Improved algorithms for scheduling on proportionate flowshop with job-rejection

The rejection method is an option commonly used in manufacturing systems as a tool to overcome overloaded assembly lines. The operations manager is given the ability to refuse to produce, or alternately to outsource some of the products. A flowshop environment is a setting of machines in series such that each job should be processed on each of the machines. In a recent paper, Shabtay and Oron (2016), the two disciplines are combined, and the makespan criteria is studied, which is one of the fundamental measures in scheduling theory that reveals the utilization of the system. The problems considered are minimizing the makespan subject to a constraint on the maximal rejection cost E, and minimizing the rejection cost given that the makespan cannot exceed a given upper bound, K. The computational complexity of the pseudo polynomial dynamic programming (DP) algorithms presented by Shabtay and Oron are O(n²E) and O(n²K), respectively, where n is the number of jobs. In this paper, we consider the same problems, and our contributions are enhanced DP algorithms, which run in O(nE) and O(nK) time, in correspondence, implying an improvement of a factor of n. Furthermore, we supply empirical results based on the experimental simulations. This study, therefore, has both theoretical significance and practical implications, as our numerical study proves that the introduced DP algorithms are capable of solving large-size instances.