Proportionate flow shop: New complexity results and models with due date assignment

Abstract

Abstract:It is known that the proportionate flow shop minimum makespan F m/prpt/C max problem is solved optimally by any permutation job sequence. We show that the F m/prpt/C max problem is at least ordinary NP-hard when missing operations are allowed and present some solvable cases. We then consider the standard proportionate flow shop problem (with no missing operations) and show that the solution algorithms for a class of single-machine due date assignment problems can be extended/generalized to the corresponding proportionate flow shop problems.

On equivalence between the proportionate flow shop and single‐machine scheduling problems

“…140). The extension of the optimal and sequences to the proportionate flow shop was recently shown by Panwalkar and Koulamas . To the best of our knowledge, all other results mentioned above (including the modifications of some of these results as outlined in Baker and Scudder and in Gordon et al ) have not been shown to extend to the proportionate flow shop.…”

“…Let [j ] denote the job in the j th position in the sequence and I [j ] denote the idle time on the last machine immediately preceding job [j ]; by assumption, p [0] = 0. Then, as shown in Panwalkar and Koulamas [31],…”

“…However, in that case, we must assume that all operations of a job are outsourced when the outsourcing option is exercised by the scheduler. This is because the partial outsourcing of some jobs will result in a proportionate flow shop with missing operations which was shown to be at least ordinary NP‐hard by Panwalkar and Koulamas .…”

“…Let denote the job in the position in the sequence and denote the idle time on the last machine immediately preceding job ; by assumption, . Then, as shown in Panwalkar and Koulamas , and …”