Abstract
The single machine scheduling problem with sum of completion times criterion (SS) can be solved easily by the Shortest Processing Time (SPT) rule. In the case of significant uncertainty of the processing times, a robustness approach is appropriate. In this paper, we show that the robust version of the (SS) problem is NP-complete even for very restricted cases. We present an algorithm for finding optimal solutions for the robust (SS) problem using dynamic programming. We also provide two polynomial time heuristics and demonstrate their effectiveness.
Similar content being viewed by others
References
A.K. Agrawala, E.G. Coffman, Jr., M.R. Garey, and S.K. Tripathi, “A static optimization algorithm minimizing expected flowtime on uniform processors,” IEEE Transactions on Computing, vol.33, pp.351–357, 1984.
A. Allahverdi and J. Mittenthal, “Two-machine ordered flow shop scheduling under random breakdowns,” Mathematical and Computer Modeling, vol. 20, pp. 9–17, 1994a.
A. Allahverdi and J. Mittenthal, “Scheduling on M parallel machines subject to random breakdowns to minimize expected mean flow time,” Naval Research Logistics, vol. 41, pp. 677–682,1994b.
A. Allahverdi and J. Mittenthal, “Scheduling on a two-machine flow shop subject to random breakdowns with a makespan objective function,” European Journal of Operational Research, vol. 81, pp. 376–387, 1995.
J. Birge, J.B.G. Frenk, J. Mittenthal, and A.H.G. Rinnooy Kan, “Single machine scheduling subject to stochastic breakdowns,” Naval Research Logistics, vol. 37, pp. 661–677, 1990.
J.R. Birge and K. Glazebrook, “Assessing the effects of machine breakdown in stochastic scheduling,” Operations Research Letters, vol. 7, pp. 267–271, 1988.
R.L. Daniels and P. Kouvelis, “Robust scheduling to hedge against processing time uncertainty in single-stage production,” Management Science, vol. 41, 2, pp. 363–376, 1995.
C. Du and M. Pinedo, “A note on minimizing the expected makespan in flowshops subject to breakdowns,” Naval Research Logistics, vol. 42, pp. 1251–1262, 1995.
H. Emmons and M. Pinedo, “Scheduling stochastic jobs with due dates on parallel machines,” European Journal of Operational Research, vol. 47, pp. 49–55, 1990.
M.R. Garey and D.S. Johnson, Computers and Intractability, W.H. Freeman: San Francisco, 1979.
K.D. Glazebrook, “Scheduling tasks with exponential service times on parallel processors,” Journal of Applied Probability, vol. 16, pp. 685–689, 1979.
K. Glazebrook, “Evaluating the effects of machine breakdowns in stochastic scheduling problems,” Naval Research Logistics, vol. 34, pp. 319–335,1987.
K. Glazebrook, “On non-preemptive policies for stochastic single machine scheduling with breakdown,” Probability in the Engineering and Informational Science, vol. 5, pp.77–87, 1991.
T. Kampke, “Optimal scheduling of jobs with exponential service times on identical parallel machines,” Operations Research, vol. 37, pp. 126–133, 1989.
P. Kouvelis and G. Yu, Robust Discrete Optimization and Its Applications,Kluwer Academic Publishers: Boston, 1997.
G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons: New York,1988.
M.L. Pinedo and S.H. Ross, “Scheduling jobs subject to non homogeneous Poisson shocks,” Management Science, vol. 26, pp. 1250–1257,1980.
G. Weiss and M. Pinedo, “Scheduling tasks with exponential service times on nonidentical processors to minimize various cost functions,” Journal of Applied Probability, vol. 17, pp. 187–202, 1980.
G. Yu and J. Yang, “On the robust shortest path problem,” Computers & Operations Research, vol. 25, no.6, pp. 457–468, 1998.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yang, J., Yu, G. On the Robust Single Machine Scheduling Problem. Journal of Combinatorial Optimization 6, 17–33 (2002). https://doi.org/10.1023/A:1013333232691
Issue Date:
DOI: https://doi.org/10.1023/A:1013333232691