Partial-Outsourcing Strategy for the Vehicle Routing Problem With Stochastic Demands

Abstract

This paper studies a combined delivery strategy involving a private vehicle and external carriers under stochastic customer demands. The routing problem focuses on a single private vehicle, while external carriers are allowed to determine their own routes independently and are compensated with a fixed price per unit demand served. A strategy incorporating routing re-optimization is proposed, along with a new recourse mechanism that leverages outsourcing through external carriers. To enable routing re-optimization, a novel approximate linear programming (ALP) approach is introduced. This offers a new pathway for addressing vehicle routing problems (VRPs) under stochastic demand considerations. The ALP approach is adapted to the specific structure of routing under stochastic demands, leading to the development of a decomposition-based ALP solution framework. This adaptation arises from changes in the decision sequence of routing and restocking at each step of the Markov decision process (MDP), which differs from previous formulations of vehicle routing under stochastic demands. Additionally, further adaptations are made to facilitate the computation of the proposed strategy by exploring the relationships among variables and constraints specific to the problem context, as well as by developing a constraint sampling procedure designed to mimic the near-optimal heuristic policy. Our numerical results show that the proposed outsourcing-based policy yields notable operating-cost savings, with an average improvement of 4.06% over the traditional recourse strategy in midpoint-depot instances. Moreover, in small instances where the optimal policy within the traditional partial re-optimization framework can be computed, the proposed price-directed (PD) policy still provides cost advantages over this re-optimization scheme, demonstrating the value of our ALP-based framework.