# A Simple and Reproducible Hybrid Solver for a Truck-Drone VRP with Recharge

arXiv: https://arxiv.org/abs/2509.18162 · HTML: https://mlsanity.com/paper/2509.18162/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 18 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Builds on (most cited first, 5 of 5)

- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [Self-critical Sequence Training for Image Captioning](https://mlsanity.com/paper/1612.00563/) (2016)
- [On the Min-cost Traveling Salesman Problem with Drone](https://mlsanity.com/paper/1512.01503/) (2015)

## Abstract (arXiv)

We study last-mile delivery with one truck and one drone under explicit battery management: the drone flies at twice the truck speed; each sortie must satisfy an endurance budget; after every delivery the drone recharges on the truck before the next launch. We introduce a hybrid reinforcement learning (RL) solver that couples an ALNS-based truck tour (with 2/3-opt and Or-opt) with a small pointer/attention policy that schedules drone sorties. The policy decodes launch-serve-rendezvous triplets with hard feasibility masks for endurance and post-delivery recharge; a fast, exact timeline simulator enforces launch/recovery handling and computes the true makespan used by masked greedy/beam decoding. On Euclidean instances with N=50, E=0.7, and R=0.1, the method achieves an average makespan of 5.203±0.093, versus 5.349±0.038 for ALNS and 5.208±0.124 for NN -- i.e., 2.73% better than ALNS on average and within 0.10% of NN. Per-seed, the RL scheduler never underperforms ALNS on the same instance and ties or beats NN on two of three seeds. A decomposition of the makespan shows the expected truck-wait trade-off across heuristics; the learned scheduler balances both to minimize the total completion time. We provide a config-first implementation with plotting and significance-test utilities to support replication.