An optimal replenishment policy for an EOQ model with partial backlogging

We study an inventory system where demand on the stockout period is partially backlogged. The backlogged demand ratio is a mixture of two exponential functions. The shortage cost has two significant costs: the unit backorder cost (which includes a fixed cost and a cost proportional to the length of time for which the backorder exists) and the cost of lost sales. A general procedure to determine the optimal policy and the minimum inventory cost for all the parameter values is developed. This model generalizes several inventory systems analyzed by different authors. Numerical examples are used to illustrate the theoretical results.