Minimum‐risk routing through a mapped minefield

Abstract

We embed a directed graph G(V, E) in a representation of a naval minefield; vertices V represent waypoints and edges E denote possible segments for ship transit. A new model identifies a simple s‐t path through the minefield that minimizes the risk of incurring unacceptable damage from threats, that is, mine detonations. Traditional “edge‐additive” models rely on shortest‐path algorithms that over‐accumulate risk along a path. Our “threat‐additive” approach accumulates risk based upon the path's closest point of approach to each mine. We formulate and solve this model (1) using an integer program (IP) and its stronger variant, and (2) via an A* search algorithm. Preprocessing routines are key to reducing run times. We investigate the relative merits, both with respect to solution quality and requisite computational effort, of two types of graphs, one based on a rectilinear scheme and one based on Voronoi diagrams. We find that graphs based on Voronoi diagrams provide higher quality solutions with less computational effort, and that the A* search procedure requires less computational effort than solving instances of our models as IPs.

Navigating free‐floating minefields via time‐varying Voronoi graphs

“…Li mitigates this concern by including and encouraging the use of longer edges, but admits that the possibility of double‐counting remains. In order to eliminate any possibility of double‐counting, we employ a “threat‐additive” model introduced by Richards et al . This formulation precludes the use of Dijkstra's algorithm, but employs an A * search and several heuristics which maintain the tractability of the problem without impact to solution quality, even for instances with extremely dense minefields.…”

“…Furthermore, Babel and Zimmermann argue that paths found in such networks may not be navigable by a ship, and focuses on developing a network with edges that better reflect the maneuvers available to seagoing vessels. Li and Richards eliminate difficult, acute turns by restricting retrograde movement and introducing distance penalties that avoid unnecessary maneuvers. For a typical minefield composed of a single mine type, Richards et al show that a Voronoi graph, generated around the locations of the mines M in a minefield, provides paths that maximize the minimum distance to each mine, limits both | V | and | E | to , and maximizes the minimum angle between edges.…”

“…We employ empirically‐gathered environmental data along with ocean‐drift models to predict the location of mines at each future time step (see Section 4.1). Second, we build upon the risk model presented in Richards et al for a static network representation of navigable space. This model (1) derived a measure of the risk incurred by each edge, (2) proposed a methodology to accurately accumulate the risk across a series of connected edges, and (3) efficiently identified minimum‐risk paths within the network.…”

“…We extend this model to account for mine movement by constructing time‐dependent “journeys” via a series of static Voronoi graphs that serve as a “sequence of footprints” to build a discrete TVG. Unlike Richards et al , journeys within such a TVG require temporal as well as physical description; we construct parametric equations that continuously define ship location, acceleration, and risk. Lastly, prior research identifies topological changes that can occur within a Voronoi‐based TVG, and defines limits for their frequency.…”

“…The inclusion of M β ensures that G τ sufficiently models the boundary of the AO, while M ensures that each edge e ∈ E τ maximizes the minimum distance to any mine m ∈ M at time τ (see Section 4.2 for specifics concerning the choice of δ and the generation of M β ). Estimating the boundary of the AO using M β is convenient as it allows G τ to be generated via a single function call and avoids much of the post‐processing required by Richards et al to calculate the intersection points of the the AO boundary and the Voronoi graph and the unique parameters for each newly truncated edge. A source vertex s is connected to each edge separating pairs of dummy mines associated with , while each edge separating pairs of dummy mines associated with is connected to a sink vertex t .…”