Annals of Operations Research

, Volume 100, Issue 1–4, pp 25–53 | Cite as

Scenarios for Multistage Stochastic Programs

  • Jitka Dupačová
  • Giorgio Consigli
  • Stein W. Wallace


A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.

scenarios and scenario trees clustering importance sampling matching moments problem oriented requirements inference and bounds 


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  1. [1]
    J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res. 33 (1985) 989–1007.Google Scholar
  2. [2]
    J.R. Birge and J.M. Mulvey, Stochastic programming, in: Mathematical Programming for Industrial Engineers, eds. M. Avriel and B. Golany (Dekker, New York, 1996) pp. 543–574.Google Scholar
  3. [3]
    J.R. Birge and C.H. Rosa, Modeling investment uncertainty in the costs of global CO2 emission policy, EJOR 83 (1995) 466–488.Google Scholar
  4. [4]
    J.R. Birge, M.A.H. Dempster, H.I. Gassmann, E.A. Gunn, A.J. King and S.W. Wallace, A standard input format for multiperiod stochastic linear programs, Mathematical Programming Society, COAL Newsletter 17 (1987) 1–20.Google Scholar
  5. [5]
    O.J. Botnen, A. Johannesen, A. Haugstad, S. Kroken and O. Frøystein, Modelling of hydropower scheduling in a national/international context, in: Hydropower '92, eds. E. Broch and D.K. Lysne (Balkema, Rotterdam, 1996) pp. 575–584.Google Scholar
  6. [6]
    S.P. Bradley and D.B. Crane, A dynamic model for bond portfolio management, Manag. Sci. 19 (1972) 139–151.Google Scholar
  7. [7]
    S.P. Bradley and D.B. Crane, Managing a bank bond portfolio over time, in: Stochastic Programming, ed. M.A.H. Dempster (Academic Press, London, 1980) pp. 449–471.Google Scholar
  8. [8]
    E. Canestrelli and S. Giove, Scenarios identification for financial time series, in: Current Topics in Quantitative Finance, ed. E. Canestrelli (Physica-Verlag, 1999) pp. 25–36.Google Scholar
  9. [9]
    D.R. Cariño, D.H. Myers and W.T. Ziemba, Concepts, technical issues and uses of the Russell–Yasuda Kasai financial planning model, Oper. Res. 46 (1998) 450–462.Google Scholar
  10. [10]
    Z. Chen, G. Consigli. M.A.H. Dempster and N. Hicks-Pedrón, Towards sequential sampling algorithms for dynamic portfolio management, in: New Operational Tools for the Management of Financial Risks, ed. C. Zopounidis (Kluwer, 1997) pp. 197–211.Google Scholar
  11. [11]
    G. Consigli, Dynamic stochastic programming for asset and liability management, Ph.D. thesis, University of Essex (1997).Google Scholar
  12. [12]
    G. Consigli and M.A.H. Dempster, Dynamic stochastic programming for asset-liability management, Annals of Oper. Res. 81 (1998) 131–161.Google Scholar
  13. [13]
    H. Dahl, A. Meeraus and S.A. Zenios, Some financial optimization models: II. Financial engineering, in: Financial Optimization, ed. S.A. Zenios (Cambridge University Press, 1993) pp. 37–71.Google Scholar
  14. [14]
    G.B. Dantzig and G. Infanger, Multi-stage stochastic linear programs for portfolio optimization, Annals of Oper. Res. 45 (1993) 59–76.Google Scholar
  15. [15]
    R.S. Dembo, Scenario optimization, Annals of Oper. Res. 30 (1991) 63–80.Google Scholar
  16. [16]
    R.S. Dembo, A. Chiarri, L. Paradinas and J. Gomez, Managing Hidroeléctrica Española's hydroelectric power system, Interfaces 20 (1990) 115–135.Google Scholar
  17. [17]
    M.A.H. Dempster, On stochastic programming II: Dynamic problems under risk, Stochastics 25 (1988) 15–42.Google Scholar
  18. [18]
    M.A.H. Dempster, Sequential importance sampling algorithms for dynamic stochastic programming, WP 32/98, The Judge Inst. of Management Studies, Cambridge Univ. (1998).Google Scholar
  19. [19]
    M.A.H. Dempster and A.M. Ireland, A financial expert decision support system, in: Mathematical Models for Decision Support, ed. G. Mitra, NATO ASI Series, Vol. F48 (Springer, Berlin, 1988) pp. 415–440.Google Scholar
  20. [20]
    M.A.H. Dempster and A.M. Ireland, Object oriented model integration in financial decision support systems, Decision Support Systems 7 (1991) 329–340.Google Scholar
  21. [21]
    M.A.H. Dempster and R.T. Thompson, EVPI-based importance sampling solution procedures for multistage stochastic linear programmes on parallel MIMD architectures, Annals of Oper. Res. (to appear).Google Scholar
  22. [22]
    M.A.H. Dempster and A.E. Thorlacius, Stochastic simulation of international economic variables and asset returns: The Falcon asset model, Preprint (1998).Google Scholar
  23. [23]
