Consider an initial population of size N subject to a pure death process. The lifetimes of members are independent exponential random variables with a common parameter. Each member of the population consumes from a commodity continuously and at a constant rate. We wish to establish a one period optimal inventory for the commodity taking into account purchasing costs, selling price, and out of stock penalty costs. For example, consider the manufacture of typewriter ribbons. As long as the typewriters are in operation, the demand for the ribbon is fairly constant per typewriter in use. However, if that particular model typewriter goes out of production and the ribbon does not match the new model, then eventually the ribbon will also go out of production. We will be facing a closing inventory decision.
