Sung-Pil Hong (홍성필), Approximability of a 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 batch 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. We focuse on the special case in which up to 2 items can be processed in a single batch. The problem is NP-hard and can not be approximated within 1.00142 of the optimum under the premise, P ≠ NP as can be shown by a polynomial reduction from the vertex cover problem with bounded degree. However, the problem admits a 3/2 -approximation. The idea is to decompose the orders of items so that a maximum matching in the graph on the vertices of the decomposed orders provides a well-consolidated batch set.