In this paper, a single facility centre location problem with a line barrier, which is uniformly distributed on a given horizontal route in the plane is proposed. The rectilinear distance metric is considered. The objective function minimizes the maximum expected barrier distance from the new facility to all demand points in the plane. An algorithm to solve the desired problem is proposed where a mixed integer nonlinear programming needs to be solved. The proposed model of this paper is solved using some already existed benchmark problem in the literature and the results are compared with other available methods.