The input of the PCTTP is the demand that takes form of the number of passengers nit that want to travel between OD pair i∈ I and that want to arrive to their destination at their desired arrival time t ∈ Ti. The actual value of their desired arrival time is stored in the parameter ait and the set Ti is only used for indexing purposes. In other words, each OD pair i is having several passenger groups with different desired arrival times (indexed by t ∈ Ti = {t : nit > 0}). Therefore, the combination of indices (i, t)
forms a unique group of passengers. Note, that the time is discretized into minutes. In PCTTP, a train is defined by its line ∈ L, i.e. the sequence of stations that it serves. Each line has a train frequency that is expressed as the number of available trains v ∈ V, e.g. if a line has a frequency of 18 trains a day v ∈ {1, ..., 18} = V. Both lines and frequencies are provided by the result of LPP. Each line can be further
decomposed into an ordered set of segments S. A segment is a part of the line between two stations, where the train does not stop. Therefore, the purpose of the segments is to verify the train capacity constraints. The segments are unique part of an infrastructure irrespective of the lines.
A timetable is defined as a set of arrival and departure times of each train v ∈ V of each line ∈ L. The combination of indices (, v) forms a unique train. The model assumes the dwell times to be fixed at the time of solving. However, they can be adjusted further on upon solving of the traditional TTP. Since the travel times of rpi consisting of dwell times and running times in between stations are deterministic, it is sufficient to decide only on the departure time dv of each train (, v) from its origin station. The model can design two types of a timetable: non-cyclic (by default) and cyclic (imposing cyclicity constraints). The size of the cycle is given by parameter c. The model does not take care of the conflicts among trains, in order to exploit the maximum impact of the passenger centric timetabling approach. In reality, the impact of such timetables might be smaller, due to the timetable shifts needed to secure the safety in the network. The safety of the network remains the task of the traditional train timetabling problems and can be removed upon solving of the model presented by Caprara et al. (2002) for the non-cyclic timetable and upon solving of the model presented by Peeters (2003) for the cyclic timetable.
Based on the set of trains, the set of paths Pi for each OD pair i is given. Each path
p between an origin and a destination consists in several attributes: a sequence of lines Lp in order that they are being traversed, travel time bpi from the origin of the line to the origin of the OD pair (where = 1), the travel time rip from an origin of the OD pair to a transferring point between two lines (where = 1), the travel time bpi from the origin of the line to the transferring point in the path (where > 1 and < |Lp|), the travel time rpi from one transferring point to another (where > 1 and < |Lp|)
and the travel time rip from the last transferring point to a destination of the OD pair (where = |Lp|). Note that different lines using the same track might have different travel times (due to the different stopping patterns outside of the stations considered by the model). For more explanations of what a path is, refer to the Appendix A.3. The set of all paths is pre-processed and can be created with an algorithm described in Appendix A.
When making a transfer from one train to another, a minimum transfer time m is always secured. Any additional time spent in the transferring stations is counted as a waiting time (wpit, where = 2 is the waiting time for a transfer between the first and the second train and = 3 is the waiting time for a transfer between the second and the third train). Given the actual departure times, the paths where a transfer is not possible are rendered infeasible.
3.1. MODEL
a decision variable xpit, we secure that each passenger group can use at most one path. All passengers, in the same passenger group, always follow exactly the same path and cannot be split. If there is no path assigned to a given passenger group (due to the limited capacity of trains given by parameter W ), it is assumed that the passenger group would follow its shortest path after the end of the planning horizon H. In such a case, the revenue this passenger group would generate, is not accounted for in the objective function and different passenger satisfaction function will be applied (detailed explanation further on in Section 3.1.3).
Within the path itself, passenger group is taking exactly one train per each line in the path (decision variable yitvp ). These decision variables, among others, allow us to backtrace the exact itinerary of each passenger group. Since we know the exact itinerary of each passenger group, we can measure the train occupation osv of each train v on each line on each of its segment s. Derived from the occupation, number of train units υv is assigned to each train. This value can be equal to zero, which means that the train is not running and the frequency of the line can be reduced. The length of a train cannot exceed the maximum allowed length G. The variable length of the trains is used, in order to exploit the maximum impact of the new passenger centric approach. However, the feasibility of such a solution should be verified upon solving of the Rolling Stock Planning Problem. The model also keeps track of the number of train drivers αv needed to realize the timetable. The model assembles the revenues generated by the passengers and the costs inflicted by the operation. The two together allow for calculation of the profit.