A Polynomial-Time Algorithm for the Routing Flow Shop Problem with Two Machines: An Asymmetric Network with a Fixed Number of Nodes

  • Ilya Chernykh
  • Alexander Kononov
  • Sergey SevastyanovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


We consider the routing flow shop problem with two machines on an asymmetric network. For this problem we discuss properties of an optimal schedule and present a polynomial time algorithm assuming the number of nodes of the network to be bounded by a constant. To the best of our knowledge, this is the first positive result on the complexity of the routing flow shop problem with an arbitrary structure of the transportation network, even in the case of a symmetric network. This result stands in contrast with the complexity of the two-machine routing open shop problem, which was shown to be NP-hard even on the two-node network.


Scheduling Flow shop Routing flow shop Polynomially-solvable case Dynamic programming 


  1. 1.
    Averbakh, I., Berman, O.: Routing two-machine flowshop problems on networks with special structure. Transp. Sci. 30(4), 303–314 (1996). Scholar
  2. 2.
    Averbakh, I., Berman, O.: A simple heuristic for \(m\)-machine flow-shop and its applications in routing-scheduling problems. Oper. Res. 47(1), 165–170 (1999). 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). Scholar
  4. 4.
    Chernykh, I., Kononov, A., Sevastyanov, S.: A polynomial-time algorithm for the routing flow shop problem with two machines: an asymmetric network with a fixed number of nodes.
  5. 5.
    Chernykh, I., Kononov, A., Sevastyanov, S.: Exact polynomial-time algorithm for the two-machine routing flow shop problem with a restricted transportation network. In: Optimization problems and their applications (OPTA-2018), Abstracts of the VII International Conference, Omsk, Russia, 8–14 July 2018, pp. 37–37. Omsk State University (2018)Google Scholar
  6. 6.
    Garey, M.R., Johnson, D.S., Sethi, R.: The complexity of flowshop and jobshop scheduling. Math. Oper. Res. 1(2), 117–129 (1976). Scholar
  7. 7.
    Johnson, S.M.: Optimal two- and three-stage production schedules with setup times included. Rand Corporation (1953).
  8. 8.
    Józefczyk, J., Markowski, M.: Heuristic solution algorithm for routing flow shop with buffers and ready times. In: Swiątek, J., Grzech, A., Swiątek, P., Tomczak, J.M. (eds.) Advances in Systems Science. AISC, vol. 240, pp. 531–541. Springer, Cham (2014). Scholar
  9. 9.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: algorithms and complexity. In: Logistics of Production and Inventory, Handbooks in Operations Research and Management Science, vol. 4, pp. 445–522. Elsevier (1993).
  10. 10.
    Yu, V.F., Lin, S., Chou, S.: The museum visitor routing problem. Appl. Math. Comput. 216(3), 719–729 (2010). Scholar
  11. 11.
    Yu, W., Liu, Z., Wang, L., Fan, T.: Routing open shop and flow shop scheduling problems. Eur. J. Oper. Res. 213(1), 24–36 (2011). Scholar
  12. 12.
    Yu, W., Zhang, G.: Improved approximation algorithms for routing shop scheduling. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 30–39. Springer, Heidelberg (2011). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

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

Personalised recommendations