Container Yard Allocation under Uncertainty
Yue Wu
Southampton Business School, University of Southampton, Univesity Road, Southmapton, SO17 1BJ, U.K.
Keywords: Container Terminal, Yard Space Allocation, Stochastic Programming.
Abstract: This paper investigates allocation of space in storage yard to export containers under uncertain shipment
information. We define two types of stacks: one is called the dedicated stack and the other is called the shared
stack. Since containers meant for the same destination are assigned to dedicated stacks in the same block, no
re-handling is required for containers in dedicated stacks. However, containers in shared stacks have different
destinations; re-handling is required. We develop a two-stage stochastic recourse programming model for
determining an optimal storage strategy, called the dual-response storage strategy. The first-stage response,
regarding the allocation of containers to dedicated stacks, is made before accurate shipment information
becomes available. The second-stage response, regarding allocation of additional containers to shared stacks,
is taken after realization of stochasticity. Then, the unused spaces in the yard area can be released for other
purposes. Computational results are provided to demonstrate the effectiveness of the proposed dual-response
storage strategy obtained from the stochastic model.
1 INTRODUCTION
Containers were first used for international sea
transportation in the 1950s, and the proportion of
containerized items has been steadily increasing since
then. Today, over 60% of the world deep sea cargo is
transported in containers, where some routes,
especially between economically strong and stable
countries, are containerized up to 100% (Steenken et
al., 2004). Containers are standardized steel boxes in
three lengths, 20, 40 and 45 feet, and 8 feet wide and
either 8.5 or 9 feet high. This standardization offers
advantages of simplified discharging and loading of
containers, protection against weather and pilferage
and improved process of scheduling and controlling
facilities, etc. Container terminals are places where
containerized cargo is temporarily stored, before
being shipped to the destination. The increased
volume of container shipments has resulted in
increased demand for seaport container terminals,
container logistics and management, and the related
technical equipment. Heightened competition
between seaports, especially between geographically
close ports, is a result of this development.
The container storage area in the terminal is
usually separated into rectangular regions, called
blocks, which are further segmented into rows, bays
and tiers. The width of a block is typically divided
into several rows, one for trucks that interact with
yard cranes and others for storing containers. Blocks
are divided along their length into bays. Each bay is
made up of several container stacks of a certain height
(3 ~ 6 tiers). Containers are stored one on top of
another to form a stack. The blocks are usually
separated into areas allocated for export, import,
special (such as reefer, dangerous, overweight/over
width), and empty containers (Steenken et al., 2004).
A vessel normally visits a sequential list of ports,
called the shipping route. A number of containers are
discharged from the vessel to the port terminals along
its shipping route. The locations occupied by these
import containers on the vessel become available for
loading new export containers from the terminals to
the vessel. Export containers are assigned to specific
locations on the vessel such that they can be easily
discharged when the vessel arrives at the ports where
they are to be discharged. However, for container
terminals, decisions are different from the vessels.
Containers pass through a terminal in three ways:
imported, exported and transhipped. Import
containers arrive in batches, in vessels, and leave the
terminal by truck or rail, while export containers
normally come at the terminals one-by-one, by trucks,
in a random manner, and leave by vessels.
Transhipped containers arrive and leave terminals in
vessels. In practice, accurate shipment information,
30
Wu, Y.
Container Yard Allocation under Uncertainty.
DOI: 10.5220/0007253700300036
In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 30-36
ISBN: 978-989-758-352-0
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
particularly for export containers, is usually difficult
to obtain because of the uncertainty involved in the
process of delivering the containers to the yard (for
example, truck delay and urgent shipment
requirement, etc.). Yard managers are increasingly
challenged by limited yard capacities, and the
uncertain and dynamic information involved in the
decision-making process. Therefore, yard managers
need new methodologies to help them make better
decisions about allocation of yard space.
