Stochastic optimization of a timetable

M.E. van Kooten Niekerk

Outline

• Timetable: theory and reality

• Time Supplements

• Optimization of Time Supplements

• Extension of model

• Theoretical results

• Practical results

• Conclusion

Timetable: Theory & Reality

• Theoretical: Minimum technical driving times

• Reality is different:– Human factor– Weather– Other

• To cover this, extra driving time is scheduled

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?

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

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

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

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

Optimization model (1)

• At every timing point: Actual departure ≥ scheduled departure. If too early, wait. No negative delay. Dn ≥ 0

• Formula: RrNtsDD trtrtrt ..1,..1for },0max{ ,,1,

Optimization model (2)

..Nts

..R..N, rtD

Ss

..R..N, rtDsD

RDwD

t

rt

t

rttrtrt

N

t

R

rrtt

1for 0

10for 0

11for

:toSubject

/min

,

N

1t

,,,1

0 1,

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

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

Extension of model

• Overtaking of trains is not possible

• Minimal Headway between trips:

hrhrd ddH ,,1,,2

~~~

hrhra aaH ,,1,,2~~~

Extension of model

• Possible conflicts on track usage

• Eg. Crossing of trains• Train t2 should wait

until t1 has arrived

0,,1,,2 hrhr ad

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

Theoretical results

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%.

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

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.

Questions?