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    <title language="eng">An Eikonal Approach for Globally Optimal Free Flight Trajectories</title>
    <abstract language="eng">We present an eikonal-based approach that is capable of finding a continuous globally optimal trajectory for an aircraft in a stationary wind field. This minimizes emissions and fuel consumption. If the destination is close to a cut locus of the associated Hamilton-Jacobi-Bellman equation, small numerical discretization errors can lead to selecting a merely locally optimal trajectory and missing the globally optimal one. Based on finite element error estimates, we construct a trust region around the cut loci in order to guarantee uniqueness of trajectories for destinations sufficiently far from cut loci.</abstract>
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    <author>Ralf Borndörfer</author>
    <submitter>Martin Weiser</submitter>
    <author>Arturas Jocas</author>
    <author>Martin Weiser</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
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    <title language="eng">Non-linear Battery Behavior in Electric Vehicle Scheduling Problems</title>
    <abstract language="eng">The currently most popular approach to handle non-linear battery behavior for electric vehicle scheduling is to use a linear spline interpolation of the charge curve. We show that this can lead to approximate models that underestimate the charge duration and overestimate the state of charge, which is not desirable. While the error is of second order with respect to the interpolation step size, the associated mixed-integer linear programs do not scale well with the number of spline segments. It is therefore recommendable to use coarse interpolation grids adapted to the curvature of the charge curve, and to include sufficient safety margins to ensure solutions of approximate models remain feasible subjected to the exact charge curve.</abstract>
    <parentTitle language="eng">Operations Research Proceedings 2023. OR 2023.</parentTitle>
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    <enrichment key="ConferencePlace">Hamburg, Germany</enrichment>
    <enrichment key="ConferenceTitle">International Conference on Operations Research (OR 2023)</enrichment>
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    <enrichment key="Series">Lecture Notes in Operations Research</enrichment>
    <author>Fabian Löbel</author>
    <editor>Guido Voigt</editor>
    <submitter>Fabian Löbel</submitter>
    <author>Ralf Borndörfer</author>
    <editor>Malte Fliedner</editor>
    <author>Steffen Weider</author>
    <editor>Knut Haase</editor>
    <editor>Wolfgang Brüggermann</editor>
    <editor>Kai Hoberg</editor>
    <editor>Joern Meissner</editor>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Electric Vehicle Scheduling</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Non-Linear Charging</value>
    </subject>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
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  <doc>
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    <language>deu</language>
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    <publisherName>Thieme</publisherName>
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    <title language="deu">Mathematische Optimierung in der OP-Planung</title>
    <abstract language="deu">Deutsche Krankenhäuser sehen sich derzeit mit enormen Schwierigkeiten konfrontiert. Ungefähr jede 2. Klinik muss drastische Sparmaßnahmen ergreifen, was auch die Allgemeinversorgung beeinträchtigt. Die Gründe dafür sind vielschichtig: stark gestiegene Sach- und Personalkosten bei gleicher Finanzierung, teilweiser Patientenrückgang, starke regionale Unterschiede in der Versorgung, Fachkräftemangel und fehlende Investitionen in Kern- und Zukunftsbereiche, insbesondere der Digitalisierung. Das belastet die Haushalte der Kliniken. Insbesondere die Digitalisierung und die Anwendung von Methoden der künstlichen Intelligenz und der mathematischen Optimierung könnten eine Schlüsselrolle spielen, um die komplexen Krankenhausprozesse mit Kennzahlen qualitativ zu bewerten und zu verbessern. In diesem Artikel stellen wir vier Praxisprobleme aus der OP-Planung vor und benennen welche Entscheidungen, Nebenbedingungen und Zielkriterien mit mathematischen Entscheidungsmodellen dargestellt und optimiert werden können. Hierzu erläutern wir das erweiterte Potenzial einer umfassenden Anwendung von mathematischer Optimierung im OP-Bereich.</abstract>
    <parentTitle language="deu">OP-Management up2date</parentTitle>
    <identifier type="doi">10.1055/a-2322-2124</identifier>
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    <enrichment key="AcceptedDate">10.03.2025</enrichment>
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    <author>Alexander Tesch</author>
    <submitter>Alexander Tesch</submitter>
    <author>Ralf Borndörfer</author>
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    <collection role="persons" number="tesch">Tesch, Alexander</collection>
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    <publishedYear>2024</publishedYear>
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    <title language="eng">Logic-Constrained Shortest Paths for Flight Planning</title>
