The capacitated multi-facility Weber problem is concerned with locating m facilities in the plane, and allocating their capacities to n customers at minimum total cost, which is a non-convex optimization problem and difficult to solve. In this work we relax the capacity constraints and solve the uncapacitated Lagrangean subproblems every step of a subgradient algorithm. Their objectives have facility dependent distance functions, which is different than the usual multifacility Weber problem. We solve them by branch-and-price using column generation with concave minimization pricing.