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textabstractLagrangian relaxation is a powerful bounding technique that has been applied successfully to manyNP-hard combinatorial optimization problems. The basic idea is to see anNP-hard problem as an easy-to-solve problem complicated by a number of nasty side constraints. We show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds. The trick is widely applicable, but we focus on a broad class of machine scheduling problems for which it is particularly useful. We provide promising computational results for three problems belonging to this class for which Lagrangian bounds have appeared in the literature: the single-machine problem of minimizing total weighted completion time subject to precedence constraints, the two-machine flow-shop problem of minimizing total completion time, and the single-machine problem of minimizing total weighted tardiness

Stronger Lagrangian bounds by use of slack variables: Applications to machine scheduling problems