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the possible presence of traffic light (Cascetta 1990). Yet, in our case, the point is to evaluate
the traffic variation due to the location of a certain facility in a certain area.
Then, the mentioned function can be linearly approximated by the tangent in the operation
point in question. More precisely, given the regular traffic in the considered link, one can
determine the corresponding travel time t 0 (i,j) . Then, the travel time after that the facility
has been located can be written as
(1)
t
(
i
,
j
)
t
(
i
,
j
)
R
[
n
(
i
,
j
)
n
(
i
,
j
)]
t
(
i
,
j
)
R
ij
n
(
i
,
j
)
0
0
0
where ∆N(i,j) is the traffic intensity induced by the considered facility and
dt
(
i
,
j
)
R ij
(2)
dn
(
i
,
j
)
0
Clearly, in searching the optimal facility location, it is the average access time to be crucial
rather than the access time of a single client. This is the reason why a macroscopic
continuous-time model turns out to be the correct choice. Moreover, since there are plenty of
efficient commercial tools for the analysis of electrical networks, it seems natural to depict
an electrical equivalent of each link of the transportation network (Figure 1) in fact
representing this latter as an electric network.
Fig. 1. The electric equivalent of a link of the transportation network
Note that, in the electric equivalent of a link, the presence of a diode guarantees that the
current flows in a unique direction, and so does the traffic. The role of the voltage generator
E ij is to polarize the diode, thus representing the original travel time t 0 (i,j). The value of the
resistor R ij describes the dependence on the current I ij , i.e., on the induced traffic intensity
∆n(i,j). Finally, the voltage V ij represents the link travel time t(i,j).
To determine the electric equivalent parameters corresponding to each link of the
transportation network the following considerations can be made. The travel time, in
seconds, along an urban road in regular traffic conditions is given by
3600
t
t
(3)
s
v
where Λ is the length in Km of the path between two subsequent road intersections, ν is the
mean speed (Km/h) of the vehicles, t s is the additional time due to the presence of a traffic
light; t s can be determined by well known empirical expressions which interpolate
experimental data (Cascetta, 1990).
3.3 The model of the clients population
Relying on the electrical modeling equivalent, the clients' population can be modeled by
means of ideal current generators which inject their current in the nodes of the
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