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Scheduling with non-renewable resources: minimizing the sum of completion times

Author

Listed:
  • Kristóf Bérczi

    (Eötvös Loránd University)

  • Tamás Király

    (Eötvös Loránd University)

  • Simon Omlor

    (TU Dortmund University)

Abstract

We consider single-machine scheduling with a non-renewable resource. In this setting, we are given a set of jobs, each characterized by a processing time, a weight, and a resource requirement. At fixed points in time, certain amounts of the resource are made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that each job has enough resource available at the start of its processing. The objective that we consider is the minimization of the sum of weighted completion times. The main contribution of the paper is a PTAS for the case of 0 processing times ( $$1|rm=1,p_j=0|\sum w_jC_j$$ 1 | r m = 1 , p j = 0 | ∑ w j C j ). In addition, we show strong NP-hardness of the case of unit resource requirements and weights ( $$1|rm=1,a_j=1|\sum C_j$$ 1 | r m = 1 , a j = 1 | ∑ C j ), thus answering an open question of Györgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a 3/2-approximation for the latter problem. Finally, we investigate a variant of the problem where processing times are 0 and the resource arrival times are unknown. We present a $$(4+\epsilon )$$ ( 4 + ϵ ) -approximation algorithm, together with a $$(4-\varepsilon )$$ ( 4 - ε ) -inapproximability result, for any $$\varepsilon >0$$ ε > 0 .

Suggested Citation

  • Kristóf Bérczi & Tamás Király & Simon Omlor, 2024. "Scheduling with non-renewable resources: minimizing the sum of completion times," Journal of Scheduling, Springer, vol. 27(2), pages 151-164, April.
  • Handle: RePEc:spr:jsched:v:27:y:2024:i:2:d:10.1007_s10951-024-00807-y
    DOI: 10.1007/s10951-024-00807-y
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    References listed on IDEAS

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    1. Alexander Grigoriev & Martijn Holthuijsen & Joris van de Klundert, 2005. "Basic scheduling problems with raw material constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(6), pages 527-535, September.
    2. Györgyi, Péter & Kis, Tamás, 2017. "Approximation schemes for parallel machine scheduling with non-renewable resources," European Journal of Operational Research, Elsevier, vol. 258(1), pages 113-123.
    3. Péter Györgyi & Tamás Kis, 2015. "Approximability of scheduling problems with resource consuming jobs," Annals of Operations Research, Springer, vol. 235(1), pages 319-336, December.
    4. Péter Györgyi & Tamás Kis, 2019. "Minimizing total weighted completion time on a single machine subject to non-renewable resource constraints," Journal of Scheduling, Springer, vol. 22(6), pages 623-634, December.
    5. Gafarov, Evgeny R. & Lazarev, Alexander A. & Werner, Frank, 2011. "Single machine scheduling problems with financial resource constraints: Some complexity results and properties," Mathematical Social Sciences, Elsevier, vol. 62(1), pages 7-13, July.
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