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Robust Tail Assignment
(2010)
We propose an efficient column generation method to minimize the probability of delay propagations along aircraft rotations. In this way, delay resistant schedules can be constructed. Computational results for large-scale real-world problems demonstrate substantial punctuality improvements. The method can be generalized to crew and integrated scheduling problems.
We propose a novel integer programming approach to transfer minimization for line planning problems in public transit. The idea is to incorporate penalties for transfers that are induced by “connection capacities” into the construction of the passenger paths. We show that such penalties can be dealt with by a combination of shortest and constrained shortest path algorithms such that the pricing problem for passenger paths can be solved efficiently. Connection capacity penalties (under)estimate the true transfer times. This error is, however, not a problem in practice. We show in a computational comparison with two standard models on a real-world scenario that our approach can be used to minimize passenger travel and transfer times for large-scale line planning problems with accurate results.
Wir stellen in dieser Arbeit ein mathematisches Optimierungsmodell zur Bestimmung eines optimalen Linienplans vor, das sowohl die Fahrzeiten und die Anzahl der Umstiege berücksichtigt als auch die Kosten des Liniennetzes. Dieses Modell deckt wichtige praktische Anforderungen ab, die in einem gemeinsamen Projekt mit den Verkehrsbetrieben in Potsdam (ViP) formuliert wurden. In diesem Projekt wurde der Linienplan 2010 für Potsdam entwickelt. Unsere Berechnungen zeigen, dass die mathematische Optimierung in nichts einer "Handplanung" des Liniennetzes nachsteht. Im Gegenteil, mit Hilfe des Optimierungsprogramms ist es möglich, durch Veränderung der Parameter mehrere verschiedene Szenarien zu berechnen, miteinander zu vergleichen und Aussagen über minimale Kosten und Fahrzeiten zu machen.
The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.
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.
This paper presents a case study on a railway timetable optimization for the very dense Simplon corridor, a major railway connection in the Alps between Switzerland and Italy. Starting from a detailed microscopic network as it is used in railway simulation, the data is transformed by an automatic procedure to a less detailed macroscopic network, that is sufficient for the purpose of capacity planning and amenable to state-of-the-art integer programming optimization methods. In this way, the macroscopic railway network is saturated with trains. Finally, the corresponding timetable is re-transformed to the microscopic level in such a way that it can be operated without any conflicts among the slots. Using this integer programming based micro-macro aggregation-disaggregation approach, it becomes for the first time possible to generate a profit maximal and conflict free timetable for the complete Simplon corridor over an entire day by a simultaneous optimization of all trains requests. This also allows to to undertake a sensitivity analysis of various problem parameters.
This cumulative thesis collects the following six papers for obtaining the
habilitation at the Technische Universität Berlin, Fakultät II – Mathematik
und Naturwissenschaften:
(1) Set packing relaxations of some integer programs.
(2) Combinatorial packing problems.
(3) Decomposing matrices into blocks.
(4) A bundle method for integrated multi-depot vehicle and duty scheduling
in public transit.
(5) Models for railway track allocation.
(6) A column-generation approach to line planning in public transport.
Some changes were made to the papers compared to the published versions.
These pertain to layout unifications, i.e., common numbering, figure, table,
and chapter head layout. There were no changes with respect to notation or
symbols, but some typos have been eliminated, references updated, and some
links and an index was added. The mathematical content is identical.
The papers are about the optimization of public transportation systems,
i.e.,
bus networks, railways, and airlines, and its mathematical foundations,
i.e.,
the theory of packing problems. The papers discuss mathematical models,
theoretical analyses, algorithmic approaches, and computational aspects of
and to problems in this area.
Papers 1, 2, and 3 are theoretical. They aim at establishing a theory of
packing problems as a general framework that can be used to study traffic
optimization problems. Indeed, traffic optimization problems can often be
modelled as path packing, partitioning, or covering problems, which lead
directly to set packing, partitioning, and covering models. Such models are
used in papers 4, 5, and 6 to study a variety of problems concerning the
planning
of line systems, buses, trains, and crews. The common aim is always
to exploit as many degrees of freedom as possible, both at the level of the
individual problems by using large-scale integer programming techniques, as
well as on a higher level by integrating hitherto separate steps in the
planning
process.