Optimization of Multi-Agent Flying Sidekick Traveling Salesman Problem over Road Networks

TL;DR

The paper addresses last-mile delivery with multiple trucks and drones operating on real road networks by formulating MA-FSTSP as a MILP and proposing a three-phase, set-based heuristic. Phase 1 partitions customers into truck groups using a set-based distance, Phase 2 solves a Set-TSP to determine visiting order of sets, and Phase 3 decodes the final synchronized truck/drone routes with dynamic programming. Key contributions include a scalable three-phase framework, acceleration strategies for Set-TSP, and extensive experiments on Manhattan and Boston networks showing significant improvements in solution quality and computation time, scaling to hundreds of customers. The work offers practical impact for large-scale drone-enabled logistics by enabling efficient multi-agent coordination on realistic road graphs.

Abstract

The mixed truck-drone delivery systems have attracted increasing attention for last-mile logistics, but real-world complexities demand a shift from single-agent, fully connected graph models to multi-agent systems operating on actual road networks. We introduce the multi-agent flying sidekick traveling salesman problem (MA-FSTSP) on road networks, extending the single truck-drone model to multiple trucks, each carrying multiple drones while considering full road networks for truck restrictions and flexible drone routes. We propose a mixed-integer linear programming model and an efficient three-phase heuristic algorithm for this NP-hard problem. Our approach decomposes MA-FSTSP into manageable subproblems of one truck with multiple drones. Then, it computes the routes for trucks without drones in subproblems, which are used in the final phase as heuristics to help optimize drone and truck routes simultaneously. Extensive numerical experiments on Manhattan and Boston road networks demonstrate our algorithm's superior effectiveness and efficiency, significantly outperforming both column generation and variable neighborhood search baselines in solution quality and computation time. Notably, our approach scales to more than 300 customers within a 5-minute time limit, showcasing its potential for large-scale, real-world logistics applications.

Figures (6)

• Figure 1: An Example of the output of each phase in the algorithm. In phase 1, we assign customers, marked by hollow circles, to their closest depots, marked by solid circles, as shown in Fig. (a). In phase 2, we collect nodes within half of the distance limit for drones as the circles shown in Fig. (b) and then apply the Set TSP method to find the shortest tour to visit all sets for each truck group starting from its depot as the colored lines shown in Fig. (b). In phase 3, based on the visiting order on routes found in phase 2, we optimize the route for truck and drone together. The final results are shown in Fig. (c), where the red and blue lines are routes for trucks of different groups, and the green lines are routes for drones when they are separated from the truck to visit customers.
• Figure 2: An example of the truck and drone routes on a directed graph with node 1$\sim$6. The solid arrows (in all colors) represent the road that the truck must follow. The truck route (A, F, D, E, A) is colored red and the drone route (A, F, C, D, E, B, A), represented as $\left<(\text{A, F, D, E, A}), \{(2,\text{C},3), (5,\text{B},6)\}\right>$, where 2, 3, 4, 5 are indices of vertices F, D, E, A in the truck route, is colored green. The solid green arrows mean the drone is carried by a truck, and the dashed green arrows mean the drone is flying freely.
• Figure 3: An example of Set TSP versus TSP. Each set represents a set of vertices in the graph, which can be arbitrarily connected inside. Sets are fully connected via vertices at the boundary. The red arrows represent the optimal TSP tour to visit the center vertices of sets in the graph, and the green arrows represent the optimal Set TSP tour to visit every set. Since Set TSP does not require visiting any exact vertex, it can pass through the set via shortcuts represented in dashed green. As a result, the visiting order of sets on the TSP tour and Set TSP tour can be different.
• Figure 4: Demonstration of acceleration strategies for the Set-TSP problem. (a) When two customers (the circle centers marked as larger vertices) are close, their neighbor sets $\mathcal{S}_c(r/2)$ overlap. Vertices appearing in both sets are highlighted in purple. (b) Our first strategy reduces redundancy by assigning vertices in the overlapped area to their nearest customer. (c) Our second strategy pre-computes optimal routes for all entering-leaving pairs on the boundary of each set, then removes all inner nodes from the neighbor sets. This significantly reduces the problem's complexity while maintaining solution quality.
• Figure 5: Box plots of scalability experiments.

Theorems & Definitions (12)

• Definition 1: Truck Route
• Definition 2: Drone Delivery Tuple
• Definition 3: Drone Route
• Definition 4: Truck Group Route
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof