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Optimizing Inventory Placement for a Downstream Online Matching Problem

Boris Epstein, Will Ma

TL;DR

This work addresses the two-stage problem of pre-allocating inventory across multiple warehouses before downstream online fulfillment decisions in e-commerce. It analyzes three surrogate-based placement strategies (Offline, Myopic, Fluid) and proves that, when paired with a high-quality fulfillment policy, Offline placement yields a constant-factor approximation to the optimal joint performance, formalized as $\alpha\big(1-(1-1/d)^d\big)$ with $d$ the maximum demand-feeder degree. The core technical contribution is a tight randomized rounding scheme for the Offline surrogate, plus a sample-average approximation to handle exponentially large demand support, and extends these results to multi-SKU scenarios. Complementing theory, extensive synthetic experiments show Offline placement with a strong fulfillment policy consistently outperforms alternatives, demonstrating practical value even when using sampling-based estimates. Overall, the paper provides scalable, provable guidance for coordinating inventory placement and fulfillment in online matching settings, with clear implications for multi-SKU networks and demand-uncertainty environments.

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 theoretically, due to the computational complexity of the downstream matching problem, and practically, as the fulfillment team continuously updates its algorithm while the placement team lacks direct evaluation of placement decisions. 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 an $α(1-(1-1/d)^d)$-approximation for the joint placement and fulfillment problem under any demand model that admits an $α$-competitive fulfillment policy. We assume $d$ is an upper bound on how many warehouses can serve any demand location. 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 further show how to extend this result to a multi-SKU setting, improving upon the best known approximation of $1/2$. We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, indeed has vanishing loss. On the experimental side, we evaluate how different combinations of placement and fulfillment procedures perform on a wide array of synthetic instances. When coupled with a good fulfillment procedure, optimizing the Offline surrogate performs best even compared to computationally-intensive simulation procedures, corroborating our theory.

Optimizing Inventory Placement for a Downstream Online Matching Problem

TL;DR

This work addresses the two-stage problem of pre-allocating inventory across multiple warehouses before downstream online fulfillment decisions in e-commerce. It analyzes three surrogate-based placement strategies (Offline, Myopic, Fluid) and proves that, when paired with a high-quality fulfillment policy, Offline placement yields a constant-factor approximation to the optimal joint performance, formalized as with the maximum demand-feeder degree. The core technical contribution is a tight randomized rounding scheme for the Offline surrogate, plus a sample-average approximation to handle exponentially large demand support, and extends these results to multi-SKU scenarios. Complementing theory, extensive synthetic experiments show Offline placement with a strong fulfillment policy consistently outperforms alternatives, demonstrating practical value even when using sampling-based estimates. Overall, the paper provides scalable, provable guidance for coordinating inventory placement and fulfillment in online matching settings, with clear implications for multi-SKU networks and demand-uncertainty environments.

Abstract

We study the inventory placement problem of splitting 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 theoretically, due to the computational complexity of the downstream matching problem, and practically, as the fulfillment team continuously updates its algorithm while the placement team lacks direct evaluation of placement decisions. 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 an -approximation for the joint placement and fulfillment problem under any demand model that admits an -competitive fulfillment policy. We assume is an upper bound on how many warehouses can serve any demand location. The crux of our theoretical contribution is to use randomized rounding to derive a tight -approximation for the integer programming problem of optimizing the Offline surrogate. We further show how to extend this result to a multi-SKU setting, improving upon the best known approximation of . We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, indeed has vanishing loss. On the experimental side, we evaluate how different combinations of placement and fulfillment procedures perform on a wide array of synthetic instances. When coupled with a good fulfillment procedure, optimizing the Offline surrogate performs best even compared to computationally-intensive simulation procedures, corroborating our theory.
Paper Structure (37 sections, 11 theorems, 64 equations, 2 figures, 2 tables)

This paper contains 37 sections, 11 theorems, 64 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Results and Techniques
  3. Numerical Results From Simulations
  4. Further Related Work
  5. Problem Statement
  6. Optimizing the Offline Surrogate
  7. Randomized Rounding
  8. Sample Average Approximation
  9. Approximating the Joint Problem
  10. Assumptions on Demand Model
  11. Approximation for the Join Fulfillment and Placement Problem
  12. Extension of Randomized Rounding to Multiple SKUs Setting
  13. Problem Setting
  14. Rounding a Fraction Solution
  15. Experiments with Synthetic Data
  16. ...and 22 more sections

Key Result

Theorem 1

Consider an instance such that the maximum degree across demand nodes is $d$. For any fractional inventory placement $x\in CH(\mathcal{X})$, and any rounding procedure $R$ that satisfies Properties (P1)-(P3), it holds that

Figures (2)

  • Figure 1: Competitive ratio as a function of total inventory, for all combinations of graphs and demand models with uniform weights. Each row corresponds to a different network structure and each column corresponds to a different demand model.
  • Figure 2: Competitive ratio as a function of total inventory for all combinations of graphs and demand models with reward-based weights. Each row corresponds to a different network structure and each column corresponds to a different demand model.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Claim 1
  • Claim 2
  • Lemma 4
  • ...and 3 more