1. Train platforming problem Ľudmila Jánošíková Michal Krempl University of Žilina, VŠB-Technical University of Ostrava, Slovak Republic Czech Republic

2. Train platforming problem The train platforming problem consists in the allocation of passenger trains to platforms in a railway station. The solution is a track occupancy plan. The plan is a dispatcher’s decision support tool. It specifies for each arriving or departing train  the platform track  the time slot during which the track will be occupied by the train.

3. Track occupancy plan

4. Benefits better management of train operation train routes are occupied shorter time workload of the infrastructure elements, such as tracks, switches, and platforms is more uniform higher service quality perceived by passengers shorter distances needed for changing trains less probability of changing the planned platform when the train delays regular platforms for the same direction meeting train operators’ requirements on arrival and departure times and platforms

5. Problem description Given  list of the trains arriving, departing, or travelling through the station  network timetable  safety rules for train movements  layout of the station The train platforming problem:  assign a route through the station to each of the trains,  adjust the arrival and departure times due to safety or capacity reasons.

6. Track layout

7. Problem description Criteria: 1.minimise deviations from the planned arrival and departure times 2.maximise preferences of trains for platforms

8. Mathematical programming model Variables  adjust the arrival and departure times u i real arrival time of train i at a platform, i  U v i real departure time of train i from a platform, i  U  assign a route through the station to each of the trains Notation U set of all arriving, departing, and transit trains K(i) set of feasible platform tracks for train i

9. Mathematical programming model and similar variables for combinations arrival – departure, departure – arrival, departure – departure. Auxiliary variables - model safety headways between conflicting trains:

10. Solution method  Mathematical programming  MIP model with a huge number of variables (> 45 000) and constraints (> 700 000)  Decomposition o 0:00 – 5:00 o 5:00 – 8:00 o 8:00 – 10:00 o 10:00 – 12:00 o 12:00 – 15:00 o 15:00 – 18:00 o 18:00 – 24:00  Local branching metaheuristics

11. Local branching framework We consider a generic MIP with 0-1 variables of the form: where B index set of the 0-1 variables G (possibly empty) index set of the general integer variables C (possibly empty) index set of the continuous variables

12. Local branching framework Given feasible reference solution Let binary support of We define the k -opt neighborhood of as the set of feasible solutions satisfying the local branching constraint:

13. Local branching framework The local branching constraint can be used as a branching criterion within an enumerative scheme. Given the incumbent solution, the solution space associated with the current branching node can be partitioned by adding the constraints: left branch: right branch:

14. Local branching scheme AB CD E F improved solution no improved solution

15. Local branching scheme AB CD E F Pink nodes are explored through a standard “tactical” branching criterion such as branching on fractional variables, i.e. they represent the application of the black-box IP solver.

16. Local branching metaheuristics  Time limit for the IP solver  Diversification

17. Prague main station Timetable 2004/2005, 4 min change time PeriodNo. of trains Deviation (min)Different platform assignments arrivaldeparture 0:00 - 5:00 20000 5:00 - 8:00 56225 8:00 - 10:00 45058 10:00 - 12:00 37025 12:00 - 15:00 52129 15:00 - 18:00 54020 18:00 - 24:007244 Total 51731 (11%)

18. The model proposes a feasible track occupancy plan that:  respects safety constraints for train movements  minimises deviations of the arrival and departure times from the timetable  maximises preferences for the platform tracks  respects relations between connecting trains:  ensures that passengers have enough time to change trains  minimises distance between connecting trains Conclusions