A novel correlation coefficient of intuitionistic fuzzysets based on the connection number of set pair analysis and its application

Document Type : Article

Authors

School of Mathematics, Thapar University Patiala 147004, Punjab, India

Abstract

Set pair analysis (SPA) is an updated theory for dealing with the uncertainty, which overlaps the other
theories of uncertainty such as probability, vague, fuzzy and intuitionistic fuzzy set (IFS). Considering the
fact that the correlation coecient plays an important role during the decision-making process, in this paper,
after pointing out the weakness of the existing correlation coecients between the IFSs, we propose a novel
correlation coecient and weighted correlation coecients formulation to measure the relative strength of the
di erent IFSs. For it, rstly corresponding to each intuitionistic fuzzy number, the connection number of the
SPA theory has been formulated in the form of the degree of identity, discrepancy and contrary and then
based on its, a novel correlation coecient measures have been de ned. Pairs of identity, discrepancy and
contrary of the connection number have been taken as a vector representation during the formulation. Lastly,
a decision-making approach based on the proposed measures has been presented which has been illustrated
by two numerical examples in pattern recognition and medical diagnosis.

Keywords

Main Subjects


References
1. Zadeh, L.A. Fuzzy sets", Inform. Control, 8, pp. 338-
353 (1965).
2. Atanassov, K.T. Intuitionistic fuzzy sets", Fuzzy Sets
Syst., 20, pp. 87-96 (1986).
3. Atanassov, K. and Gargov, G. Interval-valued intuitionistic
fuzzy sets", Fuzzy Sets Syst., 31, pp. 343-349
(1989).
4. Xu, Z.S. and Yager, R.R. Some geometric aggregation
operators based on intuitionistic fuzzy sets", Int. J. of
Gen. Syst., 35, pp. 417-433 (2006).
5. Xu, Z.S. Intuitionistic fuzzy aggregation operators",
IEEE T. Fuzzy Syst., 15, pp. 1179-1187 (2007).
6. Garg, H. Generalized intuitionistic fuzzy interactive
geometric interaction operators using Einstein t-norm
and t-conorm and their application to decision making",
Comput. Ind. Eng., 101, pp. 53-69 (2016).
7. Xu, Z. and Chen, J. Approach to group decision making
based on interval valued intuitionistic judgment
matrices", Syst. Eng. - Theory and Pract., 27(4), pp.
126-133 (2007).
8. Xu, Z.S. Methods for aggregating interval-valued
intuitionistic fuzzy information and their application
to decision making", Control and Decision, 22(2), pp.
215-219 (2007).
9. Garg, H., Agarwal, N., and Tripathi, A. Entropy
based multi-criteria decision making method under
fuzzy environment and unknown attribute weights",
Global J. Technol. Optimiz., 6, pp. 13-20 (2015).
10. Garg, H. A new generalized improved score function
of interval-valued intuitionistic fuzzy sets and applications
in expert systems", Appl. Soft Comput., 38, pp.
988-999 (2016).
11. Garg, H. Generalized Pythagorean fuzzy geometric
aggregation operators using Einstein t-norm and tconorm
for multicriteria decision-making process", Int.
J. Intell. Syst., 32(6), pp. 597-630 (2017).
12. Garg, H. A new generalized Pythagorean fuzzy information
aggregation using Einstein operations and its
application to decision making", Int. J. Intell. Syst.,
31(9), pp. 886-920 (2016).
13. Garg, H. Generalized intuitionistic fuzzy multiplicative
interactive geometric operators and their application
to multiple criteria decision making", Int. J.
Mach. Learn. Cybern., 7(6), pp. 1075-1092 (2016).
14. Garg, H. Some series of intuitionistic fuzzy interactive
averaging aggregation operators", SpringerPlus, 5(1),
p. 999 (2016). DOI: 10.1186/s40064-016-2591-9
15. Garg, H. Con dence levels based Pythagorean fuzzy
aggregation operators and its application to decisionmaking
process", Computat. Math. Organization Theory,
pp. 1-26, 23(4), pp. 546-571 (2017).
