Rail Slot Exchange Auctioning - Theory and a First Algorithm - - PowerPoint PPT Presentation

ant@wip.tu-berlin.de

Workgroup for Infrastructure Policy, TU Berlin

Andreas Tanner

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

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Overview

Introduction Rail Track Auctions Bidding Language Auction Design The (OPTRA) Model Computational Results Outlook

ZIB WIP

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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,

• ptimization module finds best allocation

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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

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Bidding language

Bid flexibility modelled by time-value specifications Examples: Formally equivalent to XOR-connected

combinatorial bid

€ Depar t ur e t i m e t _opt € Depar t ur e t i m e t _m i n t _m ax

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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

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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

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Bidding language

How to coordinate bids? Better idea: explicite formulation of connections,

rolling stock management and regular services

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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

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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 s1 → s2, the last station of s1 coincides with the first

station of s2,

for any s1, s2 in S which are incomparable in →∗, there exists s3

in S such that either s1 →∗ s3 and s2 →∗ s3, or s3 →∗ s1 and s3 →∗ s2.

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Tour bids

Example:

s1 and s2 are merged into s3 s4 uses same material as s3 s4 is split into s5 and s6

s1 s2 s3 s5 s6 s4

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Tour bids

Example: 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

s1 s2 s3 s5 s6 s4 s7 s8

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Tour bids

Example: Now ok

s1 s2 s3 s5 s6 s4 s0 s7 s8

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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

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Open points in language design

Find compromise between language expressive

power and computational complexity

Prove that coordination produces more efficient

• utcome than auctions without

Exists already very simple commuter model where

uncoordinated bids lead to schedule with lower quality

Need to advance methodology

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Auction design

Iterative, combinatorial auction similar to Parkes’

e-bundle auction

Next slide shows procedure

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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 Willingness to pay reached? no yes yes no

http://www.zib.de/schlechte schlechter@zib.de

Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Thomas Schlechte

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

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Overview

Introduction Combinatorial Auctions Rail Track Auctions Infrastructure/Bid Generator Modules Optimal Track Allocation Problem (OPTRA) Computational Results

ZIB WIP

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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

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Micro-/ Macroscopic Model

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Blocks, Standardized Dynamics and Train Types

State (i,T,t,v) Directed block i Train type Y Starting time t, velocity v

train type V max [km/ h] train length [m] security

410 LZB LZB Signal Signal Signal Signal 400 225 100 125 600 ICE 250 IC 200 RE 160 RB 120 SB 140 ICG 100

i j k

v s

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Block Conflicts

conflict conflict t s

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Multi-Commodity-Flow Model

Space-time graph G= (V,A) of railroad network Nodes z= (i,t,v) ∈ V [Possible states] Arcs a= (z1,z2) ∈ A

[State transit]

Block and capacity conflicts on arcs Timetabled track request, bid ≅ path in G Timetable ≅ set of compatible timetabled tracks

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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

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I nteger Programming Formulation

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Selected Literature

Brännlund et al. (1998) Standardized Driving Dynamics State Model, 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 corridor scenarios

std

v {0,v (i)} ∈

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A First Algorithm

Create OPTRA Instance Model Preprocessing Solve by CPLEX

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Track Allocation

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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 Willingness to pay reached? no yes yes no

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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

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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

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Experiments (A. Reuter)

Auction results

50 100 150 200 250 300 350 I C E i n d . I C i n d . R E i n d . R B i n d . S i n d . I C G i n d . # t r a i n s train types #trains

Reihe1 Reihe2 Reihe3 Reihe4 Reihe5 Reihe6 Reihe7

http://www.zib.de/schlechte schlechter@zib.de

Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Thomas Schlechte

Thank you for your attention!