MODAL-RailLab
The vision of MODAL's RailLab is to increase the attractivity and competitiveness of rail-bound traffic by developing powerful novel mathematical optimization methods in order to support the planning processes of railways. The core problem is the efficient use of the rolling stock. The RailLab focusses on long distance passenger traffic and, in particular, the use of high-speed railcars. The following research topics are investigated: • fully dated vehicle rotation planning in order to roll out a given cyclic rotation plan onto a concrete calendar period including seasonal changes, holidays, construction sites etc (phase I) • robust vehicle rotation planning in order to enhance the stability of the plan w.r.t. disruptions (phase II) • service oriented vehicle rotation planning in order to use the "the right train on the right relation" (phase III).
RailLab is a cooperation of ZIB, DB Fernverkehr AG, and LBW Optimization GmbH to develop novel optimization algorithms for vehicle rotation planning in order to automate and improve planning processes in railway operations.
Germany has the largest railway network in Europe, containing roughly 33,500 km of tracks and 5,400 stations. Every day, 27,000 long and short distance passenger trains transporting 5.5 million passengers and 5,000 cargo run on this network. This system is highly relevant for society, industry, and the environment. Its maintenance and operation is expensive, infrastructure construction is extremely long termed. Putting the available network capacity to the best possible use is therefore of utmost importance for increasing the attractivity and competitiveness of rail-bound traffic not only in Germany.
Strategic planning of ICE train rotations: Hypergraph visualization tool HyDraw of Ricardo Euler and Gerwin Gamrath based on JavaView of Konrad Polthier. In strategic train rotation planning rotation cycles for so-called standard weeks are designed in order to determine the number of trains needed for the future timetable.
Many planning processes in the railway industry are still operated in a manual way, the use of mathematical optimization technology is rare. The reason for this is the high complexity of railway operations respectively the lack of performance of the existing algorithms, which until recently thwarted all attempts to mathematical railway optimization.
Visualization of a 2016 ICE timetable. Analyzing timetables is a common task for rotation planners. This image shows the result of an automated timetable analysis tool, which has been developed in the MODAL-RailLab. The tool is already being used by practitioners and helps to make planning more efficient.
RailLab is based on the results of two forerunner projects Vehicle Rotation Planning for Long Distance Passenger Railways (DB-ROTOR) and Multi-day Cyclic Rotations for Trains (VS-RAIL). These projects developed vehicle rotation optimizers ROTOR and VS-RAIL, respectively, which allow, for the first time, to solve strategic vehicle rotation planning problems for large-scale passenger railways such as the German ICE network. At present, rotation plans for a cyclic standard week can be optimized, aiming at the most efficient use of rolling stock, taking into account maintenance and other treatments of vehicles in facilities of limited capacities as well as operational regularity requirements on the similar use of vehicles on every day of the week. It is also possible to re-optimize a given rotation in order to adapt it to a changed scenario that results, e.g., from construction sites or seasonal changes of the timetable. Such a re-optimization is supposed to preserve the original schedule, as long as this is possible and not too expensive. In this way, scheduled maintenance services and deadhead trips do not have to be planned again, there is no need to apply again for resources like track capacity, and it is easier to adapt the duty schedule in the succeeding planning step.
Now that the principle feasibility of vehicle rotation planning is proved for a significant problem with a role model character for the entire planning chain, further and more complex applications in rolling stocking planning have come within reach. RailLab investigates the following planning problems:
- fully dated vehicle rotation planning (phase I)
- robust vehicle rotation planning (phase II)
- service oriented vehicle rotation planning (phase III).
This long-term program will be carried out in three phases of 4-5 years each. In addition to basic mathematical research, the projects include software development work packages of specification, collection and analysis of data, system analysis, modelling, algorithmic development, implementation, tests, system integration, dissemination, and the transfer of results to further applications.
Publikationen
2021 |
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Ralf Borndörfer, Thomas Eßer, Patrick Frankenberger, Andreas Huck, Christoph Jobmann, Boris Krostitz, Karsten Kuchenbecker, Kai Moorhagen, Philipp Nagl, Michael Peterson, Markus Reuther, Thilo Schang, Michael Schoch, Hanno Schülldorf, Peter Schütz, Tobias Therolf, Kerstin Waas, Steffen Weider | Deutsche Bahn Schedules Train Rotations Using Hypergraph Optimization | Informs Journal on Applied Analytics, 51(1), pp. 42-62, 2021 |
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Julian Bushe | Rolling Stock Rotation Optimization with Maintenance Paths | Master's thesis, Technische Universität Berlin, Thorsten Koch, Ralf Borndörfer, Boris Grimm, Stanley Schade (Advisors), 2021 |
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2020 |
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Fabian Löbel | Implementing the Network Simplex for Hypergraphs | Master's thesis, Freie Universität Berlin, Ralf Borndörfer, Niels Lindner, Boris Grimm (Advisors), 2020 |
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Guvenc Sahin, Amin Ahmadi, Ralf Borndörfer, Thomas Schlechte | Multi-period line planning with resource transfers | Transportation Research Part C: Emerging Technologies, Vol.119, p. 102726, 2020 (in press, preprint available as ZIB-Report 19-51) |
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Isabel Beckenbach | Von Graphen zu Hypergraphen | OR News, 2020 |
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2019 |
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Boris Grimm, Ralf Borndörfer, Markus Reuther, Thomas Schlechte | A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance | 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019), Valentina Cacchiani, Alberto Marchetti-Spaccamela (Eds.), pp. 1:1-1:12, Vol.75, OpenAccess Series in Informatics (OASIcs), 2019 (epub ahead of print, preprint available as ZIB-Report 19-61) |
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Thomas Breugem, Ralf Borndörfer, Thomas Schlechte, Christof Schulz | A Three-Phase Heuristic for Cyclic Crew Rostering with Fairness Requirements | ZIB-Report 19-43 |
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Isabel Beckenbach | Matchings and Flows in Hypergraphs | Doctoral thesis, Freie Universität Berlin, Ralf Borndörfer, Winfried Hochstättler (Advisors), 2019 |
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Ralf Borndörfer, Boris Grimm, Markus Reuther, Thomas Schlechte | Optimization of handouts for rolling stock rotations | Journal of Rail Transport Planning & Management, pp. 1-8, 2019 (preprint available as ZIB-Report ZR-16-73) |
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Ralf Borndörfer, Boris Grimm, Thomas Schlechte | Re-optimizing ICE Rotations after a Tunnel Breakdown near Rastatt | Proceedings of the 8th International Conference on Railway Operations Modelling and Analysis - RailNorrköping 2019, pp. 160-168, Linköping Electronic Conference Proceedings(069), 2019, ISBN: 978-91-7929-992-7 (epub ahead of print, preprint available as ZIB-Report 19-02) |
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Timo Berthold, Boris Grimm, Markus Reuther, Stanley Schade, Thomas Schlechte | Strategic Planning of Rolling Stock Rotations for Public Tenders | Proceedings of the 8th International Conference on Railway Operations Modelling and Analysis - RailNorrköping 2019, pp. 148-159, Linköping Electronic Conference Proceedings(069), 2019, ISBN: 978-91-7929-992-7 |
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2018 |
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Isabel Beckenbach | A Hypergraph Network Simplex Algorithm | Operations Research Proceedings 2017, Natalia Kliewer, Jan Fabian Ehmke, Ralf Borndörfer (Eds.), pp. 309-316, Vol.Operations Research Proceedings 2017, 2018 (preprint available as ZIB-Report 17-36) |
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Ralf Borndörfer, Marika Karbstein, Christian Liebchen, Niels Lindner | A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable | 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018), pp. 16:1-16:15, Vol.65, OpenAccess Series in Informatics (OASIcs), 2018 (preprint available as ) |
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Brady Gilg, Torsten Klug, Rosemarie Martienssen, Joseph Paat, Thomas Schlechte, Christof Schulz, Senan Seymen, Alexander Tesch | Conflict-free railway track assignment at depots | Journal of Rail Transport Planning & Management, 2018 (epub ahead of print) |
