Railway Slot Auctioning Ralf Borndörfer joint work with Martin Grötschel Thomas Schlechte Arrival/ M ATHEON Fall School on Timetabling and Line Planning Dabendorf, 29. September 2006 � DFG Research Center M ATHEON Mathematics for Key Technologies Ralf Borndörfer � Zuse-Institute Berlin (ZIB) borndoerfer@zib.de http://www.zib.de/borndoerfer
2 Project "Trassenbörse" Multiple EVUs Routerequests, Single EVU Multiple EVUs Auctiondesgin Routerequests, Auktio Routeportal Auctiondesgin W eb- Auktio Railsys Single winner Free Many current Bids Request Routes winner current Many Bids Trackallocation, Trackallocation, Routesearch Optimization Optimization Description Railsys TS-Opt Implementation TS-Opt Infrastructure, Infrastracture Drivingdynamics Infrastructure, Adjustment Drivingdynamics Railsys I nfraGen I nfraGen Ralf Borndörfer
3 Overview � Rail Track Auctions � The Optimal Track Allocation Problem (OPTRA) � Mathematical Models � Computational Results Ralf Borndörfer
4 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, IVE, rmcon Ralf Borndörfer
5 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
6 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
7 Examples � I n 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 � I n 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
8 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 � 2 854 ( ≈ 10 257 ) 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
9 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
Borndörfer Ralf 10
Track Request Form Borndörfer Ralf 11
Rail Track Auctioning Borndörfer Ralf 12
13 Rail Track Auction BEGIN Minimum Bid = Basic Price EVUs decide on bids for bundles of timetabled tracks Bids is unchanged OPTRA finds Bids are increased by a allocation with maximum earnings minimum increment Bid is assigned All bids assigned: Bid is not assigned END Ralf Borndörfer
14 Overview � Rail Track Auctions � The Optimal Track Allocation Problem (OPTRA) � Mathematical Models � Computational Results Ralf Borndörfer
15 Optimal Track Allocation Problem (OPTRA) I nput � 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
16 Track Allocation Problem � Route/Track Ralf Borndörfer
I nfrastructure Borndörfer Ralf 17
18 Blocks and Standardized Dynamics State (i,T,t,v) � Directed block i � Train type T � Starting time t, velocity v v s i j k Ralf Borndörfer
19 Standard Train Types train V max train security … type [km/ h] length [m] ICE 250 410 LZB IC 200 400 LZB RE 160 225 Signal RB 120 100 Signal SB 140 125 Signal ICG 100 600 Signal Ralf Borndörfer
20 Track Allocation Problem � Route/Track � Route Bundle/Bid Ralf Borndörfer
21 Variable Bids Bid = Basic Bid + Departure/Arrival Time Bonus + Travel Time Bonus € € 90 4 €/min 80 Travel 12:00 12:08 12:20 Dep. time 40 60 time [min] Ralf Borndörfer
22 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
23 Track Allocation Problem � Route/Track � Route Bundle/Bid � Scheduling Graph Ralf Borndörfer
24 Track Allocation Problem � Route/Track � Route Bundle/Bid � Scheduling Graph � Conflict Ralf Borndörfer
Block Conflict Borndörfer Ralf 25
26 Track Allocation Problem � Route/Track � Route Bundle/Bid � Scheduling Graph � Conflict � Headway Times � Station Capacities � This Talk: Only Block Occupancy Conflicts Ralf Borndörfer
27 Track Allocation Problem � Route/Track � Route Bundle/Bid � Scheduling Graph � Conflict � Track Allocation (Timetable) Ralf Borndörfer
28 Track Allocation Problem � Route/Track … … � Route Bundle/Bid � Scheduling Graph � Conflict � Track Allocation (Timetable) � Optimal Track Allocation Problem (OPTRA) Ralf Borndörfer
29 Optimal Track Allocation Problem difficult! 3 x + 1 x = ??? A B C D I. variant II. III. ICE slower time ICE goes ICE drops out Ralf Borndörfer
30 Track Allocation Problem � Route/Track Proposition [Caprara, Fischetti, Toth (02)]: � Route Bundle/Bid OPTRA is NP -hard. � Scheduling Graph � Conflict Proof: � Track Allocation Reduction from Independent-Set. (Timetable) � Optimal Track Allocation Problem (OPTRA) � Complexity Ralf Borndörfer
31 Selected Literature Brännlund et al. (1998) � Standardized Driving Dynamics ∈ � States (i,T,t,v) v {0,v (i)} std � 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
32 I P Model OPTRA 1 � Arc-based � Routes: Multiflow � Conflicts: Packing (pairwise) � This talk: Block occupancy conflicts only Variables � Arc occupancy Constraints � Flow conservation � Arc conflicts (pairwise) Objective Ralf � Maximize proceedings Borndörfer
33 I P Model OPTRA 1 � Arc-based � Routes: Multiflow � Conflicts: Packing (pairwise) � Conflict Graph (Interval Graph) � Cliques � Perfectness Ralf Borndörfer
34 I P Model OPTRA 2 � Arc-based � Routes: Multiflow � Conflicts: Packing (Max. Cliques) � Proposition: The LP-relaxation of OPTRA 2 can be Variables � solved in Arc occupancy Constraints polynomial time. � Flow conservation � Arc conflicts (cliques) Objective Ralf � Maximize proceedings Borndörfer
35 Multicommodity Flow Model with Packing Constraints � Arc-based � Routes: Multiflow � Conflicts: Packing (Max. Cliques) � Proposition: The LP-relaxation of OPTRA 2 can be solved in polynomial time. � Looks like … Ralf Borndörfer
36 I P Model OPTRA 3 � Track Occupancy Configurations Ralf Borndörfer
37 I P Model OPTRA 3 � Track Occupancy Configurations Ralf Borndörfer
38 I P Model OPTRA 3 � Path-based Routes � Path-based Configs Variables � Path und config usage Constraints � Path and config choice � Path-config-coupling (track capacity) Objective Function � Ralf Maximize proceedings Borndörfer
39 I P Model OPTRA 3 � Path-based Routes � Path-based Configs � Shadow prices (useful in auction) � Slot prices σ i � Track prices τ r � Arc prices α a Ralf Borndörfer
40 I P Model OPTRA 3 � Path-based Routes � Path-based Configs � Shadow prices � Proposition: ⊇ P LP (OPTRA 1 ) ⊇ P LP (OPTRA 2 ) = P LP (OPTRA 3 ). Ralf Borndörfer
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