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Economic Warehouse Lot Scheduling: Breaking the 2-Approximation Barrier

Danny Segev

TL;DR

This work tackles the long-standing challenge of the economic warehouse lot scheduling problem (EWLP) by moving beyond the SOSI framework and directly comparing against fully dynamic replenishment policies. The authors develop a structural decomposition into volume classes and devise a novel suffix+dense coordination mechanism, including a matching-based mimicking partition and the Po2-Synchronization theorem, to balance long-run cost with capacity constraints. They prove the first sub-2-approximation for general EWLP instances by constructing a polynomial-time random capacity-feasible policy with expected cost at most (2 − 17/5000 + ε) times OPT. This result not only closes a decades-long gap but also opens avenues for strategic formulations, efficient implementations, and possible derandomization, significantly advancing capacity-constrained inventory theory and its algorithmic toolkit.

Abstract

The economic warehouse lot scheduling problem is a foundational inventory-theory model, capturing computational challenges in dynamically coordinating replenishment decisions for multiple commodities subject to a shared capacity constraint. Even though this model has generated a vast body of literature over the last six decades, our algorithmic understanding has remained surprisingly limited. Indeed, for general problem instances, the best-known approximation guarantees have remained at a factor of $2$ since the mid-1990s. These guarantees were attained by the now-classic work of Anily [Operations Research, 1991] and Gallego, Queyranne, and Simchi-Levi [Operations Research, 1996] via the highly-structured class of "stationary order sizes and stationary intervals" (SOSI) policies, thereby avoiding direct competition against fully dynamic policies. The main contribution of this paper resides in developing new analytical foundations and algorithmic techniques that enable such direct comparisons, leading to the first provable improvement over the $2$-approximation barrier. Leveraging these ideas, we design a constructive approach that allows us to balance cost and capacity at a finer granularity than previously possible via SOSI-based methods. Consequently, given any economic warehouse lot scheduling instance, we present a polynomial-time construction of a random capacity-feasible dynamic policy whose expected long-run average cost is within factor $2-\frac{17}{5000} + ε$ of optimal.

Economic Warehouse Lot Scheduling: Breaking the 2-Approximation Barrier

TL;DR

This work tackles the long-standing challenge of the economic warehouse lot scheduling problem (EWLP) by moving beyond the SOSI framework and directly comparing against fully dynamic replenishment policies. The authors develop a structural decomposition into volume classes and devise a novel suffix+dense coordination mechanism, including a matching-based mimicking partition and the Po2-Synchronization theorem, to balance long-run cost with capacity constraints. They prove the first sub-2-approximation for general EWLP instances by constructing a polynomial-time random capacity-feasible policy with expected cost at most (2 − 17/5000 + ε) times OPT. This result not only closes a decades-long gap but also opens avenues for strategic formulations, efficient implementations, and possible derandomization, significantly advancing capacity-constrained inventory theory and its algorithmic toolkit.

Abstract

The economic warehouse lot scheduling problem is a foundational inventory-theory model, capturing computational challenges in dynamically coordinating replenishment decisions for multiple commodities subject to a shared capacity constraint. Even though this model has generated a vast body of literature over the last six decades, our algorithmic understanding has remained surprisingly limited. Indeed, for general problem instances, the best-known approximation guarantees have remained at a factor of since the mid-1990s. These guarantees were attained by the now-classic work of Anily [Operations Research, 1991] and Gallego, Queyranne, and Simchi-Levi [Operations Research, 1996] via the highly-structured class of "stationary order sizes and stationary intervals" (SOSI) policies, thereby avoiding direct competition against fully dynamic policies. The main contribution of this paper resides in developing new analytical foundations and algorithmic techniques that enable such direct comparisons, leading to the first provable improvement over the -approximation barrier. Leveraging these ideas, we design a constructive approach that allows us to balance cost and capacity at a finer granularity than previously possible via SOSI-based methods. Consequently, given any economic warehouse lot scheduling instance, we present a polynomial-time construction of a random capacity-feasible dynamic policy whose expected long-run average cost is within factor of optimal.
Paper Structure (97 sections, 18 theorems, 53 equations, 6 figures)

This paper contains 97 sections, 18 theorems, 53 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model formulation
  3. The economic order quantity model.
  4. The economic warehouse lot scheduling problem.
  5. Known results and open questions
  6. Hardness results and representational issues.
  7. Constant-factor approximations via SOSI policies.
  8. Recent approximation scheme for $O(1)$ commodities.
  9. Main open questions.
  10. Main contributions
  11. Technical highlights.
  12. Background: Revisiting the Classical Approximation Framework
  13. The average-space bound and its induced relaxation
  14. The average-space bound.
  15. The resulting relaxation.
  16. ...and 82 more sections

Key Result

Theorem 1.2

For any $\epsilon \in (0,\frac{ 1 }{ 10 })$, we can compute in $O( | {\cal I} |^{\tilde{O}( 1/\epsilon^5)} \cdot 2^{ \tilde{O}( 1 / \epsilon^{35} ) } )$ time a random capacity-feasible replenishment policy ${\cal P}$, with an expected long-run average cost of Here, $| {\cal I} |$ stands for the input length in its binary specification.

Figures (6)

  • Figure 1: Case 1
  • Figure 2: Case 2
  • Figure 3: Case 3
  • Figure 4: Case 4
  • Figure 5: Case 5
  • ...and 1 more figures

Theorems & Definitions (22)

  • Claim 1.1
  • Theorem 1.2
  • Corollary 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • Lemma 4.2
  • ...and 12 more