The Competitive Ratio of Threshold Policies for Online Unit-Density Knapsack Problems

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Abstract
We study a wholesale supply chain ordering problem. In this problem, the supplier has an initial stock and faces an unpredictable stream of incoming orders, making real-time decisions on whether to accept or reject each order. What makes this wholesale supply chain ordering problem special is its knapsack constraint; that is, we do not allow partially accepting an order or splitting an order. The objective is to maximize the utilized stock. We model this wholesale supply chain ordering problem as an online unit-density knapsack problem. We study randomized threshold algorithms that accept an item as long as its size exceeds the threshold. We derive two optimal threshold distributions, the first is 0.4324-competitive relative to the optimal off-line integral packing, and the second is 0.4285-competitive relative to the optimal off-line fractional packing. Both results require optimizing the cumulative distribution function of the random threshold, which are challenging infinite-dimensional optimization problems. We also consider the generalization to multiple knapsacks, in which an arriving item has a different size in each knapsack. We derive a 0.2142-competitive algorithm for this problem. We also show that any randomized algorithm for this problem cannot be more than 0.4605-competitive. This is the first upper bound strictly less than 0.5, which implies the intrinsic challenge of the knapsack constraint. We show how to naturally implement our optimal threshold distributions in the warehouses of a Latin American chain department store. We run simulations on its order data that demonstrate the efficacy of our proposed algorithms. This paper was accepted by George Shanthikumar, data science. Funding: This work was supported by the Massachusetts Institute of Technology–Accenture Alliance for Business Analytics. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01577 .

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