Maximum Service Coverage in Business Site Selection Using Computer Geometry Software

Abstract

A planar maximum coverage location problem in a continuous formulation is considered. The demand zone and service areas are presented as geometric items of given shapes and sizes. Each service area is associated with a point (centroid), relative to which the corresponding geometric item forms. The task is to find the position of the centroids to provide an optimal service for the demand zone according to a given criterion. The mathematical model is constructed as a nonlinear optimization problem, in which the variables are the coordinates of the centroids, and the objective function is defined as the area of the demand zone covered by the services. For the formalization and calculation of the objective function, both analytical expressions and computer geometry software are used. The methodology we propose is applicable to the arbitrary shapes of both the demand zone and the service areas. Moreover, this technique does not depend on the complexity of the corresponding items, since it uses the Shapely library, which operates with the same Polygon class. An approach to solving the problem based on the consistent application of local and global optimization methods is proposed. An auxiliary problem is posed that allows one to significantly reduce the run time at the stage of local optimization. The implementation of the approach is illustrated by examples of the maximum coverage location problem when the demand zone is a polygon and the service areas have the shape of a circle and an ellipse. The innovation of this paper lies in the fact that the maximum service coverage problem in business site selection is studied in such a way that both the demand zone and the service areas have an arbitrary shape.