Robustness

In order to support the generation of cost-efficient vehicle and driver schedules, public transport companies increasingly use software tools, containing state-of-the-art optimization methods. However, using optimization methods to compute cost-optimal schedules usually results in tense schedules for both vehicles and crews without much idle or waiting time. Thus, during execution of vehicle and crew schedules delays cannot be absorbed and can propagate through the whole schedule. As a result, planned schedules can become infeasible and the operations control has to initiate expensive recovery actions.

Delays are unavoidable during operations, but schedules can be created that way that possible disruptions can be absorbed.

Schedules are called robust when effects of disruptions are less likely to be propagated into the future (Amberg, Amberg and Kliewer, 2011).

Another definitions are: the robustness of a solution is defined with respect its ability to preserve the solution quality, that is, the completion time or make span (Wu, Leon and Storer, 2004), or a timetable is robust if we can cope with unexpected troubles without significant modifications (Tomii, 2005), or a timetable can have the characteristic that delayed trains will lead to considerable knock-on effects, whereas another configuration of the timetable may be able to absorb such effects more readily, this last timetable is more robust than the other (Salido, Barber and Ingolotti, 2008).

The idea of robust schedules consists in solutions that can tolerate a certain degree of uncertainty during execution. In other words, they should be able to absorb dynamic variations in the problem due to both external reasons (exogenous events), and internal reasons (false definitions in the problem) (Policella, 2005).

How to obtain robust timetables

 Salido et al. in 2008 identified several methods that can be applied in order to obtain robust timetables.

1) To decrease optimality (by introducing buffer times).

The robustness of the timetable against small disturbances can be improved by optimally allocating time supplements and buffer times in the timetable. This is a frequent method that implies adding additional buffer times in the travel times of each train in each track section of its journey.

2) To decrease capacity

Robustness is obtained decreasing the theoretical capacity of the infrastructure. It is especially useful in the case of double tracks. Decreasing capacity is also other usual method applied by railway infrastructure managers, as the UIC methods proposed by the International Union of Railways [UIC96, UIC04] do. Railway managers know that scheduling as many trains as theoretical capacity says is not viable. However, they have to make the best use of the expensive railway infrastructures. Trade-off between capacity and reliability/ robustness, in other words, between the ’physical maximum’ level of capacity and the ’economically optimal’ level of capacity is a key point in operational management contexts (Abril et al., 2008).

Clearly, as greater capacity is used, greater will be the risk of secondary delays due to incidences.

Thus, the idea is not to use the infrastructure at the maximum level of capacity (theoretical capacity), but with a practical capacity limit.

3) To decrease heterogeneity

Railway traffic is considered to be homogeneous if all trains have similar characteristics, especially the same average speed per track segment, resulting from the running times and the stopping times. However, for large railway networks, railway traffic cannot be fully homogeneous due to freight and passenger trains share the same infrastructure. If there are large differences in the timetable characteristics of the trains on the same track, then the railway traffic is called heterogeneous (Vroman, Dekker, and Kroon, 2006).

It is generally accepted in practice that the heterogeneity resulting from the line plan and the timetable has a negative influence on the punctuality and the reliability of a railway system (UIC, 2004).

Other concepts to considerer

Salido et al. in 2008 identified other concepts related to the above methods to considerer:

Buffer time

Buffer time is the temporal interval in which a train is stopped and it is not carrying out any traffic operation. This slack is assigned by railway operators in certain stations to absorb disruptions that have been carried out in these stations or previous stations. As the number of buffer times increased the optimality decreased, but robustness must increase.

Number of trains

In timetables composed by a high number of trains is more probably that a disruption occurs. Indeed, capacity and robustness are opposed terms, so that less capacity (less number of trains) implies higher robustness.

Number of commercial stops

The number of commercial stops is directly related with the number of disruptions. As the number of commercial stops increase, the number of possible disruptions also increase.

Flow of passengers

Some stations are more important than others. Stations in large cities are used by a larger number of passengers than stations in small villages. There is more probability that disruptions occur in high loaded stations. Furthermore this parameter also depends on trains. Railway operator can give us a disruption probability for each train in each station. Thus, it is very appropriate assigning buffer times in high loaded stations.

Tightest tracks

The tightest track is considered as the longer distance track. Thus, if there exist a large single track between two consecutive stations s1 and s2, and a train is travel from s1 to s2, all trains in opposite direction must wait in station s2 for a long time. Thus, it is convenient to maintain buffer times in both stations that compose the tightest track in order to absorb disruptions.

References

Abril, M., Barber F., Ingolotti, L., Salido, M.A., Tormos, P. and Lova, A., 2008. An assessment of railway capacity, Transportation Research – Part E, to appear.

Amberg, B., Amberg, B., Kliewer, N., 2011. Increasing delay-tolerance of vehicle and crew schedules in public, Procedia Social and Behavioral Sciences, Vol. (20), 292–301, 2011.

Policella, N., 2005. Scheduling with uncertainty: A proactive approach using partial order schedules, PhD Thesis, Universita degli Studi di Roma: La Sapienza.

Salido, M. A., Barber, F., Ingolotti, L., 2008. Robustness in Railway Transportation Scheduling.

Tomii, N., 2005. Robustness indices for train rescheduling. 1st International Seminar on Railway Operations Modelling and Analysis.

Vroman, J.C.M., Dekker, R. and Kroon, L.G., 2006. Reliability and heterogeneity of railway services, European Journal of Operational Research, 172, 647–665.



Wu. S.D., Leon, V. and R.H. Storer, 2004. Robustness measures and robust scheduling for job shops, IIE Transactions,, 26, 667–682.