Lagrangian Relaxation and the Single-Source Capacitated Facility-Location Problem

Lagrangian relaxation has been widely used in solving a number of hard combinatorial optimization problems. The success of the approach depends on the structure of the problem and on the values assigned to the Lagrange multipliers. A recent paper on the single-source capacitated facility-location problem proposed the use of Lagrangian relaxation in which the capacity constraints were relaxed. In this paper, a class of such problems is defined for which the proposed relaxation is guaranteed to result in an infeasible solution, irrespective of the values assigned to the Lagrange multipliers. In these cases, the bounds on the optimal solution, obtained from the relaxation, are generally poor. It is concluded that, when using Lagrangian relaxation, it may be worthwhile carrying out a preliminary analysis to determine the potential viability of the approach before extensive development takes place.