Hump Yard Track Allocation with Temporary Car Storage RailRome 2011
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Transcript of Hump Yard Track Allocation with Temporary Car Storage RailRome 2011
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Markus BohlinSICS
Hump Yard Track Allocation with Temporary Car StorageRailRome 2011
Holger FlierJens MaueMatus MihalakETH
Funded by Swedish Transport Administration and Swiss National Science Foundation
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2 M. Bohlin
Problem definition
Complexity
The mixing problem
Experiments
Outline
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3 M. Bohlin
PROBLEM DEFINITION
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4 M. Bohlin
Hump Yard Track allocation
… …
qi CC
Roll-in Roll-out
Dep. train
formation
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5 M. Bohlin
Temporary car storage (”Mixing”)
Tracks reserved for ”mixed” use
Pull-outImmediateroll-in
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6 M. Bohlin
• Planning by hand• Default: roll-in order = arrival
order • Pull-backs are planned in advance
• Partial pull-backs
• Train formation on multiple tracks• Multiple trains on one track
Current Practice
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7 M. Bohlin
Solution Approach
Arrival & Departure yards•Track allocations•Schedule roll-ins s.t. pull-outs
•Determine roll-outs
•Minimizes ”freight-time” on class. tracks
Classification bowl•Track allocation•Mixed tracks
•Gives feasible plan•Minimizes car roll-ins
Step 1:
Heuristic A
Heuristic Impr.
Heuristic B
MIP Model
MIP Mode
l
Step 2:
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8 M. Bohlin
Temporal constraints
Arrival Roll-in Roll-out Departure
• Roll-in can start after the arrival inspection and preparations:
• Brake test can begin when all cars have arrived:
rollrollarrarr STT ii
outoutrollroll:),( STTCji ji
Brake test
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9 M. Bohlin
• Sorting of freight carsSiddiqee, 1972Dahlhaus, Horák, Miller and Ryan, 2000Dahlhaus, Manne, Miller and Ryan, 2000Gatto, Maue, Mihalak and Widmayer, 2009Jacob, Marton, Maue and Nunkesser, 2010
• Train parkingBlasum, Bussieck, Hochstättler, Moll, Scheel and Winter, 1999Di Stefano and Koci, 2004Winter and Zimmermann, 2000
• Freight yard dispatchingHe, Song, and Chaudhry, 2003
• Track assignmentCornelsen and Di Stefano, 2007
Related Work
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10 M. Bohlin
COMPLEXITY
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11 M. Bohlin
Given a mixing plan, the ”uncut” track allocation is the remaining part after mixing.
Mixing and Cutting
”local cut-off” = number of mixed cars
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12 M. Bohlin
Cutting only allowed until the last pull-out or until departure preparations begin
Mixing and Cutting
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13 M. Bohlin
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 graph coloring if all trains fit on all tracks.
Complexity Results (1)
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14 M. Bohlin
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.
Complexity Results (2)
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15 M. Bohlin
1
2
n
q1
q2
qk
1
2
n
q1
q2
qk
Arc cost = number of mixed carsArcs between trains in roll-
out order
Departingtrains
Classificationtracks
Arcs to all trains (zero
cost)
Arcs from all trains (zero
cost)
Solved as assignment problem in O(n3)
Between tracks (no allocation,
zero cost)
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16 M. Bohlin
THE MIXING PROBLEM
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Three tracks available (horizontal lines). Dark areas cannot be cut off. Pulltimes: vertical lines. Greedy coloring, by start time.
Heuristic A: Interval coloring
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Schedule needs 2 extra tracks. Find first infeasible clique. Intersection of clique members is grey.
Heuristic A: Interval coloring
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Cut off 2 intervals with least cost (here no choice)
Heuristic A: Interval coloring
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Again, greedy coloring by start time, with intervals that have been cut off.
Heuristic A: Interval coloring
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Second infeasible clique, one extra track needed.
Heuristic A: Interval coloring
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Cut off cheaper interval (let’s say it’s the violet one)
Heuristic A: Interval coloring
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Finally, a feasible schedule. This always works if a greedy coloring of the dark areas (minimal parts of the intervals) happens to be feasible.
Heuristic A: Interval coloring
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25 M. Bohlin
• Assign trains in roll-out order• Choose best track w.r.t. resulting
local cut-off• Best-fit w.r.t. length as tie-break
Heuristic B: Greedy
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Train Sizes Track Sizes
A look at the data...
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Train Sizes Track Sizes
Every train on the left fits on each track in bucket on right
A look at the data...
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Train Sizes Track Sizes
All tracks on right are longer than many trains on the left
A look at the data...
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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)
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EXPERIMENTS
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31 M. Bohlin
Hallsberg Hump Yard (Sweden)
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32 M. Bohlin
• One week of traffic (spring 2010)– Timetabled arrivals and departures– Car allocation given
• Planning for Thursday – Sunday• Two mixing tracks (necessary)• Train length up to 613 m• 80% of arrivals between 12:00 and 23:59.
Step 1: 20 minutesMIP feasibility: 30 minutesMIP min mix: 30 minutes
Experimental Setup
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33 M. Bohlin
MIP Heur. A
Heur. A+I
Heur. B
Heur. B+I
1200130014001500160017001800190020002100
Results, 2 days (mixed usage)Meters
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34 M. Bohlin
MIP Heur. A
Heur. A+I
Heur. B
Heur. B+I
0
50
100
150
200
250
300
350
Results, 2 days (extra roll-ins)Extra car-roll-ins
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35 M. Bohlin
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36 M. Bohlin
Gantt chart, 2 days
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37 M. Bohlin
• Pull-back planning• Scheduling mixed tracks
• Integrated approach
Open issues