No abstract

In research and practice, public transportation planning is executed in a series of steps, which are often divided into the strategic, the tactical, and the operational planning phase. Timetables are normally designed in the tactical phase, taking into account a given line plan, safety restrictions arising from infrastructural constraints, as well as regularity requirements and bounds on transfer times. In this paper, however, we propose a timetabling approach that is aimed at decision making in the strategic phase of public transportation planning and to determine an outline of a timetable that is good from the passengers' perspective. Instead of including explicit synchronization constraints between train runs (as most timetabling models do), we include the adaption time (waiting time at the origin station) in the objective function to ensure regular connections between passengers' origins and destinations. We model the problem as a mixed integer quadratic program and linearise it. Furthermore we propose a heuristic to generate starting solutions. We illustrate the type of solutions found by our approach on two case studies based on the Dutch railway network and analyse trade-offs that are made to balance dwell times and regularity of trains.

One of the key assumptions of timetabling algorithms is that a solution exists that meets the pre-specified constraints, like driving times, transfer constraints and headway constraints. If this assumption is satisfied, in most cases a timetable can be found rapidly. Nowadays, railways are being used more intensively, which leads to a higher utilization of the network. Due to this increased utilisation, capacity conflicts occur, so that no feasible solution to the timetabling models can be found, without making subtle but non-trivial changes to the initial input. Resolving these conflicts is essential for railway companies with high utilization of infrastructure. In this paper, we consider infeasible timetabling instances together with a list of allowed modifications of the constraints. We iteratively identify local conflicts in these instances and resolve them by adapting some of the constraints, until there are no more conflicts. The adaptations of the constraints are changes in the right-hand sides that we try to make as small as possible but that resolve the infeasibility. We empirically show that our method can be improved by enriching the initial minimal conflicts found with more constraints. In order to keep the problems tractable, an iterative procedure is used to find solutions to subproblems corresponding to conflicts in the complete timetabling instance. In a case study on instances from the Dutch railway network, we show that these instances can be made feasible within a few minutes.

In this paper, we consider the Robust Periodic Timetabling Problem (RPTP), the problem of designing an adjustable robust periodic timetable. We develop a solution method for a parametrized class of uncertainty regions. This class relates closely to uncertainty regions known in the robust optimization literature, and naturally defines a metric for the robustness of the timetable. The proposed solution method combines a linear decision rule with well-known reformulation techniques and cutting-plane methods. We show that the RPTP can be solved for practical-sized instances by applying the solution method to practical cases of Netherlands Railways (NS). In particular, we show that the trade-off between the efficiency and robustness of a timetable can be analyzed using our solution method.

We aim at constructing a timetable that minimizes average perceived passenger travel time, which, in addition to the in-train and transfer times, includes the adaption time (waiting time at the origin station). Adaption time minimization allows us to avoid strict frequency regularity constraints and, at the same time, to ensure regular connections between passengers' origins and destinations. Besides considering safety restrictions (i.e., headway times, overtaking and crossing constraints), passenger routing, based on origindestination demand pairs, must be taken into account when building the timetable. This problem can be modelled as an extension of a Periodic Event Scheduling Problem (PESP) formulation, but cannot be directly solved by a general-purpose solver for our real-size instances. In this paper, we propose a heuristic approach consisting of two phases that are executed iteratively. First, we solve a simplified timetabling model, and determine an ideal timetable that minimizes the average perceived passenger travel time but neglects safety restrictions. Then, a Lagrangian-based heuristic makes the timetable feasible by modifying train departure and arrival times as little as possible. The obtained feasible timetable is then evaluated to compute the resulting average perceived passenger travel time, and a feedback is sent to the Lagrangian-based heuristic so as to possibly improve the obtained timetable from the passenger perspective, while still respecting safety constraints. We have tested the proposed iterative heuristic approach on real-life instances of Netherlands Railways, showing that it converges to a feasible timetable very close to the ideal one.