Finding the circle or rectangle containing the minimum weight of points

This paper considers a problem applicable to both obnoxious facility location, and to problems in production where one has to select an area of material with the least ‘weight’ of defects. In the first version of the problem, a boundary circle containing weighted points is given. It is necessary to find the location of an interior covering circle of a given radius that encloses the smallest weight of points. In the second version, it is necessary to find a rectangle inside a rectangle. The paper provides algorithms for solving both problems. Two objectives are considered, minimizing the sum of weights, or minimizing the maximal weight.