Models for Railway Track Allocation Thomas Schlechte Joint work - - PowerPoint PPT Presentation

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Models for Railway Track Allocation Thomas Schlechte Joint work - - PowerPoint PPT Presentation

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Models for Railway Track Allocation Thomas Schlechte Joint work with Ralf Borndrfer Martin Grtschel 16.11.2007 ATMOS 2007 Sevilla Konrad-Zuse-Zentrum fr Informationstechnik Berlin (ZIB) Thomas Schlechte schlechte@zib.de

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http://www.zib.de/schlechte schlechte@zib.de

Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Thomas Schlechte

Models for Railway Track Allocation

Thomas Schlechte Joint work with Ralf Borndörfer Martin Grötschel

16.11.2007 ATMOS 2007 Sevilla

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Overview

  • 1. Problem Introduction
  • 2. Model Discussion
  • 3. Column Generation Approach
  • 4. Computational Results
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Overview

  • 1. Problem Introduction
  • 2. Model Discussion
  • 3. Column Generation Approach
  • 4. Computational Results
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Planning in Public Transport

Strategic Stage Tactical Stage Operational Stage

Stops Cycles Connections Duties Tracks Lines/Freq. Timetables Vehicles Crews Rotations

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The Problem (TraVis by M.Kinder)

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Schedule in 3d

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Conflict-Free-Allocation

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Railway Timetabling – State of the Art

  • Charnes and Miller (1956), Szpigel (1973), Jovanovic and

Harker (1991),

  • Cai and Goh (1994), Schrijver and Steenbeck (1994),

Carey and Lockwood (1995)

  • Nachtigall and Voget (1996), Odijk (1996) Higgings,

Kozan and Ferreira (1997)

  • Brannlund, Lindberg, Nou, Nilsson (1998), Lindner

(2000), Oliveira and Smith (2000)

  • Caprara, Fischetti and Toth (2002), Peeters (2003)
  • Kroon and Peeters (2003), Mistry and Kwan (2004)
  • Barber, Salido, Ingolotti, Abril, Lova, Tormas (2004)
  • Semet and Schoenauer (2005),
  • Caprara, Monaci, Toth and Guida (2005)
  • Kroon, Dekker and Vromans (2005),
  • Vansteenwegen and Van Oudheusden (2006),
  • Cacchiani, Caprara, T. (2006), Cachhiani (2007)
  • Caprara, Kroon, Monaci, Peeters, Toth (2006)

non-cyclic timetabling literature

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Proposition [Caprara, Fischetti, Toth (02)]:

OPTRA/TTP is NP-hard.

Proof:

Reduction from Independent-Set.

2 3 4 5

1

s (1,2) (2,3) (2,4) (3,4) (4,5) t

1 1 2 2 3 3 4 4 5 5

Complexity

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…

Train Requests

Track Allocation Problem

Scheduling Digraph Timetable

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Overview

  • 1. Problem Introduction
  • 2. Model Discussion
  • 3. Column Generation Approach
  • 4. Computational Results
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Packing Models

A B

Conflict graph Cliques Perfect

Cacchiani (2007) – Path Compatibility Graphs

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Variables

  • Arc occupancy (request i uses arc a)

Constraints

  • Flow conservation and
  • Arc conflicts (pairwise )

Objective

  • Maximize proceedings

Arc Packing Problem

(PPP) transformation from arc to path variables (see Cachhiani (2007))

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Packing Models

Proposition:

The LP- relaxation of APP can be solved in polynomial time.

… and in

practice.

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Novel Model

A B

Track Digraph Timeline(s) Config paths

Artificial arcs represent valid successors !

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Variables

  • Path und config usage (request i uses path p, track j uses config q)

Constraints

  • Path and config choice
  • Path-config-coupling (track capacity)

Objective Function

  • Maximize proceedings

Path Coupling Problem

(ACP) transformation from path to arc variables (see Borndörfer, S. (2007))

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Linear Relaxation of PCP

  • arc price

analyse network track price analyse request bundle price useful to information about dual variable

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Dualization

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Pricing of x-variables

Pricing Problem(x) : Acyclic shortest path problems for each slot request i with modified cost function c !

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Pricing of y-variables

Pricing Problem(y) : Acyclic shortest path problem for each track j with modified cost function c !

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Observation

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And analogously ...

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Pricing Upper Bound

  • Lemma [ZR-07-02]: Given (infeasible) dual

variables of PCP and let vLP(PCP) be the optimum

  • bjective value of the LP-Relaxtion of PCP,

then:

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Model Comparison

  • Theorem [ZR-07-02]:

The LP-relaxations of ACP and PCP can be solved in polynomial time.

  • Lemma [ZR-07-02]:

vLP(PCP) = vLP(ACP) = vLP(APP) = vLP(PPP) ≤ vLP(APP´) APP ACP PCP PPP APP’

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Overview

  • 1. Problem Introduction
  • 2. Model Discussion
  • 3. Column Generation Approach
  • 4. Computational Results
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TS-OPT

Two Step Approach

  • 1. LP Solving
  • 2. IP Solving

Rapid Branching Heuristic Column Generation Pricing by Dijkstra’s Shortest Path Duals by CPLEX Duals by Bundle Method

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Branch-Bound-Price

Evaluation of only few highly different sub- problems at iteration j to reach IP-Solutions fast.

  • r Dive-Generate
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Rapid Branching

Node selection of set of fixed to 1 variables by using perturbated cost function (bonus close to 1.0). Update Upper Bound Column Generation Go on if target was reached,

  • therwise pseudo-backtrack.
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Overview

  • 1. Problem Introduction
  • 2. Model Discussion
  • 3. Column Generation Approach
  • 4. Computational Results
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Results

  • Test Network
  • 45 Tracks
  • 37 Stations
  • 6 Traintypes
  • 10 Trainsets
  • 146 Nodes
  • 1480 Arcs
  • 96 Station Capacities
  • 4320 Headway Times
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Model Comparison

  • Test Scenarios
  • 146 Train Requests
  • 285 Train Requests
  • 570 Train Requests
  • Flexibility
  • 0-30 Minutes
  • earlier departure penalties
  • late arrival penalties
  • train type depending profits
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Run of TS-OPT / LP Stage

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Model Comparison

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Model Comparison

For details see [ZR-07-02, ZR-07-20].

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http://www.zib.de/schlechte schlechte@zib.de

Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Thomas Schlechte

Thank you Thank you for your attention ! for your attention !

Fon (+ 49 30) 84185-317 Fax (+ 49 30) 84185-269 schlechte@zib.de www.zib.de/ schlechte Thomas Schlechte Zuse-I nstitut Berlin (ZI B)

  • Takustr. 7, 14195 Berlin

Deutschland

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Traffic Projects @ ZI B

VS-OPT Telebus DS: BVG VS: BVG DS-OPT IS-OPT Line+ Price Planning TS-OPT

94-97 97-00 00-03 03-07

  • MCF

92-94

CS-OPT

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Strategic Stage Tactical Stage Operational Stage

Stops Cycles Connections Duties Tracks Lines/Freq. Timetables Vehicles Crews Rotations CS-OPT VS-OPT DS-OPT IS-OPT TS-OPT B1 – B15

Planning in Public Transport

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The Problem (TraVis by M.Kinder)

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Schedule in 3d

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Conflict-Free-Allocation

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Outlook

Algorithmic Developments

  • Bundle method
  • Model refinement (connections)
  • Adaptive IP Heuristics
  • Dynamic Discretization

Simulation of results by