Approximation algorithms for the joint replenishment problem with deadlines

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.