Batching for work and rework processes on dedicated facilities to minimize the makespan

We study a planning problem of an imperfect production of a single product. The product is assumed to be continuously divisible. There are two facilities: a main facility dedicated to the original production and a facility dedicated to re-manufacturing defective units coming from the main facility. Units fabricated on the main facility are inspected for quality in batches. The quality inspection requires some time and can be performed on-line or off-line. After the inspection has been completed, defective units of the inspected batch are transported to the re-manufacturing facility. The transportation also requires some time. We assume that the fraction of the defective units is the same in each batch on the manufacturing facility and that the re-manufacturing facility is perfect. Given a demand for good quality units of the product and an upper bound K on the number of batches, the problem is to find a sequence of batch sizes such that the makespan, i.e., the time of the demand satisfaction, is minimized. We suggest a linear programming formulation, prove several properties of an optimal solution, and finally develop an time solution algorithm. A similar per time unit cost minimization problem is studied as well.