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Improved Approximation Guarantees for Joint Replenishment in Continuous Time

Danny Segev

TL;DR

This work advances the theory of deterministic continuous-time joint replenishment by breaking through four decades of power-of-2 policy guarantees. It delivers a variable-base $EPTAS$ with near-optimal costs via $\Psi$-pairwise alignment, and a black-box reduction that transfer guarantees to the fixed-base setting, achieving $1/(\sqrt{2}\ln 2)+\varepsilon$. It further shows that evenly-spaced (integer-ratio) policies can achieve a better bound, $1.01915$, resolving Roundy’s conjecture in the affirmative, and extends the framework to resource-constrained JRPs with a $1.417$-approximation (and a PTAS when the number of resources $D$ is constant). Collectively, these results significantly improve prior guarantees, provide practically implementable policy families, and open new avenues for fine-grained approximations in multi-item inventory management.

Abstract

The primary objective of this work is to revisit and revitalize one of the most fundamental models in deterministic inventory management, the continuous-time joint replenishment problem. Our main contribution consists of resolving several long-standing open questions in this context. For most of these questions, we obtain the first quantitative improvement over power-of-$2$ policies and their nearby derivatives, which have been state-of-the-art in terms of provable performance guarantees since the mid-80's.

Improved Approximation Guarantees for Joint Replenishment in Continuous Time

TL;DR

This work advances the theory of deterministic continuous-time joint replenishment by breaking through four decades of power-of-2 policy guarantees. It delivers a variable-base with near-optimal costs via -pairwise alignment, and a black-box reduction that transfer guarantees to the fixed-base setting, achieving . It further shows that evenly-spaced (integer-ratio) policies can achieve a better bound, , resolving Roundy’s conjecture in the affirmative, and extends the framework to resource-constrained JRPs with a -approximation (and a PTAS when the number of resources is constant). Collectively, these results significantly improve prior guarantees, provide practically implementable policy families, and open new avenues for fine-grained approximations in multi-item inventory management.

Abstract

The primary objective of this work is to revisit and revitalize one of the most fundamental models in deterministic inventory management, the continuous-time joint replenishment problem. Our main contribution consists of resolving several long-standing open questions in this context. For most of these questions, we obtain the first quantitative improvement over power-of- policies and their nearby derivatives, which have been state-of-the-art in terms of provable performance guarantees since the mid-80's.
Paper Structure (98 sections, 15 theorems, 68 equations)

This paper contains 98 sections, 15 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Model formulation
  3. The Economic Order Quantity model.
  4. The joint replenishment problem.
  5. The joint ordering term $\boldsymbol{J(T)}$.
  6. Known results and open questions
  7. The variable-base setting: Cornerstone algorithmic methods.
  8. The fixed-base setting: Best known algorithmic methods.
  9. Integer-ratio policies and Roundy's conjecture.
  10. Incorporating resource constraints.
  11. Hardness results.
  12. Main contributions
  13. Main result 1: The variable-base setting.
  14. Main result 2: The fixed-base setting.
  15. Main result 3: Integer-ratio policies and Roundy's conjecture.
  16. ...and 83 more sections

Key Result

Lemma 1.2

$\lim_{\Delta \to \infty} \frac{ N(T,\Delta) }{ \Delta } = \sum_{{\cal N} \subseteq [n]} \frac{ (-1)^{ |{\cal N}| + 1 } }{ M_{\cal N} }$.

Theorems & Definitions (27)

  • Claim 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.3
  • proof
  • Claim 2.4
  • ...and 17 more