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Documents authored by Schöbel, Anita


Document
Recoverable Robust Periodic Timetabling

Authors: Vera Grafe and Anita Schöbel

Published in: OASIcs, Volume 115, 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)


Abstract
We apply the concept of recoverable robustness to periodic timetabling, resulting in the Recoverable Robust Periodic Timetabling Problem (RRPT), which integrates periodic timetabling and delay management. Although the computed timetable is periodic, the model is able to take the aperiodicity of the delays into account. This is an important step in finding a good trade-off between short travel times and delay resistance. We present three equivalent formulations for this problem, differing in the way the timetabling subproblem is handled, and compare them in a first experimental study. We also show that our model yields solutions of high quality.

Cite as

Vera Grafe and Anita Schöbel. Recoverable Robust Periodic Timetabling. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{grafe_et_al:OASIcs.ATMOS.2023.9,
  author =	{Grafe, Vera and Sch\"{o}bel, Anita},
  title =	{{Recoverable Robust Periodic Timetabling}},
  booktitle =	{23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)},
  pages =	{9:1--9:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-302-7},
  ISSN =	{2190-6807},
  year =	{2023},
  volume =	{115},
  editor =	{Frigioni, Daniele and Schiewe, Philine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2023.9},
  URN =		{urn:nbn:de:0030-drops-187708},
  doi =		{10.4230/OASIcs.ATMOS.2023.9},
  annote =	{Keywords: Public Transport, Recoverable Robustness, Periodic Timetabling, Delay Management, Mixed Integer Programming}
}
Document
Delay Management with Integrated Decisions on the Vehicle Circulations

Authors: Vera Grafe, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 106, 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)


Abstract
The task of delay management in public transport is to decide whether a vehicle should wait for a delayed vehicle in order to maintain the connection for transferring passengers. So far, the vehicle circulations are often ignored in the optimization process, although they have an influence on the propagation of the delay through the network. In this paper we consider different ways from literature to incorporate vehicle circulations in the delay management stage of public transport planning. Since the IP formulation for the integrated problem is hard to solve, we investigate bounds and develop several heuristics for the integrated problem. Our experiments on close-to real-world instances show that integrating delay management and decisions on vehicle circulations may reduce the overall delay by up to 39 percent. We also compare the runtimes and objective function values of the different heuristics. We conclude that we can find competitive solutions in a reasonable amount of time.

Cite as

Vera Grafe, Alexander Schiewe, and Anita Schöbel. Delay Management with Integrated Decisions on the Vehicle Circulations. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{grafe_et_al:OASIcs.ATMOS.2022.7,
  author =	{Grafe, Vera and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Delay Management with Integrated Decisions on the Vehicle Circulations}},
  booktitle =	{22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)},
  pages =	{7:1--7:18},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-259-4},
  ISSN =	{2190-6807},
  year =	{2022},
  volume =	{106},
  editor =	{D'Emidio, Mattia and Lindner, Niels},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2022.7},
  URN =		{urn:nbn:de:0030-drops-171119},
  doi =		{10.4230/OASIcs.ATMOS.2022.7},
  annote =	{Keywords: Public Transport, Delay Management, Vehicle Circulations, Integer Programming}
}
Document
The Edge Investment Problem: Upgrading Transit Line Segments with Multiple Investing Parties

Authors: Rowan Hoogervorst, Evelien van der Hurk, Philine Schiewe, Anita Schöbel, and Reena Urban

Published in: OASIcs, Volume 106, 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)


Abstract
Bus Rapid Transit (BRT) systems can provide a fast and reliable service to passengers at lower costs compared to tram, metro and train systems. Therefore, they can be of great value to attract more passengers to use public transport, which is vital in reaching the Paris Agreement Targets. However, the main advantage of BRT systems, namely their flexible implementation, also leads to the risk that the system is only implemented partially to save costs. This paper focuses therefore on the Edge Investment Problem: Which edges (segments) of a bus line should be upgraded to full-level BRT? Motivated by the construction of a new BRT line around Copenhagen, we consider a setting in which multiple parties are responsible for different segments of the line. Each party has a limited budget and can adjust its investments according to the benefits provided to its passengers. We suggest two ways to determine the number of newly attracted passengers, prove that the corresponding problems are NP-hard and identify special cases that can be solved in polynomial time. In addition, problem relaxations are presented that yield dual bounds. Moreover, we perform an extensive numerical comparison in which we evaluate the extent to which these two ways of modeling demand impact the computational performance and the choice of edges to be upgraded.

Cite as

Rowan Hoogervorst, Evelien van der Hurk, Philine Schiewe, Anita Schöbel, and Reena Urban. The Edge Investment Problem: Upgrading Transit Line Segments with Multiple Investing Parties. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{hoogervorst_et_al:OASIcs.ATMOS.2022.9,
  author =	{Hoogervorst, Rowan and van der Hurk, Evelien and Schiewe, Philine and Sch\"{o}bel, Anita and Urban, Reena},
  title =	{{The Edge Investment Problem: Upgrading Transit Line Segments with Multiple Investing Parties}},
  booktitle =	{22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)},
  pages =	{9:1--9:19},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-259-4},
  ISSN =	{2190-6807},
  year =	{2022},
  volume =	{106},
  editor =	{D'Emidio, Mattia and Lindner, Niels},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2022.9},
  URN =		{urn:nbn:de:0030-drops-171137},
  doi =		{10.4230/OASIcs.ATMOS.2022.9},
  annote =	{Keywords: Network Design, Public Transport, Bus Rapid Transit, Modeling}
}
Document
Towards Improved Robustness of Public Transport by a Machine-Learned Oracle

Authors: Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 96, 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)


Abstract
The design and optimization of public transport systems is a highly complex and challenging process. Here, we focus on the trade-off between two criteria which shall make the transport system attractive for passengers: their travel time and the robustness of the system. The latter is time-consuming to evaluate. A passenger-based evaluation of robustness requires a performance simulation with respect to a large number of possible delay scenarios, making this step computationally very expensive. For optimizing the robustness, we hence apply a machine-learned oracle from previous work which approximates the robustness of a public transport system. We apply this oracle to bi-criteria optimization of integrated public transport planning (timetabling and vehicle scheduling) in two ways: First, we explore a local search based framework studying several variants of neighborhoods. Second, we evaluate a genetic algorithm. Computational experiments with artificial and close to real-word benchmark datasets yield promising results. In all cases, an existing pool of solutions (i.e., public transport plans) can be significantly improved by finding a number of new non-dominated solutions, providing better and different trade-offs between robustness and travel time.