16. Wei, G.W. and Merigo, J.M. Methods for strategic
decision-making problems with immediate probabilities
in intuitionistic fuzzy setting", Scientia Iranica,
19(6), pp. 1936-1946 (2012).
17. Nancy and Garg, H. Novel single-valued neutrosophic
decision making operators under frank norm operations
and its application", Int. J. Uncertainty Quantif.,
6(4), pp. 361-375 (2016).
18. Nancy and Garg, H. An improved score function
for ranking neutrosophic sets and its application to
decision-making process", Int. J. Uncertainty Quantif.,
6(5), pp. 377-385 (2016).
19. Yu, S.M., Wang, J., and Wang, J.Q. An extended
TODIM approach with intuitionistic linguistic numbers",
Int. Trans. in Operational Research, 25(3), pp.
781-805 (2018).
H. Garg and K. Kumar/Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2373{2388 2387
20. Garg, H., Agarwal, N., and Tripathi, A. Some
improved interactive aggregation operators under
interval-valued intuitionistic fuzzy environment and
its application to decision making process", Scientia
Iranica: Transaction on Industrial Engineering, 24(5),
pp. 2581-2604 (2017).
21. Wei, G. Some induced geometric aggregation operators
with intuitionistic fuzzy information and their
application to group decision making", Appl. Soft
Comput., 10, pp. 423-431 (2010).
22. Wang, J.Q., Han, Z.Q., and Zhang, H.Y. Multi
criteria group decision-making method based on intuitionistic
interval fuzzy information", Group Decision
and Negotiation, 23, pp. 715-733 (2014).
23. Wang, J.Q., Yang, Y., and Li, L. Multi-criteria
decision-making method based on single-valued neutrosophic
linguistic Maclaurin symmetric mean operators",
Neural Comput. Appl., pp. 1-19 (2016). DOI:
http://dx.doi.org/10.1007/s00521-016-2747-0
24. Chiang, D.A. and Lin, N.P. Correlation of fuzzy sets",
Fuzzy Sets Syst., 102, pp. 221-226 (1999).
25. Liu, S.T. and Kao, C. Fuzzy measures for correlation
coecient of fuzzy numbers", Fuzzy Sets Syst., 128,
pp. 267-275 (2002).
26. Hong, D.H. Fuzzy measures for a correlation coe-
cient of fuzzy numbers under tw (the weakest t-norm)-
based fuzzy arithmetic operations", Inform. Sci., 176,
pp. 150-160 (2006).
27. Wang, G.J. and Li, X.P. Correlation and information
energy of interval-valued fuzzy numbers", Fuzzy Sets
Syst., 103, pp. 169-175 (1999).
28. Gerstenkorn, T. and Manko, J. Correlation of intuitionistic
fuzzy sets", Fuzzy Sets Syst., 44, pp. 39-43
(1991).
29. Bustince, H. and Burillo, P. Correlation of intervalvalued
intuitionistic fuzzy sets", Fuzzy Sets Syst., 74,
pp. 237-244 (1995).
30. Hong, D.H. A note on correlation of interval-valued
intuitionistic fuzzy sets", Fuzzy Sets Syst., 95, pp. 113-
117 (1998).
31. Xu, Z.S., Chen, J., and Wu, J.J. Cluster algorithm
for intuitionistic fuzzy sets", Inform. Sci., 178, pp.
3775-3790 (2008).
32. Xu, Z.S. On correlation measures of intuitionistic
fuzzy sets", Lect. Notes Comput. Sci., 4224, pp. 16-24
(2006).
33. Garg, H. A novel correlation coecients between
Pythagorean fuzzy sets and its applications to decisionmaking
processes", Int. J. Intell. Syst., 31(12), pp.
1234-1252 (2016).
34. Garg, H. A novel accuracy function under intervalvalued
Pythagorean fuzzy environment for solving multicriteria
decision making problem", J. Intell. Fuzzy
Syst., 31 (1), pp. 529-540 (2016).
35. Zhao, K. Set pair and set pair analysis - a new concept
and systematic analysis method", Proceedings of the
National Conference on System Theory and Regional
Planning, pp. 87-91 (1989).
36. Liu, C. , Zhang, L., and Yang, A. The fundamental
operation on connection number and its application",
J. of Theoretical & Applied Information Technology,
49(2), pp. 618-623 (2013).
37. Wang, J.Q. and Gong, L. Interval probability stochastic
multi-criteria decision-making approach based on
set pair analysis", Control and Decision, 24, pp. 1877-
1880 (2009).
38. Hu, J. and Yang, L. Dynamic stochastic multi-criteria
decision making method based on cumulative prospect
theory and set pair analysis", Systems Engineering
Procedia, 1, pp. 432-439 (2011).
39. Xie, Z., Zhang, F., Cheng, J., and Li, L. Fuzzy
multi-attribute decision making methods based on
improved set pair analysis", 6th Int. Symposium on
Computational Intelligence and Design, 2, pp. 386-389
(2013).
40. Fu, S. and Zhou, H. Triangular fuzzy number multiattribute
decision-making method based on set-pair
analysis", J. of Software Engineering, pp. 1-7 (2016).
DOI: 10.3923/jse.2016
41. Kumar, K. and Garg, H. TOPSIS method based
on the connection number of set pair analysis under
interval-valued intuitionistic fuzzy set environment",
Comput. Appl. Math., 37(2), pp. 1319-1320 (2018).
42. ChangJian, W. Application of the set pair analysis
theory in multiple attribute decision-making", J. of
Mechanical Strength, 029, pp. 1009-1012 (2007).
43. Pan, Wu, K.Y., Jin, J.L., and Liu, X.W. Assessment
model of set pair analysis for
ood loss based on triangular
fuzzy intervals under -cut", Chinese Control
and Decision Conference, pp. 3562-3567 (2009).
44. Zhang, Y., Yang, X.H., Zhang, L., Ma, W.Y., and
Qiao, L.X. Set pair analysis based on phase space
reconstruction model and its application in forecasting
extreme temperature", Math. Probl. Eng., 2013,
Article ID 516150, 7 pages (2013).
45. Zhang, Y., Wang, S.G., and Xu, Y.T. Online credit
evaluation system based on analytic hierarchy process
and set pair analysis", Int. Symposium on Computational
Intelligence and Design, 1, pp. 453-456 (2008).
46. Yang, X.H., Di, C.L., He, J., Zhang, J., and Li, Y.Q.
Integrated assessment of water resources vulnerability
under climate change in Haihe river basin", Int. J.
Numer. Method Heat and Fluid Flow, 25(8) , pp. 1834-
1844 (2015).
47. Rui, Y., Zhongbin, W., and Anhua, P. Multi-attribute
group decision making based on set pair analysis", Int.
J. of Advancements in Computing Technology, 4(10),
pp. 205-213 (2012).
48. Sun, J., Li, L., Li, Y., and Liu, B. Set pair analysis of
lattice order decision-making model and application",
J. of Chemical and Pharmaceutical Research, 6(4), pp.
52-58 (2014).
2388 H. Garg and K. Kumar/Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2373{2388
49. Cao, Y.X., Zhou, H., and Wang, J.Q. An approach to
interval-valued intuitionistic stochastic multi-criteria
decision-making using set pair analysis", Int. J. Mach.
Learn. Cybern., 9(4), pp. 629-640 (2018).
50. Yang, J., Zhou, J., Liu, L., Li, Y., and Wu, Z.,
Similarity Measures Between Connection Numbers of
Set Pair Analysis, Springer Berlin Heidelberg, Berlin,
Heidelberg, pp. 63-68 (2008).
51. Zeng, W. and Li, H. Correlation coecient of intuitionistic
fuzzy sets", J. Indust. Eng. International,
3(5), pp. 33-40 (2007).
52. Szmidt, E. and Kacprzyk, J. Correlation of intuitionistic
fuzzy sets", Lect. Notes Comput. Sci., 6178, pp.
169-177 (2010).
53. Ye, J. Cosine similarity measures for intuitionistic
fuzzy sets and their applications", Math. Comput.
Model., 53, pp. 91-97 (2011).
54. Dengfeng, L. and Chuntian, C. New similarity measure
of intuitionistic fuzzy sets and application to
pattern recognitions", Pattern Recogn. Lett., 23 , pp.
221-225 (2002).
55. Liu, H.W. New similarity measures between intuitionistic
fuzzy sets and between elements", Math. Comput.
Model., 42, pp. 61-70 (2005).
56. Ejegwa, P.A. and Modom, E.S. Diagnosis of viral
hepatitis using new distance measure of intuitionistic
fuzzy sets", Int. J. of Fuzzy Mathematical Archive,
8(1), pp. 1-7 (2015).
57. Hung, W.L. and Yang, M.S. On similarity measures
between intuitionistic fuzzy sets", Int. J. Intell. Syst.,
23(3), pp. 364-383 (2008).
58. Son, L.H. and Thong, N.T. Intuitionistic fuzzy recommender
systems: An e ective tool for medical diagnosis",
Knowl.-Based Syst., 74, pp. 133-150 (2015).