📝 SummarXiv

Abstract

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 \textbf{5.203}$\pm$0.093, versus \textbf{5.349}$\pm$0.038 for ALNS and \textbf{5.208}$\pm$0.124 for NN -- i.e., \textbf{2.73\%} better than ALNS on average and within \textbf{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.