Proceedings of the
12th IFAC Symposium on Transportation Systems
Redondo Beach, CA, USA, September 2-4, 2009
Modelling for the optimal sizing of an
automatic intermodal freight terminal
C. Caballini ∗ P.P. Puliafito ∗,∗∗ S. Sacone ∗,∗∗ S. Siri ∗
∗
Italian Centre of Excellence in Integrated Logistics
University of Genova, Italy
∗∗
Department of Communication, Computer and System Sciences
University of Genova, Italy
Email: claudia.caballini@cieli.unige.it, ppp@dist.unige.it,
simona.sacone@unige.it, silvia.siri@cieli.unige.it
Abstract: In this paper the problem of the strategic design of an automatic intermodal terminal
is addressed. In particular, the considered terminal is supposed to be provided with an innovative
transfer system (such as the Italian system Metrocargo) allowing to load/unload containers in a
fast and horizontal way under the electric line. In this system, the main strategic decisions regard
the number of shuttles, as well as the number of storage locations to be used in the terminal. At
this purpose, an optimization problem is stated for determining the optimal resources in order
to minimize the overall cost and taking into account some specific constraints. An algorithm is
also defined for solving the proposed optimization problem.
1. INTRODUCTION
dynamics of a seaport terminal and for optimizing the
number of handling resources and yard size.
Intermodal freight transportation is an interesting and
competitive alternative to road transport since it can offer
a complete door-to-door service without using trucks for
medium-long distances. Besides, the high flexibility of containers and swap bodies makes this kind of transportation
suitable for different types of cargo. However, intermodal
transport is characterized by a much higher degree of
complexity than unimodal solutions in terms of organization, administration and support technologies. Those
complexities have an impact in economic and efficiency
terms and, consequently, in the limitation to the growth of
intermodal transportation. For these reasons, the methodologies devoted to modelling, optimizing and controlling
such systems are of great interest both for researchers and
practitioners. Different research works have been devised
aiming at planning intermodal freight transportation systems at different decision levels (i.e. strategic, tactical,
operational), as described in [1, 2].
The objective of this paper is to state and solve a strategic
planning problem for a railway terminal provided with an
innovative intermodal system, such as Metrocargo. This
latter is an innovative system, patented by an Italian
transportation company, that allows to quickly and safely
move cargo on/from trains, significantly reducing the total
handling time and, consequently, the related costs. The
idea at the basis of this innovative system is to apply to
freight transport the same concept utilized for passengers.
So rail transportation, that actually works as a pointto-point service, can be considered as a network transportation, in which cargo units are progressively loaded
on different trains up to their final destinations. Thanks to
specific innovative devices, cargo units are loaded on train
cars in an horizontal way and directly under the electric
feeding line, so allowing to dramatically reduce the total
logistic cost in comparison with traditional systems, with
a consequent improvement of the overall system competitiveness.
This paper is an extended version of [3, 4] and deals with
the strategic planning of a railway terminal devoted to
container transportation. Some review works refer to optimization and planning methods for intermodal container
terminals, including also port facilities [5, 6]. With regard
to strategic planning methods for the design of container
terminals, some papers address the issue of defining the
number of handling resources, the storage space dimensions, and so on. Among the others, in [7] a decision
support system is defined for capacity planning of container terminals, both for designing a new terminal and for
tuning an existing one. In [8] a network model is presented
as a decision tool for investment appraisal of container
terminals. In [9] a planning method is proposed with the
goal of defining the optimal storage dimensions and the
optimal number of cranes in a terminal. In [10] a queuebased discrete-time model is defined for representing the
978-3-902661-50-0/09/$20.00 © 2009 IFAC
31
The paper is organized as follows. In Section 2 a description of Metrocargo system is provided; in Section 3 the
main features of the considered problem are described.
The optimization problem is reported in Section 4, a
numerical example is proposed in Section5 and, finally,
some conclusions are drawn in Section 6.
2. DESCRIPTION OF THE SYSTEM
The objective of the present paper is the strategic planning
of an automated railway terminal, such as one provided
with Metrocargo technology. The main objectives of this
innovative system are to shift big traffic volumes from road
to rail, to simplify rail transport using shuttle trains with
prefixed schedules and itineraries, to reduce environmental
10.3182/20090902-3-US-2007.0026
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
pollution and congestion. More in general the final goal
is to decrease the total handling time and the total
logistic cost related to freight intermodal transportation,
so enabling the Italian transportation system to improve
its competitiveness against the European competitors.
is grabbed by a shuttle through proper forklifts. Then,
thanks to optical means, the shuttle detects the cargo unit
outline and aligns itself to the fixing devices placed on
the wagon in the predefined position. The loading can be
made in parallel by all the shuttles that are present in
the system. Then, the unloading of cargo units from train
wagons is operated analogously to the loading phase; the
shuttle, after lifting up the containers from wagons, utilizes
the forklifts for making the unloading operations and leave
them in the automatic storage area. After a break, that can
be of few minutes or some hours, cargo units are loaded
through cranes or forklifts on trucks which provide for their
delivery.
In traditional intermodal solutions (both referring to
ports, freight villages or dry ports), terminals are off line
and trains must be shunted away from the electrified track
using diesel locomotives, pulled to a loading yard, loaded
and brought back to the regular track again by diesel
traction. The total time required to perform all these
operations ranges from 10 to 12 hours per train. With
Metrocargo system, instead, trains remain directly under
the electrical track in a sort of “buttonhole” parallel to
the main rail track, where they are automatically and
safely loaded and unloaded. This new approach, thanks
to the elimination of the onerous traction break and to
the automation of all the operations, allows a tremendous
decrease in total handling time from 10-12 hours to only
about 30-40 minutes per train.
More specifically, traditional terminals are connected to
the network through a non electrified siding track, which
branches out from the tracks bunch linked to the reference
railway station. In this situation, once arrived in the
destination station, the train must be towed, through
an operation of terminalization with a diesel locomotive
inside the terminal. In the first tract, this operation
is often managed by a shunting team of the railway
company and, in the final tract, by the terminal personnel,
with consequent additional costs and times. Besides, the
movement of cargo units, based on the lifting technique,
obliges to operate with expensive infrastructures, such
as gantry cranes or reach stackers, and it requires large
spaces for the storage and shunting of handling means.
On the other hand, Metrocargo solution is based on
an innovative horizontal handling system that utilizes
terminals equipped with automated storage areas, which
are placed contiguous to railway tracks. The whole of
operations is performed without leaving the electrical
feeding line (it is operated under electrical cables) and
without bringing the train out of the station, but simply
making it deviate on a parallel track.
The rapidity of loading/unloading operations, together
with the implementation of the Metrocargo technology,
allows to reduce the number of operative tracks to one.
Building them close to each other and in a parallel position
to the railway line, an electrified “buttonhole” is obtained.
The use of the horizontal transfer allows the entrance and
the exit of the train on the primary railway line without
the necessity of railway shunting operations.
A general scheme of the terminal layout is depicted in
Fig. 1. Trains arriving at the terminal are unloaded/loaded
by means of one or more shuttles which run parallel to
the train itself. Containers are stored in the terminal in
rows (perpendicular with respect to the train) which are
called bays, composed by a given number of slots. Note
that loading bays (storing containers to be loaded on the
train) and unloading bays (with containers to be unloaded
from the train) are separated. Moreover, we define as train
section the section of the train relevant to one bay.
train section
train
shuttle
bay
The Metrocargo terminal stocking area is completely automatic, composed of some bays perpendicular to rail tracks;
each bay is divided in a certain number of slots where
containers are stored. The activities connected to the
Metrocargo plant can be described through the following
phases:
Fig. 1. A schematic representation of a Metrocargo terminal.
With reference to the possible implementation of the
Metrocargo system, two different scenarios have been
considered:
• entrance of cargo units in the system;
• loading of cargo units on the train;
• unloading of cargo units from the train;
• exit of cargo units from the system.
As far as the first phase is concerned, cargo units arrive
in the terminal through road or maritime means; they
are placed on the receiving area of the plant linked to
the moving area inside the terminal. The arrival of each
unit is signalled by a centralized management software.
According to the indications given by the software which
plans the sequences of loading and transfer, containers
are disposed in a specific bay in the stocking area. In the
second phase, each cargo unit placed in the loading area
32
• scenario 1, in which loading and unloading operations
are performed on the same side of the train;
• scenario 2, in which loading and unloading operations
are performed on the two sides of the train, so not
interfering one with each other.
In fact, according to the available area where the terminal is placed, loading and unloading operations can
be assigned to both sides of the train with the purpose
of reducing interferences and accelerating operations, so
decreasing the total handling time. In both scenarios,
some assumptions are made. As already introduced, it is
assumed that the whole train length is ideally divided in
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
exactly as many sections - of equal length - as the number
of bays. Moreover, the number of containers placed on a
train results to be exactly equal to the sum of containers
in the bays. In this way, it is assumed that the frequency of
trains passing in the terminal is low enough to allow the
reorganization and preparation of the containers for the
next train. Besides, the possibility of using the terminal as
a buffer area is not taken into account.
Finally, the constraints to be taken into account can be
described as follows:
• the total handling time, spent by the shuttles to
unload and load a train, must be lower than the
maximum acceptable stop time of the train in the
terminal T max ;
• the total number of bays must be lower than the maximum threshold B max , due to physical constraints of
the area where the terminal is placed.
It is worth underlying that in our analysis we assume that
the time required by the automatic devices for making the
cargo units translating along the bays is always lower than
the time spent by the shuttle for handling one cargo unit.
This means that when the shuttle comes back in front of
the bay to manage the next container, this latter is ready
to be managed in the upper extreme of the bay.
Note that, in this paper, the total handling time is only a
constraint that must be respected; however, it could also
be considered as a term to be minimized on order to ensure
that the train stops in the terminal the least possible time.
In order to properly state the problem constraints, an
important distinction must be considered between scenario
1 and scenario 2. Remember that we consider as scenario 1
the case in which loading and unloading tasks are executed
on the same side of the train, while scenario 2 considers
the situation of separate loading and unloading on the two
train sides. In the case of scenario 1, the constraints to be
considered are:
T tot = T tot,L + T tot,U ≤ T max
(2)
Another basic hypothesis concerns the way in which containers to be loaded or unloaded are assumed to be placed
on trains: we assume that all the containers to be handled
are placed in the less favorable position compared to the
position of the bay (“worst case”). This means that they
are positioned in the extremes of their section.
3. PROBLEM DESCRIPTION
B tot = B L + B U ≤ B max
(3)
while, in case of scenario 2 (where loading and unloading operations corresponds to separate phases), the constraints to be taken into account are:
T tot,L ≤ T max
(4)
In this section we will go into the details of the mathematical formulation that will allow us to properly formulate
the problem, whose objective is the definition of the optimal layout of the Metrocargo terminal. In particular, the
decisions to be taken concern the number of resources, i.e.
the number of shuttles and loading/unloading bays, with
the final goal of minimizing the related costs and taking
into account some specific constraints. For the problem
formulation, the following input data are considered:
T tot,U ≤ T max
max
(6)
U
max
(7)
B ≤B
B ≤B
• C, i.e. the train capacity, in terms of number of cargo
units in a train (they can be either containers or swap
bodies of different length);
• λ, i.e. the average length of one container;
• v, i.e. the average speed of the shuttle;
• C L , i.e. the number of containers to be loaded on a
given train;
• C U , i.e. the number of containers to be unloaded form
a given train;
• T max , i.e. the maximum acceptable stop time for the
train in the terminal (this constant can assume different values according to the particular flow volume
that characterizes a specific Metrocargo terminal);
• B max , i.e. the maximum number of bays (this value
depends on a physical limit of the area in which the
terminal is placed).
(5)
L
train section
Tl
train
Tm
Tm
Ts
Tm
Ts
Tm
Ts
bay
Fig. 2. A schematic representation of a Metrocargo terminal.
The most critical and complicated aspect to be studied is
the definition of the total handling time for loading and
unloading all the required containers, i.e. the definition
of the terms T tot,L and T tot,U . The total handling time
is composed of three terms (as depicted in Fig. 2): the
moving time Tm , which is the time spent by the shuttle for
moving along the shuttle track within each train section;
the shifting time Ts that represents the time spent by the
shuttle for moving from one bay to the next one; the lifting
time Tl which is the time needed for lifting up and down
all the required containers. As it will be better clarified
in the following, Tm and Ts are functions of the number
of bays, while Tl does not depend on it. Of course, these
The decision variables in the considered framework refer
to:
• B L , i.e. the number of loading bays;
• B U , i.e. the number of unloading bays;
• N , i.e. the number of shuttles.
