@inproceedings{BortolettovanLieshoutMasingetal.2024, author = {Bortoletto, Enrico and van Lieshout, Rolf Nelson and Masing, Berenike and Lindner, Niels}, title = {Periodic Event Scheduling with Flexible Infrastructure Assignment}, volume = {123}, booktitle = {24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)}, doi = {10.4230/OASIcs.ATMOS.2024.4}, pages = {4:1 -- 4:18}, year = {2024}, abstract = {We present novel extensions of the Periodic Event Scheduling Problem (PESP) that integrate the assignment of activities to infrastructure elements. An application of this is railway timetabling, as station and platform capacities are limited and need to be taken into account. We show that an assignment of activities to platforms can always be made periodic, and that it can be beneficial to allow larger periods for the assignment than for the timetable. We present mixed-integer programming formulations for the general problem, as well as for the practically relevant case when multiple platforms can be considered equivalent, for which we present a bipartite matching approach. We finally test and compare these models on real-world instances.}, language = {en} } @misc{Masing2020, type = {Master Thesis}, author = {Masing, Berenike}, title = {Optimal Line Planning in the Parametric City}, year = {2020}, abstract = {One of the fundamental steps in the optimization of public transport is line planning. It involves determining lines and assigning frequencies of service such that costs are minimized while also maximizing passenger comfort and satisfying travel demands. We formulate the problem as a mixed integer linear program that considers all circuit-like lines in a graph and allows free passenger routing. Traveler and operator costs are included in a linear scalarization in the objective. We apply said programming problem to the Parametric City, which is a graph model introduced by Fielbaum, Jara-D{\´i}az and Gschwender that exibly represents different cities. In his dissertation, Fielbaum solved the line planning problem for various parameter choices in the Parametric City. In a first step, we therefore review his results and make comparative computations. Unlike Fielbaum we arrive at the conclusion that the optimal line plan for this model indeed depends on the demand. Consequently, we analyze the line planning problem in-depth: We find equivalent, but easier to compute formulations and provide a lower bound by LP-relaxation, which we show to be equivalent to a multi-commodity flow problem. Further, we examine what impact symmetry has on the solutions. Supported both by computational results as well as by theoretical analysis, we reach the conclusion that symmetric line plans are optimal or near-optimal in the Parametric City. Restricting the model to symmetric line plans allows for a \kappa-factor approximation algorithm for the line planning problem in the Parametric City.}, language = {en} } @article{MasingLindnerEbert2023, author = {Masing, Berenike and Lindner, Niels and Ebert, Patricia}, title = {Forward and Line-Based Cycle Bases for Periodic Timetabling}, volume = {4}, journal = {Operations Research Forum}, number = {3}, doi = {10.1007/s43069-023-00229-0}, pages = {53}, year = {2023}, abstract = {The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.}, language = {en} } @misc{BortolettoLindnerMasing2022, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {The Tropical and Zonotopal Geometry of Periodic Timetables}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/2204.13501}, doi = {https://doi.org/10.1007/s00454-024-00686-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-86708}, year = {2022}, abstract = {The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.}, language = {en} } @misc{BortolettoLindnerMasing2022, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling}, issn = {1438-0064}, doi = {10.4230/OASIcs.ATMOS.2022.3}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-87385}, year = {2022}, abstract = {Periodic timetabling is a central aspect of both the long-term organization and the day-to-day operations of a public transportation system. The Periodic Event Scheduling Problem (PESP), the combinatorial optimization problem that forms the mathematical basis of periodic timetabling, is an extremely hard problem, for which optimal solutions are hardly ever found in practice. The most prominent solving strategies today are based on mixed-integer programming, and there is a concurrent PESP solver employing a wide range of heuristics [3]. We present tropical neighborhood search (tns), a novel PESP heuristic. The method is based on the relations between periodic timetabling and tropical geometry [4]. We implement tns into the concurrent solver, and test it on instances of the benchmarking library PESPlib. The inclusion of tns turns out to be quite beneficial to the solver: tns is able to escape local optima for the modulo network simplex algorithm, and the overall share of improvement coming from tns is substantial compared to the other methods available in the solver. Finally, we provide better primal bounds for five PESPlib instances.