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Minimizing cost of empty container
repositioning in port hinterlands, while
taking repair operations into account
T. Hjortnaesa, B. Wiegmansa*, R.R. Negenborna, R.A.
Zuidwijkb, R. Klijnhoutc
!"#$%&'%
Abstract
Shipping companies are striving to optimize their empty container repositioning strategies which
also contributes to reduced congestion and environmental improvements. In this paper we
propose a multi-commodity model that makes an explicit distinction between flows of non-
damaged containers, on the one hand, and flows of damaged containers, on the other. The model
is tailored for the repositioning of these containers in the representative setting of a network of
off-dock empty depots, ocean terminals, and inland terminals. In our case study, cost savings of
up to 17% are found, depending on the composition of the network, container type, and
particular evacuation and repositioning strategy. In particular, directly transporting containers
from inland terminals to other inland terminals (direct repositioning) results in cost savings of up
to 15% for dry containers and up to 17% for reefer containers. Furthermore, the total costs might
be optimized by actually preventing the container failure from occurring possibly leading to
considerable additional cost reductions. Finally, exporting damaged containers might seem to be
the optimal solution from a regional cost perspective, but, this does not necessarily lead to total
cost optimization from the global perspective.
1
Keywords: damaged containers, repositioning, hinterland, optimization
2
1 Introduction
Shipping companies mainly focus on providing transport between major ports in a global
network. Chang et al., (2015) analyzed the minimum transportation cost for the repositioning of
empty containers for an entire shipping network. However, also a trend of incorporating the port
hinterland into a carrier’s supply chain can also be observed (Gadhia et al., 2011). In general, the
carriers’ customers are not located directly near the terminal, and it is therefore necessary for
shipping companies to transport containers between the ocean terminals and provide empty
containers at the customer’s front door in the hinterland. This is complicated due to the existence
of large trade imbalances between the continents (e.g. from Asia to Europe). These imbalances
contribute to policy requests to reduce these additional empty transport flows causing congestion
and environmental problems on a local and regional level. In an ever-growing volatile container
transport market, cost reductions and efficiency improvements are required. For container
carriers it is therefore crucial to (re)position empty containers optimally (i.e. at the lowest
possible cost). Indirectly this also contributes to reduced congestion and environmental pollution.
1.1 Empty container repositioning in a regional network
Empty container repositioning is performed at various network levels, viz. global, regional, and
local scales. The local level covers the repositioning of empty containers between inland
terminals or depots and surrounding customers. The regional level focuses on hinterland
transport between inland terminals, off-dock empty depots, and ocean terminals. In a research,
Mittal et al., (2013) determined optimal inland-empty container depot locations under stochastic
demand for the New York/New Jersey port region. The global level focuses on balancing
international trade flows between ocean terminals. Inland terminals hereby serve as nodes which
connect the regional and local-scale network, while ocean terminals serve as gateways to
interconnect the global scale with the regional-scale network. Trade imbalances can be observed
leading to regions being either surplus (i.e. import dominated) or deficit (i.e. export dominated)
regions, resulting in empty container transport. At a regional level, this results in repositioning
flows between the deficit and surplus areas on a regional scale. At a global scale, this results in
what are called ‘evacuation flows’ between continents (e.g. from Europe to Asia). In general,
approximately 20% of the exports are empties, but a wide range from 0-90% can be observed.
Overall this means that empty flows can be considerable.
3
There are various types of empty-container flows between: ocean terminals at a port
(1)
, off-
dock empty depots at the port
(2)
, inland terminals in the port hinterland regional network
(3)
, and customers
(4)
. Off-dock empty depots serve as container storage locations, from
where containers are picked up, and to where they are returned to serve export demand. As
illustrated in Figure 1 below, flow interactions exist between these different locations: repair
flows for transporting damaged empty containers to workshops located at depots
(5)
,
customer flows (6) to meet local demand, repositioning flows
(7)
to meet regional demand
and evacuation flows
(8)
towards a global network to serve overseas deficit areas
(9)
.
Meeting customers’ demand globally through the repositioning of empty containers follows a
hierarchical order from local via regional to global scale, until the costs exceed the price of
producing new containers (Theofanis & Boile, 2009). At each location what is called a ‘safety
stock’ in the form of a Target Stock Level (TSL) is maintained to meet demand. The TSL is
based on historic data and the carrier’s expert knowledge. Hardly any information regarding the
actual distribution and availability of empty containers throughout the network is available.
4
Figure 1 An overview of the regional network surrounding the ocean terminal in relation to the transport super-network
(based on Rodrigue et al., 2013)
1.2 Non-damaged versus damaged empty containers
Containers are a commodity that is not handled gently. They are built to last, however during the
transport process, containers can get damaged, which is often inflicted by careless handling on a
terminal or during transport, failure of cooling equipment, regular maintenance, etc.. This can
have a significant influence on the available supply of empty containers. Currently, as soon as a
container is damaged, it is taken out of service until it has been repaired. A repair activity is a
direct reduction of the available supply of empty containers for meeting export demand. Given
the high failure rates (20-25%), understanding the cost impact of extra container movement for
meeting demand is quite relevant. Current methods for repositioning do not take this reduction of
supplies into account, resulting in higher costs due to inefficient transport, handling, storage, and
repair operations. A more comprehensive approach that explicitly takes into account the presence
of both damaged and non-damaged empty containers is therefore desired. This article aims to fill
this research gap and investigates how the total costs of repositioning damaged and non-damaged
empty containers can be optimized, while keeping operations in the hinterland and ocean
terminals in mind. Section 2 provides an overview of the scientific literature for modeling empty
container repositioning. Section 3 gives a detailed explanation of the developed mathematical
model. In Section 4 the mathematical model is implemented for a case study in the Port of
Rotterdam. Section 5 gives the conclusions and further research opportunities.
