An Exact Algorithm for the Pickup and Delivery Problem with Time Windows and Last-in-First-out Loading

Applications of the pickup and delivery problem with time windows and last-in-first-out (LIFO) loading (PDPTWL) constraints can be found in the transportation of animals, heavyweight goods, and hazardous materials, where unloading vehicles requires more time and special handling. Examples include carrying livestock, cars, and chemical containers. Research on exact methods to solve the pickup and delivery problem with time windows (PDPTW) and its variants has mainly focused on branch-and-price-and-cut algorithms. In this paper, we propose a novel exact approach based on fragments— a series of pickup and delivery requests starting and ending with an empty vehicle. We use fragments to formulate a relaxed network flow model with side constraints. Lazy constraints are used to cut off any illegal solution that may occur while solving the integer program. Extensive computational experiments show that the proposed approach is superior to the current state-of-the-art method.