In the cost function two cost terms are considered: the
storage cost and the cost of shuttles. The total cost can be
defined as follows:
C = N · δ + B L · σL + B U · σU
(1)
where δ is the cost of one shuttle and σ L and σ U represent
the cost of a loading and unloading bay, respectively.
33
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
three terms are considered separately for the loading and
unloading phase, thus leading to the following:
L
T tot,L = Tm
+ TsL + TlL
(8)
tot,U
U
U
U
T
= Tm + Ts + Tl
(9)
TsL =
Among the three terms composing the total handling
time, the moving time Tm must be analysed in detail;
for doing this, a single train section is firstly considered.
In particular, we want to determine the time necessary
to handle a given number of containers in a section
(this handling time is the same both in the loading
case and in the unloading one). In order to do this,
we denote with Γ the maximum number of containers
in a section and with γ the number of containers to
be handled. Then, the section length is given by λΓ
(remember that λ is the average container length). In the
following, two propositions are reported relative to the
single train section (whose proofs are not reported in the
present paper for space limitations).
Proposition 1. Given a train section of λΓ length in which
γ containers must be loaded (or unloaded), the distance
covered by one shuttle Sm (γ) is given by the following
expression:
γ
X
i−2
Sm (γ) = γλΓ − 2λ
⌈
⌉
(10)
2
i=3
Proposition 2. Given a train section of λΓ length, the
optimal position of the bay is the central one (i.e. this
is the position such that the moving time is minimized).
λC
BL − 1
L
vB
(14)
and similarly, for containers to be unloaded, it is given by:
λC
TsU =
BU − 1
(15)
U
vB
Concerning the third term of the total handling time, that
is the lifting time Tl , for containers to be loaded it is
expressed by:
TlL = C L · τ + α
(16)
where τ represents the time for lifting one single container
and α is a constant that corresponds to the time needed
to approach the first container. For containers to be
unloaded, the lifting time is given by:
TlU = C U · τ + α
(17)
Let us consider the generalized case with N shuttles. In
order to avoid interferences among shuttles, we suppose
that the train length is divided in N equal parts and
each shuttle serves only one of them. In this way each
shuttle is allowed to move only within its competence
area without entering the other shuttle areas. As in the
case of one single shuttle, we want to define all the terms
composing expressions (8) and (9) in order to state the
problem constraints.
Considering that all the shuttles work at the same time,
the total time is obviously given by the time spent by the
most underprivileged shuttle. Therefore, in the following,
we will analyse the case of the most critical shuttle and
this corresponds to studying the total time. The critical
shuttle has to load and unload in a section a number of
containers given, respectively, by the following values:
Considering the results reported in the previous propositions, it is straightforward that, when all containers in the
section are handled (i.e. γ = Γ), the following applies:
Sm (Γ − 1) = Sm (Γ)
(11)
since the shuttle does not have to horizontally move
when handling the last container of the section (that is
considered to be the one in the centre of the section).
KL =
C
min( N
, CL)
BL
N
KU =
C
min( N
, CU )
BU
N
(18)
As a matter of fact, the most critical shuttle has to load
a quantity of containers that is the minimum between C L
(that is the case in which all the containers to be loaded
are in the competence area of the shuttle) and C/N . The
unloading case is analogous.
