Lot sizing with randomly graded yields

We consider a single inventory item that is graded into one of two quality levels after production. The proportion of grade one units produced in a lot is a random variable assumed to follow the lognormal distribution. Demands for grade 2 may be filled with grade 1 product, but not vice versa. Assuming that demands for the two grades are known and constant, we develop a continuous review model in which cycles are defined as times when total inventory equals zero. Since grade 1 inventory may be depicted before total stock and shortages are not permitted, cycles may consist of multiple set-ups. We show that the optimal order-to-point, S, has a form similar to the economic order quantity. Tables of the cdf of an appropriately defined unit normal multivariate distribution are incorporated into a spreadsheet to facilitate calculations for any parameter setting.