The Continuous-Time Joint Replenishment Problem: $\epsilon$-Optimal Policies via Pairwise Alignment

TL;DR

This paper introduces a novel algorithmic approach called $ ext{ extbackslash Psi}$-pairwise alignment for the continuous-time joint replenishment problem, providing the first efficient approximation scheme and improving upon longstanding policies.

Contribution

It develops a new combinatorial algorithm for approximating optimal policies and resolves the open question of the problem's computational complexity.

Findings

First polynomial-time approximation scheme (EPTAS) for the problem

Improves upon power-of-2 policies since the 1980s

Provides the first quantitative improvement over previous policies

Abstract

The main contribution of this paper resides in developing a new algorithmic approach for addressing the continuous-time joint replenishment problem, termed $\Psi$-pairwise alignment. The latter mechanism, through which we synchronize multiple Economic Order Quantity models, allows us to devise a purely-combinatorial algorithm for efficiently approximating optimal policies within any degree of accuracy. As a result, our work constitutes the first quantitative improvement over power-of-$2$ policies, which have been state-of-the-art in this context since the mid-80's. Moreover, in light of recent intractability results, by proposing an efficient polynomial-time approximation scheme (EPTAS) for the joint replenishment problem, we resolve the long-standing open question regarding the computational complexity of this classical setting.