THE LOGISTICS MANAGEMENT IN THE SIZING OF
THE FLEET OF CONTAINERS PER SHIPS IN DEDICATED ROUTE - THE USE OF COMPUTER
SIMULATION: A BRAZILIAN SHIPPING COMPANY CASE
Delmo Alves de Moura
Federal University of ABC, Brazil
E-mail: delmo.moura@ufabc.edu.br
Rui Carlos Botter
University of São Paulo, Brazil
E-mail: rcbotter@usp.br
João Ferreira Netto
Innovation Center for Logistics and Ports
Infrastructure - USP, Brazil
E-mail: joaofnetto@usp.br
Submission: 19/03/2015
Revision: 31/03/2015
Accept: 07/04/2015
ABSTRACT
The aim of this paper is provide the use of the simulation in the
discrete event to manage one important point in the logistics systems to
shipping companies that is the imbalance of containers, movement of empty
containers from surplus ports to deficit ports.
From a survey of data from a shipping company operating in Brazil, at
various ports, it was possible to model and simulate the needs in six major
domestic ports of empty and full containers and seek to meet demand in the
shipping market, reducing storage of containers and maintaining the level of
excellence in service.
Based on the discrete event simulation it was possible to analyze the
problem of empty and full containers at the ports in the maritime
transportation system. It was possible study the imbalance situation in the
ports e provide one tool the companies to manage yours service.
The data are confined to one company located in São Paulo and operating
in Brazil at maritime transportation.
The research shows that the imbalance problem between full and empty
containers is a real case to all companies in the maritime transportation and
can have effective solutions using discrete event simulation.
To
have excellent supply chain management it is important to have also one
effective transportation system. This paper contributes to research in the
inbound and outbound part of the supply chain management.
Keywords: Full and empty containers, logistics system,
sizing of the fleet of containers, simulation model.
1.
INTRODUCTION
This work addresses a topic considered of
great relevance to the shipping
companies that operate in the ordinary
market (liner) carrying laden and empty containers. It approaches the management of containers available to ships and customers, so
that the expected cargo matrix on
each port is fully answered. If there are no laden containers available
and ready to be transported when a ship berths at
one port of the route, the cargo matrix will not
be accomplished due to the unavailability of containers in the system, causing loss of
transport and hence revenue.
The raw material,
parts, components or finished products will not feed the production line of
business (inbound part), and there will not be products available to the
consumer (outbound part). The Role of Supply Chain Management is to deliver
products on time and in the correct amount to their customers. In addition,
having a proper sizing of containers in maritime transportation system is
essential for the effectiveness and efficiency of logistic system.
As the cargo matrices of cargo liners
usually present imbalance between containers delivered and received by
container ships, many ports end up accumulating many empty containers, while
others suffer from a lack thereof. To mitigate the effects of "imbalance", shipping companies seek to accomplish the repositioning of empty
containers, taking them from the surplus to the deficit ports.
An efficient repositioning
of empty spaces between the ports
allows the company to have smaller fleets of
containers, and avoid leasing
costs and loss of transportation (OHAZULIKE, et al., 2013).
The problem in the
transport chain of maritime shipping service companies is crucial for an
excellent logistics system, as far as most of the world transport occurs by
maritime shipping. So, lack of containers in the system directly influences the
efficiency of the logistics chain. If there are no empty containers to offer to
customers of its maritime transport system, there may be a stop on the
production line or the shortage in point of sales. Proper sizing of the fleet
of containers for logistics providers is an essential condition to offer their
services to the market, with levels considered good to excellent (GUAN; YANG, 2010).
The alternatives to
accomplish the repositioning of those containers have been studied and are
essential so that the total amount of containers inside the system is
minimized, while the space available for the transport of laden containers
should be maximized. The focus is to prevent a client to run out of container
to transport its products, or have empty containers available at the port with
no demand to use them, meanwhile in another port, need for empty containers to
meet the market. Balancing the need for container is the core of the problem.
Thus, through a
simulation model, this paper aims to scale a fleet in the system so that
customers have containers to be loaded on land and shipped to ports to be
boarded into vessels and then, fulfilling the cargo matrix planned by the
shipping company, involving the repositioning of empty spaces between the ports
that comprise the routes taken. Keeping the system balanced is a prerequisite
for the logistics system of a shipping enterprise. The study was conducted in a
shipping company established in Brazil and covers some of the main Brazilian
ports (HARTMANN, 2013).
The presented simulation model
was built in the ARENA software and it has the help of an interface built
in a spreadsheet for
data to be entered and then have the scenarios
configured to be simulated. Through this
interface has also been possible to
obtain the results generated by
the simulation in a clear and organized
way, allowing even comparisons between scenarios
and sensitivity analysis of
design parameters.
2.
LITERATURE REVIEW
2.1.
The problem with Empty Containers
Empty containers have
been an issue in several works by different authors. This is due to the great importance and difficulty
regarding planning their destination.
Lam, Lee and Tang (2007) pointed out in
his research that transporting a laden
container results in at least one
empty container movement. According to Di Francesco et al. (2009), the
process of repositioning empty
containers in a maritime system
can be defined as the planning and distribution of those containers so that inventories are
minimized as well as the cost of
shipping and handling, while the demand
is met in all
ports.
Braekers, Janssens and Caris (2011) defined a route commonly
performed by the container during a logistics loop. The containers
available at a port are all the stored ones,
those returning from customers, the unloaded
and the ones which can be rented and
reused. Braekers, Janssens and Caris (2011), empty containers generate costs when they are loaded and unloaded from vessels, stored,
transported, and when they are rented.
Yun et al (2011)
argue that companies often waste time
repositioning empty containers between locations in which they are stored (ports, depots, etc..) and cite that effective
management in this area can
increase company productivity. The
decisions that a company must
take are basically how the demand for empty
containers will be answered, which
route should be drawn by an empty container, how
and when the shipping will be performed. For those
decisions, it is necessary to know
the inventory of empty containers
at all ports and existing
depots and whether those can be
used (ZOU et
al. 2013; LI,
2013; NISHIMURA et al., 2009; ZHEN et al, 2011).
Those factors lead to
uncertainty in the data used as input to a mathematical model that addresses
the logistics of the empty container. Braekers, Janssens and Caris (2011) argue
that containers availability in a liner service provider is subject to a number
of uncertainties, including the demands of the ports, the time involved in
returning an empty container and available capacity on ships to transport those
containers. To face such uncertainties, the owners usually act with prudent
conservatism.
