Global optimization method for robust pricing of transportation networks under uncertain demand

We extend the existing toll pricing studies with fixed demand to stochastic demand. A new and practical second-best pricing problem with uncertain demand is proposed and formulated as a stochastic mathematical program with equilibrium constraints. In view of the problem structure, we develop a tailored global optimization algorithm. This algorithm incorporates a sample average approximation scheme, a relaxation-strengthening method, and a linearization approach. The proposed global optimization algorithm is applied to three networks: a two-link network, a seven-eleven network and the Sioux-Falls network. The results demonstrate that using a single fixed estimation of future demand may overestimate the future system performance, which is consistent with previous studies. Moreover, the optimal toll obtained by using the mean demand value may not be optimal considering demand uncertainty. The proposed global optimization algorithm explicitly captures demand uncertainty and yields solutions that outperform those without considering demand uncertainty.