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This paper formulates the dynamic signal timing optimization (DSTO) problem as a bilevel model. In the upper level, total network travel time is minimized subject to some necessary signalisation constraints. In the lower level, the dynamic user-optimal route choice is formulated as a variational inequality model, which complies with the dynamic extension of Wardrop's first principle. The sensitivity analysis using a generalized inverse approach performed in the link variable is adopted to solve the DSTO problem. It is not required to generate further a nondegenerate extreme path solution from the link flow solution obtained by the FW method in the sensitivity analysis. This is the case as any path flow solution can be readily accommodated for use. Four heuristics with the same framework of the nested diagonalization (ND) method, but differing in updating decision variable or determining step size, are proposed and compared in terms of both computational performance and precision level. Numerical examples show that all four heuristics algorithms can obtain equilibrium solutions in compliance with the dynamic extension of War drop's first principle. However, considering a balance between precision level and computational performance, the Armijo-based method is recommended.

As a commentary note, this paper attempts to elaborate and make constructive critiques on Chen et als paper in which four issues may be identified: (1) travelers rerouting behavior is not respected when degeneracy of link capacities occurs, (2) variations of link flows and capacities in time is not considered, (3) sensitivity information of network capacity with respect to link flow cannot be directly derived, and (4) the trial-and-error method used in solving the bi-level network reliability problem is computationally inefficient. A new reliability model formulation is thus proposed to resolve the first two shortcomings. The methodological framework of Chen et al is still adopted but with necessary modifications based in the last two comments. A numerical example is then provided to validate the proposed framework.

The paper incorporates both the link capacity and first-in-first-out constraints into the dynamic user equilibrium problem. The corresponding dynamic equilibrium conditions for each origin-destination pair and time interval state that the generalized route travel times experienced by travelers are equal and minimal. A nested Lagrangian method embedding the gradient projection algorithm is then proposed and demonstrated with a numerical example.