An exact decomposition approach for the optimal real-time train rescheduling problem

Trains are running through a rail network trying to meet a predened schedule, the Ocial Timetable, which species when each train enters and exists the stations on its route. When one or more trains deviate from the ocial timetable, new schedules and possibly new routes must be identied and implemented very quickly. Also, the new plan should minimize some measure of the delays. In a rst and very simplied picture, a rail network may be viewed as a set of stations connected by tracks. Each train follows a specic route in this network, namely an alternating sequence of stations and tracks. The trains run their routes trying to agree with the production plan, which species the movements (routing) and the times when a train should enter and leave the various segments of its route (schedule), including stations arrival and departure times. In principle, the production plan ensures that no two trains will occupy simultaneously the same railway resource, or incompatible resources such as a platform in a station and the track to access it. In other words, a production plan is a conict free schedule. The problem to design optimal production plans is of crucial relevance for railway operators. As pointed out in [11] optimum resource allocation can make a dier-ence between prot and loss for a railway transport company. However, due to dierent causes the actual train timetables can deviate from the ocial ones, and potential con-icts in the use of resources may arise. As a consequence, rerouting and rescheduling decisions must be taken in real-time. These decisions are still, in most cases, taken by human operators (dispatchers), and implemented by reorienting switches and by controlling the signals status (i.e. setting signalling lights to green or to red), or even by telephone connections with the drivers. The dispatchers take their decisions trying to minimize delays, typically having in mind some ranking of the trains or simply following operating rules. So, what the dispatchers are actually doing, is solving an optimization problem (and of a very tough nature). We call this problem the Real-time Trac Control in Rail Systems problem (RT C). In short, the RTC problem amounts to establish in real-time for each controlled train a route and a schedule so that no conicts occur with other trains and some