routing cars in rail freight service€¦ · in rail freight service armin fügenschuh - zuse...
TRANSCRIPT
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Routing Cars in Rail Freight Service
Armin Fügenschuh - Zuse Institute Berlin, Germany
Henning Homfeld - Technische Universität Darmstadt, GermanyAlexander Martin - Technische Universität Darmstadt, GermanyHanno Schülldorf - Technische Universität Darmstadt, Germany & Deutsche Bahn AG, Frankfurt am Main, Germany
January 7, 2010
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Deutsche Bahn (G. Pfau, J. Wolfner)• Department for Company Development (GSU1)• Interested in long-term traffic & demand forecast
& simulaton• Code testing & result benchmarking
• Schenker (A. Below, C. Liebchen, T. Klingler)• DB subsidary for freight service• Solving the operational routing problem manually• Providing real-life data
• Federal Ministry for Science and Education (BMBF)• Funding applied math projects (period 2007-2010)• OVERSYS:
• U. Zimmermann, R. Hansmann (TU Braunschweig)• U. Clausen, A. Chmielewski, J. Baudach (Fraunhofer IML)• C. Helmberg, F. Fischer (TU Chemnitz)• R. Schultz (U Duisburg), G. Reinelt (U Heidelberg)
Project Partners
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Introduction
• Railway freight transport has a market share of 20%.• 100,000 Mil. ton km, of which:
• 45% inland traffic,• 45% cross-border traffic,• 10% transit traffic.
• Deutsche Bahn offers whole trains (~30 cars) and individual cars.
• Several individual cars with different destinationsare grouped to trains at classification yards.
• At the next classification yard, the cars are re-grouped, until they reached their destinations.
• Main question: what is the „best“ path for each car?
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Facts and Figures
• Railway network length: 38,200 km• 5000 trains per day, 150,000 cars• Terminal stations: 2,200• Classification yards:
• Large („Rangierbahnhöfe“): 11• Medium („Knotenbahnhöfe“): 30• Small („Satellitenbahnhöfe“): 200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Classification Yards
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Disintegration of trains• Sorting the cars (with the help of gravity)• Assembling of new trains
Classification Yards
entry tracks hump sorting tracks exit tracks
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Survey of the Literature
• First models for the blocking problem emerged in the 1960s. Since then...
• Bodin, Schuster & Golden (1980)
• Assad (1983)
• Crainic, Ferland & Rousseau (1984)Crainic & Rousseau (1986)
• Keaton (1989, 1992)
• Newton (1997)Newton, Barnhart & Vance (1998)Barnhart, Hong & Vance (2000)
• Ahuja (2007)
• Main differences to our problem:
• Special constraints due to DB operational rules („Leitwege“).
• Costs are induced by trains (and not so much by cars).
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by cars
100
100
60
10
30
30
40
0
0
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by cars
100
100
60
10
30
30
40
0
0
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by cars
100
100
60
10
30
30
40
200
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by cars
100
100
60
10
30
30
40
200
200
![Page 12: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/12.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
0
0
![Page 13: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/13.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
0
0
![Page 14: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/14.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
70
70
![Page 15: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/15.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
70
70
![Page 16: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/16.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
130
190
![Page 17: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/17.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
130
190
![Page 18: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/18.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
170
230
![Page 19: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/19.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Bundling Effect
• Cost induced by trains
100
100
60
10
30
30
40
170
230
![Page 20: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/20.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Modes of Operation
• Three ways of sending cars from origin to destination:• Individual car routing
• Assign a sequence of yards to each car
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
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200200
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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200 200
200200
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Example: Individual Car Routing
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
500
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
500
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
500
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
700
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
700
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
700
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
900
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
900
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
900
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
1100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
1100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
1100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
1300
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Individual Car Routing
1300
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Modes of Operation
• Three ways of sending cars from origin to destination:• Individual car routing
• Assign a sequence of yards to each car• Blocking of cars
• Assign a sequence of yards to each order• All cars in an order follow the same path
through the network
![Page 46: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/46.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
0
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
0
![Page 48: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/48.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Blocking of Cars
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
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200200
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Example: Blocking of Cars
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
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Example: Blocking of Cars
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
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Example: Blocking of Cars
400
![Page 56: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/56.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
600
![Page 57: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/57.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
600
![Page 58: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/58.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
600
![Page 59: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/59.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
800
![Page 60: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/60.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
800
![Page 61: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/61.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
800
![Page 62: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/62.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
1000
![Page 63: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/63.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1000
![Page 64: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/64.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: Blocking of Cars
1000
![Page 65: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/65.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1200
![Page 66: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/66.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1200
![Page 67: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/67.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1200
![Page 68: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/68.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1400
![Page 69: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/69.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: Blocking of Cars
1400
![Page 70: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/70.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Modes of Operation
• Three ways of sending cars from origin to destination:• Individual car routing
• Assign a sequence of yards to each car• Blocking of cars
• Assign a sequence of yards to each order• All cars in an order follow the same path
through the network• „Leitwege“ - DB car routing
• For all orders with the same destination and each yard assign a successor
![Page 71: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/71.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
0
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
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Example: DB „Leitwege“ System
0
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
100
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
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Example: DB „Leitwege“ System
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
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Example: DB „Leitwege“ System
200
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
500
![Page 81: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/81.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
500
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
500
![Page 83: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/83.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
800
![Page 84: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/84.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
800
![Page 85: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/85.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
800
![Page 86: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/86.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1000
![Page 87: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/87.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1000
![Page 88: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/88.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1000
![Page 89: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/89.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1200
![Page 90: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/90.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1200
![Page 91: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/91.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1200
![Page 92: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/92.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1400
![Page 93: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/93.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1400
![Page 94: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/94.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1400
![Page 95: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/95.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1600
![Page 96: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/96.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
300
200 200
200200
100
Example: DB „Leitwege“ System
1600
![Page 97: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/97.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Modes of Operation
• Three ways of sending cars from origin to destination:• Individual car routing
• Assign a sequence of yards to each car• Blocking of cars
• Assign a sequence of yards to each order• All cars in an order follow the same path
through the network• „Leitwege“ - DB car routing
• For all orders with the same destination and each yard assign a successor
• If „Leitwege“ are so expensive, then why do they do it?• Historical reasons• More robust / errors can easily be corrected
![Page 98: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/98.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
![Page 99: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/99.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .V A := V × V K
![Page 100: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/100.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 101: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/101.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 102: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/102.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 103: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/103.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
![Page 104: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/104.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
• Time limit:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 105: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/105.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
• Time limit:
• Yards:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 106: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/106.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
• Time limit:
• Yards:• Number of sorting tracks:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V :�
j:(i,j)∈A
yi,j ≤ Yi
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 107: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/107.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
• Time limit:
• Yards:• Number of sorting tracks:
• Number of trains per track:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V :�
j:(i,j)∈A
yi,j ≤ Yi
∀(i, j) ∈ A : ni,j ≤ 6yi,j
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 108: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/108.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
• Orders:• Delivery:
• Time limit:
• Yards:• Number of sorting tracks:
• Number of trains per track:
• Hump capacity:
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀i ∈ V :�
j:(i,j)∈A
yi,j ≤ Yi
∀(i, j) ∈ A : ni,j ≤ 6yi,j
∀i ∈ V :�
k∈K,j:(i,j)∈A
vk · xki,j ≤ Hi
∀i ∈ V, k ∈ K :�
j:(i,j)∈A
xki,j −
�
j:(j,i)∈A
xkj,i =
1, i = o(k),−1, i = d(k),
0, else.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 109: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/109.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 110: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/110.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 111: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/111.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
![Page 112: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/112.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
• Max. weight:
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
∀(i, j) ∈ A :�
k∈K
wk · xki,j ≤ 1600 · ni,j
![Page 113: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/113.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
• Max. weight:
• „Leitweg“ (unique successor) constraint („plain“):
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
∀(i, j) ∈ A :�
k∈K
wk · xki,j ≤ 1600 · ni,j
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
![Page 114: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/114.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
• Max. weight:
• „Leitweg“ (unique successor) constraint („plain“):
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
∀(i, j) ∈ A :�
k∈K
wk · xki,j ≤ 1600 · ni,j
i
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
![Page 115: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/115.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
• Max. weight:
• „Leitweg“ (unique successor) constraint („plain“):
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
∀(i, j) ∈ A :�
k∈K
wk · xki,j ≤ 1600 · ni,j
i
k
l
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
![Page 116: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/116.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Trains:• Max. length:
• Max. weight:
• „Leitweg“ (unique successor) constraint („plain“):
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .• Constraints
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
∀(i, j) ∈ A :�
k∈K
lk · xki,j ≤ 700 · ni,j
∀(i, j) ∈ A :�
k∈K
wk · xki,j ≤ 1600 · ni,j
i
k
l
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
k
l
j1
j2
![Page 117: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/117.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
![Page 118: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/118.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
• Objective:
The Arc-Flow Model
• Sets: stations , precedence relations , orders .• Variables: car routes , sorting tracks , trains .
V A := V × V K
xki,j ∈ B yi,j ∈ N ni,j ∈ N
C1 ·�
(i,j)∈A
ni,j + C2 ·�
(i,j)∈A
yi,j + C3 ·�
k∈K,(i,j)∈A
xki,j → min
C1 � C2 � C3
![Page 119: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/119.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
![Page 120: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/120.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints
![Page 121: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/121.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
![Page 122: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/122.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
![Page 123: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/123.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
j1
j2
ik
l
![Page 124: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/124.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
• Improved formulation („lifted“):
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
j1
j2
ik
l
![Page 125: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/125.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
• Improved formulation („lifted“):
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
j1
j2
ik
lj1
j2
j2
![Page 126: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/126.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (1)
• Strengthening of „Leitweg“-constraints• Model formulation:
• Improved formulation („lifted“):
∀k, l ∈ K, d(k) = d(l), i ∈ V, (i, j1) �= (i, j2) ∈ A : xki,j1
+ xli,j2
≤ 1
∀k, l ∈ K, d(k) = d(l), i ∈ V, V1∪̇V2 = V :�
j1∈V1
xki,j1
+�
j2∈V2
xli,j2
≤ 1
j1
j2
ik
lj1
j2
j2
![Page 127: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/127.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
![Page 128: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/128.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
![Page 129: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/129.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
![Page 130: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/130.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
![Page 131: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/131.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
![Page 132: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/132.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.
![Page 133: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/133.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
![Page 134: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/134.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
![Page 135: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/135.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
![Page 136: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/136.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
• Cars follow trees:
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
∀(i, j) ∈ A, k ∈ K : xki,j ≤ zd(k)
i,j
![Page 137: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/137.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
• Cars follow trees:• Cons: more variables
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
∀(i, j) ∈ A, k ∈ K : xki,j ≤ zd(k)
i,j
![Page 138: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/138.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
• Cars follow trees:• Cons: more variables• Pros:
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
∀(i, j) ∈ A, k ∈ K : xki,j ≤ zd(k)
i,j
![Page 139: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/139.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
• Cars follow trees:• Cons: more variables• Pros:
• Do not need „Leitweg“-constraint in -variables.
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
∀(i, j) ∈ A, k ∈ K : xki,j ≤ zd(k)
i,j
x
![Page 140: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/140.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Improving the Arc-Flow Model (2)(communicated by A. Bley, 2008)
• Consider the structure of a feasible solution:
• The „Leitweg“-constraint (uniqueness of successor) leads to trees.• Formulate trees directly and merge them with the model:
• Variable for all and .• Unique successor:
• Cars follow trees:• Cons: more variables• Pros:
• Do not need „Leitweg“-constraint in -variables.• Better LP-relaxation.
(i, j) ∈ Azκi,j ∈ B κ ∈ K := {d(k) : k ∈ K}
∀i ∈ V, κ ∈ K :�
j:(i,j)∈A
zκi,j ≤ 1
∀(i, j) ∈ A, k ∈ K : xki,j ≤ zd(k)
i,j
x
![Page 141: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/141.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
![Page 142: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/142.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
![Page 143: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/143.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
![Page 144: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/144.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
4 4
3
2
1 1 1
2
3
![Page 145: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/145.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
![Page 146: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/146.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
4 4
2
1
3
![Page 147: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/147.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
![Page 148: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/148.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Hence a „typical“ path has the following structure:
4 4
3
2
3
![Page 149: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/149.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• The following path is considered as „untypical“:
![Page 150: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/150.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• The following path is considered as „untypical“:
4 4
3
2
1 1
2
33
![Page 151: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/151.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
![Page 152: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/152.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
![Page 153: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/153.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Analysis of historical data shows: 98% of all car paths are „monotone“.
![Page 154: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/154.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Heuristic Cuts: Hierarchy Constraints
• Observation: Large yards „attract“ cars, because• They are at central locations in the network,• They can handle more cars (capacity & speed),• They can be connected to more yards.
• Analysis of historical data shows: 98% of all car paths are „monotone“.• Separate „monotone“ paths from „zig-zag“ paths by new constraints:
∀j ∈ V, l ∈ K :�
i:(i,j)∈A,hi≤hj
xli,j +
�
k:(j,k)∈A,hj≥hk
xlj,k ≤ 1
![Page 155: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/155.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
![Page 156: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/156.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:∀k ∈ K :
�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
![Page 157: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/157.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:
• Note: The turnover time is a constant here.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
ui
![Page 158: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/158.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:
• Note: The turnover time is a constant here.• In reality, the turnover time depends on the number of trains that are
assembled on the resp. sorting track:
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
ui
![Page 159: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/159.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:
• Note: The turnover time is a constant here.• In reality, the turnover time depends on the number of trains that are
assembled on the resp. sorting track:
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
0
6
12
18
24
30
1 2 3 4 5 6
ui
![Page 160: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/160.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:
• Note: The turnover time is a constant here.• In reality, the turnover time depends on the number of trains that are
assembled on the resp. sorting track:
• In a more realistic model, is a variable, and .
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
0
6
12
18
24
30
1 2 3 4 5 6
ui
uki,j ∈ R+ uk
i,j =24ni,j
· xki,j
![Page 161: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/161.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Refining the Model: Turnover Times
• Consider again the time-limit constraint:
• Note: The turnover time is a constant here.• In reality, the turnover time depends on the number of trains that are
assembled on the resp. sorting track:
• In a more realistic model, is a variable, and .
• Problem: We now have an MINLP.
∀k ∈ K :�
(i,j)∈A
(ui + ti,j) · xki,j ≤ Tk
0
6
12
18
24
30
1 2 3 4 5 6
ui
uki,j ∈ R+ uk
i,j =24ni,j
· xki,j
![Page 162: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/162.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
![Page 163: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/163.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:u · n = 24 · x
(u, n, x)
![Page 164: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/164.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:u · n = 24 · x
(u, n, x)
![Page 165: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/165.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .u · n = 24 · x
(u, n, x)
x ∈ {0, 1} n ∈ Z+
![Page 166: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/166.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
![Page 167: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/167.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 168: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/168.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 169: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/169.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 170: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/170.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 171: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/171.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 172: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/172.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
Nu ≤ 24N + 24x − 24n
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 173: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/173.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
x ≤ 1Nu ≤ 24N + 24x − 24n
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 174: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/174.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
x ≤ 1Nu ≤ 24N + 24x − 24n
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 175: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/175.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 1
• The set of feasible points with is:
• Remeber that and .• Thus the convex hull is
• Note that all lower-bound-cuts contain the origin, like „projective cuts“, c.f.• Frangioni, Gentile (2006, 2009)• Frangioni, Gentile, Grande, Pacifici (2009)• Günlük, Linderoth (2009)
u · n = 24 · x(u, n, x)
x ∈ {0, 1} n ∈ Z+
w ≤ 24x
x ≤ 1Nu ≤ 24N + 24x − 24n
24x ≤ Nu
24(2i + 1)x − 24n ≤ i(i + 1)u, i = 1, . . . , N
![Page 176: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/176.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
January 7, 2010
![Page 177: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/177.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,
January 7, 2010
u · n = 24 · x
![Page 178: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/178.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
![Page 179: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/179.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
u · n = 24 · x
![Page 180: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/180.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
u · n = 24 · x u · n = 24 · x1
![Page 181: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/181.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
u · n = 24 · x u · n = 24 · x2
![Page 182: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/182.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
u · n = 24 · x u · n = 24 · x2
![Page 183: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/183.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2
• Consider the nonlinear relation ,and make it even „more“ nonlinear: .In the sequel we are interested in a lower bound on the waiting time:
January 7, 2010
u · n = 24 · x
u · n = 24 · x2
u · n = 24 · x u · n = 24 · x2
u · n ≥ 24 · x2.
![Page 184: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/184.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitutionu = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
![Page 186: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/186.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitutionu = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
u · n ≥ 24 · x2
![Page 187: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/187.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitutionu = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
u · n ≥ 24 · x2
![Page 188: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/188.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitutionu = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
u · n ≥ 24 · x2
![Page 189: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/189.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitutionu = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
u · n ≥ 24 · x2
![Page 190: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/190.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linearization 2 (cont.)
• On we apply the following variable substitution
• We obtain the Lorentz (second order) cone
u = τ + α,
n = τ − α,
24 · x2 = β2.
u · n ≥ 24 · x2
u · n ≥ 24 · x2
�α2 + β2 ≤ τ.
�α2 + β2 ≤ τ.
![Page 191: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/191.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linear Approximation of the SOC
![Page 192: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/192.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linear Approximation of the SOC
• The SOC can be approximated by linear inequalities (Ben-Tal, Nemirovski 1998, Glineur 2000):
![Page 193: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/193.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linear Approximation of the SOC
• The SOC can be approximated by linear inequalities (Ben-Tal, Nemirovski 1998, Glineur 2000):
ξ0 ≥ |α|η0 ≥ |β|
ξi = cos(π
2i+1)ξi−1 + sin(
π
2i+1)ηi−1
ηi ≥ | − sin(π
2i+1)ξi−1 + cos(
π
2i+1)ηi−1|
ξn ≤ τ
ηn ≤ tan(π
2n+1)ξn
i = 2, 3, . . . , M
![Page 194: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/194.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linear Approximation of the SOC
• The SOC can be approximated by linear inequalities (Ben-Tal, Nemirovski 1998, Glineur 2000):
with accuracy
ξ0 ≥ |α|η0 ≥ |β|
ξi = cos(π
2i+1)ξi−1 + sin(
π
2i+1)ηi−1
ηi ≥ | − sin(π
2i+1)ξi−1 + cos(
π
2i+1)ηi−1|
ξn ≤ τ
ηn ≤ tan(π
2n+1)ξn
i = 2, 3, . . . , M
ε(M) =1
cos( π2M+1 )
− 1 = O(1
4M)
![Page 195: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/195.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Linear Approximation of the SOC
• The SOC can be approximated by linear inequalities (Ben-Tal, Nemirovski 1998, Glineur 2000):
with accuracy
• Good news: We are back in the MILP world!
ξ0 ≥ |α|η0 ≥ |β|
ξi = cos(π
2i+1)ξi−1 + sin(
π
2i+1)ηi−1
ηi ≥ | − sin(π
2i+1)ξi−1 + cos(
π
2i+1)ηi−1|
ξn ≤ τ
ηn ≤ tan(π
2n+1)ξn
i = 2, 3, . . . , M
ε(M) =1
cos( π2M+1 )
− 1 = O(1
4M)
![Page 196: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/196.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Computational Results (1)
• Comparison of formulations on random instances • CPlex 11.2, default settings
• CPU times:
• B&B nodes:
instance plain lifted tree
1 338 47 31
2 956 136 80
3 7208 3019 6095
4 3422 796 190
5 83 43 34
instance plain lifted tree
1 557 572 628
2 1347 1811 1428
3 63893 37806 52388
4 19555 21740 5454
5 8376 3832 1884
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Computational Results (2)
• National traffic in Germany• 43 yards (medium and large)• 1,600 orders• CPlex10• 6 hours CPU time• 8% gap• Visible bundling effects -
most traffic on main axes
![Page 198: Routing Cars in Rail Freight Service€¦ · in Rail Freight Service Armin Fügenschuh - Zuse Institute Berlin, Germany Henning Homfeld - Technische Universität Darmstadt, Germany](https://reader033.vdocuments.mx/reader033/viewer/2022042804/5f53444f50c0d9319628dbfb/html5/thumbnails/198.jpg)
Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Computational Results (3)
• Cross-border traffic to France(real-world instance)
• 26 yards, ~250 orders• CPlex 11.1• 8 processors, 32 GByte RAM• 1 hours CPU time• Best feasible solution: 89,100• With only small gap (2.58%)
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Armin Fügenschuh January 7, 2010Zuse Institute BerlinDep. Optimization
Thank you for your attention!