Skip to main content
SpringerLink
Log in

Scheduling with Hierarchies and Overload Cost

  • Conference paper
  • First Online:
Theoretical Computer Science (NCTCS 2023)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1944))

Included in the following conference series:

  • 102 Accesses

Abstract

In this paper, we consider the problem of scheduling with hierarchies and overload cost (SHOC). Given a set of jobs \(\mathcal{J} =\{J_1,\ldots ,J_n\}\), a set of two hierarchical identical parallel machines \(\mathcal{M}=\{M_1,M_2\}\), a processing time function p and a hierarchy function g on the job set \(\mathcal J\), a regular working time \(L_0\) of the machines, a start-up cost \(c_0\) and a cost \(c_1\) of per unit overload, and the hierarchies of the machines \(M_1\) and \(M_2\) are 1 and 2 respectively. Each machine can only process the jobs whose hierarchies are no less than the hierarchy of this machine. We are asked to assign all jobs of \(\mathcal{J}\) to the machines \(M_1\) and \(M_2\), and if the total processing time \(L_i\,(i=1,2)\) of any machine is more than \(L_0\), a charge of \(c_1\) should be paid for per unit overload. The objective is to minimize the total cost of processing all the jobs. We design a \(1+\frac{1}{20}c_1/c_0\)-approximation algorithm to solve our problem by using the LPT method. And based on four characteristics of optimal solutions and two dynamic programmings, we give a pseudo-polynomial time algorithm to find an optimal solution in \(\mathcal{O}(nL_0)\) time.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bar-Noy, A., Freund, A., Naor, J.: On-line load balancing in a hierarchical server topology. SIAM J. Comput. 31(2), 527–549 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X., Sterna, M., Han, X., Blazewicz, J.: Scheduling on parallel identical machines with late work criterion: offline and online cases. Int. J. Sched. 19(6), 729–736 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X., Kovalev, S., Liu, Y., Sterna, M., Chalamon, I., Blazewicz, J.: Semi-online scheduling on two identical machines with a common due date to maximize total early work. Discret. Appl. Math. 290, 71–78 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coffman, E.G., Jr., Lueker, G.S.: Approximation algorithms for extensible bin packing. J. Sched. 9(1), 63–69 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dai, B., Li, J., Li, W.: Semi-online hierarchical scheduling for bag-of-tasks on two machines. In: 2nd International Conference on Computer Science and Artificial Intelligence Proceedings, pp. 609–614. Association for Computing Machinery, New York, United States (2018)

    Google Scholar 

  7. Dell’Olmo, P., Kellerer, H., Speranza, M.G., Tuza, Z.: A 13/12 approximation algorithm for bin packing with extendable bins. Inf. Process. Lett. 65(5), 229–233 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dell’Olmo, P., Speranza, M.G.: Approximation algorithms for partitioning small items in unequal bins to minimize the total size. Discret. Appl. Math. 4(1–3), 181–191 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Epstein, L., Tassa, T.: Vector assignment schemes for asymmetric settings. Acta Informatica 42, 501–514 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Graham, R.L.: Bounds on multiprocessor timing anomalies. SIAM J. Appl. Math. 17(2), 416–29 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. Graham, R.L., Lawler, E.L., Lenstra, J.K., Kan, A.R.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5, 287–326 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34(1), 144–162 (1987)

    Article  MathSciNet  Google Scholar 

  13. Hwang, H.C., Chang, S.Y., Lee, K.: Parallel machine scheduling under a grade of service provision. Comput. Oper. Res. 31(12), 2055–2061 (2004)

    Article  MATH  Google Scholar 

  14. Ji, M., Cheng, T.C.E.: An FPTAS for parallel-machine scheduling under a grade of service provision to minimize makespan. Inf. Process. Lett. 108(4), 171–174 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, Y.: Online scheduling on parallel machines with two GoS levels. J. Comb. Optim. 16(1), 28–38 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Levin, A.: Approximation schemes for the generalized extensible bin packing problem. Algorithmica 84(2), 325–343 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, J., Li, W., Li, J.: Polynomial approximation schemes for the max-min allocation problem under a grade of service provision. Discrete Math. Algorithms Appl. 1(03), 355–368 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ou, J., Leung, J.Y.T., Li, C.L.: Scheduling parallel machines with inclusive processing set restrictions. Nav. Res. Logist. 55(4), 328–338 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Park, J., Chang, S.Y., Lee, K.: Online and semi-online scheduling of two machines under a grade of service provision. Oper. Res. Lett. 34(6), 692–696 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Speranza, M.G., Tuza, Z.: On-line approximation algorithms for scheduling tasks on identical machines withextendable working time. Ann. Oper. Res. 86, 491–506 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wei, Q., Wu, Y., Cheng, T.C.E., Sun, F., Jiang, Y.: Online hierarchical parallel-machine scheduling in shared manufacturing to minimize the total completion time. J. Oper. Res. Soc. 74, 1–23 (2022)

    Google Scholar 

  22. Woeginger, G.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS J. Comput. 12(1), 57–74 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, G., Li, W.: Semi-online machine covering on two hierarchical machines with discrete processing times. In: Li, L., Lu, P., He, K. (eds.) NCTCS 2018. CCIS, vol. 882, pp. 1–7. Springer, Singapore (2018). https://doi.org/10.1007/978-981-13-2712-4_1

    Chapter  Google Scholar 

  24. Xiao, M., Liu, X., Li, W., Chen, X., Sterna, M., Blazewicz, J.: Online and semi-online scheduling on two hierarchical machines with a common due date to maximize the total early work. arXiv preprint arXiv:2209.08704, https://doi.org/10.48550/arXiv.2209.08704 (2022)

  25. Xiao, M., Li, W.: Online early work maximization on three hierarchical machines with a common due date. In: Li, M., Sun, X. (eds.) Frontiers of Algorithmic Wisdom. IJTCS-FAW 2022. LNCS, vol. 13461, pp. 99–109. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-20796-9_8

  26. Xiao, M., Du, Y., Li, W., Yang, J.: Semi-online machine covering problem on three hierarchical machines with bounded processing times. J. Oper. Res. Soc. China 1–13 (2023). https://doi.org/10.1007/s40305-023-00477-1

  27. Ye, D., Zhang, G.: On-line extensible bin packing with unequal bin sizes. Discrete Math. Theor. Comput. Sci. 11(1), 141–152 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Zhang, A., Jiang, Y., Tan, Z.: Online parallel machines scheduling with two hierarchies. Theoret. Comput. Sci. 410(38–40), 3597–3605 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work is supported by the General Program of Yunnan Province Science and Technology Department [No. 202001BB050062], and the Postgraduate Research and Innovation Foundation of Yunnan University under Grant KC-22221129.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Yunnan University, Kunming, 650504, China

    Yaru Yang, Wuyun Fu & Honglin Ding

Authors
  1. Yaru Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wuyun Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Honglin Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Honglin Ding .

Editor information

Editors and Affiliations

  1. National University of Defense Technology, Changsha, China

    Zhiping Cai

  2. University of Electronic Science and Technology, Chengdu, China

    Mingyu Xiao

  3. Chinese Academy of Sciences, Beijing, China

    Jialin Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Yang, Y., Fu, W., Ding, H. (2024). Scheduling with Hierarchies and Overload Cost. In: Cai, Z., Xiao, M., Zhang, J. (eds) Theoretical Computer Science. NCTCS 2023. Communications in Computer and Information Science, vol 1944. Springer, Singapore. https://doi.org/10.1007/978-981-99-7743-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-981-99-7743-7_8

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-99-7742-0

  • Online ISBN: 978-981-99-7743-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation