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.
Highlights
A leader in the game is a company producing a single product in large quantities in uncapacitated production facilities
Each customer j ∈ J is situated in the plane and her coordinates are given by a point x j ∈ Q2
Each customer j ∈ J announces to the company her budget b j ∈ Z+ indicating that the product will be purchased only if the sum of travel and purchase costs does not exceed the budget, i.e., d j × p + c j × ||x j − y|| ≤ b j, (1)
Summary
The company has to determine location of m facilities in a Euclidean plane and a selling price p per unit of the product, one for all facilities. Followers in this game are n customers of the company. The problem is to find a revenue maximizing strategy for the leader, i.e., to determine location of facilities and the price such that the total revenue generated from the winners is maximized. We first address the location-pricing problem with one facility and three customers, each having unit demand. This problem is solved by using a solution to the Apollonius’ problem. We were not aware of papers combining combinatorial pricing and geometric location problems
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