Improved algorithm for minimizing total late work on a proportionate flow shop and extensions to job rejection and generalized due dates

Gerstl et al (2019) studied the problem of minimizing the total late work (TLW) on an m-machine proportionate flow shop. They solved the case where the total late work refers to the last operation of the job (i.e., the operation performed on the last machine of the flow shop). As the problem is known to be NP-hard, the authors proved two crucial properties of an optimal schedule and introduced a pseudo-polynomial dynamic programming (DP) algorithm. In this research, we revisit the same problem and present enhanced algorithms by the factor of (n+m), where n is the number of jobs and m is the number of machines. Furthermore, based on the improved algorithm, we extend the fundamental problem to consider optional job rejection. We focus on minimizing the TLW subject to an upper bound on the total rejection cost and introduce DP algorithms. Next, we address the problem of minimizing the TLW with generalized due dates, with an upper bound on the permitted rejection cost, and likewise introduce DP algorithms. We conducted an extensive numerical study to evaluate the efficiency of all DP algorithms.