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Optimal Rail Track Allocation

Optimal Rail Track Allocation. Ralf Borndörfer joint work with Martin Grötschel Thomas Schlechte X Encuentro de Matem á tica, Quito, 25. Juli 2006. Overview. Rail Track Auctions The Optimal Track Allocation Problem (OPTRA) Mathematical Models Computational Results. Background.

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Optimal Rail Track Allocation

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  1. Optimal Rail Track Allocation Ralf Borndörfer joint work with Martin Grötschel Thomas Schlechte X Encuentro de Matemática, Quito, 25. Juli 2006

  2. Overview • Rail Track Auctions • The Optimal Track Allocation Problem (OPTRA) • Mathematical Models • Computational Results Ralf Borndörfer

  3. Background • Problems • Network utilization • Deficit • European Union • Establish a rail traffic market • Open the market to competition • Improve cost recovery of infrastructure provider, reduce subsidies • Deregulate/regulate this market • Project • WiP (TUB), SFWBB (TUB), I&M, Z, ZIB Ralf Borndörfer

  4. Auctioning Approach • Goals • More traffic at lower cost • Better service • How do you measure? • Possible answer: in terms of willingness to pay • What is the „commodity“ of this market? • Possible answer: timetabled track= dedicated, timetabled track section = use of railway infrastructure in time and space • How does the market work? • Possible answer: by auctioning timetabled tracks Ralf Borndörfer

  5. Arguments for Auctions • Auctions can … • resolve user conflicts in such a way that the bidder with the highest willigness to pay receives the commodity (efficient allocation, wellfare maximization) • maximize the auctioneer’s earnings • reveal the bidders’ willigness to pay • reveal bottlenecks and the added value if they are removed • Economists argue … • that a “working auctioning system” is usually superior to alternative methods such as bargaining, fixed prices, etc. Ralf Borndörfer

  6. Examples • In ancient times … • Auctions are known since 500 b.c. • March 28, 193 a.d.: The pretorians auction the Roman Emperor‘s throne to Marcus Didius Severus Iulianus, who ruled as Iulianus I. for 66 days Ralf Borndörfer

  7. The Story of Didius Iulianus(http://www.roman-emperors.org/didjul.htm) [193 A.D., March 28] When the emperor Pertinax was killed trying to quell a mutiny, no accepted successor was at hand. Pertinax's father-in-law and urban prefect, Flavius Sulpicianus, entered the praetorian camp and tried to get the troops to proclaim him emperor, but he met with little enthusiasm. Other soldiers scoured the city seeking an alternative, but most senators shut themselves in their homes to wait out the crisis. Didius Julianus, however, allowed himself to be taken to the camp, where one of the most notorious events in Roman history was about to take place. Didius Julianus was prevented from entering the camp, but he began to make promises to the soldiers from outside the wall. Soon the scene became that of an auction, with Flavius Sulpicianus and Didius Julianus outbidding each other in the size of their donatives to the troops. The Roman empire was for sale to the highest bidder. When Flavius Sulpicianus reached the figure of 20,000 sesterces per soldier, Didius Julianus upped the bid by a whopping 5,000 sesterces, displaying his outstretched hand to indicate the amount. The empire was sold, Didius Julianus was allowed into the camp and proclaimed emperor. Ralf Borndörfer

  8. Examples • In ancient times … • Auctions are known since 500 b.c. • March 28, 193 a.d.: The pretorians auction the Roman Emperor‘s throne to Marcus Didius Severus Iulianus, who ruled as Iulianus I. for 66 days • In modern times … • Traditional auctions (antiques, flowers, stamps, etc.) • Stock market • eBay etc. • Oil drilling rights, energy spot market, etc. • Procurement • Sears, Roebuck & Co. • Frequency auctions in mobile telecommunication • Regional monopolies (franchising) at British Rail Ralf Borndörfer

  9. Sears, Roebuck & Co. • 3-year contracts for transports on dedicated routes • First auction in 1994 with 854 contracts • Combinatorial auction • „And-“ and „or-“ bids allowed • 2854 (≈10257) theoretically possible combinations • Sequential auction (5 rounds, 1 month between rounds) • Results • 13% cost reduction • Extension to 1.400 contracts (14% cost reduction) Ralf Borndörfer

  10. Frequency Auctions(Cramton 2001, Spectrum Auctions, Handbook of Telecommunications Economics) • Prices for mobile telecommunication frequencies (2x10 MHz+5MHz) • Low earnings are not per se inefficient • Only min. prices => insufficient cost recovery Ralf Borndörfer

  11. Ralf Borndörfer

  12. Track Request Form Ralf Borndörfer

  13. Track Construction Ralf Borndörfer

  14. Rail Track Auctioning Ralf Borndörfer

  15. Rail Track Auction EVUs decide on bids for bundles of timetabled tracks Bid is assigned All bids assigned: END BEGINMinimum Bid = Basic Price Bids is unchanged OPTRA finds allocation withmaximum earnings Bids are increased by aminimum increment Bid is not assigned Ralf Borndörfer

  16. Overview • Rail Track Auctions • The Optimal Track Allocation Problem (OPTRA) • Mathematical Models • Computational Results Ralf Borndörfer

  17. Optimal Track Allocation Problem (OPTRA) Input • Set of bids for timetabled tracks incl. willingness to pay • Available infrastructure (space and time) Output • Assignment of bids that maximizes the total willigness to pay • Conflict free track assignments for the chosen bids Ralf Borndörfer

  18. Bids for Timetabled Tracks • Train number(s) and type(s) • Starting station, earliest starting time • Final station, latest arrival time • Basic bid (in Euro) • Intermediate stops (Station, min. stopping time, arrival interval) • Connections • Combinatorial bids Ralf Borndörfer

