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15.3.2007
Train Scheduling in a Main Station Area
© ETH Zürich | M. Fuchsberger
Martin Fuchsberger
Master thesis, Final Presentation
Zurich, March 15. 2007
215.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Outline
Introduction
Train Routing Three Conceptual Models Conflict Modeling Results
Train Scheduling Model Results
Outlook
315.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Introduction
Goal: Satisfy customers demands by finding
suitable conflict-free timetables
Restrictions: Topology, Rolling stock, service
requirements
415.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Network Density
Local LayerGlobal Layer
Bottlenecks: Main Station Areas
515.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Example Main Station Area: Bern
Radius of about 6 km
500 switches
6 main directionsOlten
Neuchatel
Fribourg
Bienne
Bern
Belp
Thun
615.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Example Main Station Area: Bern
715.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Two Problems in Main Station Areas
1. Train Routing
Input: Topology, Rolling Stock, Departure
Times at portals/platforms
Output: Train Routings
2. Train Scheduling
Input: Topology, Rolling Stock, Departure Time
Windows at portals/platforms
Output: Conflict-free Timetable
815.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Routing:Three conceptual models
Conflict Graph
Tree Conflict Graph
Resource Tree Conflict Graph
915.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Routing Model: Conflict Graph1
: Each routing of a train corresponds to a node
in the conflict graph.
: Conflict Edges model conflicts between two
routings (nodes).
: As one train can use only one routing, the
routings of a train form a clique.
1Zwaneveld et al. 1997
1015.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Routing: Conflict GraphSolution Approach: Independent Set with Cardinality equal to number of trains
Train 1
Train 2
Train 3
1115.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Conflict Graph – Mathematical Model
Only one nodefor each train
Only non-connectednodes
1215.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Conflict Graph – Solving Problems
Finding exact solutions for larger problem instances
took too much time1.
The heuristic attempt (Randomized FPI2) fails to find
solutions for big instances.
How can the model (structure) be improved?
1Zwaneveld et al. 1997
2Fixed Point Iteration, Herrmann, Burkolter et al. 2005
1315.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Improvement: Include Local Topology
0 2 2
31 3 4 4 7 7
5 5 6 6
9
8 8 10
0 23 4
5 6
7 8 10
8 10
5 6
7 8 10
8 10
7 8 10
A
B
C
D
E
31 47 9
5 6 7 9
Y
Z
1 Conflict
B C D
Y Z
A E
6 Conflicts
Conflict Graph:
Conflict
1415.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Routing Model: Tree Conflict Graph1
: For each tuple (train,time,topology-
element,velocity) a node is created.
: Red edges model conflicts between two
tuples (nodes).
: Flow edges model the routings from origin to
destination.
1Herrmann and Caimi 2005
1515.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Tree Conflict Graph - Solution Approach: Multi Commodity Flow
1. Add Sources and Sinks
2. Assign Variables xij to the flow edges
3. Flow equal to 1 from source to destination
4. Conflict Constraints
5
6
7
8
10
8
10
8
10
8
10
8
10
7 7
6
5
4
3
2
0 1
3
4
5
7
6
9
7
9
x00
S0 S1
S0 S1
1615.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Tree Conflict Graph – Mathematical Model
1715.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Allocation of a Resource
A resource is composition of track elements.
A resource can be allocated by trains for a
closed time interval („allocation time interval“).
A conflict exists, if a resource is allocated by
more than one train at the same time.
track section switch crossing single slip
1815.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Allocation Time Intervals
How are the allocation time intervals of the different
trains determined?
An allocation time interval consists of: Occupation time
Minimal braking time: Based on track signals
Additional security related times (track switch, reaction...)
1915.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Conflict Modeling for a Resource
TimeTime Interval where the Resource is occupied by a train
Conflicts between two trains
Grouped conflicts between several trains = „Cliques“
2015.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
How to gather the conflicts into cliques?
