Discrete Optimization The scheduling problem of PCBs for multiple non-identical parallel machines Nguyen Van Hop a , Nagendra N. Nagarur b, * a Industrial Engineering Program, Sirindhorn International Institute of Technology, Thammasat University, P.O. Box 22, Thammasat-Rangsit Post Office, Klong Luang, Pathumthani 12121, Thailand b Department of System Science and Industrial Engineering, Thomas Watson School of Engineering and Applied Science, Binghamton University, Binghamton, NY 13902-6000, USA Received 21 May 2001; accepted 5 May 2003 Available online 24 November 2003 Abstract The scheduling problem of n printed circuit boards (PCBs) for m non-identical parallel machines is considered in this paper. The problem has to deal with three issues: (i) classifying the PCBs into m groups corresponding to m machines, (ii) sequencing the boards for each machine, and (iii) component switching (component unloading/loading) from the machine magazine. A general objective is to minimize the total makespan, which is shown here to be the same as minimizing the maximum of number of component switches. The complete problem is complex, and is usually dealt in stages, which may not yield a good solution. We model the problem in an integrated manner using weighted multiple objectives to deal with grouping of the boards, load balancing at each machine, board sequencing and component switching at a machine. A composite genetic algorithm is developed to solve this multi-objective problem. The inte- grated solution is encoded as a string of pair values for each group of boards. The first number indicates the board membership in a group, and the second one represents the sequencing position of a board in that group. A new population of solutions is generated by using both binary genetic operators for grouping and genetic operators for board sequencing. A fitness function evaluates workload balancing, board similarities and total setup time simulta- neously. Experiments are designed and run to test the proposed methodology, and the results show that the solutions are efficient, and are obtained within a reasonable amount of time. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Scheduling; Genetic algorithm; PCB assembly; Non-identical parallel machines; Multiple criteria 1. Introduction The modern era technology is dominated by microchips, chips for computers, control systems, data communications, and many consumer products. Assembly of these chips onto printed circuit boards * Corresponding author. Tel.: +1-607-7773027; fax: +1-607-7774094. E-mail addresses: vanhop@siit.tu.ac.th (N. Van Hop), nnagarur@binghamton.edu (N.N. Nagarur). 0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00376-X European Journal of Operational Research 158 (2004) 577–594 www.elsevier.com/locate/dsw