A Robust Rolling Stock Rescheduling Approach
J.C. Wagenaara,∗ , E. van der Hurkb , R.M. Lustbyb , and M. Schmidtc
a Tilburg University, The Netherlands

j.c.wagenaar_1@tilburguniversity.edu
b Technical University of Denmark, Denmarkc Erasmus University, The Netherlands∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

April 12, 2022
Keywords: Disruption management, Railways, Rolling Stock, Robustness

1Introduction

Railway operators all over the world transport millions of passengers on a daily basis. These
railway operators schedule the timetable and associated resources during an extensive planning
period. The most common resources are the rolling stock and the crew schedule. All threeschedules are often an optimized trade-off between passenger service, efficiency and robustness.
During actual operations, railway operators are faced with all kinds of unforeseen incidents
having a negative effect on the railway schedules, which are named disruptions. For example, asystem malfunction, an accident, or the complete blockage of a track segment by a fallen tree.
In such cases, the timetable, the rolling stock schedule, and the crew schedule might be renderedinfeasible. Effective disruption management is currently an active research area in Operations
Research. This has lead to many papers on algorithmic tools for rescheduling the timetable,rolling stock and crew schedule (see Cacchiani et al. (2014)).
The focus in this paper is solely on rescheduling the rolling stock (RS) schedule, schedulingthe timetable and crew is outside the scope. Many researchers (e.g. Nielsen (2011), Wagenaar(2016)) have already conducted research on this topic. However, all these models assume theduration of a disruption to be known at the time of rescheduling. Unfortunately, in practice theduration of a disruption is often uncertain.
Assuming that the disruption duration is known may lead to new problems if the realized
duration of the disruption differs from what was assumed. Not considering the variation in
disruption duration will result in a situation in which the rescheduling must be performed allover again. Some decisions could turn out to be very costly in terms of i) operating costs, ii)
passenger service, and iii) possible communication problems.
In practice, an estimation on the disruption duration is made as soon as a disruption occurs.
This estimation gives, among other information, a minimum and maximum duration. The main
contribution of this paper is a model that is able to reschedule the RS schedule in a robust way.
This means that the RS schedule created for the minimum disruption duration requires no or
small additional changes in case the disruption duration turns out to be longer. A fully robust
solution might come at an high expense. To that end, a second contribution is that we havedeveloped a model to create a semi-robust rolling stock circulation, which includes a given levelof robustness required.
TRISTAN XI Symposium

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J.C. Wagenaar, E. van den Hurk, R. Lusby, M. Schmidt

22Methodology

Algorithm 1 presents the current common approach to deal with an uncertain disruption duration.
It starts with solving the nominal (deterministic) RS rescheduling model M 0 , described in, among
others, Fioole et al. (2006), Wagenaar et al. (2017), and Haahr et al. (2016), that assumes thedisruption duration to be known and equal to the assumed minimum duration (denoted by dmin ).
In the first iteration the planned RS schedule (R0 ) and the rescheduled timetable Tλ0 are provided
as input and a new RS schedule Rλ0 0 is constructed, allowing changes in composition assignmentsfrom the start of the disruption. When the disruption lasts longer, e.g. to λ1 (or other durations
λ2 , · · · , λn ), the RS schedule is updated, where changes to the previous RS schedule (Rλ0 0 ) areonly allowed after communicating λ1 , which is strictly after the time of communicating λ0 . Thisis repeated until no updates on the disruption duration take place. As M 0 does not consider
any information on possible alternative durations of the disruption, it cannot anticipate on thesealternative scenarios.
Algorithm 1: Nominal Approach
01 M 0 (Tλ0 , R0 ) → Rλ02 for i = 1: n do3M 0 (Tλi , Rλ0 i−1 ) → Rλ0 i
4 end

Algorithm 2 presents the strict composition robust approach to deal with a disruption. Asadditional input, the model requires the maximum duration (T+ ) of the disruption in order toproduce a new RS schedule Rλs 0 , which is robust against all possible disruption durations in
between the minimum and maximum duration.
The strict RS rescheduling model (M s ) is an extension of the nominal model existing in
the literature, which creates an RS schedule for the minimum duration (λ0 ), but restricts thecomposition assignment such that if the disruption takes longer, only very small changes arerequired, namely the trains that travel over the tracks hit by the disruption need to be cancelled.
This is represented in Algorithm 2 by the method M cancel . Such a change is easy and fast, asmajor additional shunting movements, which need to be communicated, are avoided. In order toachieve this, the M s model has additional constraints, making sure that for all possible disruption
updates in between the minimum and maximum duration of the disruption, all trips have thesame composition and composition changes. A resulting disadvantage is that the model could
be restrictive, as it needs to be robust for all possible disruption lengths, and may as a result beexpensive in terms of operating costs.
Algorithm 2: Strict Composition Robust Approachs
1 M s (Tλ0 , R0 , T+ ) → Rλ0

2 for i = 1: n do
3
M cancel (Tλi , Rλs i−1 ) → Rλs i
4 end

A third method has been developed which is more robust than the nominal approach, butless restrictive than the strict approach. Algorithm 3 presents the light trip robust approach.
This method provides the possibility to specify a level of robustness α that defines α percentageof trips that must receive the same composition irrespective of the disruption duration when
rescheduling with M α , resulting in a new RS schedule Rλα0 . A difference with M s is that newcomposition changes are permitted to ensure this consistency. Therefore, even with α = 1, thismethod is less restrictive than the strict approach, but it will likely need more changes in case of aTRISTAN XI Symposium

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J.C. Wagenaar, E. van den Hurk, R. Lusby, M. Schmidt

