## Statistics Papers

#### Document Type

Technical Report

#### Date of this Version

12-2010

#### Publication Source

Journal of Mathematical Modelling and Algorithms

#### Volume

9

#### Issue

4

#### Start Page

375

#### Last Page

392

#### DOI

10.1007/s10852-010-9142-0

#### 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.

#### Copyright/Permission Statement

This is a pre-print of an article published in the Journal of Mathematical Modelling and Algorithms. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10852-010-9142-0

#### Keywords

computational geometry, optimization, stationary and mobile facilities, Voronoi diagrams

#### Recommended Citation

Bhattacharya, B. B.
(2010).
Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location.
*Journal of Mathematical Modelling and Algorithms,*
*9*
(4),
375-392.
http://dx.doi.org/10.1007/s10852-010-9142-0

#### Included in

Applied Statistics Commons, Business Administration, Management, and Operations Commons, Business Analytics Commons, Management Sciences and Quantitative Methods Commons, Mathematics Commons

**Date Posted:** 25 October 2018

This document has been peer reviewed.