Abstract
In this paper, we study a robust single-machine scheduling problem under four alternative optimization criteria: minimizing total completion time, minimizing total weighted completion time, minimizing maximum lateness, and minimizing the number of late jobs. We assume that job processing times are subject to uncertainty. Accordingly, we construct three alternative uncertainty sets, each of which defines job processing times that can simultaneously occur. The robust optimization framework assumes that, given a job schedule, a worst-case set of processing times will be realized from among those allowed by the uncertainty set under consideration. For each combination of objective function and uncertainty set, we first analyze the problem of identifying a set of worst-case processing times with respect to a fixed schedule, and then investigate the problem of selecting a schedule whose worst-case objective is minimal.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aissi, H., Bazgan, C., & Vanderpooten, D. (2009). Min-max and min-max regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research, 197(2), 427–438.
Aloulou, M. A., & Della Croce, F. (2008). Complexity of single machine scheduling problems under scenario-based uncertainty. Operations Research Letters, 36(3), 338–342.
Averbakh, I. (2000). Minmax regret solutions for minimax optimization problems with uncertainty. Operations Research Letters, 27(2), 57–65.
Averbakh, I. (2005). Computing and minimizing the relative regret in combinatorial optimization with interval data. Discrete Optimization, 2(4), 273–287.
Ben-Tal, A., El-Ghaoui, L., & Nemirovski, A. (2009). Robust optimization. Princeton, NJ: Princeton University Press.
Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.
Birge, J. R., & Louveaux, F. V. (1997). Introduction to stochastic programming. New York: Springer.
Cormen, T. H., Leiserson, C. H., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms (3rd ed.). Cambridge, MA: The MIT Press.
Daniels, R. L., & Kouvelis, P. (1995). Robust scheduling to hedge against processing time uncertainty in single-stage production. Management Science, 41(2), 363–376.
de Farias, I. R, Jr., Zhao, H., & Zhao, M. (2010). A family of inequalities valid for the robust single machine scheduling polyhedron. Computers and Operations Research, 37(9), 1610–1614.
Graham, R. L., Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic machine scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.
Kall, P., & Wallace, S. W. (1994). Stochastic programming. Chichester, UK: Wiley.
Kasperski, A. (2005). Minimizing maximal regret in single machine sequencing problem with maximum lateness criterion. Operations Research Letters, 33(4), 431–436.
Kasperski, A. (2008). Discrete optimization with interval data: Minmax regret and fuzzy approach (studies in fuzziness and soft computing) (Vol. 228). Heidelberg: Springer.
Kasperski, A., & Zieliński, P. (2006). An approximation algorithm for interval data minmax regret combinatorial optimization problems. Information Processing Letters, 97(5), 177–180.
Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and its applications. Norwell, MA: Kluwer.
Lawler, E. L. (1973). Optimal sequencing of a single machine subject to precedence constraints. Management Science, 19(5), 544–546.
Lebedev, V., & Averbakh, I. (2006). Complexity of minimizing the total flow time with interval data and minmax regret criterion. Discrete Applied Mathematics, 154(15), 2167–2177.
Lu, C. C., Lin, S. W., & Ying, K. C. (2012). Robust scheduling on a single machine to minimize total flow time. Computers and Operations Research, 39(7), 1682–1691.
Montemanni, R. (2007). A mixed integer programming formulation for the total flow time single machine robust scheduling problem with interval data. Journal of Mathematical Modelling and Algorithms, 6(2), 287–296.
Moore, J. (1968). An \(n\) job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15(1), 102–109.
Sabuncuoglu, I., & Goren, S. (2009). Hedging production schedules against uncertainty in manufacturing environment with a review of robustness and stability research. International Journal of Computer Integrated Manufacturing, 22(2), 138–157.
Smith, J. C., & Ahmed, S. (2010). Introduction to robust optimization. In J. Cochran (Ed.), Wiley Encyclopedia of Operations Research and Management Science (pp. 2574–2581). Hoboken, NJ: Wiley.
Smith, W. E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1–2), 59–66.
Wood, R. K. (1993). Deterministic network interdiction. Mathematical and Computer Modelling, 17(2), 1–18.
Yang, J., & Yu, G. (2002). On the robust single machine scheduling problem. Journal of Combinatorial Optimization, 6(1), 17–33.
Acknowledgments
The authors sincerely appreciate the comments made by two anonymous referees. Dr. Smith gratefully acknowledges the support of the National Science Foundation under Grant CMMI-1100765, the Defense Threat Reduction Agency under Grant HDTRA-10-01-0050, the Air Force Office of Scientific Research under Grant FA9550-12-1-0353, and the Office of Naval Research under Grant N000141310036.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tadayon, B., Smith, J.C. Algorithms and Complexity Analysis for Robust Single-Machine Scheduling Problems. J Sched 18, 575–592 (2015). https://doi.org/10.1007/s10951-015-0418-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-015-0418-0