OPTIMIZATION IN RAILWAY SCHEDULING

M.A. Salido

, M. Abril

, F. Barber

, L. Ingolotti

,A.Lova

, P. Tormos

Departamento de Sistemas Inform

aticos y Computaci

Departamento de Estad

ıstica e I.O y Calidad

Universidad Polit

ecnica de Valencia

Keywords:

optimization, constraint optimization, railway scheduling problem.

Abstract:

Train scheduling has been a signiﬁcant issue in the railway industry. Over the last few years, numerous ap-

proaches and tools have been developed to aid in the management of railway infrastructure. In this paper, we

describe some techniques, which was developed in a project in collaboration with the Spanish Railway In-

fractructure Manager (ADIF). We formulate train scheduling as constraint optimization problems and present

two ﬁltering techniques for these problem types. These ﬁltering techniques are developed to speed up and

direct the search towards suboptimal solutions in periodic train scheduling problems. The feasibility of our

problem-oriented techniques are conﬁrmed with experimentation using real-life data. The results show that

these techniques enables MIP solvers such as LINGO and ILOG Concert Technology (CPLEX



) to terminate

earlier with good solutions.

1 INTRODUCTION

Railway transportation has played a major role in the

economic development of the last two centuries. It

represented a major improvement in land transport

technology and has obviously introduced important

changes in the movement of freight and passengers.

Over the last few years, railway trafﬁc has increased

considerably, which has created the need to optimize

the use of railway infrastructures. This is, however,

a very difﬁcult task. Nevertheless, numerous ap-

proaches and tools have been developed to aid in the

management of railway infrastructure. These systems

provide advanced graphical interfaces, but they still

lack of beneﬁts for automatic planning of efﬁcient

and robust scheduling. Thanks to developments in

computer science and advances in the ﬁelds of opti-

mization and intelligent resource management, rail-

way managers can optimize the use of available in-

frastructures, obtain more robust timetables and ob-

tain useful conclusions about capacity of their topol-

ogy.

We describe some results of a long-term collab-

oration between our research group and the Span-

This work has been partially supported by the grant

TIN2004-06354-C02- 01 (MEC, Spain - FEDER) and

GV04B/516 (Generalitat Valenciana, Spain).

ish Railway Infractructure Manager (ADIF) (Mom,

2005). The aim of the project is to offer assistance in

the planning of train scheduling to obtain conclusions

about the maximum capacity of the network, to iden-

tify bottlenecks, to provide support in the resolution

of incidents, etc. Besides the mathematical processes,

a high level of interaction with railway experts is re-

quired to be able to take advantage of their experi-

ence.

In this paper, we propose two problem-oriented ﬁl-

tering techniques for solving periodic train schedul-

ing. The train scheduling problem has received con-

siderable attention in the literature: (Szpigel, 1972) is

the ﬁrst to propose a branch and bound algorithm for

train scheduling; (Higgins, 1997) deﬁne local search,

tabu search, genetic and hybrid heuristics; (Cai, 1994)

illustrate a constructive greedy heuristic. Periodic

timetables for railway networks is usually modeled

by Periodic Event Scheduling Problem (PESP) (Ser-

aﬁni, 1989). It is known that the PESP is NP-hard

(Seraﬁni, 1989). Approaches to solve PESP instances

cover backtracking strategies in a branch-and-bound

context (Seraﬁni, 1989), genetic algorithms (Nachti-

gall, 1996), and some classes of cutting planes (Odijk,

1994). Furthermore, several European companies are

also working on similar systems. These systems in-

clude complex stations, rescheduling due to incidents

188

A. Salido M., Abril M., Barber F., Ingolotti L., Lova A. and Tormos P. (2005).

OPTIMIZATION IN RAILWAY SCHEDULING.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 188-195

DOI: 10.5220/0001176101880195

 SciTePress

(Chiu et al., 2002), rail network capacities (Kaas,

1998), etc. These are complex problems for which

work in network topology and heuristic-dependent

models can offer appropriate solutions.

The problem formulation is (traditionally) trans-

lated into a formal mathematical model to be solved

for optimality by means of mixed integer program-

ming (MIP) techniques. However, in realtime-

environments, with hundred of trains, in different di-

rections, along paths of dozens of stations, with con-

straints about departure and arrival times, generate

thousands of inequalities and a high number of vari-

ables take only integer values. As is well known,

this type of model is far more difﬁcult to solve than

linear programming models. In our framework, the

formal mathematical model is simpliﬁed by ﬁltering

techniques in order to speed up the efﬁciency of well-

known solvers.

2 PRELIMINARIES

2.1 Terminology

A running map contains information regarding rail-

way topology (stations, tracks, distances between sta-

tions, trafﬁc control features, etc.) and the schedules

of the trains that use this topology (arrival and depar-

ture times of trains at each station, frequency, stops,

junctions, crossings, etc,). A sample of a running map

is shown in Figure 1, where several train crossings can

be observed. On the left side of Figure 1, the names of

the stations are presented and the vertical line repre-

sents the number of tracks between stations (one-way

or two-way). The objective of our system is to obtain

a correct and optimized running map taking into ac-

count: (i) trafﬁc rules, (ii) user requirements and (iii)

the railway infrastructure topology.

A railway network is basically composed of sta-

tions and one-way or two-way tracks. A dependency

can be: Station: is a place for trains to park, stop

or pass through. Each station is associated with a

unique station identiﬁer. There are two or more tracks

in a station where crossings or overtaking can be

performed; Halt: ia a place for trains to stop, pass

through, but not park. Each halt is associated with a

unique halt identiﬁer.

In Figure 1, horizontal dotted lines represent halts,

while continuous lines represent stations. On a rail

network, the user needs to schedule the paths of n

trains going in one direction and m trains going in the

opposite direction. These trains are of a given type

and a scheduling frequency is required.

The type of trains to be scheduled determines the

time assigned for travel between two locations on the

path. The path selected by the user for a train trip

determines which stations are used and the stop time

required at each station for commercial purposes. In

order to perform crossing in a section with a one-way

track, one of the trains should wait in a station. This is

called a technical stop. One of the trains is detoured

from the main track so that the other train can cross

or continue.

2.2 Problem Statement

There are three groups of scheduling rules in our rail-

way system: trafﬁc rules, user requirements rules and

topological rules. A valid running map must satisfy

and optimize the above rules. These scheduling rules

can be modeled using the following constraints:

1. Trafﬁc rules guarantee crossing and overtaking

operations. The main constraints to take into ac-

count are:

• Crossing constraint: Any two trains going in

opposite directions must not simultaneously use

the same one-way track.

The crossing of two trains can be performed only

on two-way tracks and at stations, where one of

the two trains has been detoured from the main

track. Several crossings are shown in Figure 1.

• Expedition time constraint. There exists a given

time to put a detoured train back on the main

track and exit from a station.

• Reception time constraint. There exists a given

time to detour a train from the main track so that

crossing or overtaking can be performed.

2. User Requirements: The main constraints due to

user requirements are:

• Type and Number of trains going in each direc-

tion to be scheduled.

• Path of trains: Locations used and Stop time for

commercial purposes in each direction.

• Scheduling frequency. The frequency require-

ments of the departure of trains in both direc-

tions. This constraint is very restrictive because,

when crossing are performed, trains must wait

for a certain time interval at stations. This in-

terval must be propagated to all trains going in

the same direction in order to maintain the es-

tablished scheduling frequency. The user can

require a ﬁxed frequency, a frequency within

a minimum and maximum interval, or multiple

frequencies.

• Departure interval for the departure of the ﬁrst

trains going in both the up and down directions.

• Maximum slack. This is the maximum percent-

age δ that a train may delay with respect to the

minimum journey time.

OPTIMIZATION IN RAILWAY SCHEDULING

189

INFORM

Stations

Halts

Time

Paths

Tracks

Figure 1: A sample of a running map

3. Topological railway infrastructure and type of

trains to be scheduled give rise other constraints to

be taken into account. Some of them are:

• Number of tracks in stations (to perform techni-

cal and/or commercial operations) and the num-

ber of tracks between two locations (one-way or

two-way). No crossing or overtaking is allowed

on a one-way track,

• Time constraints, between each two contiguous

stations,

• Added Station time constraints for technical

and/or commercial purposes.

In accordance with user requirements, the system

should obtain the best solution available so that all

the above constraints are satisﬁed. Several criteria can

exist to qualify the optimality of solutions: minimize

duration and/or number of technical stops, minimize

the total time of train trips (span) of the total schedule,

giving priority to certain trains, etc.

2.3 The Formal Mathematical Model

Our formal mathematical model can be described as a

constraint optimization problem, where the main ob-

jective function is to minimize the journey time of all

trains. Variables are frequencies, arrival and departure

times of trains at stations and binary auxiliary vari-

ables generated for modelling disjunctive constraints.

Constraints are composed by user requirements, traf-

ﬁc rules, and topological constraints. These con-

straints are composed by the parameters deﬁned by

user interfaces and database accesses.

The formal mathematical model is presented in Ta-

ble 1. Let’s suppose a railway network with r sta-

tions, n trains running in the down direction, and m

trains running in the up direction. We assume that

two connected stations have only one line connect-

ing them. T

represents that train i arrives at sta-

tion k; T

means that train i departs from station k;

Timei

k−(k+1)

is the journey time of train i to travel

from station k to k +1; TSi

and CSi

represent

the technical and commercial stop times of train i in

station k, respectively; and ET

and RT

are the ex-

pedition and reception time of train i, respectively.

The main complexity of the problem derives in

solving the MIP problem due to the binary (integer)

variables. These integer variables are generated to

manage disjunctive constraints. If we are able to as-

sign values to these integer variables, the linearized

problem can be solved more efﬁciently. Therefore,

the main goal of our ﬁltering techniques is to ﬁnd val-

ues for these integer variables. This assignment will

be carried out by means of local search and railway

topological knowledge.

3 FILTERING TECHNIQUES

Given the formal mathematical model presented in

Table 1, the problem turns into a MIP problem, in

which thousands of inequalities have to be satisﬁed

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190

Table 1: Formal Mathematical Model of the railway scheduling problem.

(1) Min

i=n

i=1

− T

i=m

j=1

− T

);

Subject To

/frequency constraint ∀i =1..n, ∀k =1..r

(2) T

i+1

− T

= F r equency;

/Time Constrains ∀i =1..n, ∀k =1..r

(3.1) T

k+1

− T

= Timei

k−(k+1)

;

(3.2) T

− T

k+1

= Timei

k−(k+1)

;

/Stations Time Constrains ∀i =1..n, ∀k =1..r

(4) T

− T

− TSi

= CSi

;

/Constrains to limit journey time ∀i =1..n, ∀j =1..m

(5.1) T

− T

≤ (1 +

100

) ∗ Timei

1−r

;

(5.2) T

− T

≤ (1 +

100

) ∗ Timej

r−1

;

/Crossing Constrains ∀i =1..n, ∀j =1..m, ∀k =1..r

(6.1) T

− T

<= 86400 ∗ Y

i−j;k−(k+1)

;

(6.2) T

k+1

− T

k+1

<= 86400 ∗ (1 − Y

i,j,k

);

/Expedition time constrains ∀i =1..n, ∀j =1..m, ∀k =1..r

(7.1) T

− T

− 86400 ∗ (X

i,j

− Y

i,j,k

+ Y

i,j,k+1

− 1) + ET

<=0;

(7.2) T

− T

− 86400 ∗ (X

i,j

− Y

i,j,k

+ Y

i,j,k+1

− 2) + ET

<=0;

/Reception time constrains ∀i =1..n, ∀j =1..m, ∀k =1..r

(8.1) T

− T

− 86400 ∗ (X

i,j

− Y

i,j,k

+ Y

i,j,k+1

− 1) + RT

<=0;

(8.2) T

− T

− 86400 ∗ (X

i,j

− Y

i,j,k

+ Y

i,j,k+1

− 2) + RT

<=0;

/Binary Constraints

i−j

; ∀i =1..n, ∀j =1..m

i−j;k−(k+1)

; ∀i =1..n, ∀j =1..m, ∀k =1..r

and a high number of variables only take integer val-

ues. As is well known, this type of model is far more

difﬁcult to solve than linear programming models.

Our ﬁltering techniques work on the binary variables.

These variables are grouped into two sets:

• Variables Y



s. A variable Y

i,j,k

determines the

track between station k and station k +1in which

train i crosses with train j.IfY

i,j,h−1

=1for

h ≤ k and Y

i,j,p

=0for p ≥ k then, the crossing

between train i and train j is carried out in station

• Variables X



s. A variable X

i,j

determines which

train (i or j) arrives earlier to the crossing station.

If X

i,j

=0train i arrives at the crossing station

ﬁrst and train j arrives at the same station later.

