AbstractâTransportation networks are large-scale complex spatially distributed systems whose purpose is to deliver com- modities at the agreed time and at the ...
Proceedings of the 16th International IEEE Annual Conference on Intelligent Transportation Systems (ITSC 2013), The Hague, The Netherlands, October 6-9, 2013
MoD6.4
Hierarchical Model Predictive Control for Optimizing Intermodal Container Terminal Operations* Jo˜ao Lemos Nabais1, Rudy R. Negenborn2 and Miguel Ayala Botto3
Abstract— Transportation networks are large-scale complex spatially distributed systems whose purpose is to deliver commodities at the agreed time and at the agreed location. The network nodes (terminals, depots or warehouses) can be seen as the main decision making centers, as there the different economic actors interact with each other. In particular, the intermodal container terminal is responsible for storing containers until they are picked up for transport towards their final destination. Operations management at intermodal container terminals can be seen as a flow assignment problem. In this work we present a Hierarchical Model Predictive Control (HMPC) framework for addressing flow assignments in intermodal container terminals. The approach proposed is original due to its capability to keep track of the container class while solving a flow assignment problem respecting the available resources. However, the dimension of the problem to be solved grows with the number of container classes handled and the number of available connections at the terminal. A system decomposition inspired by container flows related to each connection served at the terminal is proposed to diminish the problem dimension to solve. The framework proposed is easily scalable to container terminals where hundreds of container classes and connections are available. The potential of the proposed framework is compared to a centralized Model Predictive Control (MPC) framework and is illustrated with a simulation study based on a 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 network 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, through Fundac¸a˜ o para a Ciˆencia e a Tecnologia, under the project PTDC/EMSCRO/2042/2012 - ORCHESTRA, through IDMEC under LAETA and by the VENI project “Intelligent multi-agent control for flexible coordination of transport hubs” (project 11210) of the Dutch Technology Foundation STW, a subdivision of the Netherlands Organisation for Scientific Research (NWO). 1 J.L. Nabais is with IDMEC, Department of Informatics and Systems Engineering, Set´ubal School of Technology, Polytechnical Institute of Set´ubal, 2910-761 Set´ubal, Portugal joao.nabais at
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. For each 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 according 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: •
estsetubal.ips.pt
•
2 R.R.
Negenborn is with Transport Engineering and Logistics, Delft University of Technology, Delft, The Netherlands r.r.negenborn
at tudelft.nl 3 M.A. Botto is with IDMEC, Instituto Superior T´ ecnico, Technical University of Lisbon, Dept. of Mechanical Engineering, Av. Rovisco Pais 1049-001 Lisbon, Portugal ayalabotto at ist.utl.pt
978-1-4799-2914-613/$31.00 ©2013 IEEE
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
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Central Yard
j
x ¯1
- j x ¯2
-
- j x ¯3
connection 1
- j x ¯4
- j x ¯5 connection 2
j x ¯6 .. . Unload Area
j
- j x ¯7 .. .
-
Import Shake Hands
.. .
- j x ¯8 .. .
- j x ¯9 .. .
- j x ¯10 .. .
Import Area
Export Area
Export Shake Load Hands Area
x ¯ny x ¯5(nc −1)+2 - j -
- j
x ¯5(nc −1)+4 - j - j x ¯5nc
x ¯5(nc −1)+1
x ¯5(nc −1)+3
connection nc Fig. 1. Container terminal graph G using nci = 5 exclusive areas per container 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 Section 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 areas inside the terminal and the links E represent admissible container flows between storage areas (Fig. 1). The model describing the terminal dynamics is based on two main features: 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 imply a 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 978-1-4799-2914-613/$31.00 ©2013 IEEE
is determined by the following parameters: number of container 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 flows inside the Pncontainer c terminal is given by np = , in accordance with n c i i=1 the terminal graph G. The total number of storage areas (nodes) inside Pnc the intermodal container terminal is given by, nci . The Import Area at the Central Yard is ny = 1 + i=1 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 x ¯j is defined. All x ¯j (j = 1, . . . , ny ) are merged to form the overall state-space vector x(k) of the complete container terminal: 1 xj (k) x ¯1 (k) x x2j (k) ¯2 (k) (1) x ¯j (k) = , , x(k) = .. .. . . x ¯ny (k) xnj t (k) 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 = nt ny . 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 2nt nc that disturbs the 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 the amount 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 u ¯j with length nt . All u ¯ j (j = 1, . . . , np ) are merged to form the overall control action vector u(k) of the complete container terminal: 1 uj (k) u ¯1 (k) u u2j (k) ¯2 (k) (2) , u(k) = u ¯j (k) = . .. .. . . unj t (k)
u ¯np (k)
with length nu = nt np . The model for the terminal dynamics can be represented in a compact form as x(k + 1) = Ax(k) + Bu u(k) + Bd d(k) y(k) = Cx(k)
(3) (4)
x(k) x(k)
≥ Pxu u(k) ∈ X
(5) (6)
u(k)
∈
(7)
U,
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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Quay
Central Yard Shake Hands
@ @ x ¯nCi +5
n
x ¯nCi +1
n
x ¯nCi +4
Export Area
pushing containers
n
- n x ¯nCi +2 .. .