2 Scientific state of the art in empty container repositioning modeling
To understand the behavior of cost induced by the interaction between damaged and non-
damaged containers while keeping different strategies of repositioning empty containers in mind
an optimization model needs to be developed. Several mathematical models have been proposed
in the literature for general network optimization problems in container liner shipping (see the
overview by Tran & Haasis (2013)), as well as more specifically for empty container
repositioning (see Braekers et al. (2011), Song & Dong (2015) for extensive overviews).
However, little research has been done that takes damaged containers into account. The goal of
this research study is to identify how to model multi-commodity container repositioning
problems and how to solve them. We use Braekers et al. (2011) as starting reference for our
literature review. Crainic et al. (1993) propose a single commodity model, which has been used
by many authors. In this article, we also use this single commodity model as a basis, as its
5
assumptions and model dynamics closely match our case. Choong et al. (2002) present a model
to investigate the impact of the length of the planning horizon. Their paper provides a better
understanding of how to implement a container allocation problem to a fixed network, yet no
damaged containers are taken into account. Olivo et al. (2005) provide a single commodity
model which takes container leasing into account. Their paper mentions the importance of the
repair of damaged containers, although it is not included in their model. The implementation of
customers and inland terminals as aggregated nodes, henceforth known as ‘macro-nodes’, will
also be used for our model. This approach is useful for a tactical model, where the focus is to
understand the impact of repositioning strategies between terminal nodes on total cost through
network optimization. The approach leaves out the interaction with and decisions by customers
in order to make optimization less computationally intensive. Jula et al. (2006) present a single
commodity model, which implements the solutions depot direct and street-turn, but leaves out
the effect of damaged containers. The methods presented are used to investigate the impact of the
solutions on total cost. Wang & Wang (2007) present a model that minimizes the cost of empty
container repositioning with a focus on inventory at locations. Empty container stock is managed
through a TSL, which is implemented as an equality constraint in their model. Furió et al. (2013)
present a Decision Support System tool that reviews street-turn in a model implemented for a
case at the Port of Valencia. No evacuation demand to locations outside the scope of their paper
is taken into account. Moon et al. (2013) present a model that investigates the influence of
foldable containers on total costs. The proposed model has inspired the formulation of our
model, yet the implementation of foldable containers is left out of the scope of our paper, since
they currently represent only a small portion of existing containers. Olivo et al. (2013) present a
deterministic multi-commodity model that takes future requirements into account.
Error: Reference source not found provides the review of the model types and solution methods
found in the scientific literature. Deciding on the model type depends on a number of choices
between alternatives, i.e. nonlinear or linear relationships; explicit or implicit system equations;
discrete or continuous states; deterministic or stochastic variables; and static or dynamic models.
The existing literature presents models that allow the addition of multiple container states and
solutions to the empty container repositioning problem. However, no papers have been found
that take into account damaged containers and the constant evacuation of empty containers. We
therefore propose a new Linear Deterministic Discrete Dynamic Mathematical Optimization
6
model for finding an optimal cost solution for the repositioning of empty containers through off-
dock depots, while taking both damaged and non-damaged containers into account. The
proposed model is then used to investigate the impact on different network topologies, and the
results of applying the following solutions to the network: street-turn; moving containers directly
between importers and exporters (Furió et al., 2013) or depot-direct; empty containers are stored
and maintained at off-dock depots next to deep-sea terminals (Jula et al., 2006). These solutions,
when applied correctly, can help to eliminate a full transport leg, as the purpose of the model is
to support decision making on empty container repositioning.
# (
)
Linear
Explicit
Continuous
Deterministic
Dynamic
Multi-commodity
Substitution
Street turn
Leasing
Depot direct
Foldable containers
Container repair
"
*"%
+,,-. / / / / / / / / "
*"%
0110. / / / / / / "
*2%
0113. / / / / / / )
4
*5%
0116. / / / / / / / / #
*%
01+-. / / / / / / "
*787
0119. / / / / / / :;
*<=%
01+-. / / / / / / /
"
4
*2%
01+-. / / / / / / /
:>
;
*(.
/ / / / / / / / /
7
Table 1 Existing literature on relevant empty container repositioning models and their relationship to our proposed model
This article contributes to the literature in three ways: (1) it provides an optimization model for
the repositioning of empty containers in the hinterland while incorporating flows of
damaged containers; (2) it applies the model to realistic instances based on real data from
Maersk Line; and (3) it provides managerial insights that support empty container
repositioning while taking into account failure of containers and repair.
3 Model incorporating ‘damaged’ and ‘non-damaged’ empty containers
The mathematical model proposed optimizes the cost of transporting damaged and non-damaged
empty containers in a network. The problem is known as a ‘minimum-cost network flow
problem’, which optimizes the objective function restricted by flow conservation constraints.
3.1 Model assumptions
A number of assumptions have been made to be able to model empty container repositioning:
A time-step of 1 week with a horizon of 52 weeks is assumed. The available dataset (provided
by Maersk Line) contains weekly data points for 52 weeks, which allows the incorporation of
seasonal effects. Also, as container repair takes roughly 1 week, the repair prices can be taken
into account by assuming a single period repair.
All transport in the model is performed by barge without capacity constraints. The model
proposed is made from the carrier’s point of view, and in this case the barge modality is the
most economic option.
Depots which offer repair facilities are limited by a repair capacity representing the number
of repairable containers per time-step. A repair facility is limited by the number of staff and
available equipment.
No capacity constraints exist at the other nodes in the network. Each inland terminal has
sufficient storage space, and is often capable of expanding its capacity.
No backlogging is permitted, and no delivery window is included. Containers are delivered
within a single time period to the location with demand. This is in alignment with the
carrier’s operational policy.
Move costs are a combination of transport costs and two handling costs for the start and end
node per arc. Each arc in the network represents the movement of a container between two
nodes, which is subject to a combination of costs.
8
Each node represents a location in a container transport network plus its surrounding
customers. Customers generate demand and supply in a container supply chain with respect
to empty containers. The physical transport to and from customers is outside the scope of our
model, which means that the various inland terminals, ocean terminals, and empty depots
represent the origin and destinations in the supply chains.
Containers can only be repaired once inside the network. A repaired container is delivered to
the customer in the following time-step and as such they leave the model. The total lifecycle
of a container is left out of scope.
Containers can also be repaired outside of the network. The container volume to be repaired
inside of the considered network (Port of Rotterdam) is assumed to be 25%. The remaining
75% of repairs is carried out outside of the considered network. Evacuation flows represent
these external repairs.
All move costs are unit costs independent of distance and time travelled. Carriers make price
agreements with inland shippers depending on the distance, travel time, and container
frequency between the legs in their network.
3.2 Network representation
9
Figure 2 Graphical representation of the scoped network of nodes and arcs, allowing for both damaged and non-damaged
container flow.
We define the network
G(N , A)
with nodes
N
and arcs
A
. We consider flows of both
damaged containers (on arcs
AD
) and non-damaged containers (on arcs
AND
), with
disjoint union
A=AD⨃AND
. As a consequence, the network consists of two separate subgraphs
GD=G(N , A D)
and
GND=G(N , AND )
. The node set
N
consists of ocean terminal
M
, off-dock empty depots
K
, workshops
K '
(located at a depot location), surplus
inland terminals
I
, deficit inland terminals
J
, virtual nodes
V1
and
V2
for damaged
flows, virtual node
V3
for non-damaged flow and node specific customers
C
. Figure 2
illustrates the structure of the network by indicating the flows between the node sets.
The network as proposed is balanced with incoming and outgoing flows. Subgraph
GD
(dashed arrows) focuses only on the damaged containers, and provides repair at a workshop
either within the region or outside the scope of the network. The
GND
subgraph (solid arrows)
focuses on the flow of non-damaged containers for the purpose of meeting shipping demand at
customers.
The flow volumes between nodes
N
are decision variables. For example, the flow volume of
damaged containers between inland terminal
i∈I
and workshops
k ' ∈K '
is given by the
non-negative real number
Xik '
D
.
The flow volumes between customers
C
and nodes
N
are input variables, where FI
represents flow ‘From Import customers’ and TE represents flow ‘To Export customers’. The
damaged container flow leaves the network through the set of virtual nodes
V2=
{
v2
}
, and the
non-damaged container flow leaves the network through the set of virtual nodes
V3=
{
v3
}
.
In network
GD
, a depot has two tasks and is represented by two separate nodes: first, serving
as a transshipment and source node
k∈K
and second, as a workshop
k ' ∈K '
. Containers
become available for repositioning or evacuation purposes after being repaired at a workshop.
Node
V1=
{
v1
}
represents a sink node with inflow equal to the sum of locally repaired
damaged containers. Node
V2=
{
v2
}
represents a sink node with inflow equal to the sum of the
evacuated damaged containers. Inland terminal nodes
I∪J
serve as a source node for repair
flows. Ocean terminal nodes
M
have two functions, being combined transshipment and
source nodes. The arcs between
K
and
K ’
represent the number of repair activities
10
Xk k'
D
of damaged containers.
Ifail ,k '
D
(
t
)
is the amount of containers originating from
workshop
k ' ∈K '
, that have failed in time-step
t
, and will re-enter the system at time-step
(t+1)
as flow of repaired containers
Ifail ,k
ND
(
t+1
)
to an off-dock empty depot
k∈K
.
Xik
D, X jk
D, Xmk
D, X km
D, Xm v2
D
(dashed arrows) represent the flows between nodes in the network of
damaged containers. Flows between the nodes
I , J , K , M
represent the transport of empty
containers to repair facilities. Flows between an ocean terminal
M
and overseas repair
facility
V2
represent the transport of empty containers to repair facilities overseas. Flows
from a repair facility
k ’∈K '
via virtual node
V1
and off-dock empty depot
k∈K
represent the repair of empty containers.
The
GND
graph is similarly constructed as the
GD
graph with some differences, but only
operates non-damaged empty containers for repositioning and evacuation purposes. The main
differences are as follows. The off-dock empty depots
K
are either sink-and-transshipment or
source-and-transshipment nodes. Inland terminal nodes are either sink nodes
I
or source
nodes
J
. An ocean terminal
m∈M
serves as a sink-and-transshipment or source-and-
transshipment node. Node
V3
has been added to serve as the location representing the
overseas location, from where containers enter or leave the network depending on the overall
state of the network at time-step t.
