On Side Constrained Models of Traffic

The basic model of traac assignment does not capture complex traac ow relationships such as interactions among vehicles on diierent road links, joint capacities on two-way streets, turning priorities in junctions, etc. Traditionally, such relationships are introduced implicitly through generalizations of the separable travel cost function of the basic model, and the reened models are usually stated as variational inequalities. However, this approach has found very little practical use, seemingly mostly because the generalized travel cost functions are not easily chosen and calibrated. An alternative strategy for reening the basic model is to capture supplementary traac ow relationships explicitly through the introduction of side constraints. This explicit approach has the advantage of always leading to optimization formulations, and the immediate physical interpretations of the side constraints introduced may make it easy to construct, calibrate and apply the resulting models compared to models with non-separable cost functions. We consider a traac equilibrium model with general side constraints and show that its optimality conditions correspond to a generalization of Wardrop's user equilibrium principle. Further, we derive a close relationship between models of traac equilibria involving side constraints and generalized travel cost functions, respectively. These results motivate a further study into the art of modelling traac assignment problems through the use of explicit side constraints. 1 INTRODUCTION AND MOTIVATION Consider a transportation network G = (N; A) of nodes and directed links. Between certain pairs of origins and destinations, (p; q) 2 C, xed positive travel demands d pq are given, and each link a 2 A is associated with a positive and strictly increasing travel cost function t a : < jAj + 7 ! < ++. The user equilibrium principle of Wardrop (1952)