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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/258915863
Hierarchical Model Predictive Control forOptimizing Intermodal Container Terminal
Operations
CONFERENCE PAPER · OCTOBER 2013
DOI: 10.1109/ITSC.2013.6728314
CITATIONS
2
READS
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3 AUTHORS:
João Miguel Lemos Chasqueira Nabais
Instituto Politécnico de Setúbal21 PUBLICATIONS 39 CITATIONS
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R.R. Negenborn
Delft University of Technology113 PUBLICATIONS 828 CITATIONS
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Miguel Ayala Botto
University of Lisbon - Instituto Superior Técn…
63 PUBLICATIONS 442 CITATIONS
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Available from: Miguel Ayala Botto
Retrieved on: 13 February 2016
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Hierarchical Model Predictive Control for Optimizing Intermodal
Container Terminal Operations*
João Lemos Nabais1, Rudy R. Negenborn2 and Miguel Ayala Botto3
Abstract— Transportation networks are large-scale complexspatially distributed systems whose purpose is to deliver com-modities at the agreed time and at the agreed location. Thenetwork nodes (terminals, depots or warehouses) can be seenas the main decision making centers, as there the differenteconomic actors interact with each other. In particular, theintermodal container terminal is responsible for storing contain-ers until they are picked up for transport towards their finaldestination. Operations management at intermodal containerterminals can be seen as a flow assignment problem. In thiswork we present a Hierarchical Model Predictive Control (H-MPC) framework for addressing flow assignments in intermodalcontainer terminals. The approach proposed is original due to
its capability to keep track of the container class while solvinga flow assignment problem respecting the available resources.However, the dimension of the problem to be solved grows withthe number of container classes handled and the number of available connections at the terminal. A system decompositioninspired by container flows related to each connection served atthe terminal is proposed to diminish the problem dimension tosolve. The framework proposed is easily scalable to containerterminals where hundreds of container classes and connectionsare available. The potential of the proposed framework iscompared to a centralized Model Predictive Control (MPC)framework and is illustrated with a simulation study based ona long-term scheduled scenario.
I. INTRODUCTION
In container transportation networks the objective is to
deliver a specific container at the agreed time and at the
agreed location. The challenge when looking at the net-
work performance is to assure the cooperation between the
different network actors towards a more sustainable and
reliable transport [1], [2]. At intermodal container terminals
containers of different classes are handled, eventually stored
or facing a transport modality change to approach the final
destination at the agreed time. The need for operations
management at intermodal container terminals arise from
*This work is supported by the Portuguese Government, throughFundaç ão para a Ciência e a Tecnologia, under the project PTDC/EMS-
CRO/2042/2012 - ORCHESTRA, through IDMEC under LAETA and bythe VENI project “Intelligent multi-agent control for flexible coordinationof transport hubs” (project 11210) of the Dutch Technology FoundationSTW, a subdivision of the Netherlands Organisation for Scientific Research(NWO).
1J.L. Nabais is with IDMEC, Department of Informatics and Sys-tems Engineering, Setúbal School of Technology, Polytechnical In-stitute of Setúbal, 2910-761 Setúbal, Portugal joao.nabais atestsetubal.ips.pt
2R.R. Negenborn is with Transport Engineering and Logistics, DelftUniversity of Technology, Delft, The Netherlands r.r.negenbornat tudelft.nl
3M.A. Botto is with IDMEC, Instituto Superior Técnico, TechnicalUniversity of Lisbon, Dept. of Mechanical Engineering, Av. Rovisco Pais1049-001 Lisbon, Portugal ayalabotto at ist.utl.pt
the demand to unload/load containers from/into connections
arriving/departing at the terminal [3]. These two types of
demand are disturbances to the terminal state and are treated
as exogenous inputs. The operations management related to
intermodal container terminals can be categorized as a flow
assignment problem and stated as: find the optimal container
flows inside the terminal such that the exogenous inputs
effects are eliminated. In a container terminal the flow of
containers is guaranteed by allocating handling resources,
such as AGV, straddle carrier, quay cranes. The terminal has
dedicated areas (called gates) where the arrival and departure
of containers in each transport modality is executed. Foreach transport modality different connections (arrival and
departure of vehicles) can be available in a single day. Take
as an example the train gate: a terminal can have three rail
tracks for serving simultaneously trains, and for each rail
track different trains can be served in a single day.
The operations management at the container terminal has
been mainly addressed by the operations research field [4].
The control field has also recently addressed attention to this
problem [5], [6] considering undistinguished containers. The
ability to distinguish containers is important to tackle the
container flows over the entire transport network, for example
the problem of repositioning empty containers in a network
of container terminals [7]. Containers are distinguished ac-cording to some relevant criteria (weight, size, destination,
volume). In this paper, the container terminal is modeled
from a push-pull container flows perspective [8]. When a
transport connection arrives at the terminal, the terminal pulls
containers from the transport (unload operation) and push
containers to the transport (load operation). The model is
used to develop a decomposition of the main flow assignment
problem into subproblems that are handled by MPC control
agents. The control agents solve their problems in a serial
procedure. The hierarchy is settled in accordance to the
priority of the related subproblem for the current container
terminal state.
The main contributions of this paper are:
• the development of a systematic and scalable framework
to model container terminals and simultaneously be able
to track different container classes. Different container
terminal structural layouts are hereby admissible;
• the development of a Hierarchical Model Predictive
Controller (H-MPC), based on a flow decomposition, to
solve in a reasonable time the flow assignment problem.
In Section II the intermodal container terminal model is
described according to a generic framework that is able to
Proceedings of the 16th International IEEE Annual Conference on
Intelligent Transportation Systems (ITSC 2013), The Hague, The
Netherlands, October 6-9, 2013
MoD6.4
978-1-4799-2914-613/$31.00 ©2013 IEEE 708
https://www.researchgate.net/publication/225323734_Voss_S_Operations_research_at_container_terminals_A_literature_update_OR_Spectrum_30_1-52?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/268001400_Intermodal_Transportation?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/224351949_Modeling_and_Feedback_Control_for_Resource_Allocation_and_Performance_Analysis_in_Container_Terminals?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/24110601_Management_of_logistics_operations_in_intermodal_terminals_by_using_dynamic_modelling_and_nonlinear_programming?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/49120721_Flow_balancing-based_empty_container_repositioning_in_typical_shipping_service_routes?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/226652669_Simulation_of_a_multiterminal_system_for_container_handling?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/224351949_Modeling_and_Feedback_Control_for_Resource_Allocation_and_Performance_Analysis_in_Container_Terminals?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/268001400_Intermodal_Transportation?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/226652669_Simulation_of_a_multiterminal_system_for_container_handling?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/49120721_Flow_balancing-based_empty_container_repositioning_in_typical_shipping_service_routes?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/24110601_Management_of_logistics_operations_in_intermodal_terminals_by_using_dynamic_modelling_and_nonlinear_programming?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==https://www.researchgate.net/publication/225323734_Voss_S_Operations_research_at_container_terminals_A_literature_update_OR_Spectrum_30_1-52?el=1_x_8&enrichId=rgreq-ede37b12-4d39-44d8-a8f8-7c125d195833&enrichSource=Y292ZXJQYWdlOzI1ODkxNTg2MztBUzoyMjEzNTU2NDg3ODY0MzJAMTQyOTc4NjcwNzMwNw==
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x̄5(nc−1)+1
x̄5(nc−1)+2
x̄5(nc−1)+3
x̄5(nc−1)+4
x̄5nc
x̄ny
UnloadArea
Import ShakeHands
ImportArea
ExportArea
Export ShakeHands
LoadArea
..
.
..
.
..
.
..
.
..
.
..
.
x̄6 x̄7 x̄8 x̄9 x̄10
x̄1 x̄2 x̄3 x̄4 x̄5
Central Yard
connection 1
connection 2
connection nc
Fig. 1. Container terminal graph G using nci = 5 exclusive areas percontainer connection flow plus a common area for all container flows.
capture different structural layouts. The control framework
for the intermodal container terminal is presented in Sec-
tion III. For comparison purposes the optimization problem
is formulated for the whole intermodal container terminal
in Section III-A. The H-MPC approach is presented in
Section III-B based on a decomposition of the container
flows associated to each connection. In Section IV numerical
results are presented, in which the H-MPC approach is
compared to the centralized MPC approach.
