In this paper, we consider the problem of minimizing the total weighted tardiness of a set of jobs processed on a single processor. First, a lower bound based on a Lagrangian decomposition is presented. The particularity of this decomposition, based on a 0–1 time indexed formulation, is to be sensitive to the domain reduction of jobs which are proposed. A branch and bound strategy including these different components is proposed. The results obtained on problems from the literature can be favourably compared to previously works and seem to prove that a trade-off between a tight lower bound and time consuming in the enumeration process can be a good strategy.