# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 6 Conclusions  
In this paper, we propose a bi-objective BACASP. To fully utilize the berth space and obtain a more realistic berthing plan, continuous berths, the dynamic arrival of vessels, and the non-crossing of QCs are considered in the proposed BACASP. Considering that shipowners are irrational and the relationship between satisfaction and delay and advancement is not linear, prospect theory is used to calculate shipowner dissatisfaction instead of the traditional approach of minimizing total delay and waiting time. Given that BACASP is an NP-hard problem, an IMOCS algorithm is proposed to solve this problem. Numerical experimental results show that the solution quality of IMOCS is close to those of the GUROBI solver on small-scale instances, with a significant advantage in computational time. For large-scale instances, IMOCS exhibits better convergence and diversity, outperforming other multi-objective optimization algorithms, such as NSGA-II, SPEA2, and MOCS, in terms of objective function values, HV values, and IGD values. Given the time constraints, two extensions can be considered for future research. The first one is to introduce tidal factors into BACASP, considering the influence of different \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Vessel**} & \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**Single-Objective**} & \multicolumn{2}{c}{**Single-Objective**} & \multicolumn{2}{c}{**Single-Objective**} & \multicolumn{2}{c}{**Bi-Objective**} & \multicolumn{2}{c}{**Bi-Objective**} \\  & & \multicolumn{2}{c}{**(\(w_{a}\)=0)**} & \multicolumn{2}{c}{**(\(w_{a}\)=0.5)**} & \multicolumn{2}{c}{**(\(w_{a}\)=1)**} & \multicolumn{2}{c}{**(Min Obj1)**} & \multicolumn{2}{c}{**(Min Obj2)**} \\ \cline{3-11}  & & **Obj1** & **Obj2** & **Obj1** & **Obj2** & **Obj1** & **Obj2** & **Obj1** & **Obj2** & **Obj1** & **Obj2** \\ \hline \multirow{4}{*}{30} & 30-1 & 13.6 & 209.6 & 9.7 & 246.5 & 5.6 & 298.1 & 6.1 & 277.3 & 13.2 & 215.7 \\  & 30-2 & 17.8 & 205.1 & 11.5 & 253.2 & 5.7 & 303.9 & 6.2 & 282.7 & 16.8 & 223.7 \\  & 30-3 & 14.0 & 164.7 & 10.2 & 203.5 & 6.4 & 253.1 & 6.9 & 235.6 & 13.4 & 172.4 \\  & 30-4 & 22.4 & 224.5 & 16.5 & 265.9 & 10.4 & 315.5 & 11.2 & 293.6 & 21.5 & 238.2 \\  & 30-5 & 10.2 & 185.8 & 6.7 & 229.4 & 3.4 & 280.6 & 3.7 & 261.0 & 9.6 & 197.8 \\ \hline \multirow{4}{*}{40} & 40-1 & 26.9 & 344.6 & 21.7 & 401.8 & 18.8 & 464.4 & 19.2 & 432.0 & 24.2 & 371.6 \\  & 40-2 & 34.9 & 354.9 & 26.4 & 395.7 & 21.2 & 454.9 & 23.7 & 423.2 & 33.1 & 368.1 \\  & 40-3 & 27.2 & 403.3 & 23.4 & 462.5 & 19.1 & 527.7 & 20.6 & 490.9 & 26.1 & 434.0 \\  & 40-4 & 41.9 & 412.7 & 34.0 & 467.1 & 27.5 & 536.7 & 29.7 & 499.5 & 38.2 & 434.7 \\  & 40-5 & 38.1 & 371.3 & 32.6 & 421.5 & 26.2 & 486.6 & 28.3 & 453.1 & 36.9 & 389.8 \\ \hline \multirow{4}{*}{50} & 50-1 & 70.4 & 516.3 & 61.4 & 559.2 & 57.0 & 627.2 & 58.3 & 583.6 & 64.5 & 534.8 \\  & 50-2 & 58.3 & 460.1 & 47.3 & 525.2 & 41.1 & 595.7 & 43.3 & 554.1 & 51.2 & 496.2 \\ \cline{1-1}  & 50-3 & 41.5 & 472.1 & 33.2 & 531.2 & 27.6 & 601.1 & 29.8 & 559.0 & 36.6 & 503.4 \\ \cline{1-1}  & 50-4 & 68.1 & 519.0 & 55.2 & 571.5 & 48.7 & 647.3 & 50.5 & 602.1 & 59.8 & 541.8 \\ \cline{1-1}  & 50-5 & 50.8 & 455.1 & 41.4 & 506.6 & 37.6 & 574.4 & 38.5 & 534.3 & 44.3 & 478.8 \\ \cline{1-1}  & Average & 35.7 & 353.3 & 28.7 & 402.7 & 23.8 & 464.5 & 25.1 & 432.1 & 32.6 & 373.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of bi-objective optimization and weighted single-objective optimization. water depths on vessels entering and leaving the channel. Another extension is to study the scenario with uncertain arrival times. Robust schedules are usually the target for the model with uncertainty, and it may be an interesting challenge. Conceptualization, Z.L. and X.Z. ; funding acquisition, X.Z. and X.Z. ; investigation, Z.L., X.Z. and J.Z. ; methodology, Z.L., J.Z. and X.Z. ; project administration, J.Z. ; supervision, J.Z. ; validation, J.Z. and Y.Z. ; writing--original draft, J.Z. ; writing--review and editing, X.Z. and C.F.