The multiple lot sizing problem of a serial production system with interrupted geometric yields, rigid demand and Pentico’s heuristic

Abstract

We consider the optimal lot sizing decision problem for a serial production system with the interrupted geometric yields and rigid demand. Such decisions are well-known for analytical difficulty due to often complicated cost expressions and necessity of deciding optimal lotsizes to stages/machines in the system. Pentico proposed a simple and effective heuristic that all usable items exiting a stage will be processed at the next stage till the end of the system. Pentico’s heuristic requires only the decision on the initial lot size. Based on Pentico’s heuristic, Bez-Zvi and Grosfeld-Nir considered the optimal “P-policies” to yield an optimal initial lot size so as to minimize the expected cost of the system while fulfilling the order. They showed the optimal initial lot size is always smaller than or equal to the outstanding demand. In this paper, we provide a finite upper bound and narrow searching range for the optimal initial lot sizes. It is well known that the worst case for minimizing the expected cost is to have an optimal initial lotsize equal to 1 for any outstanding demand. We characterize conditions in terms of the average expected costs for the worst case of the production system. An efficient algorithm for finding the optimal initial lot size is given which utilizes the recursive feature among the expected cost elements. For intellectual curiosity, we study a two-stage serial production system with a uniform yield in stage 1 and an interrupted geometric yield in stage 2. We propose an algorithm to derive an optimal initial lot size to enter this two-stage problem under Pentico’s heuristic. We show that for small outstanding demands (equal to 1 or 2) the optimal initial lot sizes are often greater than the outstanding demands. We prove that for large outstanding demands the optimal initial lot sizes are smaller than or equal to the outstanding demands. We also prove the existence of a finite upper bound for all optimal initial lot sizes. Our numerical example illustrates the existence of a threshold such that if the demand is smaller (greater) than it then the optimal lot size is larger (smaller) than the demand. Our analysis and numerical observation are very interesting to contrast with the most commonly seen binomial or uniform yield settings that the optimal lot size is always larger than the outstanding demand while for interrupted geometric yield the optimal lot size is always smaller than the outstanding demand.