Cite as

Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel. Towards Improved Robustness of Public Transport by a Machine-Learned Oracle. In 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021). Open Access Series in Informatics (OASIcs), Volume 96, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{mullerhannemann_et_al:OASIcs.ATMOS.2021.3,
  author =	{M\"{u}ller-Hannemann, Matthias and R\"{u}ckert, Ralf and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Towards Improved Robustness of Public Transport by a Machine-Learned Oracle}},
  booktitle =	{21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)},
  pages =	{3:1--3:20},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-213-6},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{96},
  editor =	{M\"{u}ller-Hannemann, Matthias and Perea, Federico},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2021.3},
  URN =		{urn:nbn:de:0030-drops-148721},
  doi =		{10.4230/OASIcs.ATMOS.2021.3},
  annote =	{Keywords: Public Transportation, Timetabling, Machine Learning, Robustness}
}
Document
A Phase I Simplex Method for Finding Feasible Periodic Timetables

Authors: Marc Goerigk, Anita Schöbel, and Felix Spühler

Published in: OASIcs, Volume 96, 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)


Abstract
The periodic event scheduling problem (PESP) with various applications in timetabling or traffic light scheduling is known to be challenging to solve. In general, it is already NP-hard to find a feasible solution. However, depending on the structure of the underlying network and the values of lower and upper bounds on activities, this might also be an easy task. In this paper we make use of this property and suggest phase I approaches (similar to the well-known phase I of the simplex algorithm) to find a feasible solution to PESP. Given an instance of PESP, we define an auxiliary instance for which a feasible solution can easily be constructed, and whose solution determines a feasible solution of the original instance or proves that the original instance is not feasible. We investigate different possibilities on how such an auxiliary instance can be defined theoretically and experimentally. Furthermore, in our experiments we compare different solution approaches for PESP and their behavior in the phase I approach. The results show that this approach can be especially helpful if the instance admits a feasible solution, while it is generally outperformed by classic mixed-integer programming formulations when the instance is infeasible.

Cite as

Marc Goerigk, Anita Schöbel, and Felix Spühler. A Phase I Simplex Method for Finding Feasible Periodic Timetables. In 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021). Open Access Series in Informatics (OASIcs), Volume 96, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{goerigk_et_al:OASIcs.ATMOS.2021.6,
  author =	{Goerigk, Marc and Sch\"{o}bel, Anita and Sp\"{u}hler, Felix},
  title =	{{A Phase I Simplex Method for Finding Feasible Periodic Timetables}},
  booktitle =	{21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)},
  pages =	{6:1--6:13},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-213-6},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{96},
  editor =	{M\"{u}ller-Hannemann, Matthias and Perea, Federico},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2021.6},
  URN =		{urn:nbn:de:0030-drops-148753},
  doi =		{10.4230/OASIcs.ATMOS.2021.6},
  annote =	{Keywords: train timetable optimization, periodic event scheduling problem, modulo simplex}
}
Document
Solving the Periodic Scheduling Problem: An Assignment Approach in Non-Periodic Networks

Authors: Vera Grafe and Anita Schöbel

Published in: OASIcs, Volume 96, 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)


Abstract
The periodic event scheduling problem (PESP) is a well researched problem used for finding good periodic timetables in public transport. While it is based on a periodic network consisting of events and activities which are repeated every period, we propose a new periodic timetabling model using a non-periodic network. This is a first step towards the goal of integrating periodic timetabling with other planning steps taking place in the aperiodic network, e.g. passenger assignment or delay management. In this paper, we develop the new model, show how we can reduce its size and prove its equivalence to PESP. We also conduct computational experiments on close-to real-world data from Lower Saxony, a region in northern Germany, and see that the model can be solved in a reasonable amount of time.

Cite as

Vera Grafe and Anita Schöbel. Solving the Periodic Scheduling Problem: An Assignment Approach in Non-Periodic Networks. In 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021). Open Access Series in Informatics (OASIcs), Volume 96, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{grafe_et_al:OASIcs.ATMOS.2021.9,
  author =	{Grafe, Vera and Sch\"{o}bel, Anita},
  title =	{{Solving the Periodic Scheduling Problem: An Assignment Approach in Non-Periodic Networks}},
  booktitle =	{21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)},
  pages =	{9:1--9:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-213-6},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{96},
  editor =	{M\"{u}ller-Hannemann, Matthias and Perea, Federico},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2021.9},
  URN =		{urn:nbn:de:0030-drops-148780},
  doi =		{10.4230/OASIcs.ATMOS.2021.9},
  annote =	{Keywords: Public Transport, Periodic Timetabling, PESP, Integer Programming}
}
Document
Cheapest Paths in Public Transport: Properties and Algorithms

Authors: Anita Schöbel and Reena Urban

Published in: OASIcs, Volume 85, 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)


Abstract
When determining the paths of the passengers in public transport, the travel time is usually the main criterion. However, also the ticket price a passenger has to pay is a relevant factor for choosing the path. The ticket price is also relevant for simulating the minimum income a public transport company can expect. However, finding the correct price depends on the fare system used (e.g., distance tariff, zone tariff with different particularities, application of a short-distance tariff, etc.) and may be rather complicated even if the path is already fixed. An algorithm which finds a cheapest path in a very general case has been provided in [R. Euler and R. Borndörfer, 2019], but its running time is exponential. In this paper, we model and analyze different fare systems, identify important properties they may have and provide polynomial algorithms for computing a cheapest path.

Cite as

Anita Schöbel and Reena Urban. Cheapest Paths in Public Transport: Properties and Algorithms. In 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020). Open Access Series in Informatics (OASIcs), Volume 85, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{schobel_et_al:OASIcs.ATMOS.2020.13,
  author =	{Sch\"{o}bel, Anita and Urban, Reena},
  title =	{{Cheapest Paths in Public Transport: Properties and Algorithms}},
  booktitle =	{20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)},
  pages =	{13:1--13:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-170-2},
  ISSN =	{2190-6807},
  year =	{2020},
  volume =	{85},
  editor =	{Huisman, Dennis and Zaroliagis, Christos D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2020.13},
  URN =		{urn:nbn:de:0030-drops-131499},
  doi =		{10.4230/OASIcs.ATMOS.2020.13},
  annote =	{Keywords: Public Transport, Fare Systems, Modeling, Cheapest Paths}
}
Document
The Trickle-In Effect: Modeling Passenger Behavior in Delay Management

Authors: Anita Schöbel, Julius Pätzold, and Jörg P. Müller

Published in: OASIcs, Volume 75, 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)


