TY - CHAP A1 - Borndörfer, Ralf A1 - Omont, Bertrand A1 - Sagnol, Guillaume A1 - Swarat, Elmar T1 - A Stackelberg game to optimize the distribution of controls in transportation networks T2 - Proceedings of the 3rd International Conference on Game Theory for Networks (GAMENETS 2012) N2 - We propose a game theoretic model for the spatial distribution of inspectors on a transportation network. The problem is to spread out the controls so as to enforce the payment of a transit toll. We formulate a linear program to find the control distribution which maximizes the expected toll revenue, and a mixed integer program for the problem of minimizing the number of evaders. Furthermore, we show that the problem of finding an optimal mixed strategy for a coalition of $N$ inspectors can be solved efficiently by a column generation procedure. Finally, we give experimental results from an application to the truck toll on German motorways. Y1 - 2012 U6 - https://doi.org/http://dx.doi.org/10.1007/978-3-642-35582-0_17 VL - 105 SP - 224 EP - 235 ER - TY - GEN A1 - Hoppmann, Heide T1 - An Extended Formulation for the Line Planning Problem N2 - In this paper we present a novel extended formulation for the line planning problem that is based on what we call “configurations” of lines and frequencies. Configurations account for all possible options to provide a required transportation capacity on an infrastructure edge. The proposed configuration model is strong in the sense that it implies several facet-defining inequalities for the standard model: set cover, symmetric band, MIR, and multicover inequalities. These theoretical findings can be confirmed in computational results. Further, we show how this concept can be generalized to define configurations for subsets of edges; the generalized model implies additional inequalities from the line planning literature. T3 - ZIB-Report - 16-08 KW - Line Planning, Extended Formulation, Public Transport Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57467 SN - 1438-0064 ER - TY - CHAP A1 - Klug, Torsten A1 - Junosza-Szaniawski, Konstanty A1 - Kwasiborski, Slawomir A1 - Fügenschuh, Armin A1 - Schlechte, Thomas T1 - Fastest, Average and Quantile Schedule T2 - SOFSEM 2015: Theory and Practice of Computer Science N2 - We consider problems concerning the scheduling of a set of trains on a single track. For every pair of trains there is a minimum headway, which every train must wait before it enters the track after another train. The speed of each train is also given. Hence for every schedule - a sequence of trains - we may compute the time that is at least needed for all trains to travel along the track in the given order. We give the solution to three problems: the fastest schedule, the average schedule, and the problem of quantile schedules. The last problem is a question about the smallest upper bound on the time of a given fraction of all possible schedules. We show how these problems are related to the travelling salesman problem. We prove NP-completeness of the fastest schedule problem, NP-hardness of quantile of schedules problem, and polynomiality of the average schedule problem. We also describe some algorithms for all three problems. In the solution of the quantile problem we give an algorithm, based on a reverse search method, generating with polynomial delay all Eulerian multigraphs with the given degree sequence and a bound on the number of such multigraphs. A better bound is left as an open question. KW - scheduling KW - eulerian multigraphs Y1 - 2015 U6 - https://doi.org/10.1007/978-3-662-46078-8_17 SP - 201 EP - 216 PB - Springer Berlin Heidelberg ER - TY - GEN A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc T1 - The Steiner Connectivity Problem N2 - The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem, namely, the Steiner cut and the Steiner partition inequalities, as well as the associated polynomial time separation algorithms, can be generalized. Similar to the Steiner tree case, a directed formulation, which is stronger than the natural undirected one, plays a central role. T3 - ZIB-Report - 09-07 KW - Steiner Tree KW - Partition Inequalities KW - Paths KW - Connectivity KW - Generalization Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11171 SN - 1438-0064 ER - TY - GEN A1 - Neumann, Marika T1 - Fare Planning for Public Transport N2 - In this paper we investigate the fare planning model for public transport, which consists in designing a system of fares maximizing the revenue. We discuss a discrete choice model in which passengers choose between different travel alternatives to express the demand as a function of fares. Furthermore, we give a computational example for the city of Potsdam and discuss some theoretical aspects. T3 - ZIB-Report - 09-04 KW - fares KW - descrete choice Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11144 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf T1 - Combinatorial Packing Problems N2 - This article investigates a certain class of combinatorial packing problems and some polyhedral relations between such problems and the set packing problem. T3 - ZIB-Report - 03-03 KW - Packing problems KW - polyhedral combinatorics Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7255 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc T1 - Optimal Fares for Public Transport N2 - The \emph{fare planning problem} for public transport is to design a system of fares that maximize the revenue. We introduce a nonlinear optimization model to approach this problem. It is based on a d iscrete choice logit model that expresses demand as a function of the fares. We illustrate our approach by computing and comparing two different fare systems for the intercity network of the Netherlands. T3 - ZIB-Report - 05-35 KW - fare planning KW - demand function KW - discrete choice model Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8698 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas T1 - A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation Problem N2 - We propose a new coarse-to-fine approach to solve certain linear programs by column generation. The problems that we address contain layers corresponding to different levels of detail, i.e., coarse layers as well as fine layers. These layers are utilized to design efficient pricing rules. In a nutshell, the method shifts the pricing of a fine linear program to a coarse counterpart. In this way, major decisions are taken in the coarse layer, while minor details are tackled within the fine layer. We elucidate our methodology by an application to a complex railway rolling stock rotation problem. We provide comprehensive computational results that demonstrate the benefit of this new technique for the solution of large scale problems. T3 - ZIB-Report - 14-26 KW - column generation KW - coarse-to-fine approach KW - multi-layer approach KW - rolling stock rotation problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-51009 SN - 1438-0064 ER - TY - THES A1 - Hoppmann, Heide T1 - A Configuration Model for the Line Planning Problem N2 - In this thesis we present a novel extended formulation for the line planning problem that is based on what we call "configurations" of lines and frequencies. Configurations are combinatorial building blocks of primal solutions; they rule out the "capacity numerics" and make the problem purely combinatorial. The concept of configurations can also be adapted to other capacitated network design problems. The configuration model is strong in the sense that it implies several facet-defining inequalities for the standard model: set cover, symmetric band, multicover, and MIR inequalities. These theoretical findings can be confirmed in computations, however, the enormous number of configurations can blow up the formulation for large instances. We propose a mixed model that enriches the standard model by a judiciously chosen subset of configurations that provide a good compromise between model strength and size. Computational results for large-scale line planning problems are presented. N2 - Das Linienplanungsproblem ist ein wichtiges Teilproblem der Angebotsplanung im öffentlichen Nahverkehr. Dabei werden Routen und Betriebsfrequenzen von Linien in einem gegebenen Infrastrukturnetzwerk gesucht, sodass ein gegebener Beförderungsbedarf gedeckt wird und die Kosten minimal sind. Praktische Ansätze zum Lösen dieser Probleme beruhen auf ganzzahliger Programmierung. Ein Schwachpunkt klassischer Modelle ist, dass schon kleine Änderungen der Eingabeparameter eine Vergrößerung oder Verkleinerung der Menge der zulässigen