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Mathematical models and an effective exact algorithm for unrelated parallel machine scheduling with family setup times and machine cost

Author

Listed:
  • Kai Li

    (Hefei University of Technology
    Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education)

  • Fulong Xie

    (Hefei University of Technology)

  • Jianfu Chen

    (Hefei University of Technology)

  • Wei Xiao

    (Hohai University)

  • Tao Zhou

    (Hefei University of Technology
    Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education)

Abstract

This paper investigates unrelated parallel machine scheduling problems, considering machine- and sequence-dependent family setup times and machine usage costs to minimise the sum of the total weighted completion time and the total machine usage cost. The machine usage cost consists of a fixed cost and a variable cost proportional to the processing times of jobs. These features align with numerous real-world applications that include machine usage costs, for example, rental fees when customers rent machines via a cloud manufacturing platform. To address the problem, five integer programming models are developed from different perspectives. Afterwards, a modified branch-and-price algorithm (B&P) based on the set-partitioning model is proposed. To enhance the performance of B&P, we introduce two dynamic programming algorithms and a heuristic pricing algorithm, and the initial solution is generated by an improved variable neighbourhood search algorithm. Extensive experimental results show that the proposed B&P outperforms state-of-the-art mathematical models and algorithms. Notably, B&P can optimally solve instances with up to 10 machines, 100 jobs, and 12 families within half an hour. Interestingly, experimental results also show that the B&P performs significantly better when total completion time is dominated by the total variable cost of machine usage.

Suggested Citation

  • Kai Li & Fulong Xie & Jianfu Chen & Wei Xiao & Tao Zhou, 2025. "Mathematical models and an effective exact algorithm for unrelated parallel machine scheduling with family setup times and machine cost," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 47(1), pages 129-176, March.
  • Handle: RePEc:spr:orspec:v:47:y:2025:i:1:d:10.1007_s00291-024-00778-8
    DOI: 10.1007/s00291-024-00778-8
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    References listed on IDEAS

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    1. Allahverdi, Ali, 2015. "The third comprehensive survey on scheduling problems with setup times/costs," European Journal of Operational Research, Elsevier, vol. 246(2), pages 345-378.
    2. Clyde L. Monma & Chris N. Potts, 1989. "On the Complexity of Scheduling with Batch Setup Times," Operations Research, INFORMS, vol. 37(5), pages 798-804, October.
    3. Nickolas K. Freeman & John Mittenthal & Sharif H. Melouk, 2014. "Parallel-machine scheduling to minimize overtime and waste costs," IISE Transactions, Taylor & Francis Journals, vol. 46(6), pages 601-618, June.
    4. Feifeng Zheng & Yuhong Chen & Ming Liu & Yinfeng Xu, 2022. "Competitive analysis of online machine rental and online parallel machine scheduling problems with workload fence," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1060-1076, September.
    5. Giorgi Tadumadze & Simon Emde & Heiko Diefenbach, 2020. "Exact and heuristic algorithms for scheduling jobs with time windows on unrelated parallel machines," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 42(2), pages 461-497, June.
    6. Jun Kim & Hyun-Jung Kim, 2021. "Parallel machine scheduling with multiple processing alternatives and sequence-dependent setup times," International Journal of Production Research, Taylor & Francis Journals, vol. 59(18), pages 5438-5453, September.
    7. Stéphane Dauzère‐Pérès & Marc Sevaux, 2003. "Using Lagrangean relaxation to minimize the weighted number of late jobs on a single machine," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(3), pages 273-288, April.
    8. Kabir Rustogi & Vitaly A. Strusevich, 2013. "Parallel Machine Scheduling: Impact of Adding Extra Machines," Operations Research, INFORMS, vol. 61(5), pages 1243-1257, October.
    9. J. M. van den Akker & J. A. Hoogeveen & S. L. van de Velde, 1999. "Parallel Machine Scheduling by Column Generation," Operations Research, INFORMS, vol. 47(6), pages 862-872, December.
    10. Söhnke Maecker & Liji Shen, 2020. "Solving parallel machine problems with delivery times and tardiness objectives," Annals of Operations Research, Springer, vol. 285(1), pages 315-334, February.
    11. A J Ruiz-Torres & F J López & P J Wojciechowski & J C Ho, 2010. "Parallel machine scheduling problems considering regular measures of performance and machine cost," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(5), pages 849-857, May.
    12. Wang, Haibo & Alidaee, Bahram, 2019. "Effective heuristic for large-scale unrelated parallel machines scheduling problems," Omega, Elsevier, vol. 83(C), pages 261-274.
    13. Xu, Jun & Wang, Jun-Qiang & Liu, Zhixin, 2022. "Parallel batch scheduling: Impact of increasing machine capacity," Omega, Elsevier, vol. 108(C).
    14. Zhi‐Long Chen & Warren B. Powell, 2003. "Exact algorithms for scheduling multiple families of jobs on parallel machines," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 823-840, October.
    15. Tadumadze, Giorgi & Emde, Simon & Diefenbach, Heiko, 2020. "Exact and heuristic algorithms for scheduling jobs with time windows on unrelated parallel machines," Publications of Darmstadt Technical University, Institute for Business Studies (BWL) 120609, Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL).
    16. J.M. van den Akker & C.A.J. Hurkens & M.W.P. Savelsbergh, 2000. "Time-Indexed Formulations for Machine Scheduling Problems: Column Generation," INFORMS Journal on Computing, INFORMS, vol. 12(2), pages 111-124, May.
    17. Vanderbeck, F. & Wolsey, L. A., 1996. "An exact algorithm for IP column generation," LIDAM Reprints CORE 1242, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    18. Kerem Bülbül & Halil Şen, 2017. "An exact extended formulation for the unrelated parallel machine total weighted completion time problem," Journal of Scheduling, Springer, vol. 20(4), pages 373-389, August.
    19. Arash Zandi & Reza Ramezanian & Leslie Monplaisir, 2020. "Green parallel machines scheduling problem: A bi-objective model and a heuristic algorithm to obtain Pareto frontier," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 71(6), pages 967-978, June.
    20. Vallada, Eva & Ruiz, Rubén, 2011. "A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times," European Journal of Operational Research, Elsevier, vol. 211(3), pages 612-622, June.
    21. Yongkui Liu & Lihui Wang & Xi Vincent Wang & Xun Xu & Lin Zhang, 2019. "Scheduling in cloud manufacturing: state-of-the-art and research challenges," International Journal of Production Research, Taylor & Francis Journals, vol. 57(15-16), pages 4854-4879, August.
    22. Axel Lopez-Esteve & Federico Perea & Juan C. Yepes-Borrero, 2023. "GRASP algorithms for the unrelated parallel machines scheduling problem with additional resources during processing and setups," International Journal of Production Research, Taylor & Francis Journals, vol. 61(17), pages 6013-6029, September.
    23. Pierre Hansen & Nenad Mladenović & Jack Brimberg & José A. Moreno Pérez, 2019. "Variable Neighborhood Search," International Series in Operations Research & Management Science, in: Michel Gendreau & Jean-Yves Potvin (ed.), Handbook of Metaheuristics, edition 3, chapter 0, pages 57-97, Springer.
    24. Heydar, Mojtaba & Mardaneh, Elham & Loxton, Ryan, 2022. "Approximate dynamic programming for an energy-efficient parallel machine scheduling problem," European Journal of Operational Research, Elsevier, vol. 302(1), pages 363-380.
    25. 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.
    26. Like Zhang & Qianwang Deng & Guiliang Gong & Wenwu Han, 2020. "A new unrelated parallel machine scheduling problem with tool changes to minimise the total energy consumption," International Journal of Production Research, Taylor & Francis Journals, vol. 58(22), pages 6826-6845, November.
    27. Yunqiang Yin & Youhua Chen & Kaida Qin & Dujuan Wang, 2019. "Two-agent scheduling on unrelated parallel machines with total completion time and weighted number of tardy jobs criteria," Journal of Scheduling, Springer, vol. 22(3), pages 315-333, June.
    28. Allahverdi, Ali & Soroush, H.M., 2008. "The significance of reducing setup times/setup costs," European Journal of Operational Research, Elsevier, vol. 187(3), pages 978-984, June.
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