Three-machine open shop with a bottleneck machine revisited


The paper considers the three-machine open shop scheduling problem to minimize the makespan. In the model, each job consists of two operations, one of which is to be processed on the bottleneck machine, which is the same for all jobs. A new linear-time algorithm to find an optimal non-preemptive schedule is developed. The suggested algorithm considerably simplifies the only previously known method as it straightforwardly exploits the structure of the problem and its key components to yield an optimal solution.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. Aksjonov, V. A. (1988). A polynomial-time algorithm of approximate solution of a scheduling problem. Upravlyaemye Sistemy, 28, 8–11. (in Russian).

    Google Scholar 

  2. Bárány, I., & Fiala, T. (1982). Nearly optimum solution of multimachine scheduling problems. Szigma, 15, 177–191. (in Hungarian).

    Google Scholar 

  3. Chen, B., & Strusevich, V. A. (1993). Approximation algorithms for three machine open shop scheduling. ORSA Journal on Computing, 5, 321–328.

    Article  Google Scholar 

  4. Chen, B., & Yu, W. (2001). How good is a dense shop schedule? Acta Mathematics Application Sinica, 17(1), 121–128.

    Article  Google Scholar 

  5. Chen, R., Huang, W., Men, Z., & Tang, G. (2012). Open-shop dense schedules: Properties and worst-case performance ratio. Journal of Scheduling, 15(1), 3–11.

    Article  Google Scholar 

  6. de Werra, D. (1989). Graph-theoretical models for preemptive scheduling. In R. Slowinski & J. Weglarz (Eds.), Advances in project scheduling (pp. 171–185). Amsterdam: Elsevier.

    Google Scholar 

  7. Drobouchevitch, I. G., & Strusevich, V. A. (1999). A polynomial algorithm for the three-machine open shop with a bottleneck machine. Annals of Operations Research, 92, 185–214.

    Article  Google Scholar 

  8. Gonzalez, T., & Sahni, S. (1976). Open shop scheduling to minimize finish time. Journal of the Association for Computing Machinery, 23, 665–679.

    Article  Google Scholar 

  9. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1993). Sequencing and scheduling: Algorithms and complexity. In S. C. Graves, et al. (Eds.), Handbook in operations research and management science (Vol. 4, pp. 445–522)., Logistics of production and inventory Amsterdam: North-Holland.

    Google Scholar 

  10. Pinedo, M., & Schrage, L. (1982). Stochastic shop scheduling: A survey. In M. A. H. Dempster, et al. (Eds.), Deterministic and stochastic scheduling (pp. 181–196). Dordrecht: Riedel.

    Google Scholar 

  11. Sevastianov, S. V., & Woeginger, G. J. (1998). Makespan minimization in open shops: A polynomial time approximation scheme. Mathematical Programming, 82, 191–198.

    Google Scholar 

  12. Soper, A. J. (2015). A cyclical search for the two machine flow shop and open shop to minimise finishing time. Journal of Scheduling, 18(3), 311–314.

    Article  Google Scholar 

  13. Williamson, D. P., Hall, L. A., Hoogeveen, J. A., Hurkens, C. A. J., Lenstra, J. K., Sevast’janov, S. V., et al. (1997). Short shop schedules. Operations Research, 45, 288–294.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Inna G. Drobouchevitch.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Drobouchevitch, I.G. Three-machine open shop with a bottleneck machine revisited. J Sched (2020).

Download citation


  • Scheduling
  • Open shop
  • Minimum makespan