Enhanced Ant Colony Algorithm for Discrete Dynamic Berth Allocation in a Case Container Terminal
Abstract
:1. Introduction
2. Literature Review
3. Problem Statement and Model Formulation
- (a)
- It is known beforehand and regarded as immutable, i.e., it is fixed.
- (b)
- It depends on where the vessels berthed.
- (c)
- It depends on the quantity of cranes serving the vessels.
- (d)
- The working schedule of the cranes is highly relied on.
- (e)
- All the mentioned conditions (b–c) need to be handled at the same time.
3.1. Notation
Notation |
Sets |
B: Set of berths, B = {1, 2, …, nb}; |
C: Set of QCs, C = {1, 2, …, nc}; |
J: Set of incoming vessels during a planning period, J = {1, 2, ..., n}; |
Model-Related Notations |
n: Number of incoming vessels within a planning period, positive integer; |
nb: Number of available berths in a port; |
nc: Number of quay cranes; |
njcap: Container capacity of vessel j; |
njl: Number of containers to be loaded/unloaded on vessel j; |
Oi: An ordered subset of vessels to be loaded/unloaded at berth i; |
tijb: Moment of vessel j starting to berth at berth i; |
tjw: Arrival moment of vessel j to the waiting area, tjw ≥ 0; |
Tjb: Time duration of a vessel to berth and leave, assumed to be fixed and the same for different vessels, Tjb > 0; |
tijfl: The finishing moment of loading/unloading vessel j at berth i, tijfl = tijl + Tijl; |
tijl: The beginning moment of loading and unloading vessel j at berth i; |
Tijl: Duration of loading and unloading vessel j at berth i; a function of njl, njcap, and njc; |
PACO-Related Notations |
dij’: Time distance between vessels j and j′; |
e(j,j′): Edge connecting i and j; |
tk(i): The length of time during which the kth ant starts from berth i to the end through all sections of the path; |
Q: Constant, the amount of information released by an ant after the complete path search is complete; |
Lk: The path length through which the kth ant passes; |
nant: Quantity of ants in a colony; |
ncon: Quantity of iterations of convergence; |
nmaxitr: Maximum allowable number of iterations; |
nitr: Current number of iterations; |
nk(s): Notation of the number of nodes passed by the current ant k in the step s; |
pjj’k(oi): Probability of ants k choosing vessel j’ from vessel j as the next candidate within the ordered subset of berth i; |
α: Adjustable parameters controlling the relative impact of pheromone trajectories ; |
β: Adjustable parameters controlling the relative impact of heuristic desirability; |
ρ: Constant, a coefficient of evaporation; 0 < ρ < 1(1 − ρ) expresses pheromone decay during t and t + 1; |
: The intensity of the trail on edge (j,j′) at time t; |
: A heuristic measure of the desirability of adding edge (j,j′) to the solution being built; |
: A total incremental quantity of the pheromones laid by all ants on edge (j,j ″); |
: An incremental quantity of pheromones laid by the kth ant on edge (j,j′) during time t and time t + 1, which is a function of the length Lk of tour Tk(t): |
Variables |
njc: Decision variable; number of quay cranes assigned to the vessel j; |
xijo: Binary decision variable, with a value of one when vessel j at berth i is served at service time t and a value of 0 otherwise; |
3.2. Single-Berth Model
3.3. Multi-Berth Model
- The vessels arrival time is random;
- The vessel loading and unloading times are derived from the number of containers, the quantity of QCs, the productivity of QCs, and other related factors;
- The mooring space must conform to the real conditions of the vessel (depth and length of water) of the constraints;
- The vessels moving from one berth to another will not be considered, where each vessel has only one berth opportunity;
- The number of assigned QCs does not exceed the number of QCs allowed to work simultaneously;
- When a large quantity of Quay Cranes (QCs) operates on a single vessel simultaneously, they can influence each other, potentially affecting the overall utilization rate of QC s. In cases where more than two terminal container cranes are operating simultaneously, one of them operates at a reduced capacity, typically around 90%.
4. Solution Algorithm
4.1. The Basic Principle of the Ant Colony Algorithm
4.2. The Basic Principle of the Parallel Ant Colony Algorithm
4.3. PACO for BAP
- (1)
- According to the concentration of hormones on the path, the ant selects the next path with the corresponding probability.
- (2)
- The ant no longer chooses nodes for the next path, which have been passed in this cycle, by using a data structure (tabu matrix) to control.
- (3)
- After completing a cycle, the ant releases pheromones in corresponding concentrations according to the length of the path and updates the pheromone concentration of the traversed path. In the initial stage, m ants are randomly placed on nodes with an equal amount of information on each path.
4.3.1. Solution Encoding
4.3.2. PACO for the Berth Allocation Problem
- (1)
- Individual ants select elements (container vessel) j to go according to the probability calculated through the state transition probability formula (3), where j ∈ tabu k, cjj′ is a time equal to vessel j’s time of completion minus vessel j’s arrival time, and lk is the path of all nodes the ant has walked, namely, the total time of all vessels staying in the port.