In this paper, we assume that yard managers can
obtain uncertain shipment information from their
customers regarding destinations and quantity of
containers to be shipped. The yard managers have to
determine the storage yard plan before accurate
shipment information is available. One of the
methods that the yard managers adopt in practice is to
place the containers heading for the same destination
in the same block. Therefore, the number of blocks is
equal to the number of destinations/ports, where the
containers are to be discharged. The advantages of
this strategy is that containers can be easily loaded
from the yard to the ship without re-handling.
However, some spaces may not be occupied at all
because of the uncertain shipment information. This
is particularly true when the possibility of high
shipment demand is low. In this paper, it is assumed
that the yard is divided into different blocks, and each
block has the same size/capacity. We conceptually
divide each block into two portions: a set of dedicated
stacks and a set of shared stacks. Containers in
dedicated stacks within the same block have the same
destination (i.e. they will go to the same port).
Containers in shared stacks have different
destinations. Therefore, rehandling or reshuffling
may be required in shared stacks. Rehandling
happens when containers placed on the top of the
required one have to be removed first. Rehandling is
one of the most unproductive operations in the yard
area. The workload at the terminals can be
significantly reduced if no or limited number of
rehandling occurs. However, containers assigned to
dedicated stacks can be loaded to the ship
sequentially, without the need of rehandling. It is
noted that dedicated and shared stacks are not divided
physically. In each block, there are two portions: one
is for dedicated stacks and the other is for shared
stacks. In addition, each block has a special stack to
be used for re-handling containers in shared stacks;
this stack can store no more than one container so that
other containers in shared stacks of this block can be
temporarily placed in the stack during the process of
re-handling.
Since only containers in the shared stacks require
re-handling, the number of containers that require re-
handling in each block is limited. Therefore,
managers reserve only one stack in each block for re-
handling. However, the traditional sharing strategy, in
which all containers are mixed up, may require more
than a stack in each block for re-handling, since re-
handling happens frequently. Sometimes, a stack in
each bay is reserved for re-handling in practice
because frequent movement within a block might
cause safety concerns.
Although the concept of separate dedicated and
shared portions has already been used in some
terminals, yard managers are increasingly facing the
challenge of determining the split between dedicated
and shared stacks under uncertain shipment
information. Steenken et al. (2004) state that the need
for optimization of container terminal operations has
become an important issue in recent years. In this
paper, we propose a dual-response storage strategy to
deal with uncertain shipment information for export
containers. At the first stage, before the accurate
shipment information is available, yard managers
need to make the first response by determining how
the dedicated stacks in each block should be allocated
for storage of containers. At the second stage, when
the uncertain shipment information is realized, yard
managers need to respond to the situation by
determining the size of the shared stacks in each block
to store extra containers. As a result, spaces still left
in the blocks will be free for use.
The main problem considered in this paper is to
determine the optimal size of spaces to be reserved
for the dedicated stacks, as well as the shared stacks,
such that the total operational cost can be minimized.
In order to obtain an optimal dual-response storage
strategy, we formulate a two-stage stochastic recourse
programming model. The rest of the paper is
organized as follows. Section 2 provides the literature
review on storage management at container terminals
and stochastic modelling for allocation problems at
container terminals. Section 3 provides notations and
definitions for modelling the storage problem.
Section 4 presents a two-stage stochastic model for
storage management under uncertainty. Section 5
shows computational results and analysis. The final
section gives the conclusions of this paper and
recommendations for future research.
