Superadditivity properties and new valid inequalities for the vehicle routing problem with stochastic demands

TL;DR

This paper investigates the superadditivity of the recourse function in the vehicle routing problem with stochastic demands, establishing conditions for the validity of the disaggregated integer L-shaped method and introducing new inequalities and algorithms that improve computational efficiency.

Contribution

It provides a necessary and sufficient condition for the validity of the DL-shaped method, rectifies previous theoretical gaps, and develops a more efficient algorithm with new valid inequalities.

Findings

The optimal restocking policy satisfies superadditivity.

The new DL-shaped algorithm outperforms existing methods.

Successfully solves challenging instances to optimality.

Abstract

Over the past thirty years, the vehicle routing problem with stochastic demands has emerged as a canonical application of the integer L-shaped method, leading to an extensive body of literature and several methodological refinements. Recently, the disaggregated integer L-shaped (DL-shaped) method, which decomposes the recourse function by customer rather than treating it as an aggregate cost, has been proposed and successfully applied under the detour-to-depot recourse policy. However, the validity of this new approach and its generalizability to other policies have not been thoroughly investigated. In this work, we provide a necessary and sufficient condition for the validity of the DL-shaped method, namely, the superadditivity of the recourse function under concatenation. We demonstrate that the optimal restocking policy satisfies this superadditivity property. Moreover, we rectify an incorrect argument from the original paper on the DL-shaped method to rigorously establish its validity under the detour-to-depot policy. We then develop a DL-shaped algorithm tailored to the optimal restocking policy. Our algorithm exploits new dynamic programming-based lower bounds on the optimal restocking recourse function. We also introduce new valid inequalities that generalize the original DL-shaped cuts and speed up computations by an order of magnitude. Computational experiments show that our DL-shaped algorithm significantly outperforms the state-of-the-art integer L-shaped algorithm from the literature. We solve several open instances to optimality, including 14 single-vehicle instances, which constitute the most challenging variant of the problem.