Location of an obnoxious facility on a network: A voting approach

We consider the location problem of an obnoxious facility with respect to a finite number of inhabitants, where the inhabitants have specified locations at vertices of a network N . A voting solution, called anti‐Condorcet point, is defined as a point of N such that no other point is farther from a strict majority of inhabitants. On a general network with an odd number of inhabitants, it is shown that there exists a finite set of points that contains all such solutions. An example shows that this result does not directly extend to an even number of inhabitants on a general network. In the special case of a tree network, one of the extreme vertices of a diameter is an anti‐Condorcet point, and a linear algorithm for finding it is presented. Finally, a bound on the maximum decrease of the total distance to the inhabitants is provided when an anti‐Condorcet point is preferred to a “maxisum” location.