Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A new picture fuzzy information measure based on Tsallis–Havrda–Charvat concept with applications in presaging poll outcome

  • 21 Accesses

Abstract

The picture fuzzy set (PFS) proposed by Cuong and Kreinovich are well suitable to capture the uncertain information in vague circumstances. The main objective of this communication is to propose a new framework as a criteria of fuzzy entropy for PFSs. Further, a new picture fuzzy information measure based on Tsallis–Havrda–Charvat entropy is proposed and validated in accordance with newly proposed framework. Besides this, some major properties of proposed information measure are also discussed. Apart from this, a new multi-criteria decision-making method using the concept of VIKOR (Vlsekriterijumska Optimizacija i Kompromisno Resenje) based on relative projection is proposed. To show the practical utility of proposed decision-making method, two numerical examples based on election forecast through opinion polls have been discussed.

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

References

  1. Aqlan F, Lam SS (2015) A fuzzy-based integrated framework for supply chain risk assessment. Int J Prod Econ 161:54–63

  2. Atanassov KT (1986) Intutionistic fuzzy sets. Fuzzy Sets Syst 20:87–96

  3. Atanassov KT (1999) Intutionistic fuzzy sets. Springer, New York

  4. Bajaj RK, Kumar T, Gupta N (2012) \(R\)-norm intutionistic fuzzy information measures and its computational applications, ICECCS 2012. CCIS 305:372–380

  5. Benayoun R, Roy B, Sussman B (1966) ELECTRE: Une méthode pour guider le choix en présence de points de vue multiples, Note de travail 49. SEMA-METRA International, Direction Scientifique, Paris

  6. Bhandari D, Pal NR (1993) Some new information measures for fuzzy sets. Inf Sci 67(3):204–228

  7. Boekee DE, Vander Lubbe JCA (1980) The \(R\)-norm information measure. Inf Control 45:136–155

  8. Brans JP, Mareschel V (1984) PROMETHEE: a new family of outranking methods in multicriteria analysis. In: Brans JP (ed) Operational research’ 84. North-Holland, New York, pp 477–490

  9. Burillo P, Bustince H (2001) Entropy on intutionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 118:305–316

  10. Chen T, Li C (2010a) Determinig objective weights with intutionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180:4207–4222

  11. Chen T, Li C (2010b) Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180:4207–4222

  12. Choo EU, Wedley WC (1985) Optimal criterian weights in repetitive multicriteria decision making. J Oper Res Soc 36:983–992

  13. Chou SY, Chang YH, Shen CY (2008) A fuzzy simple additive weighting system under group decision-making for facility location selection with objective/subjective attributes. Eur J Oper Res 189:132–145

  14. Chu ATW, Kalaba RE, Spingarn K (1979) A comparison of two methods for determining the weights of belonging to fuzzy sets. J Optimiz Theor App 27:531–538

  15. Cuong BC, Kreinovich V (2012) Picture fuzzy sets-a new concept for computational intelligence problems. In: Proceedings of of 3rd world congress on information and communication technologies (WICT), pp 1–6

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

  17. Deng Y (2012) D Numbers: theory and applications. J Inf Comput Sci 9(9):2421–2428

  18. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gener Syst 17:191–209

  19. Fan ZP (1996) Complicated multiple attribute decision making: theory and applications. Ph.D. Dissertation, Northeastern university, Shenyang, China

  20. Grattan-Guiness I (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z Math Logik Grundladen Math 22:149–160

  21. Havdra JH, Charvat F (1967) Quantification method classification process: concept of structral \(\alpha \)-entropy. Kybernetika 3:30–35

  22. Hooda DS (2004) On generalized measures of fuzzy entropy. Math Slovaca 54:315–325

  23. Hung WL, Yang MS (2006) Fuzzy entropy on intutionistic fuzzy sets. Int J Intell Syst 21:443–451

  24. Hung WL, Yang MS (2008) On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition. Inf Sci 178:1641–1650

  25. Hwang CL, Lin MJ (1987) Group decision making under multiple criteria: methods and applications. Springer, Berlin

  26. Hwang CL, Yoon KP (1981) Multiple attribute decision-making: methods and applications. Springer, New York

  27. Jahn KU (1975) Intervall-wertige Mengen. Math Nach 68:115–132

  28. Jiang YC, Tang Y, Wang J, Tang S (2009) Reasoning within intuitionistic fuzzy rough description logics. Inf Sci 179:2362–2378

  29. Joshi R (2019) A new multi-criteria decision-making method based on intuitionistic fuzzy information and its application to fault detection in a machine. J Ambient Intell Human Comput. https://doi.org/10.1007/s12652-019-01322-1

  30. Joshi R, Kumar S (2016) \((R, S)\)-norm information measure and a relation between coding and questionnaire theory. Open Syst Inf Dyn 23(3):1–12

  31. Joshi R, Kumar S (2018a) An intuitionistic fuzzy \((\delta,\gamma )\)-norm entropy with its application in supplier selection problem. Comput Appl Math. https://doi.org/10.1007/s40314-018-0656-9