.. .
Import Area
n x ¯nCi +3
-
pulling containers -
.. .
x ¯ny .. .
Fig. 2. Container flows for a barge modality connection i at the terminal having nci = 5 exclusive storage areas.
U into the state-space set X . The assumptions made in this work 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 connection available (in a flow perspective as in Fig. 1). This feature 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 x ¯j state-space vectors associated to the specific connection terminal areas plus the state of the Import Area, x ¯nCi +1 (k) i−1 .. X . nc j , 1 ≤ i ≤ nc , xi (k) = , nCi = x ¯nCi +nci (k) j=1 x ¯ny (k) (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) =
Ai xi (k) + Bu,i ui (k) nc X +Bd,i di (k) + Bu,ij uj (k) (9) j=1,j6=i
yi (k) =
Ci xi (k)
(10)
xi (k) ≥ xi (k) ∈
Pxu,i ui (k) Xi
(11) (12)
ui (k)
U i,
(13)
∈
where ui is the control action for subsystem i with length nci nt belonging to set U i , di is the exogenous input vector for subsystem i with length 2nt , yi is the quantity of 978-1-4799-2914-613/$31.00 ©2013 IEEE
containers at subsystem i storage areas, Ai , Bu,i , Bu,ij , Bd,i and Ci are the state-space matrices for subsystem i and Pxu,i is the projection from the control action set U i into the statespace set X i . The dynamic coupling between subsystems is due to the stored containers that are shared by the different subsystems at the terminal Import Area x ¯ny . 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 satisfaction 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: Np −1
min
X
f (x(k + 1 + j), u(k + j))
(14a)
s.t. x(k + 1 + j) = Ax(k + j) + Bu u(k + j) + Bd d(k + j)
(14b)
˜k u
j=0
y(k + j) = Cx(k + j), j = 0, . . . , Np − 1 (14c) x(k + 1 + j) ≥ 0 (14d) u(k + j) ≥ 0 y(k + j) ≤ ymax
(14e) (14f)
Puu u(k + j) ≤ umax x(k + j) ≥ Pxu u(k + j)
(14g) (14h)
Pdx x(k + 1 + j) ≤ dd (k + j),
(14i)
˜ k is the vector comwhere Np is the prediction horizon, u posed of the control action vectors for each time step over the prediction horizon [u(k)T , . . . , u(k + Np − 1)T ]T , ymax is the maximum storage capacity per terminal area, umax the 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 Umax . Constraints are used in this approach to guarantee a meaningful terminal behavior over time according to the structural
710
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 that different areas can be associated to the same physical location. For example, the different state-space vectors concerning Import/Export Shake Hands should be considered together as they are describing the same physical 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 action has 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]: Np −1
min ˜ k,i u
X
f (xi (k + 1 + j), ui (k + j))
(15a)
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,i is the projection matrix from the state-space set X i into the exogenous 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,
Np X
(16) di (k + j) ci (k) = x ¯nCi +1 (k) +
.
