  This study presents a Bayesian Optimization framework for area- and
distance-based time-of-day pricing (TODP) for urban networks. The road pricing
optimization problem can reach high level of complexity depending on the
pricing scheme considered, its associated detailed network properties and the
affected heterogeneous demand features. We consider heterogeneous travellers
with individual-specific trip attributes and departure-time choice parameters
together with a Macroscopic Fundamental Diagram (MFD) model for the urban
network. Its mathematical formulation is presented and an agent-based
simulation framework is constructed as evaluation function for the TODP
optimization problem. The latter becomes highly nonlinear and relying on an
expensive-to-evaluate objective function. We then present and test a Bayesian
Optimization approach to compute different time-of-day pricing schemes by
maximizing social welfare. Our proposed method learns the relationship between
the prices and welfare within a few iterations and is able to find good
solutions even in scenarios with high dimensionality in the decision variables
space, setting a path for complexity reduction in more realistic road pricing
optimization problems. Furthermore and as expected, the simulation results show
that TODP improves the social welfare against the no-pricing case.
