  Simulation-based optimization (SO or SBO) has become increasingly important
to address challenging transportation network design problems. In this paper,
we propose to solve two toll pricing problems with different levels of
complexity using the concept of the macroscopic or network fundamental diagram
(MFD or NFD), where a large-scale simulation-based dynamic traffic assignment
model of Melbourne, Australia is used. Four computationally efficient SBO
methods are applied and compared, including the proportional-integral (PI)
controller, regressing kriging (RK), DIviding RECTangles (DIRECT), and
simultaneous perturbation stochastic approximation (SPSA). The comparison
reveals that these methods work equally well on the simple problem without
exhibiting significant performance differences. But, for the complex problem,
RK manifests itself to be the best-performing method thanks to its capability
of filtering out the numerical noise arising from computer simulations (i.e.
allowing for non-smoothness of the objective function). While the PI controller
is a more competitive solution to the simple problem given its faster rate of
convergence, the poor scalability of the method in the complex problem results
in limited applicability. Two caveats, however, deserve emphasis: (i) the
chosen critical network density of the NFD does not necessarily represent a
robust network control or optimization threshold, as it might shift in the
presence of toll pricing; and (ii) re-interpolation is required as part of RK
in order to achieve global convergence.
