Conditions that cause risk pooling to increase inventory

We say product A is a partial substitute for product B if a fraction of the customers who prefer B are willing to accept A when B is out of stock. When demand is uncertain, it is intuitive and true that a larger "willing to substitute" fraction implies larger expected profits. A higher "willing to substitute" fraction allows one to pool the risk of individual products. It may also be intuitive that a larger "willing to substitute" fraction might result in lower optimal total inventory. For the full substitution structure, several researchers have shown that for certain distributions such as the exponential, this latter intuition is not true. We show that this full substitution anomaly can occur with any right skewed demand distribution. We assume i.i.d. demand distributions unless we indicate otherwise. We also show that the anomaly can occur for a number of realistic situations of partial substitution with commonly used demand distributions such as Normal, exponential, Poisson, and uniform. We also demonstrate the anomaly for more than one period, with backlogging, lost sales, more than two products, and with setup costs.