A novel probabilistic distance measure for picture fuzzy sets with its application in classification problems

Year 2020, , 2134 - 2153, 08.12.2020

Abhishek Guleria , Rakesh Kumar Bajaj

https://doi.org/10.15672/hujms.677920

Cited By: 11

Abstract

In the present communication, we propose the probabilistic distance measure for picture fuzzy sets where the probability of occurrence/non-occurrence of the picture fuzzy event have been incorporated. This framework has been clearly addressed through outline of a formulated problem and its probable solution structure along with its proof of validity. Further, the proposed probabilistic distance measure has been utilized to present an algorithm for solving some classification decision making problems in a more generalized way. Some important illustrative examples related to the problem of classification - building material classification, mineral classification and a decision making problem of financial investment risk have been worked out to exhibit the implementation of the proposed methodology. The obtained results have also been compared with the existing approaches of solving the classification problems. The uncertainty feature of the problem has been handled in a more broader sense reflecting the advantage of the introduced approach.

Keywords

Picture Fuzzy Set, Distance Measure, Probabilistic randomness, Uncertainty, Classification problem

References

• [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20, 87–96, 1986.
• [2] K.T. Atanassov, Geometrical interpretation of the elements of the intuitionistic fuzzy objects, Int. J. Bioautomation 20 (1), S27–S42, 2016.
• [3] R.K. Bajaj and A. Guleria, Dimensionality reduction technique in decision making using Pythagorean fuzzy soft matrices, Recent Advances in Computer Science and Communications (Formerly: Recent Patents on Computer Science) 13 (3), 406–413, 2020.
• [4] P. Burillo and H. Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems 78, 305–316, 1996.
• [5] B.C. Cuong, Picture fuzzy sets–first results. Part 1, in preprint of seminar on Neuro- Fuzzy Systems with Applications, Institute of Mathematics, 2013.
• [6] B.C. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics 30, 409–420, 2014.
• [7] N.V. Dinh, N.X. Thao, and N.M. Chau, Distance and dissimilarity measure of picture fuzzy sets, in Conf. FAIR 10, 104–109, 2017.
• [8] P. Dutta, Medical diagnosis via distance measures on picture fuzzy sets, Advances in Modelling and Analysis 54, 137–152, 2017.
• [9] F. Feng, H. Fujita, M. Irfan Ali, R.R. Yager, and X. Liu, Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods, IEEE Trans Fuzzy Syst 27 (3), 474–488, 2019.
• [10] F. Feng, M. Liang, H. Fujita, R.R. Yager, and X. Liu, Lexicographic orders of intuitionistic fuzzy values and their relationships, Mathematics 7 (2), 166, 2019.
• [11] F. Feng, Z. Xu, H. Fujita, and M. Liang, Enhancing PROMETHEE method with intuitionistic fuzzy soft sets, Int. J. Intell. Syst 35, 1071–1104, 2020.
• [12] F. Feng, Y. Zheng, J.C.R. Alcantud, and Q. Wang, Minkowski weighted score functions of intuitionistic fuzzy values, Mathematics 8, 1143, 2020.
• [13] H. Garg and G. Kaur, Novel distance measures for cubic intuitionistic fuzzy sets and their applications to pattern recognitions and medical diagnosis, Granular Computing 5, 169–184, 2020.
• [14] A. Guleria and R.K. Bajaj, On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis, Soft Comput. 23, 7889–7900, 2019.
• [15] A. Guleria and R.K. Bajaj, Pythagorean fuzzy (R, S)-norm discriminant measure in various decision making processes, J. Intell. Fuzzy Syst. 38, 761–777, 2020.
• [16] A. Guleria and R.K. Bajaj, A robust decision Making Approach for hydrogen power plant site selection utilizing (R, S)-norm Pythagorean fuzzy information measures based on VIKOR and TOPSIS method, Int. J. Hydrog. Energy 45 (38), 18802–18816, 2020.
