@misc{Schlechte2012, author = {Schlechte, Thomas}, title = {Railway Track Allocation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16402}, year = {2012}, abstract = {This article gives an overview of the results of the author's PhD thesis. The thesis deals with the mathematical optimization for the efficient use of railway infrastructure. We address the optimal allocation of the available railway track capacity - the track allocation problem. This track allocation problem is a major challenge for a railway company, independent of whether a free market, a private monopoly, or a public monopoly is given. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense sizes of the problem instances. Mathematical models and optimization techniques can result in huge gains for both railway customers and operators, e.g., in terms of cost reductions or service quality improvements. We tackle this challenge by developing novel mathematical models and associated innovative algorithmic solution methods for large scale instances. We made considerable progress on solving track allocation problems by two main features - a novel modeling approach for the macroscopic track allocation problem and algorithmic improvements based on the utilization of the bundle method. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland.}, language = {en} } @misc{BorndoerferSchlechteWeider2010, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Weider, Steffen}, title = {Railway Track Allocation by Rapid Branching}, organization = {Zuse Institut Berlin}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11864}, number = {10-22}, year = {2010}, abstract = {The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.}, language = {en} } @article{RaackRaymondSchlechteetal.2013, author = {Raack, Christian and Raymond, Annie and Schlechte, Thomas and Werner, Axel}, title = {Standings in sports competitions using integer programming}, volume = {10}, journal = {Journal of Quantitative Analysis in Sports}, number = {2}, publisher = {De Gruyter}, doi = {10.1515/jqas-2013-0111}, pages = {131 -- 137}, year = {2013}, language = {en} } @inproceedings{Schlechte2012, author = {Schlechte, Thomas}, title = {Railway Track Allocation}, booktitle = {Operations Research Proceedings}, doi = {10.1007/978-3-319-00795-314}, pages = {15 -- 20}, year = {2012}, language = {en} } @article{BorndoerferReutherSchlechteetal.2012, author = {Bornd{\"o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {Vehicle Rotation Planning for Intercity Railways}, journal = {Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12)}, editor = {Mu{\~n}oz, J. C. and Voß, S.}, year = {2012}, abstract = {This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.}, language = {en} } @article{BorndoerferLoebelReutheretal.2013, author = {Bornd{\"o}rfer, Ralf and L{\"o}bel, Andreas and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {Rapid Branching}, volume = {5}, journal = {Public Transport}, number = {1}, editor = {Mu{\~n}oz, J. C. and Voß, S.}, publisher = {Springer Berlin Heidelberg}, pages = {3 -- 23}, year = {2013}, abstract = {We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.}, language = {en} } @article{HeinzSchlechteStephanetal.2012, author = {Heinz, Stefan and Schlechte, Thomas and Stephan, R{\"u}diger and Winkler, Michael}, title = {Solving steel mill slab design problems}, volume = {17}, journal = {Constraints}, number = {1}, doi = {10.1007/s10601-011-9113-8}, pages = {39 -- 50}, year = {2012}, abstract = {The steel mill slab design problem from the CSPLIB is a combinatorial optimization problem motivated by an application of the steel industry. It has been widely studied in the constraint programming community. Several methods were proposed to solve this problem. A steel mill slab library was created which contains 380 instances. A closely related binpacking problem called the multiple knapsack problem with color constraints, originated from the same industrial problem, was discussed in the integer programming community. In particular, a simple integer program for this problem has been given by Forrest et al. (INFORMS J Comput 18:129-134, 2006). The aim of this paper is to bring these different studies together. Moreover, we adapt the model of Forrest et al. (INFORMS J Comput 18:129-134, 2006) for the steel mill slab design problem. Using this model and a state-of-the-art integer program solver all instances of the steel mill slab library can be solved efficiently to optimality. We improved, thereby, the solution values of 76 instances compared to previous results (Schaus et al., Constraints 16:125-147, 2010). Finally, we consider a recently introduced variant of the steel mill slab design problem, where within all solutions which minimize the leftover one is interested in a solution which requires a minimum number of slabs. For that variant we introduce two approaches and solve all instances of the steel mill slab library with this slightly changed objective function to optimality.}, language = {en} } @article{BorndoerferSchlechteSwarat2012, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Swarat, Elmar}, title = {Railway Track Allocation -- Simulation, Aggregation, and Optimization}, volume = {2}, journal = {Proc. 1st International Workshop on High-speed and Intercity Railways (IWHIR 2011)}, number = {148}, doi = {10.1007/978-3-642-27963-8}, pages = {53 -- 70}, year = {2012}, abstract = {Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.}, language = {en} } @article{BorndoerferReutherSchlechteetal.2011, author = {Bornd{\"o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {A Hypergraph Model for Railway Vehicle Rotation Planning}, volume = {OpenAccess Series in Informatics (OASIcs)}, journal = {11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, number = {20}, doi = {10.4230/OASIcs.ATMOS.2011.146}, pages = {146 -- 155}, year = {2011}, abstract = {We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the ``regularity'' of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.