Models for Railway Track Allocation

Models
for
Railway Track Allocation
Thomas Schlechte
Joint work with
Ralf Borndörfer
Martin Grötschel
16.11.2007
ATMOS 2007 Sevilla
Thomas Schlechte
ƒ Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
schlechte@zib.de
http://www.zib.de/schlechte
2
Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
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Schlechte
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Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
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Schlechte
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Planning in Public Transport
Tracks
Timetables
Vehicles
Crews
Strategic
Tactical
Operational
Stage
Stage
Stage
Stops
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Schlechte
Lines/Freq.
Cycles
Connections
Rotations
Duties
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The Problem (TraVis by M.Kinder)
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Schlechte
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Schedule in 3d
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Schlechte
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Conflict-Free-Allocation
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Schlechte
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Railway Timetabling –
State of the Art
ƒ
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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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Schlechte
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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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Schlechte
(1,2) (2,3) (2,4) (3,4) (4,5) t
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Track Allocation Problem
…
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Schlechte
Train Requests
Scheduling Digraph
Timetable
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Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
Thomas
Schlechte
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Packing Models
ƒ Conflict graph
ƒ Cliques
ƒ Perfect
Cacchiani (2007) – Path Compatibility Graphs
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Schlechte
A
B
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Arc Packing Problem
Variables
ƒ
Arc occupancy (request i uses arc a)
Constraints
ƒ
Flow conservation and
ƒ
Arc conflicts (pairwise )
Objective
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Schlechte
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Maximize proceedings
(PPP) transformation from arc to path
variables (see Cachhiani (2007))
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Packing Models
ƒ Proposition:
The LPrelaxation of
APP can be
solved in
polynomial
time.
ƒ … and in
practice.
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Schlechte
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Novel Model
ƒ Track Digraph
ƒ Timeline(s)
ƒ Config paths
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Schlechte
A
B
Artificial arcs represent valid successors !
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Path Coupling Problem
Variables
ƒ
Path und config usage (request i uses path p, track j uses config q)
Constraints
ƒ
Path and config choice
ƒ
Path-config-coupling (track capacity)
Objective Function
Thomas
Schlechte
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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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Schlechte
information about
useful to
bundle price
analyse request
track price
analyse network
arc price
-
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Dualization
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Schlechte
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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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Schlechte
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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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Schlechte
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Observation
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And analogously ...
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Schlechte
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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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Schlechte
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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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Schlechte
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Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
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Schlechte
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Two Step Approach
TS-OPT
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Schlechte
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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Schlechte
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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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Schlechte
Column
Generation
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Overview
1. Problem Introduction
2. Model Discussion
3. Column Generation Approach
4. Computational Results
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Schlechte
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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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Schlechte
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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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Schlechte
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Run of TS-OPT /LP Stage
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Schlechte
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Model Comparison
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Schlechte
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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
ƒ Konrad-Zuse-Zentrum
Informationstechnik
Berlin (ZIB)
Takustr. 7, 14195 Berlin
schlechte@zib.de
Deutschland
www.zib.de/schlechte
schlechte@zib.de
http://www.zib.de/schlechte
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Traffic Projects @ ZIB
ˆ
MCF
92-94
Thomas
Schlechte
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
Thomas
Schlechte
Lines/Freq.
TS-OPT
Cycles
Connections
Rotations
Duties
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The Problem (TraVis by M.Kinder)
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Schlechte
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Schedule in 3d
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Schlechte
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Conflict-Free-Allocation
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Schlechte
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Outlook
Algorithmic Developments
• Bundle method
• Model refinement (connections)
• Adaptive IP Heuristics
• Dynamic Discretization
Simulation of results by
Thomas
Schlechte