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 can be visited, and the agents must avoid static obstacles and satisfy speed and capacity constraints. Previously studied VRPs are often addressed using a framework called branch-and-price. This framework requires computing the cost for a single agent to visit a given sequence of target-time window pairings, for several candidate sequences. These sequences are called tours. Computing tour costs is more expensive in the MT-VRP-O than in previously studied VRPs due to the presence of both moving targets and static obstacles. Thus, we introduce a new exact algorithm, Lazy Branch-and-Price with Relaxed Continuity (Lazy BPRC), for the MT-VRP-O. The key idea in Lazy BPRC is to use cheap-to-compute lower bounds on tour costs where costs are traditionally used, and lazily update lower bounds to true costs. When computing a tour's true cost is needed, we do so by searching for a shortest path on a graph of convex sets, and we introduce a new heuristic to accelerate the search. We demonstrate that Lazy BPRC runs up to an order of magnitude faster than two ablations.
The Moving-Target Traveling Salesman Problem (MT-TSP) seeks a minimum cost trajectory for an agent that departs from a static depot, visits a set of moving targets, each within one of their assigned time windows, and returns to the depot. In this article, we study the Moving-Target Traveling Salesman Problem with Moving Obstacles (MT-TSP-MO), a generalization of the MT-TSP where the agent trajectory must avoid moving obstacles. We present a Mixed-Integer Conic Programming (MICP) formulation that can be solved using off-the-shelf solvers, as well as a fast and scalable Two-Phase Bilevel Search (TPBS) algorithm that computes high-quality feasible solutions for the problem. We evaluate our approaches against an existing baseline algorithm on a broad range of problem instances with up to 40 targets and 40 obstacles. The results demonstrate that both the proposed methods significantly outperform the baseline with respect to success rates, solution costs, and computation time.
Allen George Philip, Anoop Bhat, Sivakumar Rathinam +1
The Vehicle Routing Problem (VRP) and its variants represent some of the most practically consequential optimization challenges in modern logistics and urban mobility. In this study, we address a dynamic, online variant combining elements of the VRP and the Orienteering Problem (OP), in which a fleet of vehicles must maximise cumulative reward collected within a fixed time horizon while continuously replanning as new tasks arrive. We propose and evaluate a reward-density heuristic for dynamic multi-vehicle assignment, referred to as the Efficiency heuristic. We evaluate this formulation across two application domains: autonomous drone task allocation and urban taxi dispatch, across multiple fleet sizes and task scales. The proposed method is compared with four classical construction heuristics and three metaheuristic algorithms (Adaptive Large Neighbourhood Search, Genetic Algorithm, and Simulated Annealing), all evaluated under identical conditions. Across all tested configurations, the Efficiency heuristic matches the solution quality of the best metaheuristic algorithms while requiring two to three orders of magnitude less planning time, establishing Pareto dominance over all competing methods on the reward-versus-compute frontier. These findings suggest a practical design principle for real-time allocation and dispatch systems: in dynamic, time-constrained routing environments, carefully designed greedy heuristics can match the output of sophisticated search procedures at a fraction of the computational cost, making them preferable for online deployment.
This paper introduces the Dynamical Vehicle Orienteering Problem (DVOP), a generalization of the Orienteering Problem (OP). The OP maximizes the reward collected from spatial targets under a limited travel budget; the DVOP extends it by accounting for both external and vehicle-actuated forces. We study the DVOP in the context of multi-rotor Unmanned Aerial Vehicle (UAV) flight planning, using a three-dimensional Point-Mass Model (PMM) constrained by maximum velocity and acceleration magnitudes and subject to gravitational acceleration, with the travel budget expressed as a maximum flight time. Because the DVOP couples reward maximization with time-optimal trajectory planning, it cannot be formulated as a simple graph problem and solved exactly without relaxing or under-actuating the vehicle dynamics. We therefore propose two solution approaches: a Branch-and-Bound (BnB) procedure that combines Non-Linear Programming (NLP) and Mixed-Integer Linear Programming (MILP) to provide high-quality solutions, and a Large Neighborhood Search (LNS) metaheuristic that supplies an initial reward bound and scales to instances intractable for the BnB. The BnB relies on a novel MILP formulation of travel costs based on minimum-time trajectory primitives through target triplets, yielding a tight reward upper bound, while the LNS uses limited thrust decomposition to compute fast, high-quality PMM trajectories. Experiments on benchmark instances show improvements of up to 37 % over state-of-the-art solutions for the Kinematic Orienteering Problem, and a real-world deployment on a multi-rotor UAV verifies the proposed PMM solution trajectories.