}, language = {en} } @article{Schlechte2011, author = {Schlechte, Thomas}, title = {Railway Track Allocation - Simulation and Optimization}, volume = {4}, journal = {Proceedings of 4th International Seminar on Railway Operations Modelling and Analysis (IAROR)}, year = {2011}, abstract = {Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway infrastructure provider. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market. This leads to an increase of slot requests and then it is natural that conflicts occur among them. Furthermore, railway infrastructure networks consist of very expensive assets, even more they are rigid due to the long-term upgrade process. In order to make best use of these valuable infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithmic methodology can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and a general framework to support the integration of simulation and optimization for railway capacity allocation.}, language = {en} } @misc{BlancoSchlechte2013, author = {Blanco, Marco and Schlechte, Thomas}, title = {Analysis of Micro-Macro Transformations of Railway Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42710}, year = {2013}, abstract = {A common technique in the solution of large or complex optimization problems is the use of micro-macro transformations. In this paper, we carry out a theoretical analysis of such transformations for the track allocation problem in railway networks. We prove that the cumulative rounding technique of Schlechte et al. satisfies two of three natural optimality criteria and that this performance cannot be improved. We also show that under extreme circumstances, this technique can perform inconvieniently by underestimating the global optimal value.}, language = {en} } @misc{BorndoerferMehrgardtReutheretal.2013, author = {Bornd{\"o}rfer, Ralf and Mehrgardt, Julika and Reuther, Markus and Schlechte, Thomas and Waas, Kerstin}, title = {Re-optimization of Rolling Stock Rotations}, issn = {1438-0064}, doi = {10.1007/978-3-319-07001-8_8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42569}, year = {2013}, abstract = {The Rolling Stock Rotation Problem is to schedule rail vehicles in order to cover timetabled trips by a cost optimal set of vehicle rotations. The problem integrates several facets of railway optimization, i.e., vehicle composition, maintenance constraints, and regularity aspects. In industrial applications existing schedules often have to be re-optimized to integrate timetable changes or construction sites. We present an integrated modeling and algorithmic approach for this task as well as computational results for industrial problem instances of DB Fernverkehr AG.}, language = {en} } @article{BorndoerferFuegenschuhKlugetal.2016, author = {Bornd{\"o}rfer, Ralf and F{\"u}genschuh, Armin and Klug, Torsten and Schang, Thilo and Schlechte, Thomas and Sch{\"u}lldorf, Hanno}, title = {The Freight Train Routing Problem for Congested Railway Networks with Mixed Traffic}, volume = {50}, journal = {Transportation Science}, number = {2}, doi = {10.1287/trsc.2015.0656}, pages = {408 -- 423}, year = {2016}, abstract = {We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that the sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. In this context, macroscopic refers to an aggregation of complex and large real-world structures into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We provide a mixed-integer nonlinear programming (MINLP) formulation for the FTRP, which is a multicommodity flow model on a time-expanded graph with additional routing constraints. The model's nonlinearities originate from an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model (MILP) by piecewise linear approximation. The latter is solved by a state-of-the art MILP solver for various real-world test instances.}, language = {en} } @inproceedings{BorndoerferGrimmReutheretal.2015, author = {Bornd{\"o}rfer, Ralf and Grimm, Boris and Reuther, Markus and Schlechte, Thomas}, title = {Template based re-optimization of rolling stock rotations}, booktitle = {Proceedings of Conference on Advanced Systems in Public Transport 2015 (CASPT2015)}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57539}, year = {2015}, abstract = {Rolling stock, i.e., rail vehicles, are among the most expensive and limited assets of a railway company. They must be used efficiently applying optimization techniques. One important aspect is re-optimization, which is the topic that we consider in this paper. We propose a template concept that allows to compute cost minimal rolling stock rotations under a large variety of re-optimization requirements. Two examples, involving a connection template and a rotation template, are discussed. An implementation within the rolling stock rotation optimizer rotor and computational results for scenarios provided by DB Fernverkehr AG, one of the leading railway operators in Europe, are presented.}, language = {en} } @inproceedings{BorndoerferErolSchlechte2009, author = {Bornd{\"o}rfer, Ralf and Erol, Berkan and Schlechte, Thomas}, title = {Optimization of macroscopic train schedules via TS-OPT}, booktitle = {Proceedings of the 3rd International Seminar on Railway Operations Modelling and Analysis - Engineering and Optimisation Approaches}, editor = {Hansen, I. and Wendler, E. and Weidmann, U. and L{\"u}thi, M. and Rodriguez, J. and Ricci, S. and Kroon, Leo}, address = {Z{\"u}rich, Switzerland}, year = {2009}, language = {en} } @inproceedings{SchlechteTanner2010, author = {Schlechte, Thomas and Tanner, Andreas}, title = {Railway capacity auctions with dual prices}, booktitle = {Selected Proceedings of the 12th World Conference on Transport Research}, year = {2010}, language = {en} } @inproceedings{ErolKlemenzSchlechteetal.2008, author = {Erol, Berkan and Klemenz, Marc and Schlechte, Thomas and Schultz, S{\"o}ren and Tanner, Andreas}, title = {TTPlib 2008 - A Library for Train Timetabling Problems}, booktitle = {Computers in Railways XI}, editor = {Tomii, A. and Allan, J. and Arias, E. and Brebbia, C. and Goodman, C. and Rumsey, A. and Sciutto, G.