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Total completion time minimization in online hierarchical scheduling of unit-size jobs

Author

Listed:
  • Jueliang Hu

    (Zhejiang Sci-Tech University)

  • Yiwei Jiang

    (Zhejiang Sci-Tech University)

  • Ping Zhou

    (Zhejiang Business College)

  • An Zhang

    (Hangzhou Dianzi University)

  • Qinghui Zhang

    (Zhejiang Sci-Tech University)

Abstract

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least $$1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}$$ 1 + min { 1 m , max { 2 ⌈ x ⌉ + x ⌈ x ⌉ + 3 , 2 ⌊ x ⌋ + x ⌊ x ⌋ + 3 } , where $$x=\sqrt{2m+4}$$ x = 2 m + 4 . We then present a greedy algorithm with tight competitive ratio of $$1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}$$ 1 + 2 ( m - 1 ) m ( 4 m - 3 + 1 ) . The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when $$m=2$$ m = 2 , we propose an optimal online algorithm with competitive ratio of $$16$$ 16 $$/$$ / $$13$$ 13 , which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., $$n\rightarrow \infty $$ n → ∞ .

Suggested Citation

  • Jueliang Hu & Yiwei Jiang & Ping Zhou & An Zhang & Qinghui Zhang, 2017. "Total completion time minimization in online hierarchical scheduling of unit-size jobs," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 866-881, April.
  • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0011-2
    DOI: 10.1007/s10878-016-0011-2
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    References listed on IDEAS

    as
    1. Orion Chassid & Leah Epstein, 2008. "The hierarchical model for load balancing on two machines," Journal of Combinatorial Optimization, Springer, vol. 15(4), pages 305-314, May.
    2. Zhiyi Tan & An Zhang, 2010. "A note on hierarchical scheduling on two uniform machines," Journal of Combinatorial Optimization, Springer, vol. 20(1), pages 85-95, July.
    3. Yiwei Jiang, 2008. "Online scheduling on parallel machines with two GoS levels," Journal of Combinatorial Optimization, Springer, vol. 16(1), pages 28-38, July.
    4. An Zhang & Yiwei Jiang & Lidan Fan & Jueliang Hu, 2015. "Optimal online algorithms on two hierarchical machines with tightly-grouped processing times," Journal of Combinatorial Optimization, Springer, vol. 29(4), pages 781-795, May.
    5. Wu, Yong & Ji, Min & Yang, Qifan, 2012. "Optimal semi-online scheduling algorithms on two parallel identical machines under a grade of service provision," International Journal of Production Economics, Elsevier, vol. 135(1), pages 367-371.
    6. Li-ying Hou & Liying Kang, 2012. "Online scheduling on uniform machines with two hierarchies," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 593-612, November.
    7. Shlomo Karhi & Dvir Shabtay, 2013. "On the optimality of the TLS algorithm for solving the online-list scheduling problem with two job types on a set of multipurpose machines," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 198-222, July.
    8. Ming Liu & Chengbin Chu & Yinfeng Xu & Feifeng Zheng, 2011. "Semi-online scheduling on 2 machines under a grade of service provision with bounded processing times," Journal of Combinatorial Optimization, Springer, vol. 21(1), pages 138-149, January.
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    Cited by:

    1. Jiawei Zhang & Ling Wang & Lining Xing, 2019. "Large-scale medical examination scheduling technology based on intelligent optimization," Journal of Combinatorial Optimization, Springer, vol. 37(1), pages 385-404, January.

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