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Mathematical Problems in Engineering
Volume 2017, Article ID 6093029, 5 pages
Research Article

Batch Scheduling with Proportional-Linear Deterioration and Outsourcing

1School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
3Caoxian No. 1 Middle School, Heze, Shandong 274400, China

Correspondence should be addressed to Cuixia Miao; moc.621@aixiucoaim

Received 10 March 2017; Accepted 23 April 2017; Published 22 June 2017

Academic Editor: Marco Pizzarelli

Copyright © 2017 Cuixia Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is or . A job is either accepted and processed in batches on a single machine by manufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomial time algorithm, respectively. For the model, when the jobs have distinct ( ) release dates, we provide a dynamic programming algorithm, where is the number of jobs.