An inventory model where backordered demand ratio is exponentially decreasing with the waiting time

We analyze an inventory system with a mixture of backorders and lost sales, where the backordered demand rate is an exponential function of time the customers wait before receiving the item. Stockout costs (backorder cost and lost sales cost) include a fixed cost and a cost proportional to the length of the shortage period. A procedure for determining the optimal policy and the maximum inventory profit is presented. This work extends several inventory models of the existing literature.