A PTAS for the Multiple Parallel Identical Multi-stage Flow-Shops to Minimize the Makespan

  • Weitian Tong
  • Eiji Miyano
  • Randy Goebel
  • Guohui Lin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9711)


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.


Multiprocessor scheduling Flow-shop scheduling Makespan Linear program Polynomial-time approximation scheme 



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.


  1. 1.
    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)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)CrossRefzbMATHGoogle Scholar
  3. 3.
    Conway, R.W., Maxwell, W.L., Miller, L.W.: Theory of Scheduling. Addison-Wesley, Reading (1967)zbMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S., Sethi, R.: The complexity of flowshop and jobshop scheduling. Math. Oper. Res. 1, 117–129 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gonzalez, T., Sahni, S.: Flowshop and jobshop schedules: complexity and approximation. Oper. Res. 26, 36–52 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gupta, J.N.D.: Two-stage, hybrid flowshop scheduling problem. J. Oper. Res. Soc. 39, 359–364 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    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)CrossRefzbMATHGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Hall, L.A.: Approximability of flow shop scheduling. Math. Program. 82, 175–190 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    He, D.W., Kusiak, A., Artiba, A.: A scheduling problem in glass manufacturing. IIE Trans. 28, 129–139 (1996)CrossRefGoogle Scholar
  14. 14.
    Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34, 144–162 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)CrossRefzbMATHGoogle Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Johnson, S.M.: Optimal two- and three-stage production schedules with setup times included. Naval Res. Logistics Q. 1, 61–68 (1954)CrossRefGoogle Scholar
  18. 18.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lee, C.-Y., Vairaktarakis, G.L.: Minimizing makespan in hybrid flowshops. Oper. Res. Lett. 16, 149–158 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ruiz, R., Vázquez-Rodríguez, J.A.: The hybrid flow shop scheduling problem. Eur. J. Oper. Res. 205, 1–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sahni, S.K.: Algorithms for scheduling independent tasks. J. ACM 23, 116–127 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vairaktarakis, G., Elhafsi, M.: The use of flowlines to simplify routing complexity in two-stage flowshops. IIE Trans. 32, 687–699 (2000)Google Scholar
  24. 24.
    Wang, H.: Flexible flow shop scheduling: optimum, heuristics and artificial intelligence solutions. Expert Syst. 22, 78–85 (2005)CrossRefGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, X., van de Velde, S.: Approximation algorithms for the parallel flow shop problem. Eur. J. Oper. Res. 216, 544–552 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Weitian Tong
    • 1
    • 3
  • Eiji Miyano
    • 2
  • Randy Goebel
    • 3
  • Guohui Lin
    • 3
  1. 1.Department of Computer SciencesGeorgia Southern UniversityGeorgiaUSA
  2. 2.Department of Systems Design and InformaticsKyushu Institute of TechnologyIizuka, FukuokaJapan
  3. 3.Department of Computing ScienceUniversity of AlbertaEdmontonCanada

Personalised recommendations