A Dynamic Side-Constrained User Equilibrium Problem

Abstract: This paper incorporates two side constraints, namely, the link capacity and first-in-first-out (FIFO) constraints, into the dynamic user equilibrium (DUE) problem. Although the link capacity constraint has been addressed in the literature, its implications for the associated multiple-valued dual variables have not been properly drawn out and, thus, the matter is still subject to controversy. Since the dual variables associated with any side constraint can hardly be realized as true link queuing delays on the physical links, it is more appropriate to regard them as shadow costs (or link toll levies) for avoiding traffic congestion. Interpreting the dual variables as shadow costs (or minimum tolls) reconciles, to a great extent, the arguments that might accrue due to the multiple-valued dual variables. Moreover, this approach implies that the feasible time-space networks used in the first loop operations of the proposed nested diagonalization (ND) method can only be constructed/updated according to the actual link travel times, rather than by the generalized actual link travel times. On the other hand, the FIFO constraint is unusual because it can be activated only when the same physical links in two different time intervals have positive flows. This unusual FIFO constraint is tactically treated by introducing a set of new indicator variables to identify the incidence relationships. The new indicator variables have been incorporated into the equivalence proof between the equilibrium conditions and the variational inequality (VI) model, as well as the proposed ND method. The ND method, embedding the augmented Lagrangian method (ALM), which is in turn coupled with the path-based gradient projection (GP) algorithm, is then demonstrated with a numerical example. The results show that the corresponding dynamic generalized equilibrium conditions are satisfied, since for each origin-destination pair and time interval, the generalized route travel times incurred by the travelers on all used paths are equal and minimal. Although the two side constraints have been incorporated into the DUE problem and their physical meanings have been interpreted in a more plausible way, the interactive effects between these two side constraints are still unknown and, therefore, require more in-depth sensitivity analysis in the future.