Mixed integer formulations using natural variables for single machine scheduling around a common due date

Anne-Elisabeth Falq (1) , Pierre Fouilhoux (1) , Safia Kedad-Sidhoum (2)

1 RO - Recherche Opérationnelle
2 CEDRIC - OC - CEDRIC. Optimisation Combinatoire

While almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods.

Domaines: Informatique [cs]; Mathématique discrète [cs.DM]

Anne-Elisabeth Falq, Pierre Fouilhoux, Safia Kedad-Sidhoum. Mixed integer formulations using natural variables for single machine scheduling around a common due date. Discrete Applied Mathematics, 2021, 290, pp.36-59. ⟨10.1016/j.dam.2020.08.033⟩. ⟨hal-02074488v2⟩