@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} } @article{EulerLindnerBorndoerfer2022, author = {Euler, Ricardo and Lindner, Niels and Bornd{\"o}rfer, Ralf}, title = {Price Optimal Routing in Public Transportation}, arxiv = {http://arxiv.org/abs/2204.01326}, doi = {https://doi.org/10.48550/arXiv.2204.01326}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-86414}, year = {2022}, abstract = {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.}, language = {en} } @article{LindnerReisch2022, author = {Lindner, Niels and Reisch, Julian}, title = {An analysis of the parameterized complexity of periodic timetabling}, volume = {25}, journal = {Journal of Scheduling}, doi = {10.1007/s10951-021-00719-1}, pages = {157 -- 176}, year = {2022}, abstract = {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.}, language = {en} } @inproceedings{EulerBorndoerferStrunketal.2022, author = {Euler, Ricardo and Bornd{\"o}rfer, Ralf and Strunk, Timo and Takkula, Tuomo}, title = {ULD Build-Up Scheduling with Dynamic Batching in an Air Freight Hub}, booktitle = {Operations Research Proceedings 2021}, doi = {10.1007/978-3-031-08623-6_38}, pages = {254 -- 260}, year = {2022}, abstract = {Air freight is usually shipped in standardized unit load devices (ULDs). The planning process for the consolidation of transit cargo from inbound flights or locally emerging shipments into ULDs for outbound flights is called build-up scheduling. More specifically, outbound ULDs must be assigned a time and a workstation subject to both workstation capacity constraints and the availability of shipments which in turn depends on break-down decisions for incoming ULDs. ULDs scheduled for the same outbound flight should be built up in temporal and spatial proximity. This serves both to minimize overhead in transportation times and to allow workers to move freight between ULDs. We propose to address this requirement by processing ULDs for the same outbound flight in batches. For the above build-up scheduling problem, we introduce a multi-commodity network design model. Outbound flights are modeled as commodities; transit cargo is represented by cargo flow volume and unpack and batch decisions are represented as design variables. The model is solved with a standard MIP solver on a set of benchmark data. For instances with a limited number of resource conflicts, near-optimal solutions are found in under two hours for a whole week of operations.}, language = {en} } @misc{SchlechteBlomeGerberetal.2023, author = {Schlechte, Thomas and Blome, Christian and Gerber, Stefan and Hauser, Stefan and Kasten, Jens and M{\"u}ller, Gilbert and Schulz, Christof and Th{\"u}ring, Michel and Weider, Steffen}, title = {The Bouquet of Features in Rolling Stock Rotation Planning}, journal = {Conference Proceedings RailBelgrade 2023}, year = {2023}, abstract = {Rolling stock is one of the major assets for a railway transportation company. Hence, their utilization should be as efficiently and effectively as possible. Railway undertakings are facing rolling stock scheduling challenges in different forms - from rather idealized weekly strategic problems to very concrete operational ones. Thus, a vast of optimization models with different features and objectives exist. Thorlacius et al. (2015) provides a comprehensive and valuable collection on technical requirements, models, and methods considered in the scientific literature. We contribute with an update including recent works. The main focus of the paper is to present a classification and elaboration of the major features which our solver R-OPT is able to handle. Moreover, the basic optimization model and algorithmic ingredients of R-OPT are discussed. Finally, we present computational results for a cargo application at SBB CARGO AG and other railway undertakings for passenger traffic in Europe to show the capabilities of R-OPT.}, language = {en} } @article{MaristanydelasCasasKrausSedenoNodaetal.2023, author = {Maristany de las Casas, Pedro and Kraus, Luitgard and Sedeno-Noda, Antonio and Bornd{\"o}rfer, Ralf}, title = {Targeted multiobjective Dijkstra Algorithm}, volume = {82}, journal = {Networks}, number = {3}, arxiv = {http://arxiv.org/abs/2110.10978}, doi = {10.1002/net.22174}, pages = {277 -- 298}, year = {2023}, abstract = {We introduce the Targeted Multiobjective Dijkstra Algorithm (T-MDA), a label setting algorithm for the One-to-One Multiobjective Shortest Path (MOSP) Problem. It is based on the recently published Multiobjective Dijkstra Algorithm (MDA) and equips it with A*-like techniques. For any explored subpath, a label setting MOSP algorithm decides whether the subpath can be discarded or must be stored as part of the output. A major design choice is how to store subpaths from the moment they are first explored until the mentioned final decision can be made. The T-MDA combines the polynomially bounded size of the priority queue used in the MDA and alazy management of paths that are not in the queue. The running time bounds from the MDA remain valid. In practice, the T-MDA outperforms known algorithms from the literature and the increased memory consumption is negligible. In this paper, we benchmark the T-MDA against an improved version of the state of the art NAMOA∗drOne-to-One MOSP algorithm from the literature on a standard testbed.}, 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{SchieweGoerigkLindner2023, author = {Schiewe, Philine and Goerigk, Marc and Lindner, Niels}, title = {Introducing TimPassLib - A Library for Integrated Periodic Timetabling and Passenger Routing}, volume = {4}, journal = {Operations Research Forum}, number = {3}, doi = {10.1007/s43069-023-00244-1}, pages = {64}, year = {2023}, abstract = {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.}, language = {en} }