Given n demand points in a geographic area, the elliptical cover problem is to determine the location of p depots (anywhere in the area) so as to minimize the maximum distance of an economical delivery trip in which a delivery vehicle starts from the nearest depot to a demand point, visits the demand point and then returns to the second nearest depot to that demand point. We show that this problem is NP-hard, and adapt Cooper’s alternating locate-allocate heuristic to find locally optimal solutions for both the point-coverage and area-coverage scenarios. Experiments show that most locally optimal solutions perform similarly well, suggesting their sufficiency for practical use. The one-dimensional variant of the problem, in which the service area is reduced to a line segment, permits recursive algorithms that are more efficient than mathematical optimization approaches in practical cases. The solution also provides the best-known lower bound for the original problem at a negligible computational cost.
Industrial Engineering | Operational Research | Systems Engineering
Y. Liu, "An elliptical cover problem in drone delivery network design and its solution algorithms," European Journal of Operational Research, vol. 304, no. 3, pp. 912-925, Feb. 2023, https://doi.org/10.1016/j.ejor.2022.04.034