  32. Joshi R, Kumar S (2018b) An intuitionistic fuzzy information measure of order \((\alpha, \beta )\) with a new approach in supplier selection problems using an extended VIKOR method. J Appl Math Comput. https://doi.org/10.1007/s12190-018-1202-z

  33. Joshi R, Kumar S (2018c) A new parametric intuitionistic fuzzy entropy and its applications in multiple attribute decision making. Int J Appl Comput Math 4:52–74

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

  35. Joshi R, Kumar S (2018e) A novel fuzzy decision making method using entropy weights based correlation coefficients under intuitionistic fuzzy environment. Int J Fuzzy Syst. https://doi.org/10.1007/s40815-018-0538-8

  36. Joshi R, Kumar S (2018f) An \((R^{\prime }, S^{\prime })\)-norm fuzzy relative information measure and its application in strategic decision-making. Appl Math Comp. https://doi.org/10.1007/s40314-018-0582-x

  37. Kapur JN (1997) Measures of fuzzy information. Mathematical Sciences Trust Society, New Delhi

  38. Nei R-X, Wang J-Q, Wang T-L (2018) A hybrid outranking method for greenhouse gas emissions’ institution selection with picture 2-tuple linguistic information. Comput Appl Math 37(5):6676–6699

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

  40. Opricovic S (1998) Multi-criteria optimization of civil engineering systems. Ph.D. Thesis, University of Belgrade, Belgrade, Serbia

  41. Opricovic S, Tzeng G-H (2004) Decision aiding compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur J Oper Res 156:445–455

  42. Opricovic S, Tzeng GH (2007) Extended VIKOR method in comparison with outranking methods. Eur J Oper Res 178:514–529

  43. Pillay A, Wang J (2015) A risk ranking approach incorporating fuzzy set theory and grey theory. Reliabil Eng Syst Saf 1(10):23–25

  44. Renyi A (1961) On measures of entropy and information. In: Proceedings of 4th Barkley symposium on mathematical statistics and probability. University of California Press, vol 1, pp 547–561

  45. Saaty TL (1980) The analytical hierarchy process. McGraw-Hill, New York

  46. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423

  47. Shen K-W, Wang J-Q (2018) Z-VIKOR method Based on a new comprehensive weighted distance measure of Z-number and its application. IEEE Trans Fuzzy Syst 26(6):3232–3245

  48. Son LH (2016) Generalized picture distance measure and its application to picture fuzzy clustering. Appl Soft Comput 46:284–295

  49. Son LH (2017) Measuring analogousness in picture fuzzy sets: from picture distance measures to picture association measures. Fuzzy Optim Decis Mak 16:1–20

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

  51. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. 18th IEEE conference on fuzzy systems. Jeju Island, Korea, pp 1378–1382

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

  53. Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information–applications to pattern recognition. Pattern Recognit Lett 28(2):197–206

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

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

  56. Wang L, Zhang H-Y, Wang J-Q, Wu G-F (2018) Picture fuzzy multi-criteria group decision-making method to hotel building energy efficiency retrofit project selection. RAIRO Oper Res. https://doi.org/10.1051/ro/2019004

  57. Wang X, Wang J, Zhang H (2018) Distance-based multicriteria group decision making approach with probabilistic linguistic term sets. Expert Syst. https://doi.org/10.1111/exsy.12352

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

  59. Wang R, Wang J, Gao H, Wei G (2019) Methods for MADM with picture fuzzy muirhead mean operators and their application for evaluating the financial investment risk. Symmetry 11(1):6

  60. Wei GW (2016) Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag 17(4):491–502

  61. Wei GW (2018) Some more similarity measures for picture fuzzy sets and their applications. Iran J Fuzzy Syst 15(1):77–89

  62. Wei G (2018) TODIM method for picture fuzzy multiple attribute decision making. Informatica 29(3):555–566

  63. Wei G, Gao H (2018) The generalized dice similarity measures for picture fuzzy sets and their applications. Informatica 29(1):107–124

  64. Wei G, Alsaadi FE, Hayat T, Alsaedi A (2016) Projection models for multiple attribute decision making with picture fuzzy information. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-016-0604-1

  65. Wei G, Alsaadi FE, Hayat T, Alsaedi A (2018) Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft Comput 22(3):989–1002

  66. Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433

  67. Xu ZS, Yager RR (2008) Dynamic intuitionistic fuzzy multi-attribute decision making. Int J Approx Reason 48:246–262

  68. Yager RR (1979) On the measure of fuzziness and negation part I: membership in the unit interval. Int J Gen Syst 5(4):221–229

  69. Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18:183–190

  70. Yager RR (2004) Generalized OWA aggregation operators. Fuzzy Optim Decis Mak 3:93–107

  71. Yu PL (1973) A class of solutions for group decision making problem. Manag Sci 19:936–946

  72. Zadeh LA (1965) Fuzzy sets. Inf Comput 8(3):338–353

  73. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427

  74. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inform Sci 8:199–249

  75. Zeng W, Yu F, Yu X, Chen H, Wu S (2009) Entropy on intuitionistic fuzzy set based on similarity measure. Int J Innov Comput Inf Control 5(12):4737–4744

