This paper considers the multi-item inventory lot-sizing problem with supplier selection. The problem consists in determining an optimal purchasing plan in order to satisfy dynamic deterministic demands for multiple items over a finite planning horizon, taking into account the fact that multiple suppliers are available to purchase from... As the complexity of the problem was an open question, we show that it is NP-hard. We propose a facility location extended formulation for the problem which can be preprocessed based on the cost structure and describe new valid inequalities in the original space of variables, which we denote $(l,S_j)$-inequalities. Furthermore, we study the projection of the extended formulation into the original space and show the connection between the inequalities generated by this projection and the newly proposed $(l,S_j)$-inequalities. Additionally, we present a simple and easy to implement yet very effective MIP (mixed integer programming) heuristic using the extended formulation. Computational results show that the preprocessed facility location extended formulation outperforms all other formulations for small and medium instances, as it can solve nearly all of them to optimality within the time limit. Moreover, the presented MIP heuristic is able to obtain solutions which strictly improve those achieved by a state-of-the art method for all the large benchmark instances. (read more)