Makespan minimization in machine dominated flowshop Gopalakrishnan Easwaran a, * , Larry E. Parten a,1 , Rafael Moras a , Paul X. Uhlig b a Department of Engineering, St. Mary’s University, One Camino Santa Maria, San Antonio, TX 78228-8534, United States b Department of Mathematics, St. Mary’s University, One Camino Santa Maria, San Antonio, TX 78228-8560, United States article info Keywords: Scheduling Combinatorial optimization Flowshop Machine dominance Makespan abstract In a flowshop scheduling problem, a set of jobs is processed by a set of machines. The jobs follow the same sequence in all machines. We study the flowshop scheduling problem under a new case of machine dominance that is often found in the manufacturing of com- puters and electronic devices. We provide a formula for makespan value for a given sequence, show that the makespan value depends only on certain jobs in the sequence, and present an algorithm that finds a sequence with minimum makespan. Numerical examples of the solution approaches are provided. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Flowshops are frequently found in industry and are characterized by a set of jobs, J¼f1; 2; ... ng, and a set of machines, M¼f1; 2; ... mg. The set of n jobs is processed sequentially on m machines. In the traditional flowshop problem, we assume deterministic processing times, denoted by p ij , where indices i 2M and j 2J represent a machine and a job, respectively. Furthermore, all jobs are ready for processing at time zero and no other jobs arrive later; a job may not be preempted by another job; jobs are not allowed to pass others; no job may be processed by more than one machine; machines may process no more than one job at a time; and there are no down times due to machine breakdown or maintenance. In a flowshop problem, we usually determine the sequence of jobs to satisfy certain performance criteria including the min- imization of makespan C max (the completion time of the last job in the sequence), the sum (or the mean) of the job flowtimes (time in the system for each job), the mean tardiness or lateness, the maximum tardiness, and the number of tardy or late jobs. Most flowshop problems have been proven to be NP-hard [1]. The seminal work by Johnson [2], in which a tractable algo- rithm for the minimization of a two-machine flowshop was presented, is perhaps the most notable exception. The pervasive intractability of flowshop problems tends to justify the research efforts to find special cases that would make the solution easy (polynomially bounded), without trivializing applicability of the hypothesis. In this context, three types of machine dominance in flowshops have been proposed in the literature, which we summarize in the next section. 2. Literature review For notational convenience, we define LB i ¼ min j2J fp ij g and UB i ¼ max j2J fp ij g for each machine i 2M. The Type I dominance is characterized by one of the following nested inequalities: Case A: An increasing series of domination (idm), that is, LB 1 < UB 1 6 LB 2 < UB 2 6  6 LB m1 < UB m1 6 LB m < UB m : ð1Þ 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.022 * Corresponding author. E-mail address: geaswaran@stmarytx.edu (G. Easwaran). 1 Graduate Student. Applied Mathematics and Computation 217 (2010) 110–116 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc