To model the delay management problem and to derive solution procedures, we use the concept ofevent-activity networks as suggested in [SU89] (see also [Nac98] for the application of event-activity networks in periodic timetabling and [Sch07] for their application in delay management). An event-activity networkN = (E,A) is a directed graph whose nodes are calledeventsand whose directed edges are calledactivities. Event-activity networks are a widely used mathematical model for periodic or nonperiodic
2.2 Event-Activity Networks
scheduling of events with time constraints. In the nonperiodic case which we consider here, an activity which connects two events models a precedence constraint between those events: the start event of the activity has to take place first. Each activity has assigned a lower bound on its duration, so the scheduled time of the end event of an activity has to be larger than or equal to the scheduled time of the start event plus the lower bound. In contrast to the nonperiodic case, in periodic event-activity networks (used for example for periodic timetabling), each activity has assigned a lower and an
upper bound, modeling time window constraints.
In a railway setting (which is the main focus of our applications), the setE of events consists ofarrival events Earr, i.e. the arrivals of trains at stations, anddeparture events Edep, i.e. the departures of trains from stations. The set Aof activities consists of four different types of activities:
• Driving activities Adrive⊂ Edep× Earr model the driving of a train between two consecutive stations, so a driving activity connects a departure event of a train with its next arrival event at the subsequent station. The lower boundLa>0of a driving activity a∈ Adrive represents the minimal driving time between both stations.
• Waiting activities Await⊂ Earr× Edep represent the waiting of a train within a station, for example for the boarding and deboarding of passengers or for crew change. A waiting activity connects the arrival of a train at a station with its departure from the same station. The lower boundLa>0of a waiting activity a∈ Awaitdescribes the minimal time which is needed to let passengers get on or off and also takes into account the time for crew change or other actions.
Each activity inAdrive and Await corresponds to an action of one train; since they are all treated in the same way (e.g. the lower boundLaof each such activityaalways has to be respected), we summarize them in the set
Atrain := Adrive∪ Await.
• Changing activities Achange⊂ Earr× Edep allow passengers to transfer from one train to another one within the same station, so a changing activity connects an arrival event of some train at some station with a departure event of another train at the same station. The lower boundLa>0refers to the minimum time the passengers need when they transfer between both trains. It is one of the tasks of delay management to decide for each changing activity if the corresponding connection should be maintained or not. If a connection is maintained, the lower boundLa of the corresponding changing activityahas to be respected, otherwise it can be ignored.
• Headway activities Ahead ⊂ Edep× Edep model the limited capacity of the track system. They always appear in pairs: if (i, j)∈ Ahead, then (j, i) ∈ Ahead, too.
In contrast to the other types of activities, a single headway activity does not model a single constraint, but together with its corresponding counterpart, they model a pair of disjunctive constraints. As an arc in the event-activity network models a precedence constraint, it is not possible to satisfy both constraints resulting from a pair of headway activities at the same time. On the contrary, exactly one headway activity from each pair has to be respected. The goal of delay management hence is to choose exactly one activity of each such pair and to respect the resulting constraint, fixing the order of the two eventsiand j. If (i, j)is chosen, then eventitakes place before eventj and the lower boundLij of activity(i, j) has to be respected. If, however,(j, i) is chosen, then eventj takes place first and the lower bound Lji of activity (j, i) has to be respected. The lower boundLij >0 of a headway activity (i, j) ∈ Ahead represents a security distance: For the two departure eventsiandj, it represents the minimalheadway between the departures of the corresponding trains, i.e. the minimum time for which the train belonging to event j has to wait after the departure of the train belonging to eventito ensure safe operations. Note that those headway times need not to be symmetric; in general,Lij 6=Lji (a slow train probably will block a specific piece of track longer than a fast train). Our model covers two types of limitation: two trains driving on the same track into the same direction and two trains driving into opposite direction on a single-way track.
Summing up, we have
E = Earr∪ Edep and
A = Adrive∪ Await∪ Achange∪ Ahead.
To illustrate an event-activity network, we use the following example (which is depicted in Figure 2.1): Assume that we have five stations A, B, C, D, and E. One train drives from station A to station C and further on to station D, while a second train drives from station B to station E via station C. Within station C, passengers might transfer between both trains, and on their way to station C, both trains share a common piece of track.
In the corresponding event-activity network (see Figure 2.2 for an illustration), the arrival of the first train at station C and the departure of the second train from station C are connected by a changing activity; the same holds for the arrival of the second train and the departure of the first train. To model the limited capacity of the tracks, the departure of the first train from station A and the departure of the second train from station B are connected by a pair of disjunctive headway activities.
2.2 Event-Activity Networks
Figure 2.1: An example of five stations A, B, C, D, and E where trains driving between A and C and between B and C share a common piece of track (the solid lines between stations represent tracks).
Figure 2.2: The corresponding event-activity network if we assume that one train serves the directed line A-C-D while another one serves B-C-E. Solid arrows are activities from Atrain, dashed arrows represent changing activities, dotted arrows represent headway activities.
An equivalent model for headway activities is to connect the arrival event of one train with the departure event of the second train and vice versa, meaning that the second train’s departure is not allowed to take place before the first train’s arrival. Since in this case, four events instead of two are involved in each pair of headway activities, we do not use this model.