Routing Trains Through Railway Stations: Model Formulation and Algorithms

Abstract

In this paper we consider the problem of routing trains through railway stations. This problem occurs as a subproblem in a project which the authors are carrying out in cooperation with the Dutch railways. The project involves the analysis of future infrastructural capacity requirements in the Dutch railway network. Part of this project is the automatic generation and evaluation of timetables. To generate a timetable a hierarchical approach is followed: at the upper level in the hierarchy a tentative timetable is generated, taking into account the specific scheduling problems of the trains at the railway stations at an aggregate level. At the lower level in the hierarchy it is checked whether the tentative timetable is feasible with respect to the safety rules and the connection requirements at the stations. To carry out this consistency check, detailed schedules for the trains at the railway yards have to be generated. In this paper we present a mathematical model formulation for this detailed scheduling problem, based on the Node Packing Problem (NPP). Furthermore, we describe a solution procedure for the problem, based on a branch-and-cut approach. The approach is tested in an empirical study with data from the station ofZwolle in The Netherlands.

GRASP for set packing problems

“…In addition, modifications could be made to improve our reactive GRASP procedure and to use the learning process and reactive GRASP together more efficiently. Moreover, we could study the impact of some other preprocessing phases to reduce the size of the problem and to improve our upper bounds: reduction tests like those used by Zwaneveld et al [11] and valid inequalities as defined by Padberg [8] constitute two likely directions for research.…”

“…To design the feasibility problem, binary decision variables are introduced. These decision variables are This formulation is close to the model developed in Zwaneveld et al [11], in which the decisions of the entrance times were not included. The objective is to maximize the number of trains that can be routed through the junction without preference between trains.…”

“…R€ o onnqvist [10] worked on a cutting stock problem formulated as an SPP and solved using a Lagrangian relaxation combined with subgradient optimization. Zwaneveld et al [11] formulated a real railway feasibility problem as an SPP and solved it exactly using reduction tests and a Branch & Cut method. Kim [12] represented a ship scheduling problem as an SPP and used LINDO software to solve it.…”

“…• The values of items (c i ) are uniformly distributed in the interval [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] (weighted) or fixed to 1 (unicost).…”

“…Measuring the capacity of junctions is a matter of solving an optimization problem called the feasibility problem. As mentioned in Section 2, this problem has been formulated in terms of an SPP by Zwaneveld et al [11] and more recently by Delorme et al [3,4]. The feasibility problem of a junction can be stated as follows:…”