Uncertainty in Multi-criteria Decision-Making

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


All forest-related decisions include uncertainty. Uncertainty can be from several different sources like inventory data, growth and yield models, prices of timber or carbon, costs of operations and preferences of the forest owners. The uncertainties can be described in different ways, using a stochastic approach or a fuzzy approach, for instance. Many multi-criteria analysis methods are able to deal with uncertainty, but in each method the assumptions concerning the uncertainties are unique, and these differences may affect the results obtained. In this chapter, we present methods based on fuzzy numbers and fuzzy weighting, outranking methods based on fuzzy preference relations and stochastic multi-criteria acceptability analysis based on probability distributions. We discuss the drawbacks and benefits of these different approaches for forest management.


Fuzzy sets Fuzzy additive weighting Preference threshold Indifference threshold Outranking degree ELECTRE PROMETHEE Weight space analysis Rank probability Acceptability index 


  1. Adamo, J. M. (1980). Fuzzy decision trees. Fuzzy Sets and Systems, 4, 207–220.CrossRefGoogle Scholar
  2. Alho, J. M., & Kangas, J. (1997). Analyzing uncertainties in experts’ opinions of forest plan performance. Forest Science, 43, 521–528.Google Scholar
  3. Bana e Costa, C. (1986). A multicriteria decision-aid methodology to deal with conflicting situations on the weights. European Journal of Operational Research, 26, 22–34.CrossRefGoogle Scholar
  4. Bana e Costa, C. A. (1988). A methodology for sensitivity analysis in three criteria problems: A case study in municipal management. European Journal of Operational Research, 33, 159–173.CrossRefGoogle Scholar
  5. Bellman, R., & Zadeh, L. A. (1970). Decision making in fuzzy environment. Management Science, 17, B141–B164.CrossRefGoogle Scholar
  6. Belton, V., & Stewart, T. J. (2002). Multiple criteria decision analysis: An integrated approach. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  7. Bonissone, P. P. (1982). A fuzzy sets based linguistic approach: Theory and applications. In M. M. Gupta & E. Sanchez (Eds.), Approximate reasoning in decision analysis (pp. 329–339). New York: North-Holland Pub. Co.Google Scholar
  8. Bouyssou, D. (1992). Ranking methods based on valued preference relations: A characterization of the net flow method. European Journal of Operational Research, 60, 61–67.CrossRefGoogle Scholar
  9. Bouyssou, D., & Perny, P. (1992). Ranking methods for valued preference relations: A characterization of a method based on leaving and entering flows. European Journal of Operational Research, 61, 186–194.CrossRefGoogle Scholar
  10. Bouyssou, D., Marchant, T., Pirlot, M., Perny, P., Tsoukiàs, A., & Vincke, P. (2000). Evaluation and decision models – A critical perspective. Dordrecht: Kluwer.CrossRefGoogle Scholar
  11. Bouyssou, D., Jacquet-Lagrez, E., Perny, P., Slowinski, R., Vanderpooten, D., & Vincke, P. (Eds.). (2001). Aiding decisions with multiple criteria. Essays in honour of Bernard Roy. Dordrecht: Kluwer Academic Publishers.Google Scholar
  12. Brans, J. P., Vincke, P., & Mareschal, B. (1986). How to select and how to rank projects: The PROMETHEE method. European Journal of Operational Research, 24, 228–238.CrossRefGoogle Scholar
  13. Briggs, T., Kunsch, P. L., & Mareschal, B. (1990). Nuclear waste management: An application of the multicriteria PROMETHEE methods. European Journal of Operational Research, 44, 1–10.CrossRefGoogle Scholar
  14. Buchanan, J., Sheppard, P., & Vanderpoorten, D. (1999). Project ranking using ELECTRE III. Research report 99–01, University of Waikato, Dept. of management systems, New Zealand.Google Scholar
  15. Butler, J., Jia, J., & Dyer, J. (1997). Simulation techniques for the sensitivity analysis of multi-criteria decision models. European Journal of Operational Research, 103, 531–546.CrossRefGoogle Scholar
  16. Carlin, B. P., & Louis, T. A. (2000). Bayes and empirical Bayes methods for data analysis. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
  17. Chang, J.-R., Cheng, C.-H., & Kuo, C.-Y. (2006). Conceptual procedure for ranking fuzzy numbers based on adaptive two-dimension dominance. Soft Computing, 10, 94–103.CrossRefGoogle Scholar
  18. Charnetski, J. R., & Soland, R. M. (1978). Multiple-attribute decision making with partial information: The comparative hypervolume criterion. Naval Research Logistics Quarterly, 25, 279–288.CrossRefGoogle Scholar
  19. Chen, S.-J., & Hwang, C.-L. (1992). Fuzzy multiple attribute decision making. Methods and applications (Lecture notes in economics and mathematical systems, 375). Berlin: Springer. 536 p.CrossRefGoogle Scholar
  20. Cheng, C.-H. (1999). Evaluating weapon systems using raking fuzzy numbers. Fuzzy Sets and Systems, 107, 25–35.CrossRefGoogle Scholar
  21. Choi, D.-Y., & Oh, K.-W. (2000). ASA and its application to multi-criteria decision making. Fuzzy Sets and Systems, 114, 89–102.CrossRefGoogle Scholar
  22. d’Avignon, G. R., & Vincke, P. (1988). An outranking method under uncertainty. European Journal of Operational Research, 36, 311–321.CrossRefGoogle Scholar
  23. Despic, O., & Simonovich, S. P. (2000). Aggregation operators for soft decision making in water resources. Fuzzy Sets and Systems, 115, 11–33.CrossRefGoogle Scholar
  24. Dubois, D., & Prade, H. (1988). Possibility theory. An approach to computerized processing of uncertainty. New York: Plenum Press. 263 p.Google Scholar
  25. Dubois, D., Fargier, H., & Prade, H. (1994). Propagation and satisfaction of flexible constraints. In R. R. Yager & L. A. Zadeh (Eds.), Fuzzy sets, neural networks and soft computing (pp. 166–187). New York: Van Nostrand Reinnold.Google Scholar
  26. Dubois, D., Fargier, H., & Prade, H. (1996). Refinements of the maximin approach to decision making in a fuzzy environment. Fuzzy Sets and Systems, 81, 103–122.CrossRefGoogle Scholar
  27. Ducey, M. J., & Larson, B. C. (1999). A fuzzy set approach to the problem of sustainability. Forest Ecology and Management, 115, 29–40.CrossRefGoogle Scholar
  28. Eastman, J. R., & Jiang, H. (1996, May 21–23). Fuzzy measures in multi-criteria evaluation. In: H. Todd Mowrer, R. L. Czaplewski, & R. H. Hamre (Ed.), Proceedings of the 2nd international symposium on spatial accuracy assessment in natural resources and environmental studies (pp. 527–534, General technical report RM-GTR-277). Fort Collins: USDA Forest Service.Google Scholar
  29. Ferson, S., & Ginzburg, L. R. (1996). Different methods are needed to propagate ignorance and variability. Reliability Engineering and System Safety, 54, 133–144.CrossRefGoogle Scholar
  30. Goumas, M., & Lygerou, V. (2000). An extension of the PROMETHEE method for decision making in fuzzy environment: Ranking of alternative energy exploitation projects. European Journal of Operational Research, 123, 606–613.CrossRefGoogle Scholar
  31. Hannan, E. L. (1983). Fuzzy decision making with multiple objectives and discrete membership functions. International Journal of Man-Machine Studies, 18, 49–54.CrossRefGoogle Scholar
  32. Hellendoorn, H., & Thomas, C. (1993). Defuzzification in fuzzy controllers. Journal of Intelligent and Fuzzy Systems, 1, 109–123.Google Scholar
  33. Hokkanen, J., & Salminen, P. (1997a). Choosing a solid waste management system using multicriteria decision analysis. European Journal of Operational Research, 98, 19–36.CrossRefGoogle Scholar
  34. Hokkanen, J., & Salminen, P. (1997b). Locating a waste treatment facility by multicriteria analysis. Journal of Multi-Criteria Decision Analysis, 6, 175–184.CrossRefGoogle Scholar
  35. Hokkanen, J., & Salminen, P. (1997c). ELECTRE III and IV decision aids in an environmental problem. Journal of Multi-Criteria Decision Analysis, 6, 215–226.CrossRefGoogle Scholar
  36. Hokkanen, J., Lahdelma, R., & Salminen, P. (1999). A multiple criteria decision model for analyzing and choosing among different development patterns for the Helsinki cargo harbour. Socio-Economic Planning Sciences, 33, 1–23.CrossRefGoogle Scholar
  37. Hokkanen, J., Lahdelma, R., & Salminen, P. (2000). Multicriteria decision support in a technology competition for cleaning polluted soil in Helsinki. Journal of Environmental Management, 60, 339–348.CrossRefGoogle Scholar
  38. Kangas, A. (2006). The risk of decision making with incomplete criteria weight information. Canadian Journal of Forest Research, 36, 195–205.CrossRefGoogle Scholar
  39. Kangas, A., & Kangas, J. (2004a). Probability, possibility and evidence: Approaches to consider risk and uncertainty in forestry decision analysis. Forest Policy and Economics, 6, 169–188.CrossRefGoogle Scholar