2 LITERATURE REVIEW
Due to the growing importance of maritime
transportation, operations of sea container terminals
have received increasing attention from researchers
Container Yard Allocation under Uncertainty
31
(Froyland et al., 2008). Excellent review papers about
detailed descriptions and classification of operations
of container terminals are provided by Vis and De
Koster (2003), Steenken et al. (2004), and Stahblock
and Voß (2008). The problem to be discussed in this
paper belongs to the category of storage yard
management in the literature. Storage management
addresses the assignment of locations to containers,
which includes space allocated to containers moving
into and out of storage yards as well as
reshuffles/rehandles (Froyland et al., 2008). Chen
(1999) investigates yard operations in Taiwan, Hong
Kong, Korea and UK. Kim and Park (2003) study
storage space allocation for export containers. A
mixed-integer programming model is proposed for
efficient utilization of storage space and efficient
loading. Two heuristics approaches are designed for
solving this problem. Lee and Hsu (2007) propose an
integer programming model for the container pre-
marshalling problem with a vessel and a rail mounted
gantry crane. Jin et al. (2016) study the daily storage
yard manage problem arising in maritime container
terminals, which integrates the space allocation and
yard crane deployment decisions together with the
consideration of container traffic congestion in the
storage yard. Lin and Chiang (2017) investigate the
storage space allocation problem at a container
terminalgantry crane in Taiwan. A decision rule-
based heuristic is proposed. Extant literature on
storage management problems has mainly discussed
deterministic situations where all accurate
information is available at the time when decisions
are made. Unfortunately, storage planning based on
available (at the time of decision-making)
information seldom matches the real situation
because container delivery is a stochastic process that
cannot be exactly foreseen (Steenken et al., 2004).
Zhen et al. (2011) propose stochastic programming
models for managing container terminal operations.
A meta-heuristic approach is proposed to solve the
above problem in large-scale real environments. Zhen
(2013) exams sstorage allocation in transshipment
hubs under uncertainties and proposes a real-time
decision support system (DSS) which can act as an
ultimate solution for coping with uncertainties in yard
storage allocation process.
3 NOTATIONS AND
DEFINITIONS
3.1 Known Parameters
It is assumed that there are a total of N destinations,
indexed by n. Since the yard area is equally divided
into N blocks according to containers’ destination, n
also represents the index of blocks. Each block
consists of dedicated and shared stacks. Containers in
dedicated stacks (within the same stack) have the
same destination, while containers in shared stacks
may have different destinations. Parameters R, B, H
represent the maximum numbers of rows, bays and
tiers for any block. C represents the maximum
capacity of yard. Since each block has (H-1) spaces
for reshuffling, we have:
C = N*[R*B*H-(H-1)].
It is also assumed that the total capacity of the
storage yard is adequate to accommodate the total
quantity of export containers under any possible
scenario. If the total capacity of the yard is fully used,
further demand from customers will be either
rejected, or handled by other terminal operators. The
cost of handling these extra containers is not
considered in our model.
3.2 Stochastic Parameters
is a random vector, which represents the
stochastic nature of arrivals of export containers.
is a probability space. Let represent
the probability density function of , , for
all ,
.
is a random vector, where
represents the
random shipment demand in the nth destination
.
3.3 Decision Variables
Under uncertain shipment information, yard
managers have to decide how many containers should
be assigned to the dedicated stacks in each block.
Therefore, we have a set of decisions to be taken
without accurate information on the stochastic
demand. These decisions are called the first stage
decisions, which are represented by vector:
,
where
represents the quantity of containers to be
assigned to the dedicated stacks in the nth block
. When the full information is received on
realization of random vector , the second stage ac-
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
32
tions are taken, which are represented by vector:
,
where
represents the quantity of container to be
assigned to the shared stacks in the nth block
.
3.4 Costs
The unit cost of handling containers in the dedicated
and shared stacks plays an important role in the
model. We use the following parameters to represent
some crucial factors that have impact on handling
cost:
represents the average cost of assigning
and holding a container in the nth block. Note
that in some container terminals, the cost of
using a container location in a dedicated stack
can be bid by yard managers. It can also be
considered as the unit cost of storing containers.
Here, we write the coefficients
in a vector-
form as
.
represents the average cost of moving a
container in a dedicated stack in the nth block. The
movement includes lifting the container, putting it on
a trailer or an internal truck, and transporting it to the
quay side. This cost is calculated by yard managers
from the average movement cost for individual
containers. We write the coefficients
in a vector-
form as
.
represents the average cost of assigning and
holding a container in the shared stack. In some
container terminals, the cost of using a space in the
shared stack can be bid by yard managers. It can also
be regarded as the unit cost of storing a container in
the shared stack. One of the simplest methods to
estimate this parameter is to divide the annual cost of
assigning and holding space in the shared stacks by
the total number of all spaces used in the shared stacks
in a year.
represents the average cost of moving a
container in the shared stack. The movement includes
searching, re-handling, lifting of the container, and
then loading it on a trailer or internal truck, and
transporting it to the quay side. It is noted that the
average cost of moving a container in the shared stack
is significantly higher than the dedicated stack
because it involves searching and re-handling, which
does not occur in the dedicate stack. One of the
natural ways to estimate this parameter is to divide
the annual cost related to movements (such as
searching, re-handling, lifting, releasing and
transporting) of containers in the shared stacks by the
total number of containers stored in the shared stacks
per year
4 A TWO-STAGE STOCHASTIC
RECOURSE PROGRAMMING
MODEL FOR YARD STORAGE
ALLOCATION UNDER
UNCERTAINTY
4.1 A General Two-Stage Stochastic
Model for Uncertain Yard Storage
Allocation Problems
A two-stage stochastic recourse programming model
for determining a dual-response storage allocation
strategy is formulated as follows:
subject to
, integer for (1)
Where is the optimal solution of the
second stage problem:
In (1),
represents an appropriate
dimension. The first stage decisions (x
n
) are
independent of realization of the stochastic
variable. It means no matter what shipment demand
is realized (i.e. how containers will arrive), the first-
stage decisions remain the same. However, yard
managers can make different responses (the second
stage decision
) for any shipment situation that
might happen.
In (1), for vectors
and
, functions and
are defined in accordance with the following vector-
forms:
Here, we summarize explanations of the above
model as follows. In the objective function, a
T
x is the
total cost of assigning and holding containers in the
dedicated stacks.
represents the
Container Yard Allocation under Uncertainty
33
expectation of the overall cost caused by assigning,
holding and moving containers, plus the movement
cost in the dedicated stacks. The term
represents the expectation of the cost caused by
assigning and holding container space in the shared
stacks. The term,
, denotes the overall
expected cost of moving containers in the dedicated
stacks with the value of
being the
number of containers stored in the dedicated stacks of
the nth block. The term
,
is the cost of moving containers stored in the shared
stacks with the value of
being
the number of containers placed in the shared stacks
in the nth block.
Now, let us look at the constraints in (1). The
constraint
ensures that the quantum of
the dedicated stacks in each block does not exceed the
maximum capacity of the block, and the first stage
decision variables have to be non-negative integer.
The constraints
ensure that the total capacity of the yard is not
exceeded, and shipment demand has to be satisfied
for any scenario. It means all containers are assigned
to either the dedicated or shared stacks. The final
constraint is the non-negative and integer requirement
for all decision variables.
4.2 A Two-Stage Stochastic Model with
Finite Scenarios for Uncertain
Yard Storage Allocation Problems
In this subsection, we investigate a model (See (2)),
which is a simplified two-stage recourse model in (1)
for the uncertain storage problem. From the
definitions of
and in Section 3.4, we know that
and represent the average cost of moving a
container from the dedicated stacks and shared stack
to vessels at the quay side. It is natural to assume that
for planning purposes, both
and are dependent on
the basis of the overall capacity rather than the space
used for each individual container. In real-world
situations, both the labour costs (salary) and
equipment costs (for example, the cost of purchasing
and maintaining a vehicle) are almost fixed, even in
the case, where no service is performed in the yard
area. Compared to the cost of assigning and holding a
container, other costs related to the movement (for
example, fuel or gas used for lifting, trucking etc.) are
also relatively small. Therefore, we only focus on the
cost of assigning and holding in the yard area
(including the dedicated and shared stacks) in Model
(1).