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Malte Renken, Amin Ahmadi, Ralf Borndörfer, Guvenc Sahin, Thomas Schlechte | Demand-Driven Line Planning with Selfish Routing | Operations Research Proceedings 2017, pp. 687-692, 2018 (preprint available as ZIB-Report 17-38) |
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Ralf Borndörfer, Ricardo Euler, Marika Karbstein, Fabian Mett | Ein mathematisches Modell zur Beschreibung von Preissystemen im öV | ZIB-Report 18-47 |
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Torsten Klug | Freight Train Routing | Handbook of Optimization in the Railway Industry, Springer International Publishing, pp. 73-92, 2018, ISBN: 978-3-319-72152-1 |
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Isabel Beckenbach, Ralf Borndörfer | Hall’s and Kőnig’s theorem in graphs and hypergraphs | Discrete Mathematics, 341(10), pp. 2753-2761, 2018 (preprint available as ZIB-Report 15-45) |
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Erwin Abbink, Andreas Bärmann, Nikola Bešinovic, Markus Bohlin, Valentina Cacchiani, Gabrio Caimi, Stefano de Fabris, Twan Dollevoet, Frank Fischer, Armin Fügenschuh, Laura Galli, Rob M.P. Goverde, Ronny Hansmann, Henning Homfeld, Dennis Huisman, Marc Johann, Torsten Klug, Johanna Törnquist Krasemann, Leo Kroon, Leonardo Lamorgese, Frauke Liers, Carlo Mannino, Giorgio Medeossi, Dario Pacciarelli, Markus Reuther, Thomas Schlechte, Marie Schmidt, Anita Schöbel, Hanno Schülldorf, Anke Stieber, Sebastian Stiller, Paolo Toth, Uwe Zimmermann | Handbook of Optimization in the Railway Industry | Ralf Borndörfer, Torsten Klug, Leonardo Lamorgese, Carlo Mannino, Markus Reuther, Thomas Schlechte (Eds.), Springer Verlag, 2018, ISBN: 978-3-319-72152-1 |
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Ralf Borndörfer, Ascan Egerer, Marika Karbstein, Ralf Messerschmidt, Marc Perez, Steven Pfisterer, Petra Strauß | Kombilösung: Optimierung des Liniennetzes in Karlsruhe | ZIB-Report 18-45 (Der Nahverkehr 1-2, p. 33-38, 2019) |
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Ekaterina Oleynikova | Mathematical optimization of joint order batching and picker routing problems | Master's thesis, Freie Universität Berlin, Ralf Borndörfer (Advisor), 2018 |
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Jakob Witzig, Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber | Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes | Multiscale Modeling and Simulation, 16(1), pp. 248-265, 2018 (preprint available as ) |
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Stephan Schwartz, Ralf Borndörfer, Leonardo Balestrieri | On Finding Subpaths With High Demand | Operations Research Proceedings 2017, pp. 355-360, 2018 (preprint available as ) |
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Markus Reuther, Thomas Schlechte | Optimization of Rolling Stock Rotations | Handbook of Optimization in the Railway Industry, Springer International Publishing, pp. 213-241, 2018, ISBN: 978-3-319-72152-1 |
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Proceedings of the 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems | Ralf Borndörfer, Sabine Storandt (Eds.), Dagstuhl Publishing, Saarbrücken/Wadern, Germany, 2018, ISBN: ISBN 978-3-95977-096-5 |
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Gabrio Caimi, Frank Fischer, Thomas Schlechte | Railway Track Allocation | Handbook of Optimization in the Railway Industry, Springer International Publishing, pp. 141-160, 2018, ISBN: 978-3-319-72152-1 |
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Ralf Borndörfer, Heide Hoppmann, Marika Karbstein, Niels Lindner | Separation of Cycle Inequalities in Periodic Timetabling | ZIB-Report 18-16 (Discrete Optimization) (accepted for publication on 2019-08-13) |
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Adam Schienle | Solving the Time-Dependent Shortest Path Problem Using Super-Optimal Wind | Operations Research Proceedings 2017, Natalia Kliewer, Jan Fabian Ehmke, Ralf Borndörfer (Eds.), 2018 |
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Stanley Schade, Thomas Schlechte, Jakob Witzig | Structure-based Decomposition for Pattern-Detection for Railway Timetables | Operations Research Proceedings 2017, pp. 715-721, 2018 (preprint available as ZIB-Report 17-40) |
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Ralf Borndörfer, Stephan Schwartz, Gerald Bartz | The Graph Segmentation Problem | INOC 2017 – 8th International Network Optimization Conference, pp. 35-44, Vol.64, Electronic Notes in Discrete Mathematics, 2018 (preprint available as ) |
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Boris Grimm, Ralf Borndörfer, Christof Schulz, Steffen Weider | The Rolling Stock Rotation Planning Problem under Revenue Considerations | Proceedings of the Rail Transport Demand Management Conference, 2018 (preprint available as ZIB-Report 19-01) |
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Isabel Beckenbach, Meike Hatzel, Sebastian Wiederrecht | The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs | ZIB-Report 18-61 |
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Ralf Borndörfer, Matthias Breuer, Boris Grimm, Markus Reuther, Stanley Schade, Thomas Schlechte | Timetable Sparsification by Rolling Stock Rotation Optimization | Operations Research 2017, pp. 723-728, 2018 (preprint available as ZIB-Report 17-63) |
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2017 |
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Isabel Beckenbach, Britta Peis, Oliver Schaudt, Robert Scheidweiler | A Combinatorial Proof of a König-type Theorem for Unimodular Hypergraphs | ZIB-Report 17-27 (under review) |
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Boris Grimm, Ralf Borndörfer, Markus Reuther, Stanley Schade, Thomas Schlechte | A Propagation Approach to Acyclic Rolling Stock Rotation Optimization | Proceedings of the IAROR conference RailLille, 2017 (preprint available as ZIB-Report 17-24) |
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Frank Fischer, Boris Grimm, Torsten Klug, Thomas Schlechte | A Re-optimization Approach for Train Dispatching | Operations Research Proceedings 2016, pp. 645-651, 2017 (preprint available as ZIB-Report 16-49) |
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Cheng-Lung Wu, Stephen J. Maher | Airline scheduling and disruption management | L. Budd, S. Ison, eds., Air transportation management: an international perspective, Routledge: New York, pp. 151-167, 2017, ISBN: 9781472451064 |
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Brady Gilg, Torsten Klug, Rosemarie Martienssen, Joseph Paat, Thomas Schlechte, Christof Schulz, Sinan Seymen, Alexander Tesch | Conflict-Free Railway Track Assignment at Depots | Proceedings of the IAROR conference RailLille, 2017 (preprint available as ZIB-Report 17-23) |
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Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Pedro Maristany de las Casas, Thomas Schlechte, Swen Schlobach | Cost Projection Methods for the Shortest Path Problem with Crossing Costs | 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017), Gianlorenzo D'Angelo, Twan Dollevoet (Eds.), Vol.59, 2017 (preprint available as ) |
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Stephan Schwartz, Thomas Schlechte, Elmar Swarat | Designing Inspector Rosters with Optimal Strategies | Operations Research Proceedings 2016, pp. 217-223, 2017 (preprint available as ) |
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Ralf Borndörfer, Oytun Arslan, Ziena Elijazyfer, Hakan Güler, Malte Renken, Guvenc Sahin, Thomas Schlechte | Line Planning on Path Networks with Application to the Istanbul Metrobüs | Operations Research Proceedings 2016, pp. 235-241, 2017 (preprint available as ) |
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Markus Reuther | Mathematical Optimization of Rolling Stock Rotations | Doctoral thesis, Technische Universität Berlin, Ralf Borndörfer, Martin Grötschel (Advisors), 2017 |
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Sebastian Schenker, Ingmar Vierhaus, Ralf Borndörfer, Armin Fügenschuh, Martin Skutella | Optimisation Methods in Sustainable Manufacturing | Sustainable Manufacturing, Rainer Stark, Günther Seliger, Jérémy Bonvoisin (Eds.), Springer International Publishing, pp. 239-253, 2017, ISBN: 978-3-319-48514-0 |
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Stanley Schade, Ralf Borndörfer, Matthias Breuer, Boris Grimm, Markus Reuther, Thomas Schlechte, Patrick Siebeneicher | Pattern Detection For Large-Scale Railway Timetables | Proceedings of the IAROR conference RailLille, 2017 (preprint available as ZIB-Report 17-17) |
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Isabel Beckenbach, Robert Scheidweiler | Perfect f-Matchings and f-Factors in Hypergraphs - A Combinatorial Approach | Discrete Mathematics, 340(10), pp. 2499-2506, 2017 (preprint available as ZIB-Report 16-22) |
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Ralf Borndörfer, Torsten Klug, Leonardo Lamorgese, Carlo Mannino, Markus Reuther, Thomas Schlechte | Recent success stories on integrated optimization of railway systems | Transportation Research Part C: Emerging Technologies, 74(1), pp. 196-211, 2017 |
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Frank Fischer, Thomas Schlechte | Strong Relaxations for the Train Timetabling Problem using Connected Configurations | 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017), Gianlorenzo D'Angelo, Twan Dollevoet (Eds.), Vol.59, 2017, ISBN: 978-3-95977-042-2 (preprint available as ZIB-Report 17-46) |
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Ralf Borndörfer, Boris Grimm, Markus Reuther, Thomas Schlechte | Template-based Re-optimization of Rolling Stock Rotations | Public Transport, pp. 1-19, 2017 |
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Ralf Borndörfer, Heide Hoppmann, Marika Karbstein | Umsteigen ohne Warten | HEUREKA 2017, 2017 (epub ahead of print, preprint available as ) |
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2016 |
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Isabel Beckenbach, Ralf Borndörfer | An Approximation Result for Matchings in Partitioned Hypergraphs | Operations Research Proceedings 2014, pp. 31-36, 2016 (preprint available as ZIB-Report 14-30) |
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Ralf Borndörfer, Markus Reuther, Thomas Schlechte, Kerstin Waas, Steffen Weider | Integrated Optimization of Rolling Stock Rotations for Intercity Railways | Transportation Science, 50(3), pp. 863-877, 2016 (preprint available as ZIB-Report 12-39) |
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Boris Grimm, Ralf Borndörfer, Markus Reuther, Thomas Schlechte, Stanley Schade | Regularity patterns for rolling stock rotation optimization | 8th International Conference on Applied Operational Research, Proceedings, pp. 28-32, Vol.8, Lecture Notes in Management Science, 2016 |
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Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Adam Schienle, Thomas Schlechte, Swen Schlobach | Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind | 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016), Goerigk Marc (Ed.), Vol.54, OpenAccess Series in Informatics (OASIcs), 2016 |
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Ralf Borndörfer, Marika Karbstein, Julika Mehrgahrdt, Markus Reuther, Thomas Schlechte | The Cycle Embedding Problem | Operations Research Proceedings 2014, pp. 465-472, 2016 (preprint available as ZIB-Report 14-37) |
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Ralf Borndörfer, Armin Fügenschuh, Torsten Klug, Thilo Schang, Thomas Schlechte, Hanno Schülldorf | The Freight Train Routing Problem for Congested Railway Networks with Mixed Traffic | Transportation Science, 50(2), pp. 408-423, 2016 (in press, preprint available as ) |
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Stanley Schade, Martin Strehler | The Maximum Flow Problem for Oriented Flows | 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016), Marc Goerigk, Renato Werneck (Eds.), pp. 1-13, Vol.54, OpenAccess Series in Informatics (OASIcs), 2016, ISBN: 978-3-95977-021-7 (epub ahead of print) |
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Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Thomas Schlechte, Swen Schlobach | The Shortest Path Problem with Crossing Costs | ZIB-Report 16-70 |
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2015 |
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Frank Fischer, Thomas Schlechte | Comparing two dual relaxations of large scale train timetabling problems | Proceedings of Conference on Advanced Systems in Public Transport 2015, 2015 (preprint available as ZIB-Report 15-43) |