3.1 Filtering Technique 1

This technique carries out a ﬁltering over the set of

constraints from the formal mathematical model pre-

sented in Table 1. Many constraints of type (6) (7) and

(8) can be removed according to their departure times

and maximum slacks. If a train going in the down di-

rection arrives at the destination before a train going

in the up direction departs, then both trains will not

cross each other. Thus, a huge number of constraints

and integer variables we can eliminated. The original

problem maintains n ∗ m ∗ r integer variables (Y



and n ∗ m integer variables (X



s). A railway net-

work with 100 stations and 100 trains going in each

direction generates 1.01x10

integer variables. This

technique may signiﬁcantly reduce the problem size

with a reasonable maximum slack (α ≈ 20%) (given

by railway operator).

Theorem 2. Filtering technique 1 is sound and

complete.

Proof. Soundness. Filtering technique 1 is sound

due to the fact that the set of solutions given by Ta-

ble 1 subsumes the set of solutions obtained by ﬁlter-

ing technique 1. This is because it has removed a set

of binary variables and the constraints in which these

variables are involved.

Completeness. Filtering technique 1 does not re-

move any solution. Thus, this technique will ﬁnd the

same solution as the one obtained by Table 1. Con-

straints of type (5) make the set the of removed con-

straints redundant by ﬁltering technique 1. By contra-

diction, we assume that there is a solution that ﬁlter-

ing technique 1 does not ﬁnd. Without loss of gener-

ality, we assume a maximum slack of 20%. We can

distinguish two different cases. (1) The lost solution

falls into the maximum slack. This os a contradiction

because, under this threshold, the restricted problem

is the same as Table 1. (2) The lost solution falls out-

side the maximum slack. This solution is not valid

because it does not satisfy constraints (5.1) and (5.2).

Therefore, ﬁltering technique 1 does not lose any so-

lution.

OPTIMIZATION IN RAILWAY SCHEDULING

191

3.2 Filtering technique 2

Filtering technique 2 is a metaheuristic based on Fil-

tering technique 1. This technique carries out a

guided local search over the binary variables. Once

many integer variables have been removed by ﬁlter-

ing technique 1, a new ﬁltering process on the re-

duced problem can eliminate other integer variables

by means of a guided local search. Instead of assign-

ing a random station as a crossing station between

two opposite trains, ﬁltering technique 2 performs a

linearized execution where the integer variables have

been transformed into continues ones. Thus, the

crossing between two trains may not be assigned in

stations but on a track between two stations. This will

be the initial point to start the search to ﬁnd the sta-

tion where the crossing will ﬁnally be performed. The

search of each crossing between two opposite trains is

bounded by 2n +1contiguous tracks. This interval

is composed by n tracks located before the obtained

crossing and n tracks located after the crossing. In

this way, the resultant subproblem can be seen as a

combinatorial problem, where all combinations must

be performed for guarantee the best possible solution.

If the problem has a solution, ﬁltering technique 2

studies the arrival order to the crossing station such

as the objective function is minimized. Otherwise,

the interval is increased (n ++) and the MIP prob-

lem is again solved. This technique is useful in any-

time environment due to a solution can be found, but

ﬁltering technique 2 tries to ﬁnd a better solution in

the remaining time. To this end, each combination is

labelled with the solution obtained and the ﬁltering

technique searches neighbor combination in order to

improve the objective function.

Table 2: Pseudo-code of ﬁltering technique 2.

Filtering technique 2

{ /*Limit the stations where two trains can be crossed,

i.e., the number of integer variables*/

DeterminePossibleCrossing();

LinearSolution=SolveLinearProblem();

Crossings=DetectCrossings(LinearSolution);

n=0;

while(Not Solution) {

Solution=SearchCrossCombination(window,Crossings);

n++;

}

if (Solution)

FinalSolution=SolveCrossingOrder(Solution);

}

4 GENERAL SYSTEM

ARCHITECTURE

The general outline of our system is presented in Fig-

ure 2. It shows several steps, some of which require

the direct interaction with the human user to insert

requirement parameters, parameterize the constraint

solver for optimization, or modify a given schedule.

First of all, the user should require the parameters of

the railway network and the train type from the central

database (Figure 2).

This database stores the set of locations, lines,

tracks, trains, etc. Normally, this information does not

change, but authorized users may desire to change this

information. With the data acquired from the data-

base, the system generates the formal mathematical

model.

According to the quality of the required solution

and the problem size, a ﬁltering technique will be ex-

ecuted by one of the following ways:

1. Complete: The process is performed taking into ac-

count the entire problem. This decision is carried

out when the number of trains and stations is low

or the running time is not a important. In this case,

ﬁltering technique 1 will be selected.

2. Incremental: The process performs an incremental

coordination of trains. It can be useful in anytime

systems, where the number of trains and stations

is not very high. In this case, ﬁltering technique

1 and 2 are appropriate due to the fact that as the

number of combinations are checked, the quality

of the solution is better.

Once the problem has been ﬁltered, the optimiza-

tion process will be executed for obtaining an opti-

mal solution of the simpliﬁed problem. To this end,

CPLEX and LINGO are executed for obtaining the

optimal solution.

However, the system can also automatically recom-

mend or select the appropriate choice depending on

main parameters and the complexity of the problem.

If the mathematical model is not feasible, the user

must modify the parameters, mainly the most restric-

tive ones. If the running map is consistent, the graphic

interface plots the scheduling. Afterwards, the user

can graphically interact with the scheduling to mod-

ify the arrival or departure times. Each interaction

is automatically checked by the constraint checker in

order to guarantee the consistency of changes. The

user can ﬁnally print out the scheduling, to obtain re-

ports with the arrival and departure times of each train

in each location, or graphically observe the complete

scheduling topology.

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192

Data

Base

Starting Interface

Formal

Mathematical

Model

INFORM

Plotter

Constraint

Checker

Timetabling

Graphic Interface

General Scheme

Optimization

Process

Filtering

Technique 1

Filtering

Technique 2

Running Map

Data

Figure 2: General scheme of our tool.

Figure 3: System Interface solving and plotting instance <10,40,90>.

4.1 A Constraint-based System for

Automatic Railway Scheduling

Our techniques described here can be extended to

general problems such as: inserting a new train or

several new trains in an already compatible running

map and optimizing running maps. In this case, new

constraints must be taken into account such as clos-

ing constraints, exclusiveness constraints, precedence

constraints, etc. Figure 4 shows an example of a high

OPTIMIZATION IN RAILWAY SCHEDULING

193

loaded running map with the proposed system.

Out tool provides several beneﬁts for automatic

railway infrastructure scheduling.

• It is ﬂexible and friendly. It can be easily inte-

grated into already existing data-bases and other

computer-aided tools for railway management.

• It automatically obtains optimized and well-formed

running maps (timetables).

• It can validate complex timetables by automatically

performing all the consistency checks required for

well-formed timetables.

• It can validate and perform capacity analyses.

• It can reschedule running maps according to inci-

dences and delays in on-line, trafﬁc management.

Figure 4: Example of a high loaded running map.

5 EVALUATION

The application and performance of this system de-

pends on several factors: Railway topology (loca-

tions, distances, tracks, etc.), number and type of

trains (speeds, starting and stopping times, etc.), fre-

quency ranges, initial departure interval times, etc.

In this section, we compare the performance of

our ﬁltering techniques using some well-known con-

straint optimization problem solvers, CPLEX and

LINGO, because they are the most appropriate tools

for solving these types of problems.

This empirical evaluation was carried out on a real

railway infrastructure that joins two important Span-

ish cities (”La Coru

na” and ”Vigo”). The journey be-

tween these two cities is currently divided by 40 de-

pendencies between stations (23) and halts (17).

In our empirical evaluation, each set of instances

was deﬁned by the 3-tuple < n,s,f >, where n was

the number of trains in each direction, s the number of

stations/halts and f the frequency. The problems were

generated by modifying these parameters. Thus, each

of the tables shown sets two of the parameters and

varies the other one in order to evaluate the algorithm

performance when this parameter increases. It must

be taken into account that running time of the form

”>xh.” represents that the problem did not ﬁnish in

x hours and the best solution found up to date is pre-

sented in the journey time column. All running times

in Table 3 represent the running times of the ﬁltering

techniques plus the running times of the optimization

techniques (CPLEX or LINGO).

In Table 3 (a), we present the running time and the

journey time in problems where the number of trains

was increased from 5 to 75, and the number of sta-

tions/halts and the frequency were set at 40 and 90,

respectively: <n,40, 90 >. The results show as the

number of trains increased the running time of Filter-

ing technique 1 and 2 was worse. Filtering technique

1 obtained the optimal solution for 5,10,15 and 20

trains. However for 50 and 75 trains, Filtering tech-

nique 1 was aborted in 5 hours while Filtering tech-

nique 2 ﬁnished although with worse solutions. Fig-

ure 3 shows the system interface executing our Fil-

tering technique 2 with the instance < 10, 40, 90 >.

The ﬁrst window shows the user parameters, the sec-

ond window presents the best solution obtained at that

point, the third window presents data about the best

solution found, and ﬁnally the last window shows the

obtained running map.

Table 3 (b) shows the running time and the jour-

ney time in problems where the number of stations

was increased from 10 to 60, and the number of trains

and the frequency were set at 10 and 90, respectively:

< 10,s,90 >. In this case, only stations were in-

cluded to analyze the behavior of the techniques. It

can be observed that Filtering technique 2 was bet-

ter than Filtering technique 1 obtaining optimal solu-

tions for 10 and 20 stations in lower time. Eve for

30 stations Filtering technique 2 had better behaviour

than Filtering technique 1 (complete algorithm). It

is important to note the difference between the in-

stance < 10, 40, 90 > of the Table 3 (a) and the in-

stance < 10, 40, 90 >

in Table 3 (b). They repre-

sent the same instance; however in Table 3 (b) we

only used stations (no halts), so the number of pos-

sible crossing between trains was much larger (more

integer variables). This item reduced the journey time

from 2:20:19 to 2:20:10, using Filtering technique 1

and from 2:26:04 to 2:23:36, using CPLEX and Filter-

ing technique 2. Nevertheless, the running time also

increased from 337” to 2131 in Filtering technique 1

and from 8” to 56” in Filtering technique 2, due to the

number of integer variables was much larger.

In Table 3 (c), we present the running time and the

journey time in problems where the frequency was de-

creased from 140 to 60 and the number of trains and

the number of stations were set at 20 and 40, respec-

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194

Table 3: Running time and journey time in different problem instances.

CPLEX LINGO

(a) <n,40,90> Filtering technique 1 Filtering technique 2 Filtering technique 2

Trains running time journey running time journey running time journey

time time time

5 6” 2:19:48 4” 2:29:33 6” 2:30:54

10 337” 2:20:19 8” 2:22:08 12” 2:31:37

15 601” 2:20:29 12” 2:26:18 19” 2:31:51

20 1065” 2:20:34 16” 2:26:25 25” 2:31:58

50 > 5h. 2:20:43 43” 2:31:09 1098” 2:32:11

75 > 5h. 2:22:04 > 1h. 2:32:14 1590” 2:32:14

(b) <10,s,90>

10 3” 0:25:06 2” 0:25:06 4” 0:25:06

20 303” 1:04:11 5” 1:04:11 8” 1:04:11

30 > 1h. 1:45:38 6” 1:45:08 14” 1:45:38

40 2131” 2:20:10 56” 2:23:36 21” 2:24:36

60 > 3h. 3:33:15 217” 3:39:30 180” 3:40:30

140 15” 2:16:19 15” 2:20:18 24” 2:16:19

120 156” 2:16:17 14” 2:16:17 23” 2:18:47

100 > 5h. 2:22:55 15” 2:23:10 28” 2:22:55

90 1065” 2:20:34 15” 2:26:25 28” 2:31:58

75 > 1h. 2:29:18 > 1h. - 25” 2:24:16

60 > 1h. 2:21:23 > 1h. - > 1h. -

tively: < 20, 40,f >. As the frequency decreased,

the process solving become harder. The quality of

the solutions depends mainly of the network topology.

For this reason, Filtering technique 2 obtained the

same solutions than Filtering technique 1 but lower

running times with frequencies of 140, 120 and 100

minutes. It can be observed that depending on the

problem topology, one technique may be better than

the others. Therefore, it may be useful for the system

to automatically select the appropriate technique.

6 CONCLUSIONS

We have reported the design and development of two

ﬁltering techniques for solving periodic train schedul-

ing, which is a project in collaboration with the Na-

tional Network of Spanish Railways (RENFE), Spain.

We have formulated the train scheduling as constraint

optimization problems. Two ﬁltering techniques are

developed to speed up and direct the search towards

sub-optimal solutions. The feasibility of our algo-

rithms are conﬁrmed with experimentation using real-

life data. These techniques have been inserted into the

system to solve periodic timetables more efﬁciently.

This system is already integrated and assist to railway

managers in optimizing the use of railway infrastruc-

tures and will also help them in the resolution of com-

plex scheduling problems. It supposes the application

of methodologies of Artiﬁcial Intelligence in a prob-

lem of great scientiﬁc and commercial interest.

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