  40. Kangas, J., & Kangas, A. (2004b). Multicriteria approval and SMAA-O in natural resources decision analysis with both ordinal and cardinal criteria. Journal of Multi-Criteria Decision Analysis, 12, 3–15.CrossRefGoogle Scholar
  41. Kangas, J., Store, R., Leskinen, P., & Mehtätalo, L. (2000). Improving the quality of landscape ecological forest planning by utilizing advanced decision-support tools. Forest Ecology and Management, 132, 157–171.CrossRefGoogle Scholar
  42. Kangas, A., Kangas, J., & Pykäläinen, J. (2001a). Outranking methods as tools in strategic natural resources planning. Silva Fennica, 35, 215–227.Google Scholar
  43. Kangas, J., Kangas, A., Leskinen, P., & Pykäläinen, J. (2001b). MCDM methods in strategic planning of forestry on state-owned lands in Finland: Applications and experiences. Journal of Multi-Criteria Decision Analysis, 10, 257–271.CrossRefGoogle Scholar
  44. Kangas, J., Hokkanen, J., Kangas, A., Lahdelma, R., & Salminen, P. (2003). Applying stochastic multicriteria acceptability analysis to forest ecosystem management with both cardinal and ordinal criteria. Forest Science, 49, 928–937.Google Scholar
  45. Kangas, A., Kangas, J., Lahdelma, R., & Salminen, P. (2006a). Using SMAA methods with dependent uncertainties for strategic forest planning. Forest Policy and Economics, 9, 113–125.CrossRefGoogle Scholar
  46. Kangas, A., Kangas, J., & Laukkanen, S. (2006b). Fuzzy multicriteria approval method and its application to two forest planning problems. Forest Science, 52, 232–242.Google Scholar
  47. Kangas, A., Leskinen, P., & Kangas, J. (2007). Comparison of fuzzy and statistical approaches in multi-criteria decision making. Forest Science, 53, 37–44.Google Scholar
  48. Klir, G. J., & Harmanec, D. (1997). Types and measures of uncertainty. In J. Kacprzyk, H. Nurmi, & M. Fedrizzi (Eds.), Consensus under fuzziness (International series in intelligent technologies). Dordrecht: Kluwer Academic Publishers.Google Scholar
  49. Lahdelma, R., & Salminen, P. (2001). SMAA-2: Stochastic multicriteria acceptability analysis for group decision making. Operations Research, 49, 444–454.CrossRefGoogle Scholar
  50. Lahdelma, R., & Salminen, P. (2002). Pseudo-criteria versus linear utility function in stochastic multi-criteria acceptability analysis. European Journal of Operational Research, 141, 454–469.CrossRefGoogle Scholar
  51. Lahdelma, R., Hokkanen, J., & Salminen, P. (1998). SMAA – Stochastic multiobjective acceptability analysis. European Journal of Operational Research, 106, 137–143.CrossRefGoogle Scholar
  52. Lahdelma, R., Salminen, P., & Hokkanen, J. (2002). Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria. European Journal of Operational Research, 142, 345–356.CrossRefGoogle Scholar
  53. Lahdelma, R., Miettinen, K., & Salminen, P. (2003). Ordinal criteria in stochastic multicriteria acceptability analysis (SMAA). European Journal of Operational Research, 147, 117–127.CrossRefGoogle Scholar
  54. Lahdelma, R., Miettinen, K., & Salminen, P. (2005a). Reference point approach for multiple decision makers. European Journal of Operational Research, 164, 785–791.CrossRefGoogle Scholar
  55. Lahdelma, R., Makkonen, S., & Salminen, P. (2005b). Two ways to handle dependent uncertainties in multi-criteria decision problems. Omega. In press, 164, 785–791.Google Scholar
  56. Leskinen, P., Kangas, A., & Kangas, J. (2004). Rank-based modelling of preferences in multi-criteria decision making. European Journal of Operational Research, 158, 721–733.CrossRefGoogle Scholar
  57. Leskinen, P., Viitanen, J., Kangas, A., & Kangas, J. (2006). Alternatives to incorporate uncertainty and risk attitude into multi-criteria evaluation of forest plans. Forest Science, 52, 304–312.Google Scholar
  58. Malczewski, J. (1999). GIS and multicriteria decision analysis. New York: Wiley. 392 p.Google Scholar
  59. Martin, W. E., Schields, D. J., Tolwinski, B., & Kent, B. (1996). An application of social choice theory to U.S.D.A. Forest Service decision making. Journal of Policy Modelling, 18, 603–621.CrossRefGoogle Scholar
  60. Mattila, J. (2002). Sumean logiikan oppikirja. Helsinki: Art House. 289 p.Google Scholar
  61. Maystre, L. Y., Pictet, J., & Simos, J. (1994). Méthodes multicritères ELECTRE. Lausanne: Presses Polytechniques et Universitaires Romandes.Google Scholar
  62. Miettinen, K., & Salminen, P. (1999). Decision-aid for discrete multiple criteria decision making problems with imprecise data. European Journal of Operational Research, 119, 50–60.CrossRefGoogle Scholar
