IIE Transactions

, Volume 31, Issue 5, pp 467–477 | Cite as

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

  • SALAH E. Elmaghraby
  • KRISTIN A. Thoney


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Avis, D. and Fukuda, K. (1996) Reverse search for enumeration. Discrete Applied Mathematics, 65, 21–46.Google Scholar
  2. [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. [3]
    Makino, T. (1965) On a scheduling problem. Journal of the Operations Research Society of Japan, 8, 32–44.Google Scholar
  4. [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]
    [5]Bagga, P.C. (1970) n-Job, 2-machine sequencing problem with stochastic service. Opsearch, 7, 184–197.Google Scholar
  6. [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. [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. [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. [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. [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.Google Scholar
  11. [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. [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.Google Scholar
  13. [13]
    Kamburowski, J. (1992) Bounding the distribution of project duration in PERT networks. Operations Research Letters, 12, 17– 22.Google Scholar
  14. [14]
    Taylor, J.M. Comparisons of certain distribution functions. Math. Operations-forsch. u. Statist., Ser. Statist., 14, 397–408.Google Scholar
  15. [15]
    Hammersley, J.M. and Handscomb, D.C. (1967) Monte Carlo Methods, Methuen, London, England.Google Scholar
  16. [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. [17]
    Smith, W.E. (1956) Various optimizers for single stage production. Naval Research Logistics Quarterly, 3, 59–66.Google Scholar
  18. [18]
    Pinedo, M.L. (1995) Scheduling: Theory, Algorithms, and Systems, Prentice Hall, Englewood Cliffs, NJ 07632.Google Scholar
  19. [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. [20]
    Bland, R.G. (1977) New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2, 103–107.Google Scholar
  21. [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. [22]
    Elmaghraby, S.E. (1967) On the expected duration of PERT type networks. Management Science 13, 299–306.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • SALAH E. Elmaghraby
    • 1
    • 2
  • KRISTIN A. Thoney
    • 2
  1. 1.Department of Industrial EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.The Graduate Program in Operations ResearchNorth Carolina State UniversityRaleighUSA

Personalised recommendations