Inventory Control with Unobservable Lost Sales and Bayesian Updates. Working Paper

We study a finite-horizon lost-sales inventory model. The demand distribution is unknown and is dynamically updated based on the previous sales data in a Bayesian fashion. We derive a sample-path representation of the first order optimality condition, which characterizes the key tradeoff of the problem. The expression allows us to see why the computation of the optimal policy is difficult and why the myopic solution is not a bound on the optimal solution. It enables us to develop simpler solution bounds and approximations. It also helps us to develop cost bounds as well as cost error bounds of the approximations. Numerical examples indicate that our approximations are most effective for products with short life-cycle. Otherwise, the myopic policy may be a reasonable choice.