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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Problem definition

Complexity

The mixing problem

Experiments

Outline

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PROBLEM DEFINITION

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Hump Yard Track allocation

… …

qi CC

Roll-in Roll-out

Dep. train

formation

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Temporary car storage (”Mixing”)

Tracks reserved for ”mixed” use

Pull-outImmediateroll-in

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• 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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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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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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• 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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COMPLEXITY

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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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Cutting only allowed until the last pull-out or until departure preparations begin

Mixing and Cutting

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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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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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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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THE MIXING PROBLEM

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

Heuristic A: Interval coloring

Schedule needs 2 extra tracks. Find first infeasible clique. Intersection of clique members is grey.

Heuristic A: Interval coloring

Cut off 2 intervals with least cost (here no choice)

Heuristic A: Interval coloring

Again, greedy coloring by start time, with intervals that have been cut off.

Heuristic A: Interval coloring

Second infeasible clique, one extra track needed.

Heuristic A: Interval coloring

Cut off cheaper interval (let’s say it’s the violet one)

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.

Heuristic A: Interval coloring

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• 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

Train Sizes Track Sizes

A look at the data...

Train Sizes Track Sizes

Every train on the left fits on each track in bucket on right

A look at the data...

Train Sizes Track Sizes

All tracks on right are longer than many trains on the left

A look at the data...

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)

EXPERIMENTS

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Hallsberg Hump Yard (Sweden)

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• 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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MIP Heur. A

Heur. A+I

Heur. B

Heur. B+I

1200130014001500160017001800190020002100

Results, 2 days (mixed usage)Meters

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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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Gantt chart, 2 days

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• Pull-back planning• Scheduling mixed tracks

• Integrated approach

Open issues

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THE END

Track Allocation with Temporary Car Storage

Markus Bohlin, SICSmarkus.bohlin@sics.se