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Performance guarantees of local search for minsum scheduling problems

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Abstract

We study the worst-case performance guarantee of locally optimal solutions for the problem of minimizing the total weighted and unweighted completion time on parallel machine environments. Our method makes use of a mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the cost of the schedule. We apply the method to study the most basic local search heuristics for scheduling, namely jump and swap, and establish their worst-case performance in the case of unrelated, restricted related and restricted identical machines.

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Notes

  1. When there are no weights, these are taken to be 1, and therefore the rule is just the shortest processing time first rule.

  2. Note that it is possible that \(|{\mathcal {M}}'_j|=1\).

  3. Observe that job j has to be inserted on machine \(x_j^*\) at the appropriate position (defined by WSPT rule).

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Acknowledgements

We thank Fidaa Abed for providing the instance presented in § 5.4. We also thank two anonymous referees for many helpful suggestions that greatly improved the presentation of the paper. This work was partially supported by ANID Chile through grants BASAL AFB-180003 and BASAL AFB-170001.

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Authors and Affiliations

  1. Department of Industrial Engineering, Universidad de Chile, Santiago, Chile

    José R. Correa

  2. Department of Industrial Engineering, Universidad del Bío-Bío, Concepción, Chile

    Felipe T. Muñoz

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  1. José R. Correa
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Correspondence to José R. Correa.

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Correa, J.R., Muñoz, F.T. Performance guarantees of local search for minsum scheduling problems. Math. Program. 191, 847–869 (2022). https://doi.org/10.1007/s10107-020-01571-5

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