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

## TL;DR

This paper analyzes online threshold algorithms for unit-density knapsack problems, deriving optimal competitive ratios and demonstrating their practical effectiveness through simulations in a supply chain context.

## Contribution

It introduces optimal randomized threshold policies for online knapsack problems with proven competitive ratios and extends the analysis to multiple knapsacks, highlighting the intrinsic difficulty of the problem.

## Key findings

- Achieved a 0.4324-competitive ratio for the offline integral packing.
- Achieved a 0.4285-competitive ratio for the offline fractional packing.
- Developed a 0.2142-competitive algorithm for multiple knapsacks.

## 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.