Discrete Optimization Two-machine flow shop scheduling problem with an outsourcing option Byung-Cheon Choi a,1 , Jibok Chung b,⇑ a Department of Business Administration, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-704, Republic of Korea b Department of Business Administration, Daejeon University, 96-3 Yongun-dong, Dong-gu, Daejeon 300-716, Republic of Korea article info Article history: Received 18 June 2010 Accepted 8 March 2011 Available online 16 March 2011 Keywords: Scheduling Outsourcing Ordered flow shop Computational complexity abstract We consider a two-machine flow shop problem in which each job is processed through an in-house sys- tem or outsourced to a subcontractor. A schedule is established for the in-house jobs, and performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and total outsourcing costs. We show that the problem is NP-hard in the ordinary sense. We consider a special case in which each job has a processing require- ment, and each machine a characteristic value. In this case, the time a job occupies a machine is equal to the job’s processing requirement plus a setup time equal to the characteristic value of that machine. We introduce some optimality conditions and present a polynomial-time algorithm to solve the special case. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction In the modern business environment, many companies out- source their jobs, that is, they entrust some of their jobs to a third party instead of managing them all directly. Proper outsourcing can shorten lead times, reduce total costs, and make an organiza- tion more flexible. Thus, well-utilized outsourcing can make a company more competitive (Cachon and Harker, 2002; Webster et al., 1997), which provides the authors with motivation to con- sider a scheduling problem with an outsourcing option. This paper considers a two-stage supply chain model with an outsourcing option. Each job comprises two operations, where each job can be outsourced to an outside subcontractor or directly processed with in-house resources at each stage. For in-house operations, a schedule is constructed and its performance is mea- sured by the makespan, that is, the latest completion time of oper- ations processed in-house. Jobs processed by subcontractors require paying an outsourcing cost. The problem can therefore be regarded as two-machine flow shop scheduling with an outsourc- ing option. Our problem considers the environment wherein when a job is outsourced, two operations comprising the job must be outsourced. The objective is to minimize the sum of the makespan and total outsourcing costs. In this paper, n denotes the number of jobs; p i,j refers to the processing requirement of job j on machine i and q j is the cost for outsourcing job j; O and I are the sets of outsourced and in-house jobs, respectively; and r is a sequence of jobs belonging to I. The problem considered in this paper is defined below. Problem P. In a two-machine n-job flow shop, determine a schedule (r, O) that minimizes T(r, O), defined as T ðr; OÞ¼ C max ðrÞþ X j2O q j : Since Problem P is NP-hard (the proof will be presented in Section 3), we consider a special case with p i,j = p j + v i , where v i is the char- acteristic value of machine i. Let that special case be referred to as Problem P 0 . The motivation of Problem P 0 stems from a setting in which a machine has to be set up before it can start processing a new job. Problem P 0 is a special case of a more general type of flow shop that has already received a fair amount of attention in the lit- erature, namely, the ordered flow shop in Smith et al. (1975, 1976). A flow shop is an ordered flow shop when the following conditions hold: if, for some 1 6 k 6 m, p ki 6 p kj , then p li 6 p lj for l = 1, ... , m, and if p kl 6 p il , then p kj 6 p ij for j = 1, ... , n. The remainder of the paper is organized as follows. Section 2 discusses the related literature. Section 3 shows that Problem P is NP-hard in the ordinary sense. Section 4 shows that Problem P 0 can be solved in polynomial time. Finally, Section 5 presents our conclusions. 2. Literature review Since Johnson’s seminal paper of 1954, which analyzed the makespan minimization in a two-machine flow shop, many researchers have studied flow shop problems. For surveys of flow shop scheduling research, the reader is referred to Hejazi and Saghafian (2005) and Cheng et al. (2000). It has been shown that 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.03.017 ⇑ Corresponding author. Tel.: +82 42 280 2337; fax: +82 42 283 7171. E-mail addresses: polytime@cnu.ac.kr (B.-C. Choi), jbchung@dju.ac.kr (J. Chung). 1 This study was financially supported by the research fund of Chungnam National University in 2010. European Journal of Operational Research 213 (2011) 66–72 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor