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Minimizing Total Weighted Completion Time with Unexpected Machine Unavailability

  • Yumei Huo
  • Boris Reznichenko
  • Hairong Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

Abstract

In the past two decades, scheduling with machine availability constraints has received more and more attention. Until now most research has focused on the setting where all machine unavailability information is known at the beginning of scheduling horizon. In real world, this is impractical in some cases.

In this article, we consider the situation where the scheduler has to make scheduling decisions without any knowledge of the machine unavailable intervals. In particular, we study the problem of minimizing the total weighted completion time. When there are two or more unavailable intervals on a single machine, Fu et al. (2009) have shown that the problem is exponentially inapproximable even when jobs’ weights are equal to their processing times and one has full knowledge of unavailability. So in this paper we consider the scheduling problem on a single machine with a single unavailable period. And we assume that every job has a weight proportional to its processing time. Based on whether the unavailable interval is due to a breakdown or an emergent job, we have breakdown model and emergent job model. We first show that no \(\tfrac{\sqrt{5}+1}{2}\)-competitive online algorithm exists for breakdown model, and no \(\tfrac{11-\sqrt{2}}{7}\)-competitive online algorithm exists for emergent job model. Then we show that the simple LPT (Largest Processing Time first) rule can give a 2-competitive ratio and 9/5-competitive ratio for breakdown model and emergent job model, respectively. We show the ratios are tight by examples. For offline case, we show that First Fit LPT (FF-LPT) rule can give a tight approximation ratio of 2 and 4/3 for breakdown model and emergent job model, respectively. Finally, our experimental results show that in practice, both LPT and FF- LPT perform very well and the performance improves when the number of jobs n increases. When n ≥ 50, the worst error ratio of LPT is about 8.7 %, and the worst error ratio of FF-LPT is about 0.7%. So in both cases, the error ratio is quite far from the theoretical bound.

Keywords

Completion Time Single Machine Competitive Ratio Total Weight Completion Time Idle Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Adiri, I., Bruno, J., Frostig, E., Rinnooy Kan, A.H.G.: Single machine flowtime scheduling with a single breakdown. Acta Informatica 26, 679–696 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Albers, S., Schmidt, G.: Scheduling with Unexpected Machine Breakdowns. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 269–280. Springer, Heidelberg (1999)Google Scholar
  3. 3.
    Arkin, R., Roundy, R.: Weighted tardiness scheduling on parallel machines with proportional weights. Operations Research 39, 64–81 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Breit, J.: Improved approximation for non-preemptive single machine flowtime scheduling with an availability constraint. European Journal of Operational Research 183(3), 516–524 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Diedrich, F., Schwarz, U.M.: A Framework for Scheduling with Online Availability. In: Kermarrec, A.-M., Bougé, L., Priol, T. (eds.) Euro-Par 2007. LNCS, vol. 4641, pp. 205–213. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Diedrich, F., Jansen, K.: Improved approximation algorithms for scheduling with fixed jobs. In: Proceeding of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 675–684 (2009)Google Scholar
  7. 7.
    Fu, B., Huo, Y., Zhao, H.: Exponential inapproximability and FPTAS for scheduling with availability constraints. Theoretical Computer Science 410, 2663–2674 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fu, B., Huo, Y., Zhao, H.: Approximation schemes for parallel machine scheduling with availability constraints. Discrete Applied Mathematics 159(15), 1555–1565 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    He, Y., Zhong, W., Gu, H.: Improved algorithms for two single machine scheduling problems. Theoretical Computer Science 363, 257–265 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kacem, I.: Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Computers & Industrial Engineering 54, 401–410 (2008)CrossRefGoogle Scholar
  11. 11.
    Kacem, I., Chu, C.: Worst-case analysis of the WSPT and MWSPT rules for single machine scheduling with one planned setup period. European Journal of Operational Research 187(3), 1080–1089 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kacem, I., Mahjoub, R.: Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Computers & Industrial Engineering 56(4), 1708–1712 (2009)CrossRefGoogle Scholar
  13. 13.
    Kellerer, H., Strusevich, V.A.: Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications. Algorithmica 57(4), 769–795 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lee, C.Y.: Machine scheduling with an availability constraints. Journal of Global Optimization 9, 363–382 (1996)CrossRefGoogle Scholar
  15. 15.
    Lee, C.Y., Liman, S.D.: Single machine flow-time scheduling with scheduled maintenance. Acta Informatica 29(4), 375–382 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ma, Y., Chu, C., Zuo, C.: A survey of scheduling with deterministic machine availability constraints. Computers & Industrial Engineering 58(2), 199–211 (2010)CrossRefGoogle Scholar
  17. 17.
    Sadfi, C., Penz, B., Rapine, C., Blazewicz, J., Formanowicz, P.: An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints. European Journal of Operational Research 161, 3–10 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Scharbrodt, M., Steger, A., Weisser, H.: Approximability of scheduling with fixed jobs. Journal of Scheduling 2, 267–284 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Tan, Z., He, Y.: Optimal online algorithm for scheduling on two identical machines with machine availability constraints. Information Processing Letters 83, 323–329 (2002)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yumei Huo
    • 1
  • Boris Reznichenko
    • 1
  • Hairong Zhao
    • 2
  1. 1.Department of Computer ScienceCollege of Staten Island, CUNYStaten IslandUSA
  2. 2.Department of Mathematics, Computer Science & StatisticsPurdue University CalumetHammondUSA

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