  19. Blocks and Standardized Dynamics v State (i,T,t,v) • Directed block i • Train type T • Starting time t, velocity v s i j k Ralf Borndörfer

  20. Standard Train Types Ralf Borndörfer

  21. Infrastructure Ralf Borndörfer

  22. Block Conflicts conflict conflict t s Ralf Borndörfer

  23. Variable Bids € € 90 4 €/min 80 Travel time [min] 12:00 12:08 12:20 Dep. time 40 60 Bid = Basic Bid + Departure/Arrival Time Bonus + Travel Time Bonus Ralf Borndörfer

  24. Effects difficult! A B C D 3 x + 1 x = ??? I. variant III. II. time ICE goes ICE slower ICE drops out Ralf Borndörfer

  25. Route/Track Track Allocation Problem Ralf Borndörfer

  26. Route/Track Route Bundle/Bid Track Allocation Problem Ralf Borndörfer

  27. Route/Track Route Bundle/Bid Scheduling Graph Track Allocation Problem Ralf Borndörfer

  28. Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation Problem Ralf Borndörfer

  29. Route/Track Route Bundle/Bid Scheduling Graph Conflict Headway Times Station Capacities Track Allocation Problem Ralf Borndörfer

  30. Route/Track Route Bundle/Bid Scheduling Graph Conflict Headway Times Station Capacities This Talk: Only Block Occupancy Conflicts Track Allocation Problem Ralf Borndörfer

  31. Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation (Timetable) Track Allocation Problem Ralf Borndörfer

  32. Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation (Timetable) Optimal Track Allocation Problem (OPTRA) Track Allocation Problem … … Ralf Borndörfer

  33. Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation (Timetable) Optimal Track Allocation Problem (OPTRA) Complexity Track Allocation Problem • Proposition [Caprara, Fischetti, Toth (02)]: • OPTRA is NP-hard. • Proof: • Reduction from Independent-Set. Ralf Borndörfer

  34. Overview • Rail Track Auctions • The Optimal Track Allocation Problem (OPTRA) • Mathematical Models • Computational Results Ralf Borndörfer

  35. Arc-based Routes: Multiflow Conflicts: Packing(pairwise) This talk: Block occupancy conflicts only IP Model OPTRA1 Variables • Arc occupancy Constraints • Flow conservation • Arc conflicts (pairwise) Objective • Maximize proceedings Ralf Borndörfer

  36. Selected Literature Brännlund et al. (1998) • Standardized Driving Dynamics • States (i,T,t,v) • Path formulation • Computational experiments with 17 stations at the route Uppsala-Borlänge, 26 trains, 40,000 states Caprara, Fischetti & Toth (2002) • Multi commodity flow model • Lagrangian relaxation approach • Computational experiments on low traffic and congested scenarios Ralf Borndörfer

  37. Arc-based Routes: Multiflow Conflicts: Packing(pairwise) Conflict Graph (Interval Graph) Cliques Perfectness IP Model OPTRA1 Ralf Borndörfer

  38. Arc-based Routes: Multiflow Conflicts: Packing(Max. Cliques) Proposition: The LP-relaxation of OPTRA2 can be solved in polynomial time. IP Model OPTRA2 Variables • Arc occupancy Constraints • Flow conservation • Arc conflicts (cliques) Objective • Maximize proceedings Ralf Borndörfer

  39. Arc-based Routes: Multiflow Conflicts: Packing(Max. Cliques) Proposition: The LP-relaxation of OPTRA2 can be solved in polynomial time. In practice … IP Model OPTRA2 Ralf Borndörfer

  40. Track Occupancy Configurations IP Model OPTRA3 Ralf Borndörfer

  41. Track Occupancy Configurations IP Model OPTRA3 Ralf Borndörfer

  42. Path-based Routes Path-based Configs IP Model OPTRA3 Variables • Path and config usage Constraints • Config choice • Path-config coupling (capacities) Objective • Maximize proceedings Ralf Borndörfer

  43. Path-based Routes Path-based Configs Shadow prices (useful in auction) Arc prices a Track prices r IP Model OPTRA3 Ralf Borndörfer

  44. Path-based Routes Path-based Configs Proposition:PLP(OPTRA1) PLP(OPTRA2)= PLP(OPTRA3). IP Model OPTRA3 Ralf Borndörfer

  45. Path-based Routes Path-based Configs Proposition:PLP(OPTRA2)= PLP(OPTRA3). Proposition:Route pricing = acyclic shortest path IP Model OPTRA3 Ralf Borndörfer

  46. Path-based Routes Path-based Configs Proposition:PLP(OPTRA2)= PLP(OPTRA3). Proposition:Route pricing = acyclic shortest path Proposition: Config pricing = acyclic shortest path IP Model OPTRA3 Ralf Borndörfer

  47. Path-based Routes Path-based Configs Proposition:PLP(OPTRA2)= PLP(OPTRA3). Proposition:Route and config pricing = acyclic shortest path Proposition: The LP-relaxation of OPTRA3 can be solved in polynomial time. IP Model OPTRA3 Column Generation Begin Compute Prices Add Variables OPTRA (IP) Solve Relaxation (LP) Yes All fixed? No Unfix/Fix Variables No Stop? Yes End Ralf Borndörfer

  48. Overview • Rail Track Auctions • The Optimal Track Allocation Problem (OPTRA) • Mathematical Models • Computational Results Ralf Borndörfer

  49. Test Network 45 Tracks 32 Stations 6 Traintypes 10 Trainsets 122 Nodes 659 Arcs 3-12 Hours 96 Station Capacities 612 Headway Times Computational Results Ralf Borndörfer

  50. Test Network Preprocessing Computational Results 1486 Nodes 1881 Arcs 293 Nodes 441 Arcs Ralf Borndörfer

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