Let us consider now the case of one single shuttle. We aim
at defining all the terms composing expressions (8) and (9)
in order to state the problem constraints, i.e. (2)-(3) in case
of scenario 1, and (4)÷(7) in case of scenario 2. Note that
the case of one single shuttle corresponds to one shuttle
for scenario 1 and two shuttles (one per each train side)
in case of scenario 2. The moving time for loading a train
section is obtained from (10) (where γ is replaced with
C L /B L and Γ with C/B L ) and dividing by the shuttle
speed v. Moreover, in order to obtain the total moving
L
time for loading operations Tm
, it is necessary to multiply
by the number of loading bays B L :


⌈C L /B L ⌉
L
X
λ
C
C
i
−
2
L
Tm
=  L − 2B L
⌈
⌉
(12)
v
B
2
i=3
The moving time for loading a train section is obtained
from (10) (where γ is replaced with K L and Γ with C/B L ),
dividing by the shuttle speed v. Then, the total moving
L
time for loading operations Tm
is obtained multiplying by
the number of loading bays B L /N :


L
L
L ⌈K
X⌉ i − 2
λ
CK
B
L
Tm
= 
−2
⌈
⌉
(19)
v
N
N i=3
2
Analogously, the total moving time for unloading is the
following:


U
U
U ⌈K
X⌉ i − 2
λ
CK
B
U
Tm
= 
−2
⌈
⌉
(20)
v
N
N i=3
2
Similarly, the total moving time for unloading is the
following:


⌈C U /B U ⌉
U
X
λ
C
C
i
−
2
U
Tm
=  U − 2B U
⌈
⌉
(13)
v
B
2
i=3
The shifting time, respectively for loading and unloading,
is obtained as:
λC B L
L
Ts =
−1
(21)
vB L N
With regard to the shifting time Ts , that is the time spent
by the shuttle to move among bays, for the loading phase
it is given by the following expression:
34
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
TsU =
λC
vB U
BU
−1
N
Problem 1 cannot be solved by a mathematical programming solver, first of all because the upper term in the
sum in constraints (27) (i.e. ⌈K L ⌉ and ⌈K U ⌉) depends
on the decision variable N , according to (18). Moreover,
the terms K L and K U , which involve a min function, are
very critical for the solution of Problem 1. In order to
overcome these difficulties, an approximation is introduced
in the formulation of this problem. This is realized by
substituting the critical sum term with a polynomial of
second degree that fits the data in a least-squares sense.
More precisely, we adopt the following polynomial:
(22)
The lifting time is expressed by:
BL
·τ +α
N
BU
TlU = K U ·
·τ +α
N
TlL = K L ·
(23)
(24)
Note that the case with N shuttles also includes the case
with one shuttle. As a matter of fact, if N = 1, we obtain:
min(C, C L )
CL
K =
= L
L
B
B
min(C, C U )
CU
K =
= U
U
B
B
(25)
since C L ≤ C and C U ≤ C by definition. For this reason,
if N = 1, (12)-(13) become (19)-(20), as well as (14)-(15)
become (21)-(22).
L
a + bx + cx2 ≃
U
In this section, we want to state the optimization problem
relevant to the optimal sizing of the considered terminal.
Of course, two different approaches must be considered
for the two cases of scenario 1 and scenario 2. In this
paper, for the sake of brevity, only the case of scenario 1
will be analysed in detail. Anyway, referring to equations
(4)÷(7) instead of (2)-(3) and following the same line
of reasoning provided in this section for scenario 1, the
problem formulation for scenario 2 is straightforward.
More specifically, in the case of scenario 2, two separate
problems must be solved: one for the loading case and
the other for the unloading one. Note that there are cases
in which space limitations and specific terminal layouts
compel to choose a specific scenario. In other cases, it is
not possible to define the most suitable scenario a priori,
but the choice derives from the application of the proposed
procedure to the two cases.
By taking into account the results provided in Section 3,
the following problem can be stated for the case of scenario
1.
Problem 1. Find
min C = N · δ + B L · σ L + B U · σ U
subject to
(26)

⌈K L ⌉
λ  CK L
CK U
BL X i − 2
+
−2
⌈
⌉+
v
N
N
N i=3
2
BL
BU
+ τ KU
≤ T max
N
N
N ≥ 1, N ∈ N
+ 2α + τ K L
B L ≥ 1, B L ∈ N
U
U
B ≥ 1, B ∈ N
where K L and K U are provided by (18).
Note that ⌈x⌉ stands for ⌈K L ⌉ or ⌈K U ⌉ according to
the loading or the unloading case. Moreover, the problem
related to the min function (see (18)) can be overcome
C
by substituting ζ L = min( N
, C L ), by adding a term in
the objective function such that ζ L is maximized and by
imposing:
C
ζL ≤
(32)
N
L
L
ζ ≤C
(33)
C
Analogously, the substitution ζ U = min( N
, C U ) will be
added in the problem formulation. It is now possible to
provide another (approximate) version of the optimization
problem to be considered, as follows.
Problem 2. Find
M
M
min
Ce = N · δ + B L · σ L + B U · σ U + L + U
ζ
ζ
N, B L , B U , ζ L , ζ U
where M is a large number, subject to (26), (28), (29),
(30) and
"
!
2
λ Cζ L
Cζ U
BL
N ζL
N 2ζ L
+ U −2
a+b L +c
+
2
v BL
B
N
B
BL
!