Xmk
ND , Xik
ND , Xkj
ND , Xmj
ND , X ℑ
ND , Xij
ND , Xm v3
ND , X v3m
ND
(solid arrows) represent the flows between nodes
in the network of non-damaged containers. Flows of non-damaged containers between the nodes
I , J , K , M
represent the transport of empty containers to fulfill demand. Flows between
ocean terminals
M
and an overseas ocean terminal
V3=
{
v3
}
represent the transport of
empty containers to and from overseas locations. Flows between the virtual nodes
V1
and the
empty depots
K
represent repaired containers becoming available for empty container
demand. For a detailed description of the optimization model, see Annex 1.
4 Case study: Maersk Line
In this section, we apply the model, as sketched in the previous section and described in Annex 1,
to the Maersk Line case study. Simulations are carried out using an implementation of the model
in Matlab 2014a, in combination with the linear optimization toolbox (Kay, 2014) for
investigating the impact of various scenarios.
11
The Maersk Line dataset provides gate in (i.e. containers returning from import customers) and
gate out (i.e. empty containers sent to export customers) frequencies at inland terminals per
week. The implemented model is subject to various scenarios which show the potential of the
model and the contribution of container repair in the total repositioning cost. A set of 3 ocean
terminals, 4 off-dock empty depots, and 14 inland terminals are considered, corresponding to the
main hinterland of the Port of Rotterdam. The investigated locations are in a surplus area,
meaning that sufficient supply exists and therefore delivery time constraints are left out of the
scope of the model. The case study had the following characteristics;
Number of periods (weeks): 52
Amount of 20’dry containers (20DC): 37500
Amount of 40’ dry containers (40DC): 26500
Amount of 40’ highcube containers (40HC): 42400
Amount of 40’ highcube reefer containers (40HR): 25900
In order to explore cost-saving opportunities, a number of scenarios have been studied. Based on
Maersk Line data, empty containers feature a failure rate of 20 – 25%.
4.1 Experimental setup
The model is now used to investigate several operational scenarios. In particular the influence of
opening or closing off-dock empty depots and ocean terminals was of interest for this case study.
Moreover, some of the existing solutions found in literature that have a direct relationship to
managing empty container flow need to be considered, because they change the physical network
that drives the model. Each scenario is a combination of four scenario variables, which alter the
shape of the matrix that describes the network investigated, specifically:
1. Ocean terminal topology: see Table 2. The scenario variables change when an ocean terminal
in the network is opened or closed. Opening or closing an ocean terminal has a direct
influence on the network and on total cost, as certain network arcs can(not) be selected for
transport.
2. Off-dock empty depot topology: see Table 3. The scenario variables represent which off-dock
empty depots are open in the network.
3. Forced repositioning: see Table 4. The scenario variables allow the forced repositioning of
empty containers through depots.
12
4. Move type scenarios: see Table 5. The scenario variables allow direct transport between
inland terminals (reposition moves) or between inland terminals and ocean terminals
(evacuation moves) in the network of arcs and nodes. Indirect refers to if transport went
through an off-dock empty depot to get to its final destination and direct refers to transport
that neglects the presence of off-dock empty depots.
Table 2 Overview of the ocean terminal topology scenario variables (based on Maersk Line operations)
Ocean terminals
Scenario name: +(1)
0(1)
-(1)
‘Base’ 2# " 2#
‘Transition 1’ 2# 2# 2#
‘Transition 2’ " 2# 2#
‘Future’ " 2# "
Note: 1. For confidentiality reasons the actual terminal and depot names are left out.
Table 3 Overview of empty depot topology scenario variables (based on Maersk Line operations)
Dep
ot
Owner Size Container types
A#*+. 01?@1?8@1?
*0.
B#*+. ( 01?@1?8@1?
C#*+.
@1?
D A# 01?@1?@1?*0.
8@1?
Note: 1. For confidentiality reasons the actual terminal and depot names are left out.
2. Highcube dry containers are 1 foot taller than dry containers
Table 4 Overview of forced repositioning scenario variables (based on Maersk Line operations)
Equal depot
ow
distribution
Unequal
depot ow
distribution 1
Unequal depot
ow distribution
2
2 depots open 31B31 @1B61 C1B01
3 depots open --%-B--%-B--%- 01B01B61 01B@1B@1
4 depots open 03B03B03B03 01B01B01B@1
Table 5 Overview of move type scenario variables (based on Maersk Line operations)
Repositioning Evacuation
Move type 1 : :
Move type 2
13
Move type 3 :
Move type 4 :
The initial and current network composition per container type is referred to as the ‘base’ result.
For each scenario, the focus lies on the relative cost savings compared to the ‘base’ result. The
deterministic optimization model is applied to an instance where for each time step, failure rates
are drawn uniformly from the interval from 20% to 25%. Each scenario is run five times to
reduce variation in these randomly drawn instances. The established scenario variables result in
over 36,000 combinations to be determined by the model. To reduce the computational
complexity, a heuristic has been established that works as follows: (1) Determine the optimal
solution while varying empty depot and ocean terminal topology scenarios; (2) Use the best
solution to further investigate the influence from forced repositioning through depots; and (3)
Use the best solution to investigate the influence of direct repositioning and direct evacuation
scenarios. The main focus of the case study is to discover the cost impact of opening or closing
empty depots in various scenarios with open or closed ocean terminals in the focus area. The
application of step 2 and 3 further illustrate other means of reducing cost for the company. This
heuristic helps reduce the size of the investigation to 9,200 variants. The heuristic is applied
separately to the 4 container types investigated in the case: namely, 20 ft. dry containers (20DC),
40 ft. dry containers (40DC), and 40 ft. highcube dry containers (40HC), 40 ft. highcube reefers
(40HR). Finally, a sensitivity analysis identifies the model’s cost drivers and provides input to
the recommendations.
