# 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
For the 15 test instances, IMOCS achieves the best average Obj1, Obj2, IGD, and HV values, demonstrating its superior performance among the four algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Vessel Instance**} & \multicolumn{3}{c}{**NSGAII**} & \multicolumn{3}{c}{**SPEA2**} & \multicolumn{3}{c}{**MOCS**} & \multicolumn{3}{c}{**IMOCS**} \\ \cline{2-13}  & **Obj1** & **Obj2** & **T(s)** & **Obj1** & **Obj2** & **T(s)** & **Obj1** & **Obj2** & **T(s)** & **Obj1** & **Obj2** & **T(s)** \\ \hline \multirow{4}{*}{30} & 30-1 & 8.7 & 235.4 & 385.7 & 9.3 & 241.7 & **360.1** & 8.5 & 229.1 & 380.2 & **6.1** & **215.7** & 392.6 \\  & 30-2 & 7.6 & 244.1 & 405.9 & 7.4 & 243.6 & 415.2 & 6.6 & 245.6 & 420.3 & **6.2** & **223.7** & **390.5** \\  & 30-3 & 8.6 & 183.9 & 450.2 & 7.2 & 221.7 & 480.8 & 8.6 & 208.5 & **430.1** & **6.9** & **172.4** & 490.1 \\  & 30-4 & 13.5 & 251.8 & **315.2** & 11.8 & 261.7 & 370.5 & 11.3 & 266.0 & 405.3 & **11.2** & **238.2** & 350.9 \\  & 30-5 & 4.4 & 217.7 & 460.1 & 5.8 & 220.1 & 425.5 & 4.3 & 211.0 & **360.9** & **3.7** & **197.8** & 369.7 \\ \hline \multirow{4}{*}{40} & 40-1 & 19.4 & 433.1 & 720.5 & 20.8 & 420.4 & **485.3** & 20.7 & 428.5 & 860.1 & **19.2** & **371.6** & 590.8 \\  & 40-2 & 24.6 & 381.8 & 550.9 & 24.7 & 387.9 & 780.1 & **23.2** & 375.4 & 670.2 & 23.7 & **368.1** & **480.2** \\  & 40-3 & 25.7 & 488.6 & 890.6 & 31.3 & 462.5 & **530.4** & 28.4 & 489.4 & 620.8 & **20.6** & **434.0** & 800.6 \\  & 40-4 & 33.5 & **413.1** & 730.3 & 31.3 & 418.9 & 720.9 & 32.6 & 438.8 & **490.2** & **29.7** & 434.7 & 540.8 \\  & 40-5 & 32.1 & 391.0 & 870.2 & 31.3 & 425.3 & 760.8 & 28.9 & 443.5 & 790.9 & **28.3** & **389.8** & **750.5** \\ \hline \multirow{4}{*}{50} & 50-1 & 58.5 & 621.8 & 1124.7 & **56.8** & 647.3 & 708.4 & 57.9 & 594.8 & 1458.9 & 58.3 & **534.8** & **890.5** \\  & 50-2 & 50.1 & 532.3 & **843.2** & 45.9 & 550.8 & 679.8 & 47.3 & 587.4 & 1027.6 & **43.3** & **496.2** & 1290.3 \\ \cline{1-1}  & 50-3 & 38.9 & 530.2 & 1550.6 & 35.5 & 525.1 & 1189.3 & 37.5 & 503.6 & 1325.8 & **29.8** & **503.4** & **815.3** \\ \cline{1-1}  & 50-4 & 60.8 & 575.6 & 937.8 & 52.4 & 564.7 & **743.6** & 50.9 & 563.7 & 1364.6 & **50.5** & **541.8** & 1101.9 \\ \cline{1-1}  & 50-5 & 41.9 & 505.1 & 1290.1 & 39.1 & 491.2 & 985.4 & 38.9 & 484.4 & 1412.2 & **38.5** & **478.8** & **865.8** \\ \cline{1-1}  & Average & 28.6 & 400.4 & 768.4 & 27.4 & 405.5 & **642.4** & 27.0 & 404.6 & 801.2 & **25.1** & **373.4** & 674.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Experimental results for the large-scale instances. In order to visualize the performances of all the algorithms, Figure 7 presents the Pareto frontier plots of the three instances obtained by the compared algorithms. Figure 8 shows the boxplots of IGD and HV for all the compared algorithms across different-sized instances. One instance is chosen from each scale (30-1, 40-1, and 50-1), and the non-dominated solutions from the four algorithms are plotted in Figure 7. Based on preliminary experiments, the Pareto frontiers for the other instances display similar patterns. The IMOCS-generated Pareto frontiers are located closer to the lower left corner of the graph, indicating that IMOCS dominates the majority of solutions from the other algorithms, while only a few of the IMOCS solutions are dominated by the solutions of other algorithms. Figure 8a shows the box-and-line plots of the IGD values for the four comparison algorithms at different scales, while Figure 8b shows the box-and-line plots of the HV values for the four comparison algorithms at different scales. Each plot shows the upper and lower quartiles, the median, and the mean, to better highlight the statistical properties of the experimental outcomes. As seen in Figure 8, there is a significant difference between IMOCS and the other three algorithms. A higher HV value indicates superior performance,