolumn{4}{c}{\(\text{X}_{i\!qk}^{t}=\text{1}\)} & \multirow{2}{*}{\(\text{T1}_{i\!qk}\)} & \multirow{2}{*}{\(\text{T2}_{i\!qk}\)} & \multirow{2}{*}{**Duration**} \\ \cline{2-3} \cline{5-6}  & \(q\) & & & & & \\ \hline
1 & 1 & ship 7 & 1 & 0 & 174 & 174 \\
1 & 1 & ship 7 & 5 & 174 & 339 & 165 \\
1 & 2 & ship 7 & 7 & 0 & 282 & 282 \\
1 & 3 & ship 7 & 9 & 0 & 240 & 240 \\
1 & 3 & ship 7 & 11 & 240 & 390 & 150 \\
1 & 3 & ship 7 & 13 & 390 & 570 & 180 \\
1 & 1 & ship 8 & 1 & 570 & 744 & 174 \\
1 & 1 & ship 8 & 5 & 744 & 909 & 165 \\
1 & 2 & ship 8 & 7 & 570 & 852 & 282 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The total solution to BAP, QCAP, and QCSP  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(j\)} & \multicolumn{3}{c}{\(Y_{ijk}^{4}=1\)} & \multirow{2}{*}{\(T1_{ijk}\)} & \multirow{2}{*}{\(T2_{ijk}\)} & \multirow{2}{*}{\(Duration\)} \\ \cline{2-2} \cline{5-6}  & \(q\) & \(i\) & \(k\) & & \\ \hline  
\hline \end{tabular} \(j\): berth id; \(q\): QC id; \(j\): ship id; \(k\): task id; \(T1_{ijk}\): begin time; \(T2_{ijk}\): end time.  
\end{table}
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(j\)} & \multicolumn{3}{c}{\(Y_{iqk}^{\dagger}=1\)} & \multirow{2}{*}{\(T1_{iqk}\)} & \multirow{2}{*}{\(T2_{iqk}\)} & \multirow{2}{*}{**Duration**} \\ \cline{2-2} \cline{5-6}  & \(q\) & \(i\) & \(k\) & & \\ \hline  
\hline \hline \end{tabular} _j_: berth id; q: QC id; _j_: ship id; _k_: task id; \(T1_{iqk}\); begin time; \(T2_{iqk}\): end time.  
\end{table}
Table 2: The schedule of QCs (sorted by QC No.).  
Figure 3 shows the berthing plan for these calling ships. The total makespan of this berthing plan is 1590, and is found to be quite promising as the workloads on these berths are well balanced. As a result, the makespan on Berth 1 (1590) is very close to the makespan on Berth 2 (1524) as well as the makespan on Berth 3 (1527). In addition, it is found that the berth utilization rates for Berth 1, Berth 2, and Berth 3 reach 100%, 97%, and 100%, respectively, which indicates that these scarce resources can be efficiently utilized.