Critical paths of non-permutation and permutation flow shop scheduling problems

Abstract

The literature on flow shop scheduling has extensively analyzed two classes of problems: permutation and non-permutation ones (PFS and NPFS). Most of the papers in this field have been just devoted on comparing the solutions obtained in both approaches. Our contribution consists of analyzing the structure of the critical paths determining the makespan of both kinds of schedules for the case of 2 jobs and m machines. We introduce a new characterization of the critical paths of PFS solutions as well as a decomposition procedure, yielding a representation of NPFS solutions as sequences of partial PFS ones. In structural comparisons we find cases in which NPFS solutions are dominated by PFS solutions. Numerical comparisons indicate that a wider dispersion of processing times improves the chances of obtaining optimal non-permutation schedules, in particular when this dispersion affects only a few machines.