Uncertainty in Optimisation

  • Annika Kangas
  • Mikko Kurttila
  • Teppo Hujala
  • Kyle Eyvindson
  • Jyrki Kangas
Part of the Managing Forest Ecosystems book series (MAFE, volume 30)


Most of the applications dealing with linear programming use deterministic programming. Stochastic programming is, however, a more realistic way of formulating forest management planning problems, and it produces solutions that have a better expected value than the deterministic solutions. In this chapter, we present the linear and goal programming problems from the previous chapter using stochastic programming approach and deterministic equivalent problem formulation. We present approaches based on one stage and two stages. We describe the uncertainty using a scenario tree. We show how the value of stochastic solution and value of perfect information are calculated from the results. We also show how the computationally more easy chance-constrained and robust programming methods work in these cases. Finally, we present an approach of robust portfolio modelling for dealing with the uncertainty.


Stochastic programming with simple recourse Two-stage programming with recourse Soft and hard constraints Scenario tree Value of information Value of stochastic solution Downside risk 


  1. Alho, J. M., Kolehmainen, O., & Leskinen, P. (2001). Regression methods for pairwise comparisons data. In D. L. Schmoldt, J. Kangas, G. A. Mendoza, & M. Pesonen (Eds.), The analytic hierarchy process in natural resource and environmental decision making (pp. 235–251). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  2. Alonso-Ayoso, A., Escudero, L. F., Guignard, M., Quinteros, M., & Weintraub, A. (2011). Forestry management under uncertainty. Annals of Operations Research, 190, 18–39.Google Scholar
  3. Andalaft, N., Aldalaft, P., Guignard, M., Magendzo, A., Wainer, A., & Weintraub, A. (2003). A problem of forest harvesting and road building solved through model strengthening and lagrangean relaxation. Operations Research, 51, 613–628.CrossRefGoogle Scholar
  4. Aouni, B., Ben Abdelaziz, F., & La Torre, D. (2012). The stochastic goal programming model: Theory and applications. Journal of Multi-Criteria Decision Analysis, 19, 185–200.CrossRefGoogle Scholar
  5. Bare, B. B., & Mendoza, G. A. (1992). Timber harvest scheduling in a fuzzy decision environment. Canadian Journal of Forest Research, 22, 423–428.CrossRefGoogle Scholar
  6. Bell, D. E. (1982). Regret in decision making under uncertainty. Operations Research, 30, 961–981.CrossRefGoogle Scholar
  7. Bell, D. E. (1985). Disappointment in decision making under uncertainty. Operations Research, 33, 1–27.CrossRefGoogle Scholar
  8. Ben-Tal, A., & Nemirowski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25, 1–13.CrossRefGoogle Scholar
  9. Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming Series B, 98, 49–71.CrossRefGoogle Scholar
  10. Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52, 35–53.CrossRefGoogle Scholar
  11. Bevers, M. (2007). A chance constrained estimation approach to optimizing resource management under uncertainty. Canadian Journal of Forest Research, 37, 2270–2280.CrossRefGoogle Scholar
  12. Boychuk, D., & Martell, D. L. (1996). A multistage stochastic programming. model for sustainable forestlevel timber supply under risk of fire. Forest Science, 42(1), 10–26.Google Scholar
  13. Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer. 421 p.Google Scholar
  14. Duckstein, L., Korhonen, P., & Tecle, A. (1988). Multiobjective forest management: A visual, interactive, and fuzzy approach. In B. M. Kent & L. S. Davis (Eds.), The 1988 symposium on Systems Analysis in Forest Resources (General technical report RM-161, pp. 68–74). Fort Collins: USDA Forest Service.Google Scholar
  15. Ells, A., Bulte, E., & van Kooten, G. C. (1997). Uncertainty and forest land use allocation in British Columbia: Vague priorities and imprecise coefficients. Forest Science, 43, 509–520.Google Scholar