#
2
BU
N ζU
N 2ζ U
2C
C
C
−2
a+b U +c
+
− L− U +
2
N
B
N
B
B
BU
+ 2α + τ B L + τ B U ≤ T max
C
ζL ≤
N
ζL ≤ CL
C
ζU ≤
N
ζU ≤ CU
N, B L , B U

⌈K U ⌉
BU X i − 2
2C
C
C 
−2
⌈
⌉+
− L − U +
N i=3
2
N
B
B
(31)
where a = 0.0294, b = −0.2920, and c = 0.2508.
4. THE PLANNING PROBLEM
B L + B U ≤ B max
⌈x⌉
X
i−2
⌈
⌉
2
i=3
(34)
(35)
(36)
(37)
(38)
Problem 2 is a nonlinear integer programming problem
representing an approximate formulation of Problem 1. It
can be solved by using commercial nonlinear mathematical
programming solvers. Alternatively, Problem 1 can be
faced by applying a solution algorithm based on the
following considerations. The cost function is a simple
formulation since it considers the cost of shuttles linearly
with the number of shuttles and the cost of bays linearly
with the number of bays. Therefore, it is necessary to
minimize the number of shuttles and bays. Since the cost
of shuttles is generally much higher than the cost of bays,
the main objective is the minimization of the number of
(27)
(28)
(29)
(30)
35
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
shuttles. Then, it is possible to define a solution algorithm
based on the following approach: at first the case with one
shuttle is considered. If, in this case, it is possible to find a
number of loading and unloading bays such that conditions
(26) and (27) are met, the algorithm stops giving a feasible
solution with one shuttle and the obtained number of
bays. Otherwise, the number of shuttles is increased one
by one and, again, conditions (26) and (27) are checked.
Actually, this algorithm faces the planning problem in a
“lexicographic” way considering the minimization of the
number of shuttles as the most important goal to be
reached and the definition of the number of bays as a
secondary objective to be pursued.
Table 1. Problem data.
The proposed algorithm determines, for the minimum
acceptable number of shuttles, all the possible feasible
solutions and chooses the best among them. To do this,
a set S = {(N, B L , B U )} is defined as the set gathering all
the feasible solutions found by the algorithm. This set is in
general composed of more than one triple because there are
different pairs of B L and B U , for a fixed value of N , that
guarantee the fulfilment of the problem constraints. Then,
the optimal solution is chosen as the triple (N̂ , B̂ L , B̂ U )
corresponding to the minimum value of the cost function
C.
Quantity
Value
Description
C
λ
v
α
τ
N max
B max
δ
σL
σU
47
10.64 [m]
1 [m/s]
15 [s]
10 [s/units]
50
30
100
1
1
Train capacity
Average container length
Shuttle speed
Constant term in the lifting time
Variable term in the lifting time
Maximum number of shuttles
Maximum number of bays
Cost of one shuttle
Cost of one loading bay
Cost of one unloading bay
loaded and unloaded, i.e. C L and C U . In Table 2 the data
concerning 6 different cases are shown.
Table 2. Different cases to study.
Moreover, it is necessary to define a maximum number of
shuttles N max . The algorithm is defined as follows.
Algorithm 1 Algorithm for scenario 1.
1: Initialize N = 0, S = ∅
2: while S = ∅ and N ≤ N max do
3:
N =N +1
4:
for B L = 1 to B max − 1 do
5:
for B U = 1 to B max − B L do
6:
if (27) is met then
7:
add {N, B L , B U } in S
8:
end if
9:
end for
10:
end for
11: end while
12: (N̂ , B̂ L , B̂ U ) = arg minS C
T max [s]
CL
CU
Case 1
2700
15
20
Case 2
1800
15
20
Case 3
1200
15
20
Case 4
2700
25
20
Case 5
2700
5
5
Case 6
1000
25
22
The corresponding solutions obtained by applying Algorithm 1 are reported in Table 3. Consider for instance Case
1, that is characterized by T max = 2700, C L = 15 and
C U = 20; the optimal solution is given by N̂ = 2, B̂ L = 6
and B̂ U = 8, with the optimal value of the cost function
equal to 214. This means that with only one shuttle no
feasible solution exists, i.e. the constraints (26) and (27)
are not met.
