Discrete Optimization A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times Manuel J. Pereira Lopes a , J.M. Vale ´rio de Carvalho b, * a Dept. de Engenharia Meca ˆnica, Instituto Superior de Engenharia do Porto, Portugal b Dept. de Produc ¸a ˜o e Sistemas, Universidade do Minho, Portugal Received 30 August 2004; accepted 9 November 2005 Available online 25 January 2006 Abstract We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-depen- dent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed ‘‘primal box’’, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computa- tional time. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Parallel machines scheduling; Setup times; Column generation; Branch-and-price 1. Introduction The study of parallel machine problems is relevant from both the theoretical and the practical points of view. From the practical point of view, it is important because we can find many examples of the use of parallel machines in the real world. The motivation for this work was a real textile industry problem involv- ing the production of tissues of different colors on unrelated parallel machines (the production bottleneck) with long sequence dependent setup times and availability dates for the machines and release dates for the 0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.001 * Corresponding author. E-mail addresses: mpl@isep.ipp.pt (M.J. Pereira Lopes), vc@dps.uminho.pt (J.M. Vale ´rio de Carvalho). European Journal of Operational Research 176 (2007) 1508–1527 www.elsevier.com/locate/ejor