A Modified Pythagorean Fuzzy Similarity Operator with Application in Questionnaire Analysis
الموضوعات : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematics, College of Physical Sciences, Joseph Sarwuan Tarka University, Makurdi, Nigeria.
الکلمات المفتاحية: Decision making under uncertainty, Intuitionistic fuzzy set, Questionnaire analysis, Pythagorean fuzzy set, Similarity operator.,
ملخص المقالة :
This work presents a modified Pythagorean fuzzy similarity operator and utilizes its potential in the analysis of questionnaire. Similarity operator is a formidable methodology for decision-making under uncertain domains. Pythagorean fuzzy set is an extended form of intuitionistic fuzzy set with a better accuracy in complex real-world applications. Lots of discussions bordering on the uses of Pythagorean fuzzy sets have been explored based on Pythagorean fuzzy similarity operators. Among the extant Pythagorean fuzzy similarity operators, the work of Zhang et al. is significant but it contains some flaws which need to be corrected/modified to enhance reliable interpretation. To this end, this work explicates the Zhang et al.’s techniques of Pythagorean fuzzy similarity operator by pinpointing their drawbacks to develop an enhanced Pythagorean fuzzy similarity operator, which appropriately satisfies the similarity conditions and yields consistent results in comparison to the Zhang et al.’s techniques. Succinctly speaking, the aim of the work is to correct the flaws in Zhang et al.’s techniques via modifications. To theoretically validate the enhanced Pythagorean fuzzy similarity operator, we discuss it properties and find out that the similarity conditions are well satisfied. In addition, the enhanced PFSO and the Zhang et al.’s PFSOs are compared in the context of precision, and it is verified that the enhanced Pythagorean fuzzy similarity operator can successfully measure the similarity between vastly related but inconsistent PFSs and as well yields a very reasonable results. Furthermore, the enhanced Pythagorean fuzzy similarity operator is applied to the analysis of questionnaire on virtual library to ascertain the extent of awareness and effects of virtual library on students’ academic performance via real data collected from fieldwork. Finally, it is certified that the enhanced Pythagorean fuzzy similarity operator can handle diverse everyday problems more precisely than the Zhang et al.’s Pythagorean fuzzy similarity operators.
[1] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8: 338-353. DOI: https://doi.org/10.1016/S0019- 9958(65)90241-X
[2] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20: 87-96. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3
[3] Davvaz B, Sadrabadi EH. An application of intuitionistic fuzzy sets in medicine. International Journal of Biomathematics. 2016; 9(3): 1650037. DOI: https://doi.org/10.1142/S1793524516500376
[4] Hatzimichailidis AG, Papakostas AG, Kaburlasos VG. A novel distance measure of intuitionistic fuzzy sets and its application to pattern recognition problems. International Journal of Intelligent Systems. 2012; 27: 396-409. DOI: https://doi.org/10.1002/int.21529
[5] Wang W, Xin X. Distance measure between intuitionistic fuzzy sets. Pattern Recognition Letters. 2005; 26: 2063-2069. DOI: https://doi.org/10.1016/j.patrec.2005.03.018
[6] Liu P, Chen SM. Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers. IEEE Transaction on Cybernetics. 2017; 47(9): 2514-2530. DOI: https://doi.org/10.1109/TCYB.2016.2634599
[7] Ejegwa PA, Onyeke IC, Kausar N, Kattel P. A new partial correlation coefficient technique based on intuitionistic fuzzy information and its pattern recognition application