Berth allocation is the forefront operation performed when ships arrive at a port and is a critical task in container port optimization. Minimizing the time ships spend at berths constitutes an important objective of berth allocation problems. This study focuses on the discrete dynamic berth allocation problem (discrete DBAP), which aims to minimize total service time, and proposes an iterated greedy (IG) algorithm to solve it. The proposed IG algorithm is tested on three benchmark problem sets. Experimental results show that the proposed IG algorithm can obtain optimal solutions for all test instances of the first and second problem sets and outperforms the best-known solutions for 35 out of 90 test instances of the third problem set.
Containerization has been widely adopted in global freight transportation since the 1950s. Containerization significantly reduces shipping costs and accelerates cargo handling at ports. According to UNCTAD [
The operations of a container terminal include seaside operations, yard operations, and land-side operations [
The BAP can be categorized using temporal and spatial constraints [
In the discrete BAP, the quay is divided into a set of berths, each of which can harbor only one ship at a time. In the continuous BAP, the quay is not partitioned into definitive berths, and a vessel can occupy any arbitrary position on the quay. This improves utilization of quay space at the cost of greater computational complexity. In the hybrid BAP, the quay is partitioned into berths, but large ships may need multiple berths, while small ships require only one berth.
The DBAP is known to be NP-hard [
The remainder of the paper is organized as follows. Section
Several mixed integer programming (MIP) models for discrete DBAP have been proposed in the literature. Imai et al. [
Christensen and Holst [
Lai and Shih [
Nishimura et al. [
Mauri et al. [
Barros et al. [
A generic IG algorithm usually starts from an initial solution
Based on the framework of the generic IG algorithm, the following subsection further discusses the solution representation, the objective function calculation, and the main steps of the proposed IG algorithm.
A solution can be represented by a numerical sequence that consists of a permutation of
An example of the solution representation with 3-berth and 15-ship.
The completion time of each ship on an assigned berth is calculated according to its arrival time, the sequence in the berth, and the availability of the berth. The service time of each ship is obtained by subtracting its arrival time from its completion time. Finally, the total service time can be calculated by summing up the service times of all ships.
The main steps of the proposed IG algorithm are as follows.
The initial solution
Consider the following. Randomly choose Sequentially reinsert the ships of IF
ELSE_IF ELSE_IF
If the computational time exceeds a specified threshold, stop the algorithm.
In Step
Information of 3 berths.
|
|
|
---|---|---|
1 | 12 | 300 |
2 | 12 | 300 |
3 | 12 | 300 |
Information of 15 ships.
|
|
|
|
|
|
---|---|---|---|---|---|
1 | 71 | 300 | 20 | 20 | 40 |
2 | 90 | 300 | 44 | 44 | 88 |
3 | 39 | 300 | 22 | 22 | 44 |
4 | 17 | 300 | 34 | 34 | 68 |
5 | 12 | 300 | 12 | 12 | 24 |
6 | 117 | 300 | 30 | 30 | 60 |
7 | 94 | 300 | 28 | 28 | 56 |
8 | 29 | 300 | 6 | 6 | 12 |
9 | 43 | 300 | 26 | 26 | 52 |
10 | 79 | 300 | 22 | 22 | 44 |
11 | 2 | 300 | 20 | 20 | 40 |
12 | 129 | 300 | 16 | 16 | 32 |
13 | 123 | 300 | 26 | 26 | 52 |
14 | 43 | 300 | 14 | 14 | 28 |
15 | 5 | 300 | 18 | 18 | 36 |
An example of IG destruction phase and construction phase.
The computational complexity of the proposed IG algorithm is as follows. In Step
This section discusses the computational tests used to evaluate the performance of the proposed IG algorithm. The details of the test problems, parameters selection, and the computational results of the proposed IG algorithm are compared with those of the state-of-the-art algorithms, including
Three benchmark problem sets were used in this study. Cordeau et al. [
The proposed IG algorithm was implemented using the C language on the Windows XP operating system and run on a personal computer with an Intel Core 2 2.66 GHz CPU and 2 G RAM. Parameter selection may influence the quality of the results. One instance was randomly selected from each size in the I2 problem set and the new problem set, and three instances were randomly selected in the I3 problem set for preliminary testing. The following combinations of parameters were tested on these instances:
Tables
Computational result for I2 problem set.
