Multistage Methods for Freight Train Classification fileModel Classiﬁcation Yard Hump Yard hump track hump classi cation tracks ladder General layout: I hump track with hump I ladder:

Multistage Methods for Freight Train Classification

Riko Jacob, Peter Marton, Jens Maue, Marc Nunkesser

Institute of Theoretical Computer Science, ETH Zurich

ATMOS 2007 - 16 November 2007

Jens Maue (ETH) Multistage Train Classification ATMOS-07 1 / 14

Introduction

Train Classification

Train classification≈

Sorting railway cars

I multi-destination freight trains

I cars ordered according to destinations

I infrastructure: hump yard

I method: multistage sorting


Introduction

Outline

Introduction

ModelClassification YardClassification Process

Optimal SchedulesSchedule RepresentationOptimal Schedules

Further Results

Concluding Remarks


Model Classification Yard

Hump Yard

hump

track

hump

classification tracks

ladder

General layout:

I hump track with hump

I ladder: tree of switches

I dead-ended classification tracks


Model Classification Process

Example

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θ1θ2θ3θfin

Multistage Train Classification:

1. start: roll-in input train

2. alternately pull out and roll in

3. finish: ordered train on any track



Example

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Optimal Schedules Schedule Representation

Classification Schedules

001101

010100

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Classification schedule representation:

I efficient encoding

I applies to existing methods

I prove properties of methods

I derive optimal schedules

Efficient schedule representation:

I bitstring encodes journey of car

I logical order of tracks

I minimize length: no. of pull-out steps



Classification Schedules

θ1θ2θ3θfin

1 5 3 6

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001101

010100

100001

532416

Classification schedule representation:

I efficient encoding

I applies to existing methods

I prove properties of methods

I derive optimal schedules

Efficient schedule representation:

I bitstring encodes journey of car

I logical order of tracks

I minimize length: no. of pull-out steps



Valid Schedules

Given:

I input train Tin = τ1 . . . τn

I classification schedule B = (b1, . . . , bn)

Observation:

I bi < bj ⇒ τi goes to output track before τj does

Proof.Let bi

` = 0 < 1 = bj` and bi

k = bjk for all k > `.

Consider `-th pull-out: τi goes to track of τj .

TheoremTrain of n cars has valid schedule of length h = dlog2 ne.

Considering structure of input improves schedule!

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θ1θ2θ3θfin

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





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θ1θ2θ3θfin

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





θ1θ2θ3θfin

1 5 3 6

4

2

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





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1θ1θ2θ3θfin

4

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5

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





1

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θ1θ2θ3θfin

5

6

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





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5

4

3

2

1θ1θ2θ3θfin

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





6

5

4

3

2

1θ1θ2θ3θfin

001101

010100

100001

532416



Valid Schedules

Given:



Observation:


Proof.Let bi

` = 0 < 1 = bj` and bi





6

5

4

3

2

1θ1θ2θ3θfin

001101

010100

100001

532416


Optimal Schedules Optimal Schedules

Optimal Schedules

Consider presortedness of input to get optimal schedule:

DefinitionGiven a train Tin = τ1 . . . τn, a subsequence T ′ = τi1 . . . τim

of Tin is called a chain if τij+1= τij + 1 and T ′ maximal.

Example

Three chains: c1 = [1, 2, 3], c2 = [4, 5], and c3 = [6].

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Optimal Schedules




Example


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Optimal Schedules




Example


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Optimal Schedules




Example


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Optimal Schedules




Example


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Optimal Schedules

Given:



Observation:

I bi = bj , then τi and τj never swap relative order.

TheoremTrain of c chains has optimal schedule of length dlog2 ce.

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Optimal Schedules

Given:



Observation:

I bi = bj , then τi and τj never swap relative order.

TheoremTrain of c chains has optimal schedule of length dlog2 ce.

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Further Results

Problem Variants

Input/output specification:

I multiple output trains: essentially same problem

I multiple input trains: minimize number of chains

Restricted yard:I classification tracks of restricted capacity: NP-hard

I special case single-car chains: optimal polynomial algorithm

I bounded number of tracks: optimal polynomial algorithm


Further Results

Problem Variants

Input/output specification:

I multiple output trains: essentially same problem

I multiple input trains: minimize number of chains

Restricted yard:I classification tracks of restricted capacity: NP-hard

I special case single-car chains: optimal polynomial algorithm

I bounded number of tracks: optimal polynomial algorithm


Concluding Remarks

Conclusion

Multistage train classification:

I derive set of bitstrings

I meet restrictions of problem variant

I restrictions affect complexity θ1θ2θ3θfin

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001101

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Concluding Remarks

Future work

Further variants and practical application:

I more general output train

I other objectives

I time-dependent input and robustness

I simulation for real-world yard


Concluding Remarks

Future work



I other objectives




Concluding Remarks

Future work



I other objectives




Concluding Remarks

Future work



I other objectives




Concluding Remarks

Simulation for Real-World Yard: Lausanne Triage

Infrastructure and operation:

I two parallel humps

I time windows for multistage classification

I local freight trains (about 400 cars per day)


Concluding Remarks

Future work



I other objectives




References

Jacob, R., Marton, P., Maue, J., and Nunkesser, M. (2007).Multistage methods for freight train classification.In Proc. of the 7th Workshop on Algorithmic Methods and Models forOptimization of Railways (ATMOS-07), DROPS, pages 158–174. IBFISchloss Dagstuhl.


Documents

Multistage Methods for Freight Train Classification fileModel Classiﬁcation Yard Hump Yard hump track hump classi cation tracks ladder General layout: I hump track with hump I ladder: