A continuous location problem with different norms

This paper considers a set of problems of fixing the number and location of facilities to cover a given set of demand points, such that each demand point is served within a range of known distance standards, where each facility can be located on the entire plane and the distance can be either measured in rectilinear, Euclidean or Tchebycheff units. It is shown that an optimum solution can always be found on a small finite set of points. Thus the optimum itself can be found by either complete enumeration of these points or by zero one integer linear programming techniques. Furthermore, it is shown that the least and the largest number of facilities are required for the rectilinear and the Tchebycheff cases respectively.