  63. Miettinen, K., Lahdelma, R., & Salminen, P. (1999). SMAA-O – Stochastic multicriteria acceptability analysis with ordinal criteria. Reports of the Department of Mathematical Information Technology Series B, Scientific Computing B5. University of Jyväskylä.Google Scholar
  64. Niskanen, V. (2003). Software computing methods in human sciences (Studies in fuzziness and soft computing). Berlin: Springer. 280 p.Google Scholar
  65. Pirlot, M. (1995). A characterization of ‘min’ as a procedure for exploiting valued preference relations and related results. Journal of Multi-Criteria Decision Analysis, 4, 37–56.CrossRefGoogle Scholar
  66. Raju, K. S., & Pillai, C. R. S. (1999). Multicriteria decision making in performance evaluation of an irrigation system. European Journal of Operational Research, 112, 479–488.CrossRefGoogle Scholar
  67. Rietveld, P., & Ouwersloot, H. (1992). Ordinal data in multicriteria decision making, a stochastic dominance approach to siting nuclear power plants. European Journal of Operational Research, 56, 249–262.CrossRefGoogle Scholar
  68. Rogers, M., & Bruen, M. (1998a). A new system for weighting environmental criteria for use within ELECTRE III. European Journal of Operational Research, 107, 552–563.CrossRefGoogle Scholar
  69. Rogers, M., & Bruen, M. (1998b). Choosing realistic values of indifference, preference and veto thresholds for use with environmental criteria within ELECTRE. European Journal of Operational Research, 107, 542–551.CrossRefGoogle Scholar
  70. Rogers, M., Bruen, M., & Maystre, L.-Y. (2000). ELECTRE and decision support: Methods and applications in engineering and infrastructure investment. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  71. Rondeau, L., Ruelas, R., Levrat, L., & Lamotte, M. (1997). A defuzzification method respecting the fuzzification. Fuzzy Sets and Systems, 86, 311–320.CrossRefGoogle Scholar
  72. Roy, B. (1968). Classement et choix en présence de points de vue multiples (la méthode ELECTRE). Revue Française d’Informatique et de Recherche Opérationnelle, 8, 57–75.Google Scholar
  73. Roy, B. (1991). The outranking approach and the foundations of Electre methods. Theory and Decision, 31, 49–73.CrossRefGoogle Scholar
  74. Roy, B., Présent, M., & Silhol, D. (1986). A programming method for determining which Paris metro stations should be renovated. European Journal of Operational Research, 24, 318–334.CrossRefGoogle Scholar
  75. Salminen, P., Hokkanen, J., & Lahdelma, R. (1998). Comparing multicriteria methods in the context of environmental problems. European Journal of Operational Research, 104, 485–496.CrossRefGoogle Scholar
  76. Schafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton University Press. 297 p.Google Scholar
  77. Tervonen, T., & Lahdelma, R. (2007). Implementing stochastic multicriteria acceptability analysis. European Journal of Operational Research, 178, 500–513.CrossRefGoogle Scholar
  78. Tervonen, T., Figueira, J. R., Lahdelma, R., & Salminen, P. (2007). Towards robust ELECTRE III with simulation: Theory and software of SMAA-III. Submitted manuscript.Google Scholar
  79. Vincke, P. (1992). Multi-criteria decision aid. Chichester: Wiley. 154 p.Google Scholar
  80. Williams, D. (1991). Probability with martingales (Cambridge mathematical textbooks). Cambridge: Cambridge University Press. 251 p.CrossRefGoogle Scholar
  81. Yager, R. R. (1978). Fuzzy decision making including unequal objectives. Fuzzy Sets and Systems, 1, 87–95.CrossRefGoogle Scholar
  82. Yager, R. R. (1988). On ordered weigh averaging aggregating operators in multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 18, 183–190.CrossRefGoogle Scholar
  83. Yager, R. R. (1996). Knowledge-based defuzzification. Fuzzy Sets and Systems, 80, 177–185.CrossRefGoogle Scholar
  84. Yu, W. (1992). Aide multicritère à la décision dans le cadre de la problématique du tri: méthodes et applications. PhD thesis, LAMSADE, Université Paris-Dauphine, Paris.Google Scholar
  85. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.CrossRefGoogle Scholar
  86. Zimmermann, H.-J. (1985). Fuzzy set theory – And its applications (p. 355). Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
  87. Zimmermann, H.-J., & Zysno, P. (1980). Latest connectives in human decision making. Fuzzy Sets and Systems, 4, 37–51.CrossRefGoogle 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