Journal of Scheduling

, Volume 21, Issue 4, pp 461–482 | Cite as

Single-machine scheduling with workload-dependent tool change durations and equal processing time jobs to minimize total completion time

  • Zhijun Xu
  • Dehua XuEmail author


We consider a single-machine tool change scheduling problem where tool change durations are workload-dependent. The processing times of all the jobs are the same. The objective is to determine the number of tool change activities, the start time and the completion time of each tool change activity jointly and schedule all the jobs to the machine such that the total completion time of the jobs is minimized. For the case where the tool change duration function is concave, we present a linear time optimal algorithm. For the case where the tool change duration function is convex, we convert it into a convex integer quadratic programming problem with fixed dimension and then propose two polynomial time algorithms for it. We also study some special cases for which optimal schedules can be obtained directly. For the case where the tool change duration function is linear, we present all the optimal schedules.


Scheduling Tool change Workload-dependent Total completion time 



We thank the referees for their valuable comments which improved the paper substantially. This research was supported by the National Natural Science Foundation of China (71201022).


  1. Akturk, M. S., Ghosh, J. B., & Gunes, E. D. (2003). Scheduling with tool changes to minimize total completion time: A study of heuristics and their performance. Naval Research Logistics, 50(1), 15–30.CrossRefGoogle Scholar
  2. Akturk, M. S., Ghosh, J. B., & Gunes, E. D. (2004). Scheduling with tool changes to minimize total completion time: Basic results and SPT performance. European Journal of Operational Research, 157(3), 784–790.CrossRefGoogle Scholar
  3. Akturk, M. S., Ghosh, J. B., & Kayan, R. K. (2007). Scheduling with tool changes to minimize total completion time under controllable machining conditions. Computers & Operations Research, 34(7), 2130–2146.CrossRefGoogle Scholar
  4. Baptiste, P., & Schieber, B. (2003). A note on scheduling tall/small multiprocessor tasks with unit processing time to minimize maximum tardiness. Journal of Scheduling, 6(4), 395–404.CrossRefGoogle Scholar
  5. Baptiste, P., & Timkovsky, V. G. (2001). On preemption redundancy in scheduling unit processing time jobs on two parallel machines. Operations Research Letters, 28(5), 205–212.CrossRefGoogle Scholar
  6. Birks, M., & Fung, S. P. Y. (2013). Temperature aware online algorithms for scheduling equal length jobs. Theoretical Computer Science, 508, 54–65.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. Brucker, P., & Shakhlevich, N. V. (2016). Necessary and sufficient optimality conditions for scheduling unit time jobs on identical parallel machines. Journal of Scheduling, 19, 659–685.CrossRefGoogle Scholar
  9. Chen, J. S. (2008). Optimization models for the tool change scheduling problem. Omega, 36(5), 888–894.CrossRefGoogle Scholar
  10. Gerstl, E., & Mosheiov, G. (2013a). Due-window assignment problems with unit-time jobs. Applied Mathematics and Computation, 220, 487–495.CrossRefGoogle Scholar
  11. Gerstl, E., & Mosheiov, G. (2013b). Due-window assignment with identical jobs on parallel uniform machines. European Journal of Operational Research, 229(1), 41–47.CrossRefGoogle Scholar
  12. Gerstl, E., & Mosheiov, G. (2013c). An improved algorithm for due-window assignment on parallel identical machines with unit-time jobs. Information Processing Letters, 113(19), 754–759.CrossRefGoogle Scholar
  13. Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. R. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.CrossRefGoogle Scholar
  14. Heinz, S. (2005). Complexity of integer quasiconvex polynomial optimization. Journal of Complexity, 21(4), 543–556.CrossRefGoogle Scholar
  15. Hemmecke, R., Köppe, M., Lee, J., & Weismantel, R. (2010). Nonlinear integer programming. In 50 years of integer programming 1958–2008 (pp. 561–618). New York, NY: Springer.Google Scholar
  16. Jackson, J. R. (1956). An extension of Johnson’s results on job IDT scheduling. Naval Research Logistics Quarterly, 3(3), 201–203.CrossRefGoogle Scholar
  17. Janiak, A., Janiak, W., Kovalyov, M. Y., & Werner, F. (2012). Soft due window assignment and scheduling of unit-time jobs on parallel machines. 4OR, 10(4), 347–360.CrossRefGoogle Scholar
  18. Johnson, S. M. (1954). Optimal two-and three-stage production schedules with setup times included. Naval Research Logistics Quarterly, 1(1), 61–68.CrossRefGoogle Scholar
