Efficient Negative Cycle–Canceling Algorithm for Finding the Optimal Traffic Routing for Network Evacuation with Nonuniform Threats

A new network flow solution method is designed to determine optimal traffic routing efficiently for the evacuation of networks with several threat zones and with nonuniform threat levels across zones. The objective is to minimize total exposure (as duration and severity) to the threat for all evacuees during the evacuation. The problem is formulated as a minimum cost dynamic flow problem coupled with traffic dynamic constraints. The traffic flow dynamic constraints are enforced by the well-known point queue and spatial queue models in a time-expanded network presentation. The key to the efficiency of the proposed method is that, for any feasible solution, the algorithm can find and can cancel multiple negative cycles (including the cycle with the largest negative cost) with a single shortest path calculation made possible by applying a proposed transformation to the original problem. A cost transformation function and a multisource shortest path algorithm are proposed to facilitate the efficient detection and cancelation of negative cycles. Zone by zone, negative cycles are canceled at the border links of the zones. The solution method is proved to be optimal. The algorithm is implemented, tested, and verified to be optimal for a midsized example problem.