Instance | GSPP | SARS | IG | |||
---|---|---|---|---|---|---|
Optimal | Time | Best objective | Time to obtain the optimal | Best objective | Time to obtain the optimal | |
|
|
5.99 |
|
0.04 |
|
0.01 |
|
|
3.70 |
|
0.16 |
|
0.08 |
|
|
2.95 |
|
0.63 |
|
0.10 |
|
|
2.72 |
|
0.10 |
|
0.03 |
|
|
6.97 |
|
0.32 |
|
0.38 |
|
|
3.10 |
|
0.01 |
|
0.01 |
|
|
2.31 |
|
0.01 |
|
0.00 |
|
|
1.92 |
|
0.03 |
|
0.03 |
|
|
4.76 |
|
0.07 |
|
0.20 |
|
|
6.38 |
|
0.59 |
|
0.20 |
|
||||||
|
|
3.62 |
|
0.00 |
|
0.01 |
|
|
3.15 |
|
0.03 |
|
0.00 |
|
|
4.28 |
|
0.20 |
|
0.56 |
|
|
3.78 |
|
0.59 |
|
0.14 |
|
|
3.85 |
|
0.02 |
|
0.19 |
|
|
3.60 |
|
0.01 |
|
0.02 |
|
|
3.54 |
|
0.21 |
|
0.03 |
|
|
3.93 |
|
0.07 |
|
0.05 |
|
|
3.73 |
|
0.02 |
|
0.00 |
|
|
3.82 |
|
0.02 |
|
0.01 |
|
||||||
|
|
5.83 |
|
0.04 |
|
0.02 |
|
|
6.99 |
|
0.15 |
|
0.01 |
|
|
6.12 |
|
0.24 |
|
0.16 |
|
|
5.38 |
|
0.19 |
|
0.20 |
|
|
6.77 |
|
0.10 |
|
0.06 |
|
|
5.57 |
|
0.01 |
|
0.04 |
|
|
5.83 |
|
0.00 |
|
0.00 |
|
|
5.87 |
|
0.01 |
|
0.01 |
|
|
5.38 |
|
0.15 |
|
0.01 |
|
|
5.96 |
|
0.04 |
|
0.09 |
|
||||||
|
|
12.57 |
|
11.59 |
|
0.37 |
|
|
15.93 |
|
9.07 |
|
1.35 |
|
|
7.16 |
|
3.81 |
|
0.47 |
|
|
13.59 |
|
1.65 |
|
0.47 |
|
|
11.50 |
|
2.25 |
|
1.26 |
|
|
29.16 |
|
8.31 |
|
2.02 |
|
|
12.89 |
|
1.40 |
|
0.41 |
|
|
17.52 |
|
4.95 |
|
0.34 |
|
|
8.41 |
|
0.59 |
|
0.25 |
|
|
14.39 |
|
7.30 |
|
0.80 |
|
||||||
|
|
19.98 |
|
0.19 |
|
0.30 |
|
|
11.37 |
|
4.47 |
|
0.87 |
|
|
8.97 |
|
0.13 |
|
0.34 |
|
|
10.28 |
|
5.63 |
|
0.50 |
|
|
22.31 |
|
0.52 |
|
0.27 |
|
|
10.92 |
|
0.29 |
|
0.14 |
|
|
9.74 |
|
0.21 |
|
0.77 |
|
|
9.39 |
|
0.08 |
|
0.06 |
|
|
29.45 |
|
0.90 |
|
0.53 |
|
|
14.28 |
|
0.05 |
|
0.08 |
|
||||||
Average | 953.7 | 8.60 | 953.7 | 1.35 | 953.7 | 0.28 |
Computational result for I3 problem set.
