However, recent results have shown that competitive performance can be achieved with strongly reduced, discretized versions of the original action space (Tavakoli et al., 2018; Tang & Agrawal, 2020; Seyde et al., 2021). This opens the question whether tasks with complex high-dimensional action spaces can be solved using simpler critic-only, discrete action-space algorithms instead. A potential candidate is Q-learning which only requires learning a critic with the policy commonly following via ϵ-greedy or Boltzmann exploration (Watkins & Dayan, 1992; Mnih et al., 2013). While naive Q-learning struggles in high-dimensional action spaces due to exponential scaling of possible action combinations, the multi-agent RL literature has shown that factored value function representations in combination with centralized training can alleviate some of these challenges (Sunehag et al., 2017; Rashid et al., 2018), further inspiring transfer to single-agent control settings (Sharma et al., 2017; Tavakoli, 2021). Other methods have been shown to enable application of critic-only agents to continuous action spaces but require additional, costly, sampling-based optimization (Kalashnikov et al., 2018).

This paper studies a stochastic variant of the vehicle routing problem (VRP) where both customer locations and demands are uncertain. In particular, potential customers are not restricted to a predefined customer set but are continuously spatially distributed in a given service area. The objective is to maximize the served demands while fulfilling vehicle capacities and time restrictions. We call this problem the VRP with stochastic customers and demands (VRPSCD). 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 of the VRPSCD representing a decentralized decision-making framework, where vehicles autonomously establish their own routes. This new formulation allows us to develop several strategies to reduce the dimension of the state and action spaces, resulting in a considerably more tractable problem. We solve the decentralized problem via Reinforcement Learning, and in particular, we develop a Q-learning algorithm featuring state-of-the-art acceleration techniques such as Replay Memory and Double Q Network. Computational results show that our method considerably outperforms two commonly adopted benchmark policies (random and heuristic). Moreover, when comparing with existing literature, we show that our approach can compete with specialized methods developed for the particular case of the VRPSCD where customer locations and expected demands are known in advance. Finally, we show that the value functions and policies obtained by our algorithm can be easily embedded in Rollout algorithms, thus further improving their performances.