We consider the lp-norm multi-facility minisum location problem with linear and distance constraints, and develop the Lagrangian dual formulation for this problem. The model that we consider represents the most general location model in which the dual formulation is not found in the literature. We find that, because of its linear objective function and less number of variables, the Lagrangian dual is more useful. Additionally, the dual formulation eliminates the differentiability problem in the primal formulation. We also provide the Lagrangian dual formulation of the multi-facility minisum location problem with the lbp-norm. Finally, we provide a numerical example for solving the Lagrangian dual formulation and obtaining the optimum facility locations from the solution of the dual formulation.
