Heterogeneous locational optimisation using a generalised Voronoi partition

In this paper a generalisation of the Voronoi partition is used for locational optimisation of facilities having different service capabilities and limited range or reach. The facilities can be stationary, such as base stations in a cellular network, hospitals, schools, etc., or mobile units, such as multiple unmanned aerial vehicles, automated guided vehicles, etc., carrying sensors, or mobile units carrying relief personnel and materials. An objective function for optimal deployment of the facilities is formulated, and its critical points are determined. The locally optimal deployment is shown to be a generalised centroidal Voronoi configuration in which the facilities are located at the centroids of the corresponding generalised Voronoi cells. The problem is formulated for more general mobile facilities, and formal results on the stability, convergence and spatial distribution of the proposed control laws responsible for the motion of the agents carrying facilities, under some constraints on the agents' speed and limit on the sensor range, are provided. The theoretical results are supported with illustrative simulation results.