Introduction
I: DC J: Depot
4.5 Case Study
Based on the real data obtained from the port of Shanghai container terminal, the model proposed is used to optimize the simultaneous assignment of berths and cranes to the incoming container vessels.
The problem has previously been demonstrated by Liang et al. (2009).
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Note that, as the same dataset has later been studied by Han et al.
(2010) and Liang et al. (2011), the real case problem might be used as a benchmark. The arrival time, the total number of containers in TEU, and due dates for each vessel are given in Table 4.1.
We represent a 24 h day by 24 equal time intervals and convert all the times in Table 4.1 accordingly. Figure 4.2 illustrates the time scale used for modeling the problem. The same scaling is used for each day.
The berth structure is discrete, and the whole quay area is partitioned into four berths. Since berth lengths are not indicated in the benchmark problem, physical length restrictions are not reflected. There are seven quay cranes, with a handling rate equal to 40 TEUs/h. Due to the lack of available accurate data, the cranes are taken as identical in terms of their handling rates. That, in fact, is a generalization of our model struc- ture, as we allow for variable quay crane handling rate specification.
With more realistic crane specifications, our model can be used much more efficiently. The maximum allowable number of cranes assigned to a vessel is 4. In order to show the impact of portable and static cranes, crane ids 6 and 7 are assumed to be portable, that is, move among the berths, while five of the seven cranes are assumed to be static.
Table 4.1 Input for the Computational Study
SHIP
NAME ARRIVAL
TIME ARRIVAL TIME
(IMPLEMENTED) DUE TIME
TOTAL NUMBER OF LOADING/UNLOADING
CONTAINER (TEU)
1 MSG 9:00 10 20:00 428
2 NTD 9:00 10 21:00 455
3 CG 0:30 2 13:00 259
4 NT 21:00 22 23:50 172
5 LZ 0:30 2 23:50 684
6 XY 8:30 10 21:00 356
7 LZI 7:00 8 20:30 435
8 GC 11:30 13 23:50 350
9 LP 21:30 23 23:50 150
10 LYQ 22:00 23 23:50 150
11 CCG 9:00 10 23:50 333
(00:00–00:59) (01:00–01:59) (02:00–02:59) (22:00–22:59) (23:00–23:59)
t = 1 t = 2 t = 3 ... t = 23 t = 24
Figure 4.2 Time implementation frame.
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The model is coded in GAMS 22.5 and solved with GUROBI solver for solving integer problems. The preliminary computational experimentation is conducted on NEOS server in January 2012 (Gropp and More, 1997; Czyzyk et al. 1998; Dolan, 2001). The imple- mented model has 8058 constraints and 3950 variables of which 3749 of them are discrete. The execution of the solver for each instance is reported to have less than 1 CPU s. However, the observed real time is between 5 min (for corner points) and 2 h (for points lying in the center of the Pareto frontier).
The summary results of Pareto solutions are provided in Table 4.2, whereas Figure 4.3 illustrates the optimum Pareto efficient frontier.
Table 4.2 Summary of the Solutions
SOLUTION ID
F1: TOTAL SERVICE TIME (H)
F2: TOTAL NUMBER OF CRANE SETUP
NONDOMINATED SOLUTION (✓ IF NONDOMINATED)
1 39 42 𝟀
2 39 41 ✓
3 40 38 𝟀
4 40 37 𝟀
5 40 36 ✓
6 41 35 ✓
7 42 34 𝟀
8 42 33 ✓
9 43 32 ✓
10 44 31 ✓
11 45 30 ✓
12 46 29 ✓
13 47 28 ✓
14 48 27 ✓
15 50 26 ✓
16 52 25 ✓
17 53 24 ✓
18 56 23 ✓
19 59 22 ✓
20 63 21 ✓
21 68 20 ✓
22 72 19 ✓
23 80 18 ✓
24 89 18 𝟀
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bi- ob jeC tiVe berth – Cr ane allo Cation
Note that not all of our computed solutions contribute to the Pareto efficient frontier, since some of our solutions are dominated by the others. As in the case of solutions #3 and #4, both with total service time equal to 40 min, while total number of crane setups are 38 and 39 respectively, are dominated by solution #5 with exactly same total service time but with less total number of crane setups.
Finally, we obtain 19 nondominated solutions out of 24 solutions to form the Pareto efficient frontier. In the study by Liang et al. (2009), the Pareto efficient frontier that is provided has seven solutions. With 19 nondominated solutions, we have further developed the decision support tool by offering an extended number of alternatives to the decision maker.
