Batching work and rework processes with limited deterioration of reworkables

We study a deterministic problem of planning the production of new and recovering defective items of the same product manufactured on the same facility. Items of the product are produced in batches. The processing of a batch includes two stages. In the first work stage, all items of a batch are manufactured and good quality items go to the inventory to satisfy given demands. In the second rework stage, some of the defective items of the same batch are reworked. Each reworked item has the required good quality. While waiting for rework, defective items deteriorate. There is a given deterioration time limit. A defective item, that is decided not to be reworked or cannot be reworked because its waiting time will exceed the deterioration time limit, is disposed of immediately after its work operation completes. Deterioration results in an increase in time and cost for performing rework processes. It is assumed that the percentage of defective items is the same in each batch, and that they are evenly distributed in each batch. A setup time as well as a setup cost is required to start batch processing and to switch from production to rework. The objective is to find batch sizes and positions of items to be reworked such that a given number of good-quality items is produced and total setup, rework, inventory holding, shortage and disposal cost is minimized. A polynomial dynamic programming algorithm is presented to solve this problem.