On vehicle routing problems with stochastic demands -- Scenario-optimal recourse policies

TL;DR

This paper introduces a new framework for solving stochastic vehicle routing problems with scenario-based demands, using scenario recourse inequalities to improve solution quality and computational efficiency.

Contribution

It proposes a novel class of inequalities called scenario recourse inequalities (SRIs) that generalize existing cuts and enhance solving VRPSDs with scenario-optimal policies.

Findings

SRIs are valid for any recourse policy with mild assumptions.

SRIs dominate several known classes of integer L-shaped cuts.

The new algorithm solves 329 more instances to optimality than previous methods.

Abstract

Two-Stage Vehicle Routing Problems with Stochastic Demands (VRPSDs) form a class of stochastic combinatorial optimization problems where routes are planned in advance, demands are revealed upon vehicle arrival, and recourse actions are triggered whenever capacity is exceeded. Following recent works, we consider VRPSDs where demands are given by an empirical probability distribution of scenarios. Existing approaches rely on integer L-shaped (ILS) cuts, whose coefficients are tailored for specific recourse policies. In contrast, we propose a framework that casts recourse policies as solutions of a higher-dimensional mixed-integer program, and we characterize its convex hull in the original lower-dimensional space via a new class of inequalities called scenario recourse inequalities (SRIs). We show that SRIs are valid for any recourse policy satisfying mild assumptions and are sufficient for formulating the VRPSD under a scenario-optimal recourse policy, where the recourse actions are chosen optimally for each scenario. Under this latter policy, we also demonstrate that SRIs dominate several known classes of ILS cuts. We conduct computational experiments on the VRPSD with scenarios under both the classical and the scenario-optimal recourse policies. By using the SRIs, our algorithm solves 329 more instances to optimality than the previous state-of-the-art ILS algorithm.