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A Branch-and-Price Algorithm for Parallel Machine Scheduling Using ZDDs and Generic Branching

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  • Daniel Kowalczyk

    (ORSTAT, Faculty of Economics and Business, KU Leuven, 3000 Leuven, Belgium)

  • Roel Leus

    (ORSTAT, Faculty of Economics and Business, KU Leuven, 3000 Leuven, Belgium)

Abstract

We study the parallel machine scheduling problem to minimize the sum of the weighted completion times of the jobs to be scheduled (problem Pm ∥∑ w j C j in the standard three-field notation). We use the set covering formulation that was introduced by van den Akker et al. [van den Akker J, Hoogeveen J, van de Velde S (1999) Parallel machine scheduling by column generation. Oper. Res. 47(6):862–872.] for this problem, and we improve the computational performance of their branch-and-price (B&P) algorithm by a number of techniques, including a different generic branching scheme, zero-suppressed binary decision diagrams (ZDDs) to solve the pricing problem, dual-price smoothing as a stabilization method, and Farkas pricing to handle infeasibilities. We report computational results that show the effectiveness of the algorithmic enhancements, which depends on the characteristics of the instances. To the best of our knowledge, we are also the first to use ZDDs to solve the pricing problem in a B&P algorithm for a scheduling problem. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0809 .

Suggested Citation

  • Daniel Kowalczyk & Roel Leus, 2018. "A Branch-and-Price Algorithm for Parallel Machine Scheduling Using ZDDs and Generic Branching," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 768-782, November.
    • Handle: RePEc:inm:orijoc:v:30:y:2018:i:4:p:768-782
    DOI: 10.1287/ijoc.2018.0809
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    3. Margarita P. Castro & Andre A. Cire & J. Christopher Beck, 2022. "Decision Diagrams for Discrete Optimization: A Survey of Recent Advances," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2271-2295, July.
    4. Novak, Antonin & Sucha, Premysl & Novotny, Matej & Stec, Richard & Hanzalek, Zdenek, 2022. "Scheduling jobs with normally distributed processing times on parallel machines," European Journal of Operational Research, Elsevier, vol. 297(2), pages 422-441.
    5. Kramer, Arthur & Dell’Amico, Mauro & Iori, Manuel, 2019. "Enhanced arc-flow formulations to minimize weighted completion time on identical parallel machines," European Journal of Operational Research, Elsevier, vol. 275(1), pages 67-79.
    6. Teobaldo Bulhões & Ruslan Sadykov & Anand Subramanian & Eduardo Uchoa, 2020. "On the exact solution of a large class of parallel machine scheduling problems," Journal of Scheduling, Springer, vol. 23(4), pages 411-429, August.
    7. Kramer, Arthur & Iori, Manuel & Lacomme, Philippe, 2021. "Mathematical formulations for scheduling jobs on identical parallel machines with family setup times and total weighted completion time minimization," European Journal of Operational Research, Elsevier, vol. 289(3), pages 825-840.

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