Library

Description: Railway scheduling is based on the principle of the construction of a conflict-free timetable. This leads to a strict definition of capacity: in contrast with road transportation, it can be said in advance whether a given railway infrastructure can accommodate - at least in theory - a certain set of train requests. Consequently, auctions for railway capacity are modeled as auctions of discrete goods -- the train slots. We present estimates for the efficiency gain that may be generated by slot auctioning in comparison with list price allocation. We introduce a new class of allocation and auction problems, the feasible assignment problem, that is a proper generalization of the well-known combinatorial auction problem. The feasible assignment class was designed to cover the needs for an auction mechanism for railway slot auctions, but is of interest in its own right. As a practical instance to state and solve the railway slot allocation problem, we present an integer programming formulation, briefly the ACP, which turns out to be an instance of the feasible assignment problem and whose dual problem yields prices that can be applied to define a useful activity rule for the linearized version of the Ausubel Milgrom Proxy auction. We perform a simulation aiming to measure the impact on efficiency and convergence rate.

Description: In this paper a bottom-up approach of automatic simplification of a railway network is presented. Starting from a very detailed, microscopic level, as it is used in railway simulation, the network is transformed by an algorithm to a less detailed level (macroscopic network), that is sufficient for long-term planning and optimization. In addition running and headway times are rounded to a pre-chosen time discretization by a special cumulative method, which we will present and analyse in this paper. After the transformation we fill the network with given train requests to compute an optimal slot allocation. Then the optimized schedule is re-transformed into the microscopic level and can be simulated without any conflicts occuring between the slots. The algorithm is used to transform the network of the very dense Simplon corridor between Swiss and Italy. With our aggregation it is possible for the first time to generate a profit maximal and conflict free timetable for the corridor across a day by a simultaneously optimization run.

Description: 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.

Description: 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.

Description: 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.

Description: This article gives an overview of the results of the author's PhD thesis. The thesis deals with the 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. 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. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland.

Description: This paper provides a highly integrated solution approach for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a generic hypergraph based mixed integer programming model and an integrated algorithm for the considered rolling stock rotation planning problem. The new developed approach is able to handle a very large set of industrial railway requirements, such as vehicle composition, maintenance constraints, infrastructure capacity, and regularity aspects. By the integration of this large bundle of technical railway aspects, we show that our approach has the power to produce implementable rolling stock rotations for our industrial cooperation partner DB Fernverkehr. This is the first time that the rolling stock rotations at DB Fernverkehr could be optimized by an automated system utilizing advanced mathematical programming techniques.

Description: Sports rankings are obtained by applying a system of rules to evaluate the performance of the participants in a competition. We consider rankings that result from assigning an ordinal rank to each competitor according to their performance. We develop an integer programming model for rankings that allows us to calculate the number of points needed to guarantee a team the ith position, as well as the minimum number of points that could yield the ith place. The model is very general and can thus be applied to many types of sports. We discuss examples coming from football (soccer), ice hockey, and Formula~1. We answer various questions and debunk a few myths along the way. Are 40 points enough to avoid relegation in the Bundesliga? Do 95 points guarantee the participation of a team in the NHL playoffs? Moreover, in the season restructuration currently under consideration in the NHL, will it be easier or harder to access the playoffs? Is it possible to win the Formula~1 World Championship without winning at least one race or without even climbing once on the podium? Finally, we observe that the optimal solutions of the aforementioned model are associated to extreme situations which are unlikely to happen. Thus, to get closer to realistic scenarios, we enhance the model by adding some constraints inferred from the results of the previous years.

Description: 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, HGF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and HGF 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.