@article{HarrodSchlechte2013, author = {Harrod, Steven and Schlechte, Thomas}, title = {A Direct Comparison of Physical Block Occupancy Versus Timed Block Occupancy in Train Timetabling Formulations}, volume = {54}, journal = {Transportation Research Part E: Logistics and Transportation Review}, doi = {10.1016/j.tre.2013.04.003}, pages = {50 -- 66}, year = {2013}, abstract = {Two fundamental mathematical formulations for railway timetabling are compared on a common set of sample problems, representing both multiple track high density services in Europe and single track bidirectional operations in North America. One formulation, ACP, enforces against conflicts by constraining time intervals between trains, while the other formulation, RCHF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and RCHF return comparable solutions in the aggregate, with some significant differences in select instances, and a pattern of significant differences in performance and constraint enforcement overall.}, language = {en} } @article{BorndoerferKarbsteinPfetsch2012, author = {Bornd{\"o}rfer, Ralf and Karbstein, Marika and Pfetsch, Marc}, title = {Models for Fare Planning in Public Transport}, volume = {160}, journal = {Discrete Applied Mathematics}, number = {18}, doi = {10.1016/j.dam.2012.02.027}, pages = {2591 -- 2605}, year = {2012}, language = {en} } @article{BorndoerferGrimmReutheretal.2019, author = {Bornd{\"o}rfer, Ralf and Grimm, Boris and Reuther, Markus and Schlechte, Thomas}, title = {Optimization of handouts for rolling stock rotations}, journal = {Journal of Rail Transport Planning \& Management}, number = {10}, doi = {10.1016/j.jrtpm.2019.02.001}, pages = {1 -- 8}, year = {2019}, abstract = {A railway operator creates (rolling stock) rotations in order to have a precise master plan for the operation of a timetable by railway vehicles. A rotation is considered as a cycle that multiply traverses a set of operational days while covering trips of the timetable. As it is well known, the proper creation of rolling stock rotations by, e.g., optimization algorithms is challenging and still a topical research subject. Nevertheless, we study a completely different but strongly related question in this paper, i.e.: How to visualize a rotation? For this purpose, we introduce a basic handout concept, which directly leads to the visualization, i.e., handout of a rotation. In our industrial application at DB Fernverkehr AG, the handout is exactly as important as the rotation itself. Moreover, it turns out that also other European railway operators use exactly the same methodology (but not terminology). Since a rotation can have many handouts of different quality, we show how to compute optimal ones through an integer program (IP) by standard software. In addition, a construction as well as an improvement heuristic are presented. Our computational results show that the heuristics are a very reliable standalone approach to quickly find near-optimal and even optimal handouts. The efficiency of the heuristics is shown via a computational comparison to the IP approach.}, language = {en} } @article{SagnolBlancoSauvage2018, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {The Cone of Flow Matrices: Approximation Hierarchies and Applications}, volume = {72}, journal = {Networks}, number = {1}, doi = {10.1002/net.21820}, pages = {128 -- 150}, year = {2018}, abstract = {Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices \$\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}\$, where \$\vec{1}_P\in\RR^n\$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over \$\mathcal{K}\$. This cone is intractable: we prove that the membership problem associated to \$\mathcal{K}\$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix \$X\in \operatorname{span} \mathcal{K}\$ as a linear combination of some \$\vec{1}_P\vec{1}_P^T\$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.}, language = {en} } @article{BeckenbachBorndoerfer2018, author = {Beckenbach, Isabel and Bornd{\"o}rfer, Ralf}, title = {Hall's and Kőnig's theorem in graphs and hypergraphs}, volume = {341}, journal = {Discrete Mathematics}, number = {10}, doi = {10.1016/j.disc.2018.06.013}, pages = {2753 -- 2761}, year = {2018}, abstract = {We investigate the relation between Hall's theorem and Kőnig's theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall's theorem, thereby showing that this class strictly contains all Kőnig-Egerv{\´a}ry graphs. Furthermore, we give a generalization of Hall's theorem to normal hypergraphs.}, language = {en} } @article{BorndoerferGroetschelKlostermeieretal.1997, author = {Bornd{\"o}rfer, Ralf and Gr{\"o}tschel, Martin and Klostermeier, Fridolin and K{\"u}ttner, Christian}, title = {Berliner Telebussystem bietet Mobilit{\"a}t f{\"u}r Behinderte}, volume = {1/2}, journal = {Der Nahverkehr}, pages = {20 -- 22}, year = {1997}, language = {en} } @article{Neumann2007, author = {Neumann, Marika}, title = {Mathematische Preisplanung im {\"O}PNV}, journal = {OR News}, number = {30}, pages = {29 -- 31}, year = {2007}, language = {en} } @article{BorndoerferGroetschelKlostermeieretal.1997, author = {Bornd{\"o}rfer, Ralf and Gr{\"o}tschel, Martin and Klostermeier, Fridolin and K{\"u}ttner, Christian}, title = {Optimierung des Berliner Behindertenfahrdienstes}, volume = {2}, journal = {DMV-Mitteilungen}, pages = {38 -- 43}, year = {1997}, language = {en} } @article{Sagnol2013, author = {Sagnol, Guillaume}, title = {On the semidefinite representation of real functions applied to symmetric matrices}, volume = {439}, journal = {Linear Algebra and its Applications}, number = {10}, doi = {10.1016/j.laa.2013.08.021}, pages = {2829 -- 2843}, year = {2013}, abstract = {We present a new semidefinite representation for the trace of a real function f applied to symmetric matrices, when a semidefinite representation of the convex function f is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices that for a rational number p in the interval (0,1], the matrix X raised to the exponent p is the largest element of a set represented by linear matrix inequalities. We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation yields a speed-up factor in the order of 10.}, language = {en} } @article{LangenhanBorndoerferLoebeletal.2012, author = {Langenhan, Andreas and Bornd{\"o}rfer, Ralf and L{\"o}bel, Andreas and Schulz, Christof and Weider, Steffen}, title = {Duty Scheduling Templates}, journal = {Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12)}, editor = {Mu{\~n}oz, J. C. and Voß, S.}, year = {2012}, abstract = {We propose duty templates as a novel concept to produce similar duty schedules for similar days of operation in public transit. Duty templates can conveniently handle various types of similarity requirements, and they can be implemented with ease using standard algorithmic techniques. They have produced good results in practice.}, language = {en} } @article{HillerKlugTuchscherer2013, author = {Hiller, Benjamin and Klug, Torsten and Tuchscherer, Andreas}, title = {An Exact Reoptimization Algorithm for the Scheduling of Elevator Groups}, journal = {Flexible Services and Manufacturing Journal}, doi = {10.1007/s10696-013-9175-6}, pages = {1 -- 24}, year = {2013}, language = {en} } @article{BorndoerferKarbsteinPfetsch2013, author = {Bornd{\"o}rfer, Ralf and Karbstein, Marika and Pfetsch, Marc}, title = {The Steiner connectivity problem}, volume = {142}, journal = {Mathematical Programming A}, number = {1}, doi = {10.1007/s10107-012-0564-5}, pages = {133 -- 167}, year = {2013}, language = {en} } @article{BorndoerferReutherSchlechteetal.2012, author = {Bornd{\"o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {Vehicle Rotation Planning for Intercity Railways}, journal = {Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12)}, editor = {Mu{\~n}oz, J. C. and Voß, S.}, year = {2012}, abstract = {This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.}, language = {en} } @article{BorndoerferLoebelReutheretal.2013, author = {Bornd{\"o}rfer, Ralf and L{\"o}bel, Andreas and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {Rapid Branching}, volume = {5}, journal = {Public Transport}, number = {1}, editor = {Mu{\~n}oz, J. C. and Voß, S.}, publisher = {Springer Berlin Heidelberg}, pages = {3 -- 23}, year = {2013}, abstract = {We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.}, language = {en} } @article{Sagnol2013, author = {Sagnol, Guillaume}, title = {Approximation of a maximum-submodular-coverage problem involving spectral functions, with application to experimental designs}, volume = {161}, journal = {Discrete Applied Mathematics}, number = {1-2}, doi = {10.1016/j.dam.2012.07.016}, pages = {258 -- 276}, year = {2013}, abstract = {We study a family of combinatorial optimization problems defined by a parameter \$p\in[0,1]\$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as \$p\$ goes to \$0\$, and to a trivial instance of the knapsack problem as \$p\$ goes to \$1\$. In this article, we establish a matrix inequality which shows that the objective function is submodular for all \$p\in[0,1]\$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within \$1-1/e\$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to \$1\$ as \$p\$ goes to \$1\$.