How does inventory pooling work when product availability influences customers' purchasing decisions?

Inventory pooling is a technique that is widely used to cope with supply chain uncertainty. Naturally, however, product availability at a given store also influences the customer's purchasing decisions. Hence, a centralised inventory system with a long delivery lead time carries the risk of losing sales to customers who are product-availability-conscious. Assuming a supply chain of a supplier and multiple retailers, we study how inventory pooling works when product-availability-conscious customers abandon a given purchase if the items are not available at a given store. Developing a model with two random variables, Poisson demand and Bernoulli customer response to out-of-stock, we explore who should stock products - a supplier, a retailer, or both - in the context of a product-availability effect on sales. We then propose decision processes by which a supply chain can find the best inventory system based on supply chain size, demand rate, and inventory costs for both supplier and retailer. We also discuss the coordination required between a retailer and a supplier that must synchronise an optimal inventory system.