An exact algorithm for a mobile production vehicle routing problem

Integrating portable or vehicle-mounted additive manufacturing machines into the delivery system smoothly bridges production and delivery, exemplifying the principles of Industry 4.0 and smart logistics. This concurrent manufacturing and delivery model effectively shortens the time between production and consumption. Nevertheless, the intertwined decisions surrounding these processes pose challenges in optimizing machine schedules and vehicle routes. To address these challenges, we investigate an optimization problem called the mobile production vehicle routing problem, where each vehicle with an additive machine serves assigned customers, and each customer requires one or more items to be produced. In this paper, we first describe arc flow-based and set-partitioning-based models for the problem. We then design a branch-price-and-cut algorithm to solve the problem exactly based on the set-partitioning-based model. Specifically, we introduce a column generation algorithm relying on an adaptive adjustment mechanism of the critical resource bound within a bi-directional search framework. Finally, we perform extensive numerical experiments to validate the effectiveness of our algorithm. Additionally, we analyze the impact of batch printing capacities, production times, and machine scheduling constraints on the solution.