Mathematical Programming

, Volume 106, Issue 1, pp 137–157

The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates

  • Mabel C. Chou
  • Maurice Queyranne
  • David Simchi-Levi


Jobs arriving over time must be non-preemptively processed on one of m parallel machines, each running at its own speed, so as to minimize a weighted sum of the job completion times. In this on-line environment, the processing requirement and weight of a job are not known before the job arrives. The Weighted Shortest Processing Requirement (WSPR) heuristic is a simple extension of the well known WSPT heuristic, which is optimal for the single machine problem without release dates. According to WSPR, whenever a machine completes a job, the next job assigned to it is the one with the least ratio of processing requirement to weight among all jobs available for processing at this point in time. We analyze the performance of this heuristic and prove that its asymptotic competitive ratio is one for all instances with bounded job processing requirements and weights. This implies that the WSPR algorithm generates a solution whose relative error approaches zero as the number of jobs increases. Our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem.


Scheduling Asymptotic Performance Ratio On-line algorithms Analysis of Heuristics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bertsimas, D., Gamarnik, D., Sethruaman, J.: From Fluid Relaxations to Practical Algorithms for High-Multiplicity Job-Shop Scheduling: The Holding Cost Objective. Operations Research 51 (5), 798–813 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chakrabarti, S., Phillips, C., Schulz, A.S., Shmoys, D.B., Stein, C., Wein, J.: Improved Scheduling Algorithms for Minsum Criteria. In: Meyer auf der Heide, F., Monien, B. (eds), Automata, Languages and Programming, Lecture Notes in Computer Science Vol. 1099, Springer, Berlin, 646–657 (1996)Google Scholar
  3. 3.
    Chekuri, C., Motwani, R., Natarajan, B., Stein, C.: Approximation Techniques for Average Completion Time Scheduling. SIAM Journal on Computing 31, 146–166 (2001)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Eastman, W.L., Even, S., Isaacs, I.M.: Bounds for the Optimal Scheduling of n Jobs on m Processors. Management Science 11, 268–279 (1964)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Goemans, M. X.: A Supermodular Relaxation for Scheduling with Release Dates. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds), Integer Programming and Combinatorial Optimization Proceedings of the 5th International IPCO Conference, Lecture Notes in Computer Science Vol. 1084, Springer, Berlin, 288–300 (1996)Google Scholar
  6. 6.
    Goemans, M. X.: Improved Approximation Algorithms for Scheduling with Release Dates. Proceedings of the 8th ACM-SIAM Symposium on Discrete Algorithms, 591–598 (1997)Google Scholar
  7. 7.
    Goemans, M. X., Queyranne, M., Schulz, A. S., Skutella, M., Wang, Y.: Single Machine Scheduling with Release Dates. SIAM J. Discrete Mathematics 15, 165–192 (2002)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Graham, R.L., Lawler, E.L., Lenstra J.K., Rinnooy Kan, A.H.G.: Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey. Annals of Discrete Mathematics, 5, 287–326 (1979)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Hall, L. A., Schulz, A. S., Shmoys, D. B., Wein, J.: Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms. Mathematics of Operations Research 22 (3), 513–544 (1997)Google Scholar
  10. 10.
    Hoogeveen, J. A., Vestjens, A. P. A.: Optimal On-line Algorithms for Single-Machine Scheduling. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds), Integer Programming and Combinatorial Optimization Proceedings of the 5th International IPCO Conference, Lecture Notes in Computer Science Vol. 1084, Springer, Berlin, 404–414 (1996)Google Scholar
  11. 11.
    Kaminsky, P., Simchi-Levi, D.: Probabilistic Analysis of an On-line Algorithm for the Single Machine Mean Completion Time Problem With Release Dates. Operations Research Letters 21, 141–148 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Labetoulle, J., Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G.: Preemptive Scheduling of Uniform Machines Subject to Release Dates. In: Pulleyblank, W. R. (ed), Progress in Combinatorial Optimization, Academic Press, New York, 245–261 (1984)Google Scholar
  13. 13.
    Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., Shmoys, D. B.: Sequencing and Scheduling: Algorithms and Complexity. In: S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin (eds.), Logistics of Production and Inventory, Handbooks in Operations Research and Management Science, vol. 4, North–Holland, Amsterdam (1993)Google Scholar
  14. 14.
    Lenstra, J. K., Rinnooy Kan, A. H. G., Brucker, P.: Complexity of Machine Scheduling Problems. Annals of Discrete Math 1, 343–362 (1977)MATHMathSciNetGoogle Scholar
  15. 15.
    Phillips, C., Stein, C., Wein, J.: Minimizing Average Completion Time in the Presence of Release Dates. Mathematical Programming 82, 199–223 (1998)MATHMathSciNetGoogle Scholar
  16. 16.
    Queyranne, M., Sviridenko, M.: Approximation Algorithms for Shop Scheduling Problems with Minsum Objective. Journal of Scheduling 5, 287–305 (2002)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Schrage, L.: A Proof of the Optimality of The Shortest Remaining Processing Time Discipline. Operations Research 16, 687–690 (1968)MATHGoogle Scholar
  18. 18.
    Schulz, A.S., Skutella, M.: Scheduling – LPs Bear Probabilities: Randomized Approximations for Min – Sum Criteria. In: Burkard, R., Woeginger, G. (eds), Algorithms – ESA'97 Lecture Notes in Computer Science Vol. 1284, Springer, Berlin, 416–429 (1997)Google Scholar
  19. 19.
    Sgall, J.: On-line Scheduling –- a Survey. In: Fiat, A., Woeginger, G.J. (eds), Online Algorithms: The State of the Art, Lecture Notes in Computer Science Vol. 1442, Springer, Berlin, 196–231 (1998)Google Scholar
  20. 20.
    Shmoys, D.B., Wein, J., Williamson, D.P.: Scheduling Parallel Machines On-line. SIAM Journal on Computing 24, 1313–1331 (1995)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Smith, W.: Various Optimizers for Single-Stage Production. Naval Res. Logist. Quart. 3, 59–66 (1956)MathSciNetGoogle Scholar
  22. 22.
    Stougie, L.: Personal communication. quoted in [10] (1995)Google Scholar
  23. 23.
    Stougie, L., Vestjens, A.P.A.: Randomized Algorithms for On-line Scheduling Problems: How Low Can't You Go? Operations Research Letters 30 (2), 89–96 (2002)Google Scholar
  24. 24.
    Uma, R. N., Wein, J.: On the Relationship between Combinatorial and LP-Based Approaches to NP-hard Scheduling Problems. In: Bixby, R. E., Boyd, E. A., Rios-Mercado, R. Z. (eds), Integer Programming and Combinatorial Optimization. Proceedings of the Sixth International IPCO Conference, Lecture Notes in Computer Science Vol. 1412, Springer, Berlin, 394–408 (1998)Google Scholar
  25. 25.
    Vestjens, A.P.A.: On-line Machine Scheduling. Ph.D. Thesis, Eindhoven University of Tecnology, The Netherlands (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Mabel C. Chou
    • 1
  • Maurice Queyranne
    • 2
  • David Simchi-Levi
    • 3
  1. 1.Dept. of Decision SciencesNational University of SingaporeSingapore
  2. 2.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaCanada
  3. 3.Dept. of Civil and Environmental EngineeringMassachusetts Institute of TechnologyUSA

Personalised recommendations