Low complexity algorithms for optimal consumer push-pull partial covering in the plane

Abstract

This paper considers a model for locating a consumer within a bounded region in the plane with respect to a set of n existing pull-push suppliers. The objective is to maximize the difference of total profits and costs incurred due to the partial covering of the consumer by the suppliers pull and push influence areas. We develop efficient polynomial time algorithms for the resulting problems in the rectilinear and the Euclidean planes where the bounded region is either a rectangle or a constant size polygon, respectively. Based on these solutions, we develop algorithms for evaluating efficiently the objective function at any possible location of the consumer inside the bounded region. We also employ the algorithms for the Euclidean optimization problem and the rectilinear query computation to solve efficiently their corresponding dynamic versions, where an appearance of a new supplier or an absence of an existing one occurs. Being easy to implement due to the extensive use of simple data structures, such as the balanced and binary segment tree, and the employment of standard mechanisms, such as the sweep line, the Voronoi diagram and the circular ray shooting, our solutions potentially have wide usability.