Formulações para o problema do flow shop em duas máquinas com penalidades por atraso nas tarefas - Gonçalves, José Mauricio Brasil (2009)
Scheduling problems are very common in manufacturing and services industries. Scheduling deals with the allocation of scarce resources to jobs over time. The goal is optimizing one or more objectives, translating into the best order in with jobs must be processed by each machine. Many of scheduling problems belong to the NP-hard class of problems, this being the case of the problem studied here. The goal is to propose integer programming formulations for the deterministic two machine flow shop problem, with penalties on the jobs tardiness. In the three-field notation this problem is denoted as F2 | | Σw T . The fact of seeking to minimize the weighted sum of jobs tardiness makes the problem much more difficult when compared to the famous problem of minimizing the completion time of the last job to leave the system (the makespan), denoted as F2 | | Cmax (it can be solved in polynomial time by Johnson´s algorithm). Formulations were tested with binary variables x (that assume value one when job j completes at machine i at time t) and a formulation with binary variables x (that assume value one when job j leaves the system at time t). Computational experiments with instances with up to 50 jobs have shown that the formulation with variables obtained initial dual limits closer to the optimal value of the objective function. Finally, it is also proposed a more compact formulation for the problem, not using the index t in the decision variables and a local search heuristic for solving the problem and heuristics that support the branch and bound algorithms to find a good primal bounds.