Railway Slot Auctioning Ralf Borndrfer joint work with Martin - - PowerPoint PPT Presentation

▶
railway slot auctioning
SMART_READER_LITE
LIVE PREVIEW

Railway Slot Auctioning Ralf Borndrfer joint work with Martin - - PowerPoint PPT Presentation

Apr 04, 2023 21 likes •542 views

Railway Slot Auctioning Ralf Borndrfer joint work with Martin Grtschel Thomas Schlechte Arrival/ M ATHEON Fall School on Timetabling and Line Planning Dabendorf, 29. September 2006 DFG Research Center M ATHEON Mathematics for Key

slide-1
SLIDE 1

http://www.zib.de/borndoerfer borndoerfer@zib.de

DFG Research Center MATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB)

Ralf Borndörfer

Railway Slot Auctioning

Ralf Borndörfer

joint work with Martin Grötschel Thomas Schlechte Arrival/ MATHEON Fall School on Timetabling and Line Planning Dabendorf, 29. September 2006

slide-2
SLIDE 2

Ralf Borndörfer

2

Project "Trassenbörse"

Single Request Free Routes Routeportal Infrastracture Single EVU Routesearch Implementation Railsys W eb- Railsys Railsys Description Adjustment Many Bids current winner Trackallocation, Optimization Routerequests, Auctiondesgin Infrastructure, Drivingdynamics Multiple EVUs I nfraGen TS-Opt Auktio Many Bids current winner Trackallocation, Optimization Routerequests, Auctiondesgin Infrastructure, Drivingdynamics Multiple EVUs I nfraGen TS-Opt Auktio

slide-3
SLIDE 3

Ralf Borndörfer

3

Overview

Rail Track Auctions The Optimal Track Allocation Problem (OPTRA) Mathematical Models Computational Results

slide-4
SLIDE 4

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

slide-5
SLIDE 5

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

slide-6
SLIDE 6

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.

slide-7
SLIDE 7

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

slide-8
SLIDE 8

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 2854 (≈10257) theoretically possible combinations Sequential auction (5 rounds, 1 month between rounds)

Results

13% cost reduction Extension to 1.400 contracts (14% cost reduction)

slide-9
SLIDE 9

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
slide-10
SLIDE 10

Ralf Borndörfer

10

slide-11
SLIDE 11

Ralf Borndörfer

11

Track Request Form

slide-12
SLIDE 12

Ralf Borndörfer

12

Rail Track Auctioning

slide-13
SLIDE 13

Ralf Borndörfer

13

Rail Track Auction

All bids assigned: END Bid is assigned OPTRA finds allocation with maximum earnings EVUs decide on bids for bundles of timetabled tracks BEGIN Minimum Bid = Basic Price Bids are increased by a minimum increment Bid is not assigned Bids is unchanged

slide-14
SLIDE 14

Ralf Borndörfer

14

Overview

Rail Track Auctions The Optimal Track Allocation Problem (OPTRA) Mathematical Models Computational Results

slide-15
SLIDE 15

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

slide-16
SLIDE 16

Ralf Borndörfer

16

Track Allocation Problem

Route/Track

slide-17
SLIDE 17

Ralf Borndörfer

17

I nfrastructure

slide-18
SLIDE 18

Ralf Borndörfer

18

Blocks and Standardized Dynamics

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

i j k

v s

slide-19
SLIDE 19

Ralf Borndörfer

19

Standard Train Types

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

slide-20
SLIDE 20

Ralf Borndörfer

20

Track Allocation Problem

Route/Track Route Bundle/Bid

slide-21
SLIDE 21

Ralf Borndörfer

21

Variable Bids

Bid = Basic Bid + Departure/Arrival Time Bonus + Travel Time Bonus

€

  • Dep. time

12:00 12:20 12:08 90 80 Travel time [min] € 60 40 4 €/min

slide-22
SLIDE 22

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

slide-23
SLIDE 23

Ralf Borndörfer

23

Track Allocation Problem

Route/Track Route Bundle/Bid Scheduling Graph

slide-24
SLIDE 24

Ralf Borndörfer

24

Track Allocation Problem

Route/Track Route Bundle/Bid Scheduling Graph Conflict

slide-25
SLIDE 25

Ralf Borndörfer

25

Block Conflict

slide-26
SLIDE 26

Ralf Borndörfer

26

Track Allocation Problem

Route/Track Route Bundle/Bid Scheduling Graph Conflict

Headway Times Station Capacities This Talk: Only Block

Occupancy Conflicts

slide-27
SLIDE 27

Ralf Borndörfer

27

Track Allocation Problem

Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation

(Timetable)

slide-28
SLIDE 28

Ralf Borndörfer

28

Track Allocation Problem

… … Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation

(Timetable)

