TY - CHAP A1 - Bortoletto, Enrico A1 - van Lieshout, Rolf Nelson A1 - Masing, Berenike A1 - Lindner, Niels T1 - Periodic Event Scheduling with Flexible Infrastructure Assignment T2 - 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024) N2 - 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. Y1 - 2024 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2024.4 VL - 123 SP - 4:1 EP - 4:18 ER - TY - JOUR A1 - Euler, Ricardo A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - Price optimal routing in public transportation JF - EURO Journal on Transportation and Logistics KW - Management Science and Operations Research KW - Transportation KW - Modeling and Simulation Y1 - 2024 U6 - https://doi.org/10.1016/j.ejtl.2024.100128 SN - 2192-4376 VL - 13 SP - 1 EP - 15 PB - Elsevier BV ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Ebert, Patricia T1 - Forward and Line-Based Cycle Bases for Periodic Timetabling JF - Operations Research Forum N2 - 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. Y1 - 2023 U6 - https://doi.org/10.1007/s43069-023-00229-0 VL - 4 IS - 3 SP - 53 ER - TY - JOUR A1 - Schlechte, Thomas A1 - Borndörfer, Ralf A1 - Denißen, Jonas A1 - Heller, Simon A1 - Klug, Torsten A1 - Küpper, Michael A1 - Lindner, Niels A1 - Reuther, Markus A1 - Söhlke, Andreas A1 - Steadman, William T1 - Timetable Optimization for a Moving Block System JF - Journal of Rail Transport Planning & Management N2 - We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model is able to solve reasonably sized instances with up to hundred trains to optimality. Y1 - 2022 U6 - https://doi.org/10.1016/j.jrtpm.2022.100315 SN - 2210-9706 VL - 22 SP - 100315 ER - TY - GEN A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - The Tropical and Zonotopal Geometry of Periodic Timetables N2 - 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. T3 - ZIB-Report - 22-09 KW - periodic event scheduling KW - tropical geometry KW - zonotopal tilings Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-86708 SN - 1438-0064 ER - TY - GEN A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling N2 - 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. T3 - ZIB-Report - 22-13 KW - Periodic Timetabling KW - Tropical Geometry KW - Neighbourhood Search KW - Mixed-Integer Programming Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-87385 SN - 1438-0064 ER - TY - GEN A1 - Masing, Berenike A1 - Lindner, Niels A1 - Liebchen, Christian T1 - Periodic Timetabling with Integrated Track Choice for Railway Construction Sites N2 - 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. T3 - ZIB-Report - 22-26 KW - Railway Timetabling KW - Periodic Timetabling KW - Periodic Event Scheduling KW - Train Routing KW - Turnarounds Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-88626 SN - 1438-0064 ER - TY - JOUR A1 - Euler, Ricardo A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - Price Optimal Routing in Public Transportation N2 - We consider the price-optimal earliest arrival problem in public transit (POEAP) in which we aim to calculate the Pareto-front of journeys with respect to ticket price and arrival time in a public transportation network. Public transit fare structures are often a combination of various fare strategies such as, e.g., distance-based fares, zone-based fares or flat fares. The rules that determine the actual ticket price are often very complex. Accordingly, fare structures are notoriously difficult to model as it is in general not sufficient to simply assign costs to arcs in a routing graph. Research into POEAP is scarce and usually either relies on heuristics or only considers restrictive fare models that are too limited to cover the full scope of most real-world applications. We therefore introduce conditional fare networks (CFNs), the first framework for representing a large number of real-world fare structures. We show that by relaxing label domination criteria, CFNs can be used as a building block in label-setting multi-objective shortest path algorithms. By the nature of their extensive modeling capabilities, optimizing over CFNs is NP-hard. However, we demonstrate that adapting the multi-criteria RAPTOR (MCRAP) algorithm for CFNs yields an algorithm capable of solving POEAP to optimality in less than 400 ms on average on a real-world data set. By restricting the size of the Pareto-set, running times are further reduced to below 10 ms. Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-86414 ER - TY - JOUR A1 - Lindner, Niels A1 - Reisch, Julian T1 - An analysis of the parameterized complexity of periodic timetabling JF - Journal of Scheduling N2 - Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib. Y1 - 2022 U6 - https://doi.org/10.1007/s10951-021-00719-1 VL - 25 SP - 157 EP - 176 ER - TY - CHAP A1 - Masing, Berenike A1 - Lindner, Niels A1 - Borndörfer, Ralf ED - Trautmann, Norbert ED - Drägi, Mario T1 - Optimal Line Plans in the Parametric City and the Impact of In-Motion Costs T2 - Operations Research Proceedings 2021 N2 - 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. Y1 - 2022 SN - 978-3-031-08623-6 U6 - https://doi.org/10.1007/978-3-031-08623-6_44 SP - 295 EP - 301 PB - Springer, Cham ER - TY - JOUR A1 - Schiewe, Philine A1 - Goerigk, Marc A1 - Lindner, Niels T1 - Introducing TimPassLib – A Library for Integrated Periodic Timetabling and Passenger Routing JF - Operations Research Forum N2 - Classic models to derive a timetable for public transport often face a chicken-and-egg situation: A good timetable should offer passengers routes with small travel times, but the route choice of passengers depends on the timetable. While models that fix passenger routes were frequently considered in the literature, integrated models that simultaneously optimize timetables and passenger routes have seen increasing attention lately. This creates a growing need for a set of instances that allows to test and compare new algorithmic developments for the integrated problem. Our paper addresses this requirement by presenting TimPassLib, a new benchmark library of instances for integrated periodic timetabling and passenger routing. Y1 - 2023 U6 - https://doi.org/10.1007/s43069-023-00244-1 VL - 4 IS - 3 SP - 64 ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - The price of symmetric line plans in the Parametric City JF - Transportation Research Part B: Methodological N2 - 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. Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-88613 VL - 166 SP - 419 EP - 443 ER - TY - CHAP A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling T2 - 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022) N2 - 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 [Borndörfer 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. Y1 - 2022 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2022.3 VL - 106 SP - 3:1 EP - 3:19 ER - TY - CHAP A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - Periodic Timetabling with Cyclic Order Constraints T2 - 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023) N2 - 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. Y1 - 2023 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2023.7 VL - 115 SP - 7:1 EP - 7:18 ER - TY - CHAP A1 - Masing, Berenike A1 - Lindner, Niels A1 - Liebchen, Christian T1 - Integrating Line Planning for Construction Sites into Periodic Timetabling via Track Choice T2 - 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023) N2 - 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. Y1 - 2023 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2023.5 VL - 115 SP - 5:1 EP - 5:15 ER - TY - CHAP A1 - Lindner, Niels A1 - van Lieshout, Rolf T1 - Benders Decomposition for the Periodic Event Scheduling Problem T2 - Operations Research Proceedings 2021 N2 - The Periodic Event Scheduling Problem (PESP) is the central mathematical model behind the optimization of periodic timetables in public transport. We apply Benders decomposition to the incidence-based MIP formulation of PESP. The resulting formulation exhibits particularly nice features: The subproblem is a minimum cost network flow problem, and feasibility cuts are equivalent to the well-known cycle inequalities by Odijk. We integrate the Benders approach into a branch-and-cut framework, and assess the performance of this method on instances derived from the benchmarking library PESPlib. Y1 - 2022 U6 - https://doi.org/10.1007/978-3-031-08623-6_43 SP - 289 EP - 294 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Löbel, Fabian A1 - Lindner, Niels A1 - Borndörfer, Ralf ED - Neufeld, Janis S. ED - Buscher, Udo ED - Lasch, Rainer ED - Möst, Dominik ED - Schönberger, Jörn T1 - The Restricted Modulo Network Simplex Method for Integrated Periodic Timetabling and Passenger Routing T2 - Operations Research Proceedings 2019 N2 - The Periodic Event Scheduling Problem is a well-studied NP-hard problem with applications in public transportation to find good periodic timetables. Among the most powerful heuristics to solve the periodic timetabling problem is the modulo network simplex method. In this paper, we consider the more difficult version with integrated passenger routing and propose a refined integrated variant to solve this problem on real-world-based instances. Y1 - 2020 SN - 978-3-030-48438-5 U6 - https://doi.org/https://doi.org/10.1007/978-3-030-48439-2_92 SP - 757 EP - 763 PB - Springer International Publishing CY - Cham ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Liebchen, Christian T1 - Periodic timetabling with integrated track choice for railway construction sites JF - Journal of Rail Transport Planning & Management N2 - 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. Y1 - 2023 U6 - https://doi.org/10.1016/j.jrtpm.2023.100416 VL - 28 SP - 100416 ER -