Boundary lines between permutation flowshop problems and single machine problems

The permutation flowshop scheduling and the single machine scheduling problems are two classical problems in the scheduling literature. Since the former is known to be NP-hard for more than two machines, hundreds of algorithms have been implemented in the last decades to solve it. Furthermore, several computational studies have found that the permutation flowshop to minimise makespan is easily solvable for structured (i.e. processing times are job- and/or machine-correlated) instances and when the initial availability of all machines for processing jobs cannot be assumed. Our working hypothesis to explain this behaviour is that, under certain conditions of machine availability or structured processing times, only one stage in the flowshop layout determines the optimal sequence, approximately transforming the flowshop scheduling problem into a single machine scheduling problem. Since the single machine scheduling problem with makespan objective is a trivial problem where all feasible sequences are optimal, such transformation may explain why it is so easy to find good solutions for such flowshop scheduling problems. Therefore, the goal of this paper is to study under which assumptions a permutation flowshop scheduling problem can be reduced to a single machine scheduling problem. More specifically, we focus onto the two most common objectives in the literature (i.e. makespan and total flowtime). Our work is a combination of theoretical and computational analysis, therefore several properties are derived, together with an extensive computational evaluation.