This work considers the capacitated multi-facility Weber problem, which is concerned with locating m facilities and allocating their limited capacities to n customers in order to satisfy their demand at minimum total transportation cost. This is a non-convex optimization problem and difficult to solve. Therefore, approximate solution methods are proposed in this article. Some of them are based on the relaxation of the capacity constraints and apply the subgradient algorithm. The resulting Lagrangian subproblem is a variant of the well-known multi-facility Weber problem and can be solved using column generation and branch-and-price approach on a variant of the set covering formulation. Others are based on the approximating mixed-integer linear programming formulations obtained by exploiting norm properties and the alternate solution of the discrete location and transportation problems. The results of a detailed computational analysis are also reported.