Skip to main content
Log in

The two-machine stochastic flowshop problem with arbitrary processing time distributions

  • Published:
IIE Transactions

Abstract

We treat the two-machine flowshop problem with the objective of minimizing the expected makespan when the jobs possess stochastic durations of arbitrary distributions. We make three contributions in this paper: (1) we propose an exact approach with exponential worst-case time complexity. We also propose approximations which are computationally modest in their requirements. Experimental results indicate that our procedure is within less than 1% of the optimum; and (2) we provide a more elementary proof of the bounds on the project completion time based on the concepts of 'control networks'; and (3) we extend the 'reverse search' procedure of Avis and Fukuda [1] to the context of permutation schedules.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avis, D. and Fukuda, K. (1996) Reverse search for enumeration. Discrete Applied Mathematics, 65, 21–46.

    Google Scholar 

  2. Johnson, S.M. (1954) Optimal two-and three-stage production schedules with setup times included. Naval Research Logistics Quarterly, 1, 61–67.

    Google Scholar 

  3. Makino, T. (1965) On a scheduling problem. Journal of the Operations Research Society of Japan, 8, 32–44.

    Google Scholar 

  4. Talwar, P.P. (1967) A note on sequencing problems with uncertain job times. Journal of the Operations Research Society of Japan, 9, 65–74.

    Google Scholar 

  5. [5]Bagga, P.C. (1970) n-Job, 2-machine sequencing problem with stochastic service. Opsearch, 7, 184–197.

    Google Scholar 

  6. Cunningham, A.A. and Dutta, S.K. (1973) Scheduling jobs with exponentially distributed processing times on two machines of a flow shop. Naval Research Logistics Quarterly, 20, 69–81.

    Google Scholar 

  7. Mittal, B.S. and Bagga, P.C. (1977) A priority problem in sequencing with stochastic service times. Opsearch, 14, 19–28.

    Google Scholar 

  8. Weiss, G. (1981) Multiserver stochastic scheduling, in Deterministic and Stochastic Scheduling, Dempster, M.A.H., Lenstra, J.K. and Rinnooy Kan, A.H.G. (eds.), D. Reidel, Dordrecht, pp. 157–159.

    Google Scholar 

  9. Ku, P. and Niu, S.-C. (1986) On Johnson's two-machine flow shop with random processing times. Operations Research, 34, 130– 136.

    Google Scholar 

  10. Elmaghraby, S.E. and Thoney, K.A. (1997) The two machines stochastic flowshop revisited: the case of general distributions. Technical report, NCSU Raleigh, NC 27695-7906.

  11. Kamburowski, J. (1987) An overview of the computational complexity of the PERT, shortest route, and maximum flow problems in stochastic networks, in Proceedings of the Advanced School on Stochastics in Combinatorial Optimization, Andreatta, G., Mason, F. and Serafini, P. (eds.), World Scientific, Singapore, pp. 187–196.

    Google Scholar 

  12. Elmaghraby, S.E. (1989) The estimation of some network pa-rameters in the PERT model of activity networks: review and critique, in Advances in Project Scheduling, Słowinski, R. and Weglarz, J. (eds.), Elsevier, Part III, Ch. 1.

  13. Kamburowski, J. (1992) Bounding the distribution of project duration in PERT networks. Operations Research Letters, 12, 17– 22.

    Google Scholar 

  14. Taylor, J.M. Comparisons of certain distribution functions. Math. Operations-forsch. u. Statist., Ser. Statist., 14, 397–408.

  15. Hammersley, J.M. and Handscomb, D.C. (1967) Monte Carlo Methods, Methuen, London, England.

    Google Scholar 

  16. Bein, W.W., Kamburowski, J. and Stallmann, M.F.M. (1992) Optimal reduction of two-terminal directed acyclic graphs. SIAM Journal of Computing, 21, 1112–1129.

    Google Scholar 

  17. Smith, W.E. (1956) Various optimizers for single stage production. Naval Research Logistics Quarterly, 3, 59–66.

    Google Scholar 

  18. Pinedo, M.L. (1995) Scheduling: Theory, Algorithms, and Systems, Prentice Hall, Englewood Cliffs, NJ 07632.

    Google Scholar 

  19. Elmaghraby, S.E. (1971) A graph theoretic interpretation of the sufficient conditions for contiguous binary switching. Naval Research Logistics Quarterly, 18, 339–344.

    Google Scholar 

  20. Bland, R.G. (1977) New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2, 103–107.

    Google Scholar 

  21. Campbell, H.G., Dudek, R.A. and Smith, M.L. (1970) A heuristic algorithm for the n job m machine sequencing problem. Management Science, 16, B630–B637.

    Google Scholar 

  22. Elmaghraby, S.E. (1967) On the expected duration of PERT type networks. Management Science 13, 299–306.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elmaghraby, S.E., Thoney, K.A. The two-machine stochastic flowshop problem with arbitrary processing time distributions. IIE Transactions 31, 467–477 (1999). https://doi.org/10.1023/A:1007697625481

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007697625481

Keywords

Navigation