Stochastic optimization of a timetable

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. Dn≥ 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?