After a brief introduction on single item lot-sizing, we present tight or tighter formulations for a couple of single and multi-item constant capacity lot-sizing variants.

We then examinehow reformulation resultsforbasic(typically) single-itemprob­lems can be used or extended to tackle realistic multi-item, multi-site problems. As a first application we consider a two-level supply chain, namely a multi-item n-period model with production sites and sales areas with production at the sites and transportation to the areas. We demonstrate the effectiveness of a multi-commodity reformulation combined with the use of inequalities for family set-ups when there are capacities at the transportation level..

The second application is a multi-item parallel machine model with start-up and/orsequencedependent changeovercosts. Hereweshowhowtoextend exist­ing valid inequalities and formulations to deal with identical parallel machines. Some limited computational results are presented.