Dynamic congestion pricing with demand uncertainty: A robust optimization approach

In this paper, we consider dynamic congestion pricing in the presence of demand uncertainty. In particular, we apply a robust optimization (RO) approach based on a bi‐level cellular particle swarm optimization (BCPSO) to optimal congestion pricing problems when flows correspond to dynamic user equilibrium on the network of interest. Such a formulation is recognized as a second‐best pricing problem, and we refer to it as the dynamic optimal toll problem with equilibrium constraints (DOTPEC). We then present numerical experiments in which BCPSO is compared with two alternative robust dynamic solution approaches: bi‐level simulated annealing (BSA) and cutting plane‐based simulated annealing (CPSA), as well as a nominal dynamic solution and a robust static solution. We show that robust dynamic solutions improve the worst case, average, and stability of total travel cost in comparison with the nominal dynamic and the robust static solutions. The numerical results also show that BCPSO outperforms BSA and CPSA in terms of solution quality and computational efficiency.