Due to that, the planning and management of
a fleet of empty containers depends
on supply and demand forecasts
in all ports, which confirm the data uncertainty and the stochastic nature of the problem. What makes that problem difficult are the
uncertainties in the system and
the probability of events that interfere with the planning as, among other things, forecast
errors, damage to equipment, strikes
and delays in the return of
containers (Lai et al. , 2013). Such uncertainties lead companies to use a safety stock, avoiding
some freight to be lost. This practice leads
to increased costs, since costs related
to storage, rent and amortization of acquired new containers arise (ZOU et al.,
2013; LI, 2013;
LAI et al.,
2013; IMAI, 2007; KONINGS, 2005; KOZAN; PRESTON, 2006).
In their study Yun et al. (2011) also cite the
need for a policy solely for stock
control that mitigates the interference of empty containers (LAI et al., 2013). The distribution of empty containers is very
common among owners who choose to rent them or
transfer them from one port to another to remedy any shortcomings
in the system. The repositioning of empty containers takes into account data provided by the commercial sector of the company
with the likely future
demands and scheduling of each
ship - Schedule (LAI et al., 2013).
According Di Francesco et al. (2009), it
is common to have in large ports, specific areas for storage of empty
containers, which can be maintained for a pre-set price by the time the
shipping company deems necessary. Many companies choose to store their empty
containers in depots outside the port area by cheaper stocking prices compared
to those generally used by port companies. According to the authors, small
ports do not have areas dedicated to the storage of containers, resulting in
longer time to transport them.
2.2. Inventory Control and Repositioning
Decisions
According to Imai and Rivera (2010),
scaling the container fleet is a way to determine the quantities needed to meet
future demands, while the problems of fleet management actions generate as
repositioning or leasing of empty containers. In case a ship-owner has a wide
fleet of own containers, this probably requires performing just a few movements
with empty or rent of containers and, thus demonstrating the interdependence
between strategic and operational decisions (LAI et al., 2013).
According to Li et al.
(2007), effective management of maritime transport includes coordinating the
distribution of goods and materials between suppliers, industries, distributors
and customers through a fleet of vessels. His work aims to determine a strategy
for containers allocation in a set of ports and adjust supply and demand. The
authors describe very well the effect of the imbalance of the difference
between supply and demand in different ports and cite the container rental is
an operation frequently used by shipping companies so there is no loss of
customers due to lack of capacity.
The idea proposed by Li et al. (2007) part
from a model for determining optimal inventory policy to a port for a policy
that considers more than one port interconnected in a route. Their model
provides the maximum and minimum inventory of empty containers at a port so
that the costs are minimized. However, such inventory levels cannot be
considered as optimum values when it is analyzed a set of ports, since the
optimum amount of containers exported through a port may not be the same amount
that needs to be imported by another port. Furthermore, Li et al. (2007) found
that the minimum or maximum quantities of empty containers stored in a port
converge to the same value in case the other stock policy is kept fixed. This
enabled the development of a heuristic for determining the inventory policy
that enables cost reduction (ZOU et al., 2013; LI, 2013; LAI et al., 2013).
Li et al. (2007) concluded that their
model allows the analysis of the
limits of stock in several scenarios and routing settings, and that
the complexity of inventory policy is present in any state of the problem. Lagoudis et al. (2006) highlight approaches to the vehicles fleet sizing,
routing in travel and repositioning of empty containers
studies; and also attest to the lack of studies that address the containers fleet sizing.
In their study, Lagoudis et al. (2006) seek to determine the container
fleet that allows meeting demand
in the ports of the Mediterranean
route and minimize
what they call the "idleness"
of containers.
Imai and Rivera (2010) describe three models that address the problem of sizing the container fleet: an analytical model to treat a dry container
fleet in a balanced market, an analytical model that studies the fleet of refrigerated containers in a scenario with imbalance and
a simulation model. Noteworthy is the
fact Imai and
Rivera (2010) use
calculations in their study that do not consider deterministic and random parameters. Also, they do not use linear programming techniques
(LAI et al.,
2013; LI et al., 2004).
Yun et al. (2011) based their study on the empty containers in inventory
control and repositioning decisions, their rent and storage. They
also considered statistical distributions to obtain supply and demand and storage costs, rent
and repositioning of empty
containers (ZOU et al. 2013; LI, 2013). From the assumptions,
Yun et al.
(2011) created a simulation model
using ARENA software
that could assist
in a search process to obtain a policy that
minimizes inventory costs.
About the possible approaches to the
problem of repositioning empty containers, the authors also argue that deterministic formulations may be inefficient due to the uncertainty related to demand and future supply and therefore no differences between predicted data and the
amounts moved. Di Francesco et
al. (2009) emphasize that there
are no studies that quantify the
actual losses in profits and
efficiency from the use of a
deterministic model. Still, the authors chose to build
a deterministic model for optimizing
the reallocation of empty
containers between different ports
(LAI et al.,
2013). Di Francesco et al. (2009) used the
opinions collected in the shipping companies to determine the distributions of uncertain
parameters and, from those,
created different scenarios (multi-scenario
approach).
Song and Dong (2011) treat the repositioning of empty containers as a major problem for shipping
companies, representing many of
the movements made with containers in
the oceans. According to the authors, many factors contribute to the need for those
movements. Among them, the lack of balance between supply and demand
of containers (they cite the example of route Trans-Pacifica, in
which the volume transported from
Asia to the West is much larger and means
that there is a need to transport
empty containers in the opposite way).
Thus, for Song and Dong (2011), the
efficient repositioning of empty containers is
a "key strategy" for
the shipping companies to gain competitive advantages.
Song and Dong (2011) point out that many of the works on the shipping container
use a deterministic approach, through classical formulations of linear programming. Studies
that consider uncertainty and
stochastic factors began to draw attention from the 1990s. For the authors,
the mathematical models capture, often successfully, the stochastic and dynamic nature of
the problem, but increase the
concern that one should have some
problems such as: Choosing an appropriate time
horizon; computational complexity and difficulty of implementation of the model
and robustness on handling of uncertainties present in the problem. It should
also be noted that Song and Dong (2011) cite the interest in the development of
the analysis of qualitative characteristics of optimal policies of repositioning of containers.
3.
METHODOLOGY
The methodology of simulation was
chosen because it is an alternative to the techniques
used and provides a better
detailing of processes that involve the
movement of vessels, processes that occur on
land and inventory management.
