Task Allocation for Multi-agent Systems via Unequal-dimensional Optimal Transport

TL;DR

This paper addresses task allocation in large-scale multi-agent systems where standby agents in $\mathbb{R}^n$ must be matched to origin–destination task pairs in $\mathbb{R}^{2n}$. It develops an unequal-dimensional optimal transport formulation with a quadratic travel cost, establishes existence, uniqueness via the $x$-twist condition, and smoothness through an index-form representation, and shows the problem can be solved as a discretized linear program. The framework is illustrated with numerical examples, including a Stockholm city scenario with 30 OD requests, validating cost-efficient, smooth allocations. The work provides a scalable, probabilistic approach to drone logistics and multi-agent task assignment, with potential extensions to unbalanced supply-demand, shared rides, and real-time data integration.

Abstract

We consider a probabilistic model for large-scale task allocation problems for multi-agent systems, aiming to determine an optimal deployment strategy that minimizes the overall transport cost. Specifically, we assign transportation agents to delivery tasks with given pick-up and drop-off locations, pairing the spatial distribution of transport resources with the joint distribution of task origins and destinations. This aligns with the optimal mass transport framework where the problem and is in the unequal-dimensional setting. The task allocation problem can be thus seen as a linear programming problem that minimizes a quadratic transport cost functional, optimizing the energy of all transport units. The problem is motivated by time-sensitive medical deliveries using drones, such as emergency equipment and blood transport. In this paper, we establish the existence, uniqueness, and smoothness of the optimal solution, and illustrate its properties through numerical simulations.

Figures (4)

• Figure 1: Task allocation process: a standby agents begins at ${\mathbf y}$, where the agent picks up shipment at ${\mathbf x}^O$, delivers them to destination ${\mathbf x}^D$, and then returns to ${\mathbf y}$.
• Figure 2: Two- to one-dimensional matching with fixed drone spatial distribution $\nu({\mathbf y})$ and the OD distribution $\mu({\mathbf x}^1,{\mathbf x}^2)$ given. The size of the crosses corresponds to the mass at each point of $\pi(\mathbf{x}^1, \mathbf{x}^2, \mathbf{y})$.
• Figure 3: The drones positioned at three distinct locations are matched to three tasks with their origin-destination (OD) given.
• Figure 4: Thirty allocation tasks, with origins in blue and destinations in red, are randomly generated within specific locations in Stockholm, Sweden. Drones, represented by purple triangles, are distributed throughout the city and matched to fulfill these requests.

Theorems & Definitions (16)

• Definition 1: Pure matching
• Proposition 1: Existence
• proof
• Definition 2: x-Twist condition
• Proposition 2: Uniqueness
• proof
• Definition 3: Non-degeneracy condition chiappori2020multidimensional
• Proposition 3
• proof
• Remark 1: Smoothness