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Ralf Borndörfer, Markus Reuther, Thomas Schlechte, Christof Schulz, Elmar Swarat, Steffen Weider | Duty Rostering in Public Transport - Facing Preferences, Fairness, and Fatigue | Proceedings of Conference on Advanced Systems in Public Transport 2015 (CASPT2015), 2015 (preprint available as ZIB-Report 15-44) |
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Torsten Klug, Konstanty Junosza-Szaniawski, Slawomir Kwasiborski, Armin Fügenschuh, Thomas Schlechte | Fastest, Average and Quantile Schedule | SOFSEM 2015: Theory and Practice of Computer Science, pp. 201-216, 2015 (preprint available as ZIB-Report 14-49) |
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Charles Brett, Rebecca Hoberg, Meritxell Pacheco, Kyle Smith, Ralf Borndörfer, Ricardo Euler, Gerwin Gamrath, Boris Grimm, Olga Heismann, Markus Reuther, Thomas Schlechte, Alexander Tesch | G-RIPS 2014 RailLab - Towards robust rolling stock rotations | ZIB-Report 14-34 |
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Ralf Borndörfer, Christof Schulz, Stephan Seidl, Steffen Weider | Integration of Duty Scheduling and Rostering to Increase Driver Satisfaction | ZIB-Report 15-50 |
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Ralf Borndörfer, Leonardo Lamorgese, Torsten Klug, Carlo Mannino, Markus Reuther, Thomas Schlechte | Recent Success Stories on Optimization of Railway Systems | Proceedings of the IAROR conference RailTokyo, 2015 (preprint available as ZIB-Report 14-47) |
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Ralf Borndörfer, Markus Reuther | Regional Search for the Resource Constrained Assignment Problem | 15th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2015), pp. 111-129, Vol.48, OpenAccess Series in Informatics (OASIcs), 2015 |
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Sepideh Ahmadi, Sascha F. Gritzbach, Kathryn Lund-Nguyen, Devita McCullough-Amal | Rolling Stock Rotation Optimization in Days of Strike: An Automated Approach for Creating an Alternative Timetable | ZIB-Report 15-52 |
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Ralf Borndörfer, Boris Grimm, Markus Reuther, Thomas Schlechte | Template based re-optimization of rolling stock rotations | Proceedings of Conference on Advanced Systems in Public Transport 2015 (CASPT2015), 2015 |
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Ralf Borndörfer, Olga Heismann | The hypergraph assignment problem | Discrete Optimization, Vol.15, pp. 15-25, 2015 (preprint available as ) |
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2014 |
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Ralf Borndörfer, Markus Reuther, Thomas Schlechte | A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation Problem | 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, pp. 79-91, Vol.42, OpenAccess Series in Informatics (OASIcs), 2014 (preprint available as ZIB-Report 14-26) |
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Markus Reuther | Local Search for the Resource Constrained Assignment Problem | 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, pp. 62-78, Vol.42, OpenAccess Series in Informatics (OASIcs), 2014 |
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Ralf Borndörfer, Julika Mehrgardt, Markus Reuther, Thomas Schlechte, Kerstin Waas | Re-Optimization of Rolling Stock Rotations | pp. 49-55, Operations Research Proceedings 2013, 2014 (preprint available as ZIB-Report 13-60) |
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2013 |
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Ralf Borndörfer, Andreas Löbel, Markus Reuther, Thomas Schlechte, Steffen Weider | Rapid Branching | Public Transport, 5(1), pp. 3-23, 2013 (preprint available as ) |
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2012 |
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Ralf Borndörfer, Markus Reuther, Thomas Schlechte, Steffen Weider | Vehicle Rotation Planning for Intercity Railways | Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12), 2012 (preprint available as ZIB-Report 12-11) |
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2011 |
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Ralf Borndörfer, Markus Reuther, Thomas Schlechte, Steffen Weider | A Hypergraph Model for Railway Vehicle Rotation Planning | 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, OpenAccess Series in Informatics (OASIcs)(20), pp. 146-155, 2011 (preprint available as ZIB-Report 11-36) |
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