The relief distribution problem with trucks and drones under incomplete demand information

Disaster relief operations often take place under uncertainty regarding the extent of damage across locations. In this paper, we study the delivery of relief aid in the aftermath of disasters when delivery vehicles are assisted by surveillance drones and the demand for relief supplies is initially unknown. We introduce a stylized problem that arises in many emergency supply delivery settings – the relief distribution problem (RDP). In RDP, emergency vehicles, referred to as trucks, must distribute relief supplies on a network, starting from the depot to potential delivery locations, whose demand is initially unknown. The trucks are assisted by surveillance drones, which cannot deliver relief supplies, but scout delivery locations to see whether relief supplies are needed or not. The objective is to visit all location by any vehicle, deliver supplies to all damaged ones, and minimizing the completion time of the relief operation. We study two natural policies for the online problem RDP which we evaluate in two ways: the competitive ratio quantifies the performance in comparison to an optimal solution obtained under full information on damages, the drone-impact is the ratio of the algorithm’s performance to the best outcome achievable without drones. Through theoretical analysis and computational experiments, we characterize the operational trade-offs between these policies and derive insights for the effective deployment of drones in disaster response.

💡 Research Summary

The paper introduces the Relief Distribution Problem (RDP), an online routing problem that models the delivery of emergency supplies after a disaster when the set of damaged locations is unknown at the outset. A fleet of homogeneous trucks (speed normalized to 1) and faster surveillance drones (speed α > 0) start from a depot and must visit a known set of potential demand sites C. Only after a vehicle reaches a site is its damage status revealed; damaged sites (set D) must be serviced by a truck, while drones can only scout. The objective is to minimize the makespan, i.e., the time when the last vehicle returns to the depot, while ensuring every site is visited by at least one vehicle and all damaged sites receive supplies.

The authors first define the full‑information counterpart RDP* (where D is known) and show it reduces to a min‑max multiple‑Traveling‑Salesman Problem (Multi‑TSP), establishing NP‑hardness. In the online setting, performance is measured by two ratios: (1) the competitive ratio σ, the worst‑case ratio of an algorithm’s makespan to the optimal offline makespan; (2) the drone‑impact ratios ¯ω (worst‑case) and ω (best‑case), which compare the algorithm’s makespan to the optimal makespan of a truck‑only Multi‑TSP (i.e., the benefit or penalty of having drones).

Two natural policies are proposed:

1. Optimistic – Drones are deployed aggressively to explore as many nodes as possible early on. Trucks follow the currently known damaged nodes and re‑plan when new damage information arrives. The authors prove that when drones are sufficiently fast (α ≥ 2) and the number of drones is at least the number of trucks (k_d ≥ kₜ), Optimistic achieves a competitive ratio of at most 1 + 1/α, which approaches 1 as α grows. However, because trucks may learn about damage late, the drone‑impact ratio can exceed 1, meaning that in the worst case the presence of drones actually hurts performance compared with a truck‑only solution.

2. Regretless – This policy synchronizes scouting and delivery. Whenever a drone discovers a damaged node, a truck is immediately re‑routed to service it, ensuring that the information gained by drones is never “wasted”. Regretless guarantees the optimal drone‑impact ratios (¯ω = ω = 1) for any parameter setting, i.e., it never performs worse than the best truck‑only solution. The trade‑off is a larger competitive ratio, especially when α is close to 1, because the policy refrains from exploiting the speed advantage of drones for pure exploration.

The theoretical analysis yields a detailed trade‑off surface as a function of α, kₜ, and k_d. For example, when k_d ≫ kₜ and α ≥ 2, Optimistic dominates in competitive ratio (σ ≤ 1.5) while Regretless excels in drone‑impact (¯ω = 1). Conversely, with few drones or modest speed advantage, Regretless is preferable.

To validate the analytical bounds, the authors conduct extensive computational experiments on synthetic mountain and coastal networks as well as on real‑world disaster data (e.g., the 2015 Nepal earthquake and a 2023 Philippines typhoon). They compare Optimistic, Regretless, and a baseline truck‑only Multi‑TSP. Results show that the empirical makespans closely follow the derived theoretical bounds. Notably, in some instances the introduction of drones inflates the makespan by up to 100 % due to poor coordination between exploration and delivery, confirming the “drone‑regret” phenomenon. When drones are fast (α ≈ 3) and sufficiently numerous, Optimistic achieves the lowest makespan, whereas Regretless consistently matches the truck‑only baseline in terms of drone‑impact, never exceeding it.

The paper’s contributions are threefold: (i) formalizing RDP as the first semi‑online, exploration‑driven routing problem with heterogeneous vehicles; (ii) designing and rigorously analyzing two policies that respectively optimize competitive ratio and drone‑impact; (iii) providing empirical evidence that the theoretical trade‑offs manifest in realistic disaster logistics scenarios. The findings caution practitioners that deploying drones is not a guaranteed improvement; careful policy selection based on drone speed, fleet composition, and expected damage prevalence is essential. Future work is suggested on integrating predictive damage assessments (e.g., satellite or AI‑based image analysis), extending to multiple depots, and exploring learning‑augmented or advice‑based online algorithms.