Approximation Algorithms for the Joint Replenishment Problem with Deadlines

TL;DR

This paper investigates the approximability of the Joint Replenishment Problem with deadlines, providing new bounds on integrality gaps and approximation ratios, and establishing hardness results for special cases.

Contribution

It introduces improved bounds on the integrality gap and approximation ratio for JRP-D, and proves APX-hardness for the equal-length demand periods case.

Findings

Lower bound of 1.207 on integrality gap

Upper bound and approximation ratio of 1.574

APX-hardness for equal-length demand periods

Abstract

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness.