10. Fast Approaches to Robust Railway Timetabling

Fast Approaches to Robust Railway Timetabling Matteo Fischetti, Domenico Salvagnin, and Arrigo Zanette DEI, University of Padova, Italy Abstract. The Train Timetabling Problem (TTP) consists in finding a train schedule on a railway network that satisfies some operational constraints and maximizes a profit function which counts for the efficiency of the infrastructure usage. In practical cases, however, the maximization of the objective function is not enough and one calls for a robust solution that is capable of absorbing as much as possible delays/disturbances on the network. In this paper we propose and analyze computationally four different methods to find robust TTP solutions for the aperiodic (non cyclic) case, that combine Mixed Integer Programming (MIP) and ad-hoc Stochastic Programming/Robust Optimization techniques. We compare computationally the effectiveness and practical applicability of the four techniques under investigation on real-world test cases from the Italian railway company (Trenitalia). The outcome is that two of the proposed techniques are very fast and provide robust solutions of comparable quality with respect to the standard (but very time consuming) Stochastic Programming approach. Keywords: timetabling, integer programming, robustness, stochastic programming, robust optimization. 1 Introduction The Train Timetabling Problem (TTP) consists in finding an effective train schedule on a given railway network. The schedule needs to satisfy some operational constraints given by capacities of the network and security measures. Moreover, it is required to exploit efficiently the resources of the railway infrastructure. In many situations, the efficiency is measured as the distance of the solution from an input “ideal schedule” that optimally satisfies the network demands. In practice, however, the maximization of some objective function is not enough: the solution is also required to be robust against delays/disturbances along the network. Very often, the robustness of optimal solutions of the original problem turns out to be not enough for their practical applicability, whereas easy-to-compute robust solutions tend to be too conservative and thus unnecessarily inefficient. As a result, practitioners call for a fast yet accurate method to find the most robust timetable whose efficiency is only slightly smaller than the theoretical optimal one. The purpose of the present paper is to propose and evaluate new methods to find robust and efficient solutions to the TTP, in its aperiodic (non cyclic) ATMOS 2007 (p.142-157) 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems http://drops.dagstuhl.de/opus/volltexte/2007/1176 Fast Approaches to Robust Railway Timetabling 143 version described in [2]. Our approach combines Mixed Integer Programming (MIP) with Stochastic Programming (SP) and Robust Optimization techniques. We developed a solution framework whose main building blocks are: (1) a solver, used to obtain a tentative timetable by solving an event-based MIP model; (2) a (local) trainer that uses Stochastic Programming or Robust Optimization techniques to improve the robustness of the tentative solution by changing the train departure/arrival times without altering the combinatorial structure of the tentative timetable (train precedences being preserved); and (3) a black-box validation tool, used to quantify the robustness of the solutions found by different approaches. The paper is organized as follows. In Section 2 we present the TTP in detail and give a natural event-based MIP formulation. In Section 3 we present our overall solution framework, whose two main building blocks are described in Sections 4 and 5. Extensive computational results are given in Section 7, showing that two of the new methods we propose are very fast and provide robust solutions of comparable quality with respect to the standard (but very time consuming) Stochastic Programming approach. Finally, some conclusions are drawn in Section 8. 2 The Nominal Model In this section we describe the specific aperiodic TTP problem we consider, and give a basic event-based formulation for the “nominal” version where robustness is not taken into account. Following [2], the aperiodic TTP can be formulated as follows: Given a railway network, described as a set of stations connected by tracks, and an ideal train timetable, find an actual train schedule satisfying all the operational constraints and having a minimum distance from the ideal timetable. The entities involved in the description of the problem are the following: railway network: a graph N = (S, L), where S is the set of stations and L is the set tracks connecting them. trains: a train is a simple path on the railway network N . The set of trains is denoted by T . For each train h ∈ T we have an ideal profit πh (the profit of the train if scheduled exactly as in the ideal timetable), a stretch penalty θh (the train stretch being defined as the difference between the running times in the actual and ideal timetables) and a shift penalty σh (the train shift being defined as the absolute difference between the departure times from the first station in the actual and ideal timetables). events: arrivals and departures of the trains at the stations. The set of all the events is denoted by E. With a small abuse of notation, we will denote by thi both the i-th event of train h and its associated time. We also define – A: set of all arrival events – D: set of all departure events whereas AS , DS and ES denote the restriction of the above sets to a particular station S. Each train h is associated with an ordered sequence of length 144 Matteo Fischetti et al. len(h) of departure/arrival events thi such that thi+1 ≥ thi , the first and last event of train h being denoted by th1 and thlen(h) , respectively. (partial) schedule: a time assignment to all the events associated with a subset of trains. objective: maximize the overall profit of the scheduled trains, the profit of train h being computed as πh − σh shif th − θh stretchh i.e., the train profit decreases if the actual timetable diverges from the ideal one; trains with negative profit are intended to remain unscheduled and do not contribute to the overall profit. Operational constraints include: time window: it is possible to shift an event from its ideal time only within a given time window; headway time: for safety reasons, a minimum time distance between two consecutive arrival/departure events from the same station is imposed; track capacity: overtaking between trains is allowed only within stations (assumed of infinite capacity). Although one is allowed to leave some trains unscheduled, to simplify our presentation we consider first a non-congested network where one is required to schedule all the trains. A natural event-based model in the spirit of the Periodic Event Scheduling Problem (PESP) formulation used in the periodic (cy (...truncated)