    <abstract language="eng">The Logic-Constrained Shortest Path Problem (LCSP) combines a one-to-one shortest path problem with satisfiability constraints imposed on the routing graph. This setting arises in flight planning, where air traffic control (ATC) authorities are enforcing a set of traffic flow restrictions (TFRs) on aircraft routes in order to increase safety and throughput. We propose a new branch and bound-based algorithm for the LCSP. The resulting algorithm has three main degrees of freedom: the node selection rule, the branching rule and the conflict. While node selection and branching rules have been long studied in the MIP and SAT communities, most of them cannot be applied out of the box for the LCSP. We review the existing literature and develop tailored variants of the most prominent rules. The conflict, the set of variables to which the branching rule is applied, is unique to the LCSP. We analyze its theoretical impact on the B&amp;B algorithm. In the second part of the paper, we show how to model the Flight Planning Problem with TFRs as an LCSP and solve it using the branch and bound algorithm. We demonstrate the algorithm’s efficiency on a dataset consisting of a global flight graph and a set of around 20000 real TFRs obtained from our industry partner Lufthansa Systems GmbH. We make this dataset publicly available. Finally, we conduct an empirical in-depth analysis of node selection rules, branching rules and conflicts. Carefully choosing an appropriate combination yields an improvement of an order of magnitude compared to an uninformed choice.</abstract>
    <identifier type="arxiv">2412.13235</identifier>
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    <author>Ricardo Euler</author>
    <submitter>Ricardo Euler</submitter>
    <author>Pedro Maristany de las Casas</author>
    <author>Ralf Borndörfer</author>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="euler">Euler, Ricardo</collection>
    <collection role="persons" number="maristany">Maristany de las Casas, Pedro</collection>
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    <publishedYear>2024</publishedYear>
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    <completedDate>2024-07-19</completedDate>
    <publishedDate>2024-07-19</publishedDate>
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    <title language="eng">Electric Bus Scheduling with Non-Linear Charging, Power Grid Bottlenecks, and Dynamic Recharge Rates</title>
    <abstract language="eng">Public transport operators are gradually electrifying their bus fleets, predominantly with battery-powered drive trains. These buses commonly have to be scheduled to recharge in-service, which gives rise to a number of challenges. A major problem is that the relationship between charging time and replenished driving range is non-linear, which is often approximately modeled. We examine the associated approximation error and show how it can result in a gross over- or underestimation of the fleet size. Moreover, we demonstrate that commonly used piecewise linear underestimations of the charge curve do not result in an underestimation of the predicted charge states in electric vehicle scheduling and routing models.&#13;
Furthermore, since power grid upgrades are currently not keeping up with an ever growing electricity demand, operators are introducing active charge management tools to dynamically adjust the charging speed depending on the amount of available energy. It is therefore imperative to extend electric bus scheduling models to account for these developments.&#13;
We propose a novel mixed-integer programming formulation for the electric bus scheduling problem featuring an improved approximation of the non-linear battery charging behavior as well as dynamic recharge speeds to accommodate grid load limits. The idea is to linearly interpolate what we call the charge increment function, which is closely related to the derivative of the commonly used charge curve. This provides very good error control and integrates easily into integer programming models. We demonstrate the practical usefulness of our model on a diverse library of real-life instances.</abstract>
    <identifier type="arxiv">2407.14446</identifier>
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    <author>Fabian Löbel</author>
    <submitter>Fabian Löbel</submitter>
    <author>Ralf Borndörfer</author>
    <author>Steffen Weider</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="weider">Weider, Steffen</collection>
    <collection role="persons" number="fabian.loebel">Löbel, Fabian</collection>
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  <doc>
    <id>9795</id>
    <completedYear>2024</completedYear>
    <publishedYear>2024</publishedYear>
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    <language>eng</language>
    <pageFirst>11:1</pageFirst>
    <pageLast>11:16</pageLast>
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    <volume>123</volume>
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    <publisherName>Schloss Dagstuhl -- Leibniz-Zentrum für Informatik</publisherName>
    <publisherPlace>Dagstuhl, Germany</publisherPlace>
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    <completedDate>2024-10-07</completedDate>
    <publishedDate>2024-10-07</publishedDate>
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    <title language="eng">Solving the Electric Bus Scheduling Problem by an Integrated Flow and Set Partitioning Approach</title>
    <abstract language="eng">Attractive and cost-efficient public transport requires solving computationally difficult optimization problems from network design to crew rostering. While great progress has been made in many areas, new requirements to handle increasingly complex constraints are constantly coming up. One such challenge is a new type of resource constraints that are used to deal with the state-of-charge of battery-electric vehicles, which have limited driving ranges and need to be recharged in-service. Resource constrained vehicle scheduling problems can classically be modelled in terms of either a resource constrained (multi-commodity) flow problem or in terms of a path-based set partition problem. We demonstrate how a novel integrated version of both formulations can be leveraged to solve resource constrained vehicle scheduling with replenishment in general and the electric bus scheduling problem in particular by Lagrangian relaxation and the proximal bundle method.</abstract>
    <parentTitle language="eng">24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)</parentTitle>
    <identifier type="doi">10.4230/OASIcs.ATMOS.2024.11</identifier>
    <identifier type="urn">urn:nbn:de:0030-drops-211992</identifier>
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    <enrichment key="AcceptedDate">2024-08-07</enrichment>
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    <submitter>Fabian Löbel</submitter>
    <author>Ralf Borndörfer</author>
    <author>Andreas Löbel</author>
    <author>Fabian Löbel</author>
    <author>Steffen Weider</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="loebel">Löbel, Andreas</collection>
    <collection role="persons" number="weider">Weider, Steffen</collection>
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  <doc>
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    <publishedYear>2024</publishedYear>
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    <pageFirst>13:1</pageFirst>
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    <completedDate>2024-10-07</completedDate>
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    <title language="eng">A Bayesian Rolling Horizon Approach for Rolling Stock Rotation Planning with Predictive Maintenance</title>
    <abstract language="eng">We consider the rolling stock rotation planning problem with predictive maintenance (RSRP-PdM), where a timetable given by a set of trips must be operated by a fleet of vehicles. Here, the health states of the vehicles are assumed to be random variables, and their maintenance schedule should be planned based on their predicted failure probabilities. Utilizing the Bayesian update step of the Kalman filter, we develop a rolling horizon approach for RSRP-PdM, in which the predicted health state distributions are updated as new data become available. This approach reduces the uncertainty of the health states and thus improves the decision-making basis for maintenance planning. To solve the instances, we employ a local neighborhood search, which is a modification of a heuristic for RSRP-PdM, and demonstrate its effectiveness. Using this solution algorithm, the presented approach is compared with the results of common maintenance strategies on test instances derived from real-world timetables. The obtained results show the benefits of the rolling horizon approach.</abstract>
    <parentTitle language="eng">24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)</parentTitle>
    <identifier type="doi">10.4230/OASIcs.ATMOS.2024.13</identifier>
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    <author>Felix Prause</author>
    <submitter>Felix Prause</submitter>
    <author>Ralf Borndörfer</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
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  </doc>
  <doc>
    <id>9643</id>
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    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>259</pageFirst>
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    <pageNumber>17</pageNumber>
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    <issue/>
    <volume>14742</volume>
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    <title language="eng">ULD Build-Up Scheduling with Logic-Based Benders Decomposition</title>
    <abstract language="eng">We study a complex planning and scheduling problem arising from the build-up process of air cargo pallets and containers, collectively referred to as unit load devices (ULD), in which ULDs must be assigned to workstations for loading. Since air freight usually becomes available gradually along the planning horizon, ULD build-ups must be scheduled neither too early to avoid underutilizing ULD capacity, nor too late to avoid resource conflicts with other flights. Whenever possible, ULDs should be built up in batches, thereby giving ground handlers more freedom to rearrange cargo and utilize the ULD's capacity efficiently. The resulting scheduling problem has an intricate cost function and produces large time-expanded models, especially for longer planning horizons. We propose a logic-based Benders decomposition approach that assigns batches to time intervals and workstations in the master problem, while the actual schedule is decided in a subproblem. By choosing appropriate intervals, the subproblem becomes a feasibility problem that decomposes over the workstations. Additionally, the similarity of many batches is exploited by a strengthening procedure for no-good cuts. We benchmark our approach against a time-expanded MIP formulation from the literature on a publicly available data set. It solves 15% more instances to optimality and decreases run times by more than 50% in the geometric mean. This improvement is especially pronounced for longer planning horizons of up to one week, where the Benders approach solves over 50% instances more than the baseline</abstract>
    <parentTitle language="eng">Integration of Constraint Programming, Artificial Intelligence, and Operations Research</parentTitle>
    <identifier type="doi">10.1007/978-3-031-60597-0_17</identifier>
    <enrichment key="Series">Lecture Notes in Computer Science</enrichment>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">02.02.2024</enrichment>
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    <author>Ricardo Euler</author>
    <submitter>Ricardo Euler</submitter>
    <author>Ralf Borndörfer</author>
    <author>Christian Puchert</author>
    <author>Tuomo Takkula</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="euler">Euler, Ricardo</collection>
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  </doc>
  <doc>
    <id>9602</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
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    <language>eng</language>
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    <title language="eng">An Iterative Refinement Approach for the Rolling Stock Rotation Problem with Predictive Maintenance</title>
