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Journal of Scheduling

, Volume 17, Issue 1, pp 5–15 | Cite as

Minimizing conditional-value-at-risk for stochastic scheduling problems

  • Subhash C. Sarin
  • Hanif D. Sherali
  • Lingrui Liao
Article

Abstract

This paper introduces the use of conditional-value-at-risk (CVaR) as a criterion for stochastic scheduling problems. This criterion has the tendency of simultaneously reducing both the expectation and variance of a performance measure, while retaining linearity whenever the expectation can be represented by a linear expression. In this regard, it offers an added advantage over traditional nonlinear expectation-variance-based approaches. We begin by formulating a scenario-based mixed-integer program formulation for minimizing CVaR for general scheduling problems. We then demonstrate its application for the single machine total weighted tardiness problem, for which we present both a specialized l-shaped algorithm and a dynamic programming-based heuristic procedure. Our numerical experimental results reveal the benefits and effectiveness of using the CVaR criterion. Likewise, we also exhibit the use and effectiveness of minimizing CVaR in the context of the parallel machine total weighted tardiness problem. We believe that minimizing CVaR is an effective approach and holds great promise for achieving risk-averse solutions for stochastic scheduling problems that arise in diverse practical applications.

Keywords

Stochastic scheduling Conditional-value-at-risk Total weighted tardiness Benders decomposition Dynamic programming  

Notes

Acknowledgments

This Research has been supported by the National Science Foundation under Grant CMMI-0856270.

References

  1. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
  2. Balasubramanian, J., & Grossmann, I. E. (2003). Scheduling optimization under uncertainty—an alternative approach. Computers & Chemical Engineering, 27(4), 469–490.CrossRefGoogle Scholar
  3. Beasley, J. E. (2012). OR-Library: Weighted tardiness. http://people.brunel.ac.uk/mastjjb/jeb/orlib/wtinfo.html. Accessed 3 June 2012.
  4. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1), 238–252.CrossRefGoogle Scholar
  5. Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.Google Scholar
  6. Congram, R. K., Potts, C. N., & van de Velde, S. L. (2002). An iterated dynasearch algorithm for the single-machine total weighted tardiness scheduling problem. INFORMS Journal on Computing, 14(1), 52–67.CrossRefGoogle Scholar
  7. Daniels, R. L., & Kouvelis, P. (1995). Robust scheduling to hedge against processing time uncertainty in single-stage production. Management Science, 41(2), 363–376.CrossRefGoogle Scholar
  8. De, P., Ghosh, J. B., & Wells, C. E. (1992). Expectation-variance analyss of job sequences under processing time uncertainty. International Journal of Production Economics, 28(3), 289–297.CrossRefGoogle Scholar
  9. Grosso, A., Croce, F. D., & Tadei, R. (2004). An enhanced dynasearch neighborhood for the single-machine total weighted tardiness scheduling problem. Operations Research Letters, 32(1), 68–72.CrossRefGoogle Scholar
  10. Kouvelis, P., Daniels, R. L., & Vairaktarakis, G. (2000). Robust scheduling of a two-machine flow shop with uncertain processing times. IIE Transactions, 32(5), 421–432.Google Scholar
  11. McDaniel, D., & Devine, M. (1977). A modified Benders’ partitioning algorithm for mixed integer programming. Management Science, 24(3), 312–319.CrossRefGoogle Scholar
  12. McKay, K. N., Safayeni, F. R., & Buzacott, J. A. (1988). Job-Shop scheduling theory: What is relevant? Interfaces, 18(4), 84–90.CrossRefGoogle Scholar
  13. Ogryczak, W., & Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1), 60–78.CrossRefGoogle Scholar
  14. Pinedo, M. (2001). Scheduling: Theory, algorithms, and systems (2nd ed.). Upper Saddle, NJ: Prentice Hall.Google Scholar
  15. Porter, R. B., & Gaumnitz, J. E. (1972). Stochastic dominance vs. mean-variance portfolio analysis: An empirical evaluation. The American Economic Review, 62(3), 438–446.Google Scholar
  16. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.Google Scholar
  17. Sabuncuoglu, I., & Bayiz, M. (2000). Analysis of reactive scheduling problems in a job shop environment. European Journal of Operational Research, 126(3), 567–586.CrossRefGoogle Scholar
  18. Sarin, S., Sherali, H., & Bhootra, A. (2005). New tighter polynomial length formulations for the asymmetric traveling salesman problem with and without precedence constraints. Operations Research Letters, 33(1), 62–70.CrossRefGoogle Scholar
  19. Sarin, S. C., Nagarajan, B., & Liao, L. (2010). Stochastic scheduling: Expectation-variance analysis of a schedule (1st ed.). New York: Cambridge University Press.Google Scholar
  20. Sherali, H., Lunday, B. (2010). On generating maximal nondominated Benders cuts. Annals of Operations Research (in press). Google Scholar
  21. Skutella, M., & Uetz, M. (2005). Stochastic machine scheduling with precedence constraints. SIAM Journal on Computing, 34(4), 788.CrossRefGoogle Scholar
  22. Vieira, G. E., Herrmann, J. W., & Lin, E. (2003). Rescheduling manufacturing systems: A framework of strategies, policies, and methods. Journal of Scheduling, 6(1), 39–62.CrossRefGoogle Scholar
  23. Wang, W., & Ahmed, S. (2008). Sample average approximation of expected value constrained stochastic programs. Operations Research Letters, 36(5), 515–519.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Subhash C. Sarin
    • 1
  • Hanif D. Sherali
    • 1
  • Lingrui Liao
    • 1
  1. 1.Grado Department of Industrial and Systems EngineeringVirginia TechBlacksburgUSA

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