The Bin Packing Problem with Precedence Constraints Mauro Dell’Amico ‡ , Jos´ e Carlos D´ ıaz D´ ıaz ‡ and Manuel Iori ‡ ‡ DISMI, University of Modena and Reggio Emilia, Via Amendola 2, 42100 Reggio Emilia, Italy E-mails: {mauro.dellamico, manuel.iori, jose.diaz}@unimore.it Abstract Given a set of identical capacitated bins, a set of weighted items and a set of precedences among such items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem, denoted as the Bin Packing Problem with Precedence Constraints (BPP-P), has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge the BPP-P has received little attention in the literature, and in this paper we address it for the first time with exact so- lution methods. In particular, we develop reduction criteria, a large set of lower bounds, a Variable Neighborhood Search upper bounding technique and a branch-and-bound algorithm. We show the effectiveness of the proposed al- gorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques. Keywords: Bin Packing Problem, Precedence Constraints, Branch-and-Bound 1 Introduction Given n items each having non-negative weight w j (j =1, 2,...,n) and m bins of identical capacity C , the Bin Packing Problem (BPP) asks for the minimum number of bins that can accommodate all the items. Given a set of precedences among the items, the Bin Packing Problem with Precedence Constraints (BPP-P) is a generalization of the BPP that requires to order the bins in such a way that all precedences are satisfied. Intuitively speaking, precedences impose that the successors of an item are packed in bins that follow the bin accommodating such item, so as to model technical requirements arising in real-world applications. More formally, let us consider an acyclic digraph G 0 =(V 0 ,A 0 ) where V 0 = {1, 2,...,n} contains a vertex j for each item j of the BPP, and each arc (j, k) ∈ A 0 1