A new family of route formulations for split delivery vehicle routing problems

We propose a new family of formulations with route-based variables for the split delivery vehicle routing problem with and without time windows. Each formulation in this family is characterized by the maximum number of different quantities of demand that can be delivered to a customer during a vehicle visit. As opposed to previous formulations in the literature, the exact delivery quantities are not always explicitly known in this new family. The validity of these formulations is ensured by an exponential set of non-robust constraints. We also explore a property of optimal solutions that allows us to specify a minimum delivery quantity based on customer demand and vehicle capacity, and this number is often greater than one. We use this property to reduce the number of possible delivery quantities in our formulations, improving the solution times of the computationally strongest formulation in the family. In addition, we propose new variants of non-robust cutting planes that strengthen the formulations, which are limited-memory subset-row covering inequalities and limited-memory strong $k$-path inequalities. Finally, we develop a branch-cut-and-price algorithm to solve our formulations enriched with the proposed valid inequalities, which resorts to state-of-the-art algorithmic enhancements. We show how to effectively manage the non-robust cuts when solving the pricing problem that dynamically generates route variables. Numerical results indicate that our formulations and BCP algorithm establish new state-of-the-art results for the variant with time windows since all benchmark instances with 50 customers and many instances with 100 customers are solved to optimality for the first time. Several instances of the variant without time windows are solved to proven optimality for the first time