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Fast Approximation for Scheduling One Machine

Author

Listed:
  • Federico Alonso-Pecina

    (Faculty of Account, Administration and Informatics, Universidad Autónoma del Estado de Morelos, Cuernavaca 62209, Mexico)

  • José Alberto Hernández

    (Faculty of Account, Administration and Informatics, Universidad Autónoma del Estado de Morelos, Cuernavaca 62209, Mexico)

  • José Maria Sigarreta

    (Facultad de Matemáticas UAGro, Universidad Autónoma de Guerrero, Acapulco 39650, Mexico)

  • Nodari Vakhania

    (Centro de Investigación en Ciencias, UAEMor, On sabbatical leave at Facultad de Matemáticas UAGro, Universidad Autónoma de Guerrero, Acapulco 39650, Mexico)

Abstract

We propose an approximation algorithm for scheduling jobs with release and delivery times on a single machine with the objective to minimize the makespan. The algorithm is based on an implicit enumeration of the set of complete solutions in a search tree. By analyzing specific structural properties of the solutions created in each branch of the solution tree, a certain approximation factor of each solution from that branch is calculated. Our algorithm guarantees approximation factor 1 + 1 / κ for a generated solution if there are κ jobs with a specified property in that solution (typically, the longer the length of the path from the root to the node representing that solution in the solution tree, the larger the κ value). We have carried out an extensive computational study to verify the practical performance of our algorithm and the effectiveness of the approximation factor provided by us. While our problem instances are generated randomly, we have discarded a considerable number of the instances, ones which were already solved optimally by the earlier known dominance rules. For the vast majority of the tested problem instances, within a short running time of our algorithm parameter κ becomes sufficiently large so that the approximation factor which the algorithm guarantees becomes better than that provided by the earlier known approximation algorithms.

Suggested Citation

  • Federico Alonso-Pecina & José Alberto Hernández & José Maria Sigarreta & Nodari Vakhania, 2020. "Fast Approximation for Scheduling One Machine," Mathematics, MDPI, vol. 8(9), pages 1-18, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1524-:d:409981
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    References listed on IDEAS

    as
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    4. Graham McMahon & Michael Florian, 1975. "On Scheduling with Ready Times and Due Dates to Minimize Maximum Lateness," Operations Research, INFORMS, vol. 23(3), pages 475-482, June.
    5. Leslie A. Hall & David B. Shmoys, 1992. "Jackson's Rule for Single-Machine Scheduling: Making a Good Heuristic Better," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 22-35, February.
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