On the stationary stochastic response of an order-constrained inventory system

We investigate the stochastic response of a base stock inventory system where the order quantity is either upper- or lower-constrained. This system can represent many real-world settings: forbidden returns, minimum order quantities, and capacity constraints for example. We show that this problem can be translated into a stopping time problem where the distributions of orders and inventory can be represented by a countably infinite mixture of truncated and convoluted demand distributions. This result can be extended to the cases of arbitrary lead time and auto-correlated demand. A state-space algorithm is developed to approximate the first- and second-order moments of the order quantity and inventory level. Via a numerical analysis, we investigate the performance of the approximation, as well as the operational and economic impact of the order constraint. In particular, the constraint impacts order and inventory variances via different combinations of the mixture and truncation effects. We show how tuning the constraint can improve the operational and financial performance of the inventory system by acting as a smoothing mechanism