Abstract
Delay management is concerned with making decisions if a train should wait for passengers from delayed trains or if it should depart on time. Models for delay management exist and can be adapted to capacities of stations, capacities of tracks, or respect vehicle and driver schedules, passengers' routes and further constraints. Nevertheless, what has been neglected so far, is that a train cannot depart as planned if passengers from another train trickle in one after another such that the doors of the departing train cannot close. This effect is often observed in real-world, but has not yet been taken into account in delay management. We show the impact of this "trickle-in" effect to departure delays of trains under different conditions. We then modify existing delay management models to take the trickle-in effect into account. This can be done by forbidding certain intervals for departure. We present an integer programming formulation with these additional constraints resulting in a generalization of classic delay management models. We analyze the resulting model and identify parameters with which it can be best approximated by the classical delay management problem. Experimentally, we show that the trickle-in effect has a high impact on the overall delay of public transport systems. We discuss the impact of the trickle-in effect on the objective function value and on the computation time of the delay management problem. We also analyze the trickle-in effect for timetables which have been derived without taking this particular behavioral pattern of passengers into account.

Cite as

Anita Schöbel, Julius Pätzold, and Jörg P. Müller. The Trickle-In Effect: Modeling Passenger Behavior in Delay Management. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{schobel_et_al:OASIcs.ATMOS.2019.6,
  author =	{Sch\"{o}bel, Anita and P\"{a}tzold, Julius and M\"{u}ller, J\"{o}rg P.},
  title =	{{The Trickle-In Effect: Modeling Passenger Behavior in Delay Management}},
  booktitle =	{19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)},
  pages =	{6:1--6:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-128-3},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{75},
  editor =	{Cacchiani, Valentina and Marchetti-Spaccamela, Alberto},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2019.6},
  URN =		{urn:nbn:de:0030-drops-114187},
  doi =		{10.4230/OASIcs.ATMOS.2019.6},
  annote =	{Keywords: Public Transport Planning, Delay Management, Integer Programming}
}
Document
Robustness as a Third Dimension for Evaluating Public Transport Plans

Authors: Markus Friedrich, Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 65, 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)


Abstract
Providing attractive and efficient public transport services is of crucial importance due to higher demands for mobility and the need to reduce air pollution and to save energy. The classical planning process in public transport tries to achieve a reasonable compromise between service quality for passengers and operating costs. Service quality mostly considers quantities like average travel time and number of transfers. Since daily public transport inevitably suffers from delays caused by random disturbances and disruptions, robustness also plays a crucial role. While there are recent attempts to achieve delay-resistant timetables, comparably little work has been done to systematically assess and to compare the robustness of transport plans from a passenger point of view. We here provide a general and flexible framework for evaluating public transport plans (lines, timetables, and vehicle schedules) in various ways. It enables planners to explore several trade-offs between operating costs, service quality (average perceived travel time of passengers), and robustness against delays. For such an assessment we develop several passenger-oriented robustness tests which can be instantiated with parameterized delay scenarios. Important features of our framework include detailed passenger flow models, delay propagation schemes and disposition strategies, rerouting strategies as well as vehicle capacities. To demonstrate possible use cases, our framework has been applied to a variety of public transport plans which have been created for the same given demand for an artificial urban grid network and to instances for long-distance train networks. As one application we study the impact of different strategies to improve the robustness of timetables by insertion of supplement times. We also show that the framework can be used to optimize waiting strategies in delay management.

Cite as

Markus Friedrich, Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel. Robustness as a Third Dimension for Evaluating Public Transport Plans. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018). Open Access Series in Informatics (OASIcs), Volume 65, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{friedrich_et_al:OASIcs.ATMOS.2018.4,
  author =	{Friedrich, Markus and M\"{u}ller-Hannemann, Matthias and R\"{u}ckert, Ralf and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Robustness as a Third Dimension for Evaluating Public Transport Plans}},
  booktitle =	{18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)},
  pages =	{4:1--4:17},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-096-5},
  ISSN =	{2190-6807},
  year =	{2018},
  volume =	{65},
  editor =	{Bornd\"{o}rfer, Ralf and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2018.4},
  URN =		{urn:nbn:de:0030-drops-97097},
  doi =		{10.4230/OASIcs.ATMOS.2018.4},
  annote =	{Keywords: robustness, timetabling, vehicle schedules, delays}
}
Document
Cost-Minimal Public Transport Planning

Authors: Julius Pätzold, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 65, 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)


Abstract
In this paper we discuss what a cost-optimal public transport plan looks like, i.e., we determine a line plan, a timetable and a vehicle schedule which can be operated with minimal costs while, at the same time, allowing all passengers to travel between their origins and destinations. We are hereby interested in an exact solution of the integrated problem. In contrast to a passenger-optimal transport plan, in which there is a direct connection for every origin-destination pair, the structure or model for determining a cost-optimal transport plan is not obvious and has not been researched so far. We present three models which differ with respect to the structures we are looking for. If lines are directed and may contain circles, we prove that a cost-optimal schedule can (under weak assumptions) already be obtained by first distributing the passengers in a cost-optimal way. We are able to streamline the resulting integer program such that it can be applied to real-world instances. The model gives bounds for the general case. In the second model we look for lines operated in both directions, but allow only simplified vehicle schedules. This model then yields stronger bounds than the first one. Our most realistic model looks for lines operated in both directions, and allows all structures for the vehicle schedules. This model, however, is only computable for small instances. Finally, the results of the three models and their respective bounds are compared experimentally.

Cite as

Julius Pätzold, Alexander Schiewe, and Anita Schöbel. Cost-Minimal Public Transport Planning. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018). Open Access Series in Informatics (OASIcs), Volume 65, pp. 8:1-8:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{patzold_et_al:OASIcs.ATMOS.2018.8,
  author =	{P\"{a}tzold, Julius and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Cost-Minimal Public Transport Planning}},
  booktitle =	{18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)},
  pages =	{8:1--8:22},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-096-5},
  ISSN =	{2190-6807},
  year =	{2018},
  volume =	{65},
  editor =	{Bornd\"{o}rfer, Ralf and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2018.8},
  URN =		{urn:nbn:de:0030-drops-97138},
  doi =		{10.4230/OASIcs.ATMOS.2018.8},
  annote =	{Keywords: Public Transport Planning, Integer Optimization, Line Planning, Vehicle Scheduling}
}
Document
Integrating Passengers' Assignment in Cost-Optimal Line Planning

Authors: Markus Friedrich, Maximilian Hartl, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 59, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)


