# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 5 Numerical Experiments
### Results and Analysis
#### 5.2.2 Performance Comparison Between IMOCS and Other Algorithms
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Vessel**} & \multirow{2}{*}{**Instance**} & \multicolumn{4}{c}{**IGD**} & \multicolumn{4}{c}{**HV**} \\ \cline{3-10}  & & **NSGAII** & **SPEA2** & **MOCS** & **IMOCS** & **NSGAII** & **SPEA2** & **MOCS** & **IMOCS** \\ \hline \multirow{4}{*}{30} & 30-1 & 0.25 & 0.36 & 0.24 & **0.15** & 0.67 & 0.56 & 0.73 & **0.92** \\  & 30-2 & 0.17 & 0.37 & 0.26 & **0.09** & 0.77 & 0.53 & 0.72 & **0.83** \\  & 30-3 & 0.24 & 0.46 & **0.17** & 0.18 & 0.76 & 0.52 & 0.83 & **0.84** \\  & 30-4 & 0.61 & 0.41 & 0.48 & **0.26** & 0.82 & 0.52 & 0.84 & **0.88** \\  & 30-5 & 0.26 & 0.25 & 0.19 & **0.16** & 0.67 & 0.53 & 0.84 & **0.84** \\ \hline \multirow{4}{*}{40} & 40-1 & 0.36 & 0.34 & 0.35 & **0.15** & 0.56 & 0.38 & 0.55 & **0.64** \\  & 40-2 & 0.51 & 0.54 & 0.26 & **0.22** & 0.46 & 0.34 & 0.67 & **0.69** \\  & 40-3 & 0.19 & 0.33 & 0.18 & **0.16** & 0.56 & 0.67 & 0.52 & **0.67** \\  & 40-4 & 0.62 & 0.27 & 0.36 & **0.27** & 0.44 & 0.54 & 0.43 & **0.77** \\  & 40-5 & 0.30 & 0.47 & 0.27 & **0.12** & 0.58 & 0.59 & 0.66 & **0.73** \\ \hline \multirow{4}{*}{50} & 50-1 & 0.61 & 0.52 & 0.42 & **0.23** & 0.31 & 0.39 & 0.51 & **0.64** \\  & 50-2 & 0.44 & 0.51 & 0.36 & **0.30** & 0.45 & 0.36 & 0.52 & **0.61** \\  & 50-3 & 0.45 & 0.66 & 0.37 & **0.21** & 0.41 & 0.54 & 0.49 & **0.62** \\  & 50-4 & 0.50 & 0.53 & 0.38 & **0.29** & 0.40 & 0.51 & 0.56 & **0.66** \\  & 50-5 & 0.56 & 0.58 & 0.41 & **0.38** & 0.27 & 0.22 & 0.44 & **0.46** \\  & Average & 0.40 & 0.44 & 0.31 & **0.21** & 0.54 & 0.48 & 0.62 & **0.72** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of performance metrics for the large-scale instances. Figure 7: Pareto front diagrams for instances (**a**) 30-1, (**b**) 40-1, and (**c**) 50-1.  
while a smaller IGD value reflects better performance. Additionally, smaller quartile spacing and fewer outliers indicate more consistent performance. It is evident that IMOCS significantly outperforms the other algorithms. To further illustrate the effectiveness and applicability of the proposed algorithm, we conducted a comparative evaluation of the four algorithms using the instances from Park [44] (referred to as _GenPK_). These instances have also been utilized in prior studies, such as Reference [43], and consist of instances with complexity comparable to those presented in our work. The results of the experiment are shown in Table 8; the first column shows the size of the instances and the second column shows the evaluated metrics, including average shipowner dissatisfaction (Avg.obj1), average terminal cost (Avg.obj2), and average CPU runtime (Avg.time). As shown in Table 8, the overall average Obj1 values for NSGAII, SPEA2, MOCS, and IMOCS are 17.6, 16.7, 16.3, and 14.8, respectively. IMOCS achieves the best performance, followed by MOCS, while NSGAII and SPEA2 perform similarly but worse. For overall average Obj2 values, IMOCS again performs the best at 312.2, significantly outperforming the next-best, MOCS, at 337.8. Regarding overall average T values, SPEA2 leads with 481.1 s, followed closely by IMOCS at 517.6 s. The difference of around 36 s is minor and acceptable \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vessel** & & **NSGAII** & **SPEA2** & **MOCS** & **IMOCS** \\ \hline \multirow{3}{*}{20} & Avg.obj1 & 4.5 & 4.8 & 4.2 & **4.0** \\  & Avg.obj2 & 151.2 & 134.8 & 142.9 & **128.4** \\  & Avg.time & 125.2 & **118.4** & 131.0 & 141.8 \\ \hline \multirow{3}{*}{30} & Avg.obj1 & 8.5 & 8.3 & 7.8 & **6.8** \\  & Avg.obj2 & 226.5 & 237.7 & 232.0 & **209.5** \\  & Avg.time & **268.7** & 289.3 & 320.4 & 303.4 \\ \hline \multirow{3}{*}{40} & Avg.obj1 & 17.1 & 17.8 & 16.7 & **14.3** \\  & Avg.obj2 & 421.5 & 423.0 & 435.1 & **399.6** \\  & Avg.time & 752.5 & 655.5 & 686.4 & **632.5** \\ \hline \multirow{3}{*}{50} & Avg.obj1 & 40.2 & 35.9 & 36.5 & **34.1** \\  & Avg.obj2 & 553.0 & 555.8 & 546.7 & **511.2** \\ \cline{1-1}  & Avg.time & 1149.2 & **861.3** & 1317.8 & 992.7 \\ \hline \hline \end{tabular}
\end{table}