Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem

• Published in: European journal of operational research Vol. 233; no. 1; pp. 43 - 63
• Main Authors: Alves, Cláudio, de Carvalho, José Valério, Clautiaux, François, Rietz, Jürgen
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16.02.2014; Elsevier Sequoia S.A; Elsevier
• Subjects: Approximation; Boundaries; Branch & bound algorithms; Combinatorial analysis; Computer Science; Dual-feasible functions; Fast lower bounds; Lower bounds; Mathematical analysis; Mathematical functions; Mathematical models; Operations Research; Optimization; Optimization algorithms; Packing problem; Production planning; Studies; Transportation; Vector packing problem; Vector space; Vectors (mathematics); Vector packing problem; Dual-feasible functions; Fast lower bounds
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2013.08.011

•We propose new m-dimensional dual-feasible functions for the vector packing problem.•We describe lower bounding procedures based on these functions.•The problem arises in areas such as telecommunications, transportation and production planning.•Our procedures generate strong lower bounds and improve the convergence of branch-and-bound algorithms. In this paper, we address the 2-dimensional vector packing problem where an optimal layout for a set of items with two independent dimensions has to be found within the boundaries of a rectangle. Many practical applications in areas such as the telecommunications, transportation and production planning lead to this combinatorial problem. Here, we focus on the computation of fast lower bounds using original approaches based on the concept of dual-feasible functions. Until now, all the dual-feasible functions proposed in the literature were 1-dimensional functions. In this paper, we extend the principles of dual-feasible functions to the m-dimensional case by introducing the concept of vector packing dual-feasible function, and we propose and analyze different new families of functions. All the proposed approaches were tested extensively using benchmark instances described in the literature. Our computational results show that these functions can approximate very efficiently the best known lower bounds for this problem and improve significantly the convergence of branch-and-bound algorithms.