A linear framework for dynamic user equilibrium traffic assignment in a single origin-destination capacitated network - Hoang, Nam H.; Vu, Hai L.; Panda, Manoj; Lo, Hong K. (Transportation research. Part B: methodological, 2019)
•Develop a novel mathematical framework for studying DTA problems in a dynamic capacitated network that applies to both the SO and UE formulations in a unified manner.•Prove that in the limit, the SO objective could be used to obtain the UE solution as the time step goes to zero, given the satisfaction of the FIFO constraint.•Show that the difference between the UE and SO objectives can be used as a measure to evaluate the accuracy of the approximation.•Propose an incremental solution method to effectively solve the non-linear path-based UE-DTA problem as a sequence of linear link-based SO-DTA problems where the convergence is guaranteed with a predetermined number of iterations. The dynamic traffic assignment (DTA) problem has been studied intensively in the literature. However, there is no existing linear framework to solve the user equilibrium (UE) DTA problem. In this paper, we develop a novel linear programming framework to solve the UE-DTA problem in a dynamic capacity network that exploits the linkage between the UE and system optimal (SO) solutions underpinned by a first-in-first-out (FIFO) principle. This important property enables us to develop an incremental loading method to obtain the UE solutions efficiently by solving a sequence of linear programs. The proposed solution methodology possesses several nice properties such as a predictable number of iterations before reaching the UE solution, and a linear system of equations to be solved in each of the iterations. In contrast to the related iterative methods, such as Frank–Wolfe algorithm, successive average (MSA) or projection and their extensions where the purpose of iteration is to seek the solution convergence, whereas ours is to solve a linear problem over multiple iterations but only for a single unit of demand in each iteration. Furthermore, we provide a theoretical proof that in the limit, the SO objective can be used to obtain the UE solution as the system time step goes to zero given the satisfaction of the FIFO constraint. We show via numerical examples the significant improvements in the obtained UE solutions both in terms of accuracy and computational complexity.