    C. Dert, Asset liability management for pension funds. A multistage chance-constrained programming approach, Ph.D. thesis, Erasmus University, Rotterdam (1995).Google Scholar
  24. [24]
    J. Dupačová, Water resources system modelling using stochastic programming with recourse, in: Recent Results in Stochastic Programming, eds. P. Kall and A. Prékopa, Lecture Notes in Economics and Math. Systems, Vol. 179 (Springer, Berlin, 1980) pp. 121–133.Google Scholar
  25. [25]
    J. Dupačová, Postoptimality for multistage stochastic linear programs, Annals of Oper. Res. 56 (1995) 65–78.Google Scholar
  26. [26]
    J. Dupačová, Multistage stochastic programs: The state-of-the art and selected bibliography, Kybernetika 31 (1995) 151–174.Google Scholar
  27. [27]
    J. Dupačová, Stochastic programming: Approximation via scenarios, Apportaciones mathematicás, Ser. Communicationes 24 (1998) 77–94. Also available via Scholar
  28. [28]
    J. Dupačová, Uncertainty about input data in portfolio management, in: Modelling Techniques for Financial Markets and Bank Management, eds. M. Bertocchi et al. (Physica-Verlag, 1996) pp. 17–33.Google Scholar
  29. [29]
    O. Egeland, J. Hegge, E. Kylling and J. Nes, The extended power pool model—Operation planning of multi-river and multi-reservoir hydrodominated power production system—a hierarchial approach, Report 32.14, CIGRE (1982).Google Scholar
  30. [30]
    G.D. Eppen, R.K. Martin and L. Schrage, A scenario approach to capacity planning, Oper. Res. 37 (1989) 517–527.Google Scholar
  31. [31]
    L.F. Escudero, P.V. Kamesam, A.J. King and R.J.-B. Wets, Production planning via scenario modeling, Annals of Oper. Res. 42 (1993) 311–336.Google Scholar
  32. [32]
    S.-E. Fleten, S.W. Wallace and W.T. Ziemba, Portfolio management in a deregulated hydropowerbased electricity market, in: Hydropower '97, eds. E. Broch, D.K. Lysne, N. Flatabø and E. Helland-Hansen (Balkema, Rotterdam, 1997).Google Scholar
  33. [33]
    K. Frauendorfer, Barycentric scenario trees in convex multistage stochastic programming, Math. Progr. 75 (1996) 277–293.Google Scholar
  34. [34]
    K. Frauendorfer and C. Marohn, Refinement issues in stochastic multistage linear programming, in: Stochastic Programming Methods and Technical Applications, eds. K. Marti and P. Kall, Lecture Notes in Economics and Math. Systems, Vol. 458 (Springer, 1998) pp. 305–328.Google Scholar
  35. [35]
    H.I. Gassmann, MSLiP: A computer code for the multi-stage stochastic programming problem, Math. Progr. 47 (1990) 407–423.Google Scholar
  36. [36]
    H.I. Gassmann and A.M. Ireland, Scenario formulation in an algebraic modeling language, Annals of Oper. Res. 59 (1995) 45–75.Google Scholar
  37. [37]
    H.I. Gassmann and E. Schweitzer, A comprehensive input format for stochastic linear programs, WP–96–1, School of Business Administration, Halifax (1996).Google Scholar
  38. [38]
    N. Gröwe-Kuska, M.P. Nowak, W. Römisch and I. Wegner, Optimierung eines hydro-thermischen Kraftwerkssystems unter Ungewissheit, Optimierung in der Energieversorgung III, VDI-Berichts Nr. 1508 (Düsseldorf, 1999) pp. 147–157.Google Scholar
  39. [39]
    P. Hansen and B. Jaumard, Cluster analysis and mathematical programming, Math. Progr. 79 (1997) 191–215.Google Scholar
  40. [40]
    K. Høyland and S.W.Wallace, Generating scenario trees for multistage decision problems, Technical Report 4/97, Dept. of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim (1996). To appear in Management Science.Google Scholar
  41. [41]
    G. Infanger, Planning Under Uncertainty. Solving Large Scale Stochastic Linear Programs (Boyd and Fraser, Danvers, 1994).Google Scholar
  42. [42]
    J. Jacobs, G. Freeman, J. Grygier, D. Morton, G. Schultz, K. Staschus and J. Stedinger, SOCRATES: A system for scheduling hydroelectric generation under uncertainty, Annals of Oper. Res. 59 (1995) 99–132.Google Scholar
  43. [43]
    F. Jamshidian and Yu Zhu, Scenario simulation: Theory and methodology, Finance and Stochastics 1 (1997) 43–67.Google Scholar
  44. [44]
    D.L. Keefer, Certainty equivalents for three-point discrete-distributions approximations, Manag. Sci. 40 (1994) 760–773.Google Scholar
  45. [45]
    P. Klaassen, Discretized reality and spurious profits in stochastic programming models for asset/ liability management, EJOR 101 (1997) 374–392.Google Scholar
  46. [46]
    P. Klaassen, Financial asset-pricing theory and stochastic programming models for asset/liability management: A synthesis, Manag. Sci. 44 (1998) 31–48.Google Scholar
