A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty
Abstract
:1. Introduction
- (1)
- Novel proactive-reactive-based BACAP mathematical models are established. Different indicators are adopted for various uncertain factors to balance robustness and agility.
- (2)
- An improved genetic algorithm (IGA) is proposed, which can find a solution that is 22.86% superior to that of squawky wheel optimization within 225 s.
- (3)
- A sensitivity analysis is conducted, which implies that port operators should define the coefficient and buffer time according to the uncertainty of vessel arrival time; the more extensive the buffer time, the less significant the effect of risk resistance capability.
2. Problem Description
2.1. Maintenance and Virtual Vessels
2.2. Uncertainty and Buffer Time
3. Mathematical Formulation
3.1. Assumptions
- (a)
- All vessels met the physical or technical requirements of the berth, such as water depth, vessel draft, etc.
- (b)
- The QC assignment was time-invariant, and the number of QCs serving each vessel had upper and lower limits.
- (c)
- Each vessel had a desired berthing position.
- (d)
- No difference was present in the productivity between QCs.
- (e)
- QC assignment is time-invariant, but specific QC reallocation is allowed.
- (f)
- The PM period is predetermined, and the required PM time is known too.
3.2. Model Formulation
3.2.1. Deterministic Model
3.2.2. Baseline Planning Model
3.2.3. Reactive Model
4. Solution Methods
4.1. Chromosome Encoding
4.2. Decoding
4.2.1. QC Assignment and Partition Check
4.2.2. Partition Check-Based Local Search Algorithm
4.3. Fitness Evaluation
4.4. Selection
4.5. Crossover
4.6. Mutation
Algorithm 1. Swap strategy-based neighborhood search algorithm. |
: the set of chromosomes needing to mutate; |
: maximum number of iterations for neighborhood search; |
: number of vessels |
For |
For |
; |
; |
; |
; |
; |
For |
If |
by swapping the order of two vessels in the priority sequence; |
; |
If |
; |
; |
; |
End if |
End if |
End for |
; |
End for |
End for |
4.7. Restart Mechanism
5. Computational Experiments
5.1. Parameter Settings
5.2. Performance Analysis
5.3. The Impact of the Coefficient
5.4. Analysis of Uncertainty and Buffer Time
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Indices and sets | |
Set of actual vessels; | |
Set of virtual vessels; | |
Set of all vessels, , , ; | |
Set of QCs requiring maintenance activities; | |
Set of QCs, , , , are specific QC numbers, and assigned number of QCs is ; | |
Set of time steps in the planning time horizon,,,: the length of the time horizon; | |
Set of correspondence between a virtual vessel and maintained QC , | |
Set of scenario, . | |
Parameters | |
Length of the quay (unit: 10 m); | |
Length of the vessel (unit: 10 m); | |
Time penalty coefficient for vessel deviating from preferred berth; | |
Tesired berthing position of the vessel ; | |
Expected arrival time of a vessel , is a random parameter with a probability density function; | |
