Skip to main content

Advertisement

Springer Nature Link
Log in

A New Picture Fuzzy Entropy and Its Application Based on Combined Picture Fuzzy Methodology with Partial Weight Information

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

Abstract

Picture fuzzy set (PFS) is more comprehensive tool than intuitionistic fuzzy set (IFS) for modeling the uncertain decision-making problems. In this paper, a new picture fuzzy entropy measure is proposed and proved that the proposed measure satisfies the axiomatic definition of entropy measures for picture fuzzy sets. Besides this, the useful mathematical properties of the new entropy measure are also investigated. The justification of the proposed picture fuzzy measure is established by discussing its particular cases and compares it with the existing entropy measures. Then, for the case where criteria weights are partially known, we used an entropy-based method to produce objective weights. For the uncertain environment, TODIM (portuguese acronym for interactive multicriteria decision-making) and ELECTRE methods are useful for practical problems. Based on the advantages of PFSs,TODIM, and ELECTRE, we proposed an integrated picture fuzzy TODIM-ELECTRE to combine the prominent benefits of these theories. We present the TODIM-ELECTRE model for PFS environment and express the computing steps in brief of this new established model. Thereafter, the superiority of the new model is verified by a numerical example of supplier selection and through comparative study with other existing methods.

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

Access this article

Log in via an institution

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

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

    MATH  Google Scholar 

  2. Verma, R., Sharma, B.D.: On generalized exponential fuzzy entropy. World Acad. Sci. Eng. Technol. 60, 1402–1405 (2011)

    Google Scholar 

  3. Verma, R., Maheshwari, S.: A new measure of divergence with its application to multi-criteria decision making under fuzzy environment. Neural Comput. Appl. 28(8), 2335–2350 (2016)

    Google Scholar 

  4. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Set Syst. 20, 87–96 (1986)

    MATH  Google Scholar 

  5. Mishra, A.R., Kumari, R., Sharma, D.K.: Intuitionistic fuzzy divergence measure-based multi-criteria decision-making method. Neural Comput. Appl. (2019). https://doi.org/10.1007/s00521-017-3187-1

    Article  Google Scholar 

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

    MATH  Google Scholar 

  7. Wei, G.W.: Picture fuzzy cross-entropy for multiple attribute decision making problems. J. Bus Eco Manag. 17(4), 491–502 (2016)

    Google Scholar 

  8. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982)

    MATH  Google Scholar 

  9. Dutta, S., Ghatak, S., Dey, R., Das, A.K., Ghosh, S.: Attribute selection for improving spam classification in online social networks: a rough set theory-based approach. Soc. Netw. Anal. Min. 8, 1–16 (2018)

    Google Scholar 

  10. Singh, S., Shreevastava, S., Som, T., Somani, G.: A fuzzy similarity-based rough set approach for attribute selection in set-valued information systems. Soft. Comput. 24(6), 4675–4691 (2020)

    MATH  Google Scholar 

  11. Su, Z., Yager, R.R.: Some geometric aggregation averaging operator based on intuitionistic fuzzy sets. Int. J. Gen Syst 35, 417–433 (2006)

    MathSciNet  Google Scholar 

  12. Arya, V., Kumar, S.: A novel VIKOR-TODIM approach based on Havrda–Charvat–Tsallis Entropy of Intuitionistic fuzzy sets to evaluate Management Information System. Fuzzy Inf. Eng. (2020). https://doi.org/10.1080/16168658.2020.1840317

    Article  Google Scholar 

  13. Joshi, R., Kumar, S.: An intuitionistics fuzzy (\(\delta , \gamma\))- norm entropy with its application in supplier selection problem. Comput. Appl. Math. 37(5), 5624–5649 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Joshi, R., Kumar, S.: A new parametric intuitionistic fuzzy entropy and its applications in multiple attribute decision making. Int. J. Appl. Comput. Math. 4(1), 52 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Joshi, R., Kumar, S.: A new weighted (\(\alpha ,\beta\))-norm information measure with applications in coding theory. Phys. Stat. Mech. Appl. 510, 538–551 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Mishra, A.R., Rani, P., Pardasani, K.R., Mardani, A., Stevic, Z., Pamucar, D.: A novel entropy and divergence measures with multi-criteria service quality assessment using interval-valued intuitionistic fuzzy TODIM method. Soft. Comput. 24, 11641 (2020)

    MATH  Google Scholar 

  17. Arya, V., Kumar, S.: Extended VIKOR-TODIM approach based on entropy weight for intuitionistic fuzzy sets. Adv. Intell. Syst. Comput. 1169, 95–108 (2020)

    Google Scholar 

  18. Szmidt, E., Kacprzyk, J.: A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning. International Conference on Artificial Intelligence and Soft Computing. Springer, Berlin (2004)

  19. Zhang, Z., Yang, J., Ye, Y., Hu, Y., Zhang, Q.: A type of score function on intuitionistic fuzzy sets with double parameters and its application to pattern recognition and medical diagnosis. In: Proceedings in Engineering, pp. 4336–4342. Elsevier, Amsterdam (2012)

    Google Scholar 

  20. Jeevaraj, S.: Similarity measure on interval valued intuitionistic fuzzy numbers based on non-hesitance score and its application to pattern recognition. Comput. Appl. Math. 39(3), 1–15 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Singh, S., Ganie, A.H.: On a new picture fuzzy correlation coefficient with its applications to pattern recognition and identification of an investment sector. Comput. Appl. Math. 41(1), 1–35 (2022)

    MathSciNet  MATH  Google Scholar 

  22. Zadeh, L.A.: The concept of linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8, 199–249 (1975)

    MathSciNet  MATH  Google Scholar 

  23. Cuong, B.C.: Picture Fuzzy Sets-First results, Part 1, seminar,‘Neuro-Fuzzy Systems with Applications’. Institute of Mathematics, Hanoi (2013)

    Google Scholar 

  24. Cuong, B.C.: Picture fuzzy sets. J. Comput. Sci. Cybernet. 30(4), 409–420 (2014)

    Google Scholar 

  25. Nie, R.X., Wang, J.Q., Li, L.: A shareholder voting method for proxy advisory firm selection based on 2-tuple linguistic picture preference relation. Appl. Soft Comput. 60, 520–539 (2017)

    Google Scholar 

  26. Peng, X., Dai, J.: Algorithm for picture fuzzy multiple attribute decision making based on new distance measure. Int. J. Uncertain Quant. 7, 177–187 (2017)

    MathSciNet  Google Scholar 

  27. Joshi, R.: A novel decision-making method using R-norm concept and VIKOR approach under picture fuzzy environment. Expert Syst. Appl. 147, 113228 (2020)

    Google Scholar 

  28. Joshi, R.: A new picture fuzzy information measure based on Tsallis–Havrda–Charvat concept with applications in presaging poll outcome. Comput. Appl. Math. 39, 1–24 (2020)

    MathSciNet  MATH  Google Scholar 

  29. Kadian, R., Kumar, S.: A new picture fuzzy divergence measure based on Jensen–Tsallis information measure and its application to multicriteria decision making. Granular Comput. (2021). https://doi.org/10.1007/s41066-021-00254-6

    Article  Google Scholar 

  30. Zhang, X.Y., Wang, J.Q., Hu, J.H.: On novel operational laws and aggregation operators of picture 2-tuple linguistic information for MCDM problems. Int. J. Fuzzy Syst. 20(3), 958–969 (2018)

    MathSciNet  Google Scholar 

  31. Son, H.: Generalized Picture Distance Measure and Applications to Picture Fuzzy Clustering. Appl. Soft Comput. 46, 284–295 (2016)

    Google Scholar 

  32. Wang, C., Zhou, X., Tu, H., Tao, S.: Some geometric aggregation operators based on picture fuzzy Setsand their application in multiple attribute decision making. Ital. J. Pure Appl. Math. 37, 477–492 (2017)

    MATH  Google Scholar 

  33. Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Projection models for multiple attribute decision making with picture fuzzy information. Int. J. Mach. Learn. Cybern. 9(4), 713–719 (2018)

