Large Neighborhood-Based Metaheuristic and Branch-and-Price for the Pickup and Delivery Problem with Split Loads

We consider the multi-vehicle one-to-one pickup and delivery problem with split loads, a NP-hard problem linked with a variety of applications for bulk product transportation, bike-sharing systems and inventory re-balancing. This problem is notoriously difficult due to the interaction of two challenging vehicle routing attributes, "pickups and deliveries" and "split deliveries". This possibly leads to optimal solutions of a size that grows exponentially with the instance size, containing multiple visits per customer pair, even in the same route. To solve this problem, we propose an iterated local search metaheuristic as well as a branch-and-price algorithm. The core of the metaheuristic consists of a new large neighborhood search, which reduces the problem of finding the best insertion combination of a pickup and delivery pair into a route (with possible splits) to a resource-constrained shortest path and knapsack problem. Similarly, the branch-and-price algorithm uses sophisticated labeling techniques, route relaxations, pre-processing and branching rules for an efficient resolution. Our computational experiments on classical single-vehicle instances demonstrate the excellent performance of the metaheuristic, which produces new best known solutions for 92 out of 93 test instances, and outperforms all previous algorithms. Experimental results on new multi-vehicle instances with distance constraints are also reported. The branch-and-price algorithm produces optimal solutions for instances with up to 20 pickup-and-delivery pairs, and very accurate solutions are found by the metaheuristic.