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Making Good Use of Railroad Tracks Martin Grötschel joint work with Ralf Borndörfer and Thomas Schlechte IP@CORE May 27-29, 2009 Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel 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 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 Book Presentation on November 11, 2008 Year of Mathematics 5 Martin Grötschel Railway tracks are a valuable and costly infrastructure - not to be left empty! 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 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 Martin Grötschel 8 Where do I come from? Technische Universität Berlin Konrad-Zuse-Zentrum für Informationstechnik DFG Research Center MATHEON Mathematics for key technologies What type of problems are we aiming at? Martin Grötschel 9 Martin Grötschel 10 Martin Grötschel 11 ZIB Martin Grötschel 12 Martin Grötschel 13 MATHEON 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 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? 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. Martin Grötschel 16 Project members Kay Mitusch, Andreas Brenck, Andreas Tanner, Benedikt Peter Business consulting Multiple EVUs Routerequests, Auctiondesgin Auktio 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 Martin Grötschel Thomas Siefer, Andreas Henkel, Marc Klemenz Many Bids Gottfried Ilgmann, Klemens Polatschek current winner Economics: WIP / TU Berlin: Track allocation, Optimization TS-Opt Infrastructure, Drivingdynamics InfraGen 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) 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 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! 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? Martin Grötschel 21 Capacity allocation today A slot = right to run a train with a specified schedule on the network infrastructure Example: Berlin Hbf Berlin-Spandau Hannover Hbf dep 10:51, arr 11:03, dep 11:05, 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 Martin Grötschel Resulting network timetable is “manually optimized” 22 Martin Grötschel Capacity allocation today 23 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 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 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 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. Martin Grötschel 28 Slot request today 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 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. Martin Grötschel 31 Standardized Train Types and Standardized Train Dynamics train type velocity V max [km/h] train length [m] security ICE 250 410 LZB IC 200 400 LZB RE 160 225 Signal RB 120 100 Signal SB 140 125 Signal ICG 100 600 Signal Just like entry „Zugcharakteristik“ in today‘s „Trassenanmeldung“. 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: 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. 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 Martin Grötschel 35 Our sample network (right hand) Martin Grötschel 36 Time-value specifications Bid flexibility modelled by time-valued specifications Examples: € € base price t_opt t_min t_max Departure time € time-dependent piecewise linear price function on a time interval Martin Grötschel t_min t_opt t_max Departure time Departure time 37 Example for a slot bid Berlin Ostbahnhof depart 9.00 central Spandau (optional) core travel time 3:30 Discounts for Departure at Ostbahnhof before 9:00 Arrival at Stuttgart after 14:30 Martin Grötschel Frankfurt Hbf Stuttgart arrive 14:30 38 Martin Grötschel 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 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 s5 s3 s2 Martin Grötschel s4 s6 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? Martin Grötschel 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 EIBV and conflict resolution §9 Absatz 5 EIBV, „Höchste Summe der Regelentgelte“: Let us consider an example „(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. (Note: this is a formal definition of fair access!) …“, see http://bundesrecht.juris.de/eibv_2005/__9.html Optimization required by law! This seems to have been ignored by everyone involved! Martin Grötschel 42 Example: Bids displayed in a Time-Way-Diagram way 500 700 150 100 155 154 800 900 time Martin Grötschel 650 500 Regelentgelt base price 43 applying the EIBV rules: slots without any conflicts way 500 700 150 100 155 154 800 900 Zeit Martin Grötschel 650 500 44 applying the EIBV rules: two slots in conflict Way 500 700 150 100 155 154 800 900 Zeit Martin Grötschel 650 500 45 applying the EIBV rules: lots of conflicts, what now? way 500 700 133 100 155 154 800 900 time Martin Grötschel 657 500 46 Lots of conflicts, what now? „Bilateral conflict resolution“ way in mathematical terms: greedy heuristic 500 Greedy-Sum of base prices : 1000 700 100 800 900 time Martin Grötschel 500 47 Martin Grötschel Lots of conflicts, what now? Smart planner 48 Lots of conflicts, what now? Smart planner way 500 700 smart planner solution Greedy-Sum of base prices : 1000 Smart-Sum of base prices : 1400 100 Is that optimal, i.e., does the planner satisfy the law? time Martin Grötschel More traffic, higher network revenue 800 900 500 49 Martin Grötschel Lots of conflicts, what now? mathematical optimization 50 Lots of conflicts, what now? mathematical optimization way 500 700 Greedy-Sum of base prices : 1000 Smart-Sum of base prices : 1400 the provable optimum: 1700 100 800 900 time Martin Grötschel 500 51 Lots of conflicts, what now? mathematical optimization way the provable total optimum: 2655 500 700 150 100 155 154 800 900 time Martin Grötschel 650 500 52 Example: track bids with flexibilities way 500 700 150 100 155 154 800 900 time Martin Grötschel 650 500 53 Looking at the major conflicts: Optimumwith flexibilities way sum of base prices: 2200 > 1700 500 even more traffic, more network revenue 700 100 800 900 time Martin Grötschel 500 54 Looking at the major conflicts: Optimum with flexibilities way sum of base prices: 2200 > 1700 500 even more traffic, more network revenue 700 100 155 154 800 900 time Martin Grötschel 500 obvious case for further bidding 55 Track Allocation • Route/Track Problem Route/Track Martin Grötschel 56 Track Allocation Problem Route/Track Route Bundle/Bid Martin Grötschel 57 Track Allocation Problem Route/Track Route Bundle/Bid Scheduling Graph Martin Grötschel 58 Track Allocation Problem Route/Track Route Bundle/Bid Scheduling Graph Conflict Martin Grötschel 59 Track Allocation Problem Route/Track Route Bundle/Bid Scheduling Graph Conflict Headway Times Station Capacities Martin Grötschel 60 Track Allocation Problem Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation (Timetable) Martin Grötschel 61 Track Allocation Problem Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation (Timetable) Track Allocation Problem (OPTRA) Martin Grötschel … … 62 Track allocation problem … 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! Martin Grötschel 64 Mathematical solution approach: Integer Programming Models APP Arc-based Routes: Multiflow Conflicts: Packing (max. cliques) Proposition: The LP-relaxation of OPTRA2 can be solved in polynomial time. Variables Arc occupancy Constraints Flow conservation Arc conflicts (maximal cliques) Objective Martin Grötschel Maximize proceedings 65 IP Models PCP Path-based routes Path-based configs Variables Path und config usage Constraints Path and config choice Path-config-coupling (track capacity) Objective Function Martin Grötschel Maximize proceedings 66 two IP Models solving the track allocation problem – in priniple PCP Path-based Proposition: v (PLP(APP)) = v (PLP(PCP)) Thomas Schlechte 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 Martin Grötschel 68 Results Szenario • 324 trains • max. # trains Martin Grötschel flex/min. #var. #constr. #trains time/sec. 5 29.112 34.330 164 4,5 6 39.641 54.978 200 26,3 7 52.334 86.238 251 45,7 8 67.000 133.689 278 613,1 9 83.227 206.432 279 779,1 10 101.649 315.011 311 970,0 69 Model Comparison Scenario: Status Quo Schedule 285 Trains APP Flex. Martin Grötschel *- Runtime maximal 1h PCP LP1 IP1* LP3 IP3* 0 2351692 2080255 2234211 2125213 2 2453476 2092045 2351977 2173288 4 2453476 2092045 2426999 2234398 6 2453476 2174897 2453476 2304735 8 2453476 2282305 2453476 2304735 10 2453476 2390921 2453476 2339652 70 Line Plan Problem „China20“ Thomas Schlechte 71 Example „China20“ Which track upgrading project is more important ? upgrading tracks fixed tracks Thomas Schlechte 72 Origin Destination Matrix Estimated Passenger Demand for all pairs Thomas Schlechte 73 Optimize Cost, case (A) Cost function: 1.000.000 € per line, 100,- € per km Thomas Schlechte 74 Optimize Traveltime, case (B) Thomas Schlechte 75 Line Plan Decision ? (A cost) Thomas Schlechte (A) (B) number of lines 9 18 cost in Mio. € 238 264 traveltime in Mio. min. 383 349 (B time) 76 Timetabling periodic Passenger versus individual Cargo TS-OPT Optimization Model Train Requests Tracks Stations Thomas Schlechte maximize track utilization timetable attractiveness subject to safety requirements time windows Timetable 77 Timetable for Lineplan (A) Thomas Schlechte 78 Timetable for Lineplan (B) Thomas Schlechte 79 Saturation with Cargo Trains/Slots add cargo trains Beijing/Shanghai Thomas Schlechte 80 Timetable Decision ? (A) number of train slots passenger ICE‘s cargo trains Thomas Schlechte (A) (B) 452 462 36 18 426 444 (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 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 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 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. Martin Grötschel 85 Goal of the slot allocation auction: practical rules for an auction mechanism Components: Martin Grötschel „from coarse to fine“: … Exact mathematical optimization: … Consideration of alternatives: … Economic and technical analysis: … 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,… Martin Grötschel 87 Auction design Iterative, combinatorial auction similar to Parkes’ ibundle auction Next slide shows procedure Martin Grötschel 88 Rail Track Auction TOCs decide on bids for slots BEGIN Bid is unchanged Bid is increased by a minimum increment yes All bids Unchanged? no OPTRA model is solved with maximum earnings no yes Wish to increase bid? yes no END Martin Grötschel Bid assigned? 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? 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 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) Martin Grötschel 92 Failed railroad infrastructure planning Martin Grötschel Making Good Use of Railroad Tracks Martin Grötschel Thank you for your attention joint work with Ralf Borndörfer and Thomas Schlechte IP@CORE May 27-29, 2009 Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel