Terminal stop and node delays

4.5.2 Terminal stop and node delays

As mentioned in section 4.5, our algorithm should model two different types of delay. One is the intrinsic delay due to the passage through the node, the other derives from the stop of the train at a terminal node and should not be charged if the train does not stop in this node.

The first type of delay is intrinsic and must be paid at each passage through node i : in this case, the delay Δ i is charged on the arcs connected to node i . In particular, we

distinguish for each arc (,) i j four cases depending on the type of its extreme nodes i and j :

i) if both i and j are “not border” nodes, the transit time of arc (,) i j is set to:

t ij t ij + +

ii) if both i and j are “border” nodes (i.e. (,) i j is a border crossing arc), the transit time of arc (,) i j is set to:

t ij := t ij +Δ+Δ i j

iii) if i is a “border” node and j is not, the transit time of arc (,) i j is set to:

iv) if j is a “border” node and i is not, the transit time of arc (,) i j is set to:

Δ t ij + ij i

t :=

European Commission UMD and UOB Sixth Framework Programme

Organization Code:

Figure 4-4. The transformation of the terminal node to handle the delay due to the train stop.

The second type of delay can be modeled by adding two dummy nodes for each terminal node t 1 , called the input dummy node and the output dummy node. Each arc that in

the original graph enters t 1 , will enter the input dummy node in the modified graph. And, similarly, each arc that in the original graph exits from t 1 , will exit from the output dummy node in the modified graph. Then, three dummy arcs are added: one from the input dummy

node to the output dummy node, one from the input dummy node to t 1 and the last from t 1 to the output dummy node. In Figure 4-4 an example of this network modification is given: t 1 has three input arcs and two output arcs. We add an input dummy node and an output dummy

node that would have, respectively, three input arcs and two output arcs (the arcs that were originally connected to t 1 ). Three dummy arcs are added (the dashes arcs), one between the

two dummy nodes with transit time equal to 0, one from the input dummy node to t 1 and one Δ t

from t 1 to the output dummy node. The transit times of the last two arcs are set to 1 , where

Organization Code:

UMD and UOB

European Commission

Classification:

Unclassidied

Sixth Framework Programme

Δ t is the delay at terminal t

corresponding to the stop of the train. In this way, if the train

stops at the terminal node t 1 , the arcs selected in the path would be the one from the input dummy node to t 1 and the one from t 1 to the output dummy node (with a global delay equal

to Δ t ). Otherwise, the arc selected in the path would be the one connecting the input and the

output dummy nodes and the train would have a null delay.

Finally, the last modification of this phase is to add a source node and a destination node. In order to do this, two other dummy nodes are added. In addition, each terminal node is linked to the source node and the destination node with dummy arcs, so that we can assume that all the paths start from the dummy source node and end at the dummy destination node.

After the above modifications on the network are performed, a reduction of the network is done in order to make it easier to handle. This is obtained by dropping some useless nodes and arcs as described in the next section.