AND OPTIMIZATION
3. Yard Allocation Modeling for Export Containers 1. Problem description1.Problem description
The turnaround time of vessels consists of loading and unloading time for containers. In order to reduce loading time, the storage locations for export containers should be selected for loading onto vessels efficiently.
However, rational locations for export containers are cohesively associated with effective yard planning. In this paper, the number of containers for each vessel was determined for each block. To minimize turnaround time and handling cost of vessels, the workloads among blocks were balanced for each vessel, and the total distance of container transportation between the storage blocks and vessel berthing locations was minimized. In this regard, yard cranes in blocks were simultaneously served for vessels, and the berthing time of vessels was related to the maximal processing time of yard cranes. In general, the workload balancing on each block for vessels could reduce the completion time of vessels, and eliminate the traffic jam of equipments. In other words, the transportation distance between storage blocks and berthing locations affected directly on the turnaround time of vessels. If the transportation distance was shorter, the turnaround time would be less.
3.2. Yard allocation modeling
Owing to the uncertainty information of vessel arriving at port, a decision- making strategy, which synthetically considers all arriving vessels in 4 days, is developed using rolling-horizon approach (Zhanget al., 2003). Meanwhile, a planning horizon of 4 days, each day divided into two 12-hour periods, is set. At the beginning of the first period, a storage space allocation plan is
day1 day2 day3 day4 day5 day6 day7
P1 P2 P4 P6 P8 P10 P12 P14
Decision-making cycle of planning period2
Decision-making cycle of planning period1
Fig. 1 Rolling-horizon strategy.
formed for the 8 periods within days 1–4. Only the plan of the first period is executed and a new 4-day plan is formed at the end of the first period.
The details are shown in Fig. 1.
(i) Assumptions
The yard allocation model for export containers is developed based on the following assumptions:
1. The berth, berthing time and departing time of all arriving vessels in decision-making cycle are known;
2. It can be predicted according to statistics that the number, type and weight distribution of containers loading/unloading;
3. The number of quay crane scheduled for every vessel is estimated;
4. It accords with historical statistics that the number of export containers into yard, the retrieved number of containers and the retrieved time.
(ii) Notation
TP The total number of planning periods in a decision-making cycle.
T P = 8 Planning period is denotedt. Only the plan of the first period is executed;
NA The total number of blocks in the yard;
P The currently decision-making cycle;
V Pt The set of all vessels needing yard planning in decision-making cycleP;
V Pjt Vesselj in periodt;
NVPti The set of vessels which have been allocated in blockibefore periodt;
Bjt The berthing place of vesselVPjt;
dij Distance between blockiand the berthing place of vesselVPjt in periodt;
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178 W. Yan, J. He and D. Chang
N2jt The number of 20-foot export containers of vesselVPjt in periodt;
N4jt The number of 40-foot export containers of vesselVPjt in periodt;
Ri The lanes of blocki; Ti The tiers of blocki;
OPLj The estimated number of quay crane scheduled for vessel VPjt in periodt Kjt. Up to periodt, the periods that export containers of vesselVPjt having arrived at container terminal;
STHit The set of start times for all vessels in blocki in periodt. STHit={STHit1,STHit2, . . . ,STHitn};
ETHit The set of end times for all vessels in blockiin periodt. ETHit={ETHit1,ETHit2, . . . ,ETHitn};
STtj The loading start time of vesselVPjt in periodt; ETtj The loading end time of vesselVPjt in periodt;
N2ijtk The total number of 20-foot export containers allocated in blockithat arrive at the container terminal in period t−k;
N4ijtk The total number of 20-foot export containers allocated in blockithat arrive at the container terminal in period t−k;
NUi(t−1) The number of empty bay at the end of periodt−1;
NUBi(t−1) The number of empty 40-foot bay (Two adjacent bays) at the end of periodt−1;
λ Expansion coefficient.
(iii) Decision Variables
Aijt =
1, Blockiis allocated for vesselV Pjt
in period t
0, Blockiis not allocated for vesselV Pjt
in period t
Hijtk =
1, Blockiis allocated for vesselV Pjt
in period t−k
0, Blockiis not allocated for vesselV Pjt
in period t−k
N2ijt–The total number of 20-foot export containers allocated in block ithat arriving at the container terminal in periodt;
N4ijt–The total number of 20-foot export containers allocated in block ithat arriving at the container terminal in periodt.