}, language = {en} } @misc{MasingLindnerLiebchen2022, author = {Masing, Berenike and Lindner, Niels and Liebchen, Christian}, title = {Periodic Timetabling with Integrated Track Choice for Railway Construction Sites}, issn = {1438-0064}, doi = {10.1016/j.jrtpm.2023.100416}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88626}, year = {2022}, abstract = {We propose a mixed-integer linear programming model to generate and optimize periodic timetables with integrated track choice in the context of railway construction sites. When a section of a railway network becomes unavailable, the nearby areas are typically operated close to their capacity limits, and hence carefully modeling headways and allowing flexible routings becomes vital. We therefore discuss first how to integrate headway constraints into the Periodic Event Scheduling Problem (PESP) that do not only prevent overtaking, but also guarantee conflict-free timetables in general and particularly inside stations. Secondly, we introduce a turn-sensitive event-activity network, which is able to integrate routing alternatives for turnarounds at stations, e.g., turning at a platform vs. at a pocket track for metro-like systems. We propose several model formulations to include track choice, and finally evaluate them on six real construction site scenarios on the S-Bahn Berlin network.}, language = {en} } @inproceedings{MasingLindnerBorndoerfer2022, author = {Masing, Berenike and Lindner, Niels and Bornd{\"o}rfer, Ralf}, title = {Optimal Line Plans in the Parametric City and the Impact of In-Motion Costs}, booktitle = {Operations Research Proceedings 2021}, editor = {Trautmann, Norbert and Dr{\"a}gi, Mario}, publisher = {Springer, Cham}, isbn = {978-3-031-08623-6}, doi = {10.1007/978-3-031-08623-6_44}, pages = {295 -- 301}, year = {2022}, abstract = {Line planning in public transport involves determining vehicle routes and assigning frequencies of service such that travel demands are satisfied. We evaluate how line plans, which are optimal with respect to in-motion costs (IMC), the objective function depending purely on arc-lengths for both user and operator costs, performs with respect to the value of resources consumed (VRC). The latter is an elaborate, socio-economic cost function which includes discomfort caused by delay, boarding and alighting times, and transfers. Even though discomfort is a large contributing factor to VRC and is entirely disregarded in IMC,  we observe that the two cost functions are qualitatively comparable.}, language = {en} } @article{MasingLindnerBorndoerfer2022, author = {Masing, Berenike and Lindner, Niels and Bornd{\"o}rfer, Ralf}, title = {The price of symmetric line plans in the Parametric City}, volume = {166}, journal = {Transportation Research Part B: Methodological}, doi = {10.1016/j.trb.2022.10.009}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88613}, pages = {419 -- 443}, year = {2022}, abstract = {We consider the line planning problem in public transport in the Parametric City, an idealized model that captures typical scenarios by a (small) number of parameters. The Parametric City is rotation symmetric, but optimal line plans are not always symmetric. This raises the question to quantify the symmetry gap between the best symmetric and the overall best solution. For our analysis, we formulate the line planning problem as a mixed integer linear program, that can be solved in polynomial time if the solutions are forced to be symmetric. We prove that the symmetry gap is small when a specific Parametric City parameter is fixed, and we give an approximation algorithm for line planning in the Parametric City in this case. While the symmetry gap can be arbitrarily large in general, we show that symmetric line plans are a good choice in most practical situations.}, language = {de} } @inproceedings{BortolettoLindnerMasing2022, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling}, volume = {106}, booktitle = {22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)}, doi = {10.4230/OASIcs.ATMOS.2022.3}, pages = {3:1 -- 3:19}, year = {2022}, abstract = {Periodic timetabling is a central aspect of both the long-term organization and the day-to-day operations of a public transportation system. The Periodic Event Scheduling Problem (PESP), the combinatorial optimization problem that forms the mathematical basis of periodic timetabling, is an extremely hard problem, for which optimal solutions are hardly ever found in practice. The most prominent solving strategies today are based on mixed-integer programming, and there is a concurrent PESP solver employing a wide range of heuristics [Bornd{\"o}rfer et al., 2020]. We present tropical neighborhood search (tns), a novel PESP heuristic. The method is based on the relations between periodic timetabling and tropical geometry [Bortoletto et al., 2022]. We implement tns into the concurrent solver, and test it on instances of the benchmarking library PESPlib. The inclusion of tns turns out to be quite beneficial to the solver: tns is able to escape local optima for the modulo network simplex algorithm, and the overall share of improvement coming from tns is substantial compared to the other methods available in the solver. Finally, we provide better primal bounds for five PESPlib instances.}, language = {en} } @inproceedings{BortolettoLindnerMasing2023, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {Periodic Timetabling with Cyclic Order Constraints}, volume = {115}, booktitle = {23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)}, doi = {10.4230/OASIcs.ATMOS.2023.7}, pages = {7:1 -- 7:18}, year = {2023}, abstract = {Periodic timetabling for highly utilized railway networks is a demanding challenge. We formulate an infrastructure-aware extension of the Periodic Event Scheduling Problem (PESP) by requiring that not only events, but also activities using the same infrastructure must be separated by a minimum headway time. This extended problem can be modeled as a mixed-integer program by adding constraints on the sum of periodic tensions along certain cycles, so that it shares some structural properties with standard PESP. We further refine this problem by fixing cyclic orders at each infrastructure element. Although the computational complexity remains unchanged, the mixed-integer programming model then becomes much smaller. Furthermore, we also discuss how to find a minimal subset of infrastructure elements whose cyclic order already prescribes the order for the remaining parts of the network, and how cyclic order information can be modeled in a mixed-integer programming context. In practice, we evaluate the impact of cyclic orders on a real-world instance on the S-Bahn Berlin network, which turns out to be computationally fruitful.}, language = {en} } @inproceedings{MasingLindnerLiebchen2023, author = {Masing, Berenike and Lindner, Niels and Liebchen, Christian}, title = {Integrating Line Planning for Construction Sites into Periodic Timetabling via Track Choice}, volume = {115}, booktitle = {23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)}, doi = {10.4230/OASIcs.ATMOS.2023.5}, pages = {5:1 -- 5:15}, year = {2023}, abstract = {We consider maintenance sites for urban rail systems, where unavailable tracks typically require changes to the regular timetable, and often even to the line plan. In this paper, we present an integrated mixed-integer linear optimization model to compute an optimal line plan that makes best use of the available tracks, together with a periodic timetable, including its detailed routing on the tracks within the stations. The key component is a flexible, turn-sensitive event-activity network that allows to integrate line planning and train routing using a track choice extension of the Periodic Event Scheduling Problem (PESP). Major goals are to maintain as much of the regular service as possible, and to keep the necessary changes rather local. Moreover, we present computational results on real construction site scenarios on the S-Bahn Berlin network. We demonstrate that this integrated problem is indeed solvable on practically relevant instances.}, language = {en} } @inproceedings{Masing2022, author = {Masing, Berenike}, title = {Optimal Line Planning in the Parametric City}, booktitle = {Operations Research Proceedings 2021. OR 2021.}, editor = {Trautmann, Norbert and Dr{\"a}gi, Mario}, publisher = {Springer, Cham}, isbn = {978-3-031-08623-6}, doi = {10.1007/978-3-031-08623-6_7}, pages = {39 -- 44}, year = {2022}, abstract = {We formulate the line planning problem in public transport as a mixed integer linear program (MILP), which selects both passenger and vehicle routes, such that travel demands are met with respect to minimized travel times for both operators and users. We apply MILP to the Parametric City, a generic city model developed by Fielbaum et al. [2]. While the infrastructure graph and demand are entirely rotation symmetric, asymmetric optimal line plans can occur. Using group theory, we analyze the properties of symmetric solutions and introduce a symmetry gap to measure their deviation of the optimum. We also develop a 1+1+2√g-approximation algorithm, depending only on the cost related parameter g. Supported by computational experiments, we conclude that in practice symmetric line plans provide good solutions for the line planning problem in the Parametric City.}, language = {en} }