- (2)
- Modify the tabu search list pointer that is for ant j to move to the new element (container vessel) after choosing, and put the element (container vessel) into the individual ant’s tabu search list.
- (3)
- If the elements (container vessels) in Set C have not been traversed, which means k < m, then k = k + 1; otherwise, continue.
5. Numerical Experiments
5.1. Determination of Parameter Values
5.2. Results of Single-Berth Allocation
5.2.1. Based on ACO
5.2.2. Based on PACO
5.3. Results of Multi-Berth Allocation
5.3.1. Based on ACO
5.3.2. Based on PACO
5.4. Sensitivity Analysis
5.5. Benchmarking Analysis with Other Algorithms
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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No. of Vessels | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Inter-arrival time (h) | 0.0 | 13.5 | 8.0 | 5.0 | 2.5 | 0.5 | 0.5 | 1.5 | 5.0 | 0.0 |
Loading/unloading time (h) | 12.9 | 11.7 | 8.7 | 19. 7 | 22.0 | 11. 5 | 9.8 | 10.8 | 10.8 | 9.8 |
No. of Vessels | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Inter-arrival time (h) | 14.0 | 8.8 | 7.3 | 5.0 | 1.5 | 13.5 | 6.0 | 3.25 | 1.8 | 1.5 |
Loading/unloading time (h) | 9.7 | 9.8 | 11.0 | 12. 5 | 14.7 | 10.7 | 6. 8 | 11.3 | 10.0 | 10.3 |
Ca | The Optimal Value (the Total Residence Time of All Vessels in the Port) | Started Convergence Algebra |
---|---|---|
1 | 257.75 | 7 |
2 | 256.84 | 5 |
3 | 256.42 | 5 |
4 | 257.76 | 1 |
5 | 251.83 | 5 |
10 | 251.83 | 3 |
20 | 251.83 | 2 |
30 | 251.83 | 2 |
40 | 251.83 | 1 |
50 | 251.83 | 1 |
α | β | The Optimal Value (the Total Residence Time of All Vessels in the Port) | Started Convergence Algebra |
---|---|---|---|
0.5 | 1 | 257.35 | 53 |
0.5 | 2 | 251.83 | 15 |
0.5 | 3 | 251.83 | 21 |
0.5 | 4 | 251.83 | 3 |
0.5 | 5 | 251.83 | 3 |
0.5 | 6 | 256.84 | 6 |
1 | 1 | 262.92 | 22 |
1 | 2 | 260.33 | 21 |
1 | 3 | 255.51 | 6 |
1 | 4 | 256.60 | 5 |
1 | 5 | 251.83 | 3 |
1 | 6 | 256.76 | 2 |
α | β | The Optimal Value (the Total Residence Time of All Vessels in the Port) | Started Convergence Algebra |
---|---|---|---|
0.5 | 1 | 256.18 | 29 |
0.5 | 2 | 251.83 | 22 |
0.5 | 3 | 251.83 | 8 |
0.5 | 4 | 251.83 | 2 |
0.5 | 5 | 251.83 | 2 |
0.5 | 6 | 251.83 | 17 |
1 | 1 | 256.10 | 14 |
1 | 2 | 251.83 | 6 |
1 | 3 | 251.83 | 5 |
1 | 4 | 251.83 | 2 |
1 | 5 | 251.83 | 2 |
1 | 6 | 255.85 | 2 |
Actual | GA | PSO | ACO | PACO | |
---|---|---|---|---|---|
The total port time for the 20 vessels in the case (h) | 304.4 | 254.17 | 257.25 | 257.2 | 251.83 |
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Yu, M.; Lv, Y.; Wang, Y.; Ji, X. Enhanced Ant Colony Algorithm for Discrete Dynamic Berth Allocation in a Case Container Terminal. J. Mar. Sci. Eng. 2023, 11, 1931. https://doi.org/10.3390/jmse11101931
Yu M, Lv Y, Wang Y, Ji X. Enhanced Ant Colony Algorithm for Discrete Dynamic Berth Allocation in a Case Container Terminal. Journal of Marine Science and Engineering. 2023; 11(10):1931. https://doi.org/10.3390/jmse11101931
Chicago/Turabian StyleYu, Meng, Yaqiong Lv, Yuhang Wang, and Xiaojing Ji. 2023. "Enhanced Ant Colony Algorithm for Discrete Dynamic Berth Allocation in a Case Container Terminal" Journal of Marine Science and Engineering 11, no. 10: 1931. https://doi.org/10.3390/jmse11101931
APA StyleYu, M., Lv, Y., Wang, Y., & Ji, X. (2023). Enhanced Ant Colony Algorithm for Discrete Dynamic Berth Allocation in a Case Container Terminal. Journal of Marine Science and Engineering, 11(10), 1931. https://doi.org/10.3390/jmse11101931