j=0 1
The workload can be weighted by pi (k) to introduce priorities 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 such that po1 (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 = umax and the current prediction set for control agent decisions ˜k−1,onc }. The available handling reP 0 = {˜ uk−1,o1 , . . . , u sources θ0 are an upper bound to the flow of containers to be assigned. The control agent to start (o1 ) has all handling resources available. After the initial configuration the iterations are executed in which each control agent oi (i = 1, . . . , nc ) one after another performs the following tasks (see Fig. 3): • the maximum admissible flow for control agent oi is determined as the minimum between the subsystem i maximum handling resource consumption uomax and the handling resources not yet assigned,
j=0
i i i ), θoi−1 ; uomax = min (Pomax uomax
s. t. xi (k + 1 + j) = Ai xi (k + j) + Bu,i ui (k + j) nc X + Bd,i di (k + j) + Bu,ij uj (k + j) (15b) j=1,j6=i
yi (k + j) = Ci xi (k + j), j = 0, . . . , Np − 1 (15c) xi (k + 1 + j) ≥ 0 ui (k + j) ≥ 0
(15d) (15e)
yi (k + j) ≤ ymax,i Puu,i ui (k + j) ≤ umax,i
(15f) (15g)
xi (k + j) ≥ Pxu,i ui (k + j) Pdx,i xi (k + 1 + j) ≤ dd,i (k + j),
(15h) (15i)
where ymax,i is the maximum capacity for subsystem i ˜ k,i is the vector composed by the control storage areas, u action vectors for each time step over the prediction horizon, [ui (k)T , . . . , ui (k + Np − 1)T ]T , umax,i the maximum handling 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 978-1-4799-2914-613/$31.00 ©2013 IEEE
711
•
•
(17)
i where Pomax is the projection matrix from the global maximum handling resource set Umax to the maximum oi handling resource set Umax for subsystem oi ; in case the workload coi is nonzero the optimal control i action uoopt 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: i i (k)uoopt (k) θoi+1 = θoi − Pomu
(18)
i (k) is the projection matrix from agent oi where Pomu handling resource set U oi to the control action set Umax ; • the prediction set for control agent decisions is updated and denoted by P oi+1 replacing the control agent initial prediction u ˜k−1,oi by the new optimal sequence found u ˜ opt,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
θ0 , P 0 , o, x ? θ o1 Agento1 P o1
?uopt,o1 uopt ? Fig. 3.
the central model (3)–(7), or by 30 states per subsystem if the decomposed model (9)–(13) is used. ?? Agento2
θo2o - . . . θ nc −1 o2 P P onc −1 ?? Agentonc u uopt,onc opt,o . . . 2 ? ?
A. Simulation Setup
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 terminal allows 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 while for Barge B a maximum rate of 45 TEU/h is possible. As a 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 trains each in a single day. The maximum handling capacity for each rail track is 40 TEU/h but the train gate only offers a maximum capacity of 40 TEU/h. In this case both rail tracks will 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 composed of 4 terminals. In order to respond to the desired hinterland container flows a network of connections and weekly schedules is created [13]. For the considered terminal structural layout there are nci = 5 storage areas per transport connection available at the terminal nc = 5, the containers are categorized into nt = 5 different classes (fourPdestinations plus empty containers). A total of ny = nc 1 + i=1 nci = 26 storage areas are present at the terminal. For this setup the terminal is described by 130 states using 978-1-4799-2914-613/$31.00 ©2013 IEEE
The H-MPC architecture is compared to the centralized MPC approach using a similar optimization problem configuration. 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 possible to 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 Np = 3 steps is used. To guarantee pushing container from the Import Area to the Load Area the weights assigned should respect the relation, Np −2
− (q3+i + q4+i ) >
X
q5+i i = 0, . . . , nc − 1,
(19)
j=1
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 Operating System. 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 simultaneously; 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 case3 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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60
9150
xC 26 xH 26
V. CONCLUSIONS AND FUTURE RESEARCH
9100 9050
TEU
50
TEU
9200
xC 6 xC 10 xH 6 xH 10
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40
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0 5
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Fig. 4. Control strategies comparison (C stands for centralized MPC strategy, H stands for H-MPC strategy).
TABLE I C UMULATIVE TIME PER TIME STEP k FOR EACH CONTROL STRATEGY. Strategy
Max [s]
Mean [s]
Stdv [s]
Cost Function Performance
H-MPC1 H-MPC2 H-MPC3 MPC
4.71 8.28 7.39 367.83
2.66 2.84 2.83 118.16
1.14 1.26 1.21 67.18
−4.660 × 105 −4.660 × 105 −4.660 × 105 −4.766 × 105
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 simulation study, it is illustrated that the H-MPC approach can outperform 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 container classes at the container terminal will be used at a strategic level to developed distributed cooperative control strategies between container terminals to improve the container network throughput.
3000
R EFERENCES
2500
TEU
2000
1500
1000
500
x1i x2i x3i x4i x5i
0 20
40
60
80
100
120
140
time step k Fig. 5. Storage evolution per container class at the Import Area at the Central 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 terms of 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. 978-1-4799-2914-613/$31.00 ©2013 IEEE
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