Approximation algorithms for the joint replenishment problem with deadlines

Marcin Bienkowski, Jaroslaw Byrka, Marek Chrobak, Neil B. Dobbs, Tomasz Nowicki, Maxim Sviridenko, Grzegorz Swirszcz, Neal E. Young

Abstract: The Joint Replenishment Problem ( $${\hbox {JRP}}$$ 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 $${\hbox {JRP-D}}$$ JRP-D , the version of $${\hbox {JRP}}$$ 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 $$1.207$$ 1.207 , a stronger, computer-assisted lower bound of $$1.245$$ 1.245 , as well as an upper bound and approximation ratio of $$1.574$$ 1.574 . The best previous upper bound and approximation ratio was $$1.667$$ 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 $$1.5$$ 1.5 , a lower bound of $$1.2$$ 1.2 , and show APX-hardness.