Integrating Passengers\u27 Routes in Periodic Timetabling: A SAT approach

The periodic event scheduling problem (PESP) is a well studied problem known as intrinsically hard. Its main application is for designing periodic timetables in public transportation. To this end, the passengers\u27 paths are required as input data. This is a drawback since the final paths which are used by the passengers depend on the timetable to be designed. Including the passengers\u27 routing in the PESP hence improves the quality of the resulting timetables. However, this makes PESP even harder. Formulating the PESP as satisfiability problem and using SAT solvers for its solution has been shown to be a highly promising approach. The goal of this paper is to exploit if SAT solvers can also be used for the problem of integrated timetabling and passenger routing. In our model of the integrated problem we distribute origin-destination (OD) pairs temporally through the network by using time-slices in order to make the resulting model more realistic. We present a formulation of this integrated problem as integer program which we are able to transform to a satisfiability problem. We tested the latter formulation within numerical experiments, which are performed on Germany\u27s long-distance passenger railway network. The computation\u27s analysis in which we compare the integrated approach with the traditional one with fixed passengers\u27 weights, show promising results for future scientific investigations

The line planning routing game

In this paper, we propose a novel algorithmic approach to solve line planning problems. To this end, we model the line planning problem as a game where the passengers are players which aim at minimizing individual objective functions composed of travel time, transfer penalties, and a share of the overall cost of the solution. To find equilibria of this routing game, we use a best-response algorithm. We investigate, under which conditions on the line planning model a passenger’s best-response can be calculated efficiently and which properties are needed to guarantee convergence of the best-response algorithm. Furthermore, we determine the price of anarchy which bounds the objective value of an equilibrium with respect to a system- optimal solution of the line planning problem. For problems where best-responses cannot be found efficiently, we propose heuristic methods. We demonstrate our findings on some small computational examples

Line pool generation

Finding the lines and their frequencies in public transportation is the well-studied line planning problem. In this problem, it is common to assume that a line pool consisting of a set of potential lines is given. The goal is to choose a set of lines from the line pool that is convenient for the passengers and has low costs. The chosen lines then form the line plan to be established by the public transportation company. The line pool hence has a significant impact on the quality of the line plan. The more lines are in the line pool, the more flexible can we choose the resulting line plan and hence increase its quality. It hence would be preferable to allow all possible lines to choose from. However, the resulting instances of the line planning problem become intractable if all lines would be allowed. In this work, we study the effect of line pools for line planning models and propose an algorithm to generate ‘good’ line pools. To this end, we formally introduce the line pool generation problem and investigate its properties. The line pool generation problem asks for choosing a subset of paths (the line pool) of limited cardinality such that in a next step a good line concept can be constructed based on this subset. We show that this problem is NP-hard. We then discuss how reasonable line pools may be constructed. Our approach allows to construct line pools with different properties and even to engineer the properties of the pools to fit to the objective function of the line planning model to be used later on. Our numerical experiments on close-to real-world data show that the quality of a line plan significantly depends on the underlying line pool, and that it can be influenced by the parameters of our approach