Abstract
This paper concerns the problem of scheduling jobs with unit processing time and nonidentical sizes on single or parallel identical batch machines. The objective is to minimize the total completion time of all jobs. We show that the worst-case ratio of the algorithm based on the bin-packing algorithm First Fit Increasing lies in the interval \(\left[ \frac{109}{82}, \frac{2+\sqrt{2}}{2}\right] \approx [1.3293, 1.7071]\) for the single machine case, and is no more than \(\frac{6+\sqrt{2}}{4}\approx 1.8536\) for the parallel machines case.
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The authors would like to thank the anonymous referees for their careful reading of this article and for helpful comments.
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A preliminary version of this paper appeared in the Proceedings of the 11th International Conference on Combinatorial Optimization and Applications, Lecture Notes in Computer Science, 10627, pp. 165–179. Supported by the National Natural Science Foundation of China (11671356, 11801505).
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Li, R., Tan, Z. & Zhu, Q. Batch scheduling of nonidentical job sizes with minsum criteria. J Comb Optim 42, 543–564 (2021). https://doi.org/10.1007/s10878-019-00419-9
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DOI: https://doi.org/10.1007/s10878-019-00419-9