  47. [47]
    Y.A. Koskosidis and A.M. Duarte, A scenario-based approach to active asset allocation, J. of Portfolio Management (Winter 1997) 74–85.Google Scholar
  48. [48]
    R. Kouwenberg, Scenario generation and stochastic programming models for asset liability management, EJOR (to appear).Google Scholar
  49. [49]
    R. Kouwenberg and T. Vorst, Dynamic portfolio insurance: A stochastic programming approach, Report 9813, Erasmus Center for Financial Research, Rotterdam (1998).Google Scholar
  50. [50]
    M. Lane and P. Hutchinson, A model for managing a certificate of deposit portfolio under uncertainty, in: Stochastic Programming, ed. M.A.H. Dempster (Academic Press, London, 1980) pp. 473–495.Google Scholar
  51. [51]
    W.-K. Mak, D.P. Morton and R.K. Wood, Monte Carlo bounding techniques for determining solution quality in stochastic programs, OR Letters 24 (1999) 47–56.Google Scholar
  52. [52]
    H.M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (Yale University Press, New Haven, 1959).Google Scholar
  53. [53]
    E. Messina and G. Mitra, Modelling and analysis of multistage stochastic programming problems: A software environment, EJOR 101 (1997) 343–359.Google Scholar
  54. [54]
    J.M. Mulvey and A. Ruszczy´nski, A new scenario decomposition method for large-scale stochastic optimization, Oper. Res. 43 (1995) 477–490.Google Scholar
  55. [55]
    J.M. Mulvey and A.E. Thorlacius, The Towers Perrin global capital market scenario generation system, in [70], pp. 286–312.Google Scholar
  56. [56]
    J.M. Mulvey and H. Vladimirou, Stochastic network optimization models for investment planning, Annals of Oper. Res. 20 (1989) 187–217.Google Scholar
  57. [57]
    J.M. Mulvey and S.A. Zenios, Capturing the correlations of fixed-income instruments, Manag. Sci 40 (1994) 1329–1342.Google Scholar
  58. [58]
    J.M. Mulvey and W.T. Ziemba, Asset and liability management systems for long-term investors: discussion of the issues, in [70], pp. 3–35.Google Scholar
  59. [59]
    S.S. Nielsen and E.I. Ronn, The valuation of default risk in corporate bonds and interest rate swaps, Advances in Futures and Options Research 9 (1997) 175–196.Google Scholar
  60. [60]
    S.S. Nielsen and S.A. Zenios, Solving multistage stochastic network programs on massively parallel computers, Math. Progr. 73 (1996) 227–250.Google Scholar
  61. [61]
    G.Ch. Pflug, Scenario tree generation for multiperiod financial optimization, Math. Progr., Ser. B 89 (2001) 251–271.Google Scholar
  62. [62]
    P. Popela, An object-oriented approach to multistage stochastic programming: Models and algorithms, Ph.D. thesis, Charles University, Prague (1998).Google Scholar
  63. [63]
    A. Prékopa, Stochastic Programming (Kluwer, Dordrecht and Akadémiai Kiadó, Budapest, 1995).Google Scholar
  64. [64]
    R.T. Rockafellar and R.J.-B.Wets, The optimal recourse problem in discrete time: L 1-multipliers for inequality constraints, SIAM J. Control 16 (1978) 16–36.Google Scholar
  65. [65]
    R.T. Rockafellar and R.J.-B.Wets, Scenario and policy aggregation in optimization under uncertainty, Mathematics of OR 16 (1991) 119–147.Google Scholar
  66. [66]
    A. Ruszczy´nski, Parallel decomposition of multistage stochastic programming programs, Math. Progr. 58 (1993) 201–228.Google Scholar
  67. [67]
    J.D. Salas, G.Q. Tabios III and P. Bartolini, Approaches to multivariate modeling of water resources time series, Water Res. Bulletin 21 (1985) 683–708.Google Scholar
  68. [68]
    J.F. Shapiro, Stochastic programming models for dedicated portfolio selection, in: Mathematical Models for Decision Support, ed. B. Mitra, NATO ASI Series, Vol. F48 (Springer, 1988) pp. 587–611.Google Scholar
  69. [69]
    S. Takriti, J.R. Birge and E. Long, A stochastic model for the unit commitment problem, IEEE Transactions on Power Systems 11 (1996) 1497–1508.Google Scholar
  70. [70]
    W.T. Ziemba and J. Mulvey, eds., World Wide Asset and Liability Modeling (Cambridge University Press, 1998).Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Jitka Dupačová
    • 1
  • Giorgio Consigli
    • 2
  • Stein W. Wallace
    • 3
  1. 1.Department of Probability and Mathematical StatisticsCharles UniversityPragueCzech Republic
  2. 2.UBM UniCredit Banca MobilaireMilanoItaly
  3. 3.Department of Industrial Economics and Technology ManagementNorwegian University of Science and TechnologyTrondheimNorway

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