Among all the work and research checked, any of them used the simulation as the main tool to determine the container fleet of a shipping company. Moreover,
the problem has stochastic
nature. So, discrete event simulation is the appropriate tool for that situation. The simulation techniques used allow the modeling
of some constraints that are more difficult to be explained in an analytical modeling.
In the simulation model, it is possible
to insert rules to decide how to
reposition empty containers available at the port which can be transported to
other ports of the route, specifically those having demand for empty. Setting the appropriate
planning horizon will solve the problem
of empty containers allocation, in
order to obtain the lowest container
fleet that suits the minimum requirements of attendance in the same period.
Processing the model several times, changing the initial condition of laden or empty containers
at each port, it was obtained a configuration that meets the demand without any exaggeration
in the number of containers stored
at terminals. The model intends to obtain the optimal solution of the total cost of the system (from the point of view of the shipping company) through better composition among containers owned or leased and repositioning of empty
ones (which can be used as a great ally in the quest for more economical inventories). It is important to highlight the fact
that the model considers the transit of ships in ports to both directions (north and south). For
example, instead of having
a flow variable of laden containers to be shipped
from a port i to
port j, at a given time t, traveling towards
a direction, there must be two
different variables for transportation between those ports considering
the two destinies.
The decision variables of the model refer to the system as a whole, the handling
of laden or empty containers (owned
and leased) and vessels. In the objective function (which is the
sum of all costs involved resulting from the decisions of their own fleets, the
rental of containers and repositioning of empty
containers) are considered three situations to ensure that the resolution of the problem does not depend on a lot of information
from the previous period: a) the container fleet
must be loaded, b) full, awaiting shipment,
or c) empty, at the terminals of shipping company.
Mathematically, the objective function (total
cost) to be minimized is composed of
four different cost factors,
which are: The portion relating to the cost of the company's
own fleet; The portion relating to the
rental of containers; The portion of the cost for the transportation of their own empty containers between two ports in any direction; The
portion relating to the costs
associated with the hosting of laden containers in
ports.
The inventory policy is composed of a minimum stock
(s1 in periods of low demand and s2 in
periods of high demand) and a
ceiling (S1 in periods of low demand and S2 in
periods of high demand). If the
inventory level is less than s, the request of empty containers is made
until the level reaches
the value S. If there is a need to reposition
empty containers, but those are not available, the model
seeks to rent those containers
returning to the amount of containers leased in a
given period. As the costs of each
operation are linked to the
simulation model, they seek to reduce the amount spent by the end of the simulated period and, therefore, Yun et
al. (2011) used a
tool for finding the best solution
ARENA, called OptQuest.
Many sensitivity
analyzes are made possible by the use of the model, including the variation
limits of inventory level (s and S) and the best inventory policy to be
implemented in the terminal analysed. So as to achieve optimum inventory policy,
it has to be found values of s1, s2, S1 and S2 that minimizes the cost of the
terminal. OptQuest ARENA performs simulations and modifies the decision
variables iteratively until the stopping conditions are met.
4.
INTRODUCTION
TO SIMULATION MODELING
A model was created to simulate a route for
shipping containers between ports in a closed
loop with a given fleet of ships.
For this simulation, some assumptions are considered, as the availability of
a berth in each terminal in the simulated scenario and the possibility of system configuration,
varying the number of visited ports, the fleet size and cargo
matrices to be met.
The simulation model comprises two interconnected sub-models: the sub model movement
of ships and sub model movement of containers
in the land.
The first sub model that comprises a simulation of the transport container in a closed
loop represents the travel
of the ship and port
operations according to the
routes and matrices predetermined.
Matrices of cargo have been established in a planning phase
in order to ensure that the owner has profitability,
at least, if he complies with the
cargos represented in the matrices. In this sub model the vessels are firstly created in
quantity set in the data input interface. About those ships, should be informed
(in the interface data entry) some features like the speed and capacity, and
the matrix of loads to be served between ports and by each ship allocated on
the route.
Once created the ships of the fleet under consideration, the model assigns the specific characteristics of each, as the identification
number, the number of the trip
(in this case, the first trip of each vessel) and
the sequenced list of ports,
which will be visited on established routes. The vessels are then placed in
ports and defined as initial vessel
in each pair of ship-trip. At the beginning of the construction of the
model, it was adopted the premise
that all ships start their trips at port 1,
and must begin with an interval within the relationship between the cycle time and the estimated
fleet size (headway) in hours. From this and on, the model considers the entities ships separately.
Once defined the first port of the sequenced
list for each ship, it is also recorded the initial
instant of operations in order to
facilitate the collection of statistics
of the model. Then, the first port is allocated and there should take place port operations, according to the productivity inserted into the data input interface. The
input data are also used to
determine the initial condition for
all ports: initial stock of empty containers, initial inventory of laden containers to
be loaded on ships (out) and
initial inventory of laden containers to be shipped
to customers on land (in).
Among the operations that the model performs each time a port is
allocated by a ship, it is considered initially
unloading shipping containers bound for that port. Then,
it is considered
the loading (having enough stock of laden containers at the port) and the decision on whether or not repositioning the empty containers from
the verification performed in the
interface data entry, if a
port has supply or
demand for containers empty. This check considers
ship cargo matrices to determine if
the ports present imbalance between supply and demand for
container and a
simple calculation predicts whether there will be excess or lack of empties in each port.
Logically, during operation in ports,
an update of inventories
of laden or empty containers is carried
out according to the cargo
matrices corresponding to the ship in operation. The logic of loading ships traverses
the specific sequence and array loads, performing the sum that indicates how many containers shall be unloaded in the next port to be
visited. Thereafter, an attribute
loading is created for each of the next ports
in sequence with the amount of
containers shipped on that vessel and to be
unloaded further.
In the procedure of loading containers for
subsequent ports of the list, two situations may occur: there is stock of laden
containers (out) enough to supply the cargo matrix or containers are
insufficient, in which case shall be proportionately apportioned in the
containers available with the cargo matrix. For the latter
situation, the loss of container transport is also
calculated, given by the difference between the amount that should be loaded and
the amount that will actually be carried
by ship. Since
this value can be a decimal
number (due to the calculation
performed), is added logic
to correct such amounts
(a container is an indivisible object), in which the mass balance model is
maintained.
When operations are completed at the port,
it is recorded the instant of departure of the ship. Now this port can receive the next ship,
while the ship which had just been
serviced follows its
journey to the next port in
sequence. The seaborne
travel time to the next port
is determined by the speed of the ship
and the distance matrix between ports. Those data
filled in data
entry interface.