(iv) Mathematical Models
The decision-making objectives are presented as follows.
1. Minimizing the total distance of all vessels to transport the containers between their storage blocks and the vessel berthing locations;
2. Balancing the workload among blocks allocated vesselV Pjt in periodt; 3. Balancing the workload among all blocks.
The Mathematical models are presented as follows.
f1= Min T P t=1
j∈V Pt
NA i=1
(N2ijt+N4ijt)·Aijt·dijt, (1)
Equation 1 is the first objective to minimize the total distance of all vessels to transport the containers between their storage blocks and the vessel berthing locations, which synthetically considers all arriving vessels in 4 days.
f2= Min
Max
{i}
Kjt
k=1
(N2ijtk+ 2·N4ijtk)·Hijtk
+
8−Kjt
t=1
(N2ijt+ 2·N4ijt)·Aijt
− Min
{i}
Kjt
k=1
(N2ijtk+ 2·N4ijtk)·Hijtk
+
Kjt
k=1
(N2ijtk+ 2·N4ijtk)·Hijtk
, (2)
Equation 2 is the second objective to balance the workload among blocks allocated for vesselV Pjt, which minimizes the margin between the block with the maximum export containers ofV Pjtand the block with the
June 14, 2011 11:46 9in x 6in Advances in Maritime Logistics & Supply . . . b1185-ch08
180 W. Yan, J. He and D. Chang
minimum export containers ofV Pjt.
f3= Min
T P
t=1
Max{i}
j∈V Pt
(N2ijt+ 2·N4ijt)·Aijt
−Min
{i}
j∈V Pt
(N2ijt+ 2·N4ijt)·Aijt
, (3) Equation 3 is the third objective to balance the workload among all blocks, which minimizes the margin between the block with the maximum workloads and the block with the minimum workloads. This is used to avoid the traffic jam.
Min{ω1∗f1, ω2∗f2, ω3∗f3}, (4) Equation 4 is a multi-objective function formed via Equations 1, 2 and 3.ω1,ω2 andω3 are respectively the weights of Equations 1, 2 and 3.
t∈T P, j∈V Pt, (5)
Equation 5 is a constraint to ensure that the planning period is in the decision-making cycle and the vessel is needed for yard planning in periodt.
N2jt= NA i=1
A∗ijtN2ijt, (6) Equation 6 is a constraint to ensure that the total number of 20- foot export containers allocated in blocks i is the sum of these containers assigned to all the blocks in period t.
N4jt= NA i=1
A∗ijtN4ijt, (7) Equation 7 is a constraint ensure that the total number of 40-foot export containers allocated in blocks i is the sum of these containers assigned to all the blocks in period t.
Kjt
k=1
NA i=1
Hijtk+
8−Kjt
t=1
NA i=1
Aijt = 2·OPLjt, (8)
Equation 8 is a constraint to ensure that the total number of blocks allocated to any vessel is two times that of quay cranes scheduled.
∀t= (1,2, . . . ,8),
λ·
j∈V Pt
N2jt+ 2·
j∈V Pt
N4jt
(9)
≤NA
i=1
Aijt·N Ui(t−1)·[Ri·Ti−(Ti−1)],
Equation 9 is a constraint to ensure that the allowable stacks of all blocks will not be less than the total number of export containers of all vessels in period t. If ∃j ∈ V Pt and ALijt = 1, the constraint can be ensured. The constraint is only used for the 20-foot export containers.
As well, a special constraint for the 40-foot export containers should be given, because the 40-foot containers should be stored in two adjacent bays.
∀t= (1,2, . . . ,8), λ·
j∈V Pt
N4jt
≤ NA i=1
Aijt·NUBi(t−1)·[Ri·Ti−(Ti−1)], (10)
Equation 10 is a constraint to ensure that the allowable stacks of all blocks will not be less than the total number of 40-foot export containers of all vessels in periodt.
∀m∈NVPti, Aijt·[ST Hmt−ETjt]·[ET Hmt−STjt]>0. (11) Equation 11 is a constraint to ensure that the handling time of all vessels, for which containers stored in the same block, will not be overlapped.