Instance | GSPP | PTA/LP | CS | SARS | IG | |||||
---|---|---|---|---|---|---|---|---|---|---|
Optimal | Time | Best objective | Time | Best objective | Time | Best objective | Time to obtain the optimal | Best objective | Time to obtain the optimal | |
i01 |
|
17.92 |
|
74.61 |
|
12.47 |
|
0.51 |
|
1.53 |
i02 |
|
15.77 |
|
60.75 |
|
12.59 |
|
0.05 |
|
0.11 |
i03 |
|
13.54 |
|
135.45 |
|
12.64 |
|
0.17 |
|
0.28 |
i04 |
|
14.48 |
|
110.17 |
|
12.59 |
|
0.09 |
|
0.32 |
i05 |
|
17.21 |
|
124.70 |
|
12.68 |
|
0.07 |
|
0.07 |
i06 |
|
13.85 |
|
78.34 |
|
12.56 |
|
0.00 |
|
0.00 |
i07 |
|
14.60 |
|
114.20 |
|
12.63 |
|
0.40 |
|
0.54 |
i08 |
|
14.21 |
|
57.06 |
|
12.57 |
|
0.29 |
|
0.71 |
i09 |
|
16.51 |
|
96.47 |
|
12.58 |
|
0.21 |
|
0.57 |
i10 |
|
14.16 |
|
99.41 |
|
12.61 |
|
0.11 |
|
0.18 |
i11 |
|
14.13 |
|
99.34 |
|
12.58 |
|
1.11 |
|
3.29 |
i12 |
|
15.60 |
|
80.69 |
|
12.56 |
|
1.49 |
|
3.90 |
i13 |
|
13.87 |
|
89.94 |
|
12.61 |
|
0.04 |
|
0.07 |
i14 |
|
15.60 |
|
73.95 |
|
12.67 |
|
0.05 |
|
0.09 |
i15 |
|
13.52 |
|
74.19 |
|
13.80 |
|
0.00 |
|
0.10 |
i16 |
|
13.68 |
|
170.36 |
|
14.46 |
|
1.86 |
|
2.89 |
i17 |
|
13.37 |
|
46.58 |
|
13.73 |
|
0.02 |
|
0.07 |
i18 |
|
13.51 |
|
84.02 |
|
12.72 |
|
0.00 |
|
0.10 |
i19 |
|
14.59 |
|
123.19 |
|
13.39 |
|
3.67 |
|
4.24 |
i20 |
|
16.64 |
|
82.30 |
|
12.82 |
|
1.00 |
|
6.37 |
i21 |
|
13.37 |
|
108.08 |
|
12.68 |
|
2.06 |
|
4.27 |
i22 |
|
15.24 |
|
105.38 |
|
12.62 |
|
0.50 |
|
1.18 |
i23 |
|
13.65 |
|
43.72 |
|
12.62 |
|
0.06 |
|
0.12 |
i24 |
|
15.58 |
|
78.91 |
|
12.64 |
|
0.07 |
|
0.39 |
i25 |
|
15.80 |
|
96.58 |
|
12.62 |
|
3.60 |
|
6.45 |
i26 |
|
15.38 |
|
101.11 |
|
12.62 |
|
0.45 |
|
1.33 |
i27 |
|
15.52 |
|
82.86 |
|
12.64 |
|
0.09 |
|
0.28 |
i28 |
|
16.22 | 1360 | 52.91 |
|
12.71 |
|
11.41 |
|
11.57 |
i29 |
|
15.30 |
|
203.36 |
|
12.62 |
|
1.07 |
|
5.25 |
i30 |
|
16.52 |
|
71.02 |
|
12.58 |
|
1.86 |
|
2.33 |
|
||||||||||
Avg. | 1306.8 | 14.98 | 1306.9 | 93.99 | 1306.8 | 12.79 | 1306.8 | 1.08 | 1306.8 | 1.95 |
Computational result for new problem set with known optimal solutions.
Size | Instance | GSPP | T2S* | T2S* + PR | SARS | IG | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Optimal |
|
Best objective |
|
Gap | Best objective |
|
Gap | Best objective | Average objective |
|
Gap | Best objective | Average objective |
|
Gap | ||
|
1 |
|
11.25 |
|
0.25 | 0.00 |
|
0.25 | 0.00 |
|
1763.00 | 0.04 | 0.00 |
|
1763.00 | 0.03 | 0.00 |
2 |
|
22.59 |
|
0.26 | 0.00 |
|
0.26 | 0.00 |
|
2090.00 | 0.03 | 0.00 |
|
2090.00 | 0.05 | 0.00 | |
3 |
|
13.68 |
|
0.26 | 0.00 |
|
0.26 | 0.00 |
|
2186.00 | 0.04 | 0.00 |
|
2186.00 | 0.06 | 0.00 | |
4 |
|
12.98 |
|
0.28 | 0.00 |
|
0.28 | 0.00 |
|
1538.00 | 0.03 | 0.00 |
|
1538.00 | 0.02 | 0.00 | |
5 |
|
16.12 |
|
0.22 | 0.00 |
|
0.22 | 0.00 |
|
2114.00 | 0.05 | 0.00 |
|
2114.00 | 0.01 | 0.00 | |
6 |
|
36.97 | 2187 | 0.34 | 0.09 |
|
0.34 | 0.00 |
|
2185.00 | 0.05 | 0.00 |
|
2185.00 | 0.09 | 0.00 | |
7 |
|
23.29 | 1847 | 0.34 | 0.11 |
|
0.34 | 0.00 |
|
1845.00 | 0.31 | 0.00 |
|
1845.60 | 0.89 | 0.00 | |
8 |
|
9.77 |
|
0.31 | 0.00 |
|
0.31 | 0.00 |
|
1271.00 | 0.14 | 0.00 |
|
1271.00 | 0.13 | 0.00 | |
9 |
|
28.25 |
|
0.25 | 0.00 |
|
0.25 | 0.00 |
|
1595.00 | 0.06 | 0.00 |
|
1595.00 | 0.09 | 0.00 | |
10 |
|
9.64 |
|
0.29 | 0.00 |
|
0.29 | 0.00 |
|
2195.00 | 0.05 | 0.00 |
|
2195.00 | 0.05 | 0.00 | |
|
|||||||||||||||||
|
1 |
|
17.93 |
|
0.47 | 0.00 |
|
0.47 | 0.00 |
|
1149.53 | 1.02 | 0.00 |
|
1149.27 | 0.37 | 0.00 |
2 |
|
30.59 | 1476 | 0.47 | 0.07 |
|
0.47 | 0.00 |
|
1475.53 | 1.08 | 0.00 |
|
1475.00 | 0.33 | 0.00 | |
3 |
|
26.43 |
|
0.50 | 0.00 |
|
0.50 | 0.00 |
|
1542.00 | 0.12 | 0.00 |
|
1542.00 | 0.07 | 0.00 | |
4 |
|
15.72 |
|
0.48 | 0.00 |
|
0.48 | 0.00 |
|
1075.00 | 0.14 | 0.00 |
|
1075.00 | 0.12 | 0.00 | |
5 |
|
23.60 |
|
0.40 | 0.00 |
|
0.40 | 0.00 |
|
1463.00 | 0.43 | 0.00 |
|
1463.00 | 0.03 | 0.00 | |
6 |
|
30.84 | 1581 | 0.48 | 0.06 |
|
0.48 | 0.00 |
|
1580.00 | 0.22 | 0.00 |
|
1580.00 | 0.19 | 0.00 | |
7 |
|
19.39 |
|
0.46 | 0.00 |
|
0.46 | 0.00 |
|
1276.00 | 0.25 | 0.00 |
|
1276.00 | 0.15 | 0.00 | |
8 |
|
22.06 |
|
0.42 | 0.00 |
|
0.42 | 0.00 |
|
870.00 | 0.47 | 0.00 |
|
870.00 | 0.16 | 0.00 | |
9 |
|
22.31 | 1153 | 0.50 | 1.68 |
|
0.50 | 0.00 |
|
1136.80 | 0.90 | 0.00 |
|
1134.00 | 0.26 | 0.00 | |
10 |
|
18.80 |
|
0.46 | 0.00 |
|
0.46 | 0.00 |
|
1527.00 | 0.10 | 0.00 |
|
1527.00 | 0.07 | 0.00 | |
|
|||||||||||||||||
|
1 |
|
41.51 | 2307 | 0.90 | 0.26 | 2303 | 0.90 | 0.09 |