Crane capacity demand of vessel given as several QC hours; | |
Coefficient of increase in the QC capacity demand with deviation from desired berthing position; | |
Interference exponent for the QCs. Only effective QC hours are obtained when assigning QCs to a vessel for one hour; | |
Minimum number of QCs agreed to serve the vessel simultaneously; | |
Maximum number of QCs allowed to serve vessel simultaneously; | |
Correspondence relation between the vessel and QC ; | |
Duration of quay crane s preventive maintenance activity; | |
Working coverage width of quay crane , including the actual working range of QC and the safe working distance between QCs; | |
Earliest start time for performing quay crane s maintenance activity, ; | |
Latest start time for performing quay crane s maintenance activity, ; | |
Weight in the target function; | |
Berthing position obtained in the original robust model; | |
Berthing time obtained in the original robust model; | |
Number of QCs obtained from the original robust model; | |
Coefficient of the turnaround time of vessels in port cost; | |
Cost coefficient of deviation from the desired berth; | |
Cost coefficient, which is caused by the delay (or advance) of the vessel’s berthing time relative to the original plan in the recovery process; | |
Cost coefficient, which is caused by the right (or left) movement of the vessel’s berthing position relative to the original plan in the recovery process; | |
Cost coefficient, which is caused by the increase (or decrease) of the number of QCs relative to the original plan in the recovery process. | |
Decision variables | |
Time of starting the handling of a vessel (berthing time); | |
Time of ending the handling of a vessel (including buffer time); | |
Berthing position of a vessel ; | |
The deviation between the preferred and the actual chosen berthing position of the vessel , ; | |
The actual time at which quay crane starts maintenance, ; | |
Binary, set to 1 if the vessel is berthed below vessel , 0 otherwise 0–1, ; | |
Binary, set to 1 if the handling of the vessel ends not later than the handling of vessel starts, 0 otherwise,; | |
Binary, set to 1 if at least one QC is assigned to the vessel at time , 0 otherwise,; | |
Binary, set to 1 if QC is assigned to the vessel at time , 0 otherwise, ; | |
Buffer time of the vessel; | |
Binary, set to 1 if exactly QCs are assigned to the vessel and the buffer time is , 0 otherwise; | |
Binary, set to 1 if exactly QCs are assigned to the vessel at time , 0 otherwise,; | |
Increment (or decrement) relative to in scenario ; | |
Increment of berthing time, increment of relative to the buffer time in scenario ; | |
Increment (or decrement) relative to in scenario ; | |
Increment (or decrement) relative to in scenario . |
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Vessel Type | |||||
---|---|---|---|---|---|
Feeder | U [10, 20] | U [8, 21] | 1 | 2 | 0.2 |
Medium | U [30, 50] | U [21, 30] | 2 | 4 | 0.2 |
Jumbo | U [50, 60] | U [30, 40] | 4 | 6 | 0.2 |
