Coupled task scheduling with time-dependent processing times

Abstract

The single machine coupled task scheduling problem includes a set of jobs, each with two separated tasks, and there is an exact delay between the tasks. We investigate the single machine coupled task scheduling problem with the objective of minimizing the makespan under identical processing time for the first task and identical delay period for all jobs, and the time-dependent processing time setting for the second task. Certain healthcare appointment scheduling problems can be modeled as the coupled task scheduling problem. Also, the incorporation of time-dependent processing time for the second task lets the human resource fatigue and the deteriorating health conditions be modeled. We provide optimal solution under certain conditions. In addition, we propose a dynamic program under the condition that the majority of jobs share the same time-dependent characteristic. We develop a heuristic for the general case and show that the heuristic performs well.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Ageev, A. A. (2018). Inapproximately lower bounds for open shop problems with exact delays. Approximation and online algorithms (pp. 45–55). New York: Springer.

    Google Scholar 

  2. Ageev, A. A., & Baburin, A. E. (2007). Approximation algorithms for UET scheduling problems with exact delays. Operations Research Letters, 35(4), 533–540.

    Article  Google Scholar 

  3. Ageev, A. A., & Kononov, A. V. (2007). Approximation algorithms for scheduling problems with exact delays. Approximation and online algorithms. Berlin: Springer.

    Google Scholar 

  4. Ahr, D., Békési, J., Galambos, G., Oswald, M., & Reinelt, G. (2004). An exact algorithm for scheduling identical coupled tasks. Mathematical Methods of Operations Research, 59(2), 193–203.

    Article  Google Scholar 

  5. Azadeh, A., Farahani, M. H., Torabzadeh, S., & Baghersad, M. (2014). Scheduling prioritized patients in emergency department laboratories. Computer Methods and Programs in Biomedicine, 117(2), 61–70.

    Article  Google Scholar 

  6. Baptiste, P. (2010). A note on scheduling identical coupled tasks in logarithmic time. Discrete Applied Mathematics, 158(5), 583–587.

    Article  Google Scholar 

  7. Békési, J., Galambos, G., Jung, M. N., Oswald, M., & Reinelt, G. (2014). A branch-and-bound algorithm for the coupled task problem. Mathematical Methods of Operations Research, 80(1), 47–81.

    Article  Google Scholar 

  8. Bessy, S., & Giroudeau, R. (2019). Parameterized complexity of a coupled-task scheduling problem. Journal of Scheduling, 22(3), 305–313.

    Article  Google Scholar 

  9. Blazewicz, J., Ecker, K., Kis, T., Potts, C. N., Tanas, M., & Whitehead, J. (2010). Scheduling of coupled tasks with unit processing times. Journal of Scheduling, 13(5), 453–461.

    Article  Google Scholar 

  10. Cheng, T. C. E., Ding, Q., & Lin, B. M. T. (2004). A concise survey of scheduling with time-dependent processing times. European Journal of Operational Research, 152, 1–13.

    Article  Google Scholar 

  11. Condotta, A., & Shakhlevich, N. (2012). Scheduling coupled-operation jobs with exact time-lags. Discrete Applied Mathematics, 160(16), 2370–2388.

    Article  Google Scholar 

  12. Condotta, A., & Shakhlevich, N. (2014). Scheduling patient appointments via multilevel template: a case study in chemotherapy. Operations Research for Health Care, 3(3), 129–144.

    Article  Google Scholar 

  13. Gawiejnowicz, S. (2008). Time-dependent scheduling. New York: Springer.

    Google Scholar 

  14. Graham, R., Lawler, E., Lenstra, J., & Kan, A. R. (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics, 5, 287–326.

    Article  Google Scholar 

  15. Gupta, J. N. D., & Gupta, S. K. (1988). Single facility scheduling with nonlinear processing times. Computers and Industrial Engineering, 14(4), 387–393.

    Article  Google Scholar 

  16. Gurobi Optimization, L. (2018). Gurobi Optimizer Reference Manual.

  17. Hwang, F. J., & Lin, B. M. T. (2011). Coupled-task scheduling on a single machine subject to a fixed-job-sequence. Computers and Industrial Engineering, 60(4), 690–698.

    Article  Google Scholar 

  18. Khatami, M. and Salehipour, A. (2020). A binary search algorithm for the general coupled task scheduling problem. 4OR, 1–19.

