1. 15.3.2007 Train Scheduling in a Main Station Area © ETH Zürich | M. Fuchsberger Martin Fuchsberger Master thesis, Final Presentation Zurich, March 15. 2007

2. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 2 Outline  Introduction  Train Routing  Three Conceptual Models  Conflict Modeling  Results  Train Scheduling  Model  Results  Outlook

3. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 3 Introduction  Goal: Satisfy customers demands by finding suitable conflict-free timetables  Restrictions: Topology, Rolling stock, service requirements

4. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 4 Network Density Local LayerGlobal Layer Bottlenecks: Main Station Areas

5. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 5 Example Main Station Area: Bern  Radius of about 6 km  500 switches  6 main directions Olten Neuchatel Fribourg Bienne Bern Belp Thun

6. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 6 Example Main Station Area: Bern

7. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 7 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

8. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 8 Train Routing: Three conceptual models  Conflict Graph  Tree Conflict Graph  Resource Tree Conflict Graph

9. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 9 Train Routing Model: Conflict Graph 1 :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. 1 Zwaneveld et al. 1997

10. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 10 Train Routing: Conflict GraphSolution Approach: Independent Set with Cardinality equal to number of trains Train 1 Train 2 Train 3

11. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 11 Conflict Graph – Mathematical Model Only one node for each train Only non-connected nodes

12. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 12 Conflict Graph – Solving Problems  Finding exact solutions for larger problem instances took too much time 1.  The heuristic attempt (Randomized FPI 2 ) fails to find solutions for big instances. How can the model (structure) be improved? 1 Zwaneveld et al. 1997 2 Fixed Point Iteration, Herrmann, Burkolter et al. 2005

13. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 13 Improvement: Include Local Topology 022 3134477 5566 9 8810 02 34 56 78 8 56 78 8 78 A B C D E 314 79 5679 Y Z 1 Conflict BC D YZ A E 6 Conflicts Conflict Graph: Conflict

14. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 14 Train Routing Model: Tree Conflict Graph 1 :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. 1 Herrmann and Caimi 2005

15. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 15 Tree Conflict Graph - Solution Approach: Multi Commodity Flow 1.Add Sources and Sinks 2.Assign Variables x ij to the flow edges 3.Flow equal to 1 from source to destination 4.Conflict Constraints 5 6 7 8 10 8 8 8 8 77 6 5 4 3 2 01 3 4 5 7 6 9 7 9 x 00 S0S1 S0S1

16. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 16 Tree Conflict Graph – Mathematical Model

17. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 17 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 sectionswitchcrossingsingle slip

18. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 18 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...)

19. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 19 Conflict Modeling for a Resource Time Time Interval where the Resource is occupied by a train Conflicts between two trains Grouped conflicts between several trains = „Cliques“

20. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 20 How to gather the conflicts into cliques? Time B C F D I G 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

21. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 21 Resulting Train Routing Model: Resource Tree Conflict Graph 0 17 2 33 2 8 9 46 555 64 55 6 10 1115 8 12 13 9 14 2 17 14 13 18 1 2 3 46 55 0 1 2 3 4 Each Resource has its set of colored cliques.

22. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 22 Resource Tree Conflict Graph – Mathematical Model

23. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 23 Results for the Train Routing Problem Conflict Graph (CG&FPI) / Tree Conflict Graph (TCG) Resource Tree Conflict Graph Reduced #Conflicts! Better CPU time! Cause: Strong clique constraints!

24. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 24 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

25. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 25 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 T1Start time T2

26. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 26 Results for the Train Scheduling Problem in Bern East 2003 Adding more Start Times  Now bigger problem size compared to the train routing problem.  Still fast computable.

27. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 27 Results for the Train Scheduling Problem Bern East 2003

28. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 28 Example Clique Size Distributions Bern East 2003 One start time for each train Average Clique Size = 44 Ten selectable start times for each train Average Clique Size = 80

29. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 29 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 stations 1  Interaction with the global layer 1 Collaboration with SBB and sma (D. Burkolter)

30. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 30 Thank you for your attention!

31. 15.03.2007 M.Fuchsberger / D-INFK ETHZ / fumartin@student.ethz.ch 31 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 algorithms 1 1 Topic of the Master Thesis of Kaspar Schüpbach