Undesirable facility location with minimal covering objectives

An undesirable facility is to be located within some feasible region of any shape in the plane or on a planar network. Population is supposed to be concentrated at a finite number n of points. Two criteria are taken into account: a radius of influence to be maximised, indicating within which distance from the facility population disturbance is taken into consideration, and the total covered population, i.e. lying within the influence radius from the facility, which should be minimised. Low complexity polynomial algorithms are derived to determine all nondominated solutions, of which there are only O(n³) for a fixed feasible region or O(n²) when locating on a planar network. Once obtained, this information allows almost instant answers and a trade-off sensitivity analysis to questions such as minimising the population within a given radius (minimal covering problem) or finding the largest circle not covering more than a given total population.