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

DFG Research Center MATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW)

Ralf Borndörfer

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Ralf Borndörferjoint work with

Martin Grötschel Thomas Schlechte

X Encuentro de Matemática, Quito, 25. Juli 2006

Optimal Rail Track Allocation

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Overview

Rail Track Auctions

The Optimal Track Allocation Problem (OPTRA)

Mathematical Models

Computational Results

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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

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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

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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.

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Examples

In 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

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The Story of Didius Iulianus(http://www.roman-emperors.org/didjul.htm)

[193 A.D., March 28] When the emperor Pertinax was killed trying to quell a mutiny, no accepted successor was at hand. Pertinax's father-in-law and urban prefect, Flavius Sulpicianus, entered the praetorian camp and tried to get the troops to proclaim him emperor, but he met with little enthusiasm. Other soldiers scoured the city seeking an alternative, but most senators shut themselves in their homes to wait out the crisis. Didius Julianus, however, allowed himself to be taken to the camp, where one of the most notorious events in Roman history was about to take place. Didius Julianus was prevented from entering the camp, but he began to make promises to the soldiers from outside the wall. Soon the scene became that of an auction, with Flavius Sulpicianus and Didius Julianus outbidding each other in the size of their donatives to the troops. The Roman empire was for sale to the highest bidder. When Flavius Sulpicianus reached the figure of 20,000 sesterces per soldier, Didius Julianus upped the bid by a whopping 5,000 sesterces, displaying his outstretched hand to indicate the amount. The empire was sold, Didius Julianus was allowed into the camp and proclaimed emperor.

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Examples

In 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

In 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

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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)

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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

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Track Request Form

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Track Construction

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Rail Track Auctioning

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Rail Track Auction

All bids assigned:END

Bid is assigned

OPTRA findsallocation with

maximum earnings

EVUs decide on bids for bundles of timetabled tracks

BEGINMinimum Bid = Basic Price

Bids are increased by aminimum increment

Bid is not assigned

Bids isunchanged

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Overview

Rail Track Auctions

The Optimal Track Allocation Problem (OPTRA)

Mathematical Models

Computational Results

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Optimal Track Allocation Problem (OPTRA)

Input

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

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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

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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

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Standard Train Types

train type

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

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Infrastructure

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Block Conflicts

conflict

conflict

s

t

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Variable Bids

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

€

Dep. time12:00 12:2012:08

90

80

Traveltime

[min]

€

6040

4 €/min

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Effects

A B C D

time ICE drops out

III.

3 x + 1 x = ???

difficult!

ICE goes

I. variant

ICE slower

II.

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Track Allocation Problem

Route/Track

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Track Allocation Problem

Route/Track

Route Bundle/Bid

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

Conflict

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

Conflict Headway Times

Station Capacities

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

Conflict Headway Times

Station Capacities

This Talk: Only Block Occupancy Conflicts

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

Conflict

Track Allocation (Timetable)

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Track Allocation Problem

Route/Track

Route Bundle/Bid

Scheduling Graph

Conflict

Track Allocation (Timetable)

Optimal Track Allocation Problem (OPTRA)

… …

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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.

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Overview

Rail Track Auctions

The Optimal Track Allocation Problem (OPTRA)

Mathematical Models

Computational Results

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IP Model OPTRA1

Arc-based

Routes: Multiflow

Conflicts: Packing(pairwise)

This talk: Block occupancy conflicts only Variables

Arc occupancy

Constraints

Flow conservation

Arc conflicts (pairwise)

Objective

Maximize proceedings

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Selected LiteratureBrännlund 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

s t dv {0, v (i)}

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IP Model OPTRA1

Arc-based

Routes: Multiflow

Conflicts: Packing(pairwise)

Conflict Graph (Interval Graph)

Cliques

Perfectness

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IP Model OPTRA2

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 (cliques)

Objective

Maximize proceedings

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IP Model OPTRA2

Arc-based

Routes: Multiflow

Conflicts: Packing(Max. Cliques)

Proposition: The LP-relaxation of OPTRA2 can be

solved in polynomial time.

In practice …

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IP Model OPTRA3

Track Occupancy Configurations

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IP Model OPTRA3

Track Occupancy Configurations

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Variables

Path and config usage

Constraints

Config choice

Path-config coupling (capacities)

Objective

Maximize proceedings

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Shadow prices (useful in auction) Arc prices a

Track prices r

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Proposition: PLP(OPTRA1)

PLP(OPTRA2)

= PLP(OPTRA3).

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Proposition:PLP(OPTRA2)= PLP(OPTRA3).

Proposition: Route pricing = acyclic shortest path

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Proposition:PLP(OPTRA2)= PLP(OPTRA3).

Proposition: Route pricing = acyclic shortest path

Proposition: Config pricing = acyclic shortest path

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IP Model OPTRA3

Path-based Routes

Path-based Configs

Proposition:PLP(OPTRA2)= PLP(OPTRA3).

Proposition: Route and config pricing = acyclic shortest path

Proposition: The LP-relaxation of OPTRA3 can be solved in polynomial time.

Column Generation

Begin

OPTRA (IP)

Solve Relaxation

(LP)

Stop?

All fixed?

End

YesNo

Yes

No

AddVariables

ComputePrices

Unfix/FixVariables

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Overview

Rail Track Auctions

The Optimal Track Allocation Problem (OPTRA)

Mathematical Models

Computational Results

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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

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Computational Results

Test Network

Preprocessing

293 Nodes441 Arcs

1486 Nodes1881 Arcs

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Computational Results

Test Network

Preprocessing

293 Nodes441 Arcs

1486 Nodes1881 Arcs

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Computational Results

Test Network

Preprocessing

Degrees of Freedom 324 Trains, Profit 1

Delay in min.

#Var. #Con.#Train

sTime

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

1010164

9315011 311 970.0

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Computational Results

Test Network

Preprocessing

Degrees of Freedom

Timetables

285 Trains

255 Trains

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Computational Results

Test Network

Preprocessing

Degrees of Freedom

Timetables

Harmonization

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Tripling Experiment

Status quo 284 tracks through 6 hours in the Hannover—

Braunschweig—Fulda network, (hypothetical) total income of 28,255 €

Scenario triple requests to 946 bids

(~15 minutes alteration, identical willingness to pay)

variationcpu time (CPLEX)

earnings(% Status Quo)

trains(% Status Quo)

0 mins 6 secs 52.066 (+ 84%) 420 (+ 47%)

1 mins 8 secs 60.612 (+114%) 496 (+ 74%)

4 mins 1 days 67.069 (+137%) 617 (+117%)

5 mins 3+ days 67.975 (+140%) 737 (+159%)

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Outlook

Column Generation

Microsimulation

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

DFG Research Center MATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW)

Ralf Borndörfer

LBWLBW

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