Time
B
C
FD
IG
E
J
H
A A‘
A1 B1 A2 B2 C1 D1 E1 F1 D2 C2 G1 H1 G2 F2 E2 I1 J1 H2 I2 A‘1J2
Minimum number of cliques to cover all the edges in the corresponding circular interval graph
2115.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Resulting Train Routing Model: Resource Tree Conflict Graph
0
1 7
2
3 3
2 8
9
4 6
5 5 5
64
5 5
6
10
11 15
8
12
13 13
9
14
2
17
14
13
18
1
2
3
4 6
5 5
0
1
2
3
4
Each Resource has its set of colored cliques.
2215.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Resource Tree Conflict Graph – Mathematical Model
2315.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Results for the Train Routing Problem
Scenario # nodes # conflictBern Min Average Max Construction Solving
West 2003 12 000 700 2 12 130 <1 <1East 2003 46 500 3400 2 50 590 1 2West 2020 15 000 900 2 35 300 <1 <1East 2020 58 000 4300 2 60 1100 2 2
time [s]Clique size
Scenario # conflict # conflictBern TCG CG Construction Solving Construction Solving
West 2003 15 500 70 000 4 <1 1 5East 2003 85 500 740 000 50 4 7 15West 2020 56 500 300 000 10 2 3 7East 2020 1 110 000 7 100 000 930 180 80 2400
FPI time [s]TCG time [s]
Conflict Graph (CG&FPI) / Tree Conflict Graph (TCG)
Resource Tree Conflict Graph
Reduced #Conflicts! Better CPU time!
Cause: Strong clique constraints!
2415.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Scheduling
Discretise the time windows to create several start times
for each train (Puls 90 SBB Project).
Each train has then a set of Resource Tree Conflict
Graphs with distinct and selectable starting times.
Solve the train scheduling problem using the same
algorithms.
Train Routing
Input: Topology, Rolling Stock,
Departure Times at portals/platform
Output: Train Routings
Train Scheduling
Input: Topology, Rolling Stock, Departure Time Windows at
portals/platforms
Output:Conflict-free Timetable
2515.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Train Scheduling – Prefered Start Times
Some start times may be preferred over others.
Incorporate this idea in the model by using
weights in the objective function:Train 1
Start time T1 Start time T2
2615.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Results for the Train Scheduling Problem
in Bern East 2003#Start Times #nodes #constraints
Min Average Max Construction Solving
1 175 000 3 900 2 44 851 3 32 350 000 7 900 2 46 851 4 54 700 000 14 500 2 54 851 6 106 1 050 000 22 300 2 68 1068 9 188 1 400 000 27 200 2 74 1068 11 27
10 1 750 000 34 800 2 80 1068 14 35
Clique Size time[s]
Ad
din
g m
ore
Sta
rt T
imes
Now bigger problem size compared to the train
routing problem.
Still fast computable.
2715.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Results for the Train Scheduling Problem
Bern East 2003
05
1015
2025
3035
40
0 500000 1000000 1500000 2000000
# Nodes
Tim
e [
s]
Solving Time Construction Time (suboptimal)
2815.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Example Clique Size DistributionsBern East 2003
One start time for each trainAverage Clique Size = 44
Ten selectable start times for each trainAverage Clique Size = 80
2915.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Outlook
Further enhance the model (English track
switchs, more complex resources)
Extend testing on other main station areas
besides Bern (data from SBB is required)
Check performance on track regions between
stations1
Interaction with the global layer
1 Collaboration with SBB and sma (D. Burkolter)
3015.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Thank you for your attention!
3115.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]
Scheduling: Connecting Global Layer to Local Layer
On the global layer, the train scheduling problem is
usually modeled as a periodic event scheduling problem
(PESP)
The solution of a PESP provides input data for the
discussed train routing algorithms
A modified PESP could support time windows and hence
serve as an input for (local) train scheduling algorithms1
1Topic of the Master Thesis of Kaspar Schüpbach