An approximate periodic review stochastic inventory control system with both fixed cost and random yield

We study a periodic review stochastic inventory control system for a single product at a single location with both fixed cost and random yield. If the random yield follows either two-point or uniform distribution, then some research works have been done in the literature. For other random yield distributions, the structure of the optimal inventory control policy has been an open problem for over three decades. A Negative Dominance (ND) property has been identified for the expected total cost function, which is approximated by a piecewise linear function, in each period. Under some very mild requirements about the random yield distribution and the single-period cost function, a period-dependent lower bound for initial inventory levels in any period is provided such that the expected total cost function indeed has the ND property at any initial inventory level below this lower bound and a period-dependent upper bound for initial inventory levels in any period is also provided such that the optimal order quantity is zero at any initial inventory level above this upper bound. We are able to show that these two bounds will not tend to infinity when the number of periods tends to infinity and, through numerical experiment, we show that the gap between these two bounds is quite small. Even further, with the help of this ND property, the search of the optimal order quantity at any initial inventory level below this lower bound in any period is as simple as the case with zero fixed cost even if the random yield follows neither uniform nor two-point distribution.