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**Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead.**
*(English)*
Zbl 1237.68036

Summary: We consider online scheduling of unit length jobs on m parallel-batch machines with lookahead to minimize makespan. A parallel-batch machine can handle up to \(b\) jobs simultaneously as a batch with processing time equal to the maximum processing time of the jobs included in the batch. In the lookahead model, at a time instant \(t\), an online algorithm can foresee all the jobs that will arrive in the time segment \((t,t+\beta ]\). In this paper, we deal with two variants: the unbounded model with \(b=\infty \) and the bounded model with \(b<\infty \). For the unbounded model, we present an optimal online algorithm for \(\beta \geq 1/m\), and provide a best possible online algorithm of competitive ratio \(1+\alpha_m\) for \(0\leq \beta <1/m\), where \(\alpha_m\) is the positive root of \((1+\alpha)^{(m+1)} = \alpha + 2 - \beta\sum_{i=1} ^m (1+\alpha)^i\). For the bounded model, we establish a lower bound with a form of a piecewise function, and provide an online algorithm with competitive ratios \(1+\alpha ^{\ast}\) for \(0\leq\beta\leq\frac{1}{6}\) and \(\frac{3}{2}\) for \(\beta>\frac{1}{6}\), respectively, where \(\alpha^{\ast}\) is the positive root of \(\alpha ^{2}+(\beta +1)\alpha +\beta - 1=0\). The online algorithm is the best possible when \(0\leq\beta\leq\frac{1}{6}\).

### MSC:

68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |

68W27 | Online algorithms; streaming algorithms |

90B35 | Deterministic scheduling theory in operations research |

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\textit{W. Li} et al., Inf. Process. Lett. 112, No. 7, 292--297 (2012; Zbl 1237.68036)

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### References:

[1] | Avramidis, A. N.; Healy, K. J.; Uzsoy, R., Control of a batch-processing machine: a computational approach, International Journal of Production Research, 36, 3167-3181 (1998) · Zbl 0946.90510 |

[2] | Deng, X. T.; Poon, C. K.; Zhang, Y. Z., Approximation algorithms in batch processing, Journal of Combinatorial Optimization, 7, 247-257 (2003) · Zbl 1053.90033 |

[3] | P. Keskinocak, Online algorithms with lookahead: A survey, ISYE working paper, 1999.; P. Keskinocak, Online algorithms with lookahead: A survey, ISYE working paper, 1999. |

[4] | Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H.G.; Shmoys, D. B., Sequencing and scheduling: Algorithms and complexity, (Graves, S. C.; Zipkin, P. H.; Rinnooy Kan, A. H.G., Logistics of Production and Inventory. Logistics of Production and Inventory, Handbooks in Operations Research and Management Science, vol. 4 (1993), North-Holland: North-Holland Amsterdam), 445-522 · Zbl 0798.90028 |

[5] | Li, W. J.; Yuan, J. J.; Cao, J. F.; Bu, H. L., Online scheduling of unit length jobs on a batching machine to maximize the number of early jobs with lookahead, Theoretical Computer Science, 410, 5182-5187 (2009) · Zbl 1194.68089 |

[6] | P.H. Liu, X.W. Lu, Y. Fang, A best possible deterministic on-line algorithm for minimizing makespan on parallel batch machines, Journal of Scheduling, doi:10.1007/s10951-009-0154-4; P.H. Liu, X.W. Lu, Y. Fang, A best possible deterministic on-line algorithm for minimizing makespan on parallel batch machines, Journal of Scheduling, doi:10.1007/s10951-009-0154-4 · Zbl 1280.68298 |

[7] | Mao, W.; Kincaid, R. K., A look-ahead heuristic for scheduling jobs with release dates on a single machine, Computers and Operations Research, 21, 1041-1050 (1994) · Zbl 0812.90069 |

[8] | Mandelbaum, M.; Shabtay, D., Scheduling unit length jobs on parallel machines with lookahead information, Journal of Scheduling, 14, 335-350 (2011) · Zbl 1229.90063 |

[9] | Nong, Q. Q.; Cheng, T. C.E.; Ng, C. T., An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines, Operations Research Letters, 36, 584-588 (2008) · Zbl 1210.90094 |

[10] | Poon, C. K.; Yu, W. C., On-line scheduling algorithms for a batch machine with finite capacity, Journal of Combinatorial Optimization, 9, 167-186 (2005) · Zbl 1079.90060 |

[11] | Tian, J.; Fu, R. Y.; Yuan, J. J., A best online algorithm for scheduling on two parallel batch machines, Theoretical Computer Science, 410, 2291-2294 (2009) · Zbl 1166.90341 |

[12] | Tian, J.; Cheng, T. C.E.; Ng, C. T.; Yuan, J. J., Online scheduling on unbounded parallel-batch machines to minimize the makespan, Information Processing Letters, 109, 1211-1215 (2009) · Zbl 1206.68072 |

[13] | Uzsoy, R.; Lee, C. Y.; Martin-Vega, L. A., A review of production planning and scheduling models in the semiconductor industry, part I: System characteristics, performance evaluation and production planning, IIE Transactions on Scheduling and Logistics, 24, 47-61 (1992) |

[14] | Yuan, J. J.; Ng, C. T.; Cheng, T. C.E., Best semi-online algorithms for unbounded parallel batch scheduling, Discrete Applied Mathematics, 159, 838-847 (2011) · Zbl 1213.68714 |

[15] | Zhang, G. C.; Cai, X. Q.; Wong, C. K., Online algorithms for minimizing makespan on batch processing machines, Naval Research Logistics, 48, 241-258 (2001) · Zbl 1018.90017 |

[16] | Zhang, G. C.; Cai, X. Q.; Wong, C. K., Optimal online algorithms for scheduling on parallel batch processing machines, IIE Transactions, 35, 175-181 (2003) |

[17] | Zheng, F. F.; Xu, Y. F.; Zhang, E., How much can lookahead help in online single machine scheduling, Information Processing Letters, 106, 70-74 (2008) · Zbl 1186.68079 |

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