    Google Scholar 

  34. Arya, V., Kumar, S.: A new picture fuzzy information measure based on Shannon entropy with applications in opinion polls using extended VIKOR-TODIM approach. Comput. Appl. Math. 39(3), 1–24 (2020)

    MathSciNet  MATH  Google Scholar 

  35. Luo, S.Z., Liang, W.Z.: A hybrid TODIM approach with unknown weight information for the performance evaluation of cleaner production. Comput. Appl. Math. 40(1), 1–28 (2021)

    MathSciNet  Google Scholar 

  36. De Luca, A., Termini, S.: A definition of a non-probabilitic entropy in the setting of fuzzy set theory. Inf. Control 20, 301–312 (1972)

    MATH  Google Scholar 

  37. Arya, V., Kumar, S.: Fuzzy entropy measure with an applications in decision making under bipolar fuzzy environment based on TOPSIS method. Int. J. Inf. Manag. Sci. 31(2), 99–121 (2020)

    MATH  Google Scholar 

  38. Arya, V., Kumar, S.: Knowledge measure and entropy: a complementary concept in fuzzy theory. Granul. Comput. 6(3), 631–643 (2020)

    Google Scholar 

  39. Bhandari, D., Pal, N.R.: Some new information measures for fuzzy sets. Inf. Sci. 67(3), 204–228 (1973)

    MathSciNet  Google Scholar 

  40. Hooda, D.S.: On generalized measures of fuzzy entropy. Math. Slovaca 54(3), 315–325 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Hung, W.L., Yang, M.S.: Fuzzy entropy on intuitionistic fuzzy sets. Int. J. Intell. Syst. 21(4), 443–451 (2006)

    MATH  Google Scholar 

  42. Vlachos, I.K., Sergiadis, G.D.: Intuitionistic fuzzy information-applications to pattern recognition. Pattern Recogn. Lett. 28(2), 197–206 (2007)

    Google Scholar 

  43. Wang, J., Wang, P.: Intutionistic linguistic fuzzy multi-critria decision-making method based on intutionistic fuzzy entropy. Control Decis. 27, 1694–1698 (2012)

    MathSciNet  Google Scholar 

  44. Chatterjee, I.: Artificial intelligence and patentability: review and discussions. Int. J. Mod. Res. 1, 15–21 (2021)

    Google Scholar 

  45. Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making: Methods and Applications. Springer, Berlin (1981)

    MATH  Google Scholar 

  46. Fan, Z.P., Zhang, X., Chen, F.D., Liu, Y.: Extended TODIM method for hybrid multiple attribute decision making problems. Knowl.-Based Syst. 42, 40–48 (2013)

    Google Scholar 

  47. Lourenzutti, R., Krohling, R.A.: A study of TODIM in a intuitionistic fuzzy and random environment. Exp. Syst. Appl. 40, 6459–6468 (2013)

    Google Scholar 

  48. Krohling, R.A., Pacheco, A.G.C., Siviero, A.L.T.: IF-TODIM: an intuitionistic fuzzy TODIM to multicriteria decision making. Knowl.- Based Syst. 53, 142–146 (2013)

    Google Scholar 

  49. Konwar, N., Debnath, P.: Continuity and Banach contraction principle in intuitionistic fuzzy n-normed linear spaces. J. Intell. Fuzzy Syst. 33(4), 2363–2373 (2017)

    MATH  Google Scholar 

  50. Wei, G.W., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Int. J. Fuzzy Syst. 20(1), 1–12 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Benayoun, R., Roy, B., Sussman, B.: Electre: Une methode pour guider le choix en presence de points de vue multiples, Note de travail 49. SEMAMETRA International, Direction Scientifique (1966)

    Google Scholar 

  52. Opricovic, S.: Multicriteria Optimization of Civil Engineering Systems. Faculty of Civil Engineering, Belgrade, Serbia (1988)

  53. Brans, J.P., Mareschel, V.: PROMETHEE: a new family of outranking methods in multicriteria analysis. In: Brans, J.P. (ed.) Operational Research, vol. 84, pp. 477–490. North-Holland, New York (1984)

    Google Scholar 

  54. Arya, V., Kumar, S.: A picture fuzzy multiple criteria decision making approach based on the combined TODIM-VIKOR and entropy weighted method. Cogn. Comput. (2021). https://doi.org/10.1007/s12559-021-09892-z

