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.