II. MODELING
An intermodal container terminal is represented by a graph
G = (V , E ) [9] where the nodes V represent storage areasinside the terminal and the links E represent admissiblecontainer flows between storage areas (Fig. 1). The model
describing the terminal dynamics is based on two mainfeatures: 1) queues, to model the storage capacity related
to well defined areas inside the terminal; 2) categorization
of containers: if a container is empty or a full container,
and for a full container a division is made according to its
destination.
For each transport connection arriving at the terminal
a container flow inside the terminal is established, using
handling resources, consisting on the following operations
(see Fig. 1):
1) unload containers from the connection respecting the
unload demand;
2) transport containers from the Unload Area to the
Import Area at the container terminal (this may implya handling resource switch that will be executed in the
Import Shake Hands);
3) rehandle containers from the Import Area to the Export
Area according to the load demand;
4) take containers from the Export Area to the Load Area
(this may imply a handling resource switch that will
be executed in the Export Shake Hands);
5) load containers into the connection respecting the load
demand.
The complexity of the intermodal container terminal model
is determined by the following parameters: number of con-
tainer classes considered nt, a distinction can be made in
terms of final destination, weight, size, time or other criteria;
number of different connections simultaneously provided
at the intermodal container terminal nc, which can be of
different transport modalities; number of container terminal
areas related specifically to each connection nci .
The number of admissible container flows inside the
terminal is given by np =
nci=1 nci , in accordance with
the terminal graph G . The total number of storage areas(nodes) inside the intermodal container terminal is given by,
ny = 1 +nc
i=1 nci . The Import Area at the Central Yard is
a special area as all containers that are unloaded from each
connection pass through this area before being redirected to
the load operation. For each node in the container terminal
model a state-space vector x̄j is defined. All x̄j ( j =1, . . . , ny) are merged to form the overall state-space vectorx(k) of the complete container terminal:
x̄j(k) =
x1j (k)x2j (k)
...
xntj (k)
,x(k) =
x̄1(k)x̄2(k)
...
x̄ny(k)
, (1)
where xtj(k) is the amount of containers of type t at node j at time instant k. The state-space vector has length nx =ntny. The arrival/departure of connections at the terminal
are associated with an unload/load demand of containers,
respectively. In this work, this transport demand is seen
as an exogenous input d with length 2ntnc that disturbsthe terminal state. It is up to the terminal to allocate the
handling resources at the terminal to move containers inside
the terminal such that the unload/load operations are executed
in order to meet the transport demand. Consider utj
(k) as theamount of containers of type t to move from terminal area j
at time step k. For all admissible container flows inside the
terminal a control action vector is defined ūj with length nt.
All ūj ( j = 1, . . . , np) are merged to form the overall controlaction vector u(k) of the complete container terminal:
ūj(k) =
u1j(k)u2j(k)
...
untj (k)
,u(k) =
ū1(k)ū2(k)
...
ūnp(k)
. (2)
with length nu = ntnp.The model for the terminal dynamics can be represented
in a compact form as
x(k + 1) = Ax(k) + Buu(k) + Bdd(k) (3)
y(k) = Cx(k) (4)
x(k) ≥ Pxuu(k) (5)
x(k) ∈ X (6)
u(k) ∈ U , (7)
where y is the current amount of containers per terminal area
with dimension ny, A, Bu, Bd and C are the state-space
matrices, Pxu is the projection from the control action set
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Shake Hands
Export Area
.
..
.
..
.
..
.
..
Import Area
Central YardQuay
x̄nCi+1
x̄nCi+2
x̄nCi+3
x̄nCi+4
x̄nCi+5x̄ny
pushing containers
pulling containers
Fig. 2. Container flows for a barge modality connection i at the terminalhaving nci = 5 exclusive storage areas.
U into the state-space set X . The assumptions made in thiswork are intended to produce a general framework able to
describe different intermodal container terminals in terms of
structural layout.
