In a workshop on Integer Programming held in Germany in 1975, Geoffrion presented as an open important area for research, “What are the Cost Effective Ways to Determine the Best Multipliers for Generalized Lagrangean Relaxation?” In this paper, a method for computing the Lagrangean multipliers is presented. The method is applicable for a relatively wide class of Integer Programming problems. Four examples are presented in which, by applying this approach, it was possible to directly compute the multipliers. First, the general approach is outlined, and then the examples are presented. It is hoped that this approach and those specific examples will be of use in developing more general methods for computing the optimal Lagrangean multipliers and reduce the solution time of Integer Programming problems that fall within this class.

Preliminary computational tests have been performed on a variety of interval-bounded Zero-one Knapsack problems, using a Branch and Bound algorithm and a Lagrangean Relaxation that applied a method similar to the one presented in this research to obtain the best multipliers. The computational tests were very favorable.