Minimizing conditional-value-at-risk for stochastic scheduling problems
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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 programmingNotes
Acknowledgments
This Research has been supported by the National Science Foundation under Grant CMMI-0856270.
References
- Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
- Balasubramanian, J., & Grossmann, I. E. (2003). Scheduling optimization under uncertainty—an alternative approach. Computers & Chemical Engineering, 27(4), 469–490.CrossRefGoogle Scholar
- Beasley, J. E. (2012). OR-Library: Weighted tardiness. http://people.brunel.ac.uk/mastjjb/jeb/orlib/wtinfo.html. Accessed 3 June 2012.
- Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1), 238–252.CrossRefGoogle Scholar
- Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.Google Scholar
- 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
- 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
- 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
- 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
- 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
- McDaniel, D., & Devine, M. (1977). A modified Benders’ partitioning algorithm for mixed integer programming. Management Science, 24(3), 312–319.CrossRefGoogle Scholar
- McKay, K. N., Safayeni, F. R., & Buzacott, J. A. (1988). Job-Shop scheduling theory: What is relevant? Interfaces, 18(4), 84–90.CrossRefGoogle Scholar
- Ogryczak, W., & Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1), 60–78.CrossRefGoogle Scholar
- Pinedo, M. (2001). Scheduling: Theory, algorithms, and systems (2nd ed.). Upper Saddle, NJ: Prentice Hall.Google Scholar
- 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
- Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.Google Scholar
- 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
- 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
- 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
- Sherali, H., Lunday, B. (2010). On generating maximal nondominated Benders cuts. Annals of Operations Research (in press). Google Scholar
- Skutella, M., & Uetz, M. (2005). Stochastic machine scheduling with precedence constraints. SIAM Journal on Computing, 34(4), 788.CrossRefGoogle Scholar
- 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
- Wang, W., & Ahmed, S. (2008). Sample average approximation of expected value constrained stochastic programs. Operations Research Letters, 36(5), 515–519.CrossRefGoogle Scholar