Minimizing Total Travel Time for Collaborative Package Delivery with Heterogeneous Drones

TL;DR

This work presents a constant-factor approximation algorithm for the problem of computing the best non-preemptive schedule and shows that the best non-preemptive schedule is within a factor of three of the best preemptive schedule.

Abstract

Given a fleet of drones with different speeds and a set of package delivery requests, the collaborative delivery problem asks for a schedule for the drones to collaboratively carry out all package deliveries, with the objective of minimizing the total travel time of all drones. We show that the best non-preemptive schedule (where a package that is picked up at its source is immediately delivered to its destination by one drone) is within a factor of three of the best preemptive schedule (where several drones can participate in the delivery of a single package). Then, we present a constant-factor approximation algorithm for the problem of computing the best non-preemptive schedule. The algorithm reduces the problem to a tree combination problem and uses a primal-dual approach to solve the latter. We have implemented a version of the algorithm optimized for practical efficiency and report the results of experiments on large-scale instances with synthetic and real-world data, demonstrating that our algorithm is scalable and delivers schedules of excellent quality.

Figures (17)

• Figure 1: Consider the preemptive schedule with three drones in Figure (a), represented by solid, dotted, and dashed lines, corresponding to three different packages, respectively. Drones $2$ and $3$ are involved in parts of the dotted package schedule, while drones $1$ and $2$ are involved in parts of both the dashed and the solid package schedules. Note that points $b/g$, $c/f$, and $i/l$ are co-located. In the constructed non-preemptive schedule in Figure (b), additional travel distances are incurred for each package due to a drone deviating from its original route—either to pick up a package or to return to a point where it can resume its original schedule after a delivery. These additional distances are represented by thicker lines matching the style of the corresponding package in Figure (b). For the solid package, the fastest drone, drone $1$, first serves the sub-path $a \rightarrow b$, then the sub-path $g \rightarrow h$ to complete the delivery, and finally returns to point $b$ to resume the original schedule. For the dashed package, drone $1$ travels from point $c$ (which shares the location with $f$) to the pickup point $e$, then serves the sub-path $e \rightarrow f$ followed by the sub-path $c \rightarrow d$ to complete the delivery.
• Figure 2: An instance of the tree combination problem with nine vertices, each associated with one of three different speeds $p_1 > p_2 > p_3$. An illustration of a feasible solution shows four resulting trees.
• Figure 3: Illustration of the primal-dual algorithm
• Figure 4: The cases for determining candidate sources: (1)–(3) Step 3.1: The target $t$ lies inside the circumcircle of any Delaunay triangle. (4) Step 3.2: The target $t$ lies outside all circumcircles. (5) Step 3.3 : The corresponding source $s$ of $t$ is contained in the candidate source set.
• Figure 5: Results on the worst-case instance for the baseline algorithm with varying $n$, where $\alpha = v_{\max}/v_{\min}$ and $\epsilon$ is a small constant parameter (see Section \ref{['sec:InstanceAndBaseline']}).

Theorems & Definitions (27)

• Definition 2.1: Min-Sum Collaborative Delivery Problem (Min-Sum CD Problem)
• Definition 2.2: Preemptive vs Non-preemptive
• Theorem 3.1
• proof
• Corollary 3.2
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Theorem 4.2: Min-Sum CD with Identical Depot Locations