Hump Yard Track Allocation with Temporary Car Storage RailRome 2011

1. Hump Yard Track Allocation with Temporary Car StorageRailRome 2011 Markus Bohlin SICS • Holger Flier Jens Maue MatusMihalak ETH Funded by Swedish Transport Administration and Swiss National Science Foundation

2. Outline Problem definition Complexity The mixing problem Experiments M. Bohlin

3. Problem definition M. Bohlin

4. Hump Yard Trackallocation Roll-in Roll-out Dep. train formation … … M. Bohlin

5. Temporarycarstorage (”Mixing”) Tracksreserved for ”mixed” use Immediate roll-in Pull-out M. Bohlin

6. CurrentPractice • Planning by hand • Default: roll-in order = arrival order • Pull-backs are planned in advance • Partialpull-backs • Train formation on multiple tracks • Multiple trains on onetrack M. Bohlin

7. Solution Approach Step 1: Step 2: M. Bohlin

8. Temporal constraints Brake test Arrival Roll-in Roll-out Departure • Roll-incan start after the arrivalinspection and preparations: • Brake test canbeginwhen all cars havearrived: M. Bohlin

9. Related Work • Sortingof freight cars Siddiqee, 1972 Dahlhaus, Horák, Miller and Ryan, 2000 Dahlhaus, Manne, Miller and Ryan, 2000 Gatto, Maue, Mihalak and Widmayer, 2009 Jacob, Marton, Maue and Nunkesser, 2010 • Trainparking Blasum, Bussieck, Hochstättler, Moll, Scheel and Winter, 1999 Di Stefano and Koci, 2004 Winter and Zimmermann, 2000 • Freight yard dispatching He, Song, and Chaudhry, 2003 • Trackassignment Cornelsen and Di Stefano, 2007 M. Bohlin

10. Complexity M. Bohlin

11. Mixing and Cutting Given a mixing plan, the ”uncut” trackallocation is the remaining part after mixing. ”localcut-off” = number of mixed cars M. Bohlin

12. Mixing and Cutting Cuttingonlyalloweduntil the last pull-out or untildeparture preparations begin M. Bohlin

13. ComplexityResults (1) Unlimited mixed capacity: -coloring of interval graphs. Theorem 1. Finding a feasible track allocation for the mixing-problem is NP-complete even for instances where 1) the mixed capacity is zero, or 2) the mixed capacity is unlimited, and all intervals may have arbitrary uncutted parts. Problem reduces to interval graphcoloringif all trainsfit on all tracks. M. Bohlin

14. ComplexityResults (2) Theorem 2. In case of uniform and sufficient track lengths, the problem of finding a feasible track allocation that minimizes the number of cars sent to the mixed tracks over all time periods is solvable in polynomial time. M. Bohlin

15. Arccost = number of mixed cars Arcsbetweentrains in roll-out order Solved as assignment problem in O(n3) Departing trains Betweentracks (no allocation, zerocost) Classification tracks Arcs to all trains (zerocost) Arcs from all trains (zerocost) M. Bohlin

16. The mixing problem M. Bohlin

17. Heuristic A: Interval coloring Three tracks available (horizontal lines). Dark areas cannot be cut off. Pulltimes: vertical lines. Greedy coloring, by start time.

18. Heuristic A: Interval coloring Schedule needs 2 extra tracks. Find first infeasible clique. Intersection of clique members is grey.

19. Heuristic A: Interval coloring Cut off 2 intervals with least cost (here no choice)

20. Heuristic A: Interval coloring Again, greedy coloring by start time, with intervals that have been cut off.

21. Heuristic A: Interval coloring Second infeasible clique, one extra track needed.

22. Heuristic A: Interval coloring Cut off cheaper interval (let’s say it’s the violet one)

23. Heuristic A: Interval coloring Finally, a feasible schedule. This always works if a greedy coloring of the dark areas (minimal parts of the intervals) happens to be feasible.

24. Heuristic B: Greedy • Assigntrains in roll-out order • Choose best trackw.r.t. resultinglocalcut-off • Best-fitw.r.t. length as tie-break M. Bohlin

25. A look at the data... Train Sizes Track Sizes

26. A look at the data... Train Sizes Track Sizes Every train on the left fits on each track in bucket on right

27. A look at the data... Train Sizes Track Sizes All tracks on right are longer than many trains on the left

28. Heuristic I: Improvement • Bucket: Set of tracks and trains s.t. each train fits on each track within that bucket • Idea: build buckets from feasible schedule (length-wise) • Solve each bucket independently to optimality (total mixing usage / roll-ins) • in order of reverse length, pick tracks until some allocated train doesn’t fit on a track • selected tracks and trains  bucket (removed)

29. Experiments

30. Hallsberg Hump Yard (Sweden) M. Bohlin

31. Experimental Setup • One week of traffic (spring 2010) • Timetabledarrivals and departures • Car allocation given • Planning for Thursday – Sunday • Twomixingtracks (necessary) • Trainlength up to 613 m • 80% of arrivals between 12:00 and 23:59. Step 1: 20 minutes MIP feasibility: 30 minutes MIP min mix: 30 minutes M. Bohlin

32. Results, 2 days (mixed usage) Meters M. Bohlin

33. Results, 2 days (extra roll-ins) Extra car-roll-ins M. Bohlin

34. M. Bohlin

35. Ganttchart, 2 days M. Bohlin

36. Open issues • Pull-backplanning • Scheduling mixed tracks • Integrated approach M. Bohlin

37. TrackAllocation with Temporary Car Storage Markus Bohlin, SICS markus.bohlin@sics.se The End M. Bohlin