Multiproduct Inventory Systems with Upgrading: Replenishment, Allocation, and Online Learning

Abstract

We consider the joint optimization of ordering and upgrading decisions in a dynamic multiproduct system over a finite time horizon of T periods. Multiple types of demand arrive in each period stochastically and can be satisfied with the supply of the same type or some higher-quality product (upgrading). The overall goal is to find an optimal joint replenishment and allocation policy that maximizes the total expected profit in both the setting in which the firm knows the demand distributions a priori and the setting in which the firm needs to learn the demand distributions during the process. We first characterize the structure of the clairvoyant optimal joint ordering and allocation policy. Based on the structure of the optimal policies, we propose a new online learning algorithm termed stochastic gradient descent with perturbed gradient (SGD-PG for short), and prove that the algorithm admits a cumulative regret upper bound of $O(\sqrt{T})$, which matches the lower bound for any learning algorithms. The novelties lie in two aspects: (a) We propose a perturbation-based subroutine to compute a valid sample-path gradient of the profit function with respect to the replenishment decisions. (b) We keep track of the real-time imbalance between supply and demand to carry out the allocation decisions. We also show that SGD-PG can be extended to a nested censored demand scenario. We demonstrate the efficacy of the proposed algorithms in numerical experiments. This work provides practitioners with the optimal policy of inventory replenishment and allocation in a multiproduct system with upgrading. When the demand distribution is unknown, we propose an easy-to-implement and provably-good algorithm for demand learning. In addition, the paper numerically quantifies the value of optimal upgrading and identifies conditions under which upgrading can be the most helpful.

Keywords: multiproduct, ordering, allocation, general upgrading, online learning, censored demand