Skip to main content
SpringerLink
Log in

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

  • Published:
Fuzzy Optimization and Decision Making Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ammar E. E. (2008) On solutions of fuzzy random multiobjective quadratic programming with application in portfolio problem. Information Sciences 178: 468–484

    Article  MathSciNet  MATH  Google Scholar 

  • Campos L., Verdegay J. L. (1989) Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems 32: 1–11

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

  • 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.

  • Dubois D., Prade H. (1980) Fuzzy sets and systems: Theory and application. Academic Press, San Diego, USA

    Google Scholar 

  • Dubois D., Prade H. (1983) Ranking fuzzy numbers in the setting of possibility theory. Information Sciences 30(3): 183–224

    Article  MathSciNet  MATH  Google Scholar 

  • Farina, M., & Amato, P. (2003). Fuzzy optimality and evolutionary multiobjective optimization. In Second international conference evolutionary multi-criterion optimization, Faro, Portugal.

  • 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): 315–326

    Article  Google Scholar 

  • Hussein M. L., Abdel Aaty Maaty M. (1997) The stability notions for fuzzy nonlinear programming problem. Fuzzy Sets and Systems 85: 319–323

    Article  MathSciNet  MATH  Google Scholar 

  • 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: 59–75

    Article  MathSciNet  MATH  Google Scholar 

  • Kaufmann A., Gupta M. M. (1984) Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold, New York, USA

    Google Scholar 

  • Klir G. J., Yuan B. (1995) Fuzzy sets and fuzzy logic: Theory and applications. Prentice Hall, New Jersey, USA

    MATH  Google Scholar 

  • Köppen, M., Franke, K., Nickolay, B. (2003). Fuzzy-pareto-pominance driven multiobjective genetic algorithm. In 10th IFSA world congress, Istanbul, Turkey.

  • 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.

  • Kuwano, H. (2000). Inverse problems in fuzzy multiobjective linear programming. In Second international conference on knowledge-based intelligent eletronic systems, Adelaide, Australia.

  • Pareto V. (1897) Cours d’economique politique (Vol. 1). Macmillan, Paris, FR

    Google Scholar 

  • Pareto V. (1897) Le cours d’economique politique (Vol. 2). Macmillan, London, UK

    Google Scholar 

  • Pedrycs W., Gomide F. (1998) An introduction of fuzzy sets: Analisys and design. MIT press, Cambridge, MA

    Google Scholar 

  • Sakawa M. (1993) Fuzzy sets and interactive multiobjective optimization. Plenum Press, New York, USA

    MATH  Google Scholar 

  • Zadeh L. A. (1965) Fuzzy sets. Information and Control 8: 338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1(1): 3–28

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L. A. (2008) Is there a need for fuzzy logic?. Information Sciences 178: 2751–2779

    Article  MathSciNet  MATH  Google Scholar 

  • Zimmermann H.-J. (1996) Fuzzy set theory and its applications (3rd ed.). Kluwer Academic Publishers, Massachusetts, USA

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Telematics, School of Electrical and Computer Engineering, University of Campinas, C.P. 6101, Campinas, SP, 13083-970, Brazil

    Ricardo C. Silva & Akebo Yamakami

Authors
  1. Ricardo C. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akebo Yamakami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo C. Silva.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Silva, R.C., Yamakami, A. The use of possibility theory in the definition of fuzzy Pareto-optimality. Fuzzy Optim Decis Making 10, 11–30 (2011). https://doi.org/10.1007/s10700-010-9092-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-010-9092-z

Keywords

Search

Navigation