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The Optimality Region for a Single-Machine Scheduling Problem with Bounded Durations of the Jobs and the Total Completion Time Objective

Author

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  • Yuri N. Sotskov

    (United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova Street 6, 220012 Minsk, Belarus)

  • Natalja G. Egorova

    (United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova Street 6, 220012 Minsk, Belarus)

Abstract

We study a single-machine scheduling problem to minimize the total completion time of the given set of jobs, which have to be processed without job preemptions. The lower and upper bounds on the job duration is the only information that is available before scheduling. Exact values of the job durations remain unknown until the completion of the jobs. We use the optimality region for the job permutation as an optimality measure of the optimal schedule. We investigate properties of the optimality region and derive O ( n ) -algorithm for calculating a quasi-perimeter of the optimality set (i.e., the sum of lengths of the optimality segments for n given jobs). We develop a fast algorithm for finding a job permutation having the largest quasi-perimeter of the optimality set. The computational results in constructing such permutations show that they are close to the optimal ones, which can be constructed for the factual durations of all given jobs.

Suggested Citation

  • Yuri N. Sotskov & Natalja G. Egorova, 2019. "The Optimality Region for a Single-Machine Scheduling Problem with Bounded Durations of the Jobs and the Total Completion Time Objective," Mathematics, MDPI, vol. 7(5), pages 1-21, April.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:382-:d:226287
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    References listed on IDEAS

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    1. Jian Yang & Gang Yu, 2002. "On the Robust Single Machine Scheduling Problem," Journal of Combinatorial Optimization, Springer, vol. 6(1), pages 17-33, March.
    2. Tsung-Chyan Lai & Yuri N. Sotskov & Natalja G. Egorova & Frank Werner, 2018. "The optimality box in uncertain data for minimising the sum of the weighted job completion times," International Journal of Production Research, Taylor & Francis Journals, vol. 56(19), pages 6336-6362, October.
    3. Ozelkan, Ertunga C. & Duckstein, Lucien, 1999. "Optimal fuzzy counterparts of scheduling rules," European Journal of Operational Research, Elsevier, vol. 113(3), pages 593-609, March.
    4. Richard L. Daniels & Panagiotis Kouvelis, 1995. "Robust Scheduling to Hedge Against Processing Time Uncertainty in Single-Stage Production," Management Science, INFORMS, vol. 41(2), pages 363-376, February.
    5. Wayne E. Smith, 1956. "Various optimizers for single‐stage production," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 59-66, March.
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