Elastic-demand bi-criteria traffic assignment under the continuously distributed value of time: A two-stage gradient projection algorithm with graphical interpretations

In this paper, we study the elastic-demand bi-criteria traffic assignment problem under the continuously distributed value of time, referred to as the ED-CBiTA problem for simplicity. Specifically, the origin and destination (O–D) demand of ED-CBiTA is endogenously guided by the expected generalized travel time aggregated from all efficient paths, and the consideration of user heterogeneity regarding the tradeoff between time and toll is accomplished by incorporating a continuously distributed value of time. We present a variable demand formulation and an equivalent excess demand reformulation for the ED-CBiTA problem. Based on two types of Gauss–Seidel decomposition schemes, we propose a novel two-stage gradient projection (TSGP) algorithm, which implicitly delivers visual interpretations to depict the interplay of supply and demand interactions. The first stage, called demand equilibration, aims to adjust O–D demand and all efficient path flows “vertically upward or downward” based on the level of network congestion. The second stage, namely boundary equilibration, is to perform the boundary movements and adjust adjacent efficient flows “horizontally forward or backward”, to achieve exact positions along the Pareto frontier. Numerical results on a small network show TSGP’s features and confirm that TSGP significantly outperforms two link-based benchmark algorithms. For instances of practical network size, TSGP consistently promises to obtain high-quality solutions with a rather smaller CPU time.