  19. Khatami, M., Salehipour, A., & Cheng, T. C. E. (2020). Coupled task scheduling with exact delays: literature review and models. European Journal of Operational Research, 282(1), 19–39.

    Article  Google Scholar 

  20. Kunnathur, A. S., & Gupta, S. K. (1990). Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem. European Journal of Operational Research, 47(1), 56–64.

    Article  Google Scholar 

  21. Legrain, A., Fortin, M.-A., Lahrichi, N., Rousseau, L.-M., & Widmer, M. (2015). Stochastic optimization of the scheduling of a radiotherapy center. Journal of Physics: Conference Series. Vol. 616. 1. IOP Publishing, 012008.

  22. Lehoux-Lebacque, V., Brauner, N., & Finke, G. (2015). Identical coupled task scheduling: polynomial complexity of the cyclic case. Journal of Scheduling, 18(6), 631–644.

    Article  Google Scholar 

  23. Leung, J. Y.-T., Li, H., & Zhao, H. (2007). Scheduling two-machine flow shops with exact delays. International Journal of Foundations of Computer Science, 18(02), 341–359.

    Article  Google Scholar 

  24. Li, H., & Zhao, H. (2007). Scheduling Coupled-Tasks on a Single Machine. IEEE Symposium on Computational Intelligence in Scheduling, 137–142.

  25. Liu, Z., Lu, J., Liu, Z., Liao, G., Zhang, H. H., & Dong, J. (2019). Patient scheduling in hemodialysis service. Journal of Combinatorial Optimization, 37(1), 337–362.

    Article  Google Scholar 

  26. Marinagi, C. C., Spyropoulos, C. D., Papatheodorou, C., & Kokkotos, S. (2000). Continual planning and scheduling for managing patient tests in hospital laboratories. Artificial Intelligence in Medicine, 20(2), 139–154.

    Article  Google Scholar 

  27. Mosheiov, G. (1994). Scheduling jobs under simple linear deterioration. Computers and Operations Re-search, 21(6), 653–659.

    Article  Google Scholar 

  28. Orman, A., & Potts, C. (1997). On the complexity of coupled-task scheduling. Discrete Applied Mathematics, 72(1), 141–154.

    Article  Google Scholar 

  29. Pérez, E., Ntaimo, L., Malavé, C. O., Bailey, C., & McCormack, P. (2013). Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine. Health Care Management Science, 16(4), 281–299.

    Article  Google Scholar 

  30. Pérez, E., Ntaimo, L., Wilhelm, W. E., Bailey, C., & McCormack, P. (2011). Patient and resource scheduling of multi-step medical procedures in nuclear medicine. IIE Transactions on Healthcare Systems Engineering, 1(3), 168–184.

    Article  Google Scholar 

  31. Shapiro, R. D. (1980). Scheduling coupled tasks. Naval Research Logistics Quarterly, 27(3), 489–498.

    Article  Google Scholar 

  32. Sherali, H. D., & Smith, J. C. (2005). Interleaving two-phased jobs on a single machine. Discrete Optimization, 2(4), 348–361.

    Article  Google Scholar 

  33. Simonin, G., Darties, B., Giroudeau, R., & König, J.-C. (2011). Isomorphic coupled-task scheduling problem with compatibility constraints on a single processor. Journal of Scheduling, 14(5), 501–509.

    Article  Google Scholar 

  34. Yu, W., Hoogeveen, H., & Lenstra, J. K. (2004). Minimizing make span in a two-machine flow shop with delays and unit-time operations is NP-hard. Journal of Scheduling, 7(5), 333–348.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments. Mostafa Khatami is the recipient of the UTS International Research Scholarship (IRS) and the UTS President’s Scholarship (UTSP). Amir Salehipour is the recipient of an Australian Research Council Discovery Early Career Researcher Award (project number DE170100234) funded by the Australian Government.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Amir Salehipour.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khatami, M., Salehipour, A. Coupled task scheduling with time-dependent processing times. J Sched 24, 223–236 (2021). https://doi.org/10.1007/s10951-020-00675-2

Download citation

Keywords

  • Coupled task scheduling
  • Time-dependent processing time
  • Simple linear processing time
  • Dynamic program
  • Heuristic
  • Healthcare scheduling