Optimization and Engineering

, Volume 12, Issue 1–2, pp 5–29 | Cite as

Multicriteria analysis based on constructing payoff matrices and applying methods of decision making in fuzzy environment

  • Petr EkelEmail author
  • Illya Kokshenev
  • Reinaldo Palhares
  • Roberta Parreiras
  • Fernando Schuffner Neto


There exist two classes of problems, which need the use of a multicriteria approach: problems whose solution consequences cannot be estimated with a single criterion and problems that, initially, may require a single criterion or several criteria, but their unique solutions are unachievable, due to decision uncertainty regions, which can be contracted using additional criteria. According to this, two classes of models (〈X,M〉 and 〈X,R〉 models) can be constructed. Analysis of 〈X,M〉 and 〈X,R〉 models (based on applying the Bellman-Zadeh approach to decision making in a fuzzy environment and using fuzzy preference modeling techniques, respectively) serves as parts of a general scheme for multicriteria decision making under information uncertainty. This scheme is also associated with a generalization of the classic approach to considering the uncertainty of information (based on analyzing payoff matrices constructed for different combinations of solution alternatives and states of nature) in monocriteria decision making to multicriteria problems. The paper results are of a universal character and are illustrated by an example.


Multicriteria decision making Information uncertainty Bellman-Zadeh approach Fuzzy preference modeling Payoff matrices Particular and aggregated risks 


