Online Makespan Scheduling of Linear Deteriorating Jobs on Parallel Machines

  • Sheng Yu
  • Jude-Thaddeus Ojiaku
  • Prudence W. H. Wong
  • Yinfeng Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)

Abstract

Traditional scheduling assumes that the processing time of a job is fixed. Yet there are numerous situations that the processing time increases (deteriorates) as the start time increases. Examples include scheduling cleaning or maintenance, fire fighting, steel production and financial management. Scheduling of deteriorating jobs was first introduced on a single machine by Browne and Yechiali, and Gupta and Gupta independently. In particular, lots of work has been devoted to jobs with linear deterioration. The processing time p j of job J j is a linear function of its start time s j , precisely, p j  = a j  + b j s j , where a j is the normal or basic processing time and b j is the deteriorating rate. The objective is to minimize the makespan of the schedule.

We first consider simple linear deterioration, i.e., p j  = b j s j . It has been shown that on m parallel machines, in the online-list model, LS (List Scheduling) is \((1+b_{\rm max})^{1-\frac{1}{m}}\)-competitive. We extend the study to the online-time model where each job is associated with a release time. We show that for two machines, no deterministic online algorithm is better than (1 + b max)-competitive, implying that the problem is more difficult in the online-time model than in the online-list model. We also show that LS is \((1+b_{\rm max})^{2(1-\frac{1}{m})}\)-competitive, meaning that it is optimal when m = 2.

Keywords

Completion Time Parallel Machine Single Machine Competitive Ratio Online Algorithm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alidaee, B., Womer, N.K.: Scheduling with time dependent processing times: Review and extensions. J. of Operational Research Society 50(7), 711–720 (1999)MATHGoogle Scholar
  2. 2.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    Browne, S., Yechiali, U.: Scheduling deteriorating jobs on a single processor. Operations Research 38(3), 495–498 (1990)MATHCrossRefGoogle Scholar
  4. 4.
    Cheng, M.B., Sun, S.J.: A heuristic MBLS algorithm for the two semi-online parallel machine scheduling problems with deterioration jobs. Journal of Shanghai University 11(5), 451–456 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cheng, T.C.E., Ding, Q.: The complexity of single machine scheduling with release times. Information Processing Letters 65(2), 75–79 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheng, T.C.E., Ding, Q., Lin, B.M.T.: A concise survey of scheduling with time-dependent processing times. European J. of OR 152(1), 1–13 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  8. 8.
    Gawiejnowicz, S.: Scheduling deteriorating jobs subject to job or machine availability constraints. European J. of OR 180(1), 472–478 (2007)MATHCrossRefGoogle Scholar
  9. 9.
    Gawiejnowicz, S.: Time-Dependent Scheduling. Springer, Berlin (2008)MATHGoogle Scholar
  10. 10.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell System Technical Journal 45(9), 1563–1581 (1966)Google Scholar
  11. 11.
    Gupta, J.N.D., Gupta, S.K.: Single facility scheduling with nonlinear processing times. Computers and Industrial Engineering 14(4), 387–393 (1988)CrossRefGoogle Scholar
  12. 12.
    Ji, M., He, Y., Cheng, T.C.E.: Scheduling linear deteriorating jobs with an availability constraint on a single machine. Theoretical Computer Science 362(1-3), 115–126 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kang, L.Y., Ng, C.T.: A note on a fully polynomial-time approximation scheme for parallel-machine scheduling with deteriorating jobs. International Journal of Production Economics 109(1-2), 108–184 (2007)CrossRefGoogle Scholar
  14. 14.
    Kononov, A.: Scheduling problems with linear increasing processing times. In: Zimmermann, U., et al. (eds.) Operations Research Proceedings 1996, Berlin, pp. 208–212 (1997)Google Scholar
  15. 15.
    Kunnathur, A.S., Gupta, S.K.: Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem. European Journal of Operation Research 47(1), 56–64 (1990)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lee, C.Y.: Machine scheduling with an availability constraint. Journal of Global Optimization 9(3-4), 395–416 (1996)MATHCrossRefGoogle Scholar
  17. 17.
    Lee, C.Y.: Machine scheduling with availability constraints. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, pp. 22.1–22.13. Chapman and Hall, Boca Raton (2004)Google Scholar
  18. 18.
    Lee, C.Y., Lei, L., Pinedo, M.: Current trend in deterministic scheduling. Annals of Operations Research 70, 1–42 (1997)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Lee, W., Wu, C., Chung, Y.: Scheduling deteriorating jobs on a single machine with release times. Computers and Industrial Engineering 54(3), 441–452 (2008)CrossRefGoogle Scholar
  20. 20.
    Mosheiov, G.: V-shaped policies for scheduling deteriorating jobs. Operations Research 39, 979–991 (1991)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mosheiov, G.: Scheduling jobs under simple linear deterioration. Computers and Operations Research 21(6), 653–659 (1994)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mosheiov, G.: Multi-machine scheduling with linear deterioration. INFOR: Information Systems and Operational Research 36(4), 205–214 (1998)Google Scholar
  23. 23.
    Ng, C.T., Li, S.S., Cheng, T.C.E., Yuan, J.J.: Preemptive scheduling with simple linear deterioration on a single machine. Theoretical Computer Science 411(40-42), 3578–3586 (2010)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Pinedo, M.: Scheduling: Theory, Algorithms, and Systems. Prentice-Hall, Upper Saddle River (2002)MATHGoogle Scholar
  25. 25.
    Pruhs, K., Sgall, J., Torng, E.: Online scheduling. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, pp, 15.1–15.42. Chapman and Hall, Boca Raton (2004)Google Scholar
  26. 26.
    Ren, C.R., Kang, L.Y.: An approximation algorithm for parallel machine scheduling with simple linear deterioration. Journal of Shanghai University 11(4), 351–354 (2007)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sheng Yu
    • 1
  • Jude-Thaddeus Ojiaku
    • 2
  • Prudence W. H. Wong
    • 2
  • Yinfeng Xu
    • 3
  1. 1.School of Business AdministrationZhongnan University of Economics and LawWuhanChina
  2. 2.Department of Computer ScienceUniversity of LiverpoolUK
  3. 3.School of ManagementXi’an Jiaotong UniversityChina

Personalised recommendations