Improved linear programming relaxations for flow shop problems with makespan minimization

Machine scheduling problems with makespan minimization have been addressed in various academic and industrial fields using mixed-integer programming (MIP). In most MIP models, however, the makespan variable is poorly linked to the natural date variables of jobs. To address this, we propose novel, strengthening inequalities, derived from the single-machine scheduling polyhedron augmented by a makespan variable. While the associated optimization problem for a single machine is trivial, these inequalities can be applied as cutting planes to more complicated scheduling problems. In this work, we demonstrate their use for non-permutation flow shops. Using the Taillard benchmark set, we analyze the effect of the inequalities on the linear programming relaxations and mixed-integer programs of three commonly used MIP models. The experiments show that the inequalities significantly improve the ability of linear-ordering and time-indexed models to bound the optimum. The positive effect also extends to linear-ordering models with changeover times, demonstrating the potential of these inequalities to improve more general, application-oriented flow shop problems.