Split scheduling with uniform setup times
We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup, a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine- and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue, why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with the objective to minimise total weighted completion time, we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine- and sequence-independent setup times.
KeywordsScheduling Job splitting Setup times Complexity theory Approximation algorithms
The authors wish to acknowledge an anonymous reviewer for detailed and helpful comments on an earlier version of this manuscript.
- Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., & Sviridenko, M. (1999). Approximation schemes for minimizing average weighted completion time with release dates. In Proceedings of 40th Annual Symposium on Foundations of Computer Science (pp. 32–43).Google Scholar
- Bruno, J., Coffman, E.G., Jr., & Sethi, R. (1974). Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7), 382–387.Google Scholar
- Du, J., Leung, J.Y.-T., & Young, G.B. (1990). Minimizing mean flow time with release time constraint. Theoretical Computer Science, 75(3), 355–374.Google Scholar
- Lenstra, J.K. (1998). The mystical power of twoness: In memoriam Eugene L Lawler. Journal of Scheduling, 1(1), 3–14.Google Scholar
- Liu, Z., & Cheng, T.C.E. (2004). Minimizing total completion time subject to job release dates and preemption penalties. Journal of Scheduling, 7(4), 313–327.Google Scholar
- Schuurman, P., & Woeginger, G.J. (1999). Preemptive scheduling with job-dependent setup times. In Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms (pp. 759–767). Society for Industrial and Applied Mathematics.Google Scholar
- Van der Ster, S. (2010). The allocation of scarce resources in disaster relief, 2010. M.Sc. Thesis in Operations Research at VU University Amsterdam.Google Scholar