In case the port is not the last in the sequence, the ship should proceed
to the next port and repeat
all procedures from resource allocation at
the port until the end of all
operations. On the other hand, if
the port in question is the last in the sequence, the vessel returns to the first port
of the cycle and
the model will register every input and output to each terminal. In addition, there are
updates to inventory throughout the simulation that are recorded in
the results interface, thus
generating a very useful log for
checking the results.
It is updated, then, the number
of the next trip and
the ship will repeat the sequence,
still unloading in the sequence,
containers that were loaded at previous
ports (for example, if there is
demand from port 2 to port 1, and the
sequence of travel considers
visits to ports in
ascending order, when the ship starts the second
voyage into port 1, the demand from port 2 will be unloaded). Finally, to determine the amount of travel (cycles) of each vessel, the model calculates, first, the duration of a
round trip, recording the time elapsed since the first port of the sequence is allocated until the time when the
vessel returns to this port.
After that, it is calculated the ratio
between the duration of the simulation
and the duration of a journey,
obtaining the number of round trips
and the number of cycles for each vessel
simulated. Repositioning of containers is subject of specific studies within the shipping companies. Considering this reposition increases the complexity
existing in container
shipping, due to the large number
of variables that relate. Proper
logistic planning of empty containers
has great potential to reduce the company's costs and enable higher profits due to increased containers availability and
reducing the loss on transport.
The ability to reposition empty containers
between ports is considered in the sub model of circulation
of ships, as previously reported.
For this, the
model recognizes whether a
particular port of a sequence
has characteristics of a surplus or
deficit port for empty containers. Such recognition is performed also in the input
interface from a
calculation carried out with the
data reported in cargo matrices of the ships in the fleet.
With the values reported in the
cargo matrices, it is calculated
the amount of laden containers to
be removed from a port, and the amount to be unloaded in the
same port during a trip of the ship. Thereafter, the amount of containers unloaded is subtracted from the number of containers removed; obtaining the
balance of containers that must
remain in the vessel after the
port completes its operation. This balance determines to the model if the port has surplus or deficit
for empty containers. In ports where the
balance is positive, the
interface recognizes a surplus
port. Otherwise, if the balance
is negative, it will be a deficient port.
From an order of priority determined
by comparing the average
quantities demanded at all ports
of the route (balance), are recognized
ports of which empty
containers can be removed and
those who should receive
those objects. Note that this
balance would accumulate over time if there were no repositioning (in surplus
ports, container inventories would grow. Meanwhile the deficit ports would
accumulate losses due to lack of containers). Checked which
ports are surplus
and which ones are deficient for empty
containers, is the time to determine the amounts to be withdrawn or unloaded at each port. Such amounts are
calculated taking into account
three factors: the sequence of
ports visited, the balances of empty containers at the end of a cycle and flows between the balances.
After the ports being ordered according to
the sequence of visits, they
should be divided into groups that begin
with surplus ports. From those groups are
removed from the provider ports the maximum amount to be left in a cycle, that is, it is withdrawn
from the surplus ports a quantity equal the overplus of that port.
It is estimated, then, the percentage of the balance
that must be unloaded in the ports
that follow this surplus
port until the ship reaches
another provider port. Besides movement
of ships sub-model presented earlier, a sub-model was built to represent the
movement of containers on land. This sub-model deals with transactions with
empty containers that are shipped for loading in customers, returning filled
for shipment and transport, and with containers that are unloaded filled in
ports and are sent to recipients which empty and return them to the harbor
depots that are located in regions close to the ports used in the simulation.
The entities generated at the beginning of
the simulation through this sub-model are called entities of circulation in ports, while the number of "circulation" generated
is equal to the number of ports that comprise the
route indicated in the input
interface data. Once created
such entities, the model names each with the
number of the respective port to
which it refers, and those will be submitted to the events that may occur in the movement
of containers on land. The entities head to a module of random decision, in which there are
two different logics that can be treated as follows: drawing the daily demand
of empty containers from the port concerned which will result the generation of
laden containers to be sent to the terminal to be aboard (from a statistical
distribution that takes into account the data from the cargo matrix); and the
logic that should handle the containers that come full from ports to customers
on land and are emptied and return to depots at ports.
In the first approach, which is to determine the daily demand for empty ports, the model
checks if the stock of empty containers in port at that point of time is considered enough to meet the demand requested, by checking if
there is still a balance of
containers that was not attended previously. In
case there is not sufficient containers,
there will be an increase of an attribute named "balance to meet" specific to each
port. Such empty containers
become laden containers to be loaded at ports
and, after an interval of time,
the model updates the inventory
of laden containers out in ports. Such transactions occur daily in the model.
This time interval for updating the
inventory represents the retention time of containers
on land and incorporates the duration of all events occurring since the withdrawal of an empty container deposits
until the moment it
is delivered filled for shipment.
The model under consideration for this retention time, a triangular distribution, according to the
suggestion of the shipping company consulted. The other logic in this sub-model of containers movement
on land concerns the landing of containers that arrive full to ports under
consideration.
The entity of movement
awaits a "sign" that occurs in the sub-model of ships movement and
informs the arrival of laden containers to ports. With this sign, the sub-model
of land movement causes the variable of laden containers in (filled containers arriving at the ports from the vessels) to be
retained during the time interval that represents the retention of the
containers on land, and after that period, to become an increment variable of
stock of empty containers in warehouses located near the ports under
consideration.
It is worth noting again that the events held by the entities created in this sub-model of land movement repeat daily in
the model. The simulations of both processes occur
simultaneously and are dependent only on which refers to container availability
inventories as occurs in reality. The transport of containers between the
clients and the terminals where they are loaded or unloaded is the interface
between the two phases, causing supplies at a location to be modified by
other's interference. In other words, the demands used in each stage must be
synchronized and updated as soon as necessary, which shows the relation between
the two logics that make up the model.
The sub-model of land movement
turns empty containers into empty containers out (those that will be shipped filled) and
laden containers in to become empty
containers, always respecting the
time indicated as the retention
time on land, which includes travel
between ports and warehouses,
the time for loading and unloading
and a possible waiting time.
To illustrate the dependence between the
sub-models, the sub-model of ships movement sends a signal to the sub-model of land movement each time new containers are unloaded in ports, indicating that
there must be, then, the
"withdrawal" of laden
containers and their emptying.
