An improved cosine similarity measure for intuitionistic fuzzy sets and their applications to decision-making process

Abstract

In the present study, improved cosine similarity measure for an intuitionistic fuzzy sets (IFSs) has been proposed by considering the interaction between the pairs of the membership degrees. Pairs of membership, non-membership are to be considered as vector representation during the formulation. The shortcomings of the existing measures have been highlighted and overcome by using the proposed measure.  Also, in order to deal with the situation where the elements in a set are correlative, weighted cosine similarity measure has been defined. Finally, multi-criteria decision making (MCDM) method, based on the proposed similarity measure, has been presented under intuitionistic fuzzy environment. Numerical examples, one from the investment the money and others from the pattern recognition and medical diagnosis, have been taken to demonstrate the efficiency of the proposed approach and compared their results with the existing approaches results.

Keywords

• Cosine similarity measures,intuitionistic fuzzy set,pattern recognition,medical diagnosis,multi criteria decision making

References

1. Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20, 87--96, 1986.
2. Atanassov, K., Gargov, G., Interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst., 31, 343 -- 349, 1989.
3. Bustince, H., Burillo, P., Correlation of interval-valued intuitionistic fuzzy sets,Fuzzy Sets Syst., 74, 237 -- 244, 1995.
4. Chen, S. M., Measures of similarity between vague sets, Fuzzy Sets Syst., 74 (2), 217 -- 223, 1995.
5. Chen, S. M., Similarity measures between vague sets and between elements, IEEE Transact. Systems Man Cybernet. 27~(1), 153 -- 158, 1997.
6. Dengfeng, L., Chuntian, C., New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions, Pattern Recognition Lett. 23, 221 -- 225, 2002.
7. Fan, L., Zhangyan, X., Similarity measures between vague sets., J. Software 12~(6), 922 -- 927, 2001.
8. Garg, H., Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Comput. Industrial Eng. 101, 53 -- 69, 2016.

9. Garg, H., Generalized intuitionistic fuzzy multiplicative interactive geometric operators and their application to multiple criteria decision making, Int. J. Mach. Learn. Cybernet. 7~(6), 1075--1092, 2016.
10. Garg, H., A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems, Appl. Soft Comput. 38, 988 -- 999, 2016.
11. Garg, H., Distance and similarity measure for intuitionistic multiplicative preference relation and its application, Int. J. Uncertain. Quantif. 7~(2), 117 -- 133, 2017.
12. Garg, H., Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Eng. Appli. Artif. Intell. 60, 164 -- 174, 2017.
13. Garg, H., Agarwal, N., Choubey, A., Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights, Global J. Technol. Optim. 6~(3), 13 -- 20, 2015.
14. Garg, H., Agarwal, N., Tripathi, A., Generalized intuitionistic fuzzy entropy measure of order $\alpha$ and degree $\beta$ and its applications to multi-criteria decision making problem, Int. J. Fuzzy Syst. Appl. 6~(1), 86 -- 107, 2017.
15. Garg, H., Arora, R., Distance and similarity measures for dual hesistant fuzzy soft sets and their applications in multi criteria decision-making problem, Int. J. Uncertain. Quantif., 7 (3), 229-248, 2017.
16. Garg, H., Rani, M., Sharma, S. P., Vishwakarma, Y., Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment, Expert Systems with Applications 41, 3157 -- 3167, 2014.
17. Gerstenkorn, T., Manko, J., Correlation of intuitionistic fuzzy sets, Fuzzy Sets Syst., 44, 39 -- 43, 1991.
18. He, Y., Chen, H., Zhau, L., Liu, J., Tao, Z., Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making, Information Sci., 259, 142 -- 159, 2014.
19. He, Y., Chen, H, Zhou, L., Han, B., Zhao, Q., Liu, J., Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making, Expert Syst. Appl., 41 (0), 2484 -- 2495, 2014.
20. Hong, D. H., Kim, C., A note on similarity measures between vague sets and between elements, Information Sci., 115, 83 -- 96, 1999.
21. Hung, W. L., Yang, M. S., Similarity measures of intuitionistic fuzzy sets based on hausdorff distance, Pattern Recognition Lett. 25, 1603 -- 1611, 2004.
22. Liu, H.~W., New similarity measures between intuitionistic fuzzy sets and between elements, Math. Comput. Modelling 42, 61 -- 70, 2005.
23. Mitchell, H.~B., On the dengfeng chuntian similarity measure and its application to pattern recognition, Pattern Recognition Lett. 24, 3101 -- 3104, 2003.
24. Singh, S., Garg, H., Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process, Applied Intelligence 46~(4), 788 -- 799, 2017.
25. Szmidt, E., Kacprzyk, J., A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning, Lecture Notes in Computer Science 3070, 388 -- 393, 2004.
26. Xu, Z.~S., On correlation measures of intuitionistic fuzzy sets, Lecture Notes in Computer Science 4224, 16 -- 24, 2006.
27. Xu, Z.~S., Intuitionistic fuzzy aggregation operators, IEEE Trans. Fuzzy Systems 15, 1179 -- 1187, 2007.
28. Xu, Z.~S., Intuitionistic preference relations and their application in group decision making, Information Sci., 177, 2363 -- 2379, 2007.
29. Ye, J., Cosine similarity measures for intuitionistic fuzzy sets and their applications, Math. Comput. Modelling 53, 91 -- 97, 2011.
30. Zadeh, L. A., Fuzzy sets, Information and Control 8, 338--353, 1965.
31. Zhizhen, L., Pengfei, S., Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Lett. 24, 2687 -- 2693, 2003.