3disruption duration update. As it does take into account a possible different disruption duration,
the expectation is that fewer changes will be required in case of an update λi in comparison with
the nominal approach, at less operational costs than the strict approach.
Algorithm 3: Light Trip Robust Approach
α1 M α (Tλ0 , R0 , T+ ) → Rλ0

2 for i = 1 : n do
3
M α (Tλi , Rλi−1 , T+ ) → Rλαi
4 end

3Results and Discussion

The three approaches were tested on a case study from Netherlands Railways, where there are1078 trips in the Western part of The Netherlands. There are three different disruption locations
tested; between Utrecht and Amsterdam (Ut-Asd), between Rotterdam and The Hague (RtdGv), or between Haarlem and Amsterdam (Hlm-Asd) causing no trains being able to travel
between those stations during the disruption. At every location, the minimum disruption lasts
from 06:00-08:30, 07:00-09:00, or 09:30-12:00 in the morning. Furthermore, each disruption has
a maximum duration either one, two, three, or four hours longer than the minimum duration.
In the first case study, we test all approaches where we reschedule for the minimum duration,
where the robust approaches take into account the maximum duration. Then, we compare theresults in case the minimum duration (λ0 ) turns out to be true or in case the maximal duration(λ1 ) turns out to be true. In the latter situation, rescheduling takes place at the start of thedisruption and at time λ0 . Table 1 presents results for the light robust approach with α = 0.4, 0.8
and the strict robust approach. Each entry represents the relative objective difference with thenominal approach. For example, the strict approach is on average 22% better than the nominal
approach in case λ1 turns out to be the true disruption duration for the disruptions between
Utrecht and Amsterdam.
The last three rows represent the expected costs in case there is a 50% probability that λ0
is the correct disruption duration and a 50% probability that λ1 is the true disruption duration.
As can be seen, in case there is a 50% probability on either of the disruption durations to betrue, a robust approach outperforms the nominal approach. Furthermore, the strict robustness
approach seems to be best in case it is probable that the disruption could take longer than theminimum duration.
Disruptionλ0

λ1

Expectation

Model
0.4
0.8
Strict0.4
0.8
Strict0.4
0.8
StrictUt-Asd
0.03
0.10
0.14
-0.05
-0.07
-0.22
-0.02
0-0.07

Rtd-Gv
0.01
0.07
0.13
-0.02
-0.05
-0.18
-0.01
0-0.05

Hlm-Asd
00.12
0.15
-0.06
-0.09
-0.17
-0.04
0-0.04600-830
0.01
0.10
0.16
-0.05
-0.05
-0.17
-0.02
0.02
-0.03

700-900
0.03
0.09
0.13
-0.06
-0.08
-0.19
-0.02
-0.01
-0.06

930-1200
0.01
0.07
0.11
-0.04
-0.07
-0.20
-0.02
-0.01
-0.08

All
0.01
0.09
0.13
-0.04
-0.06
-0.19
-0.02
0-0.05

Table 1 – Results for two disruption durations. Every number represents the relative total costs
difference between rescheduling using the nominal approach and one of our approaches in casethe corresponding disruption duration turns out to be true.
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J.C. Wagenaar, E. van den Hurk, R. Lusby, M. Schmidt

4

In the second setting, we first reschedule again for the minimum duration, where the robust
approaches consider a maximum duration of four hours longer. Then, every hour an update could
be given that the disruption lasts one hour longer or that the disruption is over. After four hours
the disruption is always over. Thus, we reschedule the rolling stock potentially five times. Table
2 presents the results in case the disruption would be over after each of the potential durations.
Again, the strict robust approach performs on average best.
Disruption
λ0

λ1

λ2

λ3

λ4

Expectation

Model
0.4
0.8
Strict0.4
0.8
Strict0.4
0.8
Strict0.4
0.8
Strict0.4
0.8
Strict0.4
0.8
StrictUt-Asd
0.03
0.14
0.30
0.03
-0.02
-0.04
0.03
-0.05
-0.14
0-0.09-0.23
0-0.13
-0.30
0.01
-0.05
-0.12

Rtd-Gv
0.02
0.09
0.21
0.02
0.03
0-0.02
-0.02
-0.09
-0.02
-0.05
-0.15
-0.03
-0.13
-0.26
-0.01
-0.03
-0.08

Hlm-Asd
0.01
0.08
0.18
00.03
0.06
0-0.02
-0.05
0.02
-0.04
-0.09
-0.02
-0.11
-0.19
0-0.02-0.03

600-830
0.01
0.06
0.17
0.02
0-0.02
0.02
-0.01
-0.08
0.01
-0.03
-0.13
0-0.10
-0.22
0.01
-0.02
-0.07

700-900
0.03
0.16
0.27
0.01
0.03
0.02
-0.01
-0.05
-0.14
-0.02
-0.09
-0.19
-0.04
-0.15
-0.28
-0.01
-0.04
-0.10

930-1200
0.02
0.13
0.33
-0.01
0.03
0.06
-0.03
-0.04
-0.06
-0.01
-0.09
-0.17
-0.03
-0.15
-0.27
-0.01
-0.05
-0.07

All
0.02
0.10
0.23
0.02
0.02
00-0.03-0.09
0-0.06
-0.16
-0.02
-0.13
-0.25
0-0.03-0.08

Table 2 – Results for five disruption durations. Every number represents the relative total costs
difference between rescheduling using the nominal approach and one of our approaches in casethe corresponding disruption duration turns out to be true.
In general, we can conclude that the robust approaches work better in case uncertainty ispresent. When the estimation of the disruption duration is uncertain, the robust approaches
perform better in comparison with the nominal approach.
In future research, a sensitivity analysis must be performed in order to see whether theseresults apply in general or only in our specific case study setting.

References
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TRISTAN XI Symposium

Original abstract submittal