The second solution in Table 4.2 gives the Pareto optimal solution, which minimizes the service time of the vessels. The relative compu- tational results for the solution are given in Figure 4.4 and Table 4.3.
The service time is the difference between the departure time and the arrival time of a vessel. The waiting time is defined as the time
35 (39,41)40
38
(40,36) (41,35)
34 32 30 28 26 24 22 20
41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 (42,33)
(43,32) (44,31)
(45,30) (46,29)
(47,28) (48,27)
(50,26) (52,25)
(53,24) (56,23)
(59,22) (63,21)
(68,20)
(72,19) (80,18) Service time
Number of crane setups
Figure 4.3 Pareto efficient frontier.
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a vessel spends in the bay before being berthed; that is, the berthing time minus the arrival time. Handling time is the time vessel spends at the port. Delay is the due date minus departure time of the vessel.
Note that time scale is converted into minutes for benchmark pur- poses with earlier studies in literature. In this solution, total service time is 2165, handling time is 1555, waiting time is 610, and delay time is 0. The number of crane setups is 40.
The last nondominated solution in Table 4.2 (solution 23) gives the Pareto optimal solution, which minimizes the quay crane setups. The relative computational results, presented in a similar fashion for this solution, are given in Figure 4.5 and Table 4.4. In solution 23, total service time is 4396, handling time is 3880, waiting time is 516, and delay time is 0. The number of crane setups is 18. Note that this
08:30 09:00
04:47
Berth 1 Berth 2 Berth 3 Berth 4
02:40 23:59
22:05 Ship 4 (1)(2) (3)(7)
Ship 10
(1)(2)(3)(7) Ship 9 crane (4)(5)(6)
Ship 2 cranes:
(4)(5)(6)(7)
Ship 6 cranes:
(4)(5)(6)(7)
Ship 7 cranes:
(4)(5)(6)(7)
Ship 5 cranes:
(1)(2)(3)(7) Ship 11 cranes:
(1) (2) (3) Ship 8 cranes:
(1) (2) (3) Ship 1 cranes:
(4)(5)(6)(7)
Ship 3 cranes:
(4)(5)(6) 21:00
14.42
17:29
23:02
22:45 21:30
14:38
11:57
07:00 09:43 22:00
20:00 18:00 16:00 14:00 12:00 11:47 10:00 08:00 06:00 04:00 02:00 00:30 00:00
Figure 4.4 The Gantt chart of solution 2.
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Table 4.3 Decomposition of the Objectives for Solution 2
SHIP
NAME ASSIGNED BERTH
WAITING TIME [1]
(MIN)
HANDLING TIME [2]
(MIN)
SERVICE TIME ([1]
+ [2])
TOTAL DELAY (MIN)
NUMBER OF CRANE
SETUPS
1 MSG 2 177 161 338 0 4
2 NTD 2 338 171 509 0 4
3 CG 2 0 130 130 0 3
4 NT 2 0 65 65 0 4
5 LZ 1 0 257 257 0 4
6 XY 3 73 134 207 0 4
7 LZI 4 0 163 163 0 4
8 GC 1 17 175 250 0 3
9 LP 4 0 75 75 0 3
10 LYQ 3 5 57 62 0 4
11 CCG 1 0 167 167 0 3
Total 610 1555 2165 0 40
06:59
00:30
Berth 1 Berth 2 Berth 3 Berth 4
23.59 23:13
18:54
23:09 23:23
Ship 11 crane:
(7) Ship 9
cranes: (5) (6) Ship 10
cranes: (1) (2) Ship 4 cranes: (3) (4)
Ship 6 crane: (4) Ship 7
crane: (3) Ship 2
cranes:
(6) (7)
Ship 5 cranes:
(1) (2) Ship 3
crane: (4) Ship 1
cranes: (1) (2) Ship 8 cranes: (1) (2)
Crane: (1)
21:00 17:53
21:00 17:24
23:02
14:42
08:30 22.00
20.00 18:00 16:00 14:00 14:21
12:00
09:03 10:00 08:00 06:00 04:00 02:00 00:00
Figure 4.5 The Gantt chart of solution 23.
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solution belongs to an extreme point in the optimal Pareto efficient frontier. It is, therefore, foreseeable to have service time increased by a large extent, though it is interesting to see that the solution still does not cause any delays with respect to the due date given.
