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Economic Warehouse Lot Scheduling: Approximation Schemes via Efficiently-Representable DP-Encoded Policies

Danny Segev

TL;DR

The paper tackles the economic warehouse lot scheduling problem by developing a dynamic-programming framework that yields near-optimal, capacity-feasible replenishment policies. It introduces a novel structure of B-aligned policies and frequency classes to obtain a controlled, finite policy space, enabling a polynomial-time approximation scheme (PTAS) for a constant number of commodities with running time $O(|\mathcal{I}|^{O(n)} 2^{O(n^{6}/\epsilon^{5})})$ and efficient policy representation. The main technical milestones are the existence of low-cost short cycles (Milestone I) and the Alignment Theorem (Milestone II), which together underpin the DP-based approximation scheme and its optimal substructure. The results advance both theory and potential practice by breaking the longstanding $2$-approximation barrier for general instances in follow-up work and by offering a principled route toward scalable, provable-quality replenishment policies in multi-commodity settings.

Abstract

In this focused technical paper, we present long-awaited algorithmic advances toward the efficient construction of near-optimal replenishment policies for a true inventory management classic, the economic warehouse lot scheduling problem. While this paradigm has accumulated a massive body of surrounding literature since its inception in the late '50s, we are still very much in the dark as far as basic computational questions are concerned, perhaps due to the intrinsic complexity of dynamic policies in this context. The latter feature forced earlier attempts to either study highly-structured classes of policies or to forgo provably-good performance guarantees altogether; to this day, rigorously analyzable results have been few and far between. The current paper develops novel analytical foundations for directly competing against dynamic policies. Combined with further algorithmic progress and newly-gained insights, these ideas culminate in a polynomial-time approximation scheme for constantly-many commodities. In this regard, the efficient design of $ε$-optimal dynamic policies appeared to have been out of reach, since beyond their inherent algorithmic challenges, even the polynomial-space representation of such policies has been a fundamental open question.

Economic Warehouse Lot Scheduling: Approximation Schemes via Efficiently-Representable DP-Encoded Policies

TL;DR

The paper tackles the economic warehouse lot scheduling problem by developing a dynamic-programming framework that yields near-optimal, capacity-feasible replenishment policies. It introduces a novel structure of B-aligned policies and frequency classes to obtain a controlled, finite policy space, enabling a polynomial-time approximation scheme (PTAS) for a constant number of commodities with running time and efficient policy representation. The main technical milestones are the existence of low-cost short cycles (Milestone I) and the Alignment Theorem (Milestone II), which together underpin the DP-based approximation scheme and its optimal substructure. The results advance both theory and potential practice by breaking the longstanding -approximation barrier for general instances in follow-up work and by offering a principled route toward scalable, provable-quality replenishment policies in multi-commodity settings.

Abstract

In this focused technical paper, we present long-awaited algorithmic advances toward the efficient construction of near-optimal replenishment policies for a true inventory management classic, the economic warehouse lot scheduling problem. While this paradigm has accumulated a massive body of surrounding literature since its inception in the late '50s, we are still very much in the dark as far as basic computational questions are concerned, perhaps due to the intrinsic complexity of dynamic policies in this context. The latter feature forced earlier attempts to either study highly-structured classes of policies or to forgo provably-good performance guarantees altogether; to this day, rigorously analyzable results have been few and far between. The current paper develops novel analytical foundations for directly competing against dynamic policies. Combined with further algorithmic progress and newly-gained insights, these ideas culminate in a polynomial-time approximation scheme for constantly-many commodities. In this regard, the efficient design of -optimal dynamic policies appeared to have been out of reach, since beyond their inherent algorithmic challenges, even the polynomial-space representation of such policies has been a fundamental open question.
Paper Structure (52 sections, 4 theorems, 18 equations, 2 figures)

This paper contains 52 sections, 4 theorems, 18 equations, 2 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.
  7. Representation issues.
  8. Constant-factor approximations via SOSI policies.
  9. Main open questions.
  10. Main contributions
  11. Toward a Dynamic Programming Approach
  12. Outline.
  13. Milestone I: Existence of low-cost short-cycle policies
  14. Proof of Theorem \ref{['thm:good_cyclic']}
  15. Constructing ${\tilde{\cal P}}$.
  16. ...and 37 more sections

Key Result

Theorem 1.2

For any $\epsilon > 0$, the economic warehouse lot scheduling problem can be approximated within factor $1 + \epsilon$ of optimal. Our algorithm admits an $O( | {\cal I} |^{O(n)} \cdot 2^{ O( n^{6} / \epsilon^{5} ) } )$-time implementation, where $| {\cal I} |$ stands for the input length in its bin

Figures (2)

  • Figure 1: Visual representation of a $B$-aligned policy with respect to our nested breakpoint structure. The inventory curve reaches zero at every coarse breakpoint ($B_q^-$), placing orders only at fine breakpoints ($B_q^+$).
  • Figure 2: Visual representation of our state description $(q, {\cal I}_{q-1}, \mathrm{L}_{ \leq q-2 })$. The gray region corresponds to the space reserved by lower-frequency commodities. The blue sawtooth curve captures the aggregate inventory levels across all $\tilde{\cal F}_{q-1}$-commodities, which are replenished at $B_{q-1}^+$-points.

Theorems & Definitions (8)

  • Claim 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Claim 2.2
  • Theorem 2.3: Alignment
  • Claim 2.4
  • Claim 2.5
  • Lemma 3.1