The finite multiple lot sizing problem with interrupted geometric yield and holding costs

We consider the multiple lot sizing problem in production systems with random process yield losses governed by the interrupted geometric (IG) distribution. Our model differs from those of previous researchers which focused on the IG yield in that we consider a finite number of setups and inventory holding costs. This model particularly arises in systems with large demand sizes. The resulting dynamic programming model contains a stage variable (remaining time till due) and a state variable (remaining demand to be filled) and therefore gives considerable difficulty in the derivation of the optimal policy structure and in numerical computation to solve real application problems. We shall investigate the properties of the optimal lot sizes. In particular, we shall show that the optimal lot size is bounded. Furthermore, a dynamic upper bound on the optimal lot size is derived. An O(nD) algorithm for solving the proposed model is provided, where n and D are the two-state variables. Numerical results show that the optimal lot size, as a function of the demand, is not necessarily monotone.