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.
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\(w_{[k]}\) and \(t_{[k]}^s\) denote the weight and tardiness of the job in the \(k\mathrm{th}\) position, respectively.
References
Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.
Balasubramanian, J., & Grossmann, I. E. (2003). Scheduling optimization under uncertainty—an alternative approach. Computers & Chemical Engineering, 27(4), 469–490.
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.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.
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.
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, 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.
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.
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.
McDaniel, D., & Devine, M. (1977). A modified Benders’ partitioning algorithm for mixed integer programming. Management Science, 24(3), 312–319.
McKay, K. N., Safayeni, F. R., & Buzacott, J. A. (1988). Job-Shop scheduling theory: What is relevant? Interfaces, 18(4), 84–90.
Ogryczak, W., & Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1), 60–78.
Pinedo, M. (2001). Scheduling: Theory, algorithms, and systems (2nd ed.). Upper Saddle, NJ: Prentice Hall.
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.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.
Sabuncuoglu, I., & Bayiz, M. (2000). Analysis of reactive scheduling problems in a job shop environment. European Journal of Operational Research, 126(3), 567–586.
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.
Sarin, S. C., Nagarajan, B., & Liao, L. (2010). Stochastic scheduling: Expectation-variance analysis of a schedule (1st ed.). New York: Cambridge University Press.
Sherali, H., Lunday, B. (2010). On generating maximal nondominated Benders cuts. Annals of Operations Research (in press).
Skutella, M., & Uetz, M. (2005). Stochastic machine scheduling with precedence constraints. SIAM Journal on Computing, 34(4), 788.
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.
Wang, W., & Ahmed, S. (2008). Sample average approximation of expected value constrained stochastic programs. Operations Research Letters, 36(5), 515–519.
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This Research has been supported by the National Science Foundation under Grant CMMI-0856270.
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Sarin, S.C., Sherali, H.D. & Liao, L. Minimizing conditional-value-at-risk for stochastic scheduling problems. J Sched 17, 5–15 (2014). https://doi.org/10.1007/s10951-013-0349-6
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DOI: https://doi.org/10.1007/s10951-013-0349-6