A. Intermodal Container Terminal Decomposition
The relations between the different terminal areas inside
the container terminal lead to a highly structured model,
if nodes are numbered in sequential order per connectionavailable (in a flow perspective as in Fig. 1). This fea-
ture is used to proceed with the system decomposition
inspired by container flows associated to each connection
(see Fig. 2) [10]. Using this decomposition the overall system
is divided into smaller subsystems, each subsystem is related
to a transport connection available at the terminal. A control
agent is assigned to each subsystem. The Import Area located
at the Central Yard is a special area as it is the only area
common to all connections/subsystems: it is the area where
containers are stored and wait to be picked up for transport
over another connection.
From this new perspective, the state-space vector for a
subsystem xi will be composed of the corresponding x̄jstate-space vectors associated to the specific connection
terminal areas plus the state of the Import Area,
xi(k) =
x̄nCi+1(k)...
x̄nCi+nci (k)x̄ny(k)
, nCi =
i−1j=1
ncj , 1 ≤ i ≤ nc,
(8)
with length (nci + 1) nt belonging to state-space set X i. The
state-space model for subsystem i is given by,
xi(k + 1) = Aixi(k) + Bu,iui(k)
+Bd,idi(k) +nc
j=1,j=i
Bu,ijuj(k) (9)
yi(k) = Cixi(k) (10)
xi(k) ≥ Pxu,iui(k) (11)
xi(k) ∈ X i (12)
ui(k) ∈ U i, (13)
where ui is the control action for subsystem i with length
ncint belonging to set U i, di is the exogenous input vector
for subsystem i with length 2nt, yi is the quantity of
containers at subsystem i storage areas, Ai, Bu,i, Bu,ij, Bd,iand Ci are the state-space matrices for subsystem i and Pxu,iis the projection from the control action set U i into the state-space set X i. The dynamic coupling between subsystems isdue to the stored containers that are shared by the different
subsystems at the terminal Import Area x̄ny . Each agent
controlling a subsystem should verify if the Import Area
can receive more containers and ask for permission to take
containers of a certain class. The control action is a factor
of coupling as there is limited handling capacity that has to
be shared between all subsystems. Subsystems are coupled
in dynamics and have coupled constraints.
III. CONTROL
MPC [11] is particularly suited to deal with the control
of container terminal operations since it is able to handle
hard constraints (which include the handling capacity and
storage limits), make predictions for the future (about the
container volume per area) and include available predictions
(about the unload/load operations expected). The control goal
is to proceed with an efficient flow assignment in order to
increase the container terminal performance according to a
criteria. Common choices for evaluating container terminal
performance are the throughput [6] or the customers satis-
faction in terms of cost, time and service quality [12].
A. Centralized MPC Formulation
Terminal operations control is formulated according to
a centralized approach taking into account all container
terminal flows. The MPC problem of a centralized control
agent for the container terminal can be formulated as follows:
minũk
N p−1j=0
f (x(k + 1 + j),u(k + j)) (14a)
s.t. x(k + 1 + j) = Ax(k + j) + Buu(k + j)
+ Bdd(k + j) (14b)
y(k + j) = Cx(k + j), j = 0, . . . , N p − 1 (14c)
x(k + 1 + j) ≥ 0 (14d)
u(k + j) ≥ 0 (14e)
y(k + j) ≤ ymax (14f)
Puuu(k + j) ≤ umax (14g)
x(k + j) ≥ Pxuu(k + j) (14h)
Pdxx(k + 1 + j) ≤ dd(k + j), (14i)
where N p is the prediction horizon, ũk is the vector com-posed of the control action vectors for each time step over
the prediction horizon [u(k)T, . . . ,u(k + N p − 1)T]T, ymaxis the maximum storage capacity per terminal area, umaxthe maximum handling capacity according to the container
terminal structural layout, dd is the exogenous input vector
prediction over time, Pdx is the projection matrix from the
state-space set into the exogenous input set and Puu is
the projection matrix from the control action set into the
maximum handling capacity set U max.Constraints are used in this approach to guarantee a mean-
ingful terminal behavior over time according to the structural
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layout and unload/load demand over time. The following
constraints are included in the MPC problem formulation:
• Nonnegativity of states and control actions: negative
storage is not physically possible, imposed by (14d),
and all control actions have to be nonnegative, this is
guaranteed by (14e);
• Storage capacity: each terminal area has to respect its
own storage capacity (14f). It is important to note thatdifferent areas can be associated to the same physical
location. For example, the different state-space vectors
concerning Import/Export Shake Hands should be con-
sidered together as they are describing the same phys-
ical location, and naturally share the available storage
capacity (see Fig. 2);
• Maximum control actions: maximum flows assigned are
restricted by the available handling capacity inside the
terminal (14g). Different structural layouts can be easily
translated into the model with impact in the projection
matrix Puu;
• Feasible control actions: not all control actions that
satisfy (14e) and (14g) are allowed. The control actionhas to respect the amount of containers per container
class present at each terminal area (14h);
• Load demand: the loading request imposed by clients is
introduced in the optimization problem through (14i).
