Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Minimizing total tardiness in a two-machine flowshop scheduling problem with availability constraint on the first machine Ju-Yong Lee a, ⁎ , Yeong-Dae Kim b a System Technology Team, Semiconductor Business, Samsung Electronics Co., Ltd., Yongin-Si, Gyeonggi-Do 17113, Republic of Korea b Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Yuseong-gu, Daejeon 34141, Republic of Korea ARTICLE INFO Keywords: Scheduling Flowshop Machine availability constraint Branch and bound algorithm Total tardiness ABSTRACT This paper deals with a two-machine flowshop problem in which the machine at the first stage requires pre- ventive maintenance activities that have to be started within a given cumulative working time limit after the previous maintenance. That is, a maintenance activity can be started at any time unless the cumulative working time after the end of the previous maintenance exceeds the given limit. For the problem with the objective of minimizing total tardiness, we develop dominance properties and lower bounds for this scheduling problem as well as a heuristic algorithm, and suggest a branch and bound algorithm in which these properties, lower bounds, and heuristic algorithm are used. Computational experiments are performed to evaluate the algorithm and the results are reported. 1. Introduction Over the past decades, a large number of researchers have studied flowshop scheduling problems. In a typical flowshop problem, m ma- chines are arranged in a series and n jobs visit the machines in the same order, that is, each job consists of m operations and the kth operation of all jobs are processed on machine k for k = 1, … , m. In most research on operations scheduling problems including flowshop problems, it is assumed that machines are continuously available at all times. However, this assumption oversimplifies the scheduling problems in real manufacturing systems since in reality there should be preventive maintenance, which has a significant impact on a variety of perfor- mance aspects such as productivity, reliability, and profitability. Note that delayed maintenance may lead to machine failure or deterioration in the quality of outputs. For these reasons, preventive maintenance tasks such as inspection, repair, replacement, cleaning, lubrication, adjustment and alignment are required to be performed for keeping machines in good condition and decrease the probability of machines’ failures (Cui & Lu, 2017). Therefore, operations scheduling with main- tenance is very important in operating manufacturing systems. This study investigates a scheduling problem of minimizing total tardiness of jobs in a two-machine flowshop. In this problem, the ma- chine in the first stage requires preventive maintenance within a given cumulative working time limit after the previous maintenance (since any delay of the maintenance activity increases the risk of machine failure significantly). That is, a maintenance activity can be (or should be) started at any time before the cumulative working time after the end of the previous maintenance exceeds a given time limit. This type of maintenance is called flexible maintenance since the start times of maintenance activities are not fixed but flexible. This scheduling pro- blem can be denoted by F2/fm/T according to the nomenclature of Graham, Lawler, Lenstra, and Rinnooy-Kan (1979), where F2, fm, and T represent the two-machine flowshop, flexible maintenance, and total tardiness, respectively. In the literature, constraints on machines due to maintenance activities are treated as a machine availability constraint. For scheduling problems with preventive and flexible maintenance tasks, a large number of studies have been performed on a single-ma- chine problem with various objectives. Among others, Qi, Chen, and Tu (1999), Yang, Ma, Xu, and Yang (2011), and Mashkani and Moslehi (2016) present branch and bound (BnB) algorithms, and Akturk, Ghosh, and Gunes (2003) and Gurel and Akturk (2008) propose heuristic al- gorithms for minimizing total completion time, while Mosheiov and Sarig (2009) develop a dynamic programming (DP) algorithm and a heuristic algorithm for minimizing total weighted completion times. Also, Chen (2006) gives mixed binary-integer programming models and develops a heuristic algorithm for minimizing mean flow time, Chen (2008a, 2008b) presents mixed binary-integer programming models for minimizing the total tardiness and for minimizing makespan, respec- tively, while Sbihi and Varnier (2008) suggest a heuristic and a BnB algorithm for minimizing maximum tardiness. Recently, Luo, Cheng, and Ji (2015), Zhu, Li, and Zhou (2015) and Ying, Ju, and Chen (2016) consider problems with flexible maintenance and various scheduling http://dx.doi.org/10.1016/j.cie.2017.10.004 Received 4 March 2017; Received in revised form 2 October 2017; Accepted 3 October 2017 ⁎ Corresponding author. E-mail address: jy7433.lee@samsung.com (J.-Y. Lee). Computers & Industrial Engineering 114 (2017) 22–30 Available online 05 October 2017 0360-8352/ © 2017 Published by Elsevier Ltd. MARK