A continuous-time examination of the last buy problem

The last buy problem is a stochastic inventory management problem that occurs at the end of a product's life cycle. When production of a given product ceases, it may become necessary to shut down manufacture of all parts of the product. However, there will likely still be demand for spare parts of the product, due to part failure from the product still in use. To meet this demand, a one-time order of spare parts - a last buy - is made to satisfy the demand for all spare parts going forward. Thus, the last buy problem seeks to maximize a company's products with respect to the number of spare parts manufactured.

Several different forms of the last buy problem exist, depending on the relationship between the manufacturer and the customer and the type of cost that occurs once the inventory has been depleted. In some cases, law or contract defines and limits the costs and revenues the manufacturer incurs due to the last buy order; in other cases, a manufacturer's own policies dictate the costs and revenues involved. As a result, we explore three main types of last buy problem, and the different methods used to solve for each.

In the last buy problem with incremental replenishment, individual parts demanded beyond the last buy are fabricated individually at significantly greater cost; as the total product is concave with respect to the order amount, the optimal order amount can be found by analysis of the rate of change of the product. The last buy problem with no replenishment occurs when there is no effective way to replenish part inventory beyond the last buy; as the total product is not concave, an upper bound on the optimal order amount is determined, thus limiting the candidate solutions. Difficulties exist in calculating the optimal order amount in the last buy problem with batch replenishment, as the size of the replenishment batch is itself a last buy problem; we solve for a special case of the problem using renewal theory. We also examine the possibility of contract extensions in last buy problems, and their effect on the optimal order amount calculations.