1. Martin Grötschel  Institut für Mathematik, Technische Universität Berlin (TUB)  DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (M ATHEON )  Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) groetschel@zib.dehttp://www.zib.de/groetschel Making Good Use of Railroad Tracks Martin Grötschel joint work with Ralf Borndörfer and Thomas Schlechte IP@CORE May 27-29, 2009

2. Yesterday  Got up at 4:50 am  Left home at 5:40 am  Arrived at Brussels airport at 7:50 am  Took the train  And arrived at 10:45 am here. Martin Grötschel 2

3. The invitation Dear Martin, We aim to get "distinguished" speakers that give a 50-minute lecture on their current research (I am sure you have a nice IP application area that you can survey...). So it will be a celebration of Laurence in disguise. There will be a dinner and it will happen there.... Best, Michele Martin Grötschel 3

4. Book Presentation on November 11, 2008 Year of Mathematics

5. Martin Grötschel 5 Railway tracks are a valuable and costly infrastructure - not to be left empty!

6. Martin Grötschel 6 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

7. Martin Grötschel 7 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

8. Martin Grötschel 8 Where do I come from?  Technische Universität Berlin  Konrad-Zuse-Zentrum für Informationstechnik  DFG Research Center M ATHEON Mathematics for key technologies  What type of problems are we aiming at?

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11. Martin Grötschel 11 ZIB

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13. M ATHEON Application Area B Logistics, traffic, and telecommunication networks Scientists in charge: Martin Grötschel, Rolf Möhring, Martin Skutella  Networks, such as telephone networks, the internet, airline, railway, and bus networks are omnipresent and play a fundamental role for communication and mobility in our society. We almost take their permanent availability, reliability, and quality at low cost for granted. However, traffic jams, ill-designed train schedules, canceled flights, break-downs of telephone and computing networks, and slow internet access are reminders that networks are not automatically good networks.  In fact, designing and operating communication and traffic networks are extremely complex tasks … Martin Grötschel 13

14. Martin Grötschel 14 The project Trassenbörse: Railway Slot Auctioning  The project aims at developing new ideas to make better (or even best) use of railway tracks.  A basic assumption, always favoured by economists, is that "markets" lead to an optimal allocation of goods.  But what are the goods to be allocated in the "railway market"?  And if we can define such goods precisely, how can one introduce trade mechanisms that lead to fair competition?  In other words, is there a way to (de-)regulate the current railway system that results in a “better utilization” of the railway infrastructure?

15. Martin Grötschel 15 The project Trassenbörse: Railway Slot Auctioning  The collection of question raised calls for a multidisciplinary approach.  The project is carried out by a group of economists, mathematicians, and railway engineers from Berlin and Hannover, each group bringing in its particular expertise.

16. Martin Grötschel 16 Project members  Economics: WIP / TU Berlin:  Kay Mitusch, Andreas Brenck, Andreas Tanner, Benedikt Peter Business consulting  Gottfried Ilgmann, Klemens Polatschek  Mathematical optimization: ZIB  Ralf Borndörfer, Martin Grötschel, Thomas Schlechte  Railway engineering and timetabling: SFWGG / TU Berlin  Jürgen Siegmann, Martin Balser, Elmar Swarat IVE / Univ. Hannover, RMCon  Thomas Siefer, Andreas Henkel, Marc Klemenz Many Bids current winner Track allocation, Optimization Routerequests, Auctiondesgin Infrastructure, Drivingdynamics Multiple EVUs InfraGen TS-Opt Auktio

17. Martin Grötschel 17 The project Trassenbörse: Railway Slot Auctioning  Project funding: Bundesministerium für Bildung und Forschung, Förderungskennziffer 19M2019  Duration in three phases: 12/2002 - 4/2010 (with some interrupts, however)

18. Martin Grötschel 18 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

19. Martin Grötschel 19 Railway network as a market place The railway network manager is obliged by EU and German law to offer  as much network capacity as possible to all train operation companies (TOCs) in a non-discriminating way. → The network is a market place, but, due to the many technical and administrative constraints, not a simple one.  Our goal: We want to help impove the market design!

20. Martin Grötschel 20 A market must have goods  What are the goods of the railway network market?  The answer is clear: slots  But what is a slot precisely?

21. Martin Grötschel 21 Capacity allocation today  A slot = right to run a train with a specified schedule on the network infrastructure Example: Berlin Hbf dep 10:51, Berlin-Spandau arr 11:03, dep 11:05, Hannover Hbf arr 12:28  TOCs order specified slots.  Slot prices are fixed and regulated.  Rules to resolve conflicts: 1.Cooperatively: “Negotiations”, construction of slot alternatives 2.Non-cooperatively: Priorities, sum of regular slot prices, bidding  Resulting network timetable is “manually optimized”

22. Martin Grötschel 22 Capacity allocation today

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24. Martin Grötschel 24 Capacity allocation tomorrow: our vision  TOCs submit bids for specified slots.  “Base price” is the fixed and regulated price (necessary to maintain the network infrastructure).  Bids may already include some flexibility w.r.t. time, stops and route; also with discounts.  Conflict resolution: 1.Cooperatively: Mathematical simultaneous optimization, taking advantage of flexibility of bids 2.Non-cooperatively: An auction process (rounds of auctions)  Need to develop optimization tools and auction design

25. Martin Grötschel 25 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

26. Martin Grötschel 26 Difficulties to be considered  What is a slot precisely?  How many details can/should be taken into account?  What about track profiles?  What about engine characteristics?  Routing through stations?  Track scheduling exact with respect to switches?  Signals?  Buffer times and various slacks (path allowances)?  …  Auctioning process  Details will be explained later

27. Martin Grötschel 27 Difficulties to be considered  If we have to take all possible technical and administrative details in the general planning model into account, we can immediately give up!  Sensible complexity reduction is necessary.  Hierarchical planning is the appropriate goal.  Coarse plans first, then details to be specified, iteration of the steps, if necessary.

28. Martin Grötschel 28 Slot request today

29. Martin Grötschel 29 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

30. Martin Grötschel 30 Reduction of network complexity  Train stations become simple nodes (with capacity data)  Tracks between stations become simple directed lines (no signals, no particular switches) One has to verify that these simplifications are acceptable in practice.

31. Martin Grötschel 31 Standardized Train Types and Standardized Train Dynamics train type V max [km/h] train length [m] security ICE250410LZB IC200400LZB RE160225Signal RB120100Signal SB140125Signal ICG100600Signal velocity Just like entry „Zugcharakteristik“ in today‘s „Trassenanmeldung“.

32. Martin Grötschel 32 Discretization of time, running and waiting times of trains  Minimum time unit (interval): 1 minute (but more detail sometimes necessary)  Matrix of train types‘ running (and required waiting) times in the network:

33. Martin Grötschel 33 Further simplifications  Wherever and whenever railway engineers have no objections  Data driven model precision: do not model things precisely for which data are not available.

34. Martin Grötschel 34 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

35. Martin Grötschel 35 Our sample network (right hand)

36. Martin Grötschel 36 Time-value specifications  Bid flexibility modelled by time-valued specifications  Examples: € Departure time t_opt € Departure time t_min t_max € Departure time t_opt t_min t_max time-dependent piecewise linear price function on a time interval base price

37. Martin Grötschel 37 Example for a slot bid Berlin Frankfurt Hbf Stuttgart Ostbahnhof central Spandau (optional) depart  9.00 arrive  14:30 core travel time  3:30 Discounts for  Departure at Ostbahnhof before 9:00  Arrival at Stuttgart after 14:30

38. Martin Grötschel 38 Implicit XOR-bids: Choice of path by optimization procedure  There are many different ways to get from Hannover to Fulda  If all of them are feasible for the requested train (i.e., if the TOC does not care where exactly the train will run between Hannover and Fulda), our optimization procedure will pick one that is optimal from the network perspective.

39. Martin Grötschel 39 Tour bids: Special support for branching and merging of trains  A tour is a set of slots that are connected by a successor relation →  s1 → s2 means that s2 can use rolling stock from s1 s1 s2 s3 s5 s6 s4

40. Martin Grötschel 40 Bids  We have developed a collection of possible bids that a TOC can submit (more than I can describe here).  Suppose the TOCs have submitted their bids.  What does the network operator do?  Actually, what is the network operator supposed to do?  The network operator has to apply the „Eisenbahninfrastruktur-Benutzungsverordnung - Verordnung über den diskriminierungsfreien Zugang zur Eisenbahninfrastruktur und über die Grundsätze zur Erhebung von Entgelt für die Benutzung der Eisenbahninfrastruktur - EIBV“ vom 3. Juni 2005 (BGBl. I S. 1566), die am 1. August 2005 in Kraft getreten ist.

41. Martin Grötschel 41 EIBV and conflict resolution §9 Absatz 5 EIBV, „Höchste Summe der Regelentgelte“: „(5) Bei der Entscheidung zwischen gleichrangigen Verkehren nach Absatz 4 hat der Betreiber der Schienenwege die Entgelte für die streitigen Zugtrassen gegenüberzustellen und 1.bei einem Konflikt zwischen zwei Zugtrassen der Zugtrasse den Vorrang einzuräumen, bei der das höchste Regelentgelt zu erzielen ist, 2.bei einem Konflikt zwischen mehr als zwei Zugtrassen den Zugtrassen den Vorrang einzuräumen, bei denen in der Summe das höchste Regelentgelt zu erzielen ist. …“, see http://bundesrecht.juris.de/eibv_2005/__9.html Optimization required by law! This seems to have been ignored by everyone involved! Let us consider an example Vorrang einzuräumen, bei denen in der Summe das höchste Regelentgelt zu erzielen ist. (Note: this is a formal definition of fair access!)

42. Martin Grötschel 42 800 900 700 100 500 155 154 500 150 650 Example: Bids displayed in a Time-Way-Diagram way time Regelentgelt base price

43. Martin Grötschel 43 800 900 700 100 500 155 154 500 150 650 way Zeit applying the EIBV rules: slots without any conflicts

44. Martin Grötschel 44 800 900 700 100 500 155 154 500 150 650 Way Zeit applying the EIBV rules: two slots in conflict

45. Martin Grötschel 45 800 900 700 100 500 155 154 500 133 657 way time applying the EIBV rules: lots of conflicts, what now?

46. Martin Grötschel 46 800 900 700 100 500 way time Greedy-Sum of base prices : 1000 Lots of conflicts, what now? „Bilateral conflict resolution“ in mathematical terms: greedy heuristic

47. Martin Grötschel 47 Lots of conflicts, what now? Smart planner

48. Martin Grötschel 48 800 900 700 100 500 way time More traffic, higher network revenue smart planner solution Greedy-Sum of base prices : 1000 Smart-Sum of base prices : 1400 Is that optimal, i.e., does the planner satisfy the law? Lots of conflicts, what now? Smart planner

49. Martin Grötschel 49 Lots of conflicts, what now? mathematical optimization

50. Martin Grötschel 50 800 900 700 100 500 way time Lots of conflicts, what now? mathematical optimization Greedy-Sum of base prices : 1000 Smart-Sum of base prices : 1400 the provable optimum: 1700

51. Martin Grötschel 51 800 900 700 100 500 way time Lots of conflicts, what now? mathematical optimization the provable total optimum: 2655 155 154 150 650

52. Martin Grötschel 52 800 900 700 100 500 155 154 500 150 650 Example: track bids with flexibilities way time

53. Martin Grötschel 53 800 900 700 100 500 Looking at the major conflicts: Optimumwith flexibilities way time sum of base prices: 2200 > 1700 even more traffic, more network revenue

54. Martin Grötschel 54 800 900 700 100 500 Looking at the major conflicts: Optimum with flexibilities way time sum of base prices: 2200 > 1700 even more traffic, more network revenue 155 154 obvious case for further bidding

55. Martin Grötschel 55 Track Allocation Problem Route/Track  Route/Track

56. Martin Grötschel 56  Route/Track  Route Bundle/Bid Track Allocation Problem

57. Martin Grötschel 57 Track Allocation Problem  Route/Track  Route Bundle/Bid  Scheduling Graph

58. Martin Grötschel 58 Track Allocation Problem  Route/Track  Route Bundle/Bid  Scheduling Graph  Conflict

59. Martin Grötschel 59 Track Allocation Problem  Route/Track  Route Bundle/Bid  Scheduling Graph  Conflict  Headway Times  Station Capacities

60. Martin Grötschel 60 Track Allocation Problem  Route/Track  Route Bundle/Bid  Scheduling Graph  Conflict  Track Allocation (Timetable)

61. Martin Grötschel 61 Track Allocation Problem  Route/Track  Route Bundle/Bid  Scheduling Graph  Conflict  Track Allocation (Timetable)  Track Allocation Problem (OPTRA) ……

62. Martin Grötschel 62 … Track allocation problem

63. Martin Grötschel 63 Solution approach: What methods?  The current standard is the use of heuristics.  This is infeasible in our situation! Namely, suppose the system finds a “good” solution that rules out one bid that some TOC eagerly wants to run. And now the TOC finds a solution, including its special bid, that is overall better than the “good” solution. The TOC would declare the work of the network operator cheating.  A proof of optimality is required!

64. Martin Grötschel 64 Mathematical solution approach: Integer Programming Models  APP  Arc-based  Routes: Multiflow  Conflicts: Packing (max. cliques)  Proposition: The LP-relaxation of OPTRA 2 can be solved in polynomial time. Variables  Arc occupancy Constraints  Flow conservation  Arc conflicts (maximal cliques) Objective  Maximize proceedings

65. Martin Grötschel 65 Variables  Path und config usage Constraints  Path and config choice  Path-config-coupling (track capacity) Objective Function  Maximize proceedings IP Models  PCP  Path-based routes  Path-based configs

66. Martin Grötschel 66 two IP Models solving the track allocation problem – in priniple  PCP  Path-based  Proposition: v (P LP (APP))  = v (P LP (PCP)) Thomas Schlechte

67. Martin Grötschel 67 Results  Test Network  45 Tracks  32 Stations  6 Traintypes  10 Trainsets  122 Nodes  659 Arcs  3-12 Hours  96 Station Capacities  612 Headway Times

68. Martin Grötschel 68 Results Szenario 324 trains max. # trains flex/min.#var.#constr.#trainstime/sec. 529.11234.3301644,5 639.64154.97820026,3 752.33486.23825145,7 867.000133.689278613,1 983.227206.432279779,1 10101.649315.011311970,0

69. Martin Grötschel 69 Model Comparison Scenario: Status Quo Schedule 285 Trains Flex.LP 1 IP 1 * LP 3 IP 3 * 02351692208025522342112125213 22453476209204523519772173288 42453476209204524269992234398 62453476217489724534762304735 82453476228230524534762304735 102453476239092124534762339652 PCPAPP *- Runtime maximal 1h

70. Thomas Schlechte 70 Line Plan Problem „China20“

71. Thomas Schlechte 71 Example „China20“ Which track upgrading project is more important ? upgrading tracks fixed tracks

72. Thomas Schlechte 72 Origin Destination Matrix Estimated Passenger Demand for all pairs

73. Thomas Schlechte 73 Optimize Cost, case (A) Cost function: 1.000.000 € per line, 100,- € per km Cost function: 1.000.000 € per line, 100,- € per km

74. Thomas Schlechte 74 Optimize Traveltime, case (B)

75. Thomas Schlechte 75 Line Plan Decision ? (A)(B) number of lines918 cost in Mio. €238264 traveltime in Mio. min.383349 (A cost) (B time)

76. Thomas Schlechte 76 Timetabling Stations Tracks Train Requests TS-OPT Timetable Optimization Model maximize track utilization timetable attractiveness subject to safety requirements time windows periodic Passenger versus individual Cargo periodic Passenger versus individual Cargo

