Location of a facility in a competitive environment: A planar-based approach

The problem of locating a new facility on the plane, that will compete with a set of existing facilities, will be investigated. It is assumed that demand points (customers) patronize the facility with the highest utility value rather than the closest one. T. Drezner analyzed the theoretical background of the problem and provided simulation-based numerical experience. She proved that a demand point will patronize the new facility if and only if the new facility is located inside a certain circle. The radius of this circle depends upon the relative level of quality attributes of the new facility, the relative euclidean distance from it included. Consequently, the new facility should be located in the interior of the intersection of a set of circles, each one of them corresponding to a different demand point. In the context of the above theoretical framework, an investor may decide to locate the new facility expecting that competitors would react by upgrading quality attributes to a level determined by their available financial resources. This would lead to a shrinkage of the circles’ intersection convex ‘arc-polygon’. Given the above dynamic environment, the decision maker would ask for a central point inside the ‘arc-polygon’ to establish the new facility. The problem of locating a ‘central’ point inside a polygon (convex or not), that maximizes the distance between it and the perimeter of the polygon, has been solved optimally by J. Karkazis. The basic theoretical ideas behind the algorithm proposed for this ‘linear’ problem will be appropriately extended and adjusted in this paper to solve the problem of finding the most central point inside an ‘arc-polygon’. It is proved that the search for the optimal solution of this problem can be reduced to the ‘knots’ of a graph consisting of the ‘arc-dichotomi’ of the angles of the convex ‘arc-polygon’, and at most one ‘turning point’ on every edge.