B. Hierarchical MPC Formulation
The problem dimension to be solved in each time step is
diminished using the decomposition (9)–(13) of the overall
system into subsystems. The MPC formulation for control
agent i that is responsible for subsystem i is then stated
as [10]:
minũk,i
N p−1
j=0
f (xi(k + 1 + j),ui(k + j)) (15a)
s. t. xi(k + 1 + j) = Aixi(k + j) + Bu,iui(k + j)
+ Bd,idi(k + j) +
ncj=1,j=i
Bu,ijuj(k + j) (15b)
yi(k + j) = Cixi(k + j), j = 0, . . . , N p − 1 (15c)
xi(k + 1 + j) ≥ 0 (15d)
ui(k + j) ≥ 0 (15e)
yi(k + j) ≤ ymax,i (15f)
Puu,iui(k + j) ≤ umax,i (15g)
xi(k + j) ≥ Pxu,iui(k + j) (15h)
Pdx,ixi(k + 1 + j) ≤ dd,i(k + j), (15i)
where ymax,i is the maximum capacity for subsystem i
storage areas, ũk,i is the vector composed by the controlaction vectors for each time step over the prediction horizon,
[ui(k)T, . . . ,ui(k + N p − 1)T]T, umax,i the maximum han-
dling capacity according to the container terminal structural
layout for agent i, dd,i is the vector responsible to introduce
the load demand for agent i, Puu,i is the projection matrix
from the control action set U i into the maximum handling
capacity set for agent i, Pxu,i is the projection from the
control action set U i into the state-space set X i and Pdx,iis the projection matrix from the state-space set X i into theexogenous input set of agent i.
The order by which the control agents solve their problems
at each time step can be fixed over time or depending on the
current terminal state. In case of a time-varying order, at each
time step all control agents calculate the expected workload
operations over the prediction horizon, as the sum of the load
and unload operations,
ci(k) =
x̄nCi+1(k) +
N pj=0
di(k + j)
1
. (16)
The workload can be weighted by pi(k) to introduce priori-ties for the different connections. Each control agent shares
its workload information, for the current time step at the
terminal, with a central coordinator that sets the order o(k)in which the control agents should solve their problems,
with o(k) =
o1 . . . onc
with 1 ≤ oi ≤ nc suchthat p
o1(k
)co1(
k)
> po2(
k)
co2(
k)
> . . . > ponc (
k)
conc (
k)
.
The central coordinator also initiates the total amount of
handling resources that are available to allocate θ0 = umaxand the current prediction set for control agent decisions
P 0 = {ũk−1,o1 , . . . , ũk−1,onc}. The available handling re-sources θ0 are an upper bound to the flow of containers
to be assigned. The control agent to start (o1) has allhandling resources available. After the initial configuration
the iterations are executed in which each control agent oi(i = 1, . . . , nc) one after another performs the followingtasks (see Fig. 3):
• the maximum admissible flow for control agent oi is
determined as the minimum between the subsystem
maximum handling resource consumption uoimax and the
handling resources not yet assigned,
uoimax = min (Poimaxθ
oi−1 ;uoimax) , (17)
where Poimax is the projection matrix from the global
maximum handling resource set U max to the maximumhandling resource set U oimax for subsystem oi;
• in case the workload coi is nonzero the optimal control
action uoiopt is found solving the MPC problem (15a)–
(15i). In case the workload coi is zero the control action
is zero by default;
• the available handling resources to the next control
agent oi+1 are updated:
θoi+1 = θoi − Poimu(k)uoiopt(k) (18)
where Poimu(k) is the projection matrix from agent oihandling resource set U oi to the control action set U max;
• the prediction set for control agent decisions is updated
and denoted by P oi+1 replacing the control agent initialprediction ũk−1,oi by the new optimal sequence found
ũopt,oi .
Although no iterations are performed between control
agents a feasible solution is guaranteed by (15h). Each
control agent has as mission to move containers from a
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uopt
θ0, P 0, o,x
Agento1
uopt,o1
θo1
P o1
Agento2
uopt,o2
θo2
P o2. . .
. . .
θonc−1
P onc−1Agentonc
uopt,onc
Fig. 3. H-MPC schematics for time step k (omitted in the picture).
source node to an end node where a demand on those
containers is present. The worst scenario is to reach a solution
where no control action is applied by control agent i (no
flows between subsystem i nodes) although there is a demand
on containers. This happens when the upstream terminal area
does not have the required containers or there are no available
resources to move containers along subsystem i.
IV. NUMERICAL RESULTS
The presented H-MPC architecture will be used for
controlling a hinterland intermodal container terminal [13]
serving three transport modalities: barge, train and truck.
The terminal faces an average weekly throughput of around
16, 800 TEU (twenty-foot equivalent units). The terminalallows the berth of two barges simultaneously (Barge A and
Barge B), three connections per berth are made on a daily
basis. The quay cranes allow a maximum handling capacity
of 90 TEU/h, for Barge A this full capacity can be used whilefor Barge B a maximum rate of 45 TEU/h is possible. Asa consequence, Barge A and Barge B will be competing for
the same resource at the quay. For the train modality there
are two rail tracks (Train A and Train B) that serve 4 trainseach in a single day. The maximum handling capacity for
each rail track is 40 TEU/h but the train gate only offers amaximum capacity of 40 TEU/h. In this case both rail trackswill be competing for the gate handling capacity.
The transfer towards the Central Yard is realized by
Straddle Carriers for all transport modalities and is designed
to sustain the maximum container flow for each modality. All
containers arriving at the terminal are moved to the Import
Area at the Central Yard and all containers that departure
from the terminal by some transport modality are taken
from the Export Area at the Central Yard . The rehandling
of containers at the Central Yard from the Import Area to
the Export Area is done using Rail Mounted Gantry Cranes.
The container terminal is integrated in a network com-
posed of 4 terminals. In order to respond to the desiredhinterland container flows a network of connections and
weekly schedules is created [13]. For the considered ter-
minal structural layout there are nci = 5 storage areasper transport connection available at the terminal nc = 5,the containers are categorized into nt = 5 different classes(four destinations plus empty containers). A total of ny =1 +
nci=1 nci = 26 storage areas are present at the terminal.
For this setup the terminal is described by 130 states using
the central model (3)–(7), or by 30 states per subsystem if the decomposed model (9)–(13) is used.
A. Simulation Setup
The H-MPC architecture is compared to the centralized
MPC approach using a similar optimization problem con-
figuration. A time step of one hour is considered. A linear
function penalizing the current container volume in each
terminal area is used as the objective function. It is possibleto assign different weights to different container terminal
areas, container classes and connections depending on their
role in the container terminal dynamics and the desired
behavior. The weight assigned to the Import Area at the
Central Yard is zero as it acts as a warehouse for containers
between delivery and pick up times. The weights at the Load
Area are taken negative, such that containers are pulled from
the Import Area. The main criterion to assign weights is
related to the connection priority according to the volume
of containers to handle: the higher the volume the higher
the priority. In decreasing order: Barge A, Barge B, Train
A, Train B and Trucks. Only then, inside the Unload/Load
Area a further distinction is made to introduce priorities for
different container classes.
