A reinforced Lagrangean relaxation for non-preemptive single machine problem. Application to the earliness-tardiness common due date criterion

The time-indexed formulation [3, 6] is a well-known way to formulate a singlemachine scheduling problem as a mixed-integer scheduling problem (MIP). It is based on time discretization: time is divided into unary periods (or time slots). The main advantage of this integer program is that the linear relaxation is very good. However, as the number of time-points is pseudo-polynomial in the size of the input, the resulting MIP is usually very large and is not directly tractable by MIP solvers (such as ILOG CPLEX) for large instances. Therefore, much research effort was devoted to approaches based on this formulation, which enable to fastly derive good upper and lower bounds. Most approaches are either Lagrangean relaxations or column generation methods. In this paper, we consider the Lagrangean relaxation approach. In the litterature, it has been illustrated that we can either relax the resource constraints or the constraints that force each job to be executed once. We are going to consider this latter relaxation, which was also considered by Péridy et al. [5]. In this paper, we first present a generic approach to integrate the presence of dominance rules in the relaxation scheme. Indeed, dominance rules are very useful to solve most scheduling problems, especially when a sum of penalties has to be minimized. We also apply this approach to solve the one-machine scheduling problem with generic earliness-tardiness penalties and an arbitrary common due date. We show that all the instances of OR-Library [2] can be solved: whether the due date is large or not, instances with 1000 tasks can be solved in about 1000s. This