Parameter | Problem Scale | ||
---|---|---|---|
Small-Scale | Medium-Scale | Large-Scale | |
MAXIT | 200 | 250 | 300 |
POP | 150 | 150 | 150 |
CF | 0.7 | 0.7 | 0.75 |
MF | 0.2 | 0.25 | 0.3 |
20 | 30 | 35 |
Small | CPLEX | SWO | IGA | |||||
---|---|---|---|---|---|---|---|---|
Instances | Obj_cp | CPU (s) | Obj_SWO | CPU (s) | Gap_SWO (%) | Obj_IGA | CPU (s) | Gap_IGA (%) |
6_1 | 16.0 | 21.31 | 16.0 | 0.46 | 0.00 | 16.0 | 0.73 | 0.00 |
6_2 | 13.6 | 69.00 | 13.6 | 0.44 | 0.00 | 13.6 | 0.74 | 0.00 |
6_3 | 16.0 | 75.47 | 16.0 | 0.32 | 0.00 | 16.0 | 0.77 | 0.00 |
6_4 | 15.2 | 370.35 | 15.2 | 0.42 | 0.00 | 15.2 | 0.72 | 0.00 |
6_5 | 17.2 | 99.98 | 17.2 | 0.40 | 0.00 | 17.2 | 0.62 | 0.00 |
6_6 | 16.4 | 182.02 | 20.2 | 0.42 | 0.00 | 16.4 | 1.48 | 0.00 |
6_7 | 16.6 | 58.63 | 18.1 | 0.58 | 23.41 | 16.6 | 1.42 | 0.00 |
6_8 | 16.8 | 30.67 | 16.8 | 0.35 | 15.90 | 16.8 | 0.78 | 0.00 |
6_9 | 15.6 | 181.08 | 15.6 | 0.41 | 0.00 | 15.6 | 0.82 | 0.00 |
6_10 | 15.2 | 71.86 | 15.2 | 0.57 | 0.00 | 15.2 | 0.69 | 0.00 |
Average | 15.8 | 116.04 | 16.4 | 0.44 | 3.93 | 15.8 | 0.88 | 0.00 |
8_1 | 20.8 | 2148.05 | 24.6 | 0.64 | 18.69 | 20.8 | 1.69 | 0.39 |
8_2 | 19.8 | 867.59 | 19.8 | 0.44 | 0.40 | 19.8 | 1.00 | 0.40 |
8_3 | 23.0 | 1260.00 | 23.1 | 0.47 | 0.52 | 23.1 | 1.12 | 0.52 |
8_4 | 22.0 | 1887.00 | 22.0 | 0.42 | 0.00 | 22.0 | 1.10 | 0.00 |
8_5 | 16.8 | 895.53 | 16.8 | 0.48 | 0.00 | 16.8 | 1.10 | 0.00 |
8_6 | 20.4 | 1433.88 | 20.4 | 0.44 | 0.00 | 20.4 | 1.08 | 0.00 |
8_7 | 22.0 | 1458.45 | 22.0 | 0.56 | 0.00 | 22.0 | 1.61 | 0.00 |
8_8 | 23.0 | 707.32 | 23.0 | 0.65 | 0.35 | 23.0 | 1.22 | 0.35 |
8_9 | 22.2 | 239.86 | 22.8 | 0.35 | 2.70 | 22.2 | 1.55 | 0.00 |
8_10 | 20.0 | 149.70 | 20.0 | 0.36 | 0.00 | 20.0 | 1.09 | 0.00 |
Average | 21.0 | 1104.74 | 21.5 | 0.48 | 2.27 | 21.0 | 1.26 | 0.17 |
10_1 | 28.3 | 2673.74 | 31.1 | 0.33 | 10.04 | 28.8 | 1.95 | 1.70 |
10_2 | 26.5 | 8287.09 | 26.6 | 0.52 | 0.30 | 26.6 | 1.63 | 0.30 |
10_3 | 27.6 | 22213.89 | 27.6 | 0.31 | 0.00 | 27.6 | 1.83 | 0.00 |
10_4 | 30.0 | 19121.29 | 31.1 | 0.51 | 3.87 | 30.0 | 1.47 | 0.27 |
10_5 | 26.0 | 1567.89 | 26.1 | 0.38 | 0.31 | 26.1 | 1.65 | 0.31 |
10_6 | 25.2 | 4841.43 | 25.6 | 0.40 | 1.59 | 25.2 | 2.24 | 0.00 |
10_7 | 27.6 | 10015.62 | 27.6 | 0.55 | 0.00 | 27.6 | 1.79 | 0.00 |
10_8 | 23.2 | 3202.77 | 23.3 | 0.50 | 0.34 | 23.3 | 1.81 | 0.34 |
10_9 | 26.3 | 6486.41 | 26.7 | 0.53 | 1.67 | 26.7 | 2.73 | 1.67 |
10_10 | 24.8 | 3393.17 | 24.8 | 0.38 | 0.00 | 24.8 | 2.76 | 0.00 |
Average | 26.6 | 8180.33 | 27.1 | 0.44 | 1.81 | 26.7 | 1.99 | 0.46 |
12_1 | 33.5 | 12633.85 | 34.1 | 0.51 | 2.22 | 33.7 | 2.75 | 0.84 |
12_2 | 31.3 | 9983.61 | 32.0 | 0.72 | 2.11 | 31.7 | 2.34 | 1.15 |
12_3 | 32.7 | 14213.93 | 33.8 | 0.45 | 3.49 | 32.7 | 3.23 | 0.00 |
12_4 | 33.7 | 21123.51 | 33.9 | 0.48 | 0.65 | 33.7 | 4.56 | 0.12 |
12_5 | 30.6 | 11569.45 | 30.9 | 0.56 | 1.11 | 30.7 | 2.45 | 0.26 |
12_6 | 29.2 | 15851.87 | 30.1 | 0.63 | 3.22 | 29.2 | 3.39 | 0.00 |
12_7 | 32.4 | 18036.41 | 33.3 | 0.71 | 2.78 | 32.4 | 2.98 | 0.12 |
12_8 | 28.7 | 17201.89 | 32.0 | 0.67 | 11.43 | 29.1 | 3.81 | 1.46 |
12_9 | 32.6 | 19486.32 | 33.9 | 0.73 | 4.05 | 32.9 | 3.89 | 0.86 |
12_10 | 30.2 | 17393.67 | 31.0 | 0.59 | 2.65 | 30.2 | 2.35 | 0.00 |
Average | 31.5 | 15749.45 | 32.4 | 0.61 | 3.37 | 31.6 | 3.18 | 0.48 |
Instances | IGA | SWO | GAP | ||
---|---|---|---|---|---|
Obj_IGA | CPU(s) | Obj_SWO | CPU(s) | Gap_RE(%) | |
15_1 | 46.12 | 8.28 | 48.24 | 0.73 | 4.60 |
15_2 | 40.64 | 5.69 | 40.64 | 0.64 | 0.00 |
15_3 | 43.68 | 7.40 | 43.68 | 0.48 | 0.00 |
15_4 | 45.88 | 14.32 | 47.2 | 1.04 | 2.88 |
15_5 | 42.68 | 12.43 | 45.2 | 0.88 | 5.90 |
15_6 | 43.72 | 12.31 | 44.32 | 1.03 | 1.37 |
15_7 | 50.24 | 7.36 | 57.68 | 0.98 | 14.81 |
15_8 | 43.64 | 6.41 | 43.84 | 0.97 | 0.46 |
15_9 | 43.32 | 5.27 | 43.6 | 0.90 | 0.65 |
15_10 | 40.88 | 6.84 | 43.36 | 0.76 | 6.07 |
Average | 44.08 | 8.63 | 45.78 | 0.84 | 3.67 |
20_1 | 69.12 | 20.18 | 69.12 | 1.00 | 10.27 |
20_2 | 56.16 | 9.80 | 56.16 | 1.22 | 8.84 |
20_3 | 76.00 | 22.49 | 76.00 | 1.74 | 27.01 |
20_4 | 55.84 | 8.65 | 55.84 | 1.40 | -0.36 |
20_5 | 65.36 | 13.88 | 65.36 | 1.61 | 16.38 |
20_6 | 62.40 | 9.97 | 62.40 | 1.41 | 4.21 |
20_7 | 61.52 | 20.05 | 61.52 | 1.62 | 0.33 |
20_8 | 64.40 | 18.88 | 64.40 | 1.42 | 7.98 |
20_9 | 69.28 | 14.51 | 69.28 | 1.44 | 7.98 |
20_10 | 60.56 | 16.99 | 60.56 | 1.57 | 7.45 |
Average | 58.77 | 15.54 | 64.06 | 1.44 | 9.01 |
30_1 | 94.40 | 67.15 | 102.32 | 2.19 | 8.39 |
30_2 | 109.48 | 72.81 | 116.08 | 2.20 | 6.03 |
30_3 | 95.64 | 62.62 | 104.40 | 2.29 | 9.16 |
30_4 | 95.92 | 67.02 | 103.20 | 2.06 | 7.59 |
30_5 | 98.20 | 64.43 | 113.04 | 2.30 | 15.11 |
30_6 | 101.08 | 85.82 | 114.24 | 1.95 | 13.02 |
30_7 | 93.12 | 80.38 | 107.36 | 2.10 | 15.29 |
30_8 | 94.48 | 63.13 | 111.28 | 1.87 | 17.78 |
30_9 | 105.24 | 75.61 | 111.12 | 2.15 | 5.59 |
30_10 | 101.44 | 58.08 | 110.48 | 2.11 | 8.91 |
Average | 98.90 | 69.66 | 109.35 | 2.12 | 10.69 |
40_1 | 156.44 | 216.34 | 183.56 | 4.15 | 17.34 |
40_2 | 137.88 | 206.80 | 164.44 | 4.01 | 19.26 |
40_3 | 159.28 | 224.74 | 194.92 | 4.31 | 22.38 |
40_4 | 153.40 | 200.25 | 167.80 | 3.16 | 9.39 |
40_5 | 153.00 | 182.97 | 185.64 | 4.23 | 21.33 |
40_6 | 172.24 | 187.89 | 188.68 | 4.25 | 9.54 |
40_7 | 138.36 | 219.58 | 157.48 | 4.27 | 13.82 |
40_8 | 158.60 | 207.56 | 185.96 | 4.34 | 17.25 |
40_9 | 156.44 | 173.43 | 192.20 | 5.00 | 22.86 |
40_10 | 148.32 | 196.79 | 166.44 | 3.55 | 12.22 |
Average | 153.40 | 201.64 | 178.71 | 4.13 | 16.54 |
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Wang, Z.; Cheng, J.; Hu, H. A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty. J. Mar. Sci. Eng. 2024, 12, 182. https://doi.org/10.3390/jmse12010182
Wang Z, Cheng J, Hu H. A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty. Journal of Marine Science and Engineering. 2024; 12(1):182. https://doi.org/10.3390/jmse12010182
Chicago/Turabian StyleWang, Zhu, Junfeng Cheng, and Hongtao Hu. 2024. "A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty" Journal of Marine Science and Engineering 12, no. 1: 182. https://doi.org/10.3390/jmse12010182