Optimal size of a rental inventory with items available from a secondary source: a model with non-stationary probabilities

This article concerns operations of businesses that own inventories of rental items, and can hire additional items from secondary sources whenever they face a temporary exhaustion of their inventories. This set-up is relevant to many operations: the items may be tools, trucks, containers, communication channels, or individuals who provide services such as repairmen. A fundamental problem that emerges in the design of these operations is to determine the optimal size of the inventory of items the business should own. To solve this problem, this article takes the view of a finite horizon project and proposes an approach that chooses the inventory size that minimizes the expected present cost of the project. This approach models random times between consecutive item requests and random rental durations with corresponding expectations that may vary along the day. The expected present cost uses non-stationary transition probabilities that recent articles have computed resorting to stationary approximations. This article, in contrast, computes these probabilities faster solving a differential equation without resorting to such approximations. If the present cost is of interest, an analysis that plugs-in the optimal size into the present cost ignores the sampling variability that transfers from the traffic data to the optimal size. This article complements the analysis with simulations that provide the sampling distribution of the present cost.