77. Thomas Schlechte 77 Timetable for Lineplan (A)

78. Thomas Schlechte 78 Timetable for Lineplan (B)

79. Thomas Schlechte 79 Saturation with Cargo Trains/Slots add cargo trains Beijing/Shanghai add cargo trains Beijing/Shanghai

80. Thomas Schlechte 80 Timetable Decision ? (A)(B) number of train slots452462 passenger ICE‘s3618 cargo trains426444 (A) (B)

81. Switzerland  A real case with real data.  Micro » Macro » Micro test  Data problems, problems with the definitions, inconsistencies of simulation software systems, etc.  But we have very interesting results.  Unfortunately,… Martin Grötschel 81

82. Martin Grötschel 82 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

83. Martin Grötschel 83 Goal Development of an auction mechanism for track usage (slots): economic and technical analysis of the various track allocation rules development of a mathematical program for optimal time tabeling

84. Martin Grötschel 84 Basic idea of a slot auction  train operation companies (TOCs) deliver bids for slots (possibly including various degrees of freedom concerning willingness to pay, timing, stops, train routes)  minimum bid = base price  Auctioneer computes conflict free slot assignment (combination of bids) that maximizes the network revenue and temporarily allocates them to the bidders.  Iteration (rounds of the auction): Bids that have not won can be repeated or modified and resubmitted.  Criterion for termination of auction (# of rounds, # of changed bids,..)  The result of the process is a timetable (possibly combining slots allocated to various bidders) which then has to be refined for use in practice.

85. Martin Grötschel 85 Goal of the slot allocation auction: practical rules for an auction mechanism Components:  „from coarse to fine“: …  Exact mathematical optimization: …  Consideration of alternatives: …  Economic and technical analysis: …

86. Martin Grötschel 86 Remarks on the current EIBV  All relevant rules can be implemented, e.g.:  Priorities  „maximale Summe der Regelentgelte“  Höchstpreisverfahren  Rechte aus Rahmenverträgen  The sog. „Koordinierungsprozess“ in EIBV, i.e., the bilateral negotiation (considering also alternative options) is automatically included in the approach: no discrimination, optimality,…

87. Martin Grötschel 87 Auction design  Iterative, combinatorial auction similar to Parkes’ ibundle auction  Next slide shows procedure

88. Martin Grötschel 88 Rail Track Auction END OPTRA model is solved with maximum earnings TOCs decide on bids for slots BEGIN Bid is increased by a minimum increment Bid assigned? Bid is unchanged All bids Unchanged? yes no Wish to increase bid? yes no yes no

89. Martin Grötschel 89 There are still lots of economic issues  Auction rounds  Sequences of auctions  Informal coordination between TOCs  Use-it-or-lose-it rules  Network proceeds is operational goal  The „density“ of potential goods  Bidding strategies  How to analyse auction design?

90. Martin Grötschel 90 Contents 1.Introduction and project outline 2.What is the goal? 3.What are the problems? 4.The model: networks, tracks, trains, time, slots,… 5.Bids 6.The auction process 7.Summary

91. Martin Grötschel 91 slot allocation problem: other literature Charnes 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 & Smith (2000) Caprara, Fischetti and T. (2002), Peeters (2003) Kroon and Peeters (2003), Mistry and Kwan (2004) Barber, Salido, Ingolotti, Abril, Lova, Tormas (2004) Semet and Schoenauer (2005), Caprara, Monaci, T. and Guida (2005) Kroon, Dekker and Vromans (2005), Vansteenwegen and Van Oudheusden (2006), Cacchiani, Caprara, T. (2006) Caprara, Kroon, Monaci, Peeters, T. (2006)

92. Failed railroad infrastructure plannin g Martin Grötschel 92

93. Martin Grötschel  Institut für Mathematik, Technische Universität Berlin (TUB)  DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (M ATHEON )  Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) groetschel@zib.dehttp://www.zib.de/groetschel Making Good Use of Railroad Tracks Martin Grötschel joint work with Ralf Borndörfer and Thomas Schlechte IP@CORE May 27-29, 2009 Thank you for your attention