4.2 Results
Prior to understanding the results presented by the model, it is important to know the dispersion
of the containers through the network. Depending on the location of companies with transport
demand in the investigated network, the following statement applies; all dry container types
seem to be required more at and around inland terminals in comparison to reefer containers,
which are required more in the port area. In the Rotterdam port area a large quantity of customers
shipping perishables exists in comparison with further inland. Dry containers in comparison to
reefers thus require a different approach with respect to optimal repositioning. All numerical
values are deformed to preserve confidentiality of the data.
14
4.2.1 Step 1. Determine the optimal solution while varying the empty depot and ocean terminal
topology scenarios
Running the model with respect to Step 1 of the heuristic provides the results found in Figure 3
and Figure 4. A number of insights are obtained when investigating the results further:
1. The relative weight between transport and handling costs seems to fluctuate depending on
container type. This is caused by differences in locations where there is container
availability and locations where containers are required. Furthermore, the distance
travelled by empty containers influences the size of transport costs. When combined,
these aspects result in roughly 75% of the total cost.
2. Storage costs are constant per container type, because the pre-set TSL is constant over
time for the different locations. In the current setup, storage costs have minimal influence
on total cost, because the costs are only charged at off-dock empty depots and because
the unit costs are low.
3. The transport costs of the 40HR container scenario are significantly lower than the
transport costs in the other scenarios, as can be explained by the fact that 40HR
containers are more required in the Rotterdam port area.
15
4. The 40HR container cost composition is different compared to the cost composition of
the other containers, resulting in transport and handling cost aspects accounting for 50%
and repair cost accounting for 45%. The repair of reefers has a larger impact on the total
cost.
Some conclusions can be drawn from the impact of damaged containers on empty container
management. Empty container supply is reduced by the repair rate, resulting in extra container
moves being required in order to supply the network with empty containers. A damaged
container needs to be transported to a workshop for repair, resulting in the possibility of a
location changing from a surplus to a deficit of supply. This risk can be mitigated by
incorporating a safety factor into the TSL value. Currently, the storage cost at inland terminals is
low, resulting in almost no impact of an increased TSL on operational cost. However, a higher
16
€ 0
€ 2,000,000.00
€ 4,000,000.00
€ 6,000,000.00
€ 8,000,000.00
€ 10,000,000.00
€ 12,000,000.00
Repair cost Storage cost Handling cost Transport cost
Figure 3 Results of the first step in the heuristic
€ 0
€ 2,000,000.00
€ 4,000,000.00
€ 6,000,000.00
€ 8,000,000.00
€ 10,000,000.00
€ 12,000,000.00
Repair cost Storage cost Handling cost Transport cost
Figure 4 Unit cost per cost category and container type for the ‘base’ scenario.
TSL does mean a larger container fleet. Finally, the repair costs of containers are not directly
related to the other cost aspects, but, nevertheless, they do impact indirectly. The equipment can
be managed in a better way when knowledge of the life cycle of the empty container, whether it
is a dry or a reefer container, is used; see also (Lam and Lee, 2011). This also applies to the extra
costs that are incurred when containers are repaired outside the regional network. Damaged
empty container transport only makes sense if it results in lower total costs compared to repair
overseas.
4.2.2 Step 2. Use an optimal solution to further investigate the influence from forced
repositioning through depots
Running the model with respect to Step 2 of the empirical setup provides the results found in
Table 6. An unsteered network, in other words a solely mathematically optimal result, will send
all flows over the arc which has the lowest cost, until capacity is reached. The current model is
hardly restricted by capacity constraints. In the optimal case, the model will, therefore, send all
containers to a single cheapest depot. In real-life a carrier, such as Maersk Line, is unlikely to
consider only the cheapest depot for monopoly reasons and because of the importance of
spreading the risk of operations. A difference in cost per container type seems to impact how
containers move around in a forced network
17
Table 6 Overview of cost improvements compared to the base scenario (Step 1)
transition 1 transition 2 future
20 DC Open depots A, D A, D A, D
Steering 60/40 60/40 40/60
Cost change -3,91% -3,97% -3,92%
40DC Open depots C, D C, D A, D
Steering 60/40 60/40 20/80
Cost change -8,81% -8,90% -6,53%
40HC Open depots C, D C, D A, D
Steering 60/40 40/60 60/40
Cost change -8,23% -8,12% -6,27%
40HR Open depots A, D A, D A, D
Steering 40/60 40/60 40/60
Cost change -17,64% -17,37% -17,82%
4.2.3 Step 3. Use optimal solutions to investigate influence of direct repositioning and direct
evacuation scenarios.
Running the model with respect to Step 3 of the empirical setup provides the results found in
Table 7. Move types show an immediate advantage of using direct positioning for serving
regional balancing purposes. Even by allowing direct evacuation, a cost reduction can be
obtained. These results might give the impression that, by allowing direct connections between
export and import customers, an easy and obvious cost reduction can be made. However, it is
important to realize what the model results do not show. The implementation of direct
positioning or direct evacuation makes the planning of transport a more complex, and therefore a
more costly operation. However, customers tend to concentrate around terminals, and therefore it
might be worthwhile to further explore direct repositioning (van den Heuvel et al., 2013).
