## 5 Numerical Examples
### An Example of a Small-Sized Experiment  
Table 2 shows the original and repositioned ship data of a small-sized experiment (\(N=10\)). The field SID indicates the ID of a ship; the \(l_{j}\) shows the length of ship \(j\); the \(ETA_{j}\) is Estimated Time of Arrival of ship \(j\); \(d_{j}\) is the desired berthing position of ship \(j\); \(\Delta W_{j}\) shows the increased waiting time of ship \(j\); \(\Delta H_{j}\) shows increased handling time of ship \(j\); (\(X_{o}\), \(Y_{o}\)) and (\(X_{1}\), \(Y_{1}\)) are coordinates of left-lower corner and right-upper corners of a ship \(j\), respectively. Two ships (7 and 2) were repositioned to resolve overlaps. Ship 7 is moving towards the +Y direction to avoid overlapping with ship 9, while ship 2 is repositioning towards the -Y direction to avoid overlapping with ship 1. Figure 11 shows the berth plan with overlapping ships, while Figure 12 shows the feasible berth plan after resolving all overlaps of ships.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**Approaches**} \\ \cline{2-4}  & **FCFS** & **SFLA** & **ISFLA** \\ \hline \(F\) & - & 100 & 100 \\ \(m\) & - & 10 & 10 \\ \(n=F/m\) & - & 10 & 10 \\ \(l\_iter\) & - & 5 & 5 \\ _iterations_ & 1 & 150 & 150 \\ \(q\) & - & 5 & - \\ \(S_{m}\) & - & \(N\) & \(N\) \\ \(Rm\) & - & - & 0.3 \\ \(\delta\) & - & - & 0.5 \\ \(\theta\) & - & - & 0.5 \\ \hline \hline \end{tabular} \(F\): total number of frogs; \(m\): total number of memeplexes; \(l\_iter\): number of local search; _iterations_: total number of iterations; \(q\): number of frogs in each memeplex; \(\delta\): the rate of using TM; \(\theta\) the mutation rate; -: not used; \(N\): total number of ships.  
\end{table}
Table 1: Parameter values for different approaches.  
Figure 11: The berth plan with overlapped ships (\(N=10\) ships).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**SID**} & \multirow{2}{*}{\(I_{j}\)} & \multirow{2}{*}{\(ETA_{j}\)} & \multirow{2}{*}{\(d_{j}\)} & \multicolumn{6}{c}{**Original**} & \multicolumn{6}{c}{**Repositioned**} & \multirow{2}{*}{\(c_{1}\Delta W_{j}\)} & \multirow{2}{*}{\(c_{2}\Delta H_{j}\)} \\ \cline{5-12}  & & & & \(X_{0}\) & & \(Y_{0}\) & & \(X_{1}\) & \(Y_{1}\) & \(X_{0}\) & \(Y_{0}\) & \(X_{1}\) & \(Y_{1}\) \\ \hline
1 & 144 & 92 & 473.5 & 92 & 473.5 & 125.9 & 617.5 & 92.0 & 473.5 & 125.9 & 617.5 & 0 & 0 \\
2 & 164 & 94.4 & 452.1 & 94.4 & 452.1 & 123 & 616.1 & 94.4 & **309.5** & **123.1** & **473.5** & 0 & 55 \\
3 & 161 & 60.5 & 492.2 & 60.5 & 492.2 & 88.1 & 653.2 & 60.5 & 492.2 & 88.1 & 653.2 & 0 & 0 \\
4 & 73 & 58.1 & 864.8 & 58.1 & 864.8 & 87.2 & 937.8 & 58.1 & 864.8 & 87.2 & 937.8 & 0 & 0 \\
5 & 56 & 19.3 & 814.9 & 19.3 & 814.9 & 39.3 & 870.9 & 19.3 & 814.9 & 39.3 & 870.9 & 0 & 0 \\
6 & 112 & 55.8 & 67 & 55.8 & 67 & 71 & 179 & 55.8 & 67.0 & 71.0 & 179.0 & 0 & 0 \\
7 & 56 & 105.7 & 534.6 & 105.7 & 534.6 & 140.9 & 590.6 & 105.7 & **675.9** & **140.9** & **731.9** & 0 & 47 \\
8 & 84 & 64.1 & 249.5 & 64.1 & 249.5 & 70.9 & 333.5 & 64.1 & 249.5 & 70.9 & 333.5 & 0 & 0 \\
9 & 193 & 134.6 & 482.9 & 134.6 & 482.9 & 176.8 & 675.9 & 134.6 & 482.9 & 176.8 & 675.9 & 0 & 0 \\
10 & 126 & 10.9 & 580.5 & 10.9 & 580.5 & 56.7 & 706.5 & 10.9 & 580.5 & 56.7 & 706.5 & 0 & 0 \\ \hline Total & & & & & & & & & & & & & & & & \\ \hline \hline SID: Ship ID; \(I_{j}\); length of ship \(j\); \(ETA_{j}\): Estimated Time of Arrival of ship \(j\); \(d_{j}\): desired berthing position of ship \(j\); \(\Delta W_{j}\): increased waiting time of ship \(j\); \(\Delta H_{j}\): increased handling time of ship \(j\). & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Original and repositioning ship data after resolving overlaps (\(N=10\)).  
Figure 12: The berth plan with no overlapped ships (\(N=10\) ships).