Constraint Reformulation and a Lagrangian Relaxation–based Solution Algorithm for a Least Expected Time Path Problem

Using a sample-based representation scheme to capture spatial and temporal travel time correlations, this article constructs an integer programming model for finding the a priori least expected time paths. We explicitly consider the unique path constraint associated with the a priori path in a time-dependent and stochastic network, and propose a number of reformulations to establish linear inequalities that can be easily dualized by a Lagrangian relaxation solution approach. The relaxed model is further decomposed into two sub-problems, which can be solved directly by using a modified label-correcting algorithm and a simple single-value linear programming method. The numerical experiments investigate the quality and computational efficiency of the proposed solution approach.