• [17] A.G. Hatzimichailidis, G.A. Papakostas, and V.G. Kaburlasos, A novel distance measure of intuitionistic fuzzy sets and its application to pattern recognition problems, Int. J. Intell. Syst. 27, 396–409, 2012.
• [18] W.L. Hung and M.S. Yang, Similarity measures of intuitionistic fuzzy sets based on hausdorff distance, Pattern Recognit. Lett. 25, 1603–1611, 2004.
• [19] W.L. Hung and M.S. Yang, Similarity measures of intuitionistic fuzzy sets based on LP metric, Internat. J. Approx. Reason. 46, 120–136, 2006.
• [20] C. Jana and M. Pal, Assessment of enterprise performance based on picture fuzzy hamacher aggregation operators, Symmetry 11, 75, 2019.
• [21] C. Jana, T. Senapati, M. Pal, and R.R. Yager, Picture fuzzy Dombi aggregation operators: Application to MADM process, Appl. Soft Comput. 74, 99–109, 2019.
• [22] A.M. Khalil, S.G. Li, H. Garg, H. Li, and S. Ma, New operations on interval-valued picture fuzzy set, interval-valued picture fuzzy soft set and their applications, IEEE Access 7, 51236–51253, 2019.
• [23] D. Li and W. Zeng, Distance measure of Pythagorean fuzzy sets, Int. J. Intell. Syst. 33, 348–361, 2018.
• [24] M. Luo and R. Zhao, A distance measure between intuitionistic fuzzy sets and its application in medical diagnosis, Artif Intell Med 89, 34–39, 2018.
• [25] T. Mahmood, U. Kifayat, Q. Khan, and N. Jan, An approach toward decision making and medical diagnosis problems using the concept of spherical fuzzy sets, Neural. Comput. Appl. 31, 7041–7053, 2019.
• [26] G.A. Papakostas, A.G. Hatzimichailidis, and V.G. Kaburlasos, Distance and or similarity measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition point of view, Pattern Recognit. Lett. 34, 1609–1622, 2013.
• [27] L.H. Son, Generalized picture distance measure and applications to picture fuzzy clustering, Appl. Soft Comput. 46, 284–295, 2016.
• [28] L.H. Son, Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures, Fuzzy Optim. Decis. Mak. 16, 1–20, 2017.
• [29] E. Szmidt and J. Kacprzyk, A concept of a probability of an intuitionistic fuzzy event, 1999 IEEE International Fuzzy Systems, Conference Proceedings, Seoul, South Korea 3, 1346–1349, 1999.
• [30] E. Szmidt and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and System 114, 505–518, 2000.
• [31] P.H. Thong and L.H. Son, Picture fuzzy clustering for complex data, Eng. Appl. Artif. Intell. 56, 121–130, 2016.
• [32] P.H. Thong and L.H. Son, A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality, Knowl Based Syst. 109, 48–60, 2016.
• [33] R. Wang, J. Wang, H. Gao, and G. Wei, Methods for MADM with picture fuzzy muirhead mean operators and their application for evaluating the financial investment risk, Symmetry 11, 6, 2019.
• [34] G.W. Wei, Picture fuzzy cross-entropy for multiple attribute decision making problems, J. Bus. Econ. Manag. 17, 491–502, 2016.
• [35] G.W. Wei, Picture fuzzy aggregation operators and their application to multiple attribute decision making, J. Intell. Fuzzy Syst. 33, 713–724, 2017.
• [36] G.W.Wei, Some similarity measures for picture fuzzy sets and their applications,Iran. J. Fuzzy Syst. 15, 77–89, 2018.
• [37] G.W. Wei and H. Gao, The generalized Dice similarity measures for picture fuzzy sets and their applications, Informatica (Vilnius) 29, 1–18, 2018.
• [38] R.R. Yager, Pythagorean fuzzy subsets, in: Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 57–61, 2013.
• [39] R.R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst 22 (4), 958–965, 2014.
• [40] R.R. Yager, Generalized orthopair fuzzy sets, IEEE Trans Fuzzy Syst 25 (5), 1222– 1230, 2017.
• [41] L.A. Zadeh, Fuzzy sets, Inf. Control. 8, 338–353, 1965.
• [42] L.A. Zadeh, Probability measure of fuzzy events, J. Math. Anal. Appl. 23, 421–427, 1968.
• [43] S. Zeng, S. Asharf, M. Arif, and S. Abdullah, Application of exponential jensen picture fuzzy divergence measure in multi-criteria group decision making, Mathematics 7, 191, 2019.
• [44] X.L. Zhang and Z.S. Xu, Extension of TOPSIS to multi-criteria decisionmaking with Pythagorean fuzzy sets, Int. J. Intell. Syst. 29, 1061–1078, 2014.