Robustness of Online Inventory Balancing to Inventory Shocks

In classic adversarial online resource allocation problems such as AdWords, customers arrive online while products are given offline with a fixed initial inventory. To ensure revenue guarantees under uncertainty, the decision maker must balance consumption across products. Based on this, the prevalent policy "inventory balancing (IB)" has proved to be optimal or near-optimal competitive in almost all classic settings. However, these models do not capture various forms of inventory shocks on the supply side, which play an important role in real-world online assortment and can significantly impact the revenue performance of the IB algorithm. Motivated by this paradigm, we introduce a variant of online assortment planning with inventory shocks. Our model considers adversarial exogenous shocks (where supply increases unpredictably) and allocation-coupled endogenous shocks (where an inventory reduction is triggered by the algorithms and re-adjusted after a usage duration), whose combination leads to non-monotonic inventory fluctuations. As our main result, we show the robustness of IB-type strategies against such shocks by designing a new family of optimal competitive algorithms called "Batched Inventory Balancing (BIB)." Using a novel randomized primal-dual method, we bound the competitive ratio of BIB against optimal offline. We show that with proper choice of a certain parameter, this competitive ratio is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model. Moreover, we characterize BIB's competitive ratio parametric by its penalty function and show that it matches exactly the competitive ratio of IB without shocks. Our refined analysis reduces the dual construction to a combinatorial "interval assignment problem" whose algorithmic solution may be of independent interest.