Skip to main content
SpringerLink
Log in

Fuzzified Choquet Integral and its Applications in MADM: A Review and A New Method

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

Aggregation of information using Choquet integral method, caused to interdependent or interactive characteristics among the decision maker’s preference criteria also considered. In this paper, after introducing Choquet integral as a powerful aggregation function, some existing fuzzified Choquet integral methods will be reviewed. Then, we propose a new method for aggregation of fuzzy-valued information using Choquet integral and compare it with others. This method preserves the properties of fuzzy numbers, that is, the resulting data are the same type as the early data. So, ranking of such numbers, which is necessary in multi attribute decision-making (MADM) problems, was performed using ranking methods of fuzzy numbers. Also, we will apply the proposed method in both single and group decision-making problems to solve MADM problems, while the evaluation values and then decision matrix are fuzzy numbers.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. In [40] it is called α-cut of fuzzy-valued function, but it may be considered as α-cut of fuzzy set wrongly, then we called it α-level set as in Definition 3.1.

References

  1. Bebcakova, I.: Fuzzy methods of aggregation in decision making (In Czech). Ph.D. Thesis, Palacky University in Olomouc Department of Mathematical Analysis and Application of Mathematic (2012)

  2. Bebcakova, I., Holecek, P., Talasova, J.: On the application of the fuzzified Choquet integral to multiple criteria evaluation. Acta Polytechnica Hungarica 8(3), 65–78 (2011)

    Google Scholar 

  3. Bebcakova, I., Talasova, J., Pavalacka, O.: Fuzzification of Choquet integral and its application in multiple criteria decision making. Neural Netw. World 1(10), 125–137 (2010)

    Google Scholar 

  4. Beg, I., Rashid, T.: Multi-criteria trapezoidal valued intuitionistic fuzzy decision making with Choquet integral based TOPSIS. Oper. Res. Soc. India 51(1), 98–129 (2013)

    MathSciNet  Google Scholar 

  5. Belles-Samperaa, J., Guillena, M., Merigo, J.M., Santolinoa, M.: Indicators for the characterization of discrete Choquet integrals. Res. Inst. Appl. Econ., 1–43 (2013)

  6. Bustince, H., Fernandez, J., Mesiar, R., Kalicka, J.: Discrete interval-valued Choquet integrals. 6th International Summer School on Aggregation Operators, AGOP, 2011

  7. Bustince, H., Galar, M., Bedregal, B., Kolesarova, A., Mesiar, R.: A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Trans. Fuzzy Syst. 21(6), 1150–1162 (2013)

    Article  Google Scholar 

  8. Campos, L.M., Beolanos, M.J.: Characterization and comparison of Sugeno and Choquet integrals. Fuzzy Sets Syst. 52(1), 61–67 (1992)

    Article  MATH  Google Scholar 

  9. Cheng, C.H.: A new ranking fuzzy numbers by distance method. Fuzzy Sets Syst. 95(3), 307–317 (1998)

    Article  MATH  Google Scholar 

  10. Choquet, G.: Theory of Capacities, vol. 5, pp. 131–295. Annels Del Institut Fourier (1954)

  11. Gomesa, L.F.A.M., Machadoa, M.A.S., Costa, F.F., Rangelc, L.A.D.: Criteria interactions in multiple criteria decision aiding: a Choquet formulation for the TODIM method. Procedia Comput. Sci. 17, 324–331 (2013)

    Article  Google Scholar 

  12. Grabisch, M., Labreuche, C.: Fuzzy measures and integrals in MCDA. In: Multiple Criteria Decision Analysis, pp. 563–608. Kluwer Academic Publishers, New York (2004)

  13. Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann. Oper. Res. 175(1), 247–286 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grabisch, M., Roubens, M.: Application of the Choquet integral in multi criteria decision making. Fuzzy Meas. Integrals Theory Appl. 40, 348–374 (2000)

    MathSciNet  Google Scholar 

  15. Havens, T.C., Anderson, D.T. Keller, J.M.: A fuzzy Choquet integral with an interval type-2 fuzzy number-valued integrand. In: IEEE International Conference on Fuzzy Systems (FUZZ) 2010, vol. 1, no. 8, pp. 18–23. IEEE (2010)

  16. Herrera, F., Herrera-Viedma, E., Martinez, L.: A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114(1), 43–58 (2000)

    Article  MATH  Google Scholar 

  17. Huang, Y., Wu, C.: Real-valued Choquet integral for set-valued mappings. Int. J. Approx. Reason. 55(2), 683–688 (2013)

    Article  Google Scholar 

  18. James, S.: The use of aggregation functions in decision making. Submitted in fullment of the requirements for the degree of Doctor of Philosophy, Deakin University, December 2010

  19. Jang, L.C., Kil, B.M., Kim, Y.K., Kwon, J.S.: Some properties of Choquet integrals of set-valued functions. Fuzzy Sets Syst. 91, 95–98 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jang, L.C.: Interval-valued Choquet integral and their applications. J. Appl. Math. Comput. 16(1–2), 429–443 (2004)

    Google Scholar 

  21. Jang, L.C.: On properties of the Choquet integral of interval-valued functions. J. Appl. Math. 2011(492149), 10 (2011). doi:10.1155/2011/492149

    Google Scholar 

  22. Jang, L.C.: Note on the Choquet integral as an interval-valued aggregation operators and their application. J. Appl. Math. 2012(154670), 13 (2012). doi:10.1155/2012/154670. Hindawi

    Google Scholar 

  23. Labreuche, C., Grabisch, M.: The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets Syst. 137(1), 11–26 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Meyer, P., Roubens, M.: On the use of the Choquet integral with fuzzy numbers in multiple criteria decision support. Fuzzy Sets Syst. 157(1), 927–938 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Modave, F., Grabisch, M.: Preference representation by Choquet integral: the commensurability hypothesis. In: Proceedings of the Seventh International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems, Paris, pp. 164–171, 1998

  26. Marichal, J.L.: Aggregation of interacting criteria by means of the discrete Choquet integral. Aggreg. Oper. Stud. Fuzziness Soft Comput. 97(1), 224–244 (2002)

    MathSciNet  Google Scholar 

  27. Murofushi, T.: A technique for reading fuzzy measures (i): The Shapley value with respect to a fuzzy measure. In: 2nd Fuzzy Workshop, pp. 39–48, 1992

  28. Narukawa, Y., Torra, V., Gakuen, T.: Fuzzy measures and Choquet integral on discrete spaces. Comput. Intell. Theory Appl. Adv. Soft Comput. 33, 573–581 (2005)

    Google Scholar 

  29. Phani, P., Rao, B., Shankar, N.R.: Ranking fuzzy numbers with an area method using circumcenter of centroids. Fuzzy Inf. Eng. 5(1), 3–18 (2013)

    Article  MathSciNet  Google Scholar 

  30. Sakawa, M.: Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York (1993)

    Book  MATH  Google Scholar 

  31. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587 (1998)

    Article  MathSciNet  Google Scholar 

  32. Shapley, L.S.: A value for n-person games in Contributions to the Theory of Games II. Ann. Math. Stud. Princeton Univ. 28, 307–317 (1953)

    MathSciNet  Google Scholar 

  33. Talasova, J., Bebcakova, I.: Fuzzification of aggregation operators based on Choquet integrals. J. Appl. Math. 1(1), 463–474 (2008)

    Google Scholar 

  34. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(1), 529–539 (2010)

    MATH  Google Scholar 

  35. Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, pp. 1378–1382. IEEE (2009)

  36. Tzeng, G.H., Huang, J.J.: Multiple Attribute Decision Making Methods and Application. CRC Press, Boca Raton (2011)

    Google Scholar 

  37. Wang, H., Li, S.: Some properties and convergence theorems of set-valued Choquet integrals. Fuzzy Sets Syst. 219, 81–97 (2013)

    Article  MATH  Google Scholar 

  38. Wang, Y.M., Yanga, J.B., Xua, D.L., Chinc, K.S.: On the centroids of fuzzy numbers. Fuzzy Sets Syst. 157, 919–926 (2006)

    Article  MATH  Google Scholar 

  39. Wang, Z., Klir, G.L.: Generalized Measure Theory. Springer, New York (2008)

    Google Scholar 

  40. Wang, Z., Yang, R., Heng, P.A., Leung, K.S.: Real-valued Choquet integrals with fuzzy integrand. Fuzzy Sets Syst. 157(1), 256–269 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wang, Z., Leung, K.S., Wong, M.L., Fang, J., Xu, K.: Nonlinear nonnegative multi regressions based on Choquet integrals. Int. J. Approx. Reason. 25(1), 71–87 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang, Z., Yang, R., Leung, K.S.: Nonlinear Integrals and Their Application in Data Mining Advances in Fuzzy Systems—Applications and Theory, vol. 17. World Scientific Publishing Co., Pte. Ltd, Singapore (2010)

    Google Scholar 

  43. Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52(1), 395–407 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. Xu, Z.: Choquet integrals of weighted intuitionistic fuzzy information. Inf. Sci. 180, 726–736 (2010)

    Article  MATH  Google Scholar 

  45. Yang, R., Wang, Z., Heng, P.A., Leung, K.S.: Fuzzy numbers and fuzzification of the Choquet integral. Fuzzy Sets Syst. 153(1), 95–113 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  46. Yang, R., Wang, Z., Heng, P.A.: Fuzzified Choquet integral with a fuzzy-valued integrand and its application on temperature prediction. IEEE Trans. Syst. Man Cybern. Part B 38(2), 367–380 (2008)

    Article  Google Scholar 

  47. Yu, D., Wu, Y., Zhou, W.: Multi-criteria decision making based on Choquet integral under hesitant fuzzy environment. J. Comput. Inf. Syst. 7(12), 4506–4513 (2011)

    Google Scholar 

  48. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  49. Zhang, D., Guo, C., Liu, D.: Set- valued Choquet integral revisited. Fuzzy Sets Syst. 147, 475–485 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the associate editor and the anonymous reviewers whose comments are quite useful for us to improve the paper.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

    A. Keikha & H. Mishmast Nehi

Authors
  1. A. Keikha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Mishmast Nehi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Mishmast Nehi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keikha, A., Mishmast Nehi, H. Fuzzified Choquet Integral and its Applications in MADM: A Review and A New Method. Int. J. Fuzzy Syst. 17, 337–352 (2015). https://doi.org/10.1007/s40815-015-0037-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-015-0037-0

Keywords

Search

Navigation