Dominance conditions determination based on machine idle times for the permutation flowshop scheduling problem

•Interesting properties of positive and negative machine idle times of the flowshop.•A dominance condition independent of the preceding and subsequent partial schedules.•Three dominance conditions that improve the efficiency of branch and bound methods.•A new bounding approach based on machine idle times of the flowshop. In this study, we investigate the lower bound and dominance conditions of the N-job and M-machine permutation flowshop scheduling problem to minimize the makespan based on machine idle times. We have developed and validated three dominance conditions and a lower bound based on a new approach that minimizes the machine idle times in a flowshop to facilitate the efficiency of the branch and bound approach. Most importantly, we have developed a dominance condition that is independent of the preceding and subsequent partial schedules, which has been considered critical but has not previously been developed in flowshop scheduling research. Finally, the efficiency of our developed branch and bound approach based on machine idle times is compared with the efficiencies of some previous branch and bound algorithms from the literature. We show that the dominance conditions that we developed can significantly improve the efficiency of the branch and bound approach.