To provide an additional example, in Figure 4.6 and Table 4.5, computational results for solution 16 are reported. Here, total service time is 2842, handling time is 2578, waiting time is 264, and delay time is 0. The number of crane setups is 25.
No delay times have been encountered for the given solutions. The minimum total service time found in Liang et al. (2009) is reported as 2165 min for all vessels. Han et al. (2010) have demonstrated the minimum total service time as approximately 36 h for the case where the maximum allowable number of cranes for a vessel is set to 4. Our solution with 2165 min for the Pareto optimal solution, which mini- mizes the service time of the vessels, is equal to the value found by Liang et al. (2009). Consequently, our study proves the optimality of this solution by implementing an exact algorithm approach.
In both studies by Liang et al. (2009) and Han et al. (2010), crane movements are defined for movements among berths. In our formula- tion, however, we also take into account the movement among vessels, as the setup cost of a crane to serve a vessel is not neglected (LALB Harbor Safety Committee, 2012). Moreover, in their formulation,
Table 4.4 Decomposition of the Objectives for Solution 23
SHIP
NAME ASSIGNED BERTH
WAITING TIME [1]
(MIN)
HANDLING TIME [2]
(MIN)
SERVICE TIME ([1]
+ [2])
TOTAL DELAY (MIN)
NUMBER OF CRANE
SETUPS
1 MSG 1 3 321 324 0 2
2 NTD 4 0 342 342 0 2
3 CG 3 0 389 389 0 1
4 NT 2 0 129 129 0 2
5 LZ 1 0 513 513 0 2
6 XY 3 0 534 534 0 1
7 LZI 2 0 653 653 0 1
8 GC 1 171 273 444 0 2
9 LP 3 0 113 113 0 2
10 LYQ 1 0 113 113 0 2
11 CCG 4 342 500 842 0 1
Total 516 3880 4396 0 18
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bi- ob jeC tiVe berth – Cr ane allo Cation
22:00
23:59 23:23
23:09 23:23
22:18
21:00 21:00
14:38
13:40
15:53
11:30 11:28
08:30 21:30
Ship 10
cranes: (1) (2) Ship 4 cranes: (3) (4)
Ship 9 cranes: (5) (6)
Ship 5 cranes (1) (2) (3)
Ship 1 cranes: (3) (7)
Ship 2 cranes:
(1) (2)
Ship 8 cranes: (5) (6) Ship 11
cranes: (3) (4)
Ship 3 cranes (4)(5)
Ship 6 cranes:
(4) (6) (7) Ship 1
crane: (7)
Cranes: (1)(2)
Crane: (4) (1)(2)(3)
Ship 7 18:05
10:56
06:12 07:00
03:45
03:30 09:00
20:00 18:00 16:00 14:00 12:00 10:00 08:00 06:00 04:00 02:00 00:00
Berth 1 Berth 2 Berth 3 Berth 4
Figure 4.6 The Gantt chart of solution 16.
Table 4.5 Decomposition of the Objectives for Solution 16
SHIP
NAME ASSIGNED BERTH
WAITING TIME [1]
(MIN)
HANDLING TIME [2]
(MIN)
SERVICE TIME ([1]
+ [2])
TOTAL DELAY (MIN)
NUMBER OF CRANE
SETUPS
1 MSG 1 148 387 535 0 2
2 NTD 2 116 342 458 0 2
3 CG 3 0 195 195 0 2
4 NT 2 0 129 129 0 2
5 LZ 1 0 342 342 0 3
6 XY 4 0 178 178 0 3
7 LZI 2 0 236 236 0 3
8 GC 4 0 263 263 0 2
9 LP 3 0 113 113 0 2
10 LYQ 1 0 113 113 0 2
11 CCG 3 0 280 280 0 2
Total 264 2578 2842 0 25
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cranes are assumed as identical, and the information as to which spe- cific crane is assigned to a vessel cannot be retrieved from the solution, and this decision is left to the decision maker. We, in turn, also sup- port the decision maker by specifying the crane identities.
From the numerical results, one can conclude that when quay crane setup costs are ignored, the total service time of vessels decreases. A decision maker might choose to prefer a solution closer to the left- hand side of the Pareto efficient frontier in Figure 4.3, if the setup costs are not so significant. On the other hand, in case of extreme setup costs of quay cranes, the decision maker is directed toward the solutions in the right-hand side. The Pareto efficient frontier in this case may be used as an efficient decision support tool for decision makers.