Simulation-based optimization of toll pricing in large-scale urban networks using the network fundamental diagram: A cross-comparison of methods

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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.