Abstract
In this paper, we consider the single-machine scheduling problem with production and rejection costs to minimize the maximum earliness. If a job is accepted, then this job must be processed on the machine and a corresponding production cost needs be paid. If the job is rejected, then a corresponding rejection cost has to be paid. The objective is to minimize the sum of the maximum earliness of the accepted jobs, the total production cost of the accepted jobs and the total rejection cost of the rejected jobs. We show that this problem is equivalent to a single-machine scheduling problem to minimize the maximum earliness with two distinct rejection modes. In the latter problem, rejection cost might be negative in the rejection-award mode which is different from the traditional rejection-penalty mode in the previous literatures. We show that both of two problems are NP-hard in the ordinary sense and then provide two pseudo-polynomial-time algorithms to solve them. Finally, we also show that three special cases can be solved in polynomial time.
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References
Bartal Y, Leonardi S, Spaccamela AM, Sgall J, Stougie L (2000) Multiprocessor scheduling with rejection. SIAM J Discret Math 13:64–78
Engels DW, Karger DR, Kolliopoulos SG, Sengupta S, Uma RN, Wein J (2003) Techniques for scheduling with rejection. J Algorithms 49:175–191
Garey MR, Johnson DS (1979) Computers and intractablity: a guide to the theory of NP-completeness. Freeman, San Francisco, CA
Guinet AGP, Solomon MM (1996) Scheduling hybrid flowshops to minimize maximum tardiness or maximum completion time. Int J Prod Res 34:1643–1654
Hoogeveen H, Skutella M, Woeginger GJ (2003) Preemptive scheduling with rejection. Math Program 94:361–374
Koulamas C, Panwalkar SS (2015) On the equivalence of single machine earliness/tardiness problems with job rejection. Comput Ind Eng 87:1–3
Lu L, Zhang LQ, Yuan JJ (2008) The unbounded parallel batch machine scheduling with release dates and rejection to minimize makespan. Theor Comput Sci 396:283–289
Lu L, Cheng TCE, Yuan JJ, Zhang LQ (2009) Bounded single-machine parallel-batch scheduling with release dates and rejection. Comput Oper Res 36:2748–2751
Nie L, Cheng T (2012) Minimize the maximum earIiness for singIe-machine scheduling with due dates and rejection (in Chinese). J Zhengzhou Univ (Nature Science Edition) 44:42–45
Sengupta S (2003) Algorithms and approximation schemes for minimum lateness/tardiness scheduling with rejection. Lect Notes Comput Sci 2748:79–90
Shabtay D, Gaspar N, Kaspi M (2013) A survey on off-line scheduling with rejection. J Sched 16:3–28
Shabtay D, Gaspar N (2012) Two-machine flow-shop scheduling with rejection. Comput Oper Res 39:1087–1096
Steiner G, Zhang R (2011) Revised delivery-time quotation in scheduling with tardiness penalties. Oper Res 59:1504–1511
Woeginger GJ (2000) When does a dynamic programming formulation guarantee the exitence of an FPTAS? INFORMS J Comput 12:57–74
Zhang LQ, Lu LF, Yuan JJ (2009) Single machine scheduling with release dates and rejection. Eur J Oper Res 198:975–978
Zhang LQ, Lu LF, Yuan JJ (2010) Single-machine scheduling under the job rejection constraint. Theor Comput Sci 411:1877–1882
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This research was supported in part by NSFCs (11426094, 11571321 and U1504103).
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Lu, L., Zhang, L. Single-machine scheduling with production and rejection costs to minimize the maximum earliness. J Comb Optim 34, 331–342 (2017). https://doi.org/10.1007/s10878-016-9992-0
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DOI: https://doi.org/10.1007/s10878-016-9992-0