Exploring the nonlinear dynamics of the lost-sales order-up-to policy

Abstract

With most inventory theory investigating linear models, the dynamics of nonlinear inventory systems is not well understood. We explore the dynamics of the order-up- to policy under lost-sales for the case of i.i.d. normally distributed demand and unit lead times. We consider the ideal minimum mean squared error forecast and two alternative scenarios: partial demand observation and dynamic demand forecasting, providing a broad understanding of the operational performance of lost-sales sys- tems. In each scenario, we obtain analytical expressions for the order, inventory, and satis ed demand distributions. This allows us to quantify the Bullwhip and inven- tory variance ampli cation ratios as well as the ll rate and the inventory cover. We show the lost sales nonlinearity induces complex behaviors in inventory systems. Interestingly, lost sales smooth supply chain dynamics, signi cantly a ecting the trade-o between service level and average inventory holding. We also reveal the inventory downsides of demand censoring and the production damages induced by dynamic forecasts. We identify a key parameter, the relative safety margin, that characterizes the performance of lost-sales systems. We nish by o ering some pre- scriptive results for the optimal safety stock and capacity level in both a retail and a manufacturing lost-sales setting.