Figure 1: Nine Node Network.
4 NUMERICAL RESULTS
In this section, we present results obtained on a
benchmark network. In (Hearn and Ramana, 1998),
the authors propose different secondary objectives for
a toll problem. The problem MINSYS consists in
minimizing the total amount of toll.
In this section, we provide a comparison of our
multicommodity MINSYS algorithm, or MMINSYS,
introduced in this article, with the MINSYS algo-
rithm, on a synthetic network. We present results
for the Nine Node network described in (Hearn and
Ramana, 1998), and illustrated in figure 1. The arcs
are given “Bureau of Public Roads” (BPR) latency
functions. The tuple near an arc denotes its free flow
travel time followed by its capacity in the sense of
BPR. There are four OD pairs, with four different
travel demands.
OD pair: [1,3] [1,4] [2,3] [2,4]
Demand: 10 20 30 40
Results are presented in table 1. The aggregate
flow and commodity flows corresponding to the dif-
ferent OD pairs, at social optimum, are explicited for
each arc of the network. Origin based-arc tolls, solu-
tion of the algorithm 2, are also listed.
The Λ
k
, potential difference between the desti-
nation and the origin, are respectively 30.59, 29.21,
32.95, 31.57 for the above OD pairs. It is also an
upper bound to the tolls charged to one user of the
OD pair. As journeys for the different OD pairs are
similar in terms of travel times, the fixed part (which
does not depend on the path choosen by the user) is
approximately the same for each commodity.
A comparison of general properties of MMINSYS
program with other secondary objectives is presented
in table 2. The toll vectors are computed as explained
in the previous section.
Let us focus on the new formulation MMINSYS
compared to the other classical programs.
• MMINSYS solution gives a lower total of tolls
than MINSYS. This is mathematically evident, as
MINSYS is a problem restriction of MMINSYS.
MMINSYS total tolls are 9.5 % lower: allowing
multicommodity tolls helps minimizing the total
number of tolls raised.
• MMINSYS solution gives also interesting values
for the number of tolled arcs or for the maxi-
mum arc toll: there are respectively 4 and 5 arcs
tolled for origins 0 and 1, which represents 5 arcs
tolled to the operator point of view. The maxi-
mum arc toll is 8.00 for one commodity, 12.00 for
the other, which is greater than MINSYS solution
and MINTB solution (identical in this problem).
• The main benefit of this new algorithm is that the
pricing vector has an analytical expression and
can be computed in a linear time in the product of
the number of arcs of the network and the number
of different origins of the OD pairs. It does not
need numerical solvers. On the contrary, MIN-
SYS is the solution of a linear program with poly-
nomial complexity, not expected to be linear in
general.
5 CONCLUSIONS
In this article, we introduce the notion of commodity-
based potential pricing in order to design optimal
OD-pair differentiated congestion charges, in the con-
text of real-time GPS sensing. Our contributions in-
clude the mathematical construction and analysis of
commodity-based potential pricing schemes, the de-
sign of algorithmic methods for efficient computation
of these potentials, and the theoretical and numerical
analysis of their properties.
We show that our potential-based formulation pro-
vides a new characterization of the set of pricing
schemes such that the charge incurred on each arc is
positive, whose existence is equivalent to the acyclic
property of the commodity flows. The proof of this
equivalence result is constructive, and is based on a
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