* Corresponding author. Tel./fax: +98-21-77240482. E-mail addresses: Reza_Ramezanian@ind.iust.ac.ir (R. Ramezanian), © 2010 Growing Science Ltd. All rights reserved. doi: 10.5267/j.ijiec.2010.01.001 International Journal of Industrial Engineering Computations 1 (2010) 1–10 Contents lists available at GrowingScience International Journal of Industrial Engineering Computations homepage: www.GrowingScience.com/ijiec A discrete firefly meta-heuristic with local search for makespan minimization in permutation flow shop scheduling problems Mohammad Kazem Sayadi a , Reza Ramezanian a* and Nader Ghaffari-Nasab a a Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran A R T I C L E I N F O A B S T R A C T Article history: Received 23 January 2010 Received in revised form 23 April 2010 Accepted 26 April 2010 Available online 26 April 2010 During the past two decades, there have been increasing interests on permutation flow shop with different types of objective functions such as minimizing the makespan, the weighted mean flow-time etc. The permutation flow shop is formulated as a mixed integer programming and it is classified as NP-Hard problem. Therefore, a direct solution is not available and meta- heuristic approaches need to be used to find the near-optimal solutions. In this paper, we present a new discrete firefly meta-heuristic to minimize the makespan for the permutation flow shop scheduling problem. The results of implementation of the proposed method are compared with other existing ant colony optimization technique. The preliminary results indicate that the new proposed method performs better than the ant colony for some well known benchmark problems. © 2010 Growing Science Ltd. All rights reserved. Keywords: Meta-heuristic Firefly meta-heuristic Ant colony Permutation flow shop Scheduling Combinatorial optimization Mixed integer programming 1. Introduction The flow shop scheduling problem (FSSP) is normally classified as a complex combinatorial optimization problem, in which there is a set of n jobs (1, …, n) to be processed in a set of m machines (1, …, m) in the same order. We normally look for a special sequence of processing the jobs in the machines to minimize one or more criteria such as minimization of makespan, mean flow, etc. There are different most commonly used criteria such as the minimization of the total completion time or makespan of the schedule ( max C ) which is sometimes referred to as maximum flow time or max F . The processing times needed for the jobs on the machines are assumed to be non-negative and deterministic denoted as ij p , with i = 1 …,n and j = 1,…,m. Although the optimal solution of the flow shop scheduling problem can be determined in polynomial time when m=2 (Johnson, 1954), the general form of this kind of problem is known to be NP-Complete in the strong sense when m≥3 (see Garey et al., 1976) and generally m n ) ! ( schedules need to be considered. That is why the problem is somewhat restricted in by not allowing job passing. In this case, “only” n! schedules must be considered and the problem is then known as permutation flow shop which is classified as n/m/P/ max F or as F/prmu/ max C (see Pinedo, 2002) and the primary focus of the work of this paper is the last type of flow shop environment. Johnson (1954) is believed to be the first who introduced flow shop scheduling. Since then, flow shop scheduling has become one of the most interesting topics among researchers and practitioners. There are