|
2302.87 | 1.09 | 0.00 |
|
2301.27 | 1.01 | 0.00 |
2 |
|
59.89 | 2835 | 1.09 | 0.21 | 2834 | 1.09 | 0.18 |
|
2829.00 | 0.40 | 0.00 |
|
2829.27 | 0.97 | 0.00 | |
3 |
|
99.20 |
|
0.50 | 0.00 |
|
0.50 | 0.00 |
|
2880.67 | 1.75 | 0.00 |
|
2880.17 | 1.33 | 0.00 | |
4 |
|
39.78 |
|
0.84 | 0.00 |
|
0.84 | 0.00 |
|
2001.03 | 0.64 | 0.00 |
|
2002.33 | 1.24 | 0.00 | |
5 |
|
74.14 |
|
0.76 | 0.00 |
|
0.76 | 0.00 |
|
2816.03 | 1.00 | 0.00 |
|
2820.63 | 1.59 | 0.00 | |
6 |
|
66.46 |
|
0.87 | 0.00 |
|
0.87 | 0.00 |
|
2934.00 | 0.22 | 0.00 |
|
2934.00 | 0.20 | 0.00 | |
7 |
|
40.97 |
|
0.79 | 0.00 |
|
0.79 | 0.00 |
|
2632.07 | 0.64 | 0.00 |
|
2632.00 | 0.48 | 0.00 | |
8 |
|
40.11 | 1836 | 1.28 | 0.05 |
|
1.28 | 0.00 |
|
1835.40 | 1.18 | 0.00 |
|
1835.00 | 0.93 | 0.00 | |
9 |
|
47.70 | 2095 | 1.07 | 0.43 | 2089 | 1.07 | 0.14 |
|
2089.57 | 1.54 | 0.00 |
|
2089.37 | 1.22 | 0.00 | |
10 |
|
52.28 | 2964 | 1.06 | 0.07 |
|
1.06 | 0.00 |
|
2962.03 | 0.49 | 0.00 |
|
2962.00 | 0.20 | 0.00 |
Computational result for new problem set with unknown optimal solution.
Size | Instance | BKS | T2S* | T2S* + PR | SARS | IG | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Best objective |
|
Gap | Best objective |
|
Gap | Best objective | Average objective |
|
Gap | Best objective | Average objective |
|
Gap | |||
|
1 |
|
1467 | 1.20 | 0.62 | 1460 | 1.11 | 0.14 |
|
1464.13 | 1.79 | 0.00 |
|
1458.13 | 1.33 | 0.00 |
2 |
|
1381 | 1.01 | 0.44 |
|
1.32 | 0.00 |
|
1376.87 | 1.45 | 0.00 |
|
1375.00 | 0.56 | 0.00 | |
3 |
|
|
0.84 | 0.00 |
|
1.17 | 0.00 |
|
2128.07 | 1.98 | 0.00 |
|
2119.00 | 0.56 | 0.00 | |
4 |
|
1600 | 1.18 | 0.57 | 1597 | 1.78 | 0.38 |
|
1597.23 | 1.52 | 0.00 |
|
1592.37 | 0.36 | 0.00 | |
5 |
|
1849 | 1.11 | 0.11 |
|
1.45 | 0.00 |
|
1848.53 | 1.68 | 0.00 |
|
1847.13 | 1.04 | 0.00 | |
6 |
|
|
0.86 | 0.00 |
|
1.37 | 0.00 |
|
2080.00 | 0.21 | 0.00 |
|
2080.00 | 0.11 | 0.00 | |
7 |
|
1845 | 1.25 | 0.22 |
|
1.56 | 0.00 |
|
1841.80 | 1.36 | 0.00 |
|
1841.00 | 0.65 | 0.00 | |
8 |
|
2026 | 1.18 | 0.05 | 2026 | 1.70 | 0.05 |
|
2025.87 | 0.57 | 0.00 |
|
2025.63 | 0.49 | 0.00 | |
9 |
|
1888 | 1.06 | 0.43 |
|
1.48 | 0.00 |
|
1880.47 | 0.78 | 0.00 |
|
1880.00 | 0.25 | 0.00 | |
10 |
|
1905 | 0.71 | 1.17 | 1892 | 1.59 | 0.48 | 1884 | 1889.27 | 0.99 | 0.05 |
|
1883.10 | 0.26 | 0.00 | |
|
||||||||||||||||
|
1 |
|
4693 | 1.65 | 0.09 |
|
2.82 | 0.00 |
|
4689.13 | 1.02 | 0.00 |
