Approximating the Joint replenishment Problem with Deadlines

The objective of the classical Joint Replenishment Problem (JRP) is to minimize ordering costs by combining orders in two stages, first at some retailers, and then at a warehouse. These orders are needed to satisfy demands that appear over time at the retailers. We investigate the natural special case that each demand has a deadline until when it needs to be satisfied. For this case, we present a randomized 5/3-approximation algorithm. We moreover prove that JRP with deadlines is APX-hard. Finally, we extend the known hardness results by showing that JRP with linear delay cost functions is NP-hard, even if each retailer has to satisfy only three demands.