As previously mentioned, the simulation
model proposed in this paper considers
two distinct stages: the movement
of the fleet of ships predetermined between the ports that compose the system and the handling of containers, laden
or empty, on land, representing transport between
clients and the terminals in its
area of influence. Each one of
the steps has its peculiarities
and independent rules, although linked. This generated the
need to create the logics separately within a simulation
model, based on different concepts.
The first logic presented is the movement of vessels between
the ports that make up the system.
The main rules adopted for the creation of the conceptual model
of this logic are: Ships at the beginning of
the simulation are inserted into a certain port to fulfill, from that moment, the activities of
unloading, loading and travel according
to a predefined sequence of
ports, in a closed cycle. The vessels may have sequences
of different ports, which make the model
more flexible. The introduction of the fleet and ports sequence
is shown in the data input interface; Each ship enters a queue when arrives
in a port, unloads the laden and the empty containers and
loads other laden and empty ones,
since they available in the corresponding stocks, to be bounded to other
ports of the route.
Those actions occur according to a
cargo matrix that will appear on
the data input interface; The model examines whether the port is surplus
or deficient to
perform repositioning of empties
as efficiently as possible; If there are
not sufficient empty or laden containers to fulfill what has been programmed into the cargo matrix,
the model computes the deficit as loss of the
ship's cargo; The total number of containers moved by the
fleet is obtained when it is
inserted initially in the model
a number of laden and empty containers
in inventories in ports, large enough in order to the fleet to meet
the planning of cargo matrix. From this
number, it will be extracted the number of containers to be placed in the
stocks in ports which minimizes
the loss of transport with the fleet of containers inserted
in inventories (objective
function);
The second stage of the proposed simulation
model represents the movement of
containers on land. The rules adopted to
implement this logic in the model
are: The laden containers unloaded from ships over
time are shipped to customers
following a triangular distribution
of retention time, which includes: customer journey, emptying and return
to the stock of
empty containers at the port;
Daily, a demand for empty containers to suit export customers in the area of
influence of each port is drawn. The stock of empty containers is checked and
the total requested (or what is available) is sent to customers, clearing, this
way, the stock of empties in the harbor; It was adopted
that the balance of empty containers,
not responded, is accumulated for the next day; The retention
time of empty containers, which follows a triangular
distribution, includes travel,
carry and return to the port to be added to the inventory of laden containers, which are
available to be loaded on ships (containers
"out").
Note that is the
balance of laden and empty containers in ports that allows customer service on
land (clients which need empty containers) and service to ships passing in each
port, which need to be loaded according to the cargo matrix pre-defined. The conceptual models presented
served as the basis for encoding the
simulation model, ensuring that the
logic used contains all existing events, and that
they follow the sequence appropriately for future validation.
5.
APPLICATION OF THE SIMULATION MODEL AND THE RESULTS
OBTAINED
In order to build a base scenario, the starting point for sensitivity analysis
and validation of the simulation model,
data were requested for a large shipping company that operates in Brazil, whose identity
will be kept confidential at the request
of the company. The demands used and presented here, were multiplied by a conversion
factor to be adjusted and thus remain confidential,
keeping the same order
of magnitude and enabling realistic analysis of the results obtained.
The database provided by the
shipping company was analyzed and it was extracted the
necessary information for the construction of an initial simulation scenario. The
chosen data comprise information of a cabotage route in which ships visit six ports along the
Brazilian coast: Santos (SSZ), Sepetiba (Itaguaí) (SPB), Suape (SUA), Fortaleza
(FOR), Pecém (PEC) and Manaus (MAO). To facilitate the reading of data in the
model, the ports were numbered and all related information refers to the corresponding
number. The numbers adopted were: Port 1: SSZ (Santos); Port 2: SPB
(Sepetiba-Itaguaí); Port 3: SUA (Suape); Port 4: FOR (Fortaleza); Port 5: PEC
(Pecém) and Port 6: MAO (Manaus).
Among the data provided by the company,
are the demands of laden containers that exist between the
ports visited. Such information has been processed and then, the
cargo matrices have been obtained, considered equal for all vessels. In this cargo matrix there are quantities of laden
containers to be removed from a
port bound to each other to make up the
route for each trip. The cargo
matrix for each ship is desired by the shipping
company to maintain profitability
levels for this
type of transport project. Thus, the proposed simulation model tries to determine
which container fleet is required to meet the cargo matrix desired by the company. Actually,
this carriage is subject to changes in demand, which can overcome or not the
minimum cargo matrix required.
Once known the cargo matrix wanted,
as expected, it is presented an "imbalance"
between the ports. So, it is
calculated which ports are the surplus
and which ones
are the deficient for empty containers. The cargo matrix
considered by ships is presented in Table 1.
Table
1: Cargo matrix of vessels - baseline scenario
|
PORT 1 |
PORT 2 |
PORT 3 |
PORT4 |
PORT 5 |
PORT 6 |
PORT 1 |
0 |
7 |
503 |
39 |
303 |
332 |
PORT 2 |
15 |
0 |
221 |
9 |
51 |
300 |
PORT 3 |
61 |
10 |
0 |
5 |
9 |
400 |
PORT 4 |
20 |
11 |
100 |
0 |
5 |
18 |
PORT 5 |
128 |
15 |
273 |
50 |
0 |
299 |
PORT 6 |
1,388 |
177 |
206 |
12 |
35 |
0 |
It is noteworthy, observing Table 1, the large amount of container that
depart from port 6 to port 1, and the large amount of containers that are
unloaded at port 6. By filling the cargo
matrices, one also obtains the
balances of empty containers in
ports, presented in Figure
1. Therefore, the
model uses the proportions
calculated to determine the
amount of empty containers that
should be removed from the surplus ports and how many containers
would be unloaded at each port
in a sequence.
Figure 1: Representation of the repositioning of empty containers mechanism
In a route designed for a 15-knot speed, four identical vessels were considered in the simulation.
The capacity of those ships (3,500 TEU) is enough
to carry the container load and reposition empty containers. Figure 2 shows the amounts
of laden containers on board
at each part of the trip. Note that the maximum number of laden containers on board is 2.462, which also
allows it to have 1,038 empty containers on
the same vessel.
Figure 2: Amounts of laden containers on board at each part of the trip
The sequence of visits to ports is the same for the four ships of the fleet. The
sequence of visits: port 1 – port 2 – port 3 –
port 5 – port 6 – port 4. It is worth highlighting that at the
end of the sequence all vessels
must return to the original
port and repeat
the cycle. Figure 3 illustrates
the basic cycle trip on the route in question.
Figure 3: Representation of ships' routeing
As the speed of the ship is 15 knots (27,8 km/h),
through the distance matrix, the durations
of trips are obtained. Table 2 shows the distances,
in km, between 6
ports under consideration in the routes of ships.
Table 2: Distance matrix
between the ports under consideration
(kilometer)
|
PORT 1 |
PORT 2 |
PORT 3 |
PORT 4 |
PORT 5 |
PORT 6 |
PORT 1 |
0 |
83.4 |
695.7 |
921.7 |
1,083.7 |
1,990.8 |
PORT 2 |
83.4 |
0 |
612.3 |
838.3 |
1,000.3 |
1.907.4 |
PORT 3 |
695.7 |
612.3 |
0 |
226.0 |
388.0 |
1,457.1 |
PORT 4 |
921.7 |
838.3 |
226.0 |
0 |
162.0 |
1,069.1 |
PORT 5 |
1,083.7 |
1,000.3 |
388.0 |
162.0 |
0 |
907.1 |
PORT 6 |
1,990.8 |
1,907.4 |
1,457.1 |
1,069.1 |
907.1 |
0 |
Other data
provided by the shipping company refers
to productivity in the ports analyzed. It is considered the average duration of the operations of containers at each port, given the berthing and
unberthing of vessels. For the values, it is used a statistical triangular distribution (with 30% variation
from the average, that is, the lowest
value is 30% lower
than the average value, and
the highest, 30% larger than the same
average) which, according to the company
that supplied the data, adheres to the actual data. The average values were adopted: Port 1: 14 hours; Port 2: 30 hours; Port 3: 48
hours; Port 4: 50 hours; Port 5: 48 hours; Port 6: 30 hours.
Those figures represent the total
time of operation since the berthing to the departure of each ship and includes
loading and unloading of containers.
It can be seen that the efficiency in port 1 (SSZ - Santos) is higher than the efficiency of other container yards,
and the operation takes less than half
the time it takes ports 2 and 6, for example. To meet the cargo matrix
of containers that each vessel has to pull out of each port, there will be the
need for daily generation of empty containers compatible with the matrix, which
will be shipped to customers in order to be loaded. Those containers will
return to port after a retention period and normally, should not cause a
continuous increase in the stock of laden containers available for the vessels.
This is, then, a principle of the containers balance flow in the system.
On the other hand, as the containers are unloaded in the ports, they will be sent
to customers, and after a retention time,
they will return to stock of empty
containers. If it is a surplus
port of empty containers, this
amount will be enough to serve
customers in the region as
mentioned above, and also the balance of empties may be shipped
to the deficit ports of the
route. This should be configured directly on the model, with a variable that determines
the average daily demand of
customers for empty containers in
ports (which will return full).
The company also provided the average retention
time of containers on land. This is the time it takes
a container from the moment it is sent to the customer (whether to load or unload)
until the moment that returns to the terminal (empty or
laden). This retention time includes loading, unloading and transport of the containers. Also, can include
possible delays due to customs
procedures, inspection and cleaning of the containers
The average retention time was set at 15
days, also considering the statistical triangular distribution with 30%
variation in accordance with the informed by the shipping company. The initial inventory of laden and empty containers in ports are the decision variables of the problem of searching
for the best solution (minimization
of container fleet), because they relate to the container fleet
size of the navigation company
required to meet the cargo matrix.
At the interface it has
to be filled, for each port, at the beginning of the simulation, the
number of laden containers that can be shipped to customers on land to be emptied,
the amount of empty containers and the
amount of laden containers to be
removed by the next ship to visit the port. Note
that the sum of those stocks in each port and the sum of all ports is
the fleet of containers needed to meet the required
cargo matrix. It was necessary to divide the total stock, in each port, in those three initial stocks (full "in"
full "out" and empties), so that the simulation did not have a long transitional
period until a balance.
This total initial inventory of containers
make up a part of
the objective function one wants to minimize. An objective
function is an equation for
optimization problems to represent the
focus of the problem, for example, minimization of cost or maximization
of income. The
other part of this function to be considered is the loss of shipment due to
lack of laden containers at the port over time and lack of empty containers,
causing the non-compliance of the loading of cargo to a certain ship (not
comply the cargo matrix at each visit of a vessel to a port). It is sought a
fleet of containers that meet the required cargo matrix for each of the ships
in the fleet. In principle, the loss of cargo (no cargo matrix services) should
be zero and the fleet of containers as small as possible.
There could be
situations where a loss of cargo was solved by rental of containers (leasing)
per trip, so that the required fleet of containers would be even lower.
However, this option was not implemented in the simulation model because it
would increase complexity of the problem and a demand to generate information
not provided by the shipping company.
The responses obtained with each round of the
simulation model are: Objective function: sum of initial inventory of containers inserted in the model plus
the amount of containers not
shipped (unmet demand) over time. As one search
for optimization (or best solution) of
the responses obtained, it is suitable the use of the term "objective
function" in this simulation study; Mass balance of containers handled for each
port: demands for laden and empty
containers are loaded and unloaded from ships and
the number of containers that
circulate on land; Inventory of laden containers (in and
out) and empty at all ports; Number of containers repositioned (deficit and surplus);
Operational indicators: cycle time of vessels, time of berthing in ports,
number of cycles of each vessel.
From the results, it stands out the loss of
transport (unmet cargo matrix), one of the factors that compose the objective function to be minimized by an appropriate amount of containers in initial inventory (container fleet). Therefore, changing the initial condition
of laden and empty containers in ports will enable the evaluation of the behavior of the objective function you want to minimize.
The interface for Excel is also used to produce other results obtained with simulations of different scenarios.
Through a worksheet called "Log of ships", it is possible to fully map all travel and operations
performed by each of the ships, including
instant of operations, the
quantities of containers loaded and
unloaded at each stop of each
vessel and inventories in ports.
This tab allows extraction
of various results and allows the creation of statistics that help to interpret the results, assist in the comparison between scenarios and validate the simulation model.
You can also check the performance of stocks every trip and examine
the operation of the model, making sure that the sequences each ship follows confer with the ones that were entered in the input interface. This
first scenario was the starting
point for the simulation of
scenarios for the search of the container
fleet sizing and subsequent sensitivity analyzes according to relevant parameters.
