An analytical approach to determine the window fill rate in a repair shop with cannibalization

We consider a repair shop in which each unit comprises multiple component types and cannibalization is allowed. The shop's managers have a budget for purchasing spare components and need to decide how many spares of each component type to purchase. Customers arrive to the shop with a single unit of which at least one of its components has failed and expect to be served within a tolerable waiting time. Accordingly, the shop's performance measure is the window fill rate, that is, the fraction of customers who are served within the tolerable wait. In our analysis, we develop exact formulas for the window fill rate that comprise multiple dependent Skellam random variables. We overcome the practical complexity of evaluating these formulas by using simulation to evaluate only the random elements. We discuss run-time considerations for solving the spares allocation problem and demonstrate how the optimal solution and the window fill rate depend on the tolerable wait, budget and the customer base using an illustrative numerical example.