AbstractOptimal and Heuristic bounds are given for the optimal location to the Weber problem when the locations of demand points are not deterministic but may be within given circles. Rectilinear, Euclidean and square Euclidean types of distance measure are discussed. The exact shape of all possible optimal points is given in the rectilinear and square Euclidean cases. A heuristic method for the computation of the region of possible optimal points is developed in the case of Euclidean distance problem. The maximal distance between a possible optimal point and the deterministic solution is also computed heuristically.