|
4689.07 | 1.21 | 0.00 |
2 |
|
5483 | 1.37 | 0.29 |
|
2.81 | 0.00 |
|
5467.13 | 0.98 | 0.00 |
|
5472.53 | 2.01 | 0.00 | |
3 |
|
|
1.92 | 0.00 |
|
2.67 | 0.00 |
|
5499.00 | 0.37 | 0.00 |
|
5499.00 | 0.23 | 0.00 | |
4 |
|
4189 | 1.76 | 0.58 | 4179 | 3.65 | 0.34 |
|
4170.37 | 1.32 | 0.00 |
|
4171.80 | 1.79 | 0.00 | |
5 |
|
5484 | 1.39 | 0.11 |
|
2.73 | 0.00 |
|
5478.00 | 0.67 | 0.00 |
|
5478.07 | 0.80 | 0.00 | |
6 |
|
5599 | 1.45 | 0.07 |
|
2.56 | 0.00 |
|
5595.27 | 0.90 | 0.00 |
|
5595.00 | 0.22 | 0.00 | |
7 |
|
4902 | 1.90 | 0.66 | 4882 | 3.82 | 0.25 |
|
4878.47 | 1.34 | 0.00 |
|
4877.57 | 1.32 | 0.00 | |
8 |
|
3565 | 1.54 | 0.37 |
|
2.79 | 0.00 |
|
3552.50 | 1.25 | 0.00 |
|
3562.40 | 2.40 | 0.00 | |
9 |
|
4277 | 1.67 | 0.09 | 4275 | 2.75 | 0.05 |
|
4276.20 | 2.46 | 0.00 |
|
4273.47 | 1.41 | 0.00 | |
10 |
|
5739 | 1.85 | 0.00 |
|
2.65 | 0.00 |
|
5739.00 | 0.40 | 0.00 |
|
5739.00 | 0.22 | 0.00 | |
|
||||||||||||||||
|
1 |
|
|
3.60 | 0.00 |
|
4.78 | 0.00 |
|
2853.73 | 2.56 | 0.00 |
|
2846.97 | 2.09 | 0.00 |
2 |
|
2887 | 3.03 | 0.14 |
|
4.94 | 0.00 |
|
2888.63 | 2.70 | 0.00 |
|
2883.13 | 1.63 | 0.00 | |
3 |
|
3840 | 4.31 | 0.39 | 3833 | 5.57 | 0.21 | 3831 | 3837.07 | 2.57 | 0.16 |
|
3829.70 | 1.33 | 0.00 | |
4 |
|
2977 | 2.51 | 0.81 | 2971 | 4.31 | 0.61 | 2953 | 2966.73 | 1.64 | 0.07 |
|
2955.50 | 0.70 | 0.00 | |
5 |
|
3803 | 3.75 | 0.16 | 3801 | 3.56 | 0.11 |
|
3799.73 | 1.49 | 0.00 |
|
3797.40 | 0.44 | 0.00 | |
6 |
|
|
2.59 | 0.00 |
|
3.70 | 0.00 |
|
3783.00 | 0.68 | 0.00 |
|
3783.00 | 0.72 | 0.00 | |
7 |
|
|
2.43 | 0.00 |
|
3.84 | 0.00 |
|
3774.00 | 1.10 | 0.00 |
|
3774.00 | 0.63 | 0.00 | |
8 |
|
3864 | 2.09 | 0.05 | 3863 | 3.95 | 0.03 |
|
3864.17 | 2.18 | 0.00 |
|
3862.93 | 1.85 | 0.00 | |
9 |
|
3597 | 3.03 | 0.17 |
|
5.26 | 0.00 |
|
3592.93 | 2.39 | 0.00 |
|
3591.93 | 2.27 | 0.00 | |
10 |
|
3658 | 2.61 | 0.97 | 3635 | 4.73 | 0.33 | 3630 | 3638.07 | 2.33 | 0.19 |
|
3633.60 | 1.42 | 0.00 | |
|
||||||||||||||||
|
1 |
|
2745 | 8.09 | 0.11 | 2745 | 7.31 | 0.11 | 2744 | 2747.70 | 2.50 | 0.07 |
|
2744.90 | 1.70 | 0.00 |
2 |
|
2549 | 8.23 | 0.87 | 2534 | 6.10 | 0.28 |
|
2535.20 | 1.98 | 0.00 |
|
2528.90 | 1.06 | 0.00 | |
3 |
|
2545 | 6.20 | 0.04 | 2545 | 6.53 | 0.04 |
|
2549.57 | 2.38 | 0.00 |
|
2552.17 | 2.36 | 0.00 | |
4 |
|
|
7.00 | 0.00 |
|
5.59 | 0.00 |
|
3316.13 | 1.83 | 0.00 |
|
3321.23 | 2.63 | 0.00 | |
5 |
|
