## Abstract

In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let *ψ* be a circular path through which mobile service stations are plying, and *S* be a set of *n* stationary facilities (points) inside *ψ*. A demand point *p* is served from a mobile facility plying along *ψ* if the distance of *p* from the boundary of *ψ* is less than that from any member in *S*. On the other hand, the demand point *p* is served from a stationary facility *p*
_{
i
} ∈ *S* if the distance of *p* from *p*
_{
i
} is less than or equal to the distance of *p* from all other members in *S* and also from the boundary of *ψ*. The objective is to place the stationary facilities in *S*, inside *ψ*, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in *S* are placed equidistantly from the center *o* of *ψ*. It is shown that the maximum area is obtained when the members in *S* lie on the vertices of a regular *n*-gon, with its circumcenter at *o*. The distance of the members in *S* from *o* and the optimum area increases with *n*, and at the limit approaches the radius and the area of the circle *ψ*, respectively. We also consider another variation of this problem where a set of *n* points is placed inside *ψ*, and the task is to locate a new point *q* inside *ψ* such that the area of the Voronoi region of *q* is maximized. We give an exact solution of this problem when *n* = 1 and a (1 − *ε*)-approximation algorithm for the general case.

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Bhattacharya, B.B. Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location.
*J Math Model Algor* **9**, 375–392 (2010). https://doi.org/10.1007/s10852-010-9142-0

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DOI: https://doi.org/10.1007/s10852-010-9142-0