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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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All in-text references underlined in blue are linked to publications on ResearchGate,

letting you access and read them immediately.

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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[2] Baird, A.J.: Optimising the container transhipment hub location innorthern Europe. Journal of Transport Geography, vol. 14, no. 3, pp.195-214, 2006.

[3] Stahlbock, R., Voss, S.: Operations research at container terminals: aliterature update. OR Spectrum, vol. 30, no.1, pp. 1-52, January, 2008.

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[11] Maciejowski, J.M.: Predictive control with constraints. Harlow, UK:Prentice Hall, 2002.

[12] Wang, T.-F., Cullinane, K.: The efficiency of European containerterminals and implications for supply chain management. MaritimeEconomics and Logistics, vol. 8, pp. 82–99, 2006.

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