Optimal Delivery with a Faulty Drone

TL;DR

This paper addresses online delivery with a single faulty drone in the plane, where a finisher starting at $P=(x,y)$ must locate a starter that fails at some unknown point on $\overline{ST}$ and complete the delivery to $T=(1,0)$. It introduces three explicit online strategies with distinct trajectories: $\mathcal{A}_0: P\to S\to T$, $\mathcal{A}_1: P\to T\to S\to T$, and $\mathcal{A}_d: P\to M\to S\to T$ where $M=(d,0)$ and $d=(x^2+y^2)/(2x)$, and derives their competitive ratios as functions of the finisher’s starting position. A hybrid algorithm that selects the best among these three is proven optimal, partitioning the plane into regions where each is preferred; the worst-case competitive ratio is bounded by $3$, with a numerical maximum of about $1.74197$ at $(x,y)\approx(0.275257,0.689019)$. These results advance robust drone delivery by providing geometry-driven online strategies that closely approximate the offline optimum, and they open avenues for extensions to multiple agents, uneven speeds, and more general movement models.

Abstract

We introduce and study a new cooperative delivery problem inspired by drone-assisted package delivery. We consider a scenario where a drone, en route to deliver a package to a destination (a point on the plane), unexpectedly loses communication with its central command station. The command station cannot know whether the drone's system has wholly malfunctioned or merely experienced a communications failure. Consequently, a second, helper drone must be deployed to retrieve the package to ensure successful delivery. The central question of this study is to find the optimal trajectory for this second drone. We demonstrate that the optimal solution relies heavily on the relative spatial positioning of the command station, the destination point, and the last known location of the disconnected drone.

Figures (4)

• Figure 1: The optimal algorithm depends on the starting position of the finisher. The striped regions depict the finisher starting positions for which each of the three candidate algorithms is optimal.
• Figure 2: The worst-case fail time for the starter is $0$except in the gray shaded region, where the worst-case fail time is $t_-$ (from the proof of Theorem \ref{['thm:CR:A1']}). Recall that we only consider $\mathcal{A}_1$ when the finisher starts outside of the Disk $D(1,1)$.
• Figure 3: The worst-case fail time for the starter is $0$except in the gray shaded region, where the worst-case fail time is $t^\prime$ (from the proof of Theorem \ref{['thm:CR:Ad']}). Recall that we only consider $\mathcal{A}_d$ when the finisher starts inside the Disk $\overline{D}(1,1)$.
• Figure 4: Region plots with highlighted areas

Theorems & Definitions (8)

• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 7
• Theorem 8