A Stochastic Model for Supply Chain Risk Management Using Conditional Value at Risk

  • Mark Goh
  • Fanwen Meng


In this chapter, we establish a stochastic programming formulation for supply chain risk management using conditional value at risk. In particular, we investigate two problems on logistics under conditions of uncertainty. The sample average approximation method is introduced for solving the underlying stochastic model. Preliminary numerical results are provided.


Supply Chain Sample Average Approximation Supply Chain Risk Supply Chain Risk Management Stochastic Mathematical Program 
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  1. 1.
    F. ANDERSSON, H. MAUSSER, D. ROSEN, AND S. URYASEV. Credit risk optimization with conditional Value-at-Risk criterion, Mathematical Programming, 89(2001), pp. 273–291zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. ARTZNER, F. DELBAEN, J.M. EBER, AND D. HEATH, Coherent measures of risk, Mathematical Finance, 9(1999), pp. 203–228zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. BASTIN, C. CIRILLO, AND P.L. TOINT, Convergence theory for nonconvex stochastic programming with an application to mixed logit, Mathematical Programming, 108(2006), pp. 207–234zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    E. BOGENTOFT, H.E. ROMEIJN, AND S. URYASEV, Asset/liability management for pension funds using CVaR constraints, The Journal of Risk Finance, 3(2001), pp. 57–71CrossRefGoogle Scholar
  5. 5.
    M. GOH, J. LIM, AND F. MENG, A stochastic model for risk management in global supply chain networks, European Journal of Operational Research, 182 (2007) 164–173zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. GÜRKAN, A.Y. ÖZGE, AND S.M. ROBINSON, Sample-path solution of stochastic variational inequalities, Mathematical Programming, 84(1999), pp. 313–333zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W. HURLIMANN, Conditional Value-at-Risk bounds for compound poisson risks and a normal approximation, Journal of Applied Mathematics, 3(2003), pp. 141–153CrossRefMathSciNetGoogle Scholar
  8. 8.
    P. JORION, Value At Risk: the New Benchmark for Controlling Market Risk, McGraw-Hill, New York, 1997; McGraw-Hill International Edition, 2001Google Scholar
  9. 9.
    P. KROKHMAL, J. PALMQUIST, AND S. URYASEV, Portfolio optimization with conditional Value-At-Risk objective and constraints, The Journal of Risk, 4(2002), pp. 43–68Google Scholar
  10. 10.
    N. LARSEN, H. MAUSSER, AND S. URYASEV, Algorithms for optimization of value at risk, in P. Pardalos and V.K. Tsitsiringos, editors, Financial Engineering, E-Commerce and Supply Chain. Kluwer Academic Publishers, Norwell, 2002, pp. 129–157Google Scholar
  11. 11.
    F. MENG, J. SUN, AND M. GOH, A smoothing sample average approximation method for CVaR optimization, working paper, The Logistics Institute – Asia Pacific, School of Business, National University of Singapore, May 2008Google Scholar
  12. 12.
    F. MENG AND H. XU, A regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints, SIAM Journal on Optimization, 17(2006), pp. 891–919zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J.-S. PANG AND B.F. HOBBS, Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Operations Research, 55 (2007), pp. 113–127zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    J.-S. PANG AND S. LEYFFER, On the global minimization of the Value-at-Risk, Optimization Methods and Software, 19 (2004), pp. 611–631CrossRefMathSciNetGoogle Scholar
  15. 15.
    G. PFLUG, Some remarks on the Value-at-Risk and the conditional Value-at-Risk, in Probabilistic Constrained Optimization: Methodology and Applications, S. Uryasev (ed.) Kluwer Academic Publishers, 2000, pp. 1–10Google Scholar
  16. 16.
    L. QI, D. SUN, AND G. ZHOU, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Programming, 87(2000), pp. 1–35zbMATHMathSciNetGoogle Scholar
  17. 17.
    R.T. ROCKAFELLAR, Convex Analysis, Princeton University, 1970Google Scholar
  18. 18.
    R.T. ROCKAFELLAR AND S. URYASEV, Conditional Value-at-Risk for general loss distributions, Journal of Banking & Finance, 26(2002), pp. 1443–1471CrossRefGoogle Scholar
  19. 19.
    R.T. ROCKAFELLAR AND S. URYASEV, Optimization of conditional Value-at-Risk, Journal of Risk, 2(2000), pp. 21–41Google Scholar
  20. 20.
    A. RUSZCZYNSKI AND A. SHAPIRO, Stochastic Programming, Handbooks in Operations Research and Management Science, Volume 10, North-Holland, Amsterdam, 2003zbMATHCrossRefGoogle Scholar
  21. 21.
    T. SANTOSO, S. AHMED, M. GOETSCHALCKX, AND A. SHAPIRO, A stochastic programming approach for supply chain network design under uncertainty, European Journal of Operational Research, 67(2005), pp. 96–115MathSciNetGoogle Scholar
  22. 22.
    A. SHAPIRO, Monte Carlo sampling methods. In A. Ruszczynski, A. Shapiro (eds.), Stochastic Programming, Handbooks in OR & MS, Vol. 10, North-Holland Publishing Company, Amsterdam, 2003CrossRefGoogle Scholar
  23. 23.
    A. SHAPIRO, Stochastic mathematical programs with equilibrium constraints, Journal of Optimization Theory and Application, 128 (2006), pp. 223–243CrossRefGoogle Scholar
  24. 24.
    D. TASCHE, Expected shortfall and beyond, Journal of Banking and Finance, 6(2002), pp. 1519–1533Google Scholar
  25. 25.
    C.S. TANG, Perspectives in Supply Chain Risk Management, International Journal of Production Economics, 103 (2006), pp. 451–488CrossRefGoogle Scholar
  26. 26.
    C.-S. YU AND H. LI, A robust optimization model for stochastic logistic problems, International Journal of Production Economics, 64(2000), pp. 385–397CrossRefGoogle Scholar
  27. 27.
    S. URYASEV, Introduction to the theory of probabilistic functions and percentiles (Value-at-Risk), Probabilistic Constrained Optimization: Methodology and Applications, S.P. Uryasev (ed.), Kluwer Academic Publishers, 2000, pp. 1–25Google Scholar
  28. 28.
    S. URYASEV, Conditional Value-at-Risk: optimization algorithms and applications, Financial Engineering News, 14, 2000Google Scholar
  29. 29.
    H. XU AND F. MENG, Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints, Mathematics of Operations Research, 32(2007), pp. 648–668zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  • Mark Goh
    • 1
  • Fanwen Meng
    • 1
  1. 1.NUS Business School, The Logistics Institute – Asia PacificNational University of SingaporeSingaporeSingapore

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