# American Institute of Mathematical Sciences

November  2021, 17(6): 3551-3558. doi: 10.3934/jimo.2020132

## Research on the parallel–batch scheduling with linearly lookahead model

 College of Science, Zhongyuan University of Technology, Zhengzhou, Henan 450007, China

* Corresponding author: Chengwen Jiao

Received  April 2020 Revised  June 2020 Published  November 2021 Early access  August 2020

Fund Project: The author is supported by NSFC under grant number 11701595

In this paper, we consider the new online scheduling model with linear lookahead intervals, which has the character that at any time $t$, one can foresee the jobs that will coming in the time interval $(t, \lambda t+\beta ]$ with $\lambda\geq1, \beta\geq 0$. We consider online scheduling of unit length jobs on $m$ identical parallel-batch machines under this new lookahead model to minimize the maximum flowtime and the makespan, respectively. We give some lower bounds for these problems with the batch capacity $b = \infty$ and $b<\infty$, respectively. And for the bounded model to minimize makespan, we give an online algorithm with competitive ratio $1+\alpha$ for $1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6}$ and $\frac{3}{2}$ for $\lambda\geq1, 0\leq\beta<1$, where $\alpha$ is the positive root of $\lambda\alpha^2+(\lambda+\beta)\alpha+\beta-1 = 0$. The online algorithm is best possible when $1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6}$.

Citation: Chengwen Jiao, Qi Feng. Research on the parallel–batch scheduling with linearly lookahead model. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3551-3558. doi: 10.3934/jimo.2020132
##### References:
 [1] X. T. Deng, C. K. Poon and Y. Z. Zhang, Approximation algorithms in batch processing, Journal of Combinatorial Optimization, 7 (2003), 247-257.  doi: 10.1023/A:1027316504440. [2] C. W. Jiao, J. J. Yuan and Q. Feng, Online algorithms for scheduling unit length jobs on unbounded parallel-batch machines with linearly lookahead, Asia-Pacific Journal of Operational Research, 36 (2019), 1950024, 8 pp. doi: 10.1142/S0217595919500246. [3] P. Keskinocak, Online algorithms with lookahead: A survey, ISYE working paper, 1999. [4] W. J. Li, J. J. Yuan, J. F. Cao and H. L. Bu, Online scheduling of unit length jobs on a batching machine to maximize the number of early jobs with lookahead, Theoretical Computer Science, 410 (2009), 5182-5187.  doi: 10.1016/j.tcs.2009.07.056. [5] W. H. Li, Z. K. Zhang and S. F. Yang, Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead, Information Processing Letters, 112 (2012), 292-297.  doi: 10.1016/j.ipl.2012.01.002. [6] P. H. Liu, X. W. Lu and Y. Fang, A best possible deterministic on-line algorithm for minimizing makespan on parallel batch machines, Journal of Scheduling, 15 (2012), 77-81.  doi: 10.1007/s10951-009-0154-4. [7] M. Mandelbaum and D. Shabtay, Scheduling unit length jobs on parallel machines with lookahead information, Journal of Scheduling, 14 (2011), 335-350.  doi: 10.1007/s10951-010-0192-y. [8] W. Mao and R. K. Kincaid, A lookahead heuristic for scheduling jobs with release dates on a single machine, Computers and Operations Research, 21 (1994), 1041-1050. [9] C. K. Poon and W. C. Yu, On-line scheduling algorithms for a batch machine with finite capacity, Journal of Combinatorial Optimization, 9 (2005), 167-186.  doi: 10.1007/s10878-005-6855-5. [10] J. Tian, T. C. E. Cheng, C. T. Ng and J. J. Yuan, Online scheduling on unbound parallel-batch machines to minimize the makespan, Information Processing Letters, 109 (2009), 1211-1215.  doi: 10.1016/j.ipl.2009.08.008. [11] G. C. Zhang, X. Q. Cai and C. K. Wong, Online algorithms for minimizing makespan on batch processing machines, Naval Research Logistics, 48 (2001), 241-258.  doi: 10.1002/nav.5. [12] G. C. Zhang, X. Q. Cai and C. K. Wong, Optimal online algorithms for scheduling on parallel batch processing machines, IIE Transactions, 35 (2003), 175-181. [13] F. F. Zheng, Y. F. Xu and E. Zhang, How much can lookahead help in online single machine scheduling, Information Processing Letters, 106 (2008), 70-74.  doi: 10.1016/j.ipl.2007.10.008.

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##### References:
 [1] X. T. Deng, C. K. Poon and Y. Z. Zhang, Approximation algorithms in batch processing, Journal of Combinatorial Optimization, 7 (2003), 247-257.  doi: 10.1023/A:1027316504440. [2] C. W. Jiao, J. J. Yuan and Q. Feng, Online algorithms for scheduling unit length jobs on unbounded parallel-batch machines with linearly lookahead, Asia-Pacific Journal of Operational Research, 36 (2019), 1950024, 8 pp. doi: 10.1142/S0217595919500246. [3] P. Keskinocak, Online algorithms with lookahead: A survey, ISYE working paper, 1999. [4] W. J. Li, J. J. Yuan, J. F. Cao and H. L. Bu, Online scheduling of unit length jobs on a batching machine to maximize the number of early jobs with lookahead, Theoretical Computer Science, 410 (2009), 5182-5187.  doi: 10.1016/j.tcs.2009.07.056. [5] W. H. Li, Z. K. Zhang and S. F. Yang, Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead, Information Processing Letters, 112 (2012), 292-297.  doi: 10.1016/j.ipl.2012.01.002. [6] P. H. Liu, X. W. Lu and Y. Fang, A best possible deterministic on-line algorithm for minimizing makespan on parallel batch machines, Journal of Scheduling, 15 (2012), 77-81.  doi: 10.1007/s10951-009-0154-4. [7] M. Mandelbaum and D. Shabtay, Scheduling unit length jobs on parallel machines with lookahead information, Journal of Scheduling, 14 (2011), 335-350.  doi: 10.1007/s10951-010-0192-y. [8] W. Mao and R. K. Kincaid, A lookahead heuristic for scheduling jobs with release dates on a single machine, Computers and Operations Research, 21 (1994), 1041-1050. [9] C. K. Poon and W. C. Yu, On-line scheduling algorithms for a batch machine with finite capacity, Journal of Combinatorial Optimization, 9 (2005), 167-186.  doi: 10.1007/s10878-005-6855-5. [10] J. Tian, T. C. E. Cheng, C. T. Ng and J. J. Yuan, Online scheduling on unbound parallel-batch machines to minimize the makespan, Information Processing Letters, 109 (2009), 1211-1215.  doi: 10.1016/j.ipl.2009.08.008. [11] G. C. Zhang, X. Q. Cai and C. K. Wong, Online algorithms for minimizing makespan on batch processing machines, Naval Research Logistics, 48 (2001), 241-258.  doi: 10.1002/nav.5. [12] G. C. Zhang, X. Q. Cai and C. K. Wong, Optimal online algorithms for scheduling on parallel batch processing machines, IIE Transactions, 35 (2003), 175-181. [13] F. F. Zheng, Y. F. Xu and E. Zhang, How much can lookahead help in online single machine scheduling, Information Processing Letters, 106 (2008), 70-74.  doi: 10.1016/j.ipl.2007.10.008.
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