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This chapter shows a successful approach how to model and optimize rolling stock rotations that are required for the operation of a passenger timetable. The underlying mathematical optimization problem is described in detail and solved by RotOR, i.e., a complex optimization algorithm based on linear programming and combinatorial methods. RotOR is used by DB Fernverkehr AG (DBF) in order to optimize intercity express (ICE) rotations for the European high-speed network. We focus on main modeling and solving components, i.e. a hypergraph model and
a coarse-to-fine column generation approach. Finally, the chapter concludes with a complex industrial re-optimization application showing the effectiveness of the approach for real world challenges.
Railway Track Allocation
(2018)
This chapter addresses the classical task to decide which train runs on which track in a railway network. In this context a track allocation defines the precise routing of trains through a railway network, which usually has only a limited capacity.
Moreover, the departure and arrival times at the visited stations of each train must simultaneously meet several operational and safety requirements.
The problem to find the 'best possible' allocation for all trains
is called the track allocation problem (TTP).
Railway systems can be modeled on a very detailed scale covering the behavior of individual trains and the safety system to a large extent. However, those microscopic models are too big and not scalable to large networks, which make them inappropriate for mathematical optimization on a network wide level. Hence, most network optimization approaches consider simplified, so called macroscopic, models. In the first part we take a look at the challenge to construct a reliable and condensed macroscopic model for the associated microscopic model and to facilitate the transition between both models of different scale.
In the main part we focus on the optimization problem for macroscopic models of the railway system. Based on classical graph-theoretical tools the track allocation problem is formulated to determine conflict-free paths in corresponding time-expanded graphs. We present standard integer programming model formulations for the track allocation problem that model resource or block conflicts in terms of packing constraints. In addition, we discuss the role of maximal clique inequalities and the concept of configuration networks. We will also present classical decomposition approaches like Lagrangian relaxation and bundle methods. Furthermore, we will discuss recently developed techniques, e.g., dynamic graph generation.
Finally, we will discuss the status quo and show a vision of mathematical optimization to support real world track allocation, i.e. integrated train routing and scheduling, in a data-dominated
and digitized railway future.
Line Planning Problem
(2023)