Abstract
In the parallel k-stage flow-shops problem, we are given m identical k-stage flow-shops and a set of jobs. Each job can be processed by any one of the flow-shops but switching between flow-shops is not allowed. The objective is to minimize the makespan, which is the finishing time of the last job. This problem generalizes the classical parallel identical machine scheduling (where \(k = 1\)) and the classical flow-shop scheduling (where \(m = 1\)) problems, and thus it is \(\text {NP}\)-hard. We present a polynomial-time approximation scheme for the problem, when m and k are fixed constants. The key technique is to enumerate over schedules for big jobs, solve a linear programming for small jobs, and add the fractional small jobs at the end. Such a technique has been used in the design of similar approximation schemes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Chen, B.: Analysis of classes of heuristics for scheduling a two-stage flow shop with parallel machines at one stage. J. Oper. Res. Soc. 46, 234–244 (1995)
Chen, B., Glass, C.A., Potts, C.N., Strusevich, V.A.: A new heuristic for three-machine flow shop scheduling. Oper. Res. 44, 891–898 (1996)
Conway, R.W., Maxwell, W.L., Miller, L.W.: Theory of Scheduling. Addison-Wesley, Reading (1967)
Dong, J., Tong, W., Luo, T., Wang, X., Hu, J., Xu, Y., Lin, G.: An FPTAS for the parallel two-machine flowshop problem. Theor. Comput. Sci. (2016, in press)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Garey, M.R., Johnson, D.S., Sethi, R.: The complexity of flowshop and jobshop scheduling. Math. Oper. Res. 1, 117–129 (1976)
Gonzalez, T., Sahni, S.: Flowshop and jobshop schedules: complexity and approximation. Oper. Res. 26, 36–52 (1978)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)
Gupta, J.N.D.: Two-stage, hybrid flowshop scheduling problem. J. Oper. Res. Soc. 39, 359–364 (1988)
Gupta, J.N.D., Hariri, A.M.A., Potts, C.N.: Scheduling a two-stage hybrid flow shop with parallel machines at the first stage. Ann. Oper. Res. 69, 171–191 (1997)
Gupta, J.N.D., Tunc, E.A.: Schedules for a two-stage hybrid flowshop with parallel machines at the second stage. Int. J. Prod. Res. 29, 1489–1502 (1991)
Hall, L.A.: Approximability of flow shop scheduling. Math. Program. 82, 175–190 (1998)
He, D.W., Kusiak, A., Artiba, A.: A scheduling problem in glass manufacturing. IIE Trans. 28, 129–139 (1996)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34, 144–162 (1987)
Hoogeveen, J.A., Lenstra, J.K., Veltman, B.: Preemptive scheduling in a two-stage multiprocessor flow shop is NP-hard. Eur. J. Oper. Res. 89, 172–175 (1996)
Jansen, K., Sviridenko, M.I.: Polynomial time approximation schemes for the multiprocessor open and flow shop scheduling problem. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, p. 455. Springer, Heidelberg (2000)
Johnson, S.M.: Optimal two- and three-stage production schedules with setup times included. Naval Res. Logistics Q. 1, 61–68 (1954)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)
Lee, C.-Y., Vairaktarakis, G.L.: Minimizing makespan in hybrid flowshops. Oper. Res. Lett. 16, 149–158 (1994)
Ruiz, R., Vázquez-Rodríguez, J.A.: The hybrid flow shop scheduling problem. Eur. J. Oper. Res. 205, 1–18 (2010)
Sahni, S.K.: Algorithms for scheduling independent tasks. J. ACM 23, 116–127 (1976)
Schuurman, P., Woeginger, G.J.: A polynomial time approximation scheme for the two-stage multiprocessor flow shop problem. Theor. Comput. Sci. 237, 105–122 (2000)
Vairaktarakis, G., Elhafsi, M.: The use of flowlines to simplify routing complexity in two-stage flowshops. IIE Trans. 32, 687–699 (2000)
Wang, H.: Flexible flow shop scheduling: optimum, heuristics and artificial intelligence solutions. Expert Syst. 22, 78–85 (2005)
Williamson, D.P., Hall, L.A., Hoogeveen, J.A., Hurkens, C.A.J., Lenstra, J.K., Sevastj́anov, S.V.: Short shop schedules. Oper. Res. 45, 288–294 (1997)
Zhang, X., van de Velde, S.: Approximation algorithms for the parallel flow shop problem. Eur. J. Oper. Res. 216, 544–552 (2012)
Acknowledgments
Tong was supported by the FY16 Startup Funding from the Georgia Southern University and an Alberta Innovates Technology Futures (AITF) Graduate Student Scholarship. Miyano is supported by the Grants-in-Aid for Scientific Research of Japan (KAKENHI), Grant Number 26330017. Goebel is supported by the AITF and the Natural Sciences and Engineering Research Council of Canada (NSERC). Lin is partially supported by the NSERC and his work was mostly done during his sabbatical leave at the Kyushu Institute of Technology, Iizuka Campus.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Tong, W., Miyano, E., Goebel, R., Lin, G. (2016). A PTAS for the Multiple Parallel Identical Multi-stage Flow-Shops to Minimize the Makespan. In: Zhu, D., Bereg, S. (eds) Frontiers in Algorithmics. FAW 2016. Lecture Notes in Computer Science(), vol 9711. Springer, Cham. https://doi.org/10.1007/978-3-319-39817-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-39817-4_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39816-7
Online ISBN: 978-3-319-39817-4
eBook Packages: Computer ScienceComputer Science (R0)