Robust facility location

yLet A be a nonempty finite subset of the plane representing the geographical coordinates of a set of demand points (towns,...), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for a is an element of A if the facility is located at x is an element of S is proportional to dist(x, a) - the distance from x to a - and that demand of point a is given by omega(a), minimizing the total transportation cost TC(omega, x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector omega is not known, and only an estimator (ω) over cap can be provided. Moreover the errors in such estimation process may be non-negligible. We propose a new model for this situation: select a threshold value B > 0 representing the highest admissible transportation cost. Define the robustness rho of a location x as the minimum increase in demand needed to become inadmissible, i.e. rho(x) = min{\\omega - (ω) over cap)\\ : TC(omega, x) > B, omega greater than or equal to 0} and find the x maximizing rho to get the most robust location.