}, publisher = {WIT Press}, year = {2008}, language = {en} } @inproceedings{BorndoerferSchlechte2007, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas}, title = {Models for Railway Track Allocation}, volume = {07001}, booktitle = {ATMOS 2007 - 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems}, editor = {Liebchen, Christian and Ahuja, Ravindra and Mesa, Juan}, publisher = {Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, year = {2007}, language = {en} } @inproceedings{BorndoerferSchlechte2008, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas}, title = {Balancing Efficiency and Robustness}, booktitle = {MCDM for Sustainable Energy and Transportation Systems}, editor = {Ehrgott, Matthias and Naujoks, Boris and Stewart, Theo and Wallenius, Jyrki}, publisher = {Lecture Notes in Economics and Mathematical Systems}, year = {2008}, language = {en} } @misc{Schlechte2003, type = {Master Thesis}, author = {Schlechte, Thomas}, title = {Das Resource-Constraint-Shortest-Path-Problem und seine Anwendung in der {\"O}PNV-Dienstplanung}, year = {2003}, language = {en} } @article{GilgKlugMartienssenetal.2018, author = {Gilg, Brady and Klug, Torsten and Martienssen, Rosemarie and Paat, Joseph and Schlechte, Thomas and Schulz, Christof and Seymen, Senan and Tesch, Alexander}, title = {Conflict-free railway track assignment at depots}, journal = {Journal of Rail Transport Planning \& Management}, doi = {10.1016/j.jrtpm.2017.12.004}, year = {2018}, abstract = {Managing rolling stock with no passengers aboard is a critical component of railway operations. One aspect of managing rolling stock is to park the rolling stock on a given set of tracks at the end of a day or service. Depending on the parking assignment, shunting may be required in order for a parked train to depart or for an incoming train to park. Given a collection of tracks M and a collection of trains T with a fixed arrival-departure timetable, the train assignment problem (TAP) is to determine the maximum number of trains from T that can be parked on M according to the timetable and without the use of shunting. Hence, efficiently solving the TAP allows to quickly compute feasible parking schedules that do not require further shunting adjustments. In this paper, we show that the TAP is NP-hard and present two integer programming models for solving the TAP. We compare both models on a theoretical level. Moreover, to our knowledge, we consider the first approach that integrates track lengths along with the three most common types of parking tracks FIFO, LIFO and FREE tracks in a common model. Furthermore, to optimize against uncertainty in the arrival times of the trains we extend our models by stochastic and robust modeling techniques. We conclude by giving computational results for both models, observing that they perform well on real timetables.}, language = {en} } @incollection{ReutherSchlechte2018, author = {Reuther, Markus and Schlechte, Thomas}, title = {Optimization of Rolling Stock Rotations}, volume = {268}, booktitle = {Handbook of Optimization in the Railway Industry}, publisher = {Springer International Publishing}, isbn = {978-3-319-72152-1}, doi = {https://doi.org/10.1007/978-3-319-72153-8}, pages = {213 -- 241}, year = {2018}, abstract = {This chapter shows a successful approach how to model and optimize rolling stock rotations that are required for the operation of a passenger timetable. The underlying mathematical optimization problem is described in detail and solved by RotOR, i.e., a complex optimization algorithm based on linear programming and combinatorial methods. RotOR is used by DB Fernverkehr AG (DBF) in order to optimize intercity express (ICE) rotations for the European high-speed network. We focus on main modeling and solving components, i.e. a hypergraph model and a coarse-to-fine column generation approach. Finally, the chapter concludes with a complex industrial re-optimization application showing the effectiveness of the approach for real world challenges.}, language = {en} } @inproceedings{FischerSchlechte2017, author = {Fischer, Frank and Schlechte, Thomas}, title = {Strong Relaxations for the Train Timetabling Problem using Connected Configurations}, volume = {59}, booktitle = {17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)}, editor = {D'Angelo, Gianlorenzo and Dollevoet, Twan}, isbn = {978-3-95977-042-2}, doi = {10.4230/OASIcs.ATMOS.2017.11}, year = {2017}, abstract = {The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral.}, language = {en} } @inproceedings{BlancoBorndoerferHoangetal.2017, author = {Blanco, Marco and Bornd{\"o}rfer, Ralf and Hoang, Nam-Dung and Kaier, Anton and Maristany de las Casas, Pedro and Schlechte, Thomas and Schlobach, Swen}, title = {Cost Projection Methods for the Shortest Path Problem with Crossing Costs}, volume = {59}, booktitle = {17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)}, editor = {D'Angelo, Gianlorenzo and Dollevoet, Twan}, year = {2017}, abstract = {Real world routing problems, e.g., in the airline industry or in public and rail transit, can feature complex non-linear cost functions. An important case are costs for crossing regions, such as countries or fare zones. We introduce the shortest path problem with crossing costs (SPPCC) to address such situations; it generalizes the classical shortest path problem and variants such as the resource constrained shortest path problem and the minimum label path problem. Motivated by an application in flight trajectory optimization with overflight costs, we focus on the case in which the crossing costs of a region depend only on the nodes used to enter or exit it. We propose an exact Two-Layer-Dijkstra Algorithm as well as a novel cost-projection linearization technique that approximates crossing costs by shadow costs on individual arcs, thus reducing the SPPCC to a standard shortest path problem. We evaluate all algorithms' performance on real-world flight trajectory optimization instances, obtaining very good {\`a} posteriori error bounds.}, language = {en} } @inproceedings{BorndoerferArslanElijazyferetal.2017, author = {Bornd{\"o}rfer, Ralf and Arslan, Oytun and Elijazyfer, Ziena and G{\"u}ler, Hakan and Renken, Malte and Sahin, Guvenc and Schlechte, Thomas}, title = {Line Planning on Path Networks with Application to the Istanbul Metrob{\"u}s}, booktitle = {Operations Research Proceedings 2016}, doi = {10.1007/978-3-319-55702-1_32}, pages = {235 -- 241}, year = {2017}, abstract = {Bus rapid transit systems in developing and newly industrialized countries often consist of a trunk with a path topology. On this trunk, several overlapping lines are operated which provide direct connections. The demand varies heavily over the day, with morning and afternoon peaks typically in reverse directions. We propose an integer programming model for this problem, derive a structural property of line plans in the static (or single period) ``unimodal demand'' case, and consider approaches to the solution of the multi-period version that rely on clustering the demand into peak and off-peak service periods. An application to the Metrob{\"u}s system of Istanbul is discussed.}, language = {en} } @inproceedings{SchwartzSchlechteSwarat2017, author = {Schwartz, Stephan and Schlechte, Thomas and Swarat, Elmar}, title = {Designing Inspector Rosters with Optimal Strategies}, booktitle = {Operations Research Proceedings 2016}, doi = {10.1007/978-3-319-55702-1_30}, pages = {217 -- 223}, year = {2017}, abstract = {We consider the problem of enforcing a toll on a transportation network with limited inspection resources. We formulate a game theoretic model to optimize the allocation of the inspectors, taking the reaction of the network users into account. The model includes several important aspects for practical operation of the control strategy, such as duty types for the inspectors. In contrast to an existing formulation using flows to describe the users' strategies we choose a path formulation and identify dominated user strategies to significantly reduce the problem size. Computational results suggest that our approach is better suited for practical instances.}, language = {en} } @inproceedings{FischerGrimmKlugetal.2017, author = {Fischer, Frank and Grimm, Boris and Klug, Torsten and Schlechte, Thomas}, title = {A Re-optimization Approach for Train Dispatching}, booktitle = {Operations Research Proceedings 2016}, doi = {10.1007/978-3-319-55702-1_85}, pages = {645 -- 651}, year = {2017}, abstract = {The Train Dispatching Problem (TDP) is to schedule trains through a network in a cost optimal way. Due to disturbances during operation existing track allocations often have to be re-scheduled and integrated into the timetable. This has to be done in seconds and with minimal timetable changes to guarantee smooth and conflict free operation. We present an integrated modeling approach for the re-optimization task using Mixed Integer Programming. Finally, we provide computational results for scenarios provided by the INFORMS RAS Problem Soling Competition 2012.}, language = {en} } @misc{BlancoBorndoerferHoangetal.2016, author = {Blanco, Marco and Bornd{\"o}rfer, Ralf and Hoang, Nam-Dung and Kaier, Anton and Schlechte, Thomas and Schlobach, Swen}, title = {The Shortest Path Problem with Crossing Costs}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61240}, year = {2016}, abstract = {We introduce the shortest path problem with crossing costs (SPPCC), a shortest path problem in a directed graph, in which the objective function is the sum of arc weights and crossing costs. The former are independently paid for each arc used by the path, the latter need to be paid every time the path intersects certain sets of arcs, which we call regions. The SPPCC generalizes not only the classical shortest path problem but also variants such as the resource constrained shortest path problem and the minimum label path problem. We use the SPPCC to model the flight trajectory optimization problem with overflight costs. In this paper, we provide a comprehensive analysis of the problem. In particular, we identify efficient exact and approximation algorithms for the cases that are most relevant in practice.}, language = {en} } @misc{SchwartzSchlechteSwarat2016, author = {Schwartz, Stephan and Schlechte, Thomas and Swarat, Elmar}, title = {Designing Inspector Rosters with Optimal Strategies}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61123}, year = {2016}, abstract = {We consider the problem of enforcing a toll on a transportation network with limited inspection resources. We formulate a game theoretic model to optimize the allocation of the inspectors, taking the reaction of the network users into account. The model includes several important aspects for practical operation of the control strategy, such as duty types for the inspectors. In contrast to an existing formulation using flows to describe the users' strategies we choose a path formulation and identify dominated user strategies to significantly reduce the problem size. Computational results suggest that our approach is better suited for practical instances.}, language = {en} } @article{BorndoerferKlugLamorgeseetal.2017, author = {Bornd{\"o}rfer, Ralf and Klug, Torsten and Lamorgese, Leonardo and Mannino, Carlo and Reuther, Markus and Schlechte, Thomas}, title = {Recent success stories on integrated optimization of railway systems}, volume = {74}, journal = {Transportation Research Part C: Emerging Technologies}, number = {1}, doi = {10.1016/j.trc.2016.11.015}, pages = {196 -- 211}, year = {2017}, abstract = {Planning and operating railway transportation systems is an extremely hard task due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. Because of that, however, mathematical models and optimization techniques can result in large gains for both railway customers and operators, e.g., in terms of cost reductions or service quality improvements. In the last years a large and growing group of researchers in the OR community have devoted their attention to this domain developing mathematical models and optimization approaches to tackle many of the relevant problems in the railway planning process. However, there is still a gap to bridge between theory and practice (e.g. Cacchiani et al., 2014; Bornd{\"o}rfer et al., 2010), with a few notable exceptions. In this paper we address three individual success stories, namely, long-term freight train routing (part I), mid-term rolling stock rotation planning (part II), and real-time train dispatching (part III). In each case, we describe real-life, successful implementations. We will discuss the individual problem setting, survey the optimization literature, and focus on particular aspects addressed by the mathematical models. We demonstrate on concrete applications how mathematical optimization can support railway planning and operations. This gives proof that mathematical optimization can support the planning of railway resources. Thus, mathematical models and optimization can lead to a greater efficiency of railway operations and will serve as a powerful and innovative tool to meet recent challenges of the railway industry.}, language = {en} } @inproceedings{GrimmBorndoerferReutheretal.2016, author = {Grimm, Boris and Bornd{\"o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Schade, Stanley}, title = {Regularity patterns for rolling stock rotation optimization}, volume = {8}, booktitle = {8th International Conference on Applied Operational Research, Proceedings}, pages = {28 -- 32}, year = {2016}, abstract = {The operation of railways gives rise to many fundamental optimization problems. One of these problems is to cover a given set of timetabled trips by a set of rolling stock rotations. This is well known as the Rolling Stock Rotation Problem (RSRP). Most approaches in the literature focus primarily on modeling and minimizing the operational costs. However, an essential aspect for the industrial application is mostly neglected. As the RSRP follows timetabling and line planning, where periodicity is a highly desired property, it is also desired to carry over periodic structures to rolling stock rotations and following operations. We call this complex requirement regularity. Regularity turns out to be of essential interest, especially in the industrial scenarios that we tackle in cooperation with DB Fernverkehr AG. Moreover, regularity in the context of the RSRP has not been investigated thoroughly in the literature so far. We introduce three regularity patterns to tackle this requirement, namely regular trips, regular turns, and regular handouts. We present a two-stage approach in order to optimize all three regularity patterns. At first, we integrate regularity patterns into an integer programming approach for the minimization of the operational cost of rolling stock rotations. Afterwards regular handouts are computed. These handouts present the rotations of the first stage in the most regular way. Our computational results (i.e., rolling stock rotations evaluated by planners of DB Fernverkehr AG) show that the three regularity patterns and our concept are a valuable and, moreover, an essential contribution to rolling stock rotation optimization.}, language = {en} } @misc{SchlechteBorndoerfer2008, author = {Schlechte, Thomas and Bornd{\"o}rfer, Ralf}, title = {Balancing Efficiency and Robustness - A Bi-criteria Optimization Approach to Railway Track Allocation}, organization = {ZIB}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10763}, number = {08-22}, year = {2008}, abstract = {Technical restrictions and challenging details let railway traffic become one of the most complex transportation systems. Routing trains in a conflict-free way through a track network is one of the basic scheduling problems for any railway company. This article focuses on a robust extension of this problem, also known as train timetabling problem (TTP), which consists in finding a schedule, a conflict free set of train routes, of maximum value for a given railway network. However, timetables are not only required to be profitable. Railway companies are also interested in reliable and robust solutions. Intuitively, we expect a more robust track allocation to be one where disruptions arising from delays are less likely to be propagated causing delays of subsequent trains. This trade-off between an efficient use of railway infrastructure and the prospects of recovery leads us to a bi-criteria optimization approach. On the one hand we want to maximize the profit of a schedule, that is more or less to maximize the number of feasible routed trains. On the other hand if two trains are scheduled as tight as possible after each other it is clear that a delay of the first one always affects the subsequent train. We present extensions of the integer programming formulation in [BorndoerferSchlechte2007] for solving (TTP). These models can incorporate both aspects, because of the additional track configuration variables. We discuss how these variables can directly be used to measure a certain type of robustness of a timetable. For these models which can be solved by column generation techniques, we propose so-called scalarization techniques, see [Ehrgott2005], to determine efficient solutions. Here, an efficient solution is one which does not allow any improvement in profit and robustness at the same time. We prove that the LP-relaxation of the (TTP) including an additional \$\epsilon\$-constraint remains solvable in polynomial time. Finally, we present some preliminary results on macroscopic real-world data of a part of the German long distance railway network.}, language = {en} } @misc{ErolKlemenzSchlechteetal.2008, author = {Erol, Berkan and Klemenz, Marc and Schlechte, Thomas and Schultz, S{\"o}ren and Tanner, Andreas}, title = {TTPLIB 2008 - A Library for Train Timetabling Problems}, organization = {ZIB,}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10732}, number = {08-19}, year = {2008}, abstract = {We introduce (TTPlib), a data library for train timetabling problems that can be accessed at http://ttplib.zib.de. In version 1.0, the library contains data related to 50 scenarios. Most instances result from the combination of macroscopic railway networks and several train request sets for the German long distance area containing Hannover, Kassel and Fulda, short denoted by Ha-Ka-Fu. In this paper, we introduce the data concepts of TTPlib, describe the scenarios included in the library and provide a free visualization tool TraVis.}, language = {en} } @misc{BorndoerferGroetschelLukacetal.2005, author = {Bornd{\"o}rfer, Ralf and Gr{\"o}tschel, Martin and Lukac, Sascha and Mitusch, Kay and Schlechte, Thomas and Schultz, S{\"o}ren and Tanner, Andreas}, title = {An Auctioning Approach to Railway Slot Allocation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8786}, number = {05-45}, year = {2005}, abstract = {We present an approach to implement an auction of railway slots. Railway network, train driving characteristics, and safety requirements are described by a simplified, but still complex macroscopic model. In this environment, slots are modelled as combinations of scheduled track segments. The auction design builds on the iterative combinatorial auction. However, combinatorial bids are restricted to some types of slot bundles that realize positive synergies between slots. We present a bidding language that allows bidding for these slot bundles. An integer programming approach is proposed to solve the winner determination problem of our auction. Computational results for auction simulations in the Hannover-Fulda-Kassel area of the German railway network give evidence that auction approaches can induce a more efficient use of railway capacity.}, language = {en} } @misc{GrimmBorndoerferReutheretal.2017, author = {Grimm, Boris and Bornd{\"o}rfer, Ralf and Reuther, Markus and Schade, Stanley and Schlechte, Thomas}, title = {A Propagation Approach to Acyclic Rolling Stock Rotation Optimization}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63930}, year = {2017}, abstract = {The rolling stock, i.e., railway vehicles, are one of the key ingredients of a running railway system. As it is well known, the offer of a railway company to their customers, i.e., the railway timetable, changes from time to time. Typical reasons for that are different timetables associated with different seasons, maintenance periods or holidays. Therefore, the regular lifetime of a timetable is split into (more or less) irregular periods where parts of the timetable are changed. In order to operate a railway timetable most railway companies set up sequences that define the operation of timetabled trips by a single physical railway vehicle called (rolling stock) rotations. Not surprisingly, the individual parts of a timetable also affect the rotations. More precisely, each of the parts brings up an acyclic rolling stock rotation problem with start and end conditions associated with the beginning and ending of the corresponding period. In this paper, we propose a propagation approach to deal with large planning horizons that are composed of many timetables with shorter individual lifetimes. The approach is based on an integer linear programming formulation that propagates rolling stock rotations through the irregular parts of the timetable while taking a large variety of operational requirements into account. This approach is implemented within the rolling stock rotation optimization framework ROTOR used by DB Fernverkehr AG, one of the leading railway operators in Europe. Computational results for real world scenarios are presented to evaluate the approach.}, language = {en} } @misc{SagnolBorndoerferSchlechteetal.2014, author = {Sagnol, Guillaume and Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Swarat, Elmar}, title = {The Price of Spite in Spot-checking games}, issn = {1438-0064}, doi = {10.1007/978-3-662-44803-8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-52775}, year = {2014}, abstract = {We introduce the class of spot-checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium is more relevant for this problem, but we show that this is NP-hard. However, we give some bounds on the \emph{price of spite}, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we demonstrate the quality of these bounds for a real-world application, namely the enforcement of a truck toll on German motorways.}, language = {en} } @inproceedings{GilgKlugMartienssenetal.2017, author = {Gilg, Brady and Klug, Torsten and Martienssen, Rosemarie and Paat, Joseph and Schlechte, Thomas and Schulz, Christof and Seymen, Sinan and Tesch, Alexander}, title = {Conflict-Free Railway Track Assignment at Depots}, booktitle = {Proceedings of the IAROR conference RailLille}, year = {2017}, abstract = {Managing rolling stock with no passengers aboard is a critical component of railway operations. In particular, one problem is to park the rolling stock on a given set of tracks at the end of a day or service. Depending on the parking assignment, shunting may be required in order for a parked train to depart or for an incoming train to park. Given a collection of tracks M and a collection of trains T with fixed arrival-departure timetable, the train assignment problem (TAP) is to determine the maximum number of trains from T that can be parked on M according to the timetable and without the use of shunting. Hence, efficiently solving the TAP allows to quickly compute feasible parking schedules that do not require further shunting adjustments. In this paper, we present two integer programming models for solving the TAP. To our knowledge, this is the first integrated approach that considers track lengths along with the three most common types of parking tracks. We compare these models on a theoretical level. We also prove that a decision version of the TAP is NP-complete, justifying the use of integer programming techniques. Using stochastic and robust modelling techniques, both models produce parking assignments that are optimized and robust according to random train delays. We conclude with computational results for both models, observing that they perform well on real timetables.}, language = {en} } @misc{FischerSchlechte2017, author = {Fischer, Frank and Schlechte, Thomas}, title = {Strong Relaxations for the Train Timetabling Problem using Connected Configurations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64743}, year = {2017}, abstract = {The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral.}, language = {en} } @misc{SchadeSchlechteWitzig2017, author = {Schade, Stanley and Schlechte, Thomas and Witzig, Jakob}, title = {Structure-based Decomposition for Pattern-Detection for Railway Timetables}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64525}, year = {2017}, abstract = {We consider the problem of pattern detection in large scale railway timetables. This problem arises in rolling stock optimization planning in order to identify invariant sections of the timetable for which a cyclic rotation plan is adequate. We propose a dual reduction technique which leads to an decomposition and enumeration method. Computational results for real world instances demonstrate that the method is able to produce optimal solutions as fast as standard MIP solvers.}, language = {en} } @misc{RenkenAhmadiBorndoerferetal.2017, author = {Renken, Malte and Ahmadi, Amin and Bornd{\"o}rfer, Ralf and Sahin, Guvenc and Schlechte, Thomas}, title = {Demand-Driven Line Planning with Selfish Routing}, issn = {1438-0064}, doi = {10.1007/978-3-319-89920-6_91}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64547}, year = {2017}, abstract = {Bus rapid transit systems in developing and newly industrialized countries are often operated at the limits of passenger capacity. In particular, demand during morning and afternoon peaks is hardly or even not covered with available line plans. In order to develop demand-driven line plans, we use two mathematical models in the form of integer programming problem formulations. While the actual demand data is specified with origin-destination pairs, the arc-based model considers the demand over the arcs derived from the origin-destination demand. In order to test the accuracy of the models in terms of demand satisfaction, we simulate the optimal solutions and compare number of transfers and travel times. We also question the effect of a selfish route choice behavior which in theory results in a Braess-like paradox by increasing the number of transfers when system capacity is increased with additional lines.}, language = {en} } @misc{BrettHobergPachecoetal.2015, author = {Brett, Charles and Hoberg, Rebecca and Pacheco, Meritxell and Smith, Kyle and Bornd{\"o}rfer, Ralf and Euler, Ricardo and Gamrath, Gerwin and Grimm, Boris and Heismann, Olga and Reuther, Markus and Schlechte, Thomas and Tesch, Alexander}, title = {G-RIPS 2014 RailLab - Towards robust rolling stock rotations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53475}, year = {2015}, abstract = {The Graduate-Level Research in Industrial Projects (G-RIPS) Program provides an opportunity for high-achieving graduate-level students to work in teams on a real-world research project proposed by a sponsor from industry or the public sector. Each G-RIPS team consists of four international students (two from the US and two from European universities), an academic mentor, and an industrial sponsor. This is the report of the Rail-Lab project on the definition and integration of robustness aspects into optimizing rolling stock schedules. In general, there is a trade-off for complex systems between robustness and efficiency. The ambitious goal was to explore this trade-off by implementing numerical simulations and developing analytic models. In rolling stock planning a very large set of industrial railway requirements, such as vehicle composition, maintenance constraints, infrastructure capacity, and regularity aspects, have to be considered in an integrated model. General hypergraphs provide the modeling power to tackle those requirements. Furthermore, integer programming approaches are able to produce high quality solutions for the deterministic problem. When stochastic time delays are considered, the mathematical programming problem is much more complex and presents additional challenges. Thus, we started with a basic variant of the deterministic case, i.e., we are only considering hypergraphs representing vehicle composition and regularity. We transfered solution approaches for robust optimization from the airline industry to the setting of railways and attained a reasonable measure of robustness. Finally, we present and discuss different methods to optimize this robustness measure.}, language = {en} } @misc{FischerGrimmKlugetal.2016, author = {Fischer, Frank and Grimm, Boris and Klug, Torsten and Schlechte, Thomas}, title = {A Re-optimization Approach for Train Dispatching}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60493}, year = {2016}, abstract = {The Train Dispatching Problem (TDP) is to schedule trains through a network in a cost optimal way. Due to disturbances during operation existing track allocations often have to be re-scheduled and integrated into the timetable. This has to be done in seconds and with minimal timetable changes to guarantee smooth and conflict free operation. We present an integrated modeling approach for the re-optimization task using Mixed Integer Programming. Finally, we provide computational results for scenarios provided by the INFORMS RAS Problem Soling Competition 2012.}, language = {en} } @misc{BorndoerferKlugLamorgeseetal.2015, author = {Bornd{\"o}rfer, Ralf and Klug, Torsten and Lamorgese, Leonardo and Mannino, Carlo and Reuther, Markus and Schlechte, Thomas}, title = {Recent Success Stories on Optimization of Railway Systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53726}, year = {2015}, abstract = {Planning and operating railway transportation systems is an extremely hard task due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. Because of that, however, mathematical models and optimization techniques can result in large gains for both railway cus- tomers and operators, e.g., in terms of cost reductions or service quality improvements. In the last years a large and growing group of researchers in the OR community have devoted their attention to this domain devel- oping mathematical models and optimization approaches to tackle many of the relevant problems in the railway planning process. However, there is still a gap to bridge between theory and practice, with a few notable exceptions. In this paper we address three success stories, namely, long-term freight train routing (part I), mid-term rolling stock rotation planning (part II), and real-time train dispatching (part III). In each case, we describe real-life, successful implementations. We will dis- cuss the individual problem setting, survey the optimization literature, and focus on particular aspects addressed by the mathematical models. We demonstrate on concrete applications how mathematical optimization can support railway planning and operations. This gives proof that math- ematical optimization can support the planning of rolling stock resources. Thus, mathematical models and optimization can lead to a greater effi- ciency of railway operations and will serve as a powerful and innovative tool to meet recent challenges of the railway industry.}, language = {en} } @misc{FuegenschuhJunoszaSzaniawskiKlugetal.2015, author = {F{\"u}genschuh, Armin and Junosza-Szaniawski, Konstanty and Klug, Torsten and Kwasiborski, Slawomir and Schlechte, Thomas}, title = {Fastest, average and quantile schedule}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53592}, year = {2015}, abstract = {We consider problems concerning the scheduling of a set of trains on a single track. For every pair of trains there is a minimum headway, which every train must wait before it enters the track after another train. The speed of each train is also given. Hence for every schedule - a sequence of trains - we may compute the time that is at least needed for all trains to travel along the track in the given order. We give the solution to three problems: the fastest schedule, the average schedule, and the problem of quantile schedules. The last problem is a question about the smallest upper bound on the time of a given fraction of all possible schedules. We show how these problems are related to the travelling salesman problem. We prove NP-completeness of the fastest schedule problem, NP-hardness of quantile of schedules problem, and polynomiality of the average schedule problem. We also describe some algorithms for all three problems. In the solution of the quantile problem we give an algorithm, based on a reverse search method, generating with polynomial delay all Eulerian multigraphs with the given degree sequence and a bound on the number of such multigraphs. A better bound is left as an open question.}, language = {en} } @misc{BorndoerferSchlechte2007, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas}, title = {Models for Railway Track Allocation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9451}, number = {07-02}, year = {2007}, abstract = {This article is about the optimal track allocation problem (OPTRA) to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations: a standard formulation that models block conflicts in terms of packing constraints, and a new extended formulation that is based on additional configuration' variables. We show that the packing constraints in the standard formulation stem from an interval graph, and that they can be separated in polynomial time. It follows that the LP relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the extended formulation produces the same LP bound, and that it can also be computed with this model in polynomial time. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view, because it features a constant number of rows and is therefore amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hannover-Fulda-Kassel region of the German long distance railway network are reported.}, language = {en} } @misc{BorndoerferSchlechte2007, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas}, title = {Solving Railway Track Allocation Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9631}, number = {07-20}, year = {2007}, abstract = {The \emph{optimal track allocation problem} (\textsc{OPTRA}), also known as the train routing problem or the train timetabling problem, is to find, in a given railway network, a conflict-free set of train routes of maximum value. We propose a novel integer programming formulation for this problem that is based on additional configuration' variables. Its LP-relaxation can be solved in polynomial time. These results are the theoretical basis for a column generation algorithm to solve large-scale track allocation problems. Computational results for the Hanover-Kassel-Fulda area of the German long distance railway network involving up to 570 trains are reported.}, language = {en} } @misc{BorndoerferSchlechteSwarat2011, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Swarat, Elmar}, title = {Railway Track Allocation - Simulation, Aggregation, and Optimization}, issn = {1438-0064}, doi = {10.1007/978-3-642-27963-8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14031}, number = {11-35}, year = {2011}, abstract = {Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.}, language = {en} } @inproceedings{BorndoerferMuraSchlechte2009, author = {Bornd{\"o}rfer, Ralf and Mura, Annette and Schlechte, Thomas}, title = {Vickrey Auctions for Railway Tracks}, booktitle = {Operations Research Proceedings 2008}, editor = {Fleischmann, Bernhard and Borgwardt, Karl and Klein, Robert and Tuma, Axel}, publisher = {Springer-Verlag}, pages = {551 -- 556}, year = {2009}, language = {en} } @inproceedings{BorndoerferSchlechteWeider2010, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Weider, Steffen}, title = {Railway Track Allocation by Rapid Branching}, volume = {14}, booktitle = {Proceedings of the 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, editor = {Erlebach, Thomas and L{\"u}bbecke, Marco}, publisher = {Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik}, address = {Dagstuhl, Germany}, doi = {http://dx.doi.org/10.4230/OASIcs.ATMOS.2010.13}, pages = {13 -- 23}, year = {2010}, language = {en} } @inproceedings{BorndoerferScheltenSchlechteetal.2006, author = {Bornd{\"o}rfer, Ralf and Schelten, Uwe and Schlechte, Thomas and Weider, Steffen}, title = {A Column Generation Approach to Airline Crew Scheduling}, booktitle = {Operations Research Proceedings 2005}, editor = {Haasis, Hans-Dietrich}, publisher = {Springer-Verlag}, pages = {343 -- 348}, year = {2006}, language = {en} } @inproceedings{BorndoerferSchlechte2008, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas}, title = {Solving Railway Track Allocation Problems}, booktitle = {Operations Research Proceedings 2007}, editor = {Kalcsics, J{\"o}rg and Nickel, Stefan}, publisher = {Springer-Verlag}, pages = {117 -- 122}, year = {2008}, language = {en} }