Multistage Production to Order with Rework Capability

This study considers multistage production systems where defective units can be reworked repeatedly at every stage. Production, as well as rework, is in lots requiring set-up and variable production costs, and orders need to be filled in their entirety. The yield of each stage is uncertain, so several production runs may need to be attempted until the quantity of finished products is sufficient. The trade-off at each stage is between using small lots, possibly necessitating repeated rework set-ups and large lots, which may result in costly overproduction. Multistage manufacturing facilities with rework capabilities are quite common in practice, but their optimal operation when orders have to be met in full has been virtually unexplored. We show that a multistage system where only one of the stages requires a setup (a "single-bottleneck system") can be reduced to a single-stage system. Moreover, if it is possible to arrange the operations in any order, we prove that it is best to make the "bottleneck" the first stage of the system. We also develop recursive algorithms for solving two- and three-stage systems, where all stages require set-ups, optimally. Generalizations to systems where rework yields and costs differ from those of initial processing are also discussed.