Advertisement

On the Optima Localization for the Three-Machine Routing Open Shop

  • Ilya ChernykhEmail author
  • Olga Krivonogova
Conference paper
  • 203 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)

Abstract

A tight optima localization interval for the classical open shop scheduling problem with three machines was established by S. Sevastyanov and I. Chernykh in 1998. It was proved that for any problem instance its optimal makespan does not exceed \(\frac{4}{3}\) times the standard lower bound. The process of proof involved massive computer-aided enumeration of the subsets of instances of the problem considered and took about 200 h of the running time to complete. This makes it seemingly impossible to use the same approach for more complicated problems, i.e. the four machine open shop for which the optima localization interval is still unknown. In this paper we apply that computer-aided approach to the three-machine routing open shop problem on a two-node transportation network. For this generalization of the plain open shop problem we derive some extreme instance properties and prove that the optimal makespan does not exceed \(\frac{4}{3}\) times the standard lower bound, thus generalizing the result previously known for the three-machine open shop.

Keywords

Open shop Routing open shop Optima localization Computer-aided proof Approximation algorithm 

References

  1. 1.
    Aksyonov, V.: An approximation polynomial time algorithm for one scheduling problem. Upravlyaemye systemy 28, 8–11 (1988). (in Russian)Google Scholar
  2. 2.
    Averbakh, I., Berman, O., Chernykh, I.: A 6/5-approximation algorithm for the two-machine routing open-shop problem on a two-node network. Eur. J. Oper. Res. 166(1), 3–24 (2005).  https://doi.org/10.1016/j.ejor.2003.06.050MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Averbakh, I., Berman, O., Chernykh, I.: The routing open-shop problem on a network: complexity and approximation. Eur. J. Oper. Res. 173(2), 531–539 (2006).  https://doi.org/10.1016/j.ejor.2005.01.034MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chernykh, I., Kononov, A.V., Sevastyanov, S.: Efficient approximation algorithms for the routing open shop problem. Comput. Oper. Res. 40(3), 841–847 (2013).  https://doi.org/10.1016/j.cor.2012.01.006MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chernykh, I., Krivonogiva, O.: Optima localization for the two-machine routing open shop on a tree, submitted to Diskretnyj Analiz i Issledovanie Operacij (2019). (in Russian)Google Scholar
  6. 6.
    Chernykh, I., Lgotina, E.: The 2-machine routing open shop on a triangular transportation network. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 284–297. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_23CrossRefGoogle Scholar
  7. 7.
    Chernykh, I., Pyatkin, A.: Refinement of the optima localization for the two-machine routing open shop. In: OPTIMA 2017 Proceedings, vol. 1987, pp. 131–138 (2017)Google Scholar
  8. 8.
    Gonzalez, T.F., Sahni, S.: Open shop scheduling to minimize finish time. J. ACM 23(4), 665–679 (1976).  https://doi.org/10.1145/321978.321985MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kononov, A., Kononova, P., Gordeev, A.: Branch-and-bound approach for optima localization in scheduling multiprocessor jobs. Int. Trans. Oper. Res. 27(1), 381–393 (2017).  https://doi.org/10.1111/itor.12503CrossRefGoogle Scholar
  10. 10.
    Kononov, A.: On the routing open shop problem with two machines on a two-vertex network. J. Appl. Ind. Math. 6(3), 318–331 (2012).  https://doi.org/10.1134/s1990478912030064MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, G.B.: Sequencing and scheduling: algorithms and complexity. In: Logistics of Production and Inventory. Elsevier (1993)Google Scholar
  12. 12.
    Sevastianov, S.V., Tchernykh, I.D.: Computer-aided way to prove theorems in scheduling. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 502–513. Springer, Heidelberg (1998).  https://doi.org/10.1007/3-540-68530-8_42CrossRefGoogle Scholar
  13. 13.
    Sevastyanov, S.V.: Some positive news on the proportionate open shop problem. Sibirskie Elektronnye Matematicheskie Izvestiya 16, 406–426 (2018).  https://doi.org/10.33048/semi.2019.16.023MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations