An enhanced exact algorithm for the multi-trip vehicle routing problem with time windows and capacitated unloading station

This paper introduces a vehicle routing problem that simultaneously considers multiple trips, time windows, and a capacitated unloading station. This problem is a generalization of the multi-trip vehicle routing problem with time windows, which determines a set of least-cost vehicle routes to fulfill all customer demands while respecting the constraints of vehicle capacity and time windows. Due to restricted resources (e.g., equipment and labor force) at the depot, vehicles may need to wait in a queue for being unloaded when they arrive. This unloading capacity constraint significantly complicates the problem, as it causes a trip to involve three stages-traveling, waiting, and unloading. We formulate this problem as an arc flow model and a trip-based set partitioning model, where the latter is solved by a branch-price-and-cut (BPC) algorithm. To improve the computational aspect of the BPC framework, a two-phase column generation (CG) algorithm is designed. First, a bidirectional labeling algorithm is tailored to solve the pricing problem, where two accelerating strategies are employed to speed up the resolution process. Meanwhile, k-path inequalities and limited-memory subset row inequalities are utilized to tighten the linear relaxation of the master problem. Computational results based on the instances adapted from the well-known Solomon's benchmark show that the developed BPC algorithm can solve most instances within 50 customers to optimality in a short time frame and some instances of 100 customers to optimality within a 3-hour time limit. Moreover, our BPC algorithm performs better than exact algorithms in the literature for similar problem variants in both solution quality and computing time.