Optimal Learning Algorithms for Stochastic Inventory Systems with Random Capacities

We propose the first learning algorithm for single-product, periodic-review, backlogging inventory systems with random production capacity. Different than the existing literature on this class of problems, we assume that the firm has neither prior information about the demand distribution nor the capacity distribution, and only has access to past demand and supply realizations. The supply realizations are censored capacity realizations in periods where the policy need not produce full capacity to reach its target inventory levels. If both the demand and capacity distributions were known at the beginning of the planning horizon, the well-known target interval policies would be optimal, and the corresponding optimal cost is referred to as the clairvoyant optimal cost. When such distributional information is not available a priori to the firm, we propose a cyclic stochastic gradient descent type of algorithm whose running average cost asymptotically converges to the clairvoyant optimal cost. We prove that the rate of convergence guarantee of our algorithm is $O(1/\sqrt{T})$, which is provably tight for this class of problems. We also conduct numerical experiments to demonstrate the effectiveness of our proposed algorithms.