Advertisement

How the Difference in Travel Times Affects the Optima Localization for the Routing Open Shop

  • Ilya ChernykhEmail author
  • Ekaterina Lgotina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

The routing open shop problem, being a generalization of the metric TSP and the open shop scheduling problem, is known to be NP-hard even in case of two machines with a transportation network consisting of two nodes only. We consider a generalization of this problem with unrelated travel times of each machine. We determine a tight optima localization interval for the two-machine problem in the case when the transportation network consists of at most three nodes. As a byproduct of our research, we present a linear time \(\frac{5}{4}\)-approximation algorithm for the same problem. We prove that the algorithm has the best theoretically possible approximation ratio with respect to the standard lower bound.

Keywords

Scheduling Routing open shop Unrelated travel times Optima localization Approximation algorithm 

References

  1. 1.
    Averbakh, I., Berman, O., Chernykh, I.: A 6/5-approximation algorithm for the two-machine routing open shop problem on a 2-node network. Eur. J. Oper. Res. 166(1), 3–24 (2005).  https://doi.org/10.1016/j.ejor.2003.06.050MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Averbakh, I., Berman, O., Chernykh, I.: The routing open-shop problem on a network: complexity and approximation. Eur. J. Oper. Res. 173(2), 521–539 (2006).  https://doi.org/10.1016/j.ejor.2005.01.034MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brucker, P., Knust, S., Edwin Cheng, T.C., Shakhlevich, N.: Complexity results for flow-shop and open-shop scheduling problems with transportation delays. Ann. Oper. Res. 129, 81–106 (2004).  https://doi.org/10.1023/b:anor.0000030683.64615.c8MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chernykh, I.: Routing open shop with unrelated travel times. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 272–283. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_22CrossRefGoogle Scholar
  5. 5.
    Chernykh, I., Kononov, A., 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
  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: Proceedings of the 8th International Conference on Optimization and Applications (OPTIMA 2017), vol. 1987, pp. 131–138. CEUR Workshop Proceedings (2017)Google Scholar
  8. 8.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburg, PA (1976)Google Scholar
  9. 9.
    Gonzalez, T., Sahni, S.: Open shop scheduling to minimize finish time. J. Assoc. Comput. Mach. 23, 665–679 (1976).  https://doi.org/10.1145/321978.321985MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kononov, A., Sevastianov, S., Tchernykh, I.: When difference in machine loads leads to efficient scheduling in open shops. Ann. Oper. Res. 92, 211–239 (1999).  https://doi.org/10.1023/a:1018986731638MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R., Shmoys, G.B.: Sequencing and scheduling: algorithms and complexity. In: Graves, S.S., Rinnooy-Kan, A.H.G., Zipkin, P. (eds.) Logistics of Production and Inventory. Elsevier, Amsterdam (1993)Google Scholar
  13. 13.
    Serdyukov, A.: On some extremal routes in graphs. Upravlyaemye Sistemy 17, 76–79 (1978). (in Russian)zbMATHGoogle Scholar
  14. 14.
    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

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations