A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines, due dates and sequence-dependent setup times

Consider the problem of scheduling a set of jobs to be processed exactly once, on any machine of a set of unrelated parallel machines, without preemption. Each job has a due date, weight, and, for each machine, an associated processing time and sequence-dependent setup time. The objective function considered is to minimize the total weighted tardiness of the jobs. This work proposes a non-delayed relax-and-cut algorithm, based on a Lagrangean relaxation of a time indexed formulation of the problem. A Lagrangean heuristic is also developed to obtain approximate solutions. Using the proposed methods, it is possible to obtain optimal solutions within reasonable time for some instances with up to 180 jobs and six machines. For the solutions for which it is not possible to prove optimality, interesting gaps are obtained.