The initial
scenario is constructed from the
input data and assumptions presented
in section 5, representing a regular operation of the entire system that includes the six ports of the
route and all four ships. As
previously mentioned, the initial inventories of containers are the decision
variables that meet the objective of this work, which is the sizing of the
shipping company container fleet. Processing
the model with different levels of initial inventory at ports, aims to achieve the minimum fleet
of containers and manages a reduced loss in transport.
For this baseline scenario it was used an initial inventory of containers (container fleet) large enough for the system to operate,
ensuring that no loss occurs due to lack of containers. This initial inventory of
containers (total sum of the
stocks of laden containers "in"
stocks of containers "out"
and stocks of empties
on all ports of the route in question) was
determined from initial tests with the model.
This allowed finding a preliminary solution with zero
loss of containers, where it was also possible to evaluate what size the
transport capacity of the fleet is, once the cargo matrix is inserted. Based on
this cargo matrix, it was then calibrated the daily amount of empty containers
that customers need. The quantities of containers in initial stocks used for
the simulation of this scenario are presented in Table 3.
Table 3: Initial inventory of containers in the initial scenario
Initial
inventory of laden containers "in" at PORT 1 |
1,600 |
Initial
inventory of laden containers "in" at PORT 2 |
250 |
Initial
inventory of laden containers "in" at PORT 3 |
1,400 |
Initial
inventory of laden containers
"in" at PORT 4 |
200 |
Initial
inventory of laden containers "in" at PORT 5 |
400 |
Initial
inventory of laden containers "in" at PORT 6 |
1,300 |
Initial
inventory of laden containers "out" at PORT 1 |
8,400 |
Initial
inventory of laden containers "out" at PORT 2 |
5,400 |
Initial
inventory of laden containers "out" at PORT 3 |
4,200 |
Initial
inventory of laden containers "out" at PORT 4 |
3,000 |
Initial
inventory of laden containers "out" at PORT 5 |
4,800 |
Initial
inventory of laden containers "out" at PORT 6 |
8,400 |
Initial
inventory of empty containers at PORT 1 |
9,000 |
Initial
inventory of empty containers at PORT 2 |
8,700 |
Initial
inventory of empty containers at PORT 3 |
4,200 |
Initial
inventory of empty containers at PORT 4 |
1,800 |
Initial
inventory of empty containers at PORT 5 |
3,600 |
Initial
inventory of empty containers at PORT 6 |
9,600 |
TOTAL
Initial inventory of containers |
76,250 |
With those initial stocks of Table 3, it was simulated
the initial scenario and quantities of laden containers unloaded on the six
ports of the route were obtained, plus the total containers generated from the
cargo matrices for one year. Table 4 shows the daily and annual quantities of
laden containers required to meet the cargo matrix in ports as well as the
amount of laden containers which were unloaded at each port, showing,
therefore, the "imbalance" between received and shipped.
Table 4: Daily and annual quantities
of containers required or unloaded at each
port
ANNUAL |
DAILY |
|
Laden containers required to PORT 1 |
114,848 |
315 |
Laden containers required to PORT 2 |
57,216 |
157 |
Laden containers required to PORT 3 |
46,560 |
128 |
Laden containers required to PORT 4 |
14,322 |
39 |
Laden containers required to PORT 5 |
71,910 |
197 |
Laden containers required to PORT 6 |
170,892 |
468 |
Laden containers unloaded at PORT 1 |
149,916 |
411 |
Laden containers unloaded at PORT 2 |
20,268 |
56 |
Laden containers unloaded at PORT 3 |
122,772 |
336 |
Laden containers unloaded at PORT 4 |
10,695 |
29 |
Laden containers unloaded at PORT 5 |
37,722 |
103 |
Laden containers unloaded at PORT 6 |
126,734 |
347 |
Also, the average cycle time of vessels was obtained
in this simulation that, divided by the fleet size (four ships), provided the
average interval to which vessels can be inserted at the beginning of the
simulation process. Note that at the beginning of the simulation, each ship is
empty and begins cycle by the first port of the sequence. The average cycle is
365 hours. As the fleet is composed of four vessels, the range is 91 hours.
The results obtained in the initial scenario of the simulation
are presented in Table 5: Results
obtained in the initial scenario of the simulation.
Table 5: Results obtained in the
initial scenario of
the simulation
|
PORT 1 |
PORT 2 |
PORT 3 |
PORT 4 |
PORT5 |
PORT 6 |
1. Quantity of containers unloaded |
149,916 |
20,268 |
122,772 |
10,695 |
37,722 |
126,734 |
2.
Quantity of laden containers shipped inland |
151,516 |
20,518 |
124,172 |
10,895 |
38,122 |
128,034 |
3.
Quantity of empty containers shipped to ports |
145,068 |
19,638 |
118,960 |
10,435 |
36,510 |
122,638 |
4.
Quantity of empty containers reposit. to the port |
0 |
32,954 |
0 |
3,454 |
32,430 |
42,016 |
5. Final
inventory of empty containers at ports |
2,176 |
5,528 |
1,748 |
1,730 |
0 |
0 |
6.
Quantity of empty containers reposit. to other ports |
37,062 |
0 |
75,999 |
0 |
0 |
0 |
7.
Quantity of empty containers generated at ports |
114,830 |
55,764 |
45,413 |
13,959 |
73,717 |
176,939 |
8.
Quantity of empty containers withdrawn from ports |
114,830 |
55,764 |
45,413 |
13,959 |
72,540 |
174,254 |
9.
Quantity of laden containers delivered at portos |
110,608 |
53,648 |
43,634 |
13,451 |
69,883 |
166,982 |
10. Final
invent. of laden containers to depart at ports |
4,160 |
1,832 |
1,274 |
2,129 |
2,773 |
4,490 |
11. Loss
of transport by port |
0 |
0 |
0 |
0 |
0 |
0 |
12.
Quantity of containers generated at the port |
114,848 |
57,216 |
46,560 |
14,322 |
71,910 |
170,892 |
13.
Quantity of containers departed at the port |
114,848 |
57,216 |
46,560 |
14,322 |
71,910 |
170,892 |
From the results obtained, it is noted that the
repositioning of empty containers from the ports occurred from ports 1 and 3
(surplus) to ports 2, 4, 5 and 6 (deficit) according to the data presented in
section 5. Moreover, the initial inventory (fleet) used was absolutely
sufficient, so that there was no loss of demand in ports. With the simulation of this initial
scenario, it could also be observed that Ship 1 made 24
complete cycles, while the other three ships performed
23 cycles (due
to the gap between the positions of the
ship at the first port of the sequence, which caused the ship 1 to perform
one more complete journey). The average cycle time was 364.5
hours. It is also important to
visualize the behavior of stocks
of laden containers ("in"
and "out") and empty containers over the 365 days simulated, presented
in graphs.
6.
CONCLUSIONS
The conclusion is that
the discrete event simulation is a tool that allows to treat the problem of
container fleet sizing in order to meet a fleet of ships operating efficiently
in closed loop and obtaining satisfactory results. Moreover, with the
simulation model and the aid of the search engine OptQuest, the processing time
of the proposed scenarios was considered appropriate, which turns that into a
simulator tool able to be used in management of large shipping companies, as
well as in generation of new scenarios due to changes in the initial
assumptions adopted.
The models proposed by
the theoretical framework provided apply the knowledge needed to determine a
theoretical model that could meet the need for a shipping company to manage its
container fleet.
The initial inventory
of containers in ports (container fleet) in all scenarios specified was
obtained and the simulation model was validated. The results show consistency
of sizing in the sensitivity analysis performed. Note that the most important
parameter is the container retention time on land, which increases the required
fleet size more than the change in the ship speed, when both parameters are
high. It was also possible to notice the importance that repositioning of
containers has in system operations.
It was also presented a
way to use the model for annual planning in a shipping company using the
variation of two impacting parameters for that planning: the fleet of ships
used and the load matrix to be served (demand for containers). The model allows
variation and combination of these parameters to obtain the containers fleet
required to avoid loss of transport. Another contribution of this work was to
obtain values that relate the number of containers to be handled and the size
of the containers fleet required according to the load matrix and amount of
vessels used in the route.
The practical
contribution of this study was to verify the importance of repositioning
containers for proper operation of shipping operations. Therefore, it was
possible to use the annual planning model of a shipping company considering two
essential parameters, the fleet of ships and loading matrix (container demand).
The model contributed
to be an additional way to approach the problem of repositioning empty
containers, besides sizing of containers fleet.
REFERENCES
BRAEKERS,
K.; JANSSENS, G. K.; CARIS, A. (2011). Challenges in Managing empty container
movements at multiple levels. Transport
Reviews, v. 31, n. 6, p. 681-708.
DI
FRANCESCO, M.; CRAINIC, T. G.; ZUDDAS, P. (2009) The Effect of Multi-Scenario
Policies on Empty Container Repositioning. Transportation
Research Part E: Logistics and Transportation Review, v. 45, n. 5, p.
758-770.
GUAN,
Y.; YANG, K.-H. (2010) Analysis of berth allocation and inspection operations
in a container terminal, Maritime
Economics & Logistics, v. 12, n. 4, p. 347–369.
HARTMANN,
S. (2013) Scheduling reefer mechanics at container terminals. Transportation Research Part E: Logistics
and Transportation Review, n. 51, p. 17-27.
IMAI,
A.; RIVERA, F. (2010) Strategic fleet size planning for maritime refrigerated
containers. Maritime Policy &
Management, p. 28, n. 4, p.
361-374.
IMAI,
A.; ZHANG, J.-T.; NISHIMURA, E.; PAPADIMITRIOU, S. (2007) The berth allocation
problem with service time and delay time objectives. Maritime Economics & Logistics, v. 9, n. 4, p. 269–290.
KONINGS,
R. (2005) Foldable containers to reduce the costs of empty transport? A cost-benefit
analysis from a chain and multi-actor perspective. Maritime Economics and Logistics, v. 7, n. 3, p. 223–249.
KOZAN,
E.; PRESTON, P. (2006) Mathematical modeling of container transfers and storage
locations at seaport terminals. OR
Spectrum Quantitative Approaches in Management, v. 28, n. 4, p. 519–537.
LAGOUDIS,
I. N.; LITINAS, N. A.; FRAGKOS, S. (2006) Modelling container fleet size: the
case of a medium size container shipping company. Paper presented at the international conference Shipping in the era of
Social Responsability, Greece.
LAI,
M.; CRAINIC, T. G.; DI FRANCESCO, M. D.; ZUDDAS, P. (2013) An heuristic search
for the routing of heterogeneous trucks with single and double container loads.
Transportation Research Part E:
Logistics and Transportation Review, n. 56, p. 108-118.
LAM,
S. W.; LEE, L. L.; TANG, L. C. (2007). An approximate dynamic programming
approach for the empty container allocation problem. Transport Research Part C, 15, 265-267.
LI,
J. A.; LEUNG, S. C. H.; WU, Y., LIU, K. (2007) Allocation of empty containers
between multi-ports. European Journal of
Operational Research, v. 182, n. 1, p. 400-412.
LI,
J. A.; LIU, K.; LEUNG, S. C. H.; LAI, K. K. (2004) Empty container management
in a port with long-run average criterion. Mathematical
and Computer Modelling 40 (12),
85–100.
LI,
X. (2013) An integrated modeling framework for design of logistics networks
with expedited shipment services. Transportation
Research Part E: Logistics and Transportation Review, n. 56, p. 46-63.
NISHIMURA,
E.; IMAI, A.; JANSSENS, G. K.; PAPADIMITRIOU, S. (2009) Container storage and
transshipment marine terminals. Transportation
Research Part E: Logistics and Transportation Review, n. 45, p. 771-786.
OHAZULIKE,
A. E.; STILL, G.; KERN, W.; VAN BERKUM, E. C. (2013) An origin–destination
based road pricing model for static and multi-period traffic assignment
problems. Transportation Research Part
E: Logistics and Transportation Review, n. 58, p. 1-27.
SONG,
D.-P.; DONG, J.-X. (2011) Effectiveness of an Empty Container Repositioning
Policy With Flexible Destination Ports. Transport
Policy, v. 18, n. 1, p. 92-101.
YUN,
W. Y.; LEE, M. L.; CHOI, Y. S. (2011) Optimal inventory control of empty
containers in inland transportation system. International Journal of Production Economics, v. 133, n. 1, p. 451-457.
ZHEN,
L.; CHEW, E. P.; LEE, L. H. (2011) An Integrated Model for Berth Template and
Yard Template Planning in Transshipment Hubs. Transportation Science, v. 45, n. 4, p. 483–504.
ZOU,
L.; YU, C.; DRESNER, M. (2013) The application of inventory transshipment
modeling to air cargo revenue management. Transportation
Research Part E: Logistics and Transportation Review, n. 57, p. 27-44.