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

45 Pages Posted: 22 Jul 2019 Last revised: 7 Apr 2025

See all articles by Will Ma

Will Ma

Columbia University - Columbia Business School, Decision Risk and Operations

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering

Jinglong Zhao

Boston University - Questrom School of Business

Date Written: July 20, 2019

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.

Keywords: Wholesale, supply chain ordering, knapsack problem, online algorithms

Suggested Citation

Ma, Will and Simchi-Levi, David and Zhao, Jinglong, The Competitive Ratio of Threshold Policies for Online Unit-density Knapsack Problems (July 20, 2019). Available at SSRN: https://ssrn.com/abstract=3423199 or http://dx.doi.org/10.2139/ssrn.3423199

Will Ma

Columbia University - Columbia Business School, Decision Risk and Operations ( email )

New York, NY
United States

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering ( email )

MA
United States

Jinglong Zhao (Contact Author)

Boston University - Questrom School of Business ( email )

595 Commonwealth Avenue
Boston, MA MA 02215
United States

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