1. Stochastic optimization of a timetable M.E. van Kooten Niekerk

2. Outline Timetable: theory and reality Time Supplements Optimization of Time Supplements Extension of model Theoretical results Practical results Conclusion

3. Timetable: Theory & Reality Theoretical: Minimum technical driving times Reality is different: –Human factor –Weather –Other To cover this, extra driving time is scheduled

4. Time Supplements (1) In NL: about 5% of MTDT is added as time supplement Per trip segment, between important points How to assign time supplements?

5. Time Supplements (2) At every timing point: Actual departure ≥ scheduled departure. If too early, wait. No negative delay. D: Delay compared to timetable s: Time supplement δ: Actual delay on segment

6. Time Supplements (3) Spread evenly –1st intuition: OK –Likely to wait, so total time has larger average than necessary All at the start –Excessive waiting on the trip –No serious option All before arrival –Minimal waiting during the trip –Earliest arrival at end of trip –Too late on most timing points

7. Time supplements: Optimization Distribute time supplements s.t.: –Total supplement = constant –Average delay is minimal Problem: non-linearity of delay with respect to applied time supplements Solution: Combination of simulation and (I)LP

8. Time supplements: Optimization 1 Base-timetable Number of realizations (set of ‘random’ delays), about 1000 Goal: minimize average delay in the realizations by making changes to the base-timetable

9. Optimization model (1) At every timing point: Actual departure ≥ scheduled departure. If too early, wait. No negative delay. D n ≥ 0 Formula:

10. Optimization model (2)

11. Results Time supplements not evenly spread across trip segments Average delay is reduced for the greater part of the trip Delay at end of trip is larger

12. Extension of model Now 1 single line Reality: complex set of lines To model: –Slow and fast trains on the same track, overtaking is not possible –Conflicts when trains are crossing –Single track

13. Extension of model Overtaking of trains is not possible Minimal Headway between trips:

14. Extension of model Possible conflicts on track usage Eg. Crossing of trains Train t2 should wait until t1 has arrived

15. Extension of model Trips influence each other, delays can be propagated We should keep track of real departure time, only delay is not enough We should consider a whole day, not one hour Change: 21 hrs a day, 20 realizations Gives LP with 500.000 variables and 400.000 constraints 16 to 32 hours computation time

16. Theoretical results

17. Practical results Results were applied during 8 weeks in 2006 on the Zaanlijn Punctuality went from 79,4% to 86,5% Results on corridor Amsterdam-Eindhoven lead to theoretical reduction of average delay of 30%.

18. Conclusion Optimization of distribution of time supplements leads to a reduction of average delay without extra cost. Some stations may have more delays Method will be applied to whole network of NS

19. Literature Kroon, L.G., Dekker, R., Vromans, M.J.C.M., 2007, Cyclic railway timetabling: a stochastic optimization approach. In: Geraets, F., Kroon, L.G., Schöbel, A., Wagner, R., Zaroliagis, C. (Eds.), Algorithmic Methods in Railway Optimization. Lecture notes in Computer Science, vol. 4359. Springer, pp. 41-66 Kroon, L.G., Maróti, G., Retel Helmrich, M., Vromans, M.J.C.M, Dekker, R., 2007, Stochastic improvement of cyclic railway timetables. In: Transport Research, Part B 42, Elsevier, pp. 553-570.

20. Questions?