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

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Abstract

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.

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References

  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)

    Article  MathSciNet  Google Scholar 

  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)

  3. Chekuri, C., Motwani, R., Natarajan, B., Stein, C.: Approximation Techniques for Average Completion Time Scheduling. SIAM Journal on Computing 31, 146–166 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  14. Lenstra, J. K., Rinnooy Kan, A. H. G., Brucker, P.: Complexity of Machine Scheduling Problems. Annals of Discrete Math 1, 343–362 (1977)

    MATH  MathSciNet  Google Scholar 

  15. Phillips, C., Stein, C., Wein, J.: Minimizing Average Completion Time in the Presence of Release Dates. Mathematical Programming 82, 199–223 (1998)

    MATH  MathSciNet  Google Scholar 

  16. Queyranne, M., Sviridenko, M.: Approximation Algorithms for Shop Scheduling Problems with Minsum Objective. Journal of Scheduling 5, 287–305 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schrage, L.: A Proof of the Optimality of The Shortest Remaining Processing Time Discipline. Operations Research 16, 687–690 (1968)

    MATH  Google Scholar 

  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)

  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)

  20. Shmoys, D.B., Wein, J., Williamson, D.P.: Scheduling Parallel Machines On-line. SIAM Journal on Computing 24, 1313–1331 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Smith, W.: Various Optimizers for Single-Stage Production. Naval Res. Logist. Quart. 3, 59–66 (1956)

    MathSciNet  Google Scholar 

  22. Stougie, L.: Personal communication. quoted in [10] (1995)

  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. 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)

  25. Vestjens, A.P.A.: On-line Machine Scheduling. Ph.D. Thesis, Eindhoven University of Tecnology, The Netherlands (1997)

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Authors and Affiliations

  1. Dept. of Decision Sciences, National University of Singapore, Singapore

    Mabel C. Chou

  2. Faculty of Commerce and Business Administration, University of British Columbia, Canada

    Maurice Queyranne

  3. Dept. of Civil and Environmental Engineering, Massachusetts Institute of Technology, USA

    David Simchi-Levi

Authors
  1. Mabel C. Chou
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  2. Maurice Queyranne
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  3. David Simchi-Levi
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Additional information

Research supported in part by ONR Contracts N00014-90-J-1649 and N00014-95-1-0232, NSF Contracts DDM-9322828, DMI-9732795, DMI-0085683 and DMI-0245352, NUS Academic Research Grant R314-000-046-112, and a research grant from the Natural Sciences and Research Council of Canada (NSERC).

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Chou, M., Queyranne, M. & Simchi-Levi, D. The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates. Math. Program. 106, 137–157 (2006). https://doi.org/10.1007/s10107-005-0588-1

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