A Path Based Model

4.2 A Path Based Model

In this section we define formally the service design problem that we consider.

4.2.1 Problem input

Formally, the problem input specifies the railway network, represented by an acyclic digraph G =(,) VA , where V is the node set and A the arc set. As explained above, we are studying a corridor in the network, i.e. the design of paths in a given direction (North-South in our case), and this justifies the use of directed arcs (according to the corridor direction) and the acyclicity of the graph. By inverting the direction of all arcs, we can invert the direction of the corridor, and thus study the North-South and South-North directions as two separated

problems. Given that G is acyclic, we assume that the nodes in V are numbered according to a topological order, i.e. if (,)∈ ij

A then i < j . Each arc a =(,)∈ ij A has a transit time

f a = f (,) ij and a maximum axial load c a = c (,) ij , which cannot be exceeded by wagons moving on it. Finally, there are a maximum weight W and a maximum length L for trains

(seen as set of wagons, see below) travelling on the network.

A subset of the nodes of G represents terminals, where wagons can be loaded or unloaded, processing them before entering or exiting the rail system and moving to a different mode. This processing in each each terminal node i ∈ V is modelled by C i parallel

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processing lines with constant processing time P i , and each wagon is processed during a time

slot of length P i in one of the C i parallel lines.

Moreover, the problem input specifies a set T of wagons, which are homogenous carrying units. Each wagon t ∈ T has an origin o t and a destination d t , which are terminals

or shunting yards of the railway network, and correspond to nodes of G . Moreover, each wagon t ∈ T has a maximum transit time D t , i.e. if t is shipped from its origin to its

destination in a time longer than D t than t is considered in delay, a value e t which is earned if it is transported in time, and a sensitivity to delayed shipping, which reduces the profit

earned if the wagon is in delay, that will be defined formally in section 4.2.3. As already mentioned, the profit earned by carrying wagons may represent the monetary value that is paid for the shipping, but more generally it represents a priority which is given to wagons when, in a condition of limited capacity, one has to choose which wagons will be carried and which path will be used for their shipping. Finally, each wagon t ∈ T has a weight w t , a

length l t , and an axial load c t .

We let P denote the (exponentially large) collection of paths from some origin o s to

some destination d t > o s in G , with , st ∈ T (and possibly = s t ).

4.2.2 Problem objective

The problem aims at grouping the given wagons into trains. A train is represented by

a pair (,) p S , where p is a path in P and S ⊆ T is the set of wagons that are carried by the train. A train (,) p S is feasible if it satisfies the following technical constraints:

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• for each wagon t ∈ S , path p visits both nodes o t and d t , and for each arc a

between these two nodes in the path, the axial load constraint c t ≤ c a is respected;

• the total weight of the wagons simultaneously carried along path p does not exceed the weight capacity W ;

• the total length of the wagons simultaneously carried along path p does not exceed the length capacity L .

The profit of each train, given by the total profit of the associated wagons, will be defined formally in section 4.2.3.

We let Q denote the (exponentially large) collection of feasible trains.

The objective of the problem is to select at most k trains in Q having maximum total profit. The maximum number of trains to be designed is given as an input parameter, since it

represents a strategic decision which is taken a priori. Of course, it is natural to simulate different scenarios by changing this parameter.

4.2.3 Profits

The profit of a train q =(,) p S is the sum ∑ tS ∈ π t of the profits of the wagons carried

by q . For each wagon t , its profit π t equals the value e t when the wagon is shipped in time whereas, if the wagon is shipped in delay with respect to its transit time D t , π t is decreased by the cost of the delay, according to the sensitivity to delay of the wagon. The cost of the

delay is modelled by a piecewise linear function, composed by R pieces, which determines

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the cost (proportional to value e t ) for a percentage delay / lD t t , where l t is the absolute delay of the wagon. The function is convex, in order to penalize less a small delay than a long delay,

namely the profit of wagon t is:

t = e t − max{( µ t lD t / t + η t ) , = 1,..., } e t r R (1) where r µ

t is the slope of the r -th component of the function describing the cost of the delay for wagon t and r η

the intersection of this component with the vertical axis. We always set η 1

t =0 , so that there is no penalty when the delay is 0. An example of the cost delay function with R =3 is drown in Figure 4-1.

Figure 4-1. Cost of delay function.

4.2.4 Modelling stopping times in the network

In our model, the travel time f a for an arc a ∈ A is given by the arc length over the average speed on the arc (see section 4.6). In order to model waiting times which may occur

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at nodes of the network (e.g. waiting time at a terminal when the train stops to load or unload wagons, waiting time at a border crossing, waiting time in a station, etc.), we add dummy

arcs and nodes to graph G , and we set the length and the average speed on dummy arcs in such a way that the desired waiting time is obtained (see section 4.5 for details).