  19. Lenstra, H. W, Jr. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 538–548.CrossRefGoogle Scholar
  20. Li, C. L., Mosheiov, G., & Yovel, U. (2008). An efficient algorithm for minimizing earliness, tardiness, and due-date costs for equal-sized jobs. Computers & Operations Research, 35(11), 3612–3619.CrossRefGoogle Scholar
  21. Li, W., Yuan, J., & Yang, S. (2014). Online scheduling of incompatible unit-length job families with lookahead. Theoretical Computer Science, 543, 120–125.CrossRefGoogle Scholar
  22. Luo, T., Xu, Y., Luo, L., & He, C. (2014). Semi-online scheduling with two gos levels and unit processing time. Theoretical Computer Science, 521, 62–72.CrossRefGoogle Scholar
  23. Luo, W., Cheng, T. E., & Ji, M. (2015). Single-machine scheduling with a variable maintenance activity. Computers & Industrial Engineering, 79, 168–174.CrossRefGoogle Scholar
  24. Luo, W., & Ji, M. (2015). Scheduling a variable maintenance and linear deteriorating jobs on a single machine. Information Processing Letters, 115(1), 33–39.CrossRefGoogle Scholar
  25. Mosheiov, G., & Shadmon, M. (2001). Minmax earliness-tardiness costs with unit processing time jobs. European Journal of Operational Research, 130(3), 638–652.CrossRefGoogle Scholar
  26. Mosheiov, G., & Yovel, U. (2006). Minimizing weighted earliness-tardiness and due-date cost with unit processing-time jobs. European Journal of Operational Research, 172(2), 528–544.CrossRefGoogle Scholar
  27. Oron, D., Shabtay, D., & Steiner, G. (2015). Single machine scheduling with two competing agents and equal job processing times. European Journal of Operational Research, 244(1), 86–99.CrossRefGoogle Scholar
  28. Pinedo, M. L. (2012). Scheduling: Theory, algorithms, and systems (4th ed.). New York, NY: Springer.CrossRefGoogle Scholar
  29. Qi, X. (2007). A note on worst-case performance of heuristics for maintenance scheduling problems. Discrete Applied Mathematics, 155(3), 416–422.CrossRefGoogle Scholar
  30. Qi, X., Chen, T., & Tu, T. (1999). Scheduling the maintenance on a single machine. Journal of the Operational Research Society, 50, 1071–1078.CrossRefGoogle Scholar
  31. Quilliot, A., & Chrétienne, P. (2013). Homogeneously non-idling schedules of unit-time jobs on identical parallel machines. Discrete Applied Mathematics, 161(10), 1586–1597.CrossRefGoogle Scholar
  32. Rodriguez, C. E. P., & de Souza, G. F. M. (2010). Reliability concepts applied to cutting tool change time. Reliability Engineering & System Safety, 95(8), 866–873.CrossRefGoogle Scholar
  33. Sarin, S. C., & Prakash, D. (2004). Equal processing time bicriteria scheduling on parallel machines. Journal of Combinatorial Optimization, 8(3), 227–240.CrossRefGoogle Scholar
  34. Shabtay, D., & Karhi, S. (2012a). An asymptotically optimal online algorithm to minimize the total completion time on two multipurpose machines with unit processing times. Discrete Optimization, 9(4), 241–248.CrossRefGoogle Scholar
  35. Shabtay, D., & Karhi, S. (2012b). Online scheduling of two job types on a set of multipurpose machines with unit processing times. Computers & Operations Research, 39(2), 405–412.CrossRefGoogle Scholar
  36. Smith, W. E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1–2), 59–66.CrossRefGoogle Scholar
  37. Tuong, N. H., & Soukhal, A. (2010). Due dates assignment and JIT scheduling with equal-size jobs. European Journal of Operational Research, 205(2), 280–289.CrossRefGoogle Scholar
  38. Xu, D., Liu, M., Yin, Y., & Hao, J. (2013). Scheduling tool changes and special jobs on a single machine to minimize makespan. Omega-International Journal of Management Science, 41(2), 299–304.CrossRefGoogle Scholar
  39. Xu, D., Wan, L., Liu, A., & Yang, D. L. (2015). Single machine total completion time scheduling problem with workload-dependent maintenance duration. Omega-International Journal of Management Science, 52, 101–106.CrossRefGoogle Scholar
  40. Xu, D., Yin, Y., & Li, H. (2010). Scheduling jobs under increasing linear machine maintenance time. Journal of Scheduling, 13(4), 443–449.CrossRefGoogle Scholar
  41. Xu, Z., & Xu, D. (2015). Single-machine scheduling with preemptive jobs and workload-dependent maintenance durations. Operational Research, 15(3), 423–436.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of ScienceEast China University of TechnologyNanchangPeople’s Republic of China
  2. 2.Department of MathematicsTongji UniversityShanghaiPeople’s Republic of China
  3. 3.School of International Economics and BusinessNanjing University of Finance & EconomicsNanjingPeople’s Republic of China

Personalised recommendations