A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation Problem

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

Strong Relaxations for the Train Timetabling Problem Using Connected Configurations

The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral

A Hypergraph Model for Railway Vehicle Rotation Planning

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

Does Laziness Pay Off? - A Lazy-Constraint Approach to Timetabling

Timetabling is a classical and complex task for public transport operators as well as for railway undertakings. The general question is: Which vehicle is taking which route through the transportation network in which order? In this paper, we consider the special setting to find optimal timetables for railway systems under a moving block regime. We directly set up on our work of [T. Schlechte et al., 2022], i.e., we consider the same model formulation and real-world instances of a moving block headway system. In this paper, we present a repair heuristic and a lazy-constraint approach utilizing the callback features of Gurobi, see [Gurobi Optimization, 2022]. We provide an experimental study of the different algorithmic approaches for a railway network with 100 and up to 300 train requests. The computational results show that the lazy-constraint approach together with the repair heuristic significantly improves our previous approaches

Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind

We study the Flight Planning Problem for a single aircraft, which deals with finding a path of minimal travel time in an airway network. Flight time along arcs is affected by wind speed and direction, which are functions of time. We consider three variants of the problem, which can be modeled as, respectively, a classical shortest path problem in a metric space, a time-dependent shortest path problem with piecewise linear travel time functions, and a time-dependent shortest path problem with piecewise differentiable travel time functions. The shortest path problem and its time-dependent variant have been extensively studied, in particular, for road networks. Airway networks, however, have different characteristics: the average node degree is higher and shortest paths usually have only few arcs. We propose A* algorithms for each of the problem variants. In particular, for the third problem, we introduce an application-specific "super-optimal wind" potential function that overestimates optimal wind conditions on each arc, and establish a linear error bound. We compare the performance of our methods with the standard Dijkstra algorithm and the Contraction Hierarchies (CHs) algorithm. Our computational results on real world instances show that CHs do not perform as well as on road networks. On the other hand, A* guided by our potentials yields very good results. In particular, for the case of piecewise linear travel time functions, we achieve query times about 15 times shorter than CHs

Line planning on path networks with application to the Istanbul Metrobüs

Bus rapid transit systems in developing and newly industrialized countries often consist of a trunk with a path topology. On this trunk, several overlapping lines are operated which provide direct connections. The demand varies heavily over the day, with morning and afternoon peaks typically in reverse directions. We propose an integer programming model for this problem, derive a structural property of line plans in the static (or single period) “unimodal demand” case, and consider approaches to the solution of the multi-period version that rely on clustering the demand into peak and off-peak service periods. An application to the Metrobüs system of Istanbul is discussed

Cancer Biomarker Discovery: The Entropic Hallmark

Background: It is a commonly accepted belief that cancer cells modify their transcriptional state during the progression of the disease. We propose that the progression of cancer cells towards malignant phenotypes can be efficiently tracked using high-throughput technologies that follow the gradual changes observed in the gene expression profiles by employing Shannon's mathematical theory of communication. Methods based on Information Theory can then quantify the divergence of cancer cells' transcriptional profiles from those of normally appearing cells of the originating tissues. The relevance of the proposed methods can be evaluated using microarray datasets available in the public domain but the method is in principle applicable to other high-throughput methods. Methodology/Principal Findings: Using melanoma and prostate cancer datasets we illustrate how it is possible to employ Shannon Entropy and the Jensen-Shannon divergence to trace the transcriptional changes progression of the disease. We establish how the variations of these two measures correlate with established biomarkers of cancer progression. The Information Theory measures allow us to identify novel biomarkers for both progressive and relatively more sudden transcriptional changes leading to malignant phenotypes. At the same time, the methodology was able to validate a large number of genes and processes that seem to be implicated in the progression of melanoma and prostate cancer. Conclusions/Significance: We thus present a quantitative guiding rule, a new unifying hallmark of cancer: the cancer cell's transcriptome changes lead to measurable observed transitions of Normalized Shannon Entropy values (as measured by high-throughput technologies). At the same time, tumor cells increment their divergence from the normal tissue profile increasing their disorder via creation of states that we might not directly measure. This unifying hallmark allows, via the the Jensen-Shannon divergence, to identify the arrow of time of the processes from the gene expression profiles, and helps to map the phenotypical and molecular hallmarks of specific cancer subtypes. The deep mathematical basis of the approach allows us to suggest that this principle is, hopefully, of general applicability for other diseases