  76. Zhang XY, Wang J-Q, Hu JH (2017) On novel operational laws and aggregation operators of picture 2-tuple linguistic information for MCDM problems. Int J Fuzzy Syst 20(6):1–12

  77. Zhang X-Y, Wang X-K, Yu S-M, Wang J-Q, Wang T-L (2018) Location selection of offshore wind power station by consensus decision framework using picture fuzzy modelling. J Clean Prod 202:980–992

  78. Zhang S, Gao H, Wei G, Wei Y, Wei C (2019) Evaluation based on distance from average solution method for multiple criteria group decision making under picture 2-tuple linguistic environment. Mathematics 7(3):243

  79. Zhao N, Xu ZS (2016) Entropy measures for interval valued intuitionistic fuzzy information from a comparative perspective and their application to decision making. Informatica 27:203–228

  80. Zhu B, Xu Z (2018) Probability-Hesitant fuzzy sets and the representation of preference relations. Technol Econ Dev Econ 24(3):1029–1040

Download references

Author information

Correspondence to Rajesh Joshi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Marcos Eduardo Valle.

Appendix

Appendix

Proof of Theorem (4.3):

To prove theorem (4.3), we divide X into two parts say \(X_1\) and \(X_2\) such that

$$\begin{aligned} X_1=\{\kappa _i\in X: \vartheta _1\subseteq \vartheta _2\}, \quad \qquad X_2=\{\kappa _i\in X: \vartheta _1\supseteq \vartheta _2\}. \end{aligned}$$
(8.1)

Then for all \(\kappa _i\in X_1\),

$$\begin{aligned} \mu _{\vartheta _1} (\kappa _i)\le \mu _{\vartheta _2} (\kappa _i),~\delta _{\vartheta _1} (\kappa _i)\le \delta _{\vartheta _2} (\kappa _i),~\nu _{\vartheta _1} (\kappa _i)\ge \nu _{\vartheta _2} (\kappa _i), \end{aligned}$$
(8.2)

and for all \(\kappa _i\in X_2\),

$$\begin{aligned} \mu _{\vartheta _1} (\kappa _i)\ge \mu _{\vartheta _2} (\kappa _i),~\delta _{\vartheta _1} (\kappa _i)\ge \delta _{\vartheta _2} (\kappa _i),~\nu _{\vartheta _1} (\kappa _i)\le \nu _{\vartheta _2} (\kappa _i). \end{aligned}$$
(8.3)

Using (4.8),

$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{i=1}^n\left( \left( \mu _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _1\cup \vartheta _2} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$
(8.4)
$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{X_1}\left( \left( \mu _{\vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _2} (\kappa _i)^\tau \right) -1\right) \nonumber \\&+\frac{1}{n(1-\tau )}\sum _{X_2}\left( \left( \mu _{\vartheta _1} (\kappa _i)^\tau +\delta _{\vartheta _1} (\kappa _i)^\tau +\nu _{\vartheta _1} (\kappa _i)^\tau +\pi _{\vartheta _1} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$
(8.5)

Similarly,

$$\begin{aligned} \psi (\vartheta _1\cap \vartheta _2)=&\frac{1}{n(1-\tau )}\sum _{X_1}\left( \left( \mu _{\vartheta _1} (\kappa _i)^\tau +\delta _{\vartheta _1} (\kappa _i)^\tau +\nu _{\vartheta _1} (\kappa _i)^\tau +\pi _{\vartheta _1} (\kappa _i)^\tau \right) -1\right) \nonumber \\&+\frac{1}{n(1-\tau )}\sum _{X_2}\left( \left( \mu _{\vartheta _2} (\kappa _i)^\tau +\delta _{\vartheta _2} (\kappa _i)^\tau +\nu _{\vartheta _2} (\kappa _i)^\tau +\pi _{\vartheta _2} (\kappa _i)^\tau \right) -1\right) . \end{aligned}$$
(8.6)

From (8.5) and (8.6), we get

$$\begin{aligned} \psi (\vartheta _1\cup \vartheta _2)+\psi (\vartheta _1\cap \vartheta _2)=\psi (\vartheta _1)+\psi (\vartheta _2). \end{aligned}$$
(8.7)

Corollary: For any PFS \(\vartheta \) and its complement \(\vartheta ^c\), we have

$$\begin{aligned} \psi (\vartheta )=\psi (\vartheta ^c)=\psi (\vartheta \cup \vartheta ^c)=\psi (\vartheta \cap \vartheta ^c). \end{aligned}$$
(8.8)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Joshi, R. A new picture fuzzy information measure based on Tsallis–Havrda–Charvat concept with applications in presaging poll outcome. Comp. Appl. Math. 39, 71 (2020). https://doi.org/10.1007/s40314-020-1106-z

Download citation

Keywords

  • Picture fuzzy set
  • Tsallis–Havrda–Charvat picture fuzzy entropy
  • Hamming distance
  • VIKOR
  • TOPSIS

Mathematics Subject Classification

  • 94A15
  • 94A24
  • 26D15