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