Another aspect that should be considered is the congestion that can occur at the ocean terminal
when the off-dock depots are removed from the network. Better communication between all
18
Table 7 Cost improvements due to move type scenarios compared to ‘only individual move type’
Scenario
type ocean
topology
only indirect
move type
direct
evacuation
and direct
repositioning
only direct
repositioning
only direct
evacuation
20D
C
base 0% -13% -12% -5%
transition 1 0% -11% -10% -4%
transition 2 0% -11% -10% -4%
future 0% -11% -9% -4%
40D
C
base 0% -14% -13% -5%
transition 1 0% -10% -9% -3%
transition 2 0% -10% -9% -3%
future 0% -7% -5% -3%
40H
C
base 0% -16% -15% -5%
transition 1 0% -13% -10% -4%
transition 2 0% -12% -10% -4%
future 0% -8% -6% -3%
40H
R
base 0% -29% -29% -15%
transition 1 0% -16% -17% -1%
transition 2 0% -17% -16% 0%
future 0% -17% -16% 0%
parties is required with more streamlined planning to allow for direct transport to be
implemented.
4.3 Sensitivity of the model
4.3.1 Sensitivity to failure rate
η
The container failure rate is limited by a lower-bound and upper-bound value, which, for the
scenario evaluation in Section 4.2 were set between 20% (lower bound) and 25% (upper bound).
For the sensitivity analysis, the failure rate lower-bound is varied in steps of 1% up to the upper-
bound limit. Three upper-bound limits, i.e. 25% (Figure 5), 50% (Figure 6) and 75%, are
investigated. The model showed infeasibility when the upper-bound failure rate was set to 75%,
which occurs due to the number of damaged containers exceeding the repair capacity of the
19
0
2
4
6
8
10
12
14
16
18
20
22
24
€-
€20.00
€40.00
€60.00
€80.00
€100.00
€120.00
Handling cost
Transport cost
Storage cost
Repair cost
Lowerbound failure rate η [%]
Unit cost [euro]
Figure 5 Results from a sensitivity analysis on the failure rate
η
with a lower-bound from 1 to 24% and an upper-
bound of 25%
η
Figure 6 Results from a sensitivity analysis on the failure rate
η
with a lower-bound from 1 to 49% and an upper-
bound of 50%
workshops. The result showed that transport and handling costs decrease as the failure rate
increases. It is important to realize that the decrease depends on where containers become
damaged in the network. An increase in damaged containers could in the short term mean less
evacuation moves from inland locations, but in the long term results in more repositioning moves
from other locations to meet the demand, because the network becomes more deficit.
4.3.2 Sensitivity to local repair rate
α
The local repair rate is a factor that describes the number of containers which are repaired in
Rotterdam instead of overseas. In the model, the local repair rate has been set at 75%. To identify
the sensitivity of this factor on the model results, the local repair rate is varied between 0 and
99%. A value of 0% means that no containers are repaired in the model and the only repair cost
spent is to transport the containers to the ocean terminal. A value of 99% results in almost all
containers being repaired in the Port of Rotterdam, allowing for a maximum number of
containers, which can be used for future demand in the next time-step. Figure 7 shows the results
from this test. Notice that at a local repair rate equal to 58%, a critical point is found, after which
transport and handling costs start increasing again. The lower the local repair rate is, the more
containers are repaired elsewhere, which does not necessarily benefit the company at a global
scale.
5 Concl
usion
and
recommendations
20
Object 169
Figure 7 Sensitivity of the local repair rate
This paper focuses on empty container repositioning through off-dock empty depots located in
the port area. The main research question in this paper has been: “How can total costs be
optimized in the repositioning of empty containers through off-dock empty depots, while taking
account of operations in the hinterland and ocean terminals?” Empty container repositioning is a
non-revenue generating operation, yet it is an important and costly part of meeting customer
export demand. Large container ports where deep-sea foreland and hinterland meet occupy
important positions in empty container repositioning. In these areas, a more efficient
repositioning system for empty container movement will also contribute to reduced congestion
and emissions.
Three contributions can be distinguished. First, the article contributes to the scientific state of the
art by developing a multi-commodity model that takes into account container failure and repair.
The purpose of the model is to support decision making on empty container repositioning
through a network of inland terminals, depots, and ocean terminals. The model takes into account
all the described flows in the regional transport network of Rotterdam and its hinterland,
including different types of repair flows. The model investigates the opportunities to improve
container handling in the Port of Rotterdam and its hinterland from a network perspective, but
now for a single carrier (Konings, 2007). The proposed model is then used to investigate the
impact on different network topologies and to apply the street-turn (Furió et al., 2013) or depot-
direct (Jula et al., 2006) solutions.
Second, the performance of the model is assessed using Maersk Line reference datasets. Third, a
number of managerial takeaways can be inferred from the study, which considers a number of
scenarios based on: (1) Different terminal combinations in the network; (2) Forced repositioning;
and (3) Direct transport between terminals. These scenarios demonstrate the potential of the
model, and result in several important conclusions. First, operational costs related to empty
container repositioning are affected by many variables, of which container repair is an important
element. Given the high container failure rate, total costs might be optimized by actually
preventing the container failure from occurring possibly leading to considerable cost reductions.
