Discrete Optimization Flow shops with machine maintenance: Ordered and proportionate cases Byung-Cheon Choi a,1 , Kangbok Lee b, * , Joseph Y.-T. Leung c,2 , Michael L. Pinedo b,3 a Department of Business Administration, Chungnam National University 79 Daehangno, Yuseong-gu, Daejeon 305-704, Republic of Korea b Department of Information, Operations & Management Sciences, Stern School of Business, New York University, 44 West 4th Street, New York, NY 10012-1126, USA c Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA article info Article history: Received 7 August 2009 Accepted 13 April 2010 Available online 28 April 2010 Keywords: Ordered flow shop Proportionate flow shop Maintenance Computational complexity Approximation algorithm abstract We consider the m-machine ordered flow shop scheduling problem with machines subject to mainte- nance and with the makespan as objective. It is assumed that the maintenances are scheduled in advance and that the jobs are resumable. We consider permutation schedules and show that the problem is strongly NP-hard; it remains NP-hard in the ordinary sense even in the case of a single maintenance. We show that if the first (last) machine is the slowest and if maintenances occur only on the first (last) machine, then sequencing the jobs in the LPT (SPT) order yields an optimal schedule for the m-machine problem. As a special case of the ordered flow shop, we focus on the proportionate flow shop where the processing times of any given job on all the machines are identical. We prove that the proportionate flow shop problem with two maintenance periods is NP-hard, while the problem with a single maintenance period can be solved in polynomial time. Furthermore, we show that the optimal algorithm for the single maintenance case is a 3 2 -approximation algorithm for the two maintenance case. In our conclusion we dis- cuss also the computational complexity of other objective functions. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction We consider the m-machine, n-job flow shop problem with the machines subject to maintenance. Let p ij denote the processing time of job j on machine i. The objective is to find a sequence for the n jobs such that the overall completion time of the very last job, typically referred to as the makespan and denoted by C max , is minimized. In this paper we assume that the following relationships hold between the processing times of the jobs on the machines:  If, for some 1 6 h 6 m, p hj < p hk , then p ij 6 p ik for i = 1, ... , m.  If, for some 1 6 k 6 n, p ik < p hk , then p ij 6 p hj for j = 1, ... , n. A flow shop satisfying the above conditions is referred to as an ordered flow shop [29,30]. If the processing times of job j on all m machines are equal to p j , i.e., p ij = p j , then the flow shop is referred to as a proportionate flow shop [27]. Examples of job processing times in an ordered flow shop and in a proportionate flow shop are given below (see Table 1). We assume that a number of maintenance periods for the ma- chines have been scheduled in advance. The models considered are referred to in what follows, respectively, as ordered flow shops with machine maintenance and as proportionate flow shops with machine maintenance. There are many applications of the models described above in practice. First, consider a manufacturing environment for semicon- ductor and liquid crystal display (LCD) panels, where wafers and glasses are delivered in batches (referred to as foups in semicon- ductor manufacturing and as cassettes in LCD manufacturing). The manufacturing line consists of a series of stages and the pro- cess at each stage is executed in a batch mode. In this setting, the number of wafers belonging to a foup affects the processing time of the foup on each machine. Specifically, the processing time of a batch is proportional to the number of wafers in the batch. Therefore, the larger the number of wafers in a foup, the longer the processing time of the foup at every stage. Furthermore, each stage has a unique time for the processing of a wafer and thus, the longer the processing time of a wafer at a particular stage, the longer the processing time of a foup at that stage. This charac- teristic can be regarded as the ordered property, as defined above. For some reason, the processing time of a wafer at each stage may be identical. When the material handling time of a wafer is domi- nant in comparison to the processing time of a wafer, then the pre- vious wafer can be processed during the handling of the next wafer. Thus, the processing time of a foup is determined only by the number of wafers in a foup. This case may be considered a 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.04.018 * Corresponding author. E-mail addresses: polytime@cnu.ac.kr (B.-C. Choi), klee3@stern.nyu.edu (K. Lee), leung@cis.njit.edu (J.Y.-T. Leung), mpinedo@stern.nyu.edu (M.L. Pinedo). 1 Work supported by the Korea Research Foundation Grant KRF-2008-357-D00289. 2 Work supported in part by the NSF Grant DMI-0556010. 3 Work supported in part by the NSF Grant DMI-0555999. European Journal of Operational Research 207 (2010) 97–104 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor