Robust facility location

Let A be a nonempty nite 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 2 A if the facility is located at x 2 S is proportional to dist(x; a) | the distance from x to a | and that demand of point a is given by ! a , minimizing the total transportation cost TC(!; x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector ! is not known, and only an estimator ^ ! 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. Deene the robustness of a location x as the minimum increase in demand needed to become inadmissible, i.e. (x) = min fk! ? ^ !k : TC(! ; x) > B; ! 0g and nd the x maximizing to get the most robust location.