Rail Slot Exchange Auctioning - Theory and a First Algorithm Andreas Tanner joint work with T.Schlechte R.Borndörfer K.Mitusch 4th Conference on Applied I nfrastructure Research 8.10.2005 � Workgroup for Infrastructure Policy, TU Berlin Andreas Tanner ant@wip.tu-berlin.de
2 Overview � Introduction � Rail Track Auctions WIP � Bidding Language � Auction Design � The (OPTRA) Model � Computational Results ZIB � Outlook Andreas Tanner
3 Basic I dea � Replace fixed pricing of railway slots by an auction � A slot = right to run a train with a given schedule ( i.e. Berlin dep. 8:00 Potsdam Hbf arr. 8:20 Brandenburg arr. 8:40 Magdeburg arr. 9:10 Mo-Fr except public holidays ) � Competing train operators submit bids for slots, optimization module finds best allocation Andreas Tanner
4 Challenges � Optimization -> OPTRA module from ZIB, more about it later � Auction design � Rail schedules are precise up to the minute, but slot requests are more flexible � How to coordinate slots -> can we leave that aside? � Connections � Regular services � Management of rolling stock � Need to design a bidding language Andreas Tanner
5 Bidding language � Bid flexibility modelled by time-value specifications € � Examples: € t _m i n t _opt t _m ax Depar t ur e t i m e Depar t ur e t i m e � Formally equivalent to XOR-connected combinatorial bid Andreas Tanner
6 Bidding language � How to coordinate bids? � First idea: allow arbitrary AND-connected c.b. � -> bad: dominating market participant could bundle all his slot requests into 1 bid that would have to be accepted Andreas Tanner
7 Bidding language � How to coordinate bids? � First idea: allow arbitrary AND-connected c.b. � -> bad: dominating market participant could bundle all his slot requests into 1 bid that would have to be accepted � -> bad: connections etc are hard to formulate due to bid flexibility Andreas Tanner
8 Bidding language � How to coordinate bids? � Better idea: explicite formulation of connections, rolling stock management and regular services Andreas Tanner
9 Bidding language � How to coordinate bids? � Better idea: explicite formulation of connections, rolling stock management and regular services � Example: Rolling stock management expressible by tour bids Andreas Tanner
10 Tour bids For a successor relation → , we write → ∗ for the transitive closure of → . � s1 → s2 means s2 “right after” s1. s1 → ∗ s2 means s2 “somewhere after” s1 � s1 → s2 means that s2 uses rolling stock from s1 � Branching and merging of trains is supported � Definition A tour is a set of slots S together with a successor relation → such that � for any s 1 → s 2 , the last station of s 1 coincides with the first station of s 2 , � for any s 1 , s 2 in S which are incomparable in → ∗ , there exists s 3 in S such that either s 1 → ∗ s 3 and s 2 → ∗ s 3 , or s 3 → ∗ s 1 and s 3 → ∗ s 2 . Andreas Tanner
11 Tour bids � Example: s1 s5 s4 s3 s6 s2 � s1 and s2 are merged into s3 � s4 uses same material as s3 � s4 is split into s5 and s6 Andreas Tanner
12 Tour bids � Example: s1 s5 s4 s3 s6 s2 s7 s8 � This is not wanted: s7 and s8 don’t share stock with s1 to s6 � Condition above says that e.g. s1 and s7 should have common predecessor or successor Andreas Tanner
13 Tour bids � Example: s1 s5 s4 s3 s6 s2 s0 s7 s8 � Now ok Andreas Tanner
14 Connections and regular services � Operator-neutral connection bids � Example: bid for “at station A, train has connection to B arrival before 10:00” � Operator-neutral regular service rules � Example: a regular service rule for slot ((A,8:00),(B,8:20),(C,8:45),(D,9:00)) is ((A,C),60,-1,+ 10) � means that there are trains from A to C hourly from 7:00-7:45 to 18:00-18:45 (not necessarily by the bidding train operator) � A bid can be conditioned on multiple regular service rules Andreas Tanner
