Dynamic Order Fulfillment in Last-Mile Drone Delivery under Demand Uncertainty
Linxuan Shi, Zhengtian Xu, Miguel Lejeune, Qi Luo
TL;DR
This work studies dynamic order fulfillment in on-demand last-mile drone delivery under demand uncertainty, modeling decisions with a rolling-horizon two-stage stochastic program that links current-cycle actions to future recourse costs via the second-stage value function $\mathbb{Q}(\bm{\mathcal{X}}^0, \xi)$. An accelerated, slim Integer L-shaped method is developed, combining a reduced-cut master problem with heuristic recourse solved by Google OR-Tools to generate upper-bound cuts efficiently. Monte Carlo scenario sampling $\Omega$ enables a sample-average approximation that yields substantial computational gains (over 20x speedups) while maintaining average optimality gaps below 1% in many instances. Operational analyses show the two-stage model can achieve up to about 20% long-run cost savings when demand uncertainty is moderate, with savings diminishing as arrival uncertainty grows, illustrating the value of batching and cycle-level coordination for drone-enabled last-mile logistics.
Abstract
Drones have attracted growing interest in last-mile delivery due to their potential to significantly reduce costs and enhance operational flexibility, particularly in areas of sparse and uncertain demand where traditional truck delivery proves inefficient. This paper addresses the dynamic order fulfillment problem faced by a retailer operating a fleet of drones to service delivery requests that arrive stochastically. These delivery requests may vary in package profiles, delivery locations, and urgency. We adopt a rolling-horizon framework for order fulfillment and devise a two-stage stochastic program aimed at strategically managing existing orders while considering incoming requests that are subject to various uncertainties. A significant challenge in deploying the envisioned two-stage model lies in its incorporation of vehicle routing constraints, on which exact or brute-force methods are computationally inefficient and unsuitable for real-time operational decisions. To address this, we propose an accelerated L-shaped algorithm that (i) reduces the branching tree size, (ii) replaces exact second-stage solutions with heuristic estimates, and (iii) adapts an alternating strategy for adding optimality cuts. The proposed heuristic demonstrates remarkable performance superiority over the exact method, achieving a 20-fold reduction in average runtime while maintaining an average optimality gap of less than 1\%. We apply the algorithm to a wide range of instances to evaluate the benefits of postponing orders for batch service using the stochastic model. Our results show potential long-term cost savings of up to 20\% when demand uncertainty is explicitly considered in order fulfillment decisions. Meanwhile, the derived savings tend to diminish as the uncertainty increases in order arrivals.