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  1. Antunes C, Dias L (2007) Editorial: Managing uncertainty in decision support models. Eur J Oper Res 181:1425–1426 CrossRefGoogle Scholar
  2. Beliakov G, Warren J (2001) Appropriate choice of aggregation operators in fuzzy decision support systems. IEEE Trans Fuzzy Syst 9:773–784 CrossRefGoogle Scholar
  3. Bellman R, Giertz M (1974) On the analytic formalism of the theory of fuzzy sets. Inf Sci 5:149–157 MathSciNetCrossRefGoogle Scholar
  4. Bellman RE, Zadeh L (1970) Decision-making in a fuzzy environment. Manag Sci 17:141–164 MathSciNetCrossRefGoogle Scholar
  5. Belyaev L (1977) A practical approach to choosing alternative solutions to complex optimization problems under uncertainty. IIASA, Laxenburg Google Scholar
  6. Bernardes P, Ekel P, Parreiras R (2009) A new consensus scheme for multicriteria group decision making under linguistic assessments. In: Mathematics and mathematical logic: New research. Nova Science, Houppauge (in press) Google Scholar
  7. Berredo RC, Ekel PYa, Galperin EA, Sant’anna AS (2005) Fuzzy preference modeling and its management applications. In: Proceedings of the international conference on industrial logistics. Montevideo, pp 41–50 Google Scholar
  8. Berredo RC, Canha LN, Ekel PYa, Ferreira LCA, Maciel MVC (2008) Experimental design and models of power system optimization and control. WSEAS Trans Syst Control 8:40–49 Google Scholar
  9. Canha L, Ekel P, Queiroz J, Schuffner Neto F (2007) Models and methods of decision making in fuzzy environment and their applications to power engineering problems. Numer Linear Algebra Appl 14:369–390 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Dubov YuA, Travkin CI, Yakimetc VN (1986) Multicriteria models for forming and choosing system alternatives. Nauka, Moscow Google Scholar
  11. Ehrgott M (2005) Multicriteria optimization. Springer, Berlin zbMATHGoogle Scholar
  12. Ekel PYa (1999) Approach to decision making in fuzzy environment. Comput Math Appl 37:59–71 MathSciNetzbMATHCrossRefGoogle Scholar
  13. Ekel PYa (2002) Fuzzy sets and models of decision making. Comput Math Appl 44:863–875 MathSciNetzbMATHCrossRefGoogle Scholar
  14. Ekel PYa, Galperin EA (2003) Box-triangular multiobjective linear programs for resource allocation with application to load management and energy market problems. Math Comput Model 37:1–17 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Ekel PYa, Schuffner Neto FH (2006) Algorithms of discrete optimization and their application to problems with fuzzy coefficients. Inf Sci 176:2846–2868 MathSciNetzbMATHCrossRefGoogle Scholar
  16. Ekel PYa, Terra LDB (2002) Fuzzy preference relations: methods and power engineering applications. Opsearch 39:34–45 MathSciNetzbMATHGoogle Scholar
  17. Ekel P, Pedrycz W, Schinzinger R (1998a) A general approach to solving a wide class of fuzzy optimization problems. Fuzzy Sets Syst 97:49–66 MathSciNetzbMATHCrossRefGoogle Scholar
  18. Ekel PYa, Terra LDB, Junges MFD, Moraes MA, Popov VA, Prakhovnik AV, Razumovsky OV (1998b) Multicriteria power and energy shortage allocation using fuzzy set theory. In: Proceedings of the international symposium on bulk power systems dynamics and control IV: restructuring. Santorini, pp 311–318 Google Scholar
  19. Ekel PYa, Junges MFD, Morra JLT, Paletta FPG (2002) Fuzzy logic based approach to voltage and reactive power control in power systems. Int J Comput Res 11:159–170 Google Scholar
  20. Ekel PYa, Martins CAPS, Pereira Jr JG, Palhares RM, Canha LN (2006a) Fuzzy set based allocation of resources and its applications. Comput Math Appl 52:197–210 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Ekel PYa, Silva MR, Schuffner Neto F, Palhares RM (2006b) Fuzzy preference modeling and its application to multiobjective decision making. Comput Math Appl 52:179–196 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ekel PYa, Menezes M, Schuffner Neto FH (2007) Decision making in fuzzy environment and its application to power engineering problems. Nonlinear Anal, Hybrid Syst 1:527–536 MathSciNetzbMATHCrossRefGoogle Scholar
  23. Ekel P, Queiros J, Parreiros R, Palhares R (2009) Fuzzy set based models and methods of multicriteria group decision making. Nonlinear Anal 71:409–419 CrossRefGoogle Scholar
  24. Fodor J, Roubens M (1994) Fuzzy preference modelling and multicriteria decision support. Kluwer Academic, Boston zbMATHGoogle Scholar
  25. Hodges J Jr, Lehmann E (1952) The use of previous experience in reaching statistical decision. Ann Math Stat 23:396–407 MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kuchta D (2007) Choice of the best alternative in case of a continuous set of states of nature—application of fuzzy numbers. Fuzzy Optim Decis Mak 6:173–178 MathSciNetzbMATHCrossRefGoogle Scholar
  27. Lu J, Zhang G, Ruan D, Wu F (2007) Multi-objective group decision making: methods, software and applications with fuzzy set techniques. Imperial College Press, London zbMATHGoogle Scholar
  28. Luce R, Raiffa H (1957) Games and decisions. Wiley, New York zbMATHGoogle Scholar
  29. Orlovski SA (1978) Decision making with a fuzzy preference relation. Fuzzy Sets Syst 1:155–167 CrossRefGoogle Scholar
  30. Orlovsky SA (1981) Problems of decision making with fuzzy information. Nauka, Moscow (in Russian) Google Scholar
  31. Pareto V (1896) Cours d’économie politique. Lausanne Rouge, Lausanne Google Scholar
  32. Pedrycz W, Gomide F (1998) An introduction to fuzzy sets: analysis and design. MIT Press, Cambridge zbMATHGoogle Scholar
  33. Raiffa H (1968) Decision analysis. Wesley, Reading zbMATHGoogle Scholar
  34. Rao SS (1996) Engineering optimization. Wiley, New York Google Scholar
  35. Raskin LG (1976) Analysis of complex systems and elements of optimal control theory. Sovetskoe Radio, Moscow (in Russian) zbMATHGoogle Scholar
  36. Sobol’ IM (1979) On the systematic search in a hypercube. SIAM J Numer Anal 6:790–793 MathSciNetCrossRefGoogle Scholar
  37. Trukhaev RI (1981) Models of decision making in conditions of uncertainty. Nauka, Moscow (in Russian) Google Scholar
  38. Wen M, Iwamura K (2008) Fuzzy facility location-allocation problem under the Hurwicz criterion. Eur J Oper Res 184:627–635 MathSciNetzbMATHCrossRefGoogle Scholar
  39. Yager RR (1988) On ordered weighted averaging operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18:183–190 MathSciNetzbMATHCrossRefGoogle Scholar
  40. Yager RR (1996) Fuzzy set methods for uncertainty representation in risky financial decisions. In: Proceedings of the IEEE/IAFE conference on computational intelligence for financial engineering. New York, pp 59–65 Google Scholar
  41. Yu PL (1985) Multiple criteria decision making: concepts, techniques, and extensions. Plenum, New York zbMATHGoogle Scholar
  42. Zimmermann HJ (1990) Fuzzy set theory and its application. Kluwer, Boston Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Petr Ekel
    • 1
    Email author
  • Illya Kokshenev
    • 2
  • Reinaldo Palhares
    • 2
  • Roberta Parreiras
    • 1
  • Fernando Schuffner Neto
    • 3
  1. 1.Graduate Program in Electrical EngineeringPontifical Catholic University of Minas GeraisBelo HorizonteBrasil
  2. 2.Department of Electronics EngineeringFederal University of Minas GeraisBelo HorizonteBrazil
  3. 3.Distribution and Commercialization BoardMinas Gerais State Energy CompanyBelo HorizonteBrazil

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