An Augmented Lagrangean Scheme for

The inclusion of explicit bounds on the link ows in tra c assignment models has been proposed as a means to obtain a more accurate description of the tra c behaviour than that given by the basic uncapacitated model Although the capacitated problem has the advantage of being numerically more tractable than models involving travel cost functions which tend to in nity as ows approach the link capacities it has received very limited attention in the past The main reason for this is that the explicit capacities make the problem computationally more demanding since they destroy the Cartesian product structure of the feasible set of the uncapacitated model which enables the development of highly e cient solution procedures for that problem The availability of e cient procedures for the basic model motivates the use of dualization approaches for handling the capacity constraints of the more complex model we propose and evaluate an augmented Lagrangean method in which uncapacitated tra c assign ment subproblems are solved with the disaggregate simplicial decomposition algorithm This algorithm fully exploits the subproblems structure and also has very favourable reoptimization facilities Both these properties are of greatest importance for achieving computational e ciency of augmented Lagrangean schemes The overall method exhibits a linear rate of convergence under standard nondegeneracy assumptions Experiments with medium scale tra c networks with varying levels of capacities demon strate that the proposed algorithm is e cient and solves capacitated problems within a factor of roughly three to four of the computing times needed to solve the corre sponding uncapacitated problems also the augmented Lagrangean scheme compares very favourably with both ordinary Lagrangean dualization and penalty methods The suggested strategy can be utilized for the solution of other extensions of the basic tra c assignment problem where explicit constraints are added in order to re ne the basic model these constraints can be used to model for instance tra c ows at intersections or di erent categories of vehicles This way of extending the basic model may be an interesting alternative to the use of asymmetric tra c assignment models where such extensions are made through the development of complex travel cost functions which in practical applications are di cult to calibrate