Temporal and spatial equilibrium of flow on networks

Knowing the temporal and spatial equilibrium (TSE) flow is the basis for transportation planning and management of each mode (ocean, air, rail, road, logistics, etc.). It also helps to alleviate congestion of each mode on a transportation network. The current methods in transportation planning only consider the spatial equilibrium of flow on the network. A method for obtaining the TSE of flow on a transportation network is rare. In this paper, a one-level variational inequality (VI) model was presented for the TSE of flow on networks. A relaxation with multilevel gradient projection (GP) algorithm is proposed to solve the model, with the 1st level gradient projection for temporal equilibrium and the 2nd level gradient projection for spatial equilibrium. Given a temporal distribution of total flow for each origin-destination (OD), the spatial equilibrium can be reached with the 2nd level algorithm. The temporal equilibrium can be reached by splitting the total flow for each OD on the temporal dimension with the 1st-level algorithm. The application of the model and algorithm to two networks shows that the minimum path travel time of an OD is equal and minimal. It also shows that the total cost of all flows reaches a minimum and that flows reach equilibrium on temporal and spatial dimensions. Our paper is the first study able to show this and has great potential in transportation planning and management, or in balancing vehicles/vessels/planes, or goods on temporal and spatial dimensions to alleviate congestion. The algorithm does not need time-space network expansion and applies to real-size multiple-origin multiple-destination networks.