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
Thomas Schlechte
3
Planning in Public Transport
Tracks
Timetables
Vehicles
Crews
Strategic
Tactical
Operational
Stage
Stage
Stage
Stops
Thomas Schlechte
Lines/Freq.
Cycles
Connections
Rotations
Duties
4
Traffic Projects @ ZIB
MCF
92-94 94-97
VS-OPT
97-00
DS-OPT
VS: BVG
00-03
IS-OPT
DS: BVG
03-07 Thomas Schlechte
TS-OPT
Line+Price Planning
Telebus
CS-OPT
5
Planning in Public Transport TS-OPT
VS-OPT
DS-OPT
IS-OPT
CS-OPT
B1 – B15
Tracks
Timetables
Vehicles
Crews
Strategic
Tactical
Operational
Stage
Stage
Stage
Stops Thomas Schlechte
Lines/Freq.
Cycles
Connections
Rotations
Duties
6
Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results
Thomas Schlechte
7
The Problem (TraVis by M.Kinder)
Thomas Schlechte
8
Schedule in 3d
Thomas Schlechte
9
Conflict-Free-Allocation
Thomas Schlechte
10
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 Thomas Schlechte
11
Complexity 2
1
Proposition [Caprara, Fischetti, Toth (02)]:
5 4 3
OPTRA/TTP is NP-hard. 5 4 3 2 1
Proof: Reduction from Independent-Set. 5 4 3 2 1 s Thomas Schlechte
(1,2) (2,3) (2,4) (3,4) (4,5) t
12
Track Allocation Problem
…
Thomas Schlechte
Train Requests
Scheduling Digraph
Timetable
13
Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results
Thomas Schlechte
14
Packing Models
Conflict graph Cliques Perfect
Cacchiani (2007) – Path Compatibility Graphs Thomas Schlechte
A
B
15
Arc Packing Problem
Variables
Arc occupancy (request i uses arc a)
Constraints
Flow conservation and
Arc conflicts (pairwise )
Objective Thomas Schlechte
Maximize proceedings
(PPP) transformation from arc to path variables (see Cachhiani (2007))
16
Packing Models Proposition: The LPrelaxation of APP can be solved in polynomial time. … and in practice.
Thomas Schlechte
17
Novel Model
Track Digraph Timeline(s) Config paths
Thomas Schlechte
A
B
Artificial arcs represent valid successors !
18
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 Thomas Schlechte
Maximize proceedings
(ACP) transformation from path to arc variables (see Borndörfer, S. (2007))
19
Linear Relaxation of PCP
dual variable
Thomas Schlechte
information about
useful to
bundle price
analyse request
track price
analyse network
arc price
-
20
Dualization
Thomas Schlechte
21
Pricing of x-variables
Pricing Problem(x) : Acyclic shortest path problems for each slot request i with modified cost function c ! Thomas Schlechte
22
Pricing of y-variables
Pricing Problem(y) : Acyclic shortest path problem for each track j with modified cost function c ! Thomas Schlechte
23
Observation
Thomas Schlechte
24
And analogously ...
Thomas Schlechte
25
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:
Thomas Schlechte
26
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´)
Thomas Schlechte
27
Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results
Thomas Schlechte
28
Two Step Approach TS-OPT
Thomas 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
29
Branch-Bound-Price or Dive-Generate
Evaluation of only few highly different subproblems at iteration j to reach IP-Solutions fast. Thomas Schlechte
30
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. Thomas Schlechte
Column Generation
31
Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results
Thomas Schlechte
32
Results • Test Network • 45
Tracks
• 37
Stations
•6
Traintypes
• 10
Trainsets
• 146 Nodes • 1480 Arcs • 96
Station Capacities
• 4320 Headway Times
Thomas Schlechte
33
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
Thomas Schlechte
34
Run of TS-OPT /LP Stage
Thomas Schlechte
35
Model Comparison
Thomas Schlechte
36
Model Comparison
For details see [ZR-07-02, ZR-07-20]. Thomas Schlechte
37
Outlook Algorithmic Developments • Bundle method • Model refinement (connections) • Adaptive IP Heuristics • Dynamic Discretization
Simulation of results by
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
