Routing Trains through railway stations: model formulation and algorithms

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 cheek, 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 of Zwolle in The Netherlands

MATHEMATICAL MODELS AND LAGRANGIAN HEURISTICS FOR A TWO-LEVEL LOT-SIZING PROBLEM WITH BOUNDED INVENTORY

We consider a two-level lot-sizing problem where the first level consists of N end products competing for a single type of raw material (second level), which is supposed to be critical. In particular, the storage capacity of raw materials is limited and must be carefully managed. The goal is to simultaneously determine an optimal replenishment plan for the raw material and optimal production plans for the end products on a horizon of T periods. The problem is modeled as an integer linear program and solved using both a Lagrangian relaxation-based heuristic and a commercial optimization software. The results obtained using the Lagrangian heuristic are promising and new ideas are generated to further improve the quality of the solution