Proportionate Flow Shop Scheduling with Job-dependent Due Windows and Position-dependent Weights

In this paper, we deal with the different due windows assignment proportionate flow shop problem with position-dependent weights, where the sequence is a permutation. The objective is to determine an optimal job sequence and due windows of all jobs such that the weighted sum of earliness–tardiness, the starting time and size of all due windows is to be minimized, where the weight is not related to the job but to the position in which some job is scheduled, i.e., position-dependent weights. According to a series of optimal properties, we prove that the problem can be solved in polynomial time Formula: see text, where Formula: see text is the number of jobs.