Secondly, damaged containers account for nearly 20% for dry containers and 45% for reefer
containers of the total repositioning costs, depending on where the final customer is located. Dry
containers seem to be required more around inland terminals compared with reefer containers,
which are required more in the port area. Certain container types might thus require a dedicated
21
approach with respect to total cost optimization. Thirdly, a balance needs to be struck between
exporting damaged containers to cheap repair facilities, and repairing them within the region.
Exporting damaged containers might seem to be the optimal solution from a regional
perspective. However, this does not necessarily lead to total cost optimization from the global
perspective. Therefore, empty container management total costs need to be optimized through
collaboration on a global scale within the company. Finally, directly transporting containers from
inland terminals to other inland terminals (direct repositioning) results in cost savings of up to
15% for dry containers and up to 17% for reefer containers. However, the resulting cost-savings
tend to be overstated due to the randomness of the failure rates which makes repositioning a
challenging task leading to lower actual cost-savings. Direct repositioning from customer to
customer might lead to possibilities for further cost savings. However, direct positioning makes
the planning of transport a more complex, and therefore a more costly, operation. But customers
tend to concentrate around terminals and direct repositioning might also reduce congestion, thus
making it worthwhile to explore this further. In the end, a more efficient reposition will also
contribute to a more sustainable port area (it results in reduced congestion and emissions).
Numerous further research opportunities exist. First, categorizing container repair in types of
repairs results in more accurate determination of the repair cost drivers. Secondly, including
multiple hinterland modalities, such as barge and rail, would also improve the model’s
resemblance to reality. Thirdly, the effect of a time-step of 1 day could be implemented to
analyze its impact on cost and model accuracy. Also, the current deterministic model could be
extended by taking stochastic input into account. And, finally, the object of study could be
enlarged in order to analyze the effects of empty container repositioning in global networks.
Acknowledgement
The authors thank Maersk Line for providing access to reference datasets on empty container
flows and providing their expertise judgment on the results. The 2 anonymous reviewers are
acknowledged for their helpful comments and suggestions.
References
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23
Annex 1. Optimization model
: #
i
(#:
i∈I={1, … , ITS }
&:(
:#
j
D:
j∈J={1, … , ITD }
&:
:&D4#
k
2E4##
k∈K={1, … , OD }
&2
2E4##
k '
7#
k'∈K'={1, … , WS }
&7(
&#
m
2
m∈M={1, … , OT }
&2
2
v
FG
t
H
t∈T={1,… ,T }
&#
#I
Decision Variables #
XD
J?#>&*.
XND
J4?#>&*.
Input variables #
F In
(
t
)
2
(C , n )
&
n∈I∪J∪K∪M , c ∈C
#
#
T En
(
t
)
2
(n , C )
&
n∈I∪J∪K∪M , c ∈C
#
G#
H #
α
J#?G#
#&&#
η
J<?#J4?
J?
CA
)##D
CImp
:#&D
CFloc
##D##
CFevac
###
##
CEvac
nca p
H#E4##
Dcapk'
##E4##
Ifail
(
t
)
;##
TSL ∀i , j , k , m
4#
I
(
t
)
∀i , j, k , m
4#*4+.;
yv1
(
t
)
#
yv2(t)
24
yv3(t)
B#
Ym
J#?J?
Yk
J#?J?E4
##
Cost function
The carrier aims to minimize cost by optimizing flow in the network with limitations occurring
due to container failure. The cost function, which sums all costs for moving containers through
the network between all nodes {i,j,k,m} for each time-step is given by (1).
2 2
2
' '
1 ' 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1
OD WS OD ITS ITS OD OD OT
Floc D A D A D A D
kk kk kj kj ik ik km km
k k k j i k k m
OT OD OT OD ITD ITS OD
A D Fevac D A ND A ND
mk mk mv mv kj kj ik ik
m k m v k j i k
A
ij i
C X t C X t C X t C X t
C X t C X t C X t C X t
C X
�� �� �� ��
�� �� �� ��
3 3
3
3 3
3
1
1 1 1 1 1 1 1 1
1
1 1
ITS ITD ITS OT OT ITD OT
ND A ND A ND Imp ND
j im im mj mj mv mv
i j i m m j m v
OT
Evac ND
v m v m
v m
t C X t C X t C X t
C X t
�� �� �� ��
��
(1)
The cost function contains all decision variables X and their respective unit flow cost C. For each
time-step the optimal cost is calculated. Storage and inventory is determined at the end of each
time-step. For each time-step the network can change completely, i.e., demand patterns can
change due to seasonality resulting in a deficit location becoming surplus, resulting in the size of
the set of ITD and ITS to be dependent on time.
Equality constraints
Figure 7 Supply determination process
25
Equations (2a) until (9) provide the flow balancing per node as indicated in Figure 7. Per index
the subgraph
GD
and the subgraph
GND
equality constraints are given. Node
J
serves
in subgraph
GND
as a sink node and in subgraph
GD
as a source node, because a node with
an empty container requirement can also suffer from damage to containers. Prior to local
balancing of empty containers, available supply at source and sink nodes are divided into
‘damaged’ and ‘non-damaged’ containers per time-step through a ‘failure’ rate η (see (2)-(4)) ,
which is a randomly generated value subject to a lower bound and an upper bound limit. In each
time-step, this results in a random number of damaged containers. Damaged containers are
transported to empty depots, where workshops are located, for local repair, or to ocean terminals
for overseas repair at cheap locations at the ‘local repair’ rate
α
. Figure 2 shows both the local
repair rate and the failure rate have an influence on the available supply at the various nodes
I , J , K , M
.
