# 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  
Considering the size and complexity of large-scale instances \(|V|~{}=~{}\{30,40,50\}\), it is difficult for GUROBI to obtain a feasible solution in an acceptable time. Therefore, to validate the performance of the proposed algorithm, we select three multi-objective optimization algorithms and compare them with the proposed IMOCS algorithm. These algorithms include the Non-Dominated Sorting Genetic Algorithm II (NSGAII), the Strength \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Vessel**} & \multirow{2}{*}{**Instance**} & \multicolumn{4}{c}{**GUROBI**} & \multicolumn{4}{c}{**IMOCS**} \\ \cline{3-10}  & & **Obj\({}_{1}\)** & **Obj\({}_{2}\)** & **T(s)** & **Obj\({}_{1}\)** & **Gap\({}_{1}\)** & **Obj\({}_{2}\)** & **Gap\({}_{2}\)** & **T(s)** \\ \hline \multirow{4}{*}{10} & 10-1 & 0.00 & 56.10 & 14.35 & 0.00 & 0.00\% & 56.10 & 0.00\% & 9.66 \\  & 10-2 & 0.38 & 66.70 & 16.08 & 0.38 & 0.00\% & 66.80 & 0.15\% & 6.27 \\  & 10-3 & 0.00 & 64.70 & 12.75 & 0.00 & 0.00\% & 64.70 & 0.00\% & 6.91 \\  & 10-4 & 0.00 & 54.90 & 13.40 & 0.00 & 0.00\% & 54.90 & 0.00\% & 10.67 \\  & 10-5 & 0.00 & 68.50 & 16.66 & 0.00 & 0.00\% & 68.50 & 0.00\% & 8.19 \\ \hline \multirow{4}{*}{15} & 15-1 & 2.12 & 93.38 & 327.44 & 2.14 & 0.94\% & 94.48 & 1.18\% & 68.91 \\  & 15-2 & 6.09 & 77.01 & 569.04 & 6.16 & 1.15\% & 77.82 & 1.05\% & 99.83 \\  & 15-3 & 8.72 & 81.00 & 201.48 & 8.77 & 0.57\% & 81.29 & 0.36\% & 81.95 \\  & 15-4 & 10.55 & 86.59 & 222.40 & 10.59 & 0.38\% & 87.10 & 0.59\% & 76.75 \\  & 15-5 & 7.08 & 84.39 & 757.28 & 7.21 & 1.84\% & 85.12 & 0.87\% & 81.41 \\ \hline \multirow{4}{*}{20} & 20-1 & 14.09 & 115.69 & 4904.74 & 15.32 & 8.73\% & 118.24 & 2.20\% & 128.04 \\  & 20-2 & 14.77 & 120.05 & 2372.87 & 15.61 & 5.38\% & 125.22 & 4.31\% & 137.82 \\ \cline{1-1}  & 20-3 & 14.52 & 129.47 & 4951.24 & 14.83 & 2.14\% & 131.80 & 1.80\% & 199.66 \\ \cline{1-1}  & 20-4 & - & - & - & 11.83 & - & 153.52 & - & 163.90 \\ \cline{1-1}  & 20-5 & 16.67 & 138.21 & 5772.89 & 17.81 & 6.84\% & 141.28 & 2.22\% & 153.55 \\ \hline \end{tabular}
\end{table} 
Table 5: Experimental results for the small-scale instances. Figure 6: **(a)** The berthing plan for instance 10-4. (b) QC-time Gantt chart for instance 10-4. Pareto Evolutionary Algorithm 2 (SPEA2), and MOCS. The reasons for choosing these algorithms are as follows: (1) NSGAII and SPEA2 are classical and effective multi-objective optimization algorithms, which have been widely used in solving various types of multi-objective problems, such as multi-objective berth QC scheduling, and have shown good performance [30]; (2) NSGAII and SPEA2 are often used as comparative algorithms to evaluate the effectiveness of the proposed algorithms for multi-objective optimization problems [48]. Each instance runs independently 10 times in the same computing environment, and the average values for minimum shipowner dissatisfaction (Obj1), minimum CT operating cost (Obj2), HV, and IGD are calculated for each instance. Table 6 describes the experimental results for the large-scale instances. The overall average Obj1 values for NSGAII, SPEA2, MOCS, and IMOCS are 28.6, 27.4, 27.0, and 25.1, respectively. IMOCS achieves the best performance, followed by MOCS, while NSGAII performs the worst. For overall average Obj2 values, IMOCS again performs the best at 373.4, significantly outperforming the next-best NSGAII at 400.4. Regarding overall average T values, SPEA2 leads with 642.4 s, followed closely by IMOCS at 674.7 s. The difference of around 30 s is minor and acceptable in practical applications. Table 7 describes the comparison of performance metrics for vessel sizes \(|V|\) = 30, 40, 50. The overall average IGD values for NSGAII, SPEA2, MOCS, and IMOCS are 0.40, 0.44, 0.31, and 0.21, respectively, showing that IMOCS achieves the best performance, followed by MOCS, with NSGAII performing the worst. For the overall average HV values, IMOCS also performs best, achieving 0.72 compared to 0.62 for MOCS.