Side Constrained Traffic Equilibrium Models---Traffic Management Through
Link Tolls

Torbjörn Larsson 

Division of Optimization 
Department of Mathematics 
Linköping Institute of Technology
S-581 83 Linköping 
Sweden

and 

Michael Patriksson

Department of Mathematics 
Chalmers University of Technology
S-412 96 Göteborg
Sweden

ABSTRACT:

We propose a systematic means for achieving a set of overall traffic
management or planning goals with respect to the performance of the
traffic network through the use of link tolls.  The primary goals are
defined by a set of restrictions (constraints) on the link flows.  The
tolls that achieve these goals are obtained by solving and extracting
values of Lagrange multipliers from a generalization of the classical
user equilibrium model which includes the link flow restrictions as
side constraints.  The set of toll prices obtained is not necessarily
unique in the inelastic demand case; this fact enables the traffic
manager to choose a toll scheme which satisfies exogenous constraints
on the toll scheme implemented and which may optimize a secondary
goal, such as with respect to the toll itself.  For the elastic demand
case it is however shown that the total toll revenue is unique; an
interesting implication of this result is that if the traffic
management goal is to obtain a system-optimal flow with a minimal toll
collected, then the classical marginal cost pricing concept is
optimal.
  
The overall model is derived as a restriction of a mathematical
program with equilibrium constraints (MPEC) describing a Stackelberg
game involving the traffic manager and the users of the network.  The
model is shown to yield valuable information also in the case where
the management goals and exogenous toll constraints are inconsistent
with each other or with the underlying transportation network.  We
give several examples of possible applications and extensions of the
model, including the achievement of a system optimal flow and of
commodity- and/or user class-specific traffic management goals, and
the derivation of actions for making public transport more attractive.
We propose a conceptual algorithm for solving it, present preliminary
experience from its use, and an example that illustrates some
fundamental properties of the toll optimization problem.  It is hoped
that the paper provokes continued research in both theoretical and
applied directions.


KEY WORDS: Stackelberg game, Link Tolls, Mathematical Programs With 
Equilibrium Constraints, Side Constrained Traffic Equilibrium, 
Lagrange Multipliers, Toll Optimization, Network Design, 
System Optimal Flows, Subsidies