### Abstract

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.

Original language | English |
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Pages (from-to) | 1797–1805 |

Number of pages | 9 |

Journal | Optimization Letters |

Volume | 11 |

Issue number | 8 |

DOIs | |

Publication status | Published - Dec 2017 |

### JEL classifications

- c44 - "Operations Research; Statistical Decision Theory"

### Keywords

- Pricing problem
- facility location
- Apollonius’ problem
- exact algorithm
- Complexity

## Cite this

Berger, A., Grigoriev, A., Panin, A., & Winokurow, A. (2017). Location, pricing and the problem of Apollonius.

*Optimization Letters*,*11*(8), 1797–1805. https://doi.org/10.1007/s11590-017-1159-0