In Euclidean plane geometry, Apollonius' problem is to construct a circle in a plane that is tangent to three given circles. We will use a solution to this ancient problem to solve several versions of the following geometric optimization problem. Given is a set of customers located in the plane, each having a demand for a product and a budget. A customer is satisfied if her total, travel and purchase, costs do not exceed the budget. The task is to determine location of production facilities in the plane and one price for the product such that the revenue generated from the satisfied customers is maximized.Artem Panin: This work was partially supported by RFBR Grants 15-37-51018 mol_nr and 16-07-00319.
The single facility location problem with demand regions seeks for a facility location minimizing the sum of the distances from n demand regions to the facility. The demand regions represent sales markets where the transportation costs are negligible. In this paper, we assume that all demand regions are disks of the same radius, and the distances are measured by a rectilinear norm, e.g. 1 or ∞ . We develop an exact combinatorial algorithm running in time O(n log c n) for some c dependent only on the space dimension. The algorithm is generalizable to the other polyhedral norms.
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