IIE Transactions

, Volume 30, Issue 11, pp 1085–1097 | Cite as

Efficient Lagrangian relaxation algorithms for industry size job-shop scheduling problems

  • Christos A. Kaskavelis
  • Michael C. Caramanis
Article

Abstract

We improve the job specific decomposition Lagrangian relaxation algorithm applied to industry size job shop scheduling problems with more than 10000 resource constraints. We introduce two new features in the Lagrange multiplier updating procedure. First, the usual solution of all subproblems followed by dual cost estimation and update of multiplier values is replaced by the estimation of a surrogate dual cost function and a more frequent update of multipliers is implemented after each subproblem solution. Second, an adaptive step size in the subgradient based multiplier update is introduced. Asymptotic properties of the surrogate dual cost function are obtained and the proposed algorithmic improvements are evaluated in extensive numerical examples including published data used by other researchers, as well as extensive real industrial scheduling system data.

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References

  1. [1]
    Fisher, M.L. (1971) Optimal solution of scheduling problems using Lagrange multipliers, part I. Operations Research, 21, 1114–1127.Google Scholar
  2. [2]
    Graves, S.C. (1981) A review of production scheduling. Operations Research, 18, 841–852.Google Scholar
  3. [3]
    Everett, III, H. (1963) Generalized Lagrange multiplier method for solving problems of optimum allocation of resources. Operations Research, 11, 399–471.Google Scholar
  4. [4]
    Hoitomt, D.J., Luh, P.B., Max, E. and Pattipati, K.R. (1990) Scheduling jobs with simple precedence constraints on parallel machines. Control Systems Magazine, 10, 34–40.CrossRefGoogle Scholar
  5. [5]
    Hoitomt, D.J., Luh, P.B., Pattipati, K.R. (1993) A practical approach to job shop scheduling problems. IEEE Transactions on Robotics and Automation, 9, 1–13.CrossRefGoogle Scholar
  6. [6]
    Wang, J., Luh, P.B. and Zhao, X. (1997) “An optimization-based algorithm for job shop scheduling”, SADHANA, a Journal of Indian Academy of Sciences, a Special issue on competitive manufacturing systems, Vol. 22, Part 2, April 1997, pp. 241–256.Google Scholar
  7. [7]
    Wang, J. and Luh, P.B. (1996) Scheduling of a machining center. Mathematical and Computer Modeling, 24, (11/12), 203–214.CrossRefGoogle Scholar
  8. [8]
    Brooks, R. and Geoffrion, A. (1966) Finding Everett's Lagrange multipliers by linear programming. Operations Research, 17, 1149–1153.Google Scholar
  9. [9]
    Chen, H., Chu, C. and Proth J.M. (1995) A more efficient Lagrangian relaxation approach to job shop scheduling problems, in Proceedings of IEEE International Conference on Robotics and Automation, pp. 496–501.Google Scholar
  10. [10]
    Polyak, B.T. (1969) Minimization of unsmooth functionals. USSR Computational Mathematics and Mathematical Physics, 9, 14–29.Google Scholar
  11. [11]
    Held, M., Wolfe, P. and Crowder H.P. (1974) Validation of subgradient optimization. Mathematical Programming, 6, 62–68.Google Scholar
  12. [12]
    Goffin, J.L. (1977) On convergence rates of subgradient optimization methods. Mathematical Programming, 13, 329–347.Google Scholar
  13. [13]
    Tomastik, R.N. and Luh, P.B. (1993) The facet ascending algorithm for integer programming problems, in Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, TX, pp. 2280–2884. Dec. 1993.Google Scholar
  14. [14]
    Tomastik, R.N., Luh, P.B. and Liu, G. (1996) Scheduling flexibly manufacturing systems for apparel production. IEEE Transactions on Robotics and Automation, 12, 789–799.CrossRefGoogle Scholar
  15. [15]
    Kaskavelis, C.A. and Caramanis, M.C. (1994a) A Lagrangian relaxation based algorithm for scheduling multiple-part-type-production-systems: industrial implementation practice, in Proceedings of the 1994 Japan-USA Symposium on Flexible Automation, Kobe, Japan, pp. 173–180, SCI.Google Scholar
  16. [16]
    Kaskavelis, C.A. and Caramanis, M.C. (1994b) Application of a Lagrangian relaxation based scheduling algorithm to a semiconductor testing facility, in Proceedings of the 4th International Conference on Computer Integrated Manufacturing and Automation Technology, RPI, October 10-12, 94, IEEE Computer Society Press, Troy, NY, pp. 106–112.Google Scholar
  17. [17]
    Fisher, M.L. (1981) The Lagrangian relaxation method for solving integer programming problems. Management Science, 27, 1–18.Google Scholar
  18. [18]
    Goffin, J.L., Haurie, A. and Vial, J.P. (1992) Decomposition and the nondi.erentiable optimization with the projective algorithm. Management Science, 38, No. 2, 284–302.Google Scholar
  19. [19]
    Kaskavelis, C.A. (1994) The Lagrangian relaxation scheduling algorithm: theoretical issues and applications to industry. Masters Thesis, Dept. of Manufacturing Engineering, Boston University, Boston MA.Google Scholar
  20. [20]
    Bertsekas, D.P. (1995) Non-linear Programming, Athena Scientific, Belmont, MA.Google Scholar
  21. [21]
    Kushner, H.J. and Yin, G. (1987) Stochastic approximation algorithms for parallel and distributed processing. Stochastics, 22, 219–250.Google Scholar
  22. [22]
    Vazquez-Abad, F.J., Cassandras, C.G. and Julka, V. (1998) Centralized and decentralized asynchronous optimization of stochastic discrete event systems. IEEE Transactions on Automatic Control, AC-43, 5, pp. 631–655.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Christos A. Kaskavelis
    • 1
  • Michael C. Caramanis
    • 1
  1. 1.Department of Manufacturing EngineeringBoston UniversityBostonUSA

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