MIP models for minimizing total tardiness in a two-machine flow shop

We propose compact mixed-integer programming models for the -hard problem of minimizing tardiness in a two-machine flow shop. Also, we propose valid inequalities that aim at tightening the models’ representations. We provide empirical evidence that the linear programming relaxation of an enhanced formulation yields an excellent lower bound that consistently outperforms the best bound from the literature. We further provide the results of extensive computational experiments that attest to the efficacy of the proposed MIP models. In particular, our computational study reveals that most of the 30-job hard instances are optimally solved using the proposed MIP models. Furthermore, we found that even much larger instances, with up to 70 jobs, can be solved for several problem classes.