05. Models for Railway Track Allocation

The optimal track allocation problem (OPTRA) is to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations for this problem: a standard formulation that models block conflicts in terms of packing constraints, and a novel formulation of the `extended\u27 type that is based on additional `configuration\u27 variables. The packing constraints in the standard formulation stem from an interval graph and can therefore be separated in polynomial time. It follows that the LP-relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the LP-relaxation of the extended formulation can also be solved in polynomial time, and that it produces the same LP-bound. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view. It features a constant number of rows and is amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hanover-Fulda-Kassel region of the German long distance railway network are reported

Trassenallokation im Schienenverkehr: Modelle und Algorithmen

Diese Arbeit befasst sich mit der mathematischen Optimierung zur effizienten Nutzung der Eisenbahninfrastruktur. Wir behandeln die optimale Allokation der zur Verfügung stehenden Kapazität eines Eisenbahnschienennetzes - das Trassenallokationsproblem. Das Trassenallokationsproblem stellt eine wesentliche Herausforderung für jedes Bahnunternehmen dar, unabhängig, ob ein freier Markt, ein privates Monopol, oder ein öffentliches Monopol vorherrscht. Die Planung und der Betrieb eines Schienenverkehrssystems ist extrem schwierig aufgrund der kombinatorischen Komplexität der zugrundeliegenden diskreten Optimierungsprobleme, der technischen Besonderheiten, und der immensen Größen der Probleminstanzen. Mathematische Modelle und Optimierungstechniken können zu enormen Nutzen führen, sowohl für die Kunden der Bahn als auch für die Betreiber, z.B. in Bezug auf Kosteneinsparungen und Verbesserungen der Servicequalität.Wir lösen diese Herausforderung durch die Entwicklung neuartiger mathematischer Modelle und der dazughörigen innovativen algorithmischen Lösungsmethoden für sehr große Instanzen. Dadurch waren wir erstmals in der Lage zuverlässige ösungen für Instanzen der realen Welt, d.h. für den Simplon Korridor in der Schweiz, zu produzieren. Der erste Teil beschäftigt sich mit der Modellierung des Schienenbahnsystems unter Berücksichtigung von Kapazität und Ressourcen. Kapazität im Schienenverkehr hat grundsätzlich zwei Dimensionen, eine räumliche, welche der physischen Infrastruktur entspricht, und eine zeitliche, die sich auf die Zugbewegungen innerhalb dieser bezieht, d.h. die Belegung- und Blockierungszeiten. Sicherungssysteme im Schienenverkehr beruhen überall auf der Welt auf demselben Prinzip. Ein Zug muss Blöcke der Infrastruktur für die Durchfahrt reservieren. Das gleichzeitige Belegen eines Blockes durch zwei Züge wird Blockkonflikt genannt. Um eine konfliktfreie Belegung zu erreichen, beinhalten Modelle zur Kapazität im Schienenverkehr daher die Definition und Berechnung von angemessenen Fahrzeiten und dementsprechenden Reservierungs- oder Blockierungszeiten. Im zweiten und Hauptteil der Dissertation wird das Problem des Bestimmens optimaler Trassenallokationen für makroskopische Bahnmodelle betrachtet. Ein Literaturüberblick zu verwandten Problemen wird gegeben. Für das Trassenallokationsproblem wird ein graphentheoretisches Modell entwickelt, in dem optimale ösungen als maximal gewichtete konfliktfreie Menge von Pfaden in speziellen zeitexpandierten Graphen dargestellt werden können. Des Weiteren erreichen wir wesentliche Fortschritte beim Lösen von Trassenallokationsprobleme durch zwei Hauptbeiträge - die Entwickling einer neuartigen Modellformulierung des makroskopischen Trassenallokationsproblemes und algorithmische Verbesserungen basierend auf der Nutzung des Bündelverfahrens. Den Höhepunkt bilden Resultate für Praxisszenarios zum Simplon Korridor in der Schweiz. Nach bestem Wissen des Autors und bestätigt durch zahlreiche Eisenbahnpraktiker ist dies das erste Mal, dass auf einer makroskopischen Ebene automatisch erstellte Trassenallokationen die Bedingungen des ursprünglichen mikroskopischen Modells erfüllen und der Evaluierung innerhalb des mikroskopischen Simulationstools OpenTrack standhalten. Das dokumentiert den Erfolg unseres Ansatzes und den Nutzen and die Anwendbarkeit mathematischer Optimierung zur Allokation von Trassen im Schienenverkehr.This thesis is about mathematical optimization for the efficient use of railway infrastructure. We address the optimal allocation of the available railway track capacity the track allocation problem. This track allocation problem is a major challenge for a railway company, independent of whether a free market, a private monopoly, or a public monopoly is given. 