}, language = {en} } @article{Sagnol2011, author = {Sagnol, Guillaume}, title = {A class of Semidefinite Programs with rank-one solutions}, volume = {435}, journal = {Linear Algebra and its Applications}, number = {6}, doi = {10.1016/j.laa.2011.03.027}, pages = {1446 -- 1463}, year = {2011}, abstract = {We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most \$r\$, where \$r\$ is the rank of the matrix involved in the objective function of the SDP. The optimization problems of this class are semidefinite packing problems, which are the SDP analogs to vector packing problems. Of particular interest is the case in which our result guarantees the existence of a solution of rank one: we show that the computation of this solution actually reduces to a Second Order Cone Program (SOCP). We point out an application in statistics, in the optimal design of experiments.}, language = {en} } @article{BorndoerferSchlechteSwarat2012, author = {Bornd{\"o}rfer, Ralf and Schlechte, Thomas and Swarat, Elmar}, title = {Railway Track Allocation -- Simulation, Aggregation, and Optimization}, volume = {2}, journal = {Proc. 1st International Workshop on High-speed and Intercity Railways (IWHIR 2011)}, number = {148}, doi = {10.1007/978-3-642-27963-8}, pages = {53 -- 70}, year = {2012}, abstract = {Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.}, language = {en} } @article{BorndoerferReutherSchlechteetal.2011, author = {Bornd{\"o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Weider, Steffen}, title = {A Hypergraph Model for Railway Vehicle Rotation Planning}, volume = {OpenAccess Series in Informatics (OASIcs)}, journal = {11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, number = {20}, doi = {10.4230/OASIcs.ATMOS.2011.146}, pages = {146 -- 155}, year = {2011}, abstract = {We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the ``regularity'' of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.}, language = {en} } @article{Schlechte2011, author = {Schlechte, Thomas}, title = {Railway Track Allocation - Simulation and Optimization}, volume = {4}, journal = {Proceedings of 4th International Seminar on Railway Operations Modelling and Analysis (IAROR)}, year = {2011}, abstract = {Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway infrastructure provider. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market. This leads to an increase of slot requests and then it is natural that conflicts occur among them. Furthermore, railway infrastructure networks consist of very expensive assets, even more they are rigid due to the long-term upgrade process. In order to make best use of these valuable infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithmic methodology can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and a general framework to support the integration of simulation and optimization for railway capacity allocation.}, language = {en} } @article{BorndoerferFriedowKarbstein2012, author = {Bornd{\"o}rfer, Ralf and Friedow, Isabel and Karbstein, Marika}, title = {Optimierung des Linienplans 2010 in Potsdam}, volume = {30}, journal = {Der Nahverkehr}, number = {4}, pages = {34 -- 39}, year = {2012}, language = {de} } @article{Karbstein2014, author = {Karbstein, Marika}, title = {Ohne Umsteigen ans Ziel}, volume = {52}, journal = {OR News}, pages = {12 -- 14}, year = {2014}, abstract = {Die Anforderungen an das Liniennetz eines Verkehrsunternehmens sind vielf{\"a}ltig: kurze Fahrzeiten, viele Direktverbindungen und geringe Kosten. Mathematische Optimierungsmethoden k{\"o}nnen die Planung von Liniennetzen unterst{\"u}tzen und beste Kompromisse bzgl. aller Anforderungen berechnen.}, language = {de} } @article{BorndoerferFuegenschuhKlugetal.2016, author = {Bornd{\"o}rfer, Ralf and F{\"u}genschuh, Armin and Klug, Torsten and Schang, Thilo and Schlechte, Thomas and Sch{\"u}lldorf, Hanno}, title = {The Freight Train Routing Problem for Congested Railway Networks with Mixed Traffic}, volume = {50}, journal = {Transportation Science}, number = {2}, doi = {10.1287/trsc.2015.0656}, pages = {408 -- 423}, year = {2016}, abstract = {We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that the sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. In this context, macroscopic refers to an aggregation of complex and large real-world structures into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We provide a mixed-integer nonlinear programming (MINLP) formulation for the FTRP, which is a multicommodity flow model on a time-expanded graph with additional routing constraints. The model's nonlinearities originate from an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model (MILP) by piecewise linear approximation. The latter is solved by a state-of-the art MILP solver for various real-world test instances.}, language = {en} } @article{Hoppmann2015, author = {Hoppmann, Heide}, title = {Linienplanung und Minimale Konfigurationen}, volume = {55}, journal = {OR News}, pages = {22 -- 23}, year = {2015}, language = {de} }