Solving makespan minimization problems with Lagrangean decomposition

The problem of minimizing makespan in a single stage manufacturing process with parallel machines and multiple job types can be decomposed into as many simple plant location problems as there are job types and as many constrained 0-1 knapsack problems as there are machines. None of these subproblems has the integrality property, therefore the Lagrangean dual bound obtained is strnger than the LP relaxation bound as well as the associated Lagrangean relaxation bounds. The paper shows on an example that the bound can indeed be stronger.