A periodic review inventory model with two delivery modes, fractional lead-times, and age-and-period-dependent backlogging costs

We examine a periodic review inventory model where the buyer has access to a mix of two delivery modes for the quantity it orders in a period; these include an emergency delivery mode that delivers instantaneously and a regular delivery mode that delivers within the length of the review period. The unmet demand is backordered and charged a backlogging cost that varies with the length of the backlogging time. The excess inventory is charged a holding cost. The unit costs for the delivery modes could be different. We show that the well-known base-stock policy structure remains optimal for this problem under mild conditions.