Optimising halting station of passenger railway lines Jan-Willem Goossens ∗ Stan van Hoesel † Leo Kroon ‡ April 9, 2004 Abstract In many real life passenger railway networks, the types of stations and lines characterise the halting stations of the train lines. Common types are Regional, Interregional or Intercity. This paper considers the problem of altering the halts of lines by both upgrading and down- grading stations, such that this results in less total travel time. We propose a combination of reduction methods, Lagrangian relaxation, and a problem-specific multiplier adjustment algorithm to solve the presented mixed integer linear programming formulation. A compu- tational study of several real-life instances based on problem data of the Dutch passenger railway operator NS Reizigers is included. Keywords : Mixed integer programming; Railway problems; Lagrangian relaxation 1 Introduction The planning problem faced by railway operators is a complex multi-staged problem. The individ- ual decision problems range from the strategic line planning problem (see Bussieck [2], Goossens et al. [6, 7], Zwaneveld [16]) via the construction of timetables (see Nachtigall [10], Odijk [12], Peeters [13], Schrijver and Steenbeek [15]), to traffic planning (see Zwaneveld [16]), rolling stock planning (see Schrijver [14]) and personnel (see Caprara et al. [3]), and shunting planning (see Gallo and Di Miele [4]). The line planning problems in Bussieck [2], Goossens et al. [6], Zwaneveld [16] and Goossens et al. [7] describe the decision problem of finding routes in the railway network on which trains are to be operated. However, the stations at which these lines halt are dictated by the types of the stations and lines, as we will describe later. These types are part of the problem input. In contrast, this paper considers the line plan to be given, but concentrates on altering the stops of the lines along their route to decrease the total travel time of passengers through the network. First, in §2, we introduce new concepts such as the line-event graph. Then, in §3 we show how to formulate the problem of optimising halting station of passenger railway lines as a multi-commodity network flow problem with additional constraints and variables. Using Lagrangian relaxation, we show in §4 how to find lower bounds for this problem. To effectively use these bounds in a branch- and-bound framework, as described in §5, we introduce a number of preprocessing and tree search techniques, together with a problem-specific multiplier adjustment algorithm. Finally, in §6, we describe a computational study based on instances of the Dutch passenger railway operator NSR. ∗ Dept of Quantitative Economics, University of Maastricht, P.O. Box 616, 6200 MD Maastricht, The Nether- lands. E-mail: j.goossens@t75.nl † Dept of Quantitative Economics, University of Maastricht, P.O. Box 616, 6200 MD Maastricht, The Nether- lands. E-mail: s.vanhoesel@ke.unimaas.nl ‡ Rotterdam School of Management, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands. E-mail: l.kroon@fbk.eur.nl 1