Interactive High-low Search: The Case of Lost Sales

We consider the problem of supplying perishable goods to disloyal customers. As in traditional stock-control literature, a penalty is incurred whenever there is a stockout. However, in contrast to mainstream models, loss of goodwill is explicitly treated by incorporating the behavioural assumption that a fixed proportion of unsatisfied demand is lost forever after each stockout. The problem consists of finding supply levels which minimize costs of over- and underproduction, given unknown but deterministic demand. We derive optimal adaptive search procedures under varying assumptions of a priori knowledge about demand. Optimal strategies are compared to myopic strategies. Our methodology extends the mathematical theory of ‘high-low search’ for a hidden point to incorporate the ‘Heisenberg principle’: the position of the hidden point (demand) is directly influenced by the actions (supply levels) of the searcher.