1 Introduction
Railway systems are highly interdependent, making them susceptible to delays that can easily
propagate and degrade service. Therefore, transportation plans must perform well under nor-
mal conditions and when faced with uncertainties. Since operational recovery must be fast,
feasible solutions are often prioritized over exact optimization, incurring higher recovery costs.
Consequently, this challenge led operators to gain interest in robustness, defined as the plan's
ability to absorb minor delays and enable rapid and less costly operational recovery.
Robustness is commonly achieved by incorporating buffer times in the timetable, such as
extending running and dwell times. A less explored yet critical methodology is to incorporate
buffers directly into rolling stock circulations [1]. Imbalanced dwell times can make the nominal
plan vulnerable [2]: tight dwell times increase the risk of delay propagation and rescheduling,
while excessively long dwell times prevent efficient use of available capacity. While scenario-
based planning is often used to anticipate delays and enable quick recovery, its stochastic
nature presents challenges, motivating the proactive insertion of buffers as an alternative. A
detailed review of robustness in railway is presented by [3].
Traditionally, buffer times are uniformly distributed across the plan. The objective is, for a
given timetable, to strategically distribute the total fixed available dwell time across the train
units to enhance robustness while maintaining equal or near nominal operational cost. The
challenge lies in the placement and size of the buffers, since misallocation may require additional
train units or costly shunting movements. The proposed balanced allocation mitigates the risk
of short dwell times while utilizing the inefficiency of existing long ones.

2 Methodology and Modeling
Freight railway provides greater flexibility than that of passengers, since the service quality is
primarily based on final delivery time rather than connection times. This flexibility enables
exploring different dwell time distributions within rolling stock circulations that maintain the
same or near nominal operational cost while offering a higher level of robustness. To explore
different alternatives, we embed our approach in a column-generation framework for the rolling
stock planning problem, corresponding to the rolling stock subproblem within the integrated
train path and rolling stock routing model proposed by [4]. Circulations are generated using
a label propagation algorithm that is extended with a dwell time penalty component. Each
station i has an operational dwell time interval [tmin
i , tmax
i ], representing the minimum time
required for operations before departure (e.g.: turnarounds, cleaning, (un)loading, etc.) and
the maximum time allowed to avoid platform or depot over utilization. This interval is adjusted
using buffers, resulting in [tmin
i +bi, tmax
i −bi]. The penalty component encourages dwell times to
remain within this desired interval by applying higher penalties at the operational bounds and
progressively decreasing penalties while approaching the buffered time window. The penalty
for a dwell time (di) is expressed as follows, where αmin and αmax are associated weights:
penalty(di) =



αmin (tmin
i + bi − di)2 if di < tmin
i + bi,
0 if tmin
i + bi ≤ di ≤ tmax
i − bi,
αmax (di − tmax + bi)2 if di > tmax
i − bi
Consequently, the dwell times of the selected columns are more evenly distributed across the
trips, while still allowing deviations from the desired interval when needed to maintain feasibil-
ity or cost optimality. This strategic distribution inserts buffers only where they meaningfully
enhance robustness, avoiding an unnecessary increase in costs that can easily result from the
traditional way of adding equal buffers at all stations. This method avoids tight dwell times
whenever possible which improves delay absorption. The robustness objective in our study is
therefore minimizing the sum of the penalty costs of the rolling stock circulations.

3 Results and Conclusion
Optimizing the rolling stock planning problem may lead to tight dwell times which increase the
risk of delay propagation and therefore degrade the service quality. Incorporating the dwell
times directly into the optimization process would greatly increase its complexity; however,
guiding the column generation process with the dwell time penalty provides an effective alter-
native. We evaluated this approach using instances derived from the 2022-2023 transportation
plan of Fret SNCF. For small instances, we validated the method as a proof of concept: the
same coverage of trips was achieved while using the same number of rolling stock units. Hence,
the same or near operational cost was maintained, while the robustness objective improved
significantly. This confirms that introducing the penalty successfully drives the algorithm to-
ward more robust circulations. Without the penalty, the label propagation algorithm has no
incentive to differentiate between the feasible dwell times.
The preliminary results show that creating strategic buffers through dwell time balancing
can enhance robustness without compromising its overall nominal efficiency. These findings
motivate further investigation on larger instances and sensitivity analysis, in order to develop
plans capable of absorbing delays and preserving service quality. A potential trade off may
emerge between preserving the same coverage with additional train units or accepting a reduced
coverage with the same number of units. This trade off depends on the level of robustness that
operators wish to guarantee.

References
[1] Matthias Rößler, et al. Simulation and optimization of traction unit circulations. In 2020
Winter Simulation Conference (WSC), IEEE, 2020.
[2] Sabine Tréfond, et al. Optimization and simulation for robust railway rolling-stock planning.
Journal of Rail Transport Planning & Management, 7(1-2):33–49, 2017.
[3] Richard M. Lusby, Jesper Larsen, and Simon Bull. A survey on robustness in railway
planning. European Journal of Operational Research, 266(1):1–15, 2018.
[4] Louis Fourcade, Stéphane Dauzère-Pérès, and Juliette Pouzet. Combining Lagrangian re-
laxation and a two-set column generation model for integrated railway freight planning.
2025.