Fuzzy Optimization and Decision Making

, Volume 10, Issue 1, pp 11–30 | Cite as

The use of possibility theory in the definition of fuzzy Pareto-optimality

  • Ricardo C. Silva
  • Akebo Yamakami


Pareto-optimality conditions are crucial when dealing with classic multi-objective optimization problems. Extensions of these conditions to the fuzzy domain have been discussed and addressed in recent literature. This work presents a novel approach based on the definition of a fuzzily ordered set with a view to generating the necessary conditions for the Pareto-optimality of candidate solutions in the fuzzy domain. Making use of the conditions generated, one can characterize fuzzy efficient solutions by means of carefully chosen mono-objective problems and Karush-Kuhn-Tucker conditions to fuzzy non-dominated solutions. The uncertainties are inserted into the formulation of the studied fuzzy multi-objective optimization problem by means of fuzzy coefficients in the objective function. Some numerical examples are analytically solved to illustrate the efficiency of the proposed approach.


Fuzzy logic Possibility theory Multi-objective optimization Fuzzy Pareto-optimality conditions Fuzzy optimization 


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  1. Ammar E. E. (2008) On solutions of fuzzy random multiobjective quadratic programming with application in portfolio problem. Information Sciences 178: 468–484MathSciNetMATHCrossRefGoogle Scholar
  2. Campos L., Verdegay J. L. (1989) Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems 32: 1–11MathSciNetMATHCrossRefGoogle Scholar
  3. Cantão, L. A. P., & Yamakami, A. (2003). Nonlinear programming with fuzzy parameters: Theory and applications. In International conference on computational intelligence for modelling, control and automation, CIMCA.Google Scholar
  4. Chankong, V., & Haimes, Y. Y. (1983). Multiobjective decision making: Theory and methodology, Vol. 8 of North Hollando series in system science and engineering. North Holland, New York, USA.Google Scholar
  5. Dubois D., Prade H. (1980) Fuzzy sets and systems: Theory and application. Academic Press, San Diego, USAGoogle Scholar
  6. Dubois D., Prade H. (1983) Ranking fuzzy numbers in the setting of possibility theory. Information Sciences 30(3): 183–224MathSciNetMATHCrossRefGoogle Scholar
  7. Farina, M., & Amato, P. (2003). Fuzzy optimality and evolutionary multiobjective optimization. In Second international conference evolutionary multi-criterion optimization, Faro, Portugal.Google Scholar
  8. Farina M., Amato P. (2004) A fuzzy definition of”optimalit” for many-criteria optimization problems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 34(3): 315–326CrossRefGoogle Scholar
  9. Hussein M. L., Abdel Aaty Maaty M. (1997) The stability notions for fuzzy nonlinear programming problem. Fuzzy Sets and Systems 85: 319–323MathSciNetMATHCrossRefGoogle Scholar
  10. Jiménez F., Cadenas J. M., Sánchez G., Gómez-skarmeta A. F., Verdegay J. L. (2006) Multi-objective evolutionary computation and fuzzy optimization. International Journal of Approximate Reasoning 43: 59–75MathSciNetMATHCrossRefGoogle Scholar
  11. Kaufmann A., Gupta M. M. (1984) Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold, New York, USAGoogle Scholar
  12. Klir G. J., Yuan B. (1995) Fuzzy sets and fuzzy logic: Theory and applications. Prentice Hall, New Jersey, USAMATHGoogle Scholar
  13. Köppen, M., Franke, K., Nickolay, B. (2003). Fuzzy-pareto-pominance driven multiobjective genetic algorithm. In 10th IFSA world congress, Istanbul, Turkey.Google Scholar
  14. Köppen, M., Garcia, R. V., Nickolay, B. (2005). Fuzzy-pareto-dominance and its application in evolutionary multi-objective optimization. In Evolutionary multi-criterion optimization, third international conference, Guanajuato, Mexico.Google Scholar
  15. Kuwano, H. (2000). Inverse problems in fuzzy multiobjective linear programming. In Second international conference on knowledge-based intelligent eletronic systems, Adelaide, Australia.Google Scholar
  16. Pareto V. (1897) Cours d’economique politique (Vol. 1). Macmillan, Paris, FRGoogle Scholar
  17. Pareto V. (1897) Le cours d’economique politique (Vol. 2). Macmillan, London, UKGoogle Scholar
  18. Pedrycs W., Gomide F. (1998) An introduction of fuzzy sets: Analisys and design. MIT press, Cambridge, MAGoogle Scholar
  19. Sakawa M. (1993) Fuzzy sets and interactive multiobjective optimization. Plenum Press, New York, USAMATHGoogle Scholar
  20. Zadeh L. A. (1965) Fuzzy sets. Information and Control 8: 338–353MathSciNetMATHCrossRefGoogle Scholar
  21. Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1(1): 3–28MathSciNetMATHCrossRefGoogle Scholar
  22. Zadeh L. A. (2008) Is there a need for fuzzy logic?. Information Sciences 178: 2751–2779MathSciNetMATHCrossRefGoogle Scholar
  23. Zimmermann H.-J. (1996) Fuzzy set theory and its applications (3rd ed.). Kluwer Academic Publishers, Massachusetts, USAMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Telematics, School of Electrical and Computer EngineeringUniversity of CampinasCampinasBrazil

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