Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute...
-
date post
19-Dec-2015 -
Category
Documents
-
view
216 -
download
2
Transcript of Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute...
![Page 1: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/1.jpg)
http://www.zib.de/[email protected]
DFG Research Center MATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW)
Ralf Borndörfer
LBWLBW
Ralf Borndörferjoint work with
Martin Grötschel Thomas Schlechte
X Encuentro de Matemática, Quito, 25. Juli 2006
Optimal Rail Track Allocation
![Page 2: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/2.jpg)
Ralf Borndörfer
2
LBWLBW
Overview
Rail Track Auctions
The Optimal Track Allocation Problem (OPTRA)
Mathematical Models
Computational Results
![Page 3: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/3.jpg)
Ralf Borndörfer
3
LBWLBW
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
![Page 4: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/4.jpg)
Ralf Borndörfer
4
LBWLBW
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
![Page 5: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/5.jpg)
Ralf Borndörfer
5
LBWLBW
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.
![Page 6: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/6.jpg)
Ralf Borndörfer
6
LBWLBW
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
![Page 7: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/7.jpg)
Ralf Borndörfer
7
LBWLBW
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.
![Page 8: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/8.jpg)
Ralf Borndörfer
8
LBWLBW
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
![Page 9: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/9.jpg)
Ralf Borndörfer
9
LBWLBW
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)
![Page 10: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/10.jpg)
Ralf Borndörfer
10
LBWLBW
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
![Page 11: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/11.jpg)
Ralf Borndörfer
11
LBWLBW
![Page 12: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/12.jpg)
Ralf Borndörfer
12
LBWLBW
Track Request Form
![Page 13: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/13.jpg)
Ralf Borndörfer
13
LBWLBW
Track Construction
![Page 14: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/14.jpg)
Ralf Borndörfer
14
LBWLBW
Rail Track Auctioning
![Page 15: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/15.jpg)
Ralf Borndörfer
15
LBWLBW
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
![Page 16: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/16.jpg)
Ralf Borndörfer
16
LBWLBW
Overview
Rail Track Auctions
The Optimal Track Allocation Problem (OPTRA)
Mathematical Models
Computational Results
![Page 17: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/17.jpg)
Ralf Borndörfer
17
LBWLBW
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
![Page 18: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/18.jpg)
Ralf Borndörfer
18
LBWLBW
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
![Page 19: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/19.jpg)
Ralf Borndörfer
19
LBWLBW
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
![Page 20: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/20.jpg)
Ralf Borndörfer
20
LBWLBW
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
![Page 21: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/21.jpg)
Ralf Borndörfer
21
LBWLBW
Infrastructure
![Page 22: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/22.jpg)
Ralf Borndörfer
22
LBWLBW
Block Conflicts
conflict
conflict
s
t
![Page 23: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/23.jpg)
Ralf Borndörfer
23
LBWLBW
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
![Page 24: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/24.jpg)
Ralf Borndörfer
24
LBWLBW
Effects
A B C D
time ICE drops out
III.
3 x + 1 x = ???
difficult!
ICE goes
I. variant
ICE slower
II.
![Page 25: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/25.jpg)
Ralf Borndörfer
25
LBWLBW
Track Allocation Problem
Route/Track
![Page 26: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/26.jpg)
Ralf Borndörfer
26
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
![Page 27: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/27.jpg)
Ralf Borndörfer
27
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
![Page 28: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/28.jpg)
Ralf Borndörfer
28
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict
![Page 29: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/29.jpg)
Ralf Borndörfer
29
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict Headway Times
Station Capacities
![Page 30: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/30.jpg)
Ralf Borndörfer
30
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict Headway Times
Station Capacities
This Talk: Only Block Occupancy Conflicts
![Page 31: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/31.jpg)
Ralf Borndörfer
31
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict
Track Allocation (Timetable)
![Page 32: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/32.jpg)
Ralf Borndörfer
32
LBWLBW
Track Allocation Problem
Route/Track
Route Bundle/Bid
Scheduling Graph
Conflict
Track Allocation (Timetable)
Optimal Track Allocation Problem (OPTRA)
… …
![Page 33: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/33.jpg)
Ralf Borndörfer
33
LBWLBW
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.
![Page 34: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/34.jpg)
Ralf Borndörfer
34
LBWLBW
Overview
Rail Track Auctions
The Optimal Track Allocation Problem (OPTRA)
Mathematical Models
Computational Results
![Page 35: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/35.jpg)
Ralf Borndörfer
35
LBWLBW
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
![Page 36: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/36.jpg)
Ralf Borndörfer
36
LBWLBW
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)}
![Page 37: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/37.jpg)
Ralf Borndörfer
37
LBWLBW
IP Model OPTRA1
Arc-based
Routes: Multiflow
Conflicts: Packing(pairwise)
Conflict Graph (Interval Graph)
Cliques
Perfectness
![Page 38: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/38.jpg)
Ralf Borndörfer
38
LBWLBW
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
![Page 39: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/39.jpg)
Ralf Borndörfer
39
LBWLBW
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 …
![Page 40: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/40.jpg)
Ralf Borndörfer
40
LBWLBW
IP Model OPTRA3
Track Occupancy Configurations
![Page 41: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/41.jpg)
Ralf Borndörfer
41
LBWLBW
IP Model OPTRA3
Track Occupancy Configurations
![Page 42: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/42.jpg)
Ralf Borndörfer
42
LBWLBW
IP Model OPTRA3
Path-based Routes
Path-based Configs
Variables
Path and config usage
Constraints
Config choice
Path-config coupling (capacities)
Objective
Maximize proceedings
![Page 43: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/43.jpg)
Ralf Borndörfer
43
LBWLBW
IP Model OPTRA3
Path-based Routes
Path-based Configs
Shadow prices (useful in auction) Arc prices a
Track prices r
![Page 44: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/44.jpg)
Ralf Borndörfer
44
LBWLBW
IP Model OPTRA3
Path-based Routes
Path-based Configs
Proposition: PLP(OPTRA1)
PLP(OPTRA2)
= PLP(OPTRA3).
![Page 45: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/45.jpg)
Ralf Borndörfer
45
LBWLBW
IP Model OPTRA3
Path-based Routes
Path-based Configs
Proposition:PLP(OPTRA2)= PLP(OPTRA3).
Proposition: Route pricing = acyclic shortest path
![Page 46: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/46.jpg)
Ralf Borndörfer
46
LBWLBW
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
![Page 47: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/47.jpg)
Ralf Borndörfer
47
LBWLBW
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
![Page 48: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/48.jpg)
Ralf Borndörfer
48
LBWLBW
Overview
Rail Track Auctions
The Optimal Track Allocation Problem (OPTRA)
Mathematical Models
Computational Results
![Page 49: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/49.jpg)
Ralf Borndörfer
49
LBWLBW
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
![Page 50: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/50.jpg)
Ralf Borndörfer
50
LBWLBW
Computational Results
Test Network
Preprocessing
293 Nodes441 Arcs
1486 Nodes1881 Arcs
![Page 51: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/51.jpg)
Ralf Borndörfer
51
LBWLBW
Computational Results
Test Network
Preprocessing
293 Nodes441 Arcs
1486 Nodes1881 Arcs
![Page 52: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/52.jpg)
Ralf Borndörfer
52
LBWLBW
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
![Page 53: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/53.jpg)
Ralf Borndörfer
53
LBWLBW
Computational Results
Test Network
Preprocessing
Degrees of Freedom
Timetables
285 Trains
255 Trains
![Page 54: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/54.jpg)
Ralf Borndörfer
54
LBWLBW
Computational Results
Test Network
Preprocessing
Degrees of Freedom
Timetables
Harmonization
![Page 55: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/55.jpg)
Ralf Borndörfer
55
LBWLBW
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%)
![Page 56: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/56.jpg)
Ralf Borndörfer
56
LBWLBW
Outlook
Column Generation
Microsimulation
![Page 57: Ralf Borndörfer DFG Research Center M ATHEON Mathematics for Key Technologies Zuse-Institute Berlin (ZIB) Löbel, Borndörfer & Weider GbR (LBW) borndoerfer@zib.de.](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d3f5503460f94a18f95/html5/thumbnails/57.jpg)
http://www.zib.de/[email protected]
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 !