  16. Eriksson, L. O. (2006). Planning under uncertainty at the forest level: A systems approach. Scandinavian Journal of Forest Research, 21, 111–117.CrossRefGoogle Scholar
  17. Eyvindson, K., & Kangas, A. (2014). Stochastic goal programming in forest planning. Canadian Journal of Forest Research, 44, 1274–1280.Google Scholar
  18. Eyvindson, K., & Kangas, A. (2015). Integrating risk preferences in forest harvest scheduling. Annals of Forest Science.Google Scholar
  19. Fischetti, M., & Monaci, M. (2009). Light robustness. In R. K. Ahuja, R. H. Möhring, & C. H. Zaroliagis (Eds.), Robust and online large-scale optimization. Models and techniques for transportation systems (pp. 64–81). Berlin: Springer. 421 p.Google Scholar
  20. Gassmann, H. I. (1989). Optimal harvest of a forest in the presence of uncertainty. Canadian Journal of Forest Research, 19, 1267–1274.CrossRefGoogle Scholar
  21. Gilabert, H., & McDill, M. (2011). Optimizing inventory and yield data collection for forest management planning. Forest Science, 56, 578–591.Google Scholar
  22. Haight, R. G., & Travis, L. E. (1997). Wildlife conservation planning using stochastic optimization and importance sampling. Forest Science, 43, 129–139.Google Scholar
  23. Hobbs, B. F., & Hepenstal, A. (1989). Is optimization optimistically biased? Water Resources Research, 25, 152–160.CrossRefGoogle Scholar
  24. Hof, J. G., & Pickens, J. B. (1991). Chance-constrained and chance-maximizing mathematical programs in renewable resource management. Forest Science, 37(1), 308–325.Google Scholar
  25. Hof, J. G., Kent, B. M., & Pickens, J. B. (1992). Chance constraints and chance maximization with random yield coefficients in renewable resource optimization. Forest Science, 38, 305–323.Google Scholar
  26. Hof, J. G., Pickens, J. B., & Bartlett, E. T. (1995). Pragmatic approaches to optimization with random yield coefficients. Forest Science, 41, 501–512.Google Scholar
  27. Hof, J. G., Bevers, M., & Pickens, J. (1996). Chance-constrained optimization with spatially autocorrelated forest yields. Forest Science, 42, 118–123.Google Scholar
  28. Itami, H. (1974). Expected objective value of a stochastic linear program and the degree of uncertainty of parameters. Management Science, 21, 291–301.CrossRefGoogle Scholar
  29. Kangas, A. (2010). Value of forest information. European Journal of Forest Research, 129, 863–874.CrossRefGoogle Scholar
  30. Kangas, A., & Eyvindson K. (2013). Modelling forest planning problems with stochastic optimization. In R. Kiriş (Ed.), Proceedings of the international symposium for the 50th anniversary of the forestry sector planning in Turkey (pp. 89–97), 26–28 November 2013 Antalya. 816 p. Ankara 2013.Google Scholar
  31. Kangas, A., & Kangas, J. (1999). Optimization bias in forest management planning solutions due to errors in forest variables. Silva Fennica, 33, 303–315.Google Scholar
  32. Kangas, A., Hurttala, H., Mäkinen, H., & Lappi, J. (2012). Value of quality information in selecting stands to be purchased. Canadian Journal of Forest Research, 42, 1347–1358.CrossRefGoogle Scholar
  33. Kangas, A., Hartikainen, M., & Miettinen, K. (2013). Simultaneous optimization of harvest schedule and measurement strategy. Scandinavian Journal of Forest Research, 29, 224–233.CrossRefGoogle Scholar
  34. King, A. J., & Wallace, S. W. (2012). Modelling with stochastic programming. New York: Springer. 173 p.CrossRefGoogle Scholar
  35. Krcmar, E., Stennes, B., van Kooten, G. C., & Vertinsky, I. (2001). Carbon sequestration and land management under uncertainty. European Journal of Operational Research, 135, 616–629.CrossRefGoogle Scholar
  36. Krzemienowski, A., & Ogryczak, W. (2005). On extending the LP computable risk measures to account downside risk. Computational Optimization and Applications, 32, 133–160.CrossRefGoogle Scholar
  37. Kurttila, M., Muinonen, E., Leskinen, P., Kilpeläinen, H., & Pykäläinen, J. (2009). An approach for examining the effects of preferential uncertainty on the contents of forest management plan at stand and holding level. European Journal of Forest Research, 128, 37–50.CrossRefGoogle Scholar
