Optimal Solutions for the Moving Target Vehicle Routing Problem with Obstacles via Lazy Branch and Price

TL;DR

This paper introduces Lazy BPRC, an optimization method for the Moving Target Vehicle Routing Problem with Obstacles, which efficiently finds optimal agent trajectories by lazily evaluating motion planning costs.

Contribution

The paper presents Lazy BPRC, a novel branch-and-price framework that reduces computational effort by delaying exact cost calculations using relaxed motion planning bounds.

Findings

Lazy BPRC is up to ten times faster than alternative methods.

The approach effectively handles collision-free motion planning for moving targets.

Lazy evaluation significantly reduces computational complexity in VRP with obstacles.

Abstract

The Moving Target Vehicle Routing Problem with Obstacles (MT-VRP-O) seeks trajectories for several agents that collectively intercept a set of moving targets. Each target has one or more time windows where it must be visited, and the agents must avoid static obstacles and satisfy speed and capacity constraints. We introduce Lazy Branch-and-Price with Relaxed Continuity (Lazy BPRC), which finds optimal solutions for the MT-VRP-O. Lazy BPRC applies the branch-and-price framework for VRPs, which alternates between a restricted master problem (RMP) and a pricing problem. The RMP aims to select a sequence of target-time window pairings (called a tour) for each agent to follow, from a limited subset of tours. The pricing problem adds tours to the limited subset. Conventionally, solving the RMP requires computing the cost for an agent to follow each tour in the limited subset. Computing these costs in the MT-VRP-O is computationally intensive, since it requires collision-free motion planning between moving targets. Lazy BPRC defers cost computations by solving the RMP using lower bounds on the costs of each tour, computed via motion planning with relaxed continuity constraints. We lazily evaluate the true costs of tours as-needed. We compute a tour's cost by searching for a shortest path on a Graph of Convex Sets (GCS), and we accelerate this search using our continuity relaxation method. We demonstrate that Lazy BPRC runs up to an order of magnitude faster than two ablations.