Minimizing makespan in a two-machine flowshop problem with processing time linearly dependent on job waiting time

Abstract

This paper is aimed at studying a two-machine flowshop scheduling where the processing times are linearly dependent on the waiting times of the jobs prior to processing on the second machine. That is, when a job is processed completely on the first machine, a certain delay time is required before its processing on the second machine. If we would like to reduce the actual waiting time, the processing time of the job on the second machine increases. The objective is to minimize the makespan. When the processing time is reduced, it implies that the consumption of energy is reduced. It is beneficial to environmental sustainability. We show that the proposed problem is NP-hard in the strong sense. A 0-1 mixed integer programming and a heuristic algorithm with computational experiment are proposed. Some cases solved in polynomial time are also provided.