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

Models for Railway Track Allocation Thomas Schlechte Joint work with Ralf Borndörfer Martin Grötschel 16.11.2007 ATMOS 2007 Sevilla � Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) Thomas Schlechte schlechte@zib.de http://www.zib.de/schlechte

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

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

4 Planning in Public Transport Tracks Lines/Freq. Timetables Vehicles Crews Strategic Tactical Operational Stage Stage Stage Stops Cycles Connections Rotations Duties Thomas Schlechte

5 The Problem (TraVis by M.Kinder) Thomas Schlechte

Schedule in 3d Schlechte Thomas 6

Conflict-Free-Allocation Schlechte Thomas 7

8 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

9 Complexity 2 5 4 1 3 Proposition [Caprara, Fischetti, Toth (02)]: OPTRA/TTP is NP -hard. 5 4 3 Proof: 2 1 Reduction from Independent-Set. 5 4 3 2 1 s (1,2) (2,3) (2,4) (3,4) (4,5) t Thomas Schlechte

Timetable Track Allocation Problem Scheduling Digraph Train Requests … Schlechte Thomas 10

11 Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results Thomas Schlechte

12 Packing Models � Conflict graph � Cliques � Perfect Cacchiani (2007) – Path Compatibility Graphs A B Thomas Schlechte

13 Arc Packing Problem Variables � Arc occupancy (request i uses arc a) Constraints � Flow conservation and (PPP) transformation from arc to path � Arc conflicts (pairwise ) variables (see Cachhiani (2007)) Objective � Maximize proceedings Thomas Schlechte

14 Packing Models � Proposition: The LP- relaxation of APP can be solved in polynomial time. � … and in practice. Thomas Schlechte

15 Novel Model � Track Digraph � Timeline(s) � Config paths Artificial arcs represent valid successors ! A B Thomas Schlechte

16 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) (ACP) transformation from path to arc Objective Function variables (see Borndörfer, S. (2007)) Thomas � Maximize proceedings Schlechte

17 Linear Relaxation of PCP dual variable information about useful to bundle price analyse request track price analyse network arc price - Thomas Schlechte

Dualization Schlechte Thomas 18

19 Pricing of x-variables Pricing Problem(x) : Acyclic shortest path problems for each slot request i with modified cost function c ! Thomas Schlechte

20 Pricing of y-variables Pricing Problem(y) : Acyclic shortest path problem for each track j with modified cost function c ! Thomas Schlechte

Observation Schlechte Thomas 21

And analogously ... Schlechte Thomas 22

23 Pricing Upper Bound • Lemma [ZR-07-02]: Given (infeasible) dual variables of PCP and let v LP (PCP) be the optimum objective value of the LP-Relaxtion of PCP, then: Thomas Schlechte

24 Model Comparison • Theorem [ZR-07-02]: The LP-relaxations of ACP APP’ and PCP can be solved in polynomial time. • Lemma [ZR-07-02]: APP PPP v LP (PCP) = v LP (ACP) = v LP (APP) = v LP (PPP) ≤ v LP (APP ´ ) ACP PCP Thomas Schlechte

25 Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results Thomas Schlechte

26 Two Step Approach Duals by Duals by Bundle TS-OPT CPLEX Method Pricing by Column 1. LP Solving Dijkstra’s Generation Shortest Path Rapid Branching 2. IP Solving Heuristic Thomas Schlechte

27 Branch-Bound-Price or Dive-Generate Evaluation of only few highly different sub- problems at iteration j to reach IP-Solutions fast. Thomas Schlechte

28 Rapid Branching Node selection of set of fixed to 1 variables by using perturbated cost function (bonus close to 1.0). Update Column Upper Bound Generation Go on if target was reached, otherwise pseudo-backtrack. Thomas Schlechte

29 Overview 1. Problem Introduction 2. Model Discussion 3. Column Generation Approach 4. Computational Results Thomas Schlechte

30 Results • Test Network • 45 Tracks • 37 Stations • 6 Traintypes • 10 Trainsets • 146 Nodes • 1480 Arcs • 96 Station Capacities • 4320 Headway Times Thomas Schlechte

31 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

Run of TS-OPT / LP Stage Schlechte Thomas 32

Model Comparison Schlechte Thomas 33

For details see [ZR-07-02, ZR-07-20]. Model Comparison Schlechte Thomas 34

Thank you Thank you for your attention ! for your attention ! Thomas Schlechte Fon (+ 49 30) 84185-317 Zuse-I nstitut Berlin (ZI B) Fax (+ 49 30) 84185-269 � Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) Thomas Schlechte Takustr. 7, 14195 Berlin schlechte@zib.de Deutschland www.zib.de/ schlechte schlechte@zib.de http://www.zib.de/schlechte

36 Traffic Projects @ ZI B � MCF 92-94 Telebus VS-OPT 94-97 DS-OPT VS: BVG 97-00 IS-OPT DS: BVG 00-03 Line+ Price Planning TS-OPT 03-07 CS-OPT Thomas Schlechte

37 Planning in Public Transport TS-OPT VS-OPT DS-OPT B1 – B15 IS-OPT CS-OPT Tracks Lines/Freq. Timetables Vehicles Crews Strategic Tactical Operational Stage Stage Stage Stops Cycles Connections Rotations Duties Thomas Schlechte

38 The Problem (TraVis by M.Kinder) Thomas Schlechte

Schedule in 3d Schlechte Thomas 39

Conflict-Free-Allocation Schlechte Thomas 40

41 Outlook Algorithmic Developments • Bundle method • Model refinement (connections) • Adaptive IP Heuristics • Dynamic Discretization Simulation of results by Thomas Schlechte