Optimal Track Allocation

Problem (OPTRA)

slide-29
SLIDE 29

Ralf Borndörfer

29

Optimal Track Allocation Problem

A B C D

time ICE drops out III. 3 x + 1 x = ??? difficult! ICE goes

  • I. variant

ICE slower II.

slide-30
SLIDE 30

Ralf Borndörfer

30

Track Allocation Problem

Route/Track Route Bundle/Bid Scheduling Graph Conflict Track Allocation

(Timetable)

Optimal Track Allocation

Problem (OPTRA)

Complexity

Proposition [Caprara, Fischetti, Toth (02)]:

OPTRA is NP-hard.

Proof:

Reduction from Independent-Set.

slide-31
SLIDE 31

Ralf Borndörfer

31

Selected Literature

Brännlund et al. (1998) Standardized Driving Dynamics States (i,T,t,v) 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 std

v {0,v (i)} ∈

slide-32
SLIDE 32

Ralf Borndörfer

32

I P Model OPTRA1

Variables

  • Arc occupancy

Constraints

  • Flow conservation
  • Arc conflicts (pairwise)

Objective

  • Maximize proceedings

Arc-based Routes: Multiflow Conflicts: Packing

(pairwise)

This talk: Block

  • ccupancy conflicts
  • nly
slide-33
SLIDE 33

Ralf Borndörfer

33

I P Model OPTRA1

Arc-based Routes: Multiflow Conflicts: Packing

(pairwise)

Conflict Graph

(Interval Graph)

Cliques Perfectness

slide-34
SLIDE 34

Ralf Borndörfer

34

I P Model OPTRA2

Variables

  • Arc occupancy

Constraints

  • Flow conservation
  • Arc conflicts (cliques)

Objective

  • Maximize proceedings

Arc-based Routes: Multiflow Conflicts: Packing

(Max. Cliques)

Proposition: The

LP-relaxation of OPTRA2 can be solved in polynomial time.

slide-35
SLIDE 35

Ralf Borndörfer

35

Multicommodity Flow Model with Packing Constraints

Arc-based Routes: Multiflow Conflicts: Packing

(Max. Cliques)

Proposition: The

LP-relaxation of OPTRA2 can be solved in polynomial time.

Looks like …

slide-36
SLIDE 36

Ralf Borndörfer

36

I P Model OPTRA3

Track Occupancy

Configurations

slide-37
SLIDE 37

Ralf Borndörfer

37

I P Model OPTRA3

Track Occupancy

Configurations

slide-38
SLIDE 38

Ralf Borndörfer

38

Variables

  • Path und config usage

Constraints

  • Path and config choice
  • Path-config-coupling (track capacity)

Objective Function

  • Maximize proceedings

I P Model OPTRA3

Path-based Routes Path-based Configs

slide-39
SLIDE 39

Ralf Borndörfer

39

I P Model OPTRA3

Path-based Routes Path-based Configs Shadow prices

(useful in auction)

Slot prices σi Track prices τr Arc prices αa

slide-40
SLIDE 40

Ralf Borndörfer

40

I P Model OPTRA3

Path-based Routes Path-based Configs Shadow prices Proposition: ⊇ PLP(OPTRA1) ⊇ PLP(OPTRA2)

= PLP(OPTRA3).

slide-41
SLIDE 41

Ralf Borndörfer

41

I P Model OPTRA3

Path-based Routes Path-based Configs Shadow prices Proposition:

PLP(OPTRA2)= PLP(OPTRA3).

Proposition:

Route pricing = acyclic shortest path

slide-42
SLIDE 42

Ralf Borndörfer

42

I P Model OPTRA3

Path-based Routes Path-based Configs Shadow prices Proposition:

PLP(OPTRA2)= PLP(OPTRA3).

Proposition:

Route pricing = acyclic shortest path

Proposition:

Config pricing = acyclic shortest path

slide-43
SLIDE 43

Ralf Borndörfer

43

I P Model OPTRA3

Column Generation Begin OPTRA (IP) Solve Relaxation (LP) Stop? All fixed? End

Yes No Yes No

Add Variables Compute Prices Unfix/Fix Variables

  • Path-based Routes
  • Path-based Configs
  • Shadow prices
  • Proposition:

PLP(OPTRA2)= PLP(OPTRA3).

  • Proposition:

Route/config pricing = acyclic shortest path

  • Proposition: The LP-

relaxation of OPTRA3 can be solved in polynomial time.

slide-44
SLIDE 44

Ralf Borndörfer

44

Computational Results

Test Network

45 Tracks 32 Stations 6 Traintypes 10 Trainsets 122 Nodes 659 Arcs 3-12 Hours 96 Station Capacities 612 Headway Times

slide-45
SLIDE 45

Ralf Borndörfer

45

Computational Results

Test Network Preprocessing 293 Nodes 441 Arcs 1486 Nodes 1881 Arcs

slide-46
SLIDE 46

Ralf Borndörfer

46

Computational Results

255 Trains 285 Trains Test Network Preprocessing Degrees of Freedom Timetables

slide-47
SLIDE 47

Ralf Borndörfer

47

Computational Results

Test Network Preprocessing Degrees of Freedom Timetables Harmonization

slide-48
SLIDE 48

Ralf Borndörfer

48

Computational Results

Delay in min. # Var. # Con. # Trains Time in sec.. 5 29112 34330 164 4.5 6 39641 54978 200 26.3 7 52334 86238 251 45.7 8 67000 133689 278 613.1 9 83227 206432 279 779.1 10 101649 315011 311 970.0

Test Network Preprocessing Degrees of Freedom

324 Trains, Profit 1

slide-49
SLIDE 49

Ralf Borndörfer

49

Auctioning Experiments (A. Reuter)

I CE I C RE RB S I CG

ind sync ind 28 25 26 27 + 27 R* /S sync 27 27 36 19 83 32 62 27 30 + 15 ICG 27 27 38 19 87 23 61 42 19 + 66 * 28 25 3 38 25 85 29 2 55 31 29 61 61 58 5 58

#

# Trains/Type ind sync ind sync ind sync ind sync Timetable 27 27 38 19 87 23 — + 24 IC/ICE ind 30 29 38 19 85 23 18 + 24 IC/ICE sync 24 9 27 9 36 19 83 19 22 + 27 R* /S ind 27 25 44 19 89 23 20

slide-50
SLIDE 50

Ralf Borndörfer

50

Auctioning Experiments (A. Reuter)

I CE I C RE RB S I CG

ind sync ind

1.12 1.10 1.14

+ 27 R* /S sync

2.31 1.34 1.02 1.04 0.88 1.41 1.06 0.98 33663

+ 15 ICG

1.45 1.44 1.08 1.08 0.87 0.90 0.88 1.03 32994

+ 66 *

2.21 1.88 2.87 1.03 1.10 0.89 1.11 1.53 1.47 1.60 41263 0.98 0.90 1.15 1.10

Σ

€/km ind sync ind sync ind sync ind sync Timetable + 24 IC/ICE ind

2.04 1.78 1.24 1.07 0.93 0.90 34421

+ 24 IC/ICE sync

1.89 1.94 1.45 3.27 1.14 1.10 0.89 0.83 36031

+ 27 R* /S ind

1.74 1.41 1.23 1.08 0.91 0.90 31180

slide-51
SLIDE 51

Ralf Borndörfer

51

Outlook

Column Generation Microsimulation

slide-52
SLIDE 52

http://www.zib.de/borndoerfer borndoerfer@zib.de

DFG Research Center MATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB)

Ralf Borndörfer

Thank you Thank you for your attention ! for your attention !