In addition, we notice that one of the difficulties in
solving the two stage stochastic programming model
(See (1)) is the continuity of the scenario set Ω. There
are three reasons for the difficulties caused: 1) It is
almost impossible to obtain a continuous distribution
of the uncertainty in a real-world process. However,
in most of cases, a set of finite scenarios and
approximate discretized distribution functions can be
easily obtained from historical data; 2) In a real-world
situation, the number of containers is normally finite,
and hence it is reasonable to use a finite scenario
model to capture uncertain situations that might
happen in the future. 3) Even for the case where the
number of scenarios is infinite and the distribution is
available, it is difficult to integrate the expectation of
the objective function, due to the complexity of the
distribution function. Therefore, in this subsection,
we proceed to analyse a variation of stochastic model
with a finite scenario set. It is assumed that a support
set Ω with finite number of scenarios, denoted by
, where
,
, and K is the maximum number of
scenarios. Note that this assumption holds true in the
real problem, where the number of containers arrived
is normally finite. We have the following notation
related to a finite set of scenarios:
k: Index different scenarios for demand. (
: Probability of scenario k.
: Realization of demand for containers in the
nth destination under scenario k.
: Number of containers in the shared stacks of
the nth block under scenario k.
We write the second stage decision variables
in vector-form, as
. Now, the two
stage stochastic optimisation for the storage
management problem in Model (1) can be
equivalently reformulated as the following algebraic
equivalent linear programming form:
subject to
, for
(2)
, for
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
34
and integer for
5 COMPUTATIONAL RESULTS
AND ANALYSIS
In order to illustrate the effectiveness of the proposed
two stage model in Section 4 for the uncertain storage
management problem of container terminals, we use
data provided by a container terminal in Hong Kong.
Located at the mouth of the Pearl River with a deep
natural harbour, Hong Kong is geographically and
strategically important as a gateway to China and
trans-shipment port for intra-Asian and world trade.
Hong Kong is the largest container port serving
southern China and one of the busiest ports in the
world. Consider export containers stacking for a
vessel visiting 10 ports for discharge of containers.
There are 10 blocks are reserved to hold the
containers in the yard area. Each block has 6 rows and
8 bays. Since the maximum height for stacking is 5
tiers, saving (5-1) = 4 free spaces for re-handling, the
maximum number of containers that can be
accommodated is C = 10*[6*8*5-4] = 2360. It is
assumed that the demand for containers is uncertain
for different ports (See Table 1). In general, there are
five different scenarios representing the trend for the
shipment demand in the future. The demand
quantities and likelihood of each scenario are
estimated by a yard storage planner (Table 1).
Table 1: Known data.
Scenario
Port
Likelihood
1
2
3
4
5
6
7
8
9
10
1
200
150
270
200
240
220
180
150
160
140
0.1
3
200
250
160
245
280
180
180
120
250
240
0.3
4
240
180
270
240
260
250
180
200
250
210
0.1
5
200
100
250
260
240
230
200
240
270
260
0.2
The data in Table 1 pertain to a real situation. For
simplicity, we assume the cost of assigning and
holding a container in the dedicated stacks is 1 unit,
while the cost in the shared stacks it is 3.5 units. We
run the two stage stochastic programming model in
(1) with optimization software Xpress IVE. The
results of yard space allocation are as shown in Table
2. The total cost of the dual-response storage plan is
2570.25 units. Table 2 gives the dual-response
storage strategy. The row “Dedicated Stack” indicates
the first-stage response, which is the predetermined
number of containers for dierent destinations to be
allocated to the dedicated stacks. The row “Shared
Stack” under “Scenario k” indicates the second-stage
response, which is the reactive number of containers
for dierent destinations to be allocated to the shared
stacks according to the real demand under scenario k.
