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Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
Making Good Use of Railroad Tracks
Martin Grötscheljoint work with Ralf Borndörfer and Thomas
Schlechte
IP@COREMay 27-29, 2009
Yesterday Got up at 4:50 am
Left home at 5:40 am
Arrived at Brussels airport at 7:50 am
Took the train
And arrived at 10:45 am here.
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The invitationDear Martin,
We aim to get "distinguished" speakers that give a 50-minute lecture on their current research (I am sure you have a nice IP application area
that you can survey...).
So it will be a celebration of Laurence in disguise. There will be a dinner and it will happen there....
Best,
Michele
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Book Presentation onNovember 11, 2008Year of Mathematics
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Railway tracks are a valuable and costly infrastructure - not to be left empty!
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Where do I come from? Technische Universität Berlin
Konrad-Zuse-Zentrum für Informationstechnik
DFG Research Center MATHEON
Mathematics for key technologies
What type of problems are we aiming at?
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ZIB
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MATHEON Application Area BLogistics, traffic, and telecommunication networks
Scientists in charge: Martin Grötschel, Rolf Möhring, Martin Skutella
Networks, such as telephone networks, the internet, airline, railway, and bus networks are omnipresent and play a fundamental role for communication and mobility in our society. We almost take their permanent availability, reliability, and quality at low cost for granted. However, traffic jams, ill-designed train schedules, canceled flights, break-downs of telephone and computing networks, and slow internet access are reminders that networks are not automatically good networks.
In fact, designing and operating communication and traffic networks are extremely complex tasks …
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The projectTrassenbörse: Railway Slot Auctioning The project aims at developing new ideas to make
better (or even best) use of railway tracks.
A basic assumption, always favoured by economists, is that "markets" lead to an optimal allocation of goods.
But what are the goods to be allocated in the "railway market"?
And if we can define such goods precisely, how can one introduce trade mechanisms that lead to fair competition?
In other words, is there a way to (de-)regulate the current railway system that results in a “better utilization” of the railway infrastructure?
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The projectTrassenbörse: Railway Slot Auctioning The collection of question raised calls for a
multidisciplinary approach.
The project is carried out by a group of economists, mathematicians, and railway engineers from Berlin and Hannover, each group bringing in its particular expertise.
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Project members Economics:
WIP / TU Berlin: Kay Mitusch, Andreas Brenck,
Andreas Tanner, Benedikt Peter
Business consulting Gottfried Ilgmann, Klemens
Polatschek Mathematical optimization:
ZIB Ralf Borndörfer, Martin Grötschel,
Thomas Schlechte
Railway engineering and timetabling:SFWGG / TU Berlin
Jürgen Siegmann, Martin Balser, Elmar Swarat
IVE / Univ. Hannover, RMCon Thomas Siefer, Andreas Henkel,
Marc Klemenz
Many
Bid
s
curr
ent
w
inner
Track allocation,Optimization
Routerequests,Auctiondesgin
Infrastructure, Drivingdynamics
Multiple EVUs
InfraGen
TS-Opt
Auktio
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The projectTrassenbörse: Railway Slot Auctioning Project funding: Bundesministerium für Bildung und
Forschung, Förderungskennziffer 19M2019
Duration in three phases: 12/2002 - 4/2010
(with some interrupts, however)
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Railway network as a market place
The railway network manager is obliged by EU and German law to offer
as much network capacity as possible to all train operation companies (TOCs) in a non-discriminating way.
→ The network is a market place,
but, due to the many technical and administrative constraints, not a simple one.
Our goal: We want to help impove the market design!
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A market must have goods What are the goods of the railway network market?
The answer is clear: slots
But what is a slot precisely?
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Capacity allocation today A slot = right to run a train with a specified schedule
on the network infrastructure Example: Berlin Hbf dep 10:51, Berlin-Spandau arr 11:03, dep 11:05, Hannover Hbf arr 12:28
TOCs order specified slots.
Slot prices are fixed and regulated.
Rules to resolve conflicts:
1. Cooperatively: “Negotiations”, construction of slot alternatives
2. Non-cooperatively: Priorities, sum of regular slot prices, bidding
Resulting network timetable is “manually optimized”
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Capacity allocation today
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Capacity allocation tomorrow: our vision TOCs submit bids for specified slots.
“Base price” is the fixed and regulated price(necessary to maintain the network infrastructure).
Bids may already include some flexibility w.r.t. time, stops and route; also with discounts.
Conflict resolution:1. Cooperatively: Mathematical simultaneous optimization,
taking advantage of flexibility of bids
2. Non-cooperatively: An auction process (rounds of auctions)
Need to develop optimization tools and auction design
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Difficulties to be considered What is a slot precisely?
How many details can/should be taken into account? What about track profiles?
What about engine characteristics?
Routing through stations?
Track scheduling exact with respect to switches?
Signals?
Buffer times and various slacks (path allowances)?
…
Auctioning process Details will be explained later
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Difficulties to be considered If we have to take all possible technical and
administrative details in the general planning model into account, we can immediately give up!
Sensible complexity reduction is necessary.
Hierarchical planning is the appropriate goal.
Coarse plans first, then details to be specified,iteration of the steps, if necessary.
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Slot request today
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Reduction of network complexity
Train stations become simple nodes (with capacity data)
Tracks between stations become simple directed lines (no signals, no particular switches)
One has to verify that these simplifications are acceptable in practice.
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Standardized Train Types and Standardized Train Dynamics
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
velocity
Just like entry „Zugcharakteristik“in today‘s „Trassenanmeldung“.
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Discretization of time, running and waiting times of trains
Minimum time unit (interval): 1 minute (but more detail sometimes necessary)
Matrix of train types‘ running (and required waiting) times in the network:
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Further simplifications Wherever and whenever railway engineers have no
objections
Data driven model precision: do not model things precisely for which data are not available.
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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Our sample network (right hand)
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Time-value specifications
Bid flexibility modelled by time-valued specifications
Examples:€
Departure timet_opt
€
Departure timet_min t_max
€
Departure timet_optt_min t_max
time-dependentpiecewise linearprice function ona time interval
base price
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Example for a slot bid
Berlin Frankfurt Hbf StuttgartOstbahnhof central Spandau (optional)
depart 9.00 arrive 14:30 core travel time 3:30
Discounts for Departure at Ostbahnhof before 9:00 Arrival at Stuttgart after 14:30
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Implicit XOR-bids: Choice of path by optimization procedure
There are many different ways to get from Hannover to Fulda
If all of them are feasible for the requested train (i.e., if the TOC does not care where exactly the train will run between Hannover and Fulda), our optimization procedure will pick one that is optimal from the network perspective.
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Tour bids: Special support for branching and merging of trains
A tour is a set of slots that are connected by a successor relation →
s1→s2 means that s2 can use rolling stock from s1
s1
s2
s3
s5
s6
s4
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Bids We have developed a collection of possible bids that a
TOC can submit (more than I can describe here).
Suppose the TOCs have submitted their bids.
What does the network operator do?
Actually, what is the network operator supposed to do?
The network operator has to apply the „Eisenbahninfrastruktur-Benutzungsverordnung - Verordnung über den diskriminierungsfreien Zugang zur Eisenbahninfrastruktur und über die Grundsätze zur Erhebung von Entgelt für die Benutzung der Eisenbahninfrastruktur - EIBV“ vom 3. Juni 2005 (BGBl. I S. 1566), die am 1. August 2005 in Kraft getreten ist.
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EIBV and conflict resolution§9 Absatz 5 EIBV, „Höchste Summe der Regelentgelte“:
„(5) Bei der Entscheidung zwischen gleichrangigen Verkehren nach Absatz 4 hat der Betreiber der Schienenwege die Entgelte für die streitigen Zugtrassen gegenüberzustellen und
1. bei einem Konflikt zwischen zwei Zugtrassen der Zugtrasse den Vorrang einzuräumen, bei der das höchste Regelentgelt zu erzielen ist,
2. bei einem Konflikt zwischen mehr als zwei Zugtrassen den Zugtrassen den Vorrang einzuräumen, bei denen in der Summe das höchste Regelentgelt zu erzielen ist.
…“, see http://bundesrecht.juris.de/eibv_2005/__9.htmlOptimization required by law!
This seems to have been ignored by everyone involved!
Let us consider an Let us consider an exampleexample
Vorrang einzuräumen, bei denen in der Summe das höchste Regelentgelt zu erzielen ist. (Note: this is a formal definition of fair access!)
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800
900
700
100
500
155
154
500
150
650
Example: Bids displayed in a Time-Way-Diagram
way
timeRegelentgeltbase price
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800
900
700
100
500
155
154
500
150
650
way
Zeit
applying the EIBV rules:slots without any conflicts
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800
900
700
100
500
155
154
500
150
650
Way
Zeit
applying the EIBV rules:two slots in conflict
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800
900
700
100
500
155
154
500
133
657
way
time
applying the EIBV rules:lots of conflicts, what now?
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800
900
700
100
500
500
way
time
Greedy-Sum of base prices : 1000
Lots of conflicts, what now?„Bilateral conflict resolution“
in mathematical terms: greedy heuristic
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Lots of conflicts, what now?Smart planner
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800
900
700
100
500
500
way
time
More traffic, higher network revenue
smart planner solutionGreedy-Sum of base prices : 1000
Smart-Sum of base prices : 1400
Is that optimal, i.e.,does the planner
satisfy the law?
Lots of conflicts, what now?Smart planner
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Lots of conflicts, what now?mathematical optimization
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800
900
700
100
500
500
way
time
Lots of conflicts, what now?mathematical optimization
Greedy-Sum of base prices : 1000Smart-Sum of base prices : 1400
the provable optimum: 1700
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800
900
700
100
500
500
way
time
Lots of conflicts, what now?mathematical optimization
the provable total optimum: 2655
155
154
150
650
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800
900
700
100
500
155
154
500
150
650
Example: track bids with flexibilities
way
time
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800
900
700
100
500
500
Looking at the major conflicts: Optimumwith flexibilities
way
time
sum of base prices: 2200 > 1700even more traffic, more network revenue
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800
900
700
100
500
500
Looking at the major conflicts: Optimum with flexibilities
way
time
sum of base prices: 2200 > 1700even more traffic, more network revenue
155
154
obvious casefor further bidding
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Track Allocation Problem • Route/Track
Route/Track
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Route/Track
Route Bundle/Bid
Track Allocation Problem
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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
Track Allocation (Timetable)
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Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict
Track Allocation (Timetable)
Track Allocation Problem (OPTRA)
… …
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…
Track allocation problem
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Solution approach:What methods?
The current standard is the use of heuristics.
This is infeasible in our situation!Namely, suppose the system finds a “good” solution that rules out one bid that some TOC eagerly wants to run.And now the TOC finds a solution, including its special bid, that is overall better than the “good” solution.The TOC would declare the work of the network operator cheating.
A proof of optimality is required!
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Mathematical solution approach:Integer Programming Models
APP
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 (maximal cliques)
Objective
Maximize proceedings
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Variables
Path und config usage
Constraints Path and config choice Path-config-coupling (track capacity)
Objective Function
Maximize proceedings
IP Models
PCP
Path-based routes
Path-based configs
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two IP Models solving the track allocation problem – in priniple
PCP
Path-based
Proposition: v (PLP(APP))
= v (PLP(PCP))
Thomas Schlechte
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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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Results
Szenario
• 324 trains
• max. # trains
flex/min. #var. #constr.#train
stime/sec.
5 29.112 34.330 164 4,5
6 39.641 54.978 200 26,3
7 52.334 86.238 251 45,7
8 67.000 133.689 278 613,1
9 83.227 206.432 279 779,1
10 101.649 315.011 311 970,0
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Model Comparison
Scenario:Status Quo Schedule 285 Trains
Flex. LP1 IP1* LP3 IP3
*
0 2351692 2080255 2234211 2125213
2 2453476 2092045 2351977 2173288
4 2453476 2092045 2426999 2234398
6 2453476 2174897 2453476 2304735
8 2453476 2282305 2453476 2304735
10 2453476 2390921 2453476 2339652
PCPAPP
*- Runtime maximal 1h
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Line Plan Problem „China20“
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Example „China20“
Which track upgrading project
is more important ?
Which track upgrading project
is more important ?
upgrading tracks
fixed tracks
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Origin Destination Matrix
Estimated Passenger Demand
for all pairs
Estimated Passenger Demand
for all pairs
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Optimize Cost, case (A)
Cost function:
1.000.000 € per line, 100,- € per km
Cost function:
1.000.000 € per line, 100,- € per km
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Optimize Traveltime, case (B)
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Line Plan Decision ?
(A) (B)
number of lines 9 18
cost in Mio. € 238 264
traveltime in Mio. min.
383 349
(A cost)
(B time)
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Timetabling
Stations
Tracks
Train Request
s
TS-OPT
Timetable
Optimization Model
maximize
track utilization
timetable attractiveness
subject to
safety requirements
time windows
periodic Passenger versus
individual Cargo
periodic Passenger versus
individual Cargo
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Timetable for Lineplan (A)
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Timetable for Lineplan (B)
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Saturation with Cargo Trains/Slots
add cargo trains
Beijing/Shanghai
add cargo trains
Beijing/Shanghai
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Timetable Decision ?
(A) (B)
number of train slots 452 462
passenger ICE‘s 36 18
cargo trains 426 444
(A)
(B)
Switzerland A real case with real data.
Micro » Macro » Micro test
Data problems, problems with the definitions, inconsistencies of simulation software systems, etc.
But we have very interesting results.
Unfortunately,…
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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GoalDevelopment of an auction mechanism for track usage
(slots):
economic and technical analysis of the various track allocation rules
development of a mathematical program for optimal time tabeling
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Basic idea of a slot auction train operation companies (TOCs) deliver bids for slots
(possibly including various degrees of freedom concerning willingness to pay, timing, stops, train routes)
minimum bid = base price Auctioneer computes conflict free slot assignment
(combination of bids) that maximizes the network revenue and temporarily allocates them to the bidders.
Iteration (rounds of the auction): Bids that have not won can be repeated or modified and resubmitted.
Criterion for termination of auction (# of rounds, # of changed bids,..)
The result of the process is a timetable (possibly combining slots allocated to various bidders) which then has to be refined for use in practice.
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Goal of the slot allocation auction:practical rules for an auction mechanism
Components:
„from coarse to fine“: …
Exact mathematical optimization: …
Consideration of alternatives: …
Economic and technical analysis: …
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Remarks on the current EIBV All relevant rules can be implemented, e.g.:
Priorities
„maximale Summe der Regelentgelte“
Höchstpreisverfahren
Rechte aus Rahmenverträgen
The sog. „Koordinierungsprozess“ in EIBV, i.e., the bilateral negotiation (considering also alternative options) is automatically included in the approach: no discrimination, optimality,…
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Auction design Iterative, combinatorial auction similar to Parkes’
ibundle auction
Next slide shows procedure
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Rail Track Auction
END
OPTRA model is solved withmaximum earnings
TOCs decide on bids for slots
BEGIN
Bid is increased by aminimum increment
Bid assigned?
Bid isunchanged
All bidsUnchanged?
yes
no
Wish to increase bid?
yes no
yes
no
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There are still lots of economic issues Auction rounds
Sequences of auctions
Informal coordination between TOCs
Use-it-or-lose-it rules
Network proceeds is operational goal
The „density“ of potential goods
Bidding strategies
How to analyse auction design?
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Contents1. Introduction and project outline
2. What is the goal?
3. What are the problems?
4. The model: networks, tracks, trains, time, slots,…
5. Bids
6. The auction process
7. Summary
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91 slot allocation problem: other literatureCharnes Miller (1956), Szpigel (1973), Jovanovic and Harker (1991),
Cai and Goh (1994), Schrijver and Steenbeck (1994), Carey and Lockwood (1995)
Nachtigall and Voget (1996), Odijk (1996) Higgings, Kozan and Ferreira (1997)
Brannlund, Lindberg, Nou, Nilsson (1998) Lindner (2000), Oliveira & Smith (2000)
Caprara, Fischetti and T. (2002), Peeters (2003)
Kroon and Peeters (2003), Mistry and Kwan (2004)
Barber, Salido, Ingolotti, Abril, Lova, Tormas (2004)
Semet and Schoenauer (2005),
Caprara, Monaci, T. and Guida (2005)
Kroon, Dekker and Vromans (2005),
Vansteenwegen and Van Oudheusden (2006),
Cacchiani, Caprara, T. (2006)
Caprara, Kroon, Monaci, Peeters, T. (2006)
Failed railroad infrastructure planning
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Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
Making Good Use of Railroad Tracks
Martin Grötscheljoint work with Ralf Borndörfer and Thomas
Schlechte
IP@COREMay 27-29, 2009
Thank you for Thank you for your attention your attention