3147 | 7.66 | 1.22 | 3123 | 6.12 | 0.45 | 3111 | 3120.10 | 1.57 | 0.06 |
|
3109.20 | 0.24 | 0.00 | |
6 |
|
|
6.48 | 0.00 |
|
6.54 | 0.00 |
|
2283.07 | 0.84 | 0.00 |
|
2283.00 | 0.38 | 0.00 | |
7 |
|
2146 | 5.04 | 0.09 | 2146 | 9.17 | 0.09 |
|
2150.67 | 2.70 | 0.00 |
|
2144.00 | 0.83 | 0.00 | |
8 |
|
2743 | 4.98 | 0.85 | 2726 | 5.18 | 0.22 |
|
2725.13 | 1.38 | 0.00 |
|
2720.03 | 0.73 | 0.00 | |
9 |
|
2162 | 6.50 | 0.60 | 2162 | 6.50 | 0.60 | 2152 | 2158.53 | 1.35 | 0.14 |
|
2154.13 | 0.42 | 0.00 | |
10 |
|
2815 | 5.45 | 0.04 | 2815 | 6.05 | 0.04 |
|
2814.57 | 1.08 | 0.00 |
|
2820.07 | 1.86 | 0.00 | |
|
||||||||||||||||
|
1 |
|
5761 | 1.99 | 0.14 |
|
3.12 | 0.00 |
|
5753.70 | 1.57 | 0.00 |
|
5753.00 | 0.96 | 0.00 |
2 |
|
|
2.67 | 0.00 |
|
3.20 | 0.00 |
|
6884.00 | 0.81 | 0.00 |
|
6884.00 | 0.42 | 0.00 | |
3 |
|
6782 | 2.17 | 0.03 |
|
4.25 | 0.00 |
|
6780.00 | 0.82 | 0.00 |
|
6780.00 | 0.74 | 0.00 | |
4 |
|
5105 | 2.30 | 0.26 | 5105 | 2.30 | 0.26 |
|
5101.87 | 1.75 | 0.00 |
|
5100.57 | 2.08 | 0.00 | |
5 |
|
|
2.47 | 0.00 |
|
3.18 | 0.00 |
|
6715.00 | 0.60 | 0.00 |
|
6715.00 | 0.66 | 0.00 | |
6 |
|
6618 | 2.45 | 0.03 |
|
3.53 | 0.00 |
|
6616.33 | 1.36 | 0.00 |
|
6616.00 | 0.50 | 0.00 | |
7 |
|
|
2.66 | 0.00 |
|
4.75 | 0.00 |
|
6011.00 | 0.60 | 0.00 |
|
6014.53 | 1.73 | 0.00 | |
8 |
|
4406 | 2.64 | 0.48 |
|
3.77 | 0.00 |
|
4385.00 | 0.54 | 0.00 |
|
4396.90 | 2.36 | 0.00 | |
9 |
|
|
2.17 | 0.00 |
|
3.99 | 0.00 |
|
5238.20 | 2.84 | 0.00 |
|
5235.73 | 1.93 | 0.00 | |
10 |
|
7281 | 2.22 | 0.36 | 7281 | 3.62 | 0.36 |
|
7255.00 | 0.12 | 0.00 |
|
7255.00 | 0.06 | 0.00 | |
|
||||||||||||||||
|
1 |
|
3724 | 4.40 | 0.46 | 3715 | 9.26 | 0.22 | 3709 | 3714.73 | 2.12 | 0.05 |
|
3707.77 | 0.74 | 0.00 |
2 |
|
4191 | 6.39 | 1.06 | 4172 | 6.70 | 0.60 | 4150 | 4166.30 | 1.64 | 0.07 |
|
4151.87 | 0.94 | 0.00 | |
3 |
|
4290 | 6.78 | 0.40 | 4281 | 5.90 | 0.19 | 4275 | 4282.30 | 2.14 | 0.05 |
|
4274.90 | 1.56 | 0.00 | |
4 |
|
3916 | 5.32 | 0.15 | 3916 | 7.15 | 0.15 | 3911 | 3914.00 | 1.68 | 0.03 |
|
3911.67 | 0.73 | 0.00 | |
5 |
|
4264 | 3.99 | 0.31 | 4261 | 6.23 | 0.24 |
|
4259.67 | 1.86 | 0.00 |
|
4261.93 | 2.20 | 0.00 | |
6 |
|
5731 | 6.56 | 0.07 | 5729 | 4.39 | 0.03 |
|
5729.07 | 1.87 | 0.00 |
|
5727.57 | 0.78 | 0.00 | |
7 |
|
3749 | 6.61 | 0.81 | 3743 | 8.28 | 0.65 | 3721 | 3739.57 | 1.71 | 0.05 |
|
3726.47 | 0.42 | 0.00 | |
8 |
|
4600 | 6.96 | 0.39 | 4586 | 6.96 | 0.09 |
|
4588.17 | 2.34 | 0.00 |
|
4584.97 | 1.49 | 0.00 | |
9 |
|
4011 | 5.39 | 0.80 | 4004 | 5.39 | 0.63 | 3985 | 3995.47 | 1.04 | 0.15 |
|
3985.07 | 0.47 | 0.00 | |
10 |
|
4125 | 5.66 | 0.44 | 4115 | 7.37 | 0.19 |
|
4116.83 | 2.60 | 0.00 |
|
4109.67 | 1.20 | 0.00 |
In Table
Table
In Table
Table
The running time of the IG algorithm depends on various factors, including CPU, operating system, compiler, computer program, and the precision used during execution. Therefore, the relative efficiency of the algorithms is hard to determine. Tables
To verify the effectiveness of the proposed IG algorithm, the proposed algorithm is compared with PTA/LP, CS, SARS,
At a confidence level of 95%, Tables
Paired
IG vs. | SARS |
---|---|
Difference | 0.0000 |
Degree of freedom | 49 |
|
0.0000 |
One-tailed significance | 0.5000 |
Paired
IG vs. | PTA/LP | CS | SARS |
---|---|---|---|
Difference | 2.5212 | 0.7791 | 0.0000 |
Degree of freedom | 29 | 29 | 29 |
|
13.2444 | 6.5597 | 0.0000 |
One-tailed significance | <0.0001 | <0.0001 | 0.5000 |
Paired
IG vs. | T2S* | T2S* + PR | SARS |
---|---|---|---|
Difference | 0.2403 | 0.0984 | 0.0128 |
Degree of freedom | 89 | 89 | 89 |
|
6.6505 | 5.3869 | 3.3017 |
One-tailed significance | <0.0001 | <0.0001 | 0.0007 |
This paper studies the berth allocation problem with dynamic arrival time. Because the berth allocation problem is NP-hard, exact solution approaches cannot optimally solve realistic large-scale problems while maintaining acceptable computational complexity. An IG algorithm is proposed as an alternative method to the problem. The proposed IG algorithm is tested using three benchmark problem sets and compared with the optimal solutions (or best known solutions) from the literature. Computational results indicate that the proposed IG algorithm is effective. The proposed IG algorithm obtains all the optimal solutions of the discrete DBAP instances for the first and the second problem sets, as well as exhibiting best-known solutions for 35 out of 90 test instances in the third problem set. Future research can further examine the integration of the berth allocation and quay crane assignment problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the the Ministry of Science and Technology, Taiwan and the Chang Gung Memorial Hospital for financially supporting this research’s Grants MOST103-2410-H-182-006 and CARPD3B0012, respectively.