Models for Railway Track Allocation

Transcription

Models
for
Railway Track Allocation
Thomas Schlechte
Joint work with
Ralf Borndörfer
Martin Grötschel
16.11.2007
ATMOS 2007 Sevilla
Thomas Schlechte
Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected]
http://www.zib.de/schlechte
2
Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
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Overview
2. Model Discussion
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Planning in Public Transport
Tracks
Timetables
Vehicles
Crews
Strategic
Tactical
Operational
Stage
Stage
Stage
Stops
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Lines/Freq.
Cycles
Connections
Rotations
Duties
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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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Complexity
2
1
Proposition [Caprara, Fischetti, Toth (02)]:
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4
3
OPTRA/TTP is NP-hard.
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4
3
2
1
Proof:
Reduction from Independent-Set.
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4
3
2
1
s
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(1,2) (2,3) (2,4) (3,4) (4,5) t
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Track Allocation Problem
…
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Train Requests
Scheduling Digraph
Timetable
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Overview
2. Model Discussion
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Packing Models
Conflict graph
Cliques
Perfect
Cacchiani (2007) – Path Compatibility Graphs
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A
B
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Arc Packing Problem
Variables

Arc occupancy (request i uses arc a)
Constraints

Flow conservation and

Arc conflicts (pairwise )
Objective
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Maximize proceedings
(PPP) transformation from arc to path
variables (see Cachhiani (2007))
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Packing Models
Proposition:
The LPrelaxation of
APP can be
solved in
polynomial
time.
… and in
practice.
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Novel Model
Track Digraph
Timeline(s)
Config paths
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A
B
Artificial arcs represent valid successors !
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Path Coupling Problem
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
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Maximize proceedings
(ACP) transformation from path to arc
variables (see Borndörfer, S. (2007))
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Linear Relaxation of PCP
dual variable
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information about
useful to
bundle price
analyse request
track price
analyse network
arc price
-
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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
objective 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.
APP’
• Lemma [ZR-07-02]:
vLP(PCP) = vLP(ACP)
APP
PPP
ACP
PCP
= vLP(APP) = vLP(PPP)
≤ vLP(APP´)
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Overview
2. Model Discussion
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Two Step Approach
TS-OPT
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Duals
by
Duals
by Bundle
CPLEX
Method
1. LP Solving
Column
Generation
2. IP Solving
Rapid Branching
Heuristic
Pricing by
Dijkstra’s
Shortest Path
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Branch-Bound-Price
or Dive-Generate
Evaluation of only few highly different subproblems at iteration j to reach IP-Solutions fast.
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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
Go on if target was reached,
otherwise pseudo-backtrack.
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Column
Generation
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Overview
2. Model Discussion
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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
• 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].
Thomas
Schlechte
Thank you
for your attention !
Thomas Schlechte
Fon (+49 30) 84185-317
Zuse-Institut
Berlin (ZIB) für
Fax
(+49 30) 84185-269
Thomas Schlechte
Konrad-Zuse-Zentrum
Informationstechnik
Berlin (ZIB)
Takustr. 7, 14195 Berlin
[email protected]
Deutschland
www.zib.de/schlechte
[email protected]
http://www.zib.de/schlechte
36
Traffic Projects @ ZIB

MCF
92-94
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94-97
VS-OPT
97-00
DS-OPT
VS: BVG
00-03
IS-OPT
DS: BVG
03-07
TS-OPT
Line+Price
Planning
Telebus
CS-OPT
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Planning in Public Transport
B1 – B15
Tracks
Timetables
VS-OPT
DS-OPT
IS-OPT
CS-OPT
Vehicles
Crews
Strategic
Tactical
Operational
Stage
Stage
Stage
Stops
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Lines/Freq.
TS-OPT
Cycles
Connections
Rotations
Duties
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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
Thomas
Schlechte

Models for Railway Track Allocation

Transcription

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