Abstract
Finding a line plan with corresponding frequencies is an mportant stage of planning a public transport system. A line plan should permit all passengers to travel with an appropriate quality at appropriate costs for the public transport operator. Traditional line planning procedures proceed sequentially: In a first step a traffic assignment allocates passengers to routes in the network, often by means of a shortest path assignment. The resulting traffic loads are used in a second step to determine a cost-optimal line concept. It is well known that travel time of the resulting line concept depends on the traffic assignment. In this paper we investigate the impact of the assignment on the operating costs of the line concept. We show that the traffic assignment has significant influence on the costs even if all passengers are routed on shortest paths. We formulate an integrated model and analyze the error we can make by using the traditional approach and solve it sequentially. We give bounds on the error in special cases. We furthermore investigate and enhance three heuristics for finding an initial passengers’ assignment and compare the resulting line concepts in terms of operating costs and passengers’ travel time. It turns out that the costs of a line concept can be reduced significantly if passengers are not necessarily routed on shortest paths and that it is beneficial for the travel time and the costs to include knowledge on the line pool already in the assignment step.

Cite as

Markus Friedrich, Maximilian Hartl, Alexander Schiewe, and Anita Schöbel. Integrating Passengers' Assignment in Cost-Optimal Line Planning. In 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017). Open Access Series in Informatics (OASIcs), Volume 59, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{friedrich_et_al:OASIcs.ATMOS.2017.5,
  author =	{Friedrich, Markus and Hartl, Maximilian and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Integrating Passengers' Assignment in Cost-Optimal Line Planning}},
  booktitle =	{17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)},
  pages =	{5:1--5:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-042-2},
  ISSN =	{2190-6807},
  year =	{2017},
  volume =	{59},
  editor =	{D'Angelo, Gianlorenzo and Dollevoet, Twan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2017.5},
  URN =		{urn:nbn:de:0030-drops-79015},
  doi =		{10.4230/OASIcs.ATMOS.2017.5},
  annote =	{Keywords: Line Planning, Integrated Public Transport Planning, Integer Programming, Passengers' Routes}
}
Document
Robustness Tests for Public Transport Planning

Authors: Markus Friedrich, Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 59, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)


Abstract
The classical planning process in public transport planning focuses on the two criteria operating costs and quality for passengers. Quality mostly considers quantities like average travel time and number of transfers. Since public transport often suffers from delays caused by random disturbances, we are interested in adding a third dimension: robustness. We propose passenger-oriented robustness indicators for public transport networks and timetables. These robustness indicators are evaluated for several public transport plans which have been created for an artificial urban network with the same demand. The study shows that these indicators are suitable to measure the robustness of a line plan and a timetable. We explore different trade-offs between operating costs, quality (average travel time of passengers), and robustness against delays. Our results show that the proposed robustness indicators give reasonable results.

Cite as

Markus Friedrich, Matthias Müller-Hannemann, Ralf Rückert, Alexander Schiewe, and Anita Schöbel. Robustness Tests for Public Transport Planning. In 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017). Open Access Series in Informatics (OASIcs), Volume 59, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{friedrich_et_al:OASIcs.ATMOS.2017.6,
  author =	{Friedrich, Markus and M\"{u}ller-Hannemann, Matthias and R\"{u}ckert, Ralf and Schiewe, Alexander and Sch\"{o}bel, Anita},
  title =	{{Robustness Tests for Public Transport Planning}},
  booktitle =	{17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)},
  pages =	{6:1--6:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-042-2},
  ISSN =	{2190-6807},
  year =	{2017},
  volume =	{59},
  editor =	{D'Angelo, Gianlorenzo and Dollevoet, Twan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2017.6},
  URN =		{urn:nbn:de:0030-drops-78904},
  doi =		{10.4230/OASIcs.ATMOS.2017.6},
  annote =	{Keywords: robustness measure, timetabling, line planning, delays, passenger-orientation}
}
Document
Look-Ahead Approaches for Integrated Planning in Public Transportation

Authors: Julius Pätzold, Alexander Schiewe, Philine Schiewe, and Anita Schöbel

Published in: OASIcs, Volume 59, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)


Abstract
In this paper we deal with three consecutive planning stages in public transportation: Line planning (including line pool generation), timetabling, and vehicle scheduling. These three steps are traditionally performed one after another in a sequential way often leading to high costs in the (last) vehicle scheduling stage. In this paper we propose three different ways to "look ahead", i.e., to include aspects of vehicle scheduling already earlier in the sequential process: an adapted line pool generation algorithm, a new cost structure for line planning, and a reordering of the sequential planning stages. We analyze these enhancements experimentally and show that they can be used to decrease the costs significantly.

Cite as

Julius Pätzold, Alexander Schiewe, Philine Schiewe, and Anita Schöbel. Look-Ahead Approaches for Integrated Planning in Public Transportation. In 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017). Open Access Series in Informatics (OASIcs), Volume 59, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{patzold_et_al:OASIcs.ATMOS.2017.17,
  author =	{P\"{a}tzold, Julius and Schiewe, Alexander and Schiewe, Philine and Sch\"{o}bel, Anita},
  title =	{{Look-Ahead Approaches for Integrated Planning in Public Transportation}},
  booktitle =	{17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)},
  pages =	{17:1--17:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-042-2},
  ISSN =	{2190-6807},
  year =	{2017},
  volume =	{59},
  editor =	{D'Angelo, Gianlorenzo and Dollevoet, Twan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2017.17},
  URN =		{urn:nbn:de:0030-drops-78944},
  doi =		{10.4230/OASIcs.ATMOS.2017.17},
  annote =	{Keywords: line pool generation, line planning, vehicle scheduling, integrated planning, public transport}
}
Document
Algorithmic Methods for Optimization in Public Transport (Dagstuhl Seminar 16171)

Authors: Leo G. Kroon, Anita Schöbel, and Dorothea Wagner

Published in: Dagstuhl Reports, Volume 6, Issue 4 (2016)


Abstract
This report documents the talks and discussions at the Dagstuhl seminar 16171 “Algorithmic Methods for Optimization in Public Transport”. The seminar brought together researchers from algorithm, algorithm engineering, operations research, mathematical optimization and engineering, all interested in algorithms in public transportation. Also several practitioners were able to join the group and brought valuable insights on current practice and challenging problems.

Cite as

Leo G. Kroon, Anita Schöbel, and Dorothea Wagner. Algorithmic Methods for Optimization in Public Transport (Dagstuhl Seminar 16171). In Dagstuhl Reports, Volume 6, Issue 4, pp. 139-160, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@Article{kroon_et_al:DagRep.6.4.139,
  author =	{Kroon, Leo G. and Sch\"{o}bel, Anita and Wagner, Dorothea},
  title =	{{Algorithmic Methods for Optimization in Public Transport (Dagstuhl Seminar 16171)}},
  pages =	{139--160},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2016},
  volume =	{6},
  number =	{4},
  editor =	{Kroon, Leo G. and Sch\"{o}bel, Anita and Wagner, Dorothea},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.6.4.139},
  URN =		{urn:nbn:de:0030-drops-66949},
  doi =		{10.4230/DagRep.6.4.139},
  annote =	{Keywords: delay and disruption management, dynamic passenger information, public transportation, resource scheduling, timetabling}
}
Document
A Matching Approach for Periodic Timetabling

Authors: Julius Pätzold and Anita Schöbel

Published in: OASIcs, Volume 54, 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)


Abstract
The periodic event scheduling problem (PESP) is a well studied problem known as intrinsically hard, but with important applications mainly for finding good timetables in public transportation. In this paper we consider PESP in public transportation, but in a reduced version (r-PESP) in which the driving and waiting times of the vehicles are fixed to their lower bounds. This results in a still NP-hard problem which has less variables, since only one variable determines the schedule for a whole line. We propose a formulation for r-PESP which is based on scheduling the lines. This enables us on the one hand to identify a finite candidate set and an exact solution approach. On the other hand, we use this formulation to derive a matching-based heuristic for solving PESP. Our experiments on close to real-world instances from LinTim show that our heuristic is able to compute competitive timetables in a very short runtime.

Cite as

Julius Pätzold and Anita Schöbel. A Matching Approach for Periodic Timetabling. In 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016). Open Access Series in Informatics (OASIcs), Volume 54, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{patzold_et_al:OASIcs.ATMOS.2016.1,
  author =	{P\"{a}tzold, Julius and Sch\"{o}bel, Anita},
  title =	{{A Matching Approach for Periodic Timetabling}},
  booktitle =	{16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)},
  pages =	{1:1--1:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-021-7},
  ISSN =	{2190-6807},
  year =	{2016},
  volume =	{54},
  editor =	{Goerigk, Marc and Werneck, Renato F.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2016.1},
  URN =		{urn:nbn:de:0030-drops-65251},
  doi =		{10.4230/OASIcs.ATMOS.2016.1},
  annote =	{Keywords: PESP, Timetabling, Public Transport, Matching, Finite Dominating Set}
}
Document
Integrating Passengers' Routes in Periodic Timetabling: A SAT approach

Authors: Philine Gattermann, Peter Großmann, Karl Nachtigall, and Anita Schöbel

Published in: OASIcs, Volume 54, 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)


Abstract
The periodic event scheduling problem (PESP) is a well studied problem known as intrinsically hard. Its main application is for designing periodic timetables in public transportation. To this end, the passengers' paths are required as input data. This is a drawback since the final paths which are used by the passengers depend on the timetable to be designed. Including the passengers' routing in the PESP hence improves the quality of the resulting timetables. However, this makes PESP even harder. Formulating the PESP as satisfiability problem and using SAT solvers for its solution has been shown to be a highly promising approach. The goal of this paper is to exploit if SAT solvers can also be used for the problem of integrated timetabling and passenger routing. In our model of the integrated problem we distribute origin-destination (OD) pairs temporally through the network by using time-slices in order to make the resulting model more realistic. We present a formulation of this integrated problem as integer program which we are able to transform to a satisfiability problem. We tested the latter formulation within numerical experiments, which are performed on Germany's long-distance passenger railway network. The computation's analysis in which we compare the integrated approach with the traditional one with fixed passengers' weights, show promising results for future scientific investigations.

Cite as

Philine Gattermann, Peter Großmann, Karl Nachtigall, and Anita Schöbel. Integrating Passengers' Routes in Periodic Timetabling: A SAT approach. In 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016). Open Access Series in Informatics (OASIcs), Volume 54, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{gattermann_et_al:OASIcs.ATMOS.2016.3,
  author =	{Gattermann, Philine and Gro{\ss}mann, Peter and Nachtigall, Karl and Sch\"{o}bel, Anita},
  title =	{{Integrating Passengers' Routes in Periodic Timetabling: A SAT approach}},
  booktitle =	{16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)},
  pages =	{3:1--3:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-021-7},
  ISSN =	{2190-6807},
  year =	{2016},
  volume =	{54},
  editor =	{Goerigk, Marc and Werneck, Renato F.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2016.3},
  URN =		{urn:nbn:de:0030-drops-65279},
  doi =		{10.4230/OASIcs.ATMOS.2016.3},
  annote =	{Keywords: PESP, Timetabling, Public Transport, Passengers' Routes, SAT}
}
Document
Recoverable Robust Timetable Information

Authors: Marc Goerigk, Sacha Heße, Matthias Müller-Hannemann, Marie Schmidt, and Anita Schöbel

Published in: OASIcs, Volume 33, 13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2013)


Abstract
Timetable information is the process of determining a suitable travel route for a passenger. Due to delays in the original timetable, in practice it often happens that the travel route cannot be used as originally planned. For a passenger being already en route, it would hence be useful to know about alternatives that ensure that his/her destination can be reached. In this work we propose a recoverable robust approach to timetable information; i.e., we aim at finding travel routes that can easily be updated when delays occur during the journey. We present polynomial-time algorithms for this problem and evaluate the performance of the routes obtained this way on schedule data of the German train network of 2013 and simulated delay scenarios.

Cite as

Marc Goerigk, Sacha Heße, Matthias Müller-Hannemann, Marie Schmidt, and Anita Schöbel. Recoverable Robust Timetable Information. In 13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 33, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{goerigk_et_al:OASIcs.ATMOS.2013.1,
  author =	{Goerigk, Marc and He{\ss}e, Sacha and M\"{u}ller-Hannemann, Matthias and Schmidt, Marie and Sch\"{o}bel, Anita},
  title =	{{Recoverable Robust Timetable Information}},
  booktitle =	{13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{1--14},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-58-3},
  ISSN =	{2190-6807},
  year =	{2013},
  volume =	{33},
  editor =	{Frigioni, Daniele and Stiller, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2013.1},
  URN =		{urn:nbn:de:0030-drops-42407},
  doi =		{10.4230/OASIcs.ATMOS.2013.1},
  annote =	{Keywords: timetable information, recoverable robustness, delay scenarios}
}
Document
The Stop Location Problem with Realistic Traveling Time

Authors: Emilio Carrizosa, Jonas Harbering, and Anita Schöbel

Published in: OASIcs, Volume 33, 13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2013)


Abstract
In this paper we consider the location of stops along the edges of an already existing public transportation network. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The positive effect of new stops is given by the better access of the customers to the public transport network, while the traveling time increases due to the additional stopping activities of the trains which is a negative effect for the customers. Our goal is to locate new stops minimizing a realistic traveling time which takes acceleration and deceleration of the vehicles into account. We distinguish two variants: in the first (academic) version we locate $p$ stops, in the second (real-world applicable) version the goal is to cover all demand points with a minimal amount of realistic traveling time. As in other works on stop location, covering may be defined with respect to an arbitrary norm. For the first version, we present a polynomial approach while the latter version is NP-hard. We derive a finite candidate set and an IP formulation. We discuss the differences to the model neglecting the realistic traveling time and provide a case study showing that our procedures are applicable in practice and do save in average more than 3% of traveling time for the passengers.

Cite as

Emilio Carrizosa, Jonas Harbering, and Anita Schöbel. The Stop Location Problem with Realistic Traveling Time. In 13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 33, pp. 80-93, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{carrizosa_et_al:OASIcs.ATMOS.2013.80,
  author =	{Carrizosa, Emilio and Harbering, Jonas and Sch\"{o}bel, Anita},
  title =	{{The Stop Location Problem with Realistic Traveling Time}},
  booktitle =	{13th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{80--93},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-58-3},
  ISSN =	{2190-6807},
  year =	{2013},
  volume =	{33},
  editor =	{Frigioni, Daniele and Stiller, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2013.80},
  URN =		{urn:nbn:de:0030-drops-42467},
  doi =		{10.4230/OASIcs.ATMOS.2013.80},
  annote =	{Keywords: Stop Location, Realistic Traveling Time, IP Formulation}
}
Document
The Price of Robustness in Timetable Information

Authors: Marc Goerigk, Martin Knoth, Matthias Müller-Hannemann, Marie Schmidt, and Anita Schöbel

Published in: OASIcs, Volume 20, 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2011)


Abstract
In timetable information in public transport the goal is to search for a good passenger's path between an origin and a destination. Usually, the travel time and the number of transfers shall be minimized. In this paper, we consider robust timetable information, i.e. we want to identify a path which will bring the passenger to the planned destination even in the case of delays. The classic notion of strict robustness leads to the problem of identifying those changing activities which will never break in any of the expected delay scenarios. We show that this is in general a strongly NP-hard problem. Therefore, we propose a conservative heuristic which identifies a large subset of these robust changing activities in polynomial time by dynamic programming and so allows us to find strictly robust paths efficiently. We also transfer the notion of light robustness, originally introduced for timetabling, to timetable information. In computational experiments we then study the price of strict and light robustness: How much longer is the travel time of a robust path than of a shortest one according to the published schedule? Based on the schedule of high-speed trains within Germany of 2011, we quantitatively explore the trade-off between the level of guaranteed robustness and the increase in travel time. Strict robustness turns out to be too conservative, while light robustness is promising: a modest level of guarantees is achievable at a reasonable price for the majority of passengers.

Cite as

Marc Goerigk, Martin Knoth, Matthias Müller-Hannemann, Marie Schmidt, and Anita Schöbel. The Price of Robustness in Timetable Information. In 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 20, pp. 76-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{goerigk_et_al:OASIcs.ATMOS.2011.76,
  author =	{Goerigk, Marc and Knoth, Martin and M\"{u}ller-Hannemann, Matthias and Schmidt, Marie and Sch\"{o}bel, Anita},
  title =	{{The Price of Robustness in Timetable Information}},
  booktitle =	{11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{76--87},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-33-0},
  ISSN =	{2190-6807},
  year =	{2011},
  volume =	{20},
  editor =	{Caprara, Alberto and Kontogiannis, Spyros},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2011.76},
  URN =		{urn:nbn:de:0030-drops-32680},
  doi =		{10.4230/OASIcs.ATMOS.2011.76},
  annote =	{Keywords: strict and light robustness, delay scenarios, experimental study}
}
Document
Delay Management including Capacities of Stations

Authors: Twan Dollevoet, Marie Schmidt, and Anita Schöbel

Published in: OASIcs, Volume 20, 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2011)


Abstract
The question of delay management (DM) is whether trains should wait for delayed feeder trains or should depart on time. Solutions to this problem strongly depend on the capacity constraints of the tracks making sure that no two trains can use the same piece of track at the same time. While these capacity constraints have been included in integer programming formulations for DM, the capacity constraints of the stations (only offering a limited number of platforms) have been neglected so far. This can lead to highly infeasible solutions. In order to overcome this problem we suggest two new formulations for DM both including the stations' capacities. We present numerical results showing that the assignment-based formulation is clearly superior to the packing formulation. We furthermore propose an iterative algorithm in which we improve the platform assignment with respect to the current delays of the trains at each station in each step. We will show that this subproblem asks for coloring the nodes of a graph with a given number of colors while minimizing the weight of the conflicts. We show that the graph to be colored is an interval graph and that the problem can be solved in polynomial time by presenting a totally unimodular IP formulation.

Cite as

Twan Dollevoet, Marie Schmidt, and Anita Schöbel. Delay Management including Capacities of Stations. In 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 20, pp. 88-99, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{dollevoet_et_al:OASIcs.ATMOS.2011.88,
  author =	{Dollevoet, Twan and Schmidt, Marie and Sch\"{o}bel, Anita},
  title =	{{Delay Management including Capacities of Stations}},
  booktitle =	{11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{88--99},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-33-0},
  ISSN =	{2190-6807},
  year =	{2011},
  volume =	{20},
  editor =	{Caprara, Alberto and Kontogiannis, Spyros},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2011.88},
  URN =		{urn:nbn:de:0030-drops-32699},
  doi =		{10.4230/OASIcs.ATMOS.2011.88},
  annote =	{Keywords: Delay management, station capacities}
}
Document
Vertex Disjoint Paths for Dispatching in Railways

Authors: Holger Flier, Matús Mihalák, Anita Schöbel, Peter Widmayer, and Anna Zych

Published in: OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)


Abstract
We study variants of the vertex disjoint paths problem in planar graphs where paths have to be selected from a given set of paths. We study the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are NP-hard in planar graphs, restrictions on the location of the terminals, motivated by railway applications, lead to polynomially solvable cases for the decision and maximization versions of the problem, and to a $p$-approximation algorithm for the routing-in-rounds problem, where $p$ is the maximum number of alternative paths for a terminal pair.

Cite as

Holger Flier, Matús Mihalák, Anita Schöbel, Peter Widmayer, and Anna Zych. Vertex Disjoint Paths for Dispatching in Railways. In 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, pp. 61-73, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{flier_et_al:OASIcs.ATMOS.2010.61,
  author =	{Flier, Holger and Mihal\'{a}k, Mat\'{u}s and Sch\"{o}bel, Anita and Widmayer, Peter and Zych, Anna},
  title =	{{Vertex Disjoint Paths for Dispatching in Railways}},
  booktitle =	{10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)},
  pages =	{61--73},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-20-0},
  ISSN =	{2190-6807},
  year =	{2010},
  volume =	{14},
  editor =	{Erlebach, Thomas and L\"{u}bbecke, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010.61},
  URN =		{urn:nbn:de:0030-drops-27508},
  doi =		{10.4230/OASIcs.ATMOS.2010.61},
  annote =	{Keywords: algorithms, approximation, complexity, graph theory, railways, routing, transportation}
}
Document
An Empirical Analysis of Robustness Concepts for Timetabling

Authors: Marc Goerigk and Anita Schöbel

Published in: OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)


Abstract
Calculating timetables that are insensitive to disturbances has drawn considerable research efforts due to its practical importance on the one hand and its hard tractability by classical robustness concepts on the other hand. Many different robustness concepts for timetabling have been suggested in the literature, some of them very recently. In this paper we compare such concepts on real-world instances. We also introduce a new approach that is generically applicable to any robustness problem. Nevertheless it is able to adapt the special characteristics of the respective problem structure and hence generates solutions that fit to the needs of the respective problem.

Cite as

Marc Goerigk and Anita Schöbel. An Empirical Analysis of Robustness Concepts for Timetabling. In 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, pp. 100-113, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{goerigk_et_al:OASIcs.ATMOS.2010.100,
  author =	{Goerigk, Marc and Sch\"{o}bel, Anita},
  title =	{{An Empirical Analysis of Robustness Concepts for Timetabling}},
  booktitle =	{10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)},
  pages =	{100--113},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-20-0},
  ISSN =	{2190-6807},
  year =	{2010},
  volume =	{14},
  editor =	{Erlebach, Thomas and L\"{u}bbecke, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010.100},
  URN =		{urn:nbn:de:0030-drops-27537},
  doi =		{10.4230/OASIcs.ATMOS.2010.100},
  annote =	{Keywords: Timetabling, Robust Optimization, Algorithm Engineering}
}
Document
The Complexity of Integrating Routing Decisions in Public Transportation Models

Authors: Marie Schmidt and Anita Schöbel

Published in: OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)


Abstract
To model and solve optimization problems arising in public transportation, data about the passengers is necessary and has to be included in the models in any phase of the planning process. Many approaches assume a two-step procedure: in a first step, the data about the passengers is distributed over the public transportation network using traffic-assignment procedures. In a second step, the actual planning of lines, timetables, etc. takes place. This approach ignores that for most passengers there are many possible ways to reach their destinations in the public transportation network, thus the actual connections the passengers will take depend strongly on the decisions made during the planning phase. In this paper we investigate the influence of integrating the traffic assignment procedure in the optimization process on the complexity of line planning and aperiodic timetabling. In both problems, our objective is to maximize the passengers' benefit, namely to minimize the overall travel time of the passengers in the network. We present new models, analyze NP-hardness results arising from the integration of the routing decisions in the traditional models, and derive polynomial algorithms for special cases.

Cite as

Marie Schmidt and Anita Schöbel. The Complexity of Integrating Routing Decisions in Public Transportation Models. In 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, pp. 156-169, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{schmidt_et_al:OASIcs.ATMOS.2010.156,
  author =	{Schmidt, Marie and Sch\"{o}bel, Anita},
  title =	{{The Complexity of Integrating Routing Decisions in Public Transportation Models}},
  booktitle =	{10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)},
  pages =	{156--169},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-20-0},
  ISSN =	{2190-6807},
  year =	{2010},
  volume =	{14},
  editor =	{Erlebach, Thomas and L\"{u}bbecke, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010.156},
  URN =		{urn:nbn:de:0030-drops-27575},
  doi =		{10.4230/OASIcs.ATMOS.2010.156},
  annote =	{Keywords: Line Planning, Timetabling, Routing}
}
Document
Dynamic Algorithms for Recoverable Robustness Problems

Authors: Serafino Cicerone, Gabriele Di Stefano, Michael Schachtebeck, and Anita Schöbel

Published in: OASIcs, Volume 9, 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08) (2008)


Abstract
Recently, the recoverable robustness model has been introduced in the optimization area. This model allows to consider disruptions (input data changes) in a unified way, that is, during both the strategic planning phase and the operational phase. Although the model represents a significant improvement, it has the following drawback: we are typically not facing only one disruption, but many of them might appear one after another. In this case, the solutions provided in the context of the recoverable robustness are not satisfying. In this paper we extend the concept of recoverable robustness to deal not only with one single recovery step, but with arbitrarily many recovery steps. To this aim, we introduce the notion of dynamic recoverable robustness problems. We apply the new model in the context of timetabling and delay management problems. We are interested in finding efficient dynamic robust algorithms for solving the timetabling problem and in evaluating the price of robustness of the proposed solutions.

Cite as

Serafino Cicerone, Gabriele Di Stefano, Michael Schachtebeck, and Anita Schöbel. Dynamic Algorithms for Recoverable Robustness Problems. In 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08). Open Access Series in Informatics (OASIcs), Volume 9, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{cicerone_et_al:OASIcs.ATMOS.2008.1587,
  author =	{Cicerone, Serafino and Di Stefano, Gabriele and Schachtebeck, Michael and Sch\"{o}bel, Anita},
  title =	{{Dynamic Algorithms for Recoverable Robustness Problems}},
  booktitle =	{8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08)},
  pages =	{1--20},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-07-1},
  ISSN =	{2190-6807},
  year =	{2008},
  volume =	{9},
  editor =	{Fischetti, Matteo and Widmayer, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2008.1587},
  URN =		{urn:nbn:de:0030-drops-15876},
  doi =		{10.4230/OASIcs.ATMOS.2008.1587},
  annote =	{Keywords: Robustness, optimization problems, dynamic algorithms, timetabling, delay management}
}
Document
IP-based Techniques for Delay Management with Priority Decisions

Authors: Michael Schachtebeck and Anita Schöbel

Published in: OASIcs, Volume 9, 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08) (2008)


Abstract
Delay management is an important issue in the daily operations of any railway company. The task is to update the planned timetable to a disposition timetable in such a way that the inconvenience for the passengers is as small as possible. The two main decisions that have to be made in this respect are the wait-depart decisions to decide which connections should be maintained in case of delays and the priority decisions that determine the order in which trains are allowed to pass a specific piece of track. They later are necessary in the capacitated case due to the limited capacity of the track system and are crucial to ensure that the headways between different trains are respected and that single-track traffic is routed correctly. While the wait-depart decisions have been intensively studied in literature (e.g. [Sch06,Gat07]), the priority decisions in the capacitated case have been neglected so far in delay management optimization models. In the current paper, we add the priority decisions to the integer programming formulation of the delay management problem and are hence able to deal with the capacitated case. Unfortunately, these constraints are disjunctive constraints that make the resulting event activity network more dense and destroy the property that it does not contain any directed cycle. Nevertheless, we are able to derive reduction techniques for the network which enable us to extend the formulation of the never-meet property from the uncapacitated delay management problem to the capacitated case. We then use our results to derive exact and heuristic solution procedures for solving the delay management problem. The results of the algorithms are evaluated both from a theoretical and a numerical point of view. The latter has been done within a case study using the railway network in the region of Harz, Germany.

Cite as

Michael Schachtebeck and Anita Schöbel. IP-based Techniques for Delay Management with Priority Decisions. In 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08). Open Access Series in Informatics (OASIcs), Volume 9, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{schachtebeck_et_al:OASIcs.ATMOS.2008.1586,
  author =	{Schachtebeck, Michael and Sch\"{o}bel, Anita},
  title =	{{IP-based Techniques for Delay Management with Priority Decisions}},
  booktitle =	{8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08)},
  pages =	{1--16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-07-1},
  ISSN =	{2190-6807},
  year =	{2008},
  volume =	{9},
  editor =	{Fischetti, Matteo and Widmayer, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2008.1586},
  URN =		{urn:nbn:de:0030-drops-15862},
  doi =		{10.4230/OASIcs.ATMOS.2008.1586},
  annote =	{Keywords: Public transportation, delay, integer programming, never-meet property, heuristics, preprocessing}
}
Document
A Game-Theoretic Approach to Line Planning

Authors: Anita Schöbel and Silvia Schwarze

Published in: OASIcs, Volume 5, 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06) (2006)


Abstract
We present a game-theoretic model for the line planning problem in public transportation, in which each line acts as player and aims to minimize a cost function which is related to the traffic along its edges. We analyze the model and in particular show that a potential function exists. Based on this result, we present a method for calculating equilibria and present first numerical results using the railway network of {it Deutsche Bahn}.

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Anita Schöbel and Silvia Schwarze. A Game-Theoretic Approach to Line Planning. In 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06). Open Access Series in Informatics (OASIcs), Volume 5, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)


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@InProceedings{schobel_et_al:OASIcs.ATMOS.2006.688,
  author =	{Sch\"{o}bel, Anita and Schwarze, Silvia},
  title =	{{A Game-Theoretic Approach to Line Planning}},
  booktitle =	{6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06)},
  pages =	{1--16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-01-9},
  ISSN =	{2190-6807},
  year =	{2006},
  volume =	{5},
  editor =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2006.688},
  URN =		{urn:nbn:de:0030-drops-6887},
  doi =		{10.4230/OASIcs.ATMOS.2006.688},
  annote =	{Keywords: Line Planning, Network Game, Equilibrium}
}
Document
Line Planning with Minimal Traveling Time

Authors: Anita Schöbel and Susanne Scholl

Published in: OASIcs, Volume 2, 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05) (2006)


Abstract
An important strategic element in the planning process of public transportation is the development of a line concept, i.e. to find a set of paths for operating lines on them. So far, most of the models in the literature aim to minimize the costs or to maximize the number of direct travelers. In this paper we present a new approach minimizing the travel times over all customers including penalties for the transfers needed. This approach maximizes the comfort of the passengers and will make the resulting timetable more reliable. To tackle our problem we present integer programming models and suggest a solution approach using Dantzig-Wolfe decomposition for solving the LP-relaxation. Numerical results of real-world instances are presented.

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Anita Schöbel and Susanne Scholl. Line Planning with Minimal Traveling Time. In 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05). Open Access Series in Informatics (OASIcs), Volume 2, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)


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@InProceedings{schobel_et_al:OASIcs.ATMOS.2005.660,
  author =	{Sch\"{o}bel, Anita and Scholl, Susanne},
  title =	{{Line Planning with Minimal Traveling Time}},
  booktitle =	{5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05)},
  pages =	{1--16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-00-2},
  ISSN =	{2190-6807},
  year =	{2006},
  volume =	{2},
  editor =	{Kroon, Leo G. and M\"{o}hring, Rolf H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2005.660},
  URN =		{urn:nbn:de:0030-drops-6601},
  doi =		{10.4230/OASIcs.ATMOS.2005.660},
  annote =	{Keywords: Line planning, real-world problem, integer programming, Dantzig-Wolfe decomposition}
}
Document
Station Location - Complexity and Approximation

Authors: Steffen Mecke, Anita Schöbel, and Dorothea Wagner

Published in: OASIcs, Volume 2, 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05) (2006)


Abstract
We consider a geometric set covering problem. In its original form it consists of adding stations to an existing geometric transportation network so that each of a given set of settlements is not too far from a station. The problem is known to be NP-hard in general. However, special cases with certain properties have been shown to be efficiently solvable in theory and in practice, especially if the covering matrix has (almost) consecutive ones property. In this paper we are narrowing the gap between intractable and efficiently solvable cases of the problem. We also present an approximation algorithm for cases with almost consecutive ones property.

Cite as

Steffen Mecke, Anita Schöbel, and Dorothea Wagner. Station Location - Complexity and Approximation. In 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05). Open Access Series in Informatics (OASIcs), Volume 2, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)


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@InProceedings{mecke_et_al:OASIcs.ATMOS.2005.661,
  author =	{Mecke, Steffen and Sch\"{o}bel, Anita and Wagner, Dorothea},
  title =	{{Station Location - Complexity and Approximation}},
  booktitle =	{5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'05)},
  pages =	{1--11},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-00-2},
  ISSN =	{2190-6807},
  year =	{2006},
  volume =	{2},
  editor =	{Kroon, Leo G. and M\"{o}hring, Rolf H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2005.661},
  URN =		{urn:nbn:de:0030-drops-6612},
  doi =		{10.4230/OASIcs.ATMOS.2005.661},
  annote =	{Keywords: Station Location, facility location, complexity, approximation}
}