A prediction horizon of N p = 3 steps is used. To guaranteepushing container from the Import Area to the Load Area the
weights assigned should respect the relation,
− (q3+i + q4+i) >
N p−2j=1
q5+i i = 0, . . . , nc − 1, (19)
where q3, q4 and q5 are the weights associated to containers
located at the Export Area, Export Shake Hands and Load
Area for the first connection respectively.
The simulation is performed by MatLab R2009b on a
personal computer with a processor Intel(R) Core(TM) i7
at 1.60 GHz with 8 Gb RAM memory in a 64-bit OperatingSystem. The optimization problem is solved at each time step
of the simulation using the MPT v2.6.3 toolbox [14] with the
CDD Criss–Cross solver for linear programming problems.
B. Test Scenario
The scenario presents one week. In this scenario it is
assumed, with a sustainable environment attitude, that all
trains arriving and departing use the maximum transport
capacity. Trucks also deliver and pick up cargo simulta-
neously; there are no empty travels starting or ending at
the terminal. This just requires more coordination from the
terminal management and no loss of generality is produced.
Concerning barges, the call size is fixed but the unload/load
demand is assumed random.
Different criteria to establish the order by which the
control agents should be solved in the H-MPC approach
were tested; case 1 the call size p =
1 1 1 1 1
;
case 2 benefiting from sustainable transport modalities p = 2 2 1 1 0.5
and case 3 inverting the order consid-
ered in the MPC strategy p =
1 1 1.5 1.5 2
.
Control strategies are compared using two criteria: 1) the
sum of the cost function over the entire simulation and
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5 10 15 20 25 30 35 400
10
20
30
40
50
60
70
time step k
T E U
xC
6
xC 10
xH 6
xH
10
5 10 15 20 25 30 35 408800
8850
8900
8950
9000
9050
9100
9150
9200
time step k
T E U
xC 26
xH 26
Fig. 4. Control strategies comparison (C stands for centralized MPCstrategy, H stands for H-MPC strategy).
TABLE I
CUMULATIVE TIME PER TIME STEP k FOR EACH CONTROL S TRATEGY.
Strategy Max [s] Mean [s] Stdv [s] Cost Function Performance
H-MPC1 4.71 2.66 1.14 −4.660 × 105
H-MPC2 8.28 2.84 1.26 −4.660 × 105
H-MPC3 7.39 2.83 1.21 −4.660 × 105
MPC 367.83 118.16 67.18 −4.766 × 105
20 40 60 80 100 120 1400
500
1000
1500
2000
2500
3000
time step k
T E U
x1
i
x2
i
x3i
x4
i
x5
i
Fig. 5. Storage evolution per container class at the Import Area at theCentral Yard for the H-MPC1 strategy.
2) the computation time. In Fig. 4 it is clear that both
strategies lead to almost the same terminal behavior over
time. This similarity can be confirmed by the cost function
performance indicated in Table I. Both strategies achieve
a similar performance with a slightly better score for the
centralized approach. Interesting to note that all H-MPC
strategies tested achieved the same performance. In termsof computation time, the H-MPC approach outperforms the
MPC strategy, Table I.
In Fig. 5 it is possible to monitor the volume by container
class at the Import Area in the Central Yard . This ability
is partially responsible for the large problem dimension.
However, when looking to the total amount of containers at
the terminal it is almost constant (around 9000 TEU, Fig. 4).The model complexity is the price to pay to have more
information regarding the state of the terminal which is a
key element for the transportation network.
V. CONCLUSIONS AND FUTURE RESEARCH
In this paper we have presented a Hierarchical Model
Predictive Control (H-MPC) approach for solving the flow
assignment problem at intermodal container terminals while
monitoring different container classes. The architecture is
based on a decomposition that follows the container flows
associated to each connection at the terminal. In a simu-
lation study, it is illustrated that the H-MPC approach canoutperform the centralized approach in terms of computation
time with almost the same terminal behavior over time. The
approach is easily scalable to a large number of connections
at the terminal.
Using this approach, it is possible to access at any time
the exact quantity per container class at the terminal. This
information can be shared with the rest of the container
transportation network to access the effective volume and
container type in the network. The knowledge about con-
tainer classes at the container terminal will be used at a
strategic level to developed distributed cooperative control
strategies between container terminals to improve the con-
tainer network throughput.
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