fraktionalen Lösungen bewirken können, auch wenn die Menge der ganzzahligen Lösungen unverändert bleibt. Wir betrachten in dieser Arbeit einen neuen kombinatorischen Ansatz, welcher die Möglichkeiten den Bedarf auf einer Verbindung im Netzwerk mit Frequenzen zu überdecken mit Hilfe von diskreten "Konfigurationen" beschreibt und dieses Problem eliminiert. Um die Vorteile dieses Konfigurationsmodells aufzuzeigen, vergleichen wir es mit einem klassischen Ansatz, den wir Standardmodell nennen. Wir zeigen, dass das Konfigurationsmodell mehrere facettendefinierende Ungleichungen des Standardmodells impliziert: Setcover-, symmetrische Band-, Multicover- und MIR-Ungleichungen. Diese theoretischen Ergebnisse können wir mit Rechenergebnissen bestätigen. Um die Anzahl dieser Variablen auch für große Instanzen zu beschränken, schlagen wir ein weiteres Modell vor, welches nur eine Teilmenge der Konfigurationsvariablen enthält. Dieses stellt sich als ein sehr guter Kompromiss zwischen der Größe und der Stärke des Modells heraus. Da das Linienplanungsproblem eine Spezialisierung des Netzwerkdesignproblems ist, können unsere Methoden auch auf andere Problemklassen übertragen werden. Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-51610 ER - TY - CHAP A1 - Blanco, Marco A1 - Schlechte, Thomas T1 - Analysis of Micro–Macro Transformations of Railway Networks T2 - Operations Research Proceedings 2013 N2 - A common technique in the solution of large or complex optimization problems is the use of micro–macro transformations. In this paper, we carry out a theoretical analysis of such transformations for the track allocation problem in railway networks. We prove that the cumulative rounding technique of Schlechte et al. satisfies two of three natural optimality criteria and that this performance cannot be improved. We also show that under extreme circumstances, this technique can perform inconveniently by underestimating the global optimal value. Y1 - 2014 U6 - https://doi.org/10.1007/978-3-319-07001-8_6 SP - 37 EP - 42 ER - TY - CHAP A1 - Heismann, Olga A1 - Borndörfer, Ralf T1 - A Generalization of Odd Set Inequalities for the Set Packing Problem T2 - Operations Research Proceedings 2013 N2 - The set packing problem, sometimes also called the stable set problem, is a well-known NP-hard problem in combinatorial optimization with a wide range of applications and an interesting polyhedral structure, that has been the subject of intensive study. We contribute to this field by showing how, employing cliques, odd set inequalities for the matching problem can be generalized to valid inequalities for the set packing polytope with a clear combinatorial meaning. Y1 - 2014 U6 - https://doi.org/10.1007/978-3-319-07001-8_26 SP - 193 EP - 199 ER - TY - GEN A1 - Grötschel, Martin A1 - Borndörfer, Ralf T1 - Mathematik im Verkehr N2 - Nach einem kurzen Abriss über die Bedeutung des Verkehrssektors als eine wichtige Schlüsseltechnologie im gesamten Verlauf der Menschheitsgeschichte skizzieren wir die Rolle der Mathematik für Verkehr und Transport. Wir spekulieren dann über zukünftige Entwicklungen, insbesondere im Bereich des öffentlichen Personenverkehrs, und begründen, dass die in diesem Bereich anstehenden Herausforderungen nur mit dem Einsatz mathematischer Methoden angemessen bewältigt werden können. Die demographischen Prozesse, die in verschiedenen Teilen der Welt unterschiedlich verlaufen, wie z.B. Überalterung in Europa oder dynamische Trends zu Megastädten in Entwicklungsländern, sich ändernde Lebens- und Produktionsverhältnisse, stark wachsender Bedarf nach Mobilität und enormes Anwachsen der Komplexität der Verkehrsplanung und -durchführung, machen einen verstärkten Zugriff auf mathematische Modellierung, Simulation und Optimierung notwendig. Diese Entwicklung stellt sowohl große Herausforderungen an die Mathematik, wo vielfach noch keine geeigneten Methoden vorhanden sind, als auch an die Praktiker im Bereich von Verkehr und Transport, die sich mit neuen Planungs- und Steuerungstechnologien befassen und diese effizient einsetzen müssen. Hier wird intensive Kooperation zwischen vielen beteiligten Akteuren gefragt sein. T3 - ZIB-Report - 14-03 KW - Optimierung KW - öffentlicher Verkehr Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-50287 SN - 1438-0064 ER -