5. EXAMPLE
In order to clarify the concepts proposed above for scenario
1, a numerical example is provided. In the considered terminal the trains to be loaded and unloaded are, on average,
characterized by a length of 500 [m] and a capacity of
70 [T EU ]. Assuming that the twenty-foot and forty-foot
containers are almost present in the same quantity, 70
TEUs are obtained as the sum of 23 forty-foot containers
and 24 twenty-foot containers, thus C = 47. Moreover,
the average length λ of one container is fixed so that λC
is equal to the total length of the train; then, in this case,
λ = 10.64 [m]. All the problem data characterizing the
terminal are reported in Table 1. Note that the considered
cost terms are realistic and, in particular, the proportion
between the shuttle cost and the bay cost is the real one.
We have implemented Algorithm 1 by using Matlab software. In particular, we have considered different cases by
varying the values of the maximum stopping time for
a train, i.e. T max, and the number of containers to be
36
Table 3. Optimal solutions.
N̂
B̂ L
B̂ U
Cˆ
Case 1
2
6
8
214
Case 2
3
9
8
317
Case 3
4
12
12
424
Case 4
2
2
8
10
10
8
218
218
Case 5
1
1
3
5
5
3
108
108
Case 6
5
10
10
520
12th IFAC CTS (CTS 2009)
Redondo Beach, CA, USA, September 2-4, 2009
order to minimize the overall cost and taking into account
a set of specific constraints. The problem has been posed in
the form of a nonlinear constrained optimization problem
for which a specific algorithm has also been defined. A
numerical example has been reported at the end of the
paper in order to clarify the proposed procedure.
If for the same terminal and for the same number of
containers to be loaded and unloaded, the constraint (27)
becomes stricter, i.e. T max is lower, the number of shuttles
must be increased. In Case 2, when the maximum stopping
time for a train is T max = 1800, the obtained optimal
solution is characterized by N̂ = 3, B̂ L = 9, B̂ U = 8, with
the optimal value of the cost function equal to 317. In Case
3, the value of T max is further reduced and, as a result,
in the optimal solution the number of shuttles and bays is
further increased (with an increase of the cost value).
REFERENCES
Analogously, if the value of T max remains unchanged and
the values of C L and C U increase, this means that more
containers must be handled and, thus, more bays and/or
shuttles are needed. As a matter of fact, in Case 4 more
containers must be handled than in Case 1 and this results
in a solution with a higher number of bays (and a higher
cost). Note that, for Case 4 two equal solutions are found
(i.e., N̂ = 2, B̂ L = 8, B̂ U = 10, on one hand, and N̂ = 2,
B̂ L = 10, B̂ U = 8, on the other hand, both corresponding
to a cost value Cˆ = 218). In Case 5, instead, the opposite
situation is verified, i.e. fewer containers are to be handled
than in Case 1. Finally, if the stopping time for a train
decreases and the number of containers increases, as in
Case 6, it is necessary to provide the terminal with a higher
number of shuttles and bays.
Similar results can be obtained also by solving Problem 2
with a nonlinear programming solver (in particular we
used NPSOL solver). Nevertheless, it must be noted that
the obtained solution is not always completely reliable
since in Problem 2 the constraint (27) is partly approximated by means of a polynomial of second degree. Of
course, the approximation provided by this polynomial is
in some cases by defect and, in some others, by excess, and
therefore this is also true for the found solution. Moreover,
the dimensions of the problem instances are quite small,
not only in the proposed example but also in general for
realistic cases, meaning that the computational times are
not relevant. In any case, Algorithm 1 – implemented in
Matlab – always finds the solution in less than 1 second,
while some seconds are needed by the nonlinear solver (and
the computational time is not always the same). For this
reason, it is possible to conclude that the proposed solution
algorithm provides very satisfactory results and it is the
preferable way for solving Problem 1.
6. CONCLUSIONS
The strategic design of an automatic intermodal terminal
for logistic networks has been the objective of the present
paper. The considered kind of terminal is innovative because of the technology used to transfer containers in the
terminal. This new technology is provided by the Italian
system Metrocargo, that allows to load/unload containers
to/from trains in a horizontal way and operating under
the electric line. In this specific case, the main strategic
decision regards the number of elements composing the
transfer system acting in the terminal. These elements are
the shuttles, as well as the storage locations to be used in
the terminal.
An optimization problem has been stated in the paper with
the aim of determining the optimal number of resources in
37
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