    Article  Google Scholar 

  55. Xu, D.S., Wei, X.L., Ding, H., Bin, H.Q.: A new method based on PROMETHEE and TODIM for multi-attribute decision-making with single-valued neutrosophic sets. Mathematics 8, 1816 (2020). https://doi.org/10.3390/math8101816

    Article  Google Scholar 

  56. Cuong, B.C., Kreinovich, V.: Picture Fuzzy Sets-a new concept for computational intelligence problems. In: Third World Congress on Information and Communication Technologies, vol. 809 (2013)

  57. Havrda, J.H., Charvat, F.: Quantification method classification process: concept of structural \(\alpha\)-entropy. Kybernetika 3, 30–35 (1967)

    MathSciNet  MATH  Google Scholar 

  58. Shannon, C.E.: The mathematical theory of communication. Bell Syst. Technol. J. 27(3), 379–423 (1948)

    MathSciNet  MATH  Google Scholar 

  59. Tsallis, C.: Possible generalization of Boltzman–Gibbs statistics. J. Stat. Phys. 52, 480–487 (1988)

    Google Scholar 

  60. Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Set Syst. 114, 505–518 (2000)

    MathSciNet  MATH  Google Scholar 

  61. Dhiman, G., Kumar, V.: Emperor penguin optimizer: a bio-inspired algorithm for engineering problems. Knowl. Based Syst. 159, 20–50 (2018)

    Google Scholar 

  62. Dhiman, G., Kaur, A.: STOA: a bio-inspired based optimization algorithm for industrial engineering problems. Eng. Appl. Artif. Intell. 82, 148–174 (2019)

    Google Scholar 

  63. Vaishnav, P.K., Sharma, S., Sharma, P.: Analytical review analysis for screening COVID-19. Int. J. Mod. Res. 1, 22–29 (2021)

    Google Scholar 

  64. Dhiman, G., Kumar, V.: Seagull optimization algorithm: theory and its applications for large-scale industrial engineering problems. Knowl. Based Syst. 165, 169–196 (2019)

    Google Scholar 

  65. Wang, L., Peng, J.J., Wang, J.Q.: A multi-criteria decision-making framework for risk ranking of energy performance contracting project under picture fuzzy environment. J. Clean. Product. 191, 105–118 (2018)

    Google Scholar 

  66. Tian, C., Peng, J.J., Zhang, S., Zhang, W.Y., Wang, J.Q.: Weighted picture fuzzy aggregation operators and their applications to multi-criteria decision-making problems. Comput. Ind. Eng. 137, 106037 (2019)

    Google Scholar 

  67. Tolga, A.C., Parlak, I.B., Castillo, O.: Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem. Eng. Appl. Artif. Intell. 87, 103352 (2020)

    Google Scholar 

  68. Wu, X., Liao, H., Xu, Z., Hafezalkotob, A., Herrera, F.: Probabilistic linguistic MULTIMOORA: a multicriteria decision making method based on the probabilistic linguistic expectation function and the improved Borda rule. IEEE Trans. Fuzzy Syst. 26(6), 3688–3702 (2018)

    Google Scholar 

  69. Wang, L., Zhang, H.Y., Wang, J.Q., Li, L.: Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project. Appl. Soft Comput. 64, 216–226 (2018)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chandigarh University Gharuan, Punjab, India

    Sunit Kumar

  2. Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana-Ambala, 133207, India

    Vikas Arya & Satish Kumar

  3. Department of Mathematics, D.A.V. College, Pehowa, Kurukshetra, India

    Ashok Dahiya

Authors
  1. Sunit Kumar
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Vikas Arya
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Satish Kumar
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Ashok Dahiya
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Sunit Kumar.

Ethics declarations

Conflicts of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Arya, V., Kumar, S. et al. A New Picture Fuzzy Entropy and Its Application Based on Combined Picture Fuzzy Methodology with Partial Weight Information. Int. J. Fuzzy Syst. 24, 3208–3225 (2022). https://doi.org/10.1007/s40815-022-01332-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-022-01332-w

Keywords