Off-line approximate dynamic programming for the vehicle routing problem with a highly variable customer basis and stochastic demands

We study a stochastic variant of the vehicle routing problem arising in the context of domestic donor collection services. The problem we consider combines the following attributes. Customers requesting services are variable, in the sense that the customers are stochastic but are not restricted to a predefined set, as they may appear anywhere in a given service area. Furthermore, demand volumes are stochastic and observed upon visiting the customer. The objective is to maximize the expected served demands while meeting vehicle capacity and time restrictions. We call this problem the VRP with a highly Variable Customer basis and Stochastic Demands (VRP-VCSD). For this problem, we first propose a Markov Decision Process (MDP) formulation representing the classical centralized decision-making perspective where one decision-maker establishes the routes of all vehicles. While the resulting formulation turns out to be intractable, it provides us with the ground to develop a new MDP formulation, which we call partially decentralized. In this formulation, the action-space is decomposed by vehicle. However, the decentralization is incomplete as we enforce identical vehicle-specific policies while optimizing the collective reward. We propose several strategies to reduce the dimension of the state and action spaces associated with the partially decentralized formulation. These yield a considerably more tractable problem, which we solve via Reinforcement Learning. In particular, we develop a Q-learning algorithm called DecQN, featuring state-of-the-art acceleration techniques. We conduct a thorough computational analysis. Results show that DecQN considerably outperforms three benchmark policies. Moreover, we show that our approach can compete with specialized methods developed for the particular case of the VRP-VCSD, where customer locations and expected demands are known in advance.