Multiple lot-sizing, and value of probabilistic information, in production to order of an uncertain size

A make-to-order batch manufacturer knows that an order for items with some particular specifications is forthcoming, but does not yet know the exact size of the order. Due to production lead times, work needs to start immediately. But some units produced may fail to meet required tolerances; that is, the yield of each batch is random. Each new run involves a costly setup, and each unit attempted involves material and other variable costs. If a subjective probability distribution for the order size can be formed, the following related questions arise: (a) What is the optimal run size? (b) When should production stop? To answer these questions, this paper formulates a new model, whose main new feature is the ability to handle uncertain demand within a multiple lot-sizing setting. It is proved that the optimal policy for this scenario is one of control limit – stop if and only if the stock of good units is larger than some critical value. A computer program was developed for solving the problem for binomial yields, and optimal policies for several examples are reported. Due to the effort required to form and communicate probabilistic forecasts, a common practice is to express a forecast as a single number, corresponding, perhaps, to the mean or median of the unarticulated distribution. We compute the extra profit expected from forming, and using, probabilistic forecasts. We also show how to compute the profit's variance for any policy.