: 1 ( ) 1
D
i ik
i
i i
i
ND ND ND
i i ij ik im
i i i
FI t X t
i I FI t I t
I t TE t X t X t X t
�
�
�
� ��
�
�
�
�
�
�
� � �
(
2a)
(2b
)
: 1 ( ) 1
D
j jk
k
ND ND ND
j j ij kj mj
i k m
j j
FI t X t
j J FI t I t X t X t X t
I t TE t
�
�
� ��
�
�
�
�
� � �
(
3a)
(3b
)
'
'
: 1 ( ) 1 1
D D D D D
k jk ik mk km kk
j i m m k
ND ND
k k fail mk ik
m i
ND
k k kj
j
FI t X t X t X t X t X t
k K FI t I t I t X t X t
I t TE t X t
�
�
�
� ��
�
�
�
�
� � � � �
� �
�
(
4a)
(4b
)
, ' '
'
, ' ,
'
' ' : 1
D D
fail k kk
k k
D ND
fail k fail k
k k
I t X t
k K I t I t
�
�
��
�
�
� �
� �
(
5a)
(5b
)
26
2
3
3
: 1 ( ) 1
D D D
m km mv mk
k k
ND ND
m m v m im
i
ND ND ND
m m mk mv mj
k j
FI t X t X t X t
m M FI t I t X t X t
I t TE t X t X t X t
�
�
�
� ��
�
�
�
�
� �
�
� �
(
6a)
(6b
)
Equation (7) through (9) illustrate how the model determines the amount of containers that go to
the three virtual nodes. Equation (7) determines the total flow of damaged containers repaired in
time step t by multiplying the total amount of supply by the failure rate and the local repair rate.
Equation (8) calculates the total flow of damaged containers repaired overseas in time step t by
multiplying the total supply by the failure rate multiplied by the evacuation rate, which is the
amount of containers not repaired locally. Equation (9) calculates the total flow of non-damaged
containers and repaired containers evacuated in time step t by adding the non damaged supply to
the required storage plus the amount of containers repaired in the previous time step minus the
previous storage and minus the required demand. Determining these three sink flows the model
becomes balanced and therefore solvable.
1
( ) ( ) ( ) ( ) ( )
v i j k m
i j k m
y t FI t FI t FI t FI t
� �
� �
� �
� � � �
(
7)
2
1 ( ) ( ) ( ) ( )
v i j k m
i j k m
y t FI t FI t FI t FI t
� �
� �
� �
� � � �
(
8)
3
,
1
1 1 1 1 1
v i j k m
i j k m
i j k m i j k m
i j k m i j k m
ND
i j k m fail k
i j k m k
y t FI t FI t FI t FI t
TE t TE t TE t TE t I t I t I t I t
I t I t I t I t I t
� �
� �
� �
� � � �
� � � �
� � � �
� �
� �
� �
� � � �
� � � � � � � �
� � � � �
(
9)
Capacity constraint
Containers that are repaired at a workshop in the previous time-step become available for
repositioning or evacuation in the current time-step. Each node in the
GND
subgraph is limited
by a capacity constraint as found in equations (10)-(13).
,
0
ND ND ND
ik ij im cap i
k j m
X t X t X t n � �
���
(10)
27
,
0
ND ND ND
kj mj ij cap j
k m i
X t X t X t n � �
� � �
(11)
,
0
ND ND ND
kj ik mk cap k
j i m
X t X t X t n � �
� � �
(12)
3 3
,
0
ND ND ND ND ND ND
km mk im mj mv v m cap m
k k i j
X t X t X t X t X t X t n � �
� � � �
(13)
Repair constraint
Equation (14) ensures no more is repaired at a repair shop than the facility allows., given by
'
'
k
0
D
kk capk
X t D� �
�
(14)
Topology selection
Equations (15) and (16) are used for the investigation of open/closed corridors in the model, i.e.
setting
Yk
to 0 a set of arcs are closed and by setting it to 1 that set of arcs are open. M
represents the total flow on the affected arcs for time step t.
' '
k k fail
k
' '
fail
I t 1
D D D D D
ik j m
kk k
i j k m k
ND ND ND ND
ik kj km mk k
i j m m
X t X t X t X t X t I t
X t X t X t X t Y M
�
� � � � �
� � � �
(15)
2
3 3
D D D ND ND ND
km mk mv im mj km
k k i j k
ND ND ND
mk mv v m m
k
X t X t X t X t X t X t
X t X t X t Y M
�
� � � � �
�
(16)
Non-negativity constraint
Equation (17) is the non-negativity integer constraint.
'
D
2 fail
3 3 fail
, , , , , , I , , , , ,
, , , , , I 1 0 , , , , 1, 2, 3
D D D D D ND ND ND ND
ik jk km mk mv ik ij kj km
kk
ND ND ND ND ND
mk im mj mv v m
X t X t X t X t X t X t t X t X t X t X t
X t X t X t X t X t t i j k m v v v � �N
(17)
In our case, we consider a flow perspective, and consequently we can relax the integer restraint
on the variables representing the arcs. Furthermore by comparing results between two cases any
existing error due to considering non-integer values is minimized.
28
