Information Technology Reference
In-Depth Information
If we replace
q
1
=
Q − q
2
in Eq. 4, we can calculate the derivative of
Θ
with respect to
q
1
and equalling the derivative to 0 we obtain:
β
1
+
β
2
+
1
β
2
Q
α
2
− α
1
β
1
+
β
2
β
1
+
β
2
−
1
β
1
Q
α
2
− α
1
β
1
+
β
2
q
1
=
q
2
=
(4)
2
2
q
1
q
2
The above values
represent the
system optimum assignment
, based on
Wardrop's second principle [6], which states that road users cooperate with one another
in order to minimise the social transportation cost.
In order to force the attainment of a socially optimal user equilibrium we conceive
the entire urban road network as a market of competitive agents that “produce mobility”
on a portion of the network. Following this metaphor, for each market agent the output
q
is the number of road users that the agent can accommodate in its controlled area. The
price
p
is the price charged for the “production” of a road user that travels in the area
controlled by the agent. Since producing mobility is costly for the society, we assume
that each agent is characterised by a cost function
c
. If the cost function takes into
account exclusively the impact of congestion, it can be modelled as the delay caused by
the presence of road users:
and
c
(
q
)=
v
(
z
(
q
)
− z
(0))
(5)
where
z
(
q
)
is the travel time when
q
road users travel on the link,
z
(0)
is the travel time
at free flow, and
v
is the value of time.
For agent 1, the cost function becomes:
c
1
(
q
1
)=
v
(
z
(
q
1
)
− z
(0)) =
v
(
α
1
+
β
1
q
1
− α
1
)=
β
1
vq
1
(6)
If agent 1 produces mobility for
q
1
road users, the total production cost
TC
1
is:
TC
1
=
β
1
vq
1
· q
1
=
β
1
vq
1
(7)
Since agent 1 is operating in a market, it charges a price
p
1
(
q
1
)
to the
q
1
road users that
travel on its link. The goal is raising a total revenue
TR
1
which covers the total cost
and generates an arbitrary profit
π
1
:
TR
1
=
p
1
(
q
1
)
· q
1
=
TC
1
+
π
1
=
β
1
vq
1
+
π
1
(8)
The profit maximising price is therefore:
p
1
(
q
1
)=
β
1
vq
1
+
π
1
/q
1
(9)
Similarly, for agent 2, the profit maximising price is:
p
2
(
q
2
)=
β
2
vq
2
+
π
2
/q
2
(10)
Given that the links are now priced, the transportation cost for road users is now the
travel time cost plus the mobility cost. At equilibrium, the cost of travelling on link 1
equals the cost of travelling on link 2:
(
α
1
+
β
1
q
1
)
v
+
β
1
vq
1
+
π
1
/q
1
=(
α
2
+
β
2
q
2
)
v
+
β
2
vq
2
+
π
2
/q
2
s. t.
(11)
q
1
+
q
2
=
Q
