Abstract
In this paper, we consider four semi-online early work maximization problems on two hierarchical uniform machines \(M_1\) and \(M_2\), where machine \(M_1\) with speed \(s>0\) is available for all jobs and machine \(M_2\) with speed 1 is only available for high-hierarchy jobs. When the total size of all jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{1+s,\frac{2+2s}{1+2s}\}\). When the total size of low-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min {\{1+s, \frac{\sqrt{9\,s^2+10\,s+1}-s-1}{2\,s}}\}\). When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{\sqrt{s+1}, \sqrt{s^2+2\,s+2}-s\}\). When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound \(\frac{2s+2}{s+2}\) for the case \(s\le \frac{2}{3}\), and an optimal online algorithm with a competitive ratio of \(\frac{3s+3}{3s+2}\) for the case \(s>\frac{2}{3}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
No datasets were generated or analyzed during the current study.
References
Chen X, Sterna M, Han X, Blazewicz J (2016) Scheduling on parallel identical machines with late work criterion: offline and online cases. J Sched 19(6):729–736
Chen X, Kovalev S, Liu Y, Sterna M, Chalamon I, Blazewicz J (2021) Semi-online scheduling on two identical machines with a common due date to maximize total early work. Discret Appl Math 290:71–78
Dosa G, Speranza MG, Tuza Z (2011) Two uniform machines with nearly equal speeds: unified approach to known sum and known optimum in semi-online scheduling. J Comb Optim 21(4):458–480
Dosa G, Fugenschuh A, Tan Z, Tuza Z, Wesek K (2019) Tight lower bounds for semi-online scheduling on two uniform machines with known optimum. CEJOR 27(4):1107–1130
Dwibedy D, Mohanty R (2022) Semi-online scheduling: a survey. Comput Oper Res 139:105646
Epstein L, Noga J, Seiden S, Sgall J, Woeginger G (2001) Randomized on-line scheduling on two uniform machines. J Sched 4:71–92
Faigle U, Kern W, Turan G (1989) On the performance of on-line algorithms for partition problems. Acta Cybern 9(2):107–119
Kellerer H, Kotov V, Speranza MG, Tuza Z (1997) Semi on-line algorithms for the partition problem. Oper Res Lett 21(5):235–242
Lin L, Tan Z (2020) Online scheduling on parallel machines: a survey. Sci Sin Math 50(9):1183–1200
Lu X, Liu Z (2013) Semi-online scheduling problems on two uniform machines under a grade of service provision. Theor Comput Sci 489–490:58–66
Luo T, Xu Y (2014) Semi-online scheduling on two machines with GoS levels and partial information of processing time. Sci World J 2:576234
Ng CT, Tan Z, He Y, Cheng TCE (2009) Two semi-online scheduling problems on two uniform machines. Theor Comput Sci 410:776–792
Park J, Chang SY, Lee K (2006) Online and semi-online scheduling of two machines under a grade of service provision. Oper Res Lett 34(6):692–696
Sterna M (2021) Late and early work scheduling: a survey. Omega 104:102453
Tan Z, Zhang A (2010) A note on hierarchical scheduling on two uniform machines. J Comb Optim 20:85–95
Xiao M, Liu X, Li W (2021) Semi-online early work maximization problem on two hierarchical machines with partial information of processing time. In: Proceeding of the 15th international conference on algorithmic aspects in information and management, pp 146–156
Xiao M, Bai X, Li W (2022a) Online early work maximization problem on two hierarchical machines with buffer or rearrangements. In: Proceeding of the 16th international conference on algorithmic aspects in information and management, pp 46–54
Xiao M, Liu X, Li W, Chen X, Sterna M, Blazewicz J (2022b) Online and semi-online scheduling on two hierarchical machines with a common due date to maximize the total early work. arXiv:2209.08704
Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments and suggestions.
Funding
The work is supported in part by the National Natural Science Foundation of China [No. 12071417].
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The preliminary result of this paper has been published in the proceedings of the 15th International Conference on Algorithmic Aspects in Information and Management.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xiao, M., Liu, X. & Li, W. Semi-online early work maximization problems on two hierarchical uniform machines with partial information of processing time. J Comb Optim 46, 21 (2023). https://doi.org/10.1007/s10878-023-01086-7
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01086-7