    <abstract language="eng">The rolling stock rotation problem with predictive maintenance (RSRP-PdM) involves the assignment of trips to a fleet of vehicles with integrated maintenance scheduling based on the predicted failure probability of the vehicles. These probabilities are determined by the health states of the vehicles, which are considered to be random variables distributed by a parameterized family of probability distribution functions. During the operation of the trips, the corresponding parameters get updated. In this article, we present a dual solution approach for RSRP-PdM and generalize a linear programming based lower bound for this problem to families of probability distribution functions with more than one parameter. For this purpose, we define a rounding function that allows for a consistent underestimation of the parameters and model the problem by a state-expanded event-graph in which the possible states are restricted to a discrete set. This induces a flow problem that is solved by an integer linear program. We show that the iterative refinement of the underlying discretization leads to solutions that converge from below to an optimal solution of the original instance. Thus, the linear relaxation of the considered integer linear program results in a lower bound for RSRP-PdM. Finally, we report on the results of computational experiments conducted on a library of test instances.</abstract>
    <identifier type="arxiv">2404.08367</identifier>
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    <enrichment key="PeerReviewed">no</enrichment>
    <author>Felix Prause</author>
    <submitter>Felix Prause</submitter>
    <author>Ralf Borndörfer</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
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  </doc>
  <doc>
    <id>9556</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>100434</pageFirst>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>30</volume>
    <type>article</type>
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    <title language="eng">Approximating rolling stock rotations with integrated predictive maintenance</title>
    <abstract language="eng">We study the solution of the rolling stock rotation problem with predictive maintenance (RSRP-PdM) by an iterative refinement approach that is based on a state-expanded event-graph. In this graph, the states are parameters of a failure distribution, and paths correspond to vehicle rotations with associated health state approximations. An optimal set of paths including maintenance can be computed by solving an integer linear program. Afterwards, the graph is refined and the procedure repeated. An associated linear program gives rise to a lower bound that can be used to determine the solution quality. Computational results for six instances derived from real-world timetables of a German railway company are presented. The results show the effectiveness of the approach and the quality of the solutions.</abstract>
    <parentTitle language="eng">Journal of Rail Transport Planning &amp; Management</parentTitle>
    <identifier type="doi">10.1016/j.jrtpm.2024.100434</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2024-02-28</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-89531</enrichment>
    <author>Felix Prause</author>
    <submitter>Felix Prause</submitter>
    <author>Ralf Borndörfer</author>
    <author>Boris Grimm</author>
    <author>Alexander Tesch</author>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="grimm">Grimm, Boris</collection>
    <collection role="persons" number="tesch">Tesch, Alexander</collection>
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  </doc>
  <doc>
    <id>9505</id>
    <completedYear/>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>106186</pageFirst>
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    <pageNumber/>
    <edition/>
    <issue/>
    <volume>154</volume>
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    <completedDate>2023-02-24</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A three-phase heuristic for the Fairness-Oriented Crew Rostering Problem</title>
    <abstract language="eng">The Fairness-Oriented Crew Rostering Problem (FCRP) considers the joint optimization of attractiveness and fairness in cyclic crew rostering. Like many problems in scheduling and logistics, the combinatorial complexity of cyclic rostering causes exact methods to fail for large-scale practical instances. In case of the FCRP, this is accentuated by the additionally imposed fairness requirements. Hence, heuristic methods are necessary. We present a three-phase heuristic for the FCRP combining column generation techniques with variable-depth neighborhood search. The heuristic exploits different mathematical formulations to find feasible solutions and to search for improvements. We apply our methodology to practical instances from Netherlands Railways (NS), the main passenger railway operator in the Netherlands Our results show the three-phase heuristic finds good solutions for most instances and outperforms a state-of-the-art commercial solver.</abstract>
    <parentTitle language="eng">Computers &amp; Operations Research</parentTitle>
    <identifier type="doi">10.1016/j.cor.2023.106186</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="SubmissionStatus">epub ahead of print</enrichment>
    <enrichment key="AcceptedDate">2023-03-06</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <author>Thomas Breugem</author>
    <submitter>Thomas Schlechte</submitter>
    <author>Thomas Schlechte</author>
    <author>Christof Schulz</author>
    <author>Ralf Borndörfer</author>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
    <collection role="persons" number="schlechte">Schlechte, Thomas</collection>
    <collection role="persons" number="schulz">Schulz, Christof</collection>
    <collection role="projects" number="WS-OPT">WS-OPT</collection>
    <collection role="institutes" number="neo">Network Optimization</collection>
  </doc>
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    <issue/>
    <volume>13</volume>
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    <title language="eng">Price optimal routing in public transportation</title>
    <parentTitle language="eng">EURO Journal on Transportation and Logistics</parentTitle>
    <identifier type="doi">10.1016/j.ejtl.2024.100128</identifier>
    <identifier type="issn">2192-4376</identifier>
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