Vehicle routing problem using deep reinforcement learning - A case study about truck planning in the industry

Vehicle Routing Problems (VRPs) are fundamental combinatorial optimization problems with widespread applications in various scenarios. The advanced optimization solvers can effectively solve such problems. However, modeling complex VRP variants for solvers often requires substantial domain expertise, which limits the accessibility of advanced optimization technologies. In this paper, we propose Reinforcement Learning Enhanced LLMAgents(RLEA), a multi-agent framework designed to automate the modeling of complex VRPs. RLEA introduces a lightweight neural Planner trained with Soft Q-learning to efficiently orchestrate the actions of LLM-based agents. In addition, we equip the system with an evolutionary memory module and retrieval-augmented generation, enabling the agent to leverage both accumulated experience and external solver knowledge during program generation and refinement for solving VRPs. We evaluated 48 distinct VRP variants across various solvers. The experimental results demonstrate that RLEA outperforms the previous state-of-the-ar method, achieving a 16.67% higher success rate while significantly reducing runtime errors. These results validate that integrating reinforcement learning with LLM-based reasoning is highly effective for automated optimization modeling. The appendix is available at: https://doi.org/10.5281/zenodo.19134435.

We introduce the vehicle routing problem with stochastic demands and outsourcing options (VRP-SDO), in which a logistics service provider partitions customer requests into customers outsourced to a common carrier and customers committed to its fixed fleet. The latter induces a vehicle routing problem with stochastic demands (VRP-SD), solved dynamically. Demands are revealed upon visit; residual demand may be served by other vehicles or after restocking at the depot. Work beyond the regular shift incurs overtime costs, and the unit outsourcing cost decreases with the expected outsourced demand. The objective is to minimize expected travel, overtime, and outsourcing costs. We propose an iterative two-level methodology whose first level partitions customers into committed and outsourced subsets, while the second level estimates the expected VRP-SD routing cost. To avoid solving this problem from scratch at every iteration, we learn an offline routing policy that estimates costs almost instantly for any committed subset. An iterated local search establishes the first-level partitions. We formulate the second level as a Markov decision process and solve it with a deep Q-network whose state is represented by a graph attention network aggregating customer and vehicle information by relevance to the acting vehicle. Trained offline on instances with variable customer cardinality and locations, the policy applies to any daily customer realization; online fine-tuning improves the cost approximation. Experiments show that our policy reduces routing costs by 19.6% relative to a state-of-the-art method and by at least 29.6% over classical heuristics. Our overall algorithm saves 13.7% on average over the version without the attention-based representation and generates high-quality decisions within minutes, whereas benchmarks without an offline-trained estimator require over an hour.

In recent years, Deep Reinforcement Learning (DRL) has achieved substantial progress on Vehicle Routing Problems (VRPs). However, existing DRL-based methods are typically trained on instances generated from a uniform distribution, which limits their performance under real-world distribution shifts. In this paper, we aim to develop a generalization-oriented model that partitions the policy network into multiple modules and adaptively recombines modules to form specific policies during inference. Specifically, we propose Residual Refined Experts with Instance-level Gating (R2E-IG) to improve cross-distribution generalization. Our contributions are threefold: (1) We introduce a Residual Refined Expert (R2E) architecture that enhance expert expressiveness via residual refinement; (2) We design an instance-level gating mechanism that learns distribution-aware instance representations and routes inputs to suitable modules; (3) We propose a mixed-distribution training mechanism equipped with Dynamic Weight Adaption (DWA), which dynamically reweights training data from different distributions to emphasize more informative ones. Extensive experiments show that R2E-IG achieves competitive performance against state-of-the-art baselines on both in-distribution and out-of-distribution instances across synthetic and benchmark datasets. Moreover, R2E-IG is generic and can be easily integrated into existing DRL-based methods to further improve performance.