Nominal timetable

Using the minimum necessary process times turns constraint (2.1) into an equality constraint for each ride, dwell, or transfer action; for each (i, j) that

SOLUTION APPROACHES FOR THE TRAIN TIMETABLING PROBLEM 19

represents such an action, lij becomes equal to uijin (2.1). In order to maintain

feasibility, scheduled waiting time implying that lij < uij can be necessary in

some cases. Planning the ideal process times gives the fastest schedule in theory, however, this is very bad since the smallest delays will cause a knock-on effect.

2.3.4

Robust optimization

The next possibility is the opposite of the nominal solution; a schedule that remains conflict-free, even in the worst-case scenario. This approach is called robust optimization and is considered by, for example, Ben-Tal et al. (1998) and Bertsimas et al. (2004). In order to anticipate all worst-case scenarios, a lot of time reserves are needed. Although timetables that are constructed this way are very stable, they are not attractive since they are mostly much too slow. This is an important remark: although a timetable is insensitive to delays, it can still be very unattractive.

2.3.5

Light robustness

A solution of the TTP that lays somewhere in between the nominal solution and the timetable that results from robust optimization is possible. The idea of light robustness (Fischetti et al. 2009a,b) is to allow a worsening in objective function value compared to the nominal timetable and to use this worsening to improve the timetable by, for example, the stochastic programming approach of Section 2.3.7.

Let z∗ be the optimal objective value of the nominal solution with objective function (2.2), then the upper bound for the planned travel time of the light robustness solution is modeled as

X

i,j

wij(πj− πi) ≤ (1 + δ)z∗. (2.3)

Notice that for δ = 0, the nominal solution is optimal, and that for δ = ∞, this approach is equivalent to robust optimization. By adding supplements, the timetable becomes slower and the left-hand side of (2.3) increases. This is the price for fewer conflicts.

2.3.6

Recoverable robustness

The method of recoverable robustness is introduced by Cicerone et al. (2009) and Liebchen et al. (2009). It is initially developed to deal with large delays, but

the approach can even be used for general scheduling purposes. The idea comes from the observation that, when constructing a timetable, schedulers do not account for repair strategies like how to restore feasibility in case a train breaks down in the middle of a bottleneck. The method of recoverable robustness works as follows. In the TTP, a timetable and a recovery algorithm to solve conflicts are determined such that, when operating the schedule, the recovery algorithm can be used to regain feasibility as soon as possible. The goal is to find a timetable and a recovery algorithm such that a conflict-free timetable can be obtained in all considered scenarios.

Recoverable robustness can be applied to various railway planning problems. For example, it is applied to solve the train platforming problem (TPP), see below, by Liebchen et al. (2009) and for scheduling the rolling stock like in Cacchiani et al. (2008a, 2014). Another version of recoverable robustness for the TTP, called recover to optimality, is presented by Goerigk et al. (2010). The idea of recover to optimality is to find the timetable for which the recovery costs are as small as possible. Just like recoverable robustness, the dynamic traffic management system of Schaafsma (2001) and Schaafsma et al. (2007) leaves certain decisions for the operational phase. In Caimi et al. (2011b) and D’Ariano et al. (2008b), flexibility is inserted in the timetable, for example, to postpone ordering decisions to the point in time where more information from the operations becomes available.

2.3.7

Stochastic programming

Another approach to schedule the arrival and departure activities of all trains in a network is stochastic programming (Kroon et al. 2007a, 2008d; Vromans 2005). Stochastic programming makes use of a distribution of delays and returns the timetable that gives the best results on average, for example, the timetable with the lowest delays on average. In general, there are two types of stochastic programming: chance constrained and multi-stage programming with recourse. In chance constraint programming the solution that is conflict-free in as many scenarios as possible is searched, whereas in each stage of multi-stage programming, a solution is evaluated and adapted using information that was not known before.

The most used approach for the TTP is two-stage programming with recourse. In the first stage, a timetable is constructed, and in the second stage, delays are added to the model and used to evaluate the solution of the previous stage. Doing so, there is a kind of simulation process built into the model. As input data, a set of primary delays is used. In Kroon et al. (2008d), it is shown that the results are rather robust against small changes in input data. The main

SOLUTION APPROACHES FOR THE TRAIN TIMETABLING PROBLEM 21

drawback of this method is its complexity that causes very large computation times. This is due to delay propagation computations that are required to evaluate the objective function.

The technique of stochastic programming is used to find the optimal allocation of supplements for a case study in the Netherlands (Kroon et al. 2007a, 2008d; Vromans 2005). Starting from an input timetable, the model searches for the reallocation of the available supplements to minimize the average delays. To avoid the usage of phase shift variables like in the PESP, the cyclic order of events could not be changed during this reallocation process. In Kroon et al. (2008d), some results are presented for a small part of the network. The average

punctuality, on a 3 minutes base, increased from 86.7% to about 93%.

2.3.8

Goal programming

Goal programming is used by Vansteenwegen (2008) and Vansteenwegen et al. (2006, 2007) to construct timetables with an improved transfer schedule. Instead

of adding transfer buffers, each transfer is optimized individually by assessing the pros and cons of adding an RTS on the trip leading to the transfer station. Afterwards, a schedule for the whole network is constructed for which the planned travel times are as close as possible to the optimal ones determined in the previous step. Using goal programming, partial schedules can easily be combined in a timetable for the whole network.

Providing an RTS on the track section leading to the transfer station increases the punctuality of the feeder train. Thus, the transferring passengers benefit since the available time to make a transfer grows and the probability of a successful transfer increases. For the passengers who reach their destination, their actual arriving time does not change, but their feeling of punctuality changes; since the planned arrival time is now postponed with the supplement, they experience less delays and are more satisfied. For through passengers this is different. They may experience disadvantages since their train may dwell longer than necessary. The same holds for departing passengers in the opposite direction. Due to symmetry, the RTS in one direction turns into a maximum holding time for the same train in the opposite direction which is then the connector. Notice that this implies that only one supplement per transfer is needed. Computational experiments on a real network resulted in a reduction in waiting cost by 40%.