  38. Lappi, J., & Siitonen, M. (1985). A utility model for timber production based on different interest rates for loans and savings. Silva Fennica, 19, 271–280.CrossRefGoogle Scholar
  39. Lawrence, D. B. (1999). The economic value of information. New York: Springer. 393 p.CrossRefGoogle Scholar
  40. Leskinen, P. (2001). Statistical methods for measuring preferences (Publications in social sciences, 48). Joensuu: University of Joensuu.Google Scholar
  41. Liebchen, C., Lübbecke, M., Möhring, R., & Stiller, S. (2009). The concept of recoverable robustness, linear programming recovery, and railway applications. In R. K. Ahuja, R. H. Möhring, & C. H. Zaroliagis (Eds.), Robust and online large-scale optimization. Models and techniques for transportation systems (pp. 1–27). Berlin: Springer-Verlag. 421 p.CrossRefGoogle Scholar
  42. Liesiö, J., Mild, P., & Salo, A. (2007). Preference programming for robust portfolio modeling and project selection. European Journal of Operational Research, 181, 1488–1505.CrossRefGoogle Scholar
  43. Liesiö, J., Mild, P., & Salo, A. (2008). Robust portfolio modelling with incomplete cost information and project interdependencies. European Journal of Operational Research, 190(3), 679–695.Google Scholar
  44. Mendoza, G. A., & Sprouse, W. (1989). Forest planning and decision making under fuzzy environments: An overview and illustration. Forest Science, 33, 458–468.Google Scholar
  45. Mendoza, G. A., Bare, B. B., & Zhou, Z. (1993). A fuzzy multiple objective linear programming approach to forest planning under uncertainty. Agricultural Systems, 41, 257–274.CrossRefGoogle Scholar
  46. Ntaimo, L., Gallego-Arrubla, J. A., Gan, J., Stripling, C., Young, J., & Spencer, T. (2013). A simulation and stochastic integer programming approach to wildfire initial attack planning. Forestry Sciences, 59, 105–117.Google Scholar
  47. Palma, C. D., & Nelson, J. D. (2009). A robust optimization approach protected harvest scheduling decisions against uncertainty. Canadian Journal of Forest Research, 39, 342–355.CrossRefGoogle Scholar
  48. Pickens, J. B., & Dress, P. E. (1988). Use of stochastic production coefficients in linear programming models: Objective function distribution, feasibility, and dual activities. Forest Science, 34, 574–591.Google Scholar
  49. Pickens, J. B., & Hof, J. G. (1991). Fuzzy goal programming in forestry – An application with special solution problems. Fuzzy Sets and Systems, 39, 239–246.CrossRefGoogle Scholar
  50. Pickens, J. B., Hof, J. G., & Kent, B. M. (1991). Use of chance-constrained programming to account for stochastic variation in the A-matrix of large-scale linear programs. A forestry application. Annales of Operations Research, 31, 511–526.CrossRefGoogle Scholar
  51. Punkka, A. (2006). Luonnonarvokaupan tukeminen monikriteerisillä päätösmalleilla. Systems Analysis Laboratory, Helsinki University of Technology. (In Finnish).Google Scholar
  52. Salo, A., & Punkka, A. (2005). Rank inclusion in criteria hierarchies. European Journal of Operational Research, 163, 338–356.CrossRefGoogle Scholar
  53. Tecle, A., Duckstein, L., & Korhonen, P. (1994). Interactive multiobjective programming for forest resources management. Applied Mathematics and Computing, 63, 75–93.CrossRefGoogle Scholar
  54. Weintraub, A., & Abramovich, A. (1995). Analysis of uncertainty of future timber yields in forest management. Forest Science, 41, 217–234.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Annika Kangas
    • 1
  • Mikko Kurttila
    • 2
  • Teppo Hujala
    • 3
  • Kyle Eyvindson
    • 4
  • Jyrki Kangas
    • 5
  1. 1.Economics and SocietyNatural Resources Institute Finland (Luke)JoensuuFinland
  2. 2.Bio-based Business and IndustryNatural Resources Institute Finland (Luke)JoensuuFinland
  3. 3.Bio-based Business and IndustryNatural Resources Institute Finland (Luke)HelsinkiFinland
  4. 4.Department of Forest SciencesUniversity of HelsinkiHelsinkiFinland
  5. 5.School of Forest SciencesUniversity of Eastern FinlandJoensuuFinland

Personalised recommendations