Advertisement

Lot-size scheduling of a single product on unrelated parallel machines

  • Anton V. Eremeev
  • Mikhail Y. Kovalyov
  • Pavel M. Kuznetsov
Original Paper
  • 12 Downloads

Abstract

We study a problem in which at least a given quantity of a single product has to be partitioned into lots, and lots have to be assigned to the unrelated parallel machines for processing so that the maximum machine completion time or the sum of machine completion times is minimized. Machine dependent lower and upper bounds on the lot size are given. The product can be continuously divisible or discrete. We derive optimal polynomial time algorithms for several special cases of the problem. For other cases we provide NP-hardness proofs and demonstrate existence of fully polynomial time approximation schemes.

Keywords

Scheduling Parallel machines Lot-sizing Computational complexity Approximation 

Notes

Acknowledgements

The research presented in Sect. 2 is supported by the Russian Science Foundation Grant 15-11-10009. Research presented in Sect. 3 is supported by the program of fundamental scientific researches of the SB RAS I.5.1., Project 0314-2016-0019.

References

  1. 1.
    Allahverdi, A., Ng, C.T., Cheng, T.C.E., Kovalyov, M.Y.: A survey of scheduling problems with setup times or costs. Eur. J. Oper. Res. 187, 985–1032 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Austin, L.M., Hogan, W.W.: Optimizing the procurement of aviation fuels. Manag. Sci. 22(5), 515–527 (1976)CrossRefGoogle Scholar
  3. 3.
    Borisovsky, P.A.: A genetic algorithm for one scheduling problem with changeovers. In: Proceedings of XIV Baikal International School-Seminar Optimization methods and their applications, vol. 4, pp. 166–173. Melentiev Energy Systems Institute SB RAS, Irkutsk (2008). (In Russian)Google Scholar
  4. 4.
    Chauhan, S.S., Eremeev, A.V., Romanova, A.A., Servakh, V.V.: Approximation of linear cost supply management problem with lower-bounded demands. In: Proceedings of Discrete Optimization Methods in Production and Logistics, DOM’2004, pp. 16–21. Nasledie Dialog-Sibir Pbs., Omsk (2004)Google Scholar
  5. 5.
    Chauhan, S.S., Eremeev, A.V., Romanova, A.A., Servakh, V.V., Woeginger, G.J.: Approximation of the supply scheduling problem. Oper. Res. Lett. 33(3), 249–254 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, B., Ye, Y., Zhang, J.: Lot-sizing scheduling with batch setup times. J. Sched. 9(3), 299–310 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dolgui, A., Eremeev, A.V., Kovalyov, M.Y., Kuznetsov, P.M.: Multi-product lot sizing and scheduling on unrelated parallel machines. IIE Tans. 42(7), 514–524 (2010)CrossRefGoogle Scholar
  8. 8.
    Eremeev, A.V., Kovalyov, M.Y., Kuznetsov, P.M.: Approximate solution of the control problem of supplies with many intervals and concave cost functions. Autom. Remote Control. 69(7), 1181–1187 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eremeev, A.V., Romanova, A.A., Servakh, V.V., Chauhan, S.S.: Approximation solution of the supply management problem. J. Appl. Ind. Math. 1(4), 433–441 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco (1979)MATHGoogle Scholar
  11. 11.
    Kovalyov, M.Y., Portmann, M.-C., Oulamara, A.: Optimal testing and repairing a failed series system. J. Comb. Optim. 12, 279–95 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Logendran, A., Subur, F.: Unrelated parallel machine scheduling with job splitting. IIE Trans. 36(4), 359–372 (2004)CrossRefGoogle Scholar
  13. 13.
    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, Hoboken (1990)MATHGoogle Scholar
  14. 14.
    Ng, C.T., Kovalyov, M.Y., Cheng, T.C.E.: An FPTAS for a supply scheduling problem with non-monotone cost functions. Naval Res. Logist. 55, 194–199 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Potts, C.N., Kovalyov, M.Y.: Scheduling with batching: a review. Eur. J. Oper. Res. 120, 228–249 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Potts, C.N., Van Wassenhove, L.N.: Integrating scheduling with batching and lot-sizing: a review of algorithms and complexity. J. Oper. Res. Soc. 43(5), 395–406 (1992)CrossRefMATHGoogle Scholar
  17. 17.
    Shaik, M.A., Floudas, C.A., Kallrath, J., Pitz, H.-J.: Production scheduling of a large-scale industrial continuous plant: short-term and medium-term scheduling. Comput. Chem. Eng. 33(8), 670–686 (2009)CrossRefGoogle Scholar
  18. 18.
    Shor, N.Z., Stecuk, P.I.: Piecewise concave problem of knapsack type. In: Mikhalevich, V.S. (ed.) Methods of Research of Extremal Problems, pp. 21–28. V.M. Glushkov Institute of Cybernetics N.A., Kiev (1994). (In Russian)Google Scholar
  19. 19.
    Tahar, D.N., Yalaoui, F., Chu, C., Amodeo, L.: A linear programming approach for identical parallel machine scheduling with job splitting and sequence-dependent setup times. Int. J. Prod. Econ. 99, 63–73 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsSiberian Branch of Russian Academy of SciencesNovosibirskRussia
  2. 2.United Institute of Informatics ProblemsNational Academy of Sciences of BelarusMinskBelarus
  3. 3.Dostoevsky Omsk State UniversityOmskRussia

Personalised recommendations