The Flow Shop Scheduling Polyhedron with Setup Times

Abstract

This paper addresses the problem of improving the polyhedral representation of a certain class of machine scheduling problems. Despite the poor polyhedral representation of many such problems in general, it is shown that notably tighter linear programming representations can be obtained for many important models. In particular, we study the polyhedral structure of two different mixed-integer programming formulations of the flow shop scheduling problem with sequence-dependent setup times, denoted by SDST flow shop. The first is related to the asymmetric traveling salesman problem (ATSP) polytope. The second is less common and is derived from a model proposed by Srikar and Ghosh based on the linear ordering problem (LOP) polytope. The main contribution of this work is the proof that any facet-defining inequality (facet) of either of these polytopes (ATSP and LOP) induces a facet for the corresponding SDST flow shop polyhedron. The immediate benefit of this result is that all developments to date on facets and valid inequalities for both the ATSP and the LOP can be applied directly to the machine scheduling polytope. In addition, valid mixed-integer inequalities based on variable upper-bound flow inequalities for either model are developed as well. The derived cuts are evaluated within a branch-and-cut framework.