15 Open points in language design � Find compromise between language expressive power and computational complexity � Prove that coordination produces more efficient outcome than auctions without � Exists already very simple commuter model where uncoordinated bids lead to schedule with lower quality � Need to advance methodology Andreas Tanner
16 Auction design � Iterative, combinatorial auction similar to Parkes’ e-bundle auction � Next slide shows procedure Andreas Tanner
17 Rail Track Auction TOCs decide on bids for slots BEGIN All bids Unchanged? Bid is no Bid is increased by a unchanged minimum increment OPTRA no model is solved with yes maximum earnings yes Willingness to pay yes reached? no Bid assigned? END Andreas Tanner
Rail Slot Exchange Auctioning - Theory and a First Algorithm Part I I T.Schlechte joint work with R. Borndörfer M. Grötschel S. Lukac K. Mitusch A. Tanner 4th Conference on Applied I nfrastructure Research 8.10.2005 � Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) Thomas Schlechte schlechter@zib.de http://www.zib.de/schlechte
2 Overview � Introduction � Combinatorial Auctions WIP � Rail Track Auctions � Infrastructure/Bid Generator Modules � Optimal Track Allocation Problem (OPTRA) ZIB � Computational Results Thomas Schlechte
3 Optimal Track Allocation Problem (OPTRA) I nput � Set of timetabled tracks incl. objective � Available infrastructure (space and time) Output � Track assignment � Conflict free � Maximum objective value Thomas Schlechte
Micro-/ Macroscopic Model Schlechte Thomas 4
5 Blocks, Standardized Dynamics and Train Types train V max train security State (i,T,t,v) type [km/ h] length [m] � Directed block i ICE 250 410 LZB � Train type Y IC 200 400 LZB � Starting time t, velocity v RE 160 225 Signal RB 120 100 Signal SB 140 125 Signal v ICG 100 600 Signal s i j k Thomas Schlechte
s conflict conflict Block Conflicts t Schlechte Thomas 6
7 Multi-Commodity-Flow Model � Space-time graph G= (V,A) of railroad network � Nodes z= (i,t,v) ∈ V [Possible states] � Arcs a= (z 1 ,z 2 ) ∈ A [State transit] � Block and capacity conflicts on arcs � Timetabled track request, bid ≅ path in G � Timetable ≅ set of compatible timetabled tracks Thomas Schlechte
8 I nteger Programming Formulation Variables � train request uses arc Constraints � Multicommodity flow conditions for train request set , � Set packing conditions for block conflicting sets Objective � Maximize total earnings of scheduled train request Thomas Schlechte
9 I nteger Programming Formulation Thomas Schlechte
10 Selected Literature Brännlund et al. (1998) � Standardized Driving Dynamics ∈ v {0,v (i)} � State Model, Path formulation std � 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 corridor scenarios Thomas Schlechte
11 A First Algorithm Create OPTRA Instance Model Preprocessing Solve by CPLEX Thomas Schlechte
Track Allocation Schlechte Thomas 12
13 Rail Track Auction TOCs decide on bids for slots BEGIN All bids Unchanged? Bid is no Bid is increased by a unchanged minimum increment OPTRA no model is solved with yes maximum earnings yes Willingness to pay yes reached? no Bid assigned? END Thomas Schlechte
14 Simple Auction (A. Reuter) Round 9 Round Earnings Round Earnings 1 44563 9 46575 2 44563 10 47051 3 44598 11 48096 4 44799 12 48253 5 44799 13 48337 6 44972 14 48391 7 45551 15 48513 8 46375 Thomas Schlechte
15 Test Network � Selection Criteria � Contains railway specific aspects � Important subnet � Used in earlier studies � Data � 45 arcs (track sections) = 1176 km � 31 nodes (stations) � 6 train types Thomas Schlechte
16 Experiments (A. Reuter) Auction results #trains 350 300 250 Reihe1 200 Reihe2 150 Reihe3 100 Reihe4 Reihe5 50 Reihe6 0 Reihe7 . . . s . . . d d d d d d n n n n n n n i a i i train types i i i i E C G r E B S t # C R R C I I I Thomas Schlechte
Thank you for your attention! � Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) Thomas Schlechte schlechter@zib.de http://www.zib.de/schlechte
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