Optimal Solutions for the Moving Target Vehicle Routing Problem via Branch-and-Price with Relaxed Continuity

TL;DR

This work introduces an exact algorithm, Branch-and-Price with Relaxed Continuity (BPRC), for the MT-VRP, and introduces a new labeling algorithm that solves this subproblem by means of a novel dominance criterion tailored for problems with moving targets.

Abstract

The Moving Target Vehicle Routing Problem (MT-VRP) seeks trajectories for several agents that intercept a set of moving targets, subject to speed, time window, and capacity constraints. We introduce an exact algorithm, Branch-and-Price with Relaxed Continuity (BPRC), for the MT-VRP. The main challenge in a branch-and-price approach for the MT-VRP is the pricing subproblem, which is complicated by moving targets and time-dependent travel costs between targets. Our key contribution is a new labeling algorithm that solves this subproblem by means of a novel dominance criterion tailored for problems with moving targets. Numerical results on instances with up to 25 targets show that our algorithm finds optimal solutions more than an order of magnitude faster than a baseline based on previous work, showing particular strength in scenarios with limited agent capacities.

Figures (4)

• Figure 1: Targets move along piecewise-linear trajectories with time windows shown in bold lines. Two agents begin at the depot and collectively intercept all targets. Trajectories of agents are shown in blue.
• Figure 2: Example where existing VRP dominance checks fail for the MT-VRP. Since each target has one time window in this example, we represent a target-window simply using the index of the target. We treat the depot as a fictitious target 0. (a) ${\tau_\textnormal{a}}$ is an optimal (i.e. minimum-distance) trajectory visiting the sequence of targets $\Gamma = (0, 3, 1, 2)$, and ${\tau_\textnormal{a}}'$ is an optimal trajectory visiting the sequence $\Gamma' = (0, 1, 3, 2)$. Conventional VRP dominance checks would conclude that $\Gamma$ dominates $\Gamma'$, since ${\tau_\textnormal{a}}$ and ${\tau_\textnormal{a}}'$ have the same cost. (b) $\bar{\tau}_\text{a}$ optimally visits the sequence $\bar{\Gamma} = (0, 3, 1, 2, 4, 0)$ obtained by appending 4 and 0 to $\Gamma$, and $\bar{\tau}_\text{a}'$ optimally visits the sequence $\bar{\Gamma}' = (0, 1, 3, 2, 4, 0)$ obtained by appending 4 and 0 to $\Gamma'$. $\bar{\tau}_\text{a}$ travels more distance than $\bar{\tau}_\text{a}'$, so $\bar{\Gamma}$ is worse than $\bar{\Gamma}'$. Thus $\Gamma$ does not dominate $\Gamma'$.
• Figure 3: (a) and (b) show two partial tours $\Gamma$ and $\Gamma'$ ending with $\gamma_{1,2}$, where we want to check if $\Gamma$ dominates $\Gamma'$. (c) We check dominance by comparing an upper-bounding trajectory $\tau_\text{a,ub}$ for $\Gamma$ and a lower-bounding trajectory $\tau_\text{a,lb}'$ for $\Gamma'$, as described in Section \ref{['sec:solving_pricing_problem']}. (d) We strengthen our dominance check by dividing $\gamma_{1,2}$ into segments and applying the check from (c) for each segment; an example is shown for the 5th segment of $\gamma_{1,2}$, denoted as $\xi_{1,2,5}$.
• Figure 4: (a) Varying the number of targets. The gap in computation time between BPRC and the baseline and ablations widens as we increase the number of targets up to 20. The gap shrinks at 25 targets because BPRC begins reaching the time limit in many instances. We do not show a computation time for BPRC-ablate-dominance for 25 targets because it reached the memory limit in every instance. (b) Varying the capacity. BPRC shows advantage for small capacities. (c) %Gap (as defined in Experiment 2) of each method's relaxation solution. BPRC's relaxation has a smaller %Gap, and is thus tighter, in every instance. (d) Varying the number of agents. BPRC shows advantage when the number of agents is 3 or larger. (e) Varying the time window lengths. BPRC's performance advantage widens as we increase time window length up to 50, then shrinks because BPRC reaches the time limit in many instances. (f) Varying the number of segments in BPRC. Horizontal and vertical axes are log-scaled. MICP's min, median, and max are shown as flat lines without scatter points because the quantity being varied is specific to BPRC. We only show MICP's computation time to indicate that BPRC has smaller median computation time for a wide range of parameter values.

Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof