Times are discretized and expressed as integers from 1 to

1440 (minutes in a day).

set of stations

set of trains

set of stations visited by train j

The timetable indicates the ideal departure time from the ideal arrival time at and the ideal

arrival and departure times for each intermediate station .

},...,1{ sS

},...,1{ tT },...,{ jj

j lfS

,jfjl

1,...,1 jj lf

Each train j is also assigned an ideal profit depending on the type of the

train (eurostar, freight, etc).

If the train is shifted and/or stretched the profit is decreased.

If the profit becomes null or negative the train is cancelled.

The objective is to maximize the overall profit of the (scheduled) trains.

jjjjj

j

j shift

j stretch (i.e. sum of the stretches in all stations)

A graph formulation(Time-Space Graph)

G=(V,A) directed acyclic graph

V set of nodes:

A set of arcs:

)...()...(},{ 112 ss WWUU

iU Arrival nodes at station iiW Departure nodes from station i

,......1 tj AAA TjTwo types of arcs:

Segment arcs

Station arcs

Train j

jf

jl

dep_node_1

dep_node_3

dep_node_2

arr_node_2

arr_node_3

arr_node_4

segment_arc_2

segment_arc_3

station_arc_2

station_arc_3

segment_arc_1

time

station

stat 1

stat 2

dep_1 dep_2

Departure Constraints:

Arrival Constraints:

stat 1

stat 2

arr_1 arr_2

Overtaking Constraints:stat 1

stat 2

jA

Timetable for train j“Feasible” path from tousing arcs in

Overall timetableSet of “feasible” paths, at most one path for each train.

Tj

jViW

notation

iU

jP

tPPP ...1

p PpuPujP

SiUuTj i ,,

SiUu i ,

Set of nodes belonging to train j

Set of departure nodes from station i

Set of arrival nodes to station i

Set of feasible paths for train

Set of feasible paths for all the trains

Actual profit for path

Set of paths that visit node u,

Set of paths belonging to train j, that visit node u

for the first arc leaving

= profit of arc a:

for the segment arcs

for station arcs corresponding to stretch

a

)( jj corresponding to shift

j

0

Profit of a path :jPp

pa

ap For each train Tj

1,0px Pp

1px path p in the solution

0px otherwise

jP

tPPP ...1

Set of feasible paths for train

We introduce a binary variable for each feasible path for a train.

Set of feasible paths for all the trains

Tj

MODEL B

Pp

ppxmax

jPppx ,1 Tj

dwwwWw Pp

pi w

x11:

,1 iWwsSi 1,\

auuuUu Pp

pi u

x11:

,1 iUuSi 1,1\

Tj wwwVWw Pp

pjjji w

j

x21:

,1

,1,\ ij WwsSi

,1,0px Pp

Tjwwgw tjj ),,...,( 1112

Solution method:

branch-and-cut-and-price

constructive heuristics

The solution method uses the LP-relaxation for:

obtaining an upper bound on the optimum solution value

suggesting good choices in the construction of the solution.

LP-relaxation

is solved using

column generation

and

constraint separation

as the number of variables and constraints is huge for real-

world instances.

Column generation

Add variables with positive reduced profits.

For each train, find the maximum profit path in an acyclic graph.

Reduced profit of path p in jP

pv

vpa

a

v Sum of dual variables of constraints involving node jVv

Constraint separation

Add constraints violated by the current solution.

departure/arrival constraints

overtaking constraints

Departure/arrival constraint separation

For each station, sum the values of the variables corresponding to paths that visit nodes in each

window smaller than minimum time interval between 2 departures

(analogously for arrivals).

minimum time interval between 2 departures

Overtaking constraint separation

(overtaking between 2 trains j,k)

1,..., 21 jj ww

1,..., 21 kk ww

jw1kw1i

1i

For each ,Si for eachjij VWw 1

andkik VWw 1 let :

},max{)( 11112j

ikk

ikkjj tatwdwwgw

kw2 analogous

jw2kw2

sum the values of the variables of train j in the window:

sum the values of the variables of train k in the window:

Overtaking constraint separation

(overtaking among 2 or more trains)

Order paths by decreasing speed.

Exact method (dynamic programming).

b

c

Clique: set of incompatible paths.Status: b,c.

bdu i 111 iazc

u

z