From the results in Table 2, we can see that only 35
spaces in Block 2 and 10 spaces in Block 5 are
required for storing containers in the shared stacks if
Scenario 1 happens. The capacity for each block is
236. 235 spaces in Block 3 and 230 spaces in Block 5
are allocated to the dedicated stacks. Therefore, the
managers can use 35 spaces in Block 1, 5 spaces in
Block 4, and 5 spaces in Block 5 for storing extra
containers in the shared stacks. As a result, the spaces
that have not been allocated to either the dedicated or
shared stacks are free for other purposes. For
example, there are total 135 spaces left if Scenario 1
happens. By adopting the dual-response strategy, the
managers do not need to hold all spaces until the
containers are loaded into the ship. As soon as the
managers have full information about the shipment
demand, i.e. the stochastic demand is realized, the
managers can make the corresponding response by
deciding the size of the shared stacks and releasing
the spaces that will not be required, simultaneously.
Releasing the unused space is very important in
practice because the unused spaces make no profit,
which will potentially increase the total operations
cost of the terminal. This is particularly true for
terminals with limited yard space, like the Hong Kong
container terminal.
Table 2: Yard Allocation under Uncertainty.
It is noted that when the number of containers
cannot be divided by the number of tiers, which is five
Port
1
2
3
4
5
6
7
8
9
10
Dedicated
Stack
200
210
235
230
230
230
180
195
235
235
Shared
Stack
Scenario 1
0
0
35
0
10
0
0
0
0
0
Scenario 2
20
40
65
0
0
0
0
0
5
15
Scenario 3
0
40
0
15
50
0
0
0
15
10
Scenario 4
40
0
35
10
30
20
0
5
15
0
Scenario 5
0
0
15
25
10
0
20
45
35
25
Container Yard Allocation under Uncertainty
35
in this example, we need to move some containers
(less than five) from the dedicated stacks to the shared
stacks so that re-handling is not required in the
dedicated stacks. Whereas, if there are no dedicated
stacks allocated, all containers will be mixed in the
shared stacks. This is a traditional sharing strategy.
We can still use the model proposed in Section 4 to
obtain the solution for the traditional sharing strategy.
Since no container is assigned to the dedicated stacks,
, for . We run this model again
with optimization software Xpress IVE and obtain the
cost of using the traditional sharing strategy as
7320.25 units. Compared with this traditional sharing
strategy, the average savings in cost for the dual
response-strategy proposed in this paper is 64.89%.
6 CONCLUSIONS
This paper investigates storage yard allocation
problems for export containers under uncertainty. The
yard storage is physically divided into blocks and
each block is conceptually divided into dedicated and
shared stacks. The dedicated stacks in the same block
have the same destination/port. The shared stack has
mixed containers, destined for dierent ports. As a
result, no re-handling is required for containers stored
in the dedicated stacks but containers in the shared
stacks need re-handling. We propose the dual-
response storage policy to decide how containers are
allocated to the two dierent types of stacks under
uncertain shipment information. At the first stage,
when accurate shipment information is not available,
we need to decide how containers are to be allocated
to the dedicated stacks. At the second-stage, when the
uncertainty is realized, we need to respond to the
dierent possible shipment scenarios that might
happen. The decision at the second stage includes
determining how additional containers are allocated
to the shared stacks. As only a small number of
containers are allocated to the shared stacks, re-
handling is significantly reduced. In addition, we
develop the two-stage stochastic recourse
programming model to obtain the optimal dual-
response storage plan. The computational results
show the eectiveness of the two-stage stochastic
model for storage problems under uncertain shipment
information. Compared with the traditional sharing
strategy (in which all containers are mixed up) and
the non-sharing strategy (in which no containers are
mixed up), the dual-response storage strategy can
significantly reduce operations cost and, therefore,
enhance productivity of container terminals. Future
research might consider a situation in which both yard
space and shipment demand are uncertain. In
addition, how to precisely determine the placement of
containers in the shared stacks is a potential area for
future research. The yard storage problem for import
containers under uncertainty is also a potential area to
explore.
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