Augmented Lagrangian relaxation approach for logistics vehicle routing problem with mixed backhauls and time windows

•Study the vehicle routing problem with mixed backhauls and time windows for the city logistics.•Formulate a network flow optimization model in an extended state-space-time network.•Implement an augmented Lagrangian relaxation technique to construct the quadratic 0–1 programming model.•Decompose the model into a series of sub-problems, solved in a block Gauss–Seidel framework. This paper studies the vehicle routing problem with mixed backhauls and time windows (VRPMBTW) for city logistics. A time-discretized multi-commodity network flow optimization model is proposed in an extended state-space-time network representation, where the time-dependent pickups and deliveries can be depicted by extending the state dimensions. By implementing an augmented Lagrangian relaxation technique, the VRPMBTW is reformulated as a quadratic 0–1 programming model, which is further decomposed into the standard least-cost-path sub-problems, and iteratively solved by dynamic programming in a block nonlinear Gauss-Seidel framework. The proposed approach is tested on the simple 9-node network and the real-world Chicago sketch network.