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

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.

Figures (3)

• Figure 1: Targets move through obstacle environment and must be intercepted within time windows, shown in bold-colored lines. Agents begin and end at depot, intercepting targets while avoiding obstacles.
• Figure 2: Computing bounds on the cost of an example tour $\Gamma$. To compute the lower bound $\underline{c}(\Gamma)$, we divide each target-window visited by $\Gamma$ into segments. We then construct a segment-graph$\mathcal{G}_\text{seg}$, where the nodes are the segments, and an edge connects every segment of $\Gamma[n]$ to every segment of $\Gamma[n + 1]$. The edge cost from segment $\xi$ to $\xi'$ is the distance traveled along the shortest path in space from $\xi$ to $\xi'$, if this path satisfies the relaxed timing constraints in Section \ref{['sec:lower_bound_on_tour_cost']}, and $\infty$ otherwise. $\underline{c}(\Gamma)$ is the cost of the shortest path in $\mathcal{G}_\text{seg}$ from $\xi_0$ to $\xi_0$ visiting all target-windows in $\Gamma$. To compute the upper bound $\overline{c}$, we construct a segment-start-graph$\mathcal{G}_\text{start}$, where the nodes are the starting points of the segments, and an edge connects every segment-start of $\Gamma[n]$ to every segment-start of $\Gamma[n + 1]$. The edge cost from $s$ to $s'$ is distance traveled by a feasible minimum-distance agent trajectory from $s$ to $s'$, if such a trajectory exists or if $s' = s_0$, and $\infty$ otherwise. Our upper bound is the cost of the shortest path in $\mathcal{G}_\text{start}$ from $s_0$ to $s_0$ that visits all target-windows in $\Gamma$.
• Figure 3: (a) Varying the number of targets. Lazy BPRC shows smaller median runtime than the ablations, particularly for larger numbers of targets. (b) Varying the map resolution. Lazy BPRC's advantage in median runtime grows as we increase the map resolution. (c) Varying the capacity. Lazy BPRC has smaller median runtime than the ablations for all tested capacities.