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 sizes of the problem instances. Mathematical models and optimization techniques can result in huge gains for both railway customers and operators, e.g., in terms of cost reductions or service quality improvements. We tackle this challenge by developing novel mathematical models and associated innovative algorithmic solution methods for large scale instances. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland. The opening chapter gives a comprehensive overview on railway planning problems. This provides insights into the regulatory and technical framework, it discusses the interaction of several planning steps, and identifies optimization potentials in railway transportation. The remainder of the thesis is comprised of two major parts. The first part is concerned with modeling railway systems to allow for resource and capacity analysis. Railway capacity has basically two dimensions, a space dimension which are the physical infrastructure elements as well as a time dimension that refers to the train movements, i.e., occupation or blocking times, on the physical infrastructure. Railway safety systems operate on the same principle all over the world. A train has to reserve infrastructure blocks for some time to pass through. Two trains reserving the same block of the infrastructure within the same point in time is called block conflict. Therefore, models for railway capacity involve the definition and calculation of reasonable running and associated reservation and blocking times to allow for a conflict free allocation. In the second and main part of the thesis, the optimal track allocation problem for macroscopic models of the railway system is considered. The literature for related problems is surveyed. A graph-theoretic model for the track allocation problem is developed. In that model optimal track allocations correspond to conflict-free paths in special time-expanded graphs. Furthermore, we made considerable progress on solving track allocation problems by two main features - a novel modeling approach for the macroscopic track allocation problem and algorithmic improvements based on the utilization of the bundle method. Finally, we go back to practice and present in the last chapter several case studies using the tools netcast and tsopt. We provide a computational comparison of our new models and standard packing models used in the literature. Our computational experience indicates that our approach, i.e., configuration models'', outperforms other models. Moreover, the rapid branching heuristic and the bundle method enable us to produce high quality solutions for very large scale instances, which has not been possible before. In addition, we present results for a theoretical and rather visionary auction framework for track allocation. We discuss several auction design questions and analyze experiments of various auction simulations. The highlights are results for the Simplon corridor in Switzerland. We optimized the train traffic through this tunnel using our models and software tools. To the best knowledge of the author and confirmed by several railway practitioners this was the first time that fully automatically produced track allocations on a macroscopic scale fulfill the requirements of the originating microscopic model, withstand the evaluation in the microscopic simulation tool OpenTrack, and exploit the infrastructure capacity. This documents the success of our approach in practice and the usefulness and applicability of mathematical optimization to railway track allocation