The Competitive Ratio of Threshold Policies for Online Unit-density Knapsack Problems
Will Ma, David Simchi-Levi, Jinglong Zhao
TL;DR
This paper introduces online unit-density knapsack problems with a non-splittable constraint and analyzes threshold-based online policies. By optimizing random threshold distributions, the authors derive two tight competitive guarantees for the single-knapsack case relative to offline fractional and integer packings, achieving $0.4285$ and $0.4324$, respectively, and extend the framework to multi-knapsack scenarios with a $0.2142$-competitive algorithm while proving a hard upper bound of $0.4605$. The multi-knapsack results exploit a phantom-capacity routing approach that combines AdWords-style routing with single-knapsack thresholds, and the analysis highlights fundamental gaps between knapsack-constrained and unconstrained online settings. The work further validates its methods through simulations on data from a Latin American department store, demonstrating practical implementability, explainability, and robustness of threshold policies in real-world supply chains. Overall, the findings quantify the costs and benefits of threshold-based decision rules under knapsack constraints and provide actionable guidance for inventory management in dynamic environments.
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 offline integral packing, and the second is 0.4285-competitive relative to the optimal offline 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, where 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 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 their order data, which demonstrate the efficacy of our proposed algorithms.