Spare Parts Inventory Systems under an Increasing Failure Rate Demand Interval Distribution

Managing spare parts inventories is a challenging task and can benefit considerably from any information on the failure rate of the parts. It has often been considered in the stock control literature that parts' failures are only random, caused by external events which results in the assumption of constant failure rate and therefore the consideration of the Poisson process to represent spare parts demand. This is obviously a restricted and an unrealistic modeling assumption. In fact, a spare part recently replaced will unlikely fail shortly again but generally the older it gets, the greater this probability. Moreover, it has been shown that models with a Poisson demand process assumption advise more stock than necessary. That it is to say that it is more appropriate to model spare parts demand by a renewal process with an increasing failure rate demand interval distribution. For a continuous treatment of time, an Erlang distribution covers a large spectrum of spare parts demand interval distributions. In this paper, we develop a model for a spare part inventory system where demand is a renewal process and the supply lead-time is constant. Demand intervals follow an Erlang distribution and the demand sizes follow a Gamma distribution. The stock is controlled according to a base-stock policy which is often used in spare parts inventory control. We determine analytically the expected total inventory cost and the optimal base-stock level for a cost oriented inventory system where unfilled demands are backordered. A numerical investigation is also conducted to analyze the inventory system with respect to the different system parameters.