Optimizing Inventory Placement for a Downstream Online Matching Problem

2024-03-07

arXiv.org

AI Summary

This paper studies the NP-hard problem of offline allocation of Q units of a product across multiple warehouses to maximize the subsequent online order fulfillment rate in e-commerce. We propose the first tight randomized rounding algorithm for the offline surrogate function, achieving a $(1-(1-1/d)^d)$-approximation ratio; under stochastically independent and time-homogeneous demand, it attains a $(1-(1-1/d)^d)/2$-approximation guarantee for the joint allocation-and-fulfillment problem. Our method integrates integer programming modeling, sample average approximation (SAA), and statistical learning-based generalization analysis. Empirical evaluation on real-order sequences from JD.com demonstrates that the proposed offline strategy significantly outperforms myopic and fluid baselines—and even surpasses simulation-based optimization methods. Moreover, the theoretical approximation bound closely matches empirical performance, validating both the efficacy and practicality of our approach.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the inventory placement problem of splitting $Q$ units of a single item across warehouses, in advance of a downstream online matching problem that represents the dynamic fulfillment decisions of an e-commerce retailer. This is a challenging problem both in theory, because the downstream matching problem itself is computationally hard, and in practice, because the fulfillment team is constantly updating its algorithm and the placement team cannot directly evaluate how a placement decision would perform. We compare the performance of three placement procedures based on optimizing surrogate functions that have been studied and applied: Offline, Myopic, and Fluid placement. On the theory side, we show that optimizing inventory placement for the Offline surrogate leads to a $(1-(1-1/d)^d)/2$-approximation for the joint placement and fulfillment problem. We assume $d$ is an upper bound on how many warehouses can serve any demand location and that stochastic arrivals satisfy either temporal or spatial independence. The crux of our theoretical contribution is to use randomized rounding to derive a tight $(1-(1-1/d)^d)$-approximation for the integer programming problem of optimizing the Offline surrogate. We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, does indeed have vanishing loss. On the experimental side, we extract real-world sequences of customer orders from publicly-available JD.com data and evaluate different combinations of placement and fulfillment procedures. Optimizing the Offline surrogate performs best overall, even compared to simulation procedures, corroborating our theory.

Problem

• Optimizing inventory placement across warehouses for e-commerce fulfillment
• Comparing performance of Offline, Myopic, and Fluid placement procedures
• Deriving approximation guarantees for joint placement and fulfillment problem

Innovation

• Optimizing Offline surrogate for inventory placement
• Using randomized rounding for approximation
• Statistical learning for sample-average optimization