An inventory system with partial backlogging modeled according to a linear function

In this paper, we study an inventory system with partial backlogging, in which the backlogging rate is a continuous and nonincreasing two piece function. The total shortage cost includes three significant costs: the unit backorder cost (depending on the backorder time), the opportunity cost, and the goodwill cost. This model generalizes several inventory systems studied by other authors. The optimal policy is characterized through several results, which depend on the values of the known input parameters. Illustrative examples and a sensitivity analysis, which help us understand the theoretical results, are also given.