SeMA Journal

, Volume 76, Issue 2, pp 343–364 | Cite as

On identifying fuzzy knees in fuzzy multi-criteria optimization problems

  • Debdas GhoshEmail author


This paper introduces and analyzes the idea of fuzzy knee in fuzzy multi-criteria optimization problems. The fuzzy decision feasible region of the problem is constructed under a fuzzy inequality relation that is defined with the help of same points in fuzzy geometry. Then, fuzzy criteria feasible region is obtained through the image of the fuzzy decision feasible region by the criteria-vector-valued mapping. For the constructed fuzzy criteria feasible region, we define fuzzy knee and then propose a method to capture the fuzzy knee regions, along with the complete fuzzy Pareto set. All the studied ideas and methodologies are supported with suitable examples and pictorial illustrations. An engineering application of the presented method is also given.


Fuzzy multi-criteria optimization Same points Fuzzy inequality Fuzzy knee 

Mathematics Subject Classification

90C70 90C29 



The author is truly thankful to the anonymous reviewers and editors for their valuable comments and suggestions to improve the paper. The author gratefully acknowledges the financial support through Early Career Research Award (ECR/2015/000467), Science & Engineering Research Board, Government of India.


  1. 1.
    Bector, C.R., Chandra, S.: Fuzzy Mathematical Programming and Fuzzy Matrix Games, vol. 169. Springer, New York (2005)zbMATHGoogle Scholar
  2. 2.
    Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manag. Sci. 17(4), B141–B164 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. Lect. Notes Comput. Sci. 3242, 722–731 (2004)CrossRefGoogle Scholar
  4. 4.
    Carlsson, C., Fullér, R.: Fuzzy multiple criteria decision making: recent developments. Fuzzy Sets Syst. 78(2), 139–153 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carothers, N.L.: Real Analysis. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–109 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Das, I.: On characterizing the knee of the pareto curve based on normal-boundary intersection. Struct. Multidiscip. Optim. 18(2), 107–115 (1999)CrossRefGoogle Scholar
  8. 8.
    Deb, K.: Multi-objective evolutionary algorithms: introducing bias among pareto-optimal solutions. In: Advances in Evolutionary Computing, pp. 263–292. Springer, New York (2003)Google Scholar
  9. 9.
    Deb, K., Gupta, S.: Understanding knee points in bicriteria problems and their implications as preferred solution principles. Eng. Optim. 43(11), 1175–1204 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ehrgott, M.: Multicriteria Optimization, vol. 491. Springer, New York (2005)zbMATHGoogle Scholar
  11. 11.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ghosh, D., Chakraborty, D.: Ideal cone: a new method to generate complete pareto set of multi-criteria optimization problems. In: Mathematics and Computing 2013, pp. 171–190. Springer, New York (2014)Google Scholar
  13. 13.
    Ghosh, D., Chakraborty, D.: A new pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42(8), 514–521 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ghosh, D., Chakraborty, D.: A direction based classical method to obtain complete pareto set of multi-criteria optimization problems. Opsearch 52(2), 340–366 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III. Fuzzy Sets Syst. 283, 83–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kahraman, C.: Fuzzy Multi-criteria Decision Making: Theory and Applications with Recent Developments, vol. 16. Springer, Berlin (2008)zbMATHGoogle Scholar
  17. 17.
    Lai, Y.-J., Hwang, C.-L.: Fuzzy Multiple Objective Decision Making. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming: Methods and Applications, vol. 169. Springer, New York (1995)zbMATHGoogle Scholar
  19. 19.
    Li, X., Zhang, B., Li, H.: Computing efficient solutions to fuzzy multiple objective linear programming problems. Fuzzy Sets Syst. 157(10), 1328–1332 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pareto, V.: Cours d’économie politique, vol. 1. Librairie Droz, Paris (1964)CrossRefGoogle Scholar
  21. 21.
    Rachmawati, L., Srinivasan, D.: A multi-objective evolutionary algorithm with weighted-sum niching for convergence on knee regions. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, pp. 749–750. ACM, New York (2006)Google Scholar
  22. 22.
    Ramík, J.: Optimal solutions in optimization problem with objective function depending on fuzzy parameters. Fuzzy Sets Syst. 158(17), 1873–1881 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rommelfanger, H.: Fuzzy linear programming and applications. Eur. J. Oper. Res. 92(3), 512–527 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, X., Ruan, D., Kerre, E.E.: Mathematics of Fuzziness. Basic Issues, vol. 245. Springer, New York (2009)Google Scholar
  25. 25.
    Wu, H.C.: Using the technique of scalarization to solve the multiobjective programming problems with fuzzy coefficients. Math. Comput. Model. 48(1–2), 232–248 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1(1), 45–55 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zimmermann, H.J., Zadeh, L.A., Gaines, B.R.: Fuzzy Sets and Decision Analysis, vol. 20. North Holland, Amsterdam (1984)zbMATHGoogle Scholar
  28. 28.
    Zimmermann, H.J.: Fuzzy Set Theory—And Its Applications, 4th edn. Springer, New York (2001)CrossRefGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia

Personalised recommendations