An overview of nontraditional formulations of static and dynamic equilibrium network design

This paper reviews two classes of nontraditional models for the (dis)equilibrium network design problem and uses these to describe research needed to advance the state-of-the art in the design of both static and dynamic networks. The static equilibrium design model emphasized herein recalls an important earlier result that allows the equilibrium network design problem to be stated as a single level mathematical program (SMP), a result which is surprisingly little known; it also introduces for the first time nonseparable elastic transportation demands and attendant difficulties in evaluating the consumers' surplus line integral. The dynamic, disequilibrium network design model emphasized herein maintains the usual design objective of maximizing some measure of social welfare, but recognizes that traffic on a network is not necessarily in equilibrium and that capacity changes to the network must induce transient phenomena not captured by invocation of the static version of Wardrop's first principle (user equilibrium). It is argued that such disequilibrium models by their very nature avoid temporal versions of Braess' paradox familiar from static equilibrium design and are naturally formulated as optimal control problems. Moreover, properly formulated disequilibrium design models are shown to overcome difficulties associated with evaluating the consumers' surplus line integral. Furthermore, when the associated disequilibrium dynamics are stable, these optimal control formulations are observed to be capable of computing static equilibrium network designs.