Approximation of k-batch consolidation problem

We consider a problem of minimizing the number of batches of a fixed capacity processing the orders of various sizes on a finite set of items. This batch consolidation problem is motivated by the production system typical in raw material industries in which multiple items can be processed in the same batch in case they share sufficiently close production parameters. If the number of items processed in a batch is restricted up to some fixed integer k, the problem is referred to as the k-batch consolidation problem. We will show that the k-batch consolidation problem admits an approximation whose factor is twice that of the k-set cover problem. In particular, this implies an upperbound on the approximation factor, 2Hk − 1, where Hk = 1 + 1 2 + · · ·+ 1 k .