Optimal ordering policies with convertible lead times

A firm receives orders that will be required at an uncertain time given by an Erlang distribution, and over time observes the associated independent exponential events. The firm, in turn, places orders at a linear cost from a supplier with fixed lead time l and has the option of converting (expediting) each order, at a cost, over a certain time interval after the order is originally placed. A converted order arrives le < l units of time after it is converted. We show that a threshold policy is optimal. Under such a policy the firm places an order after a certain number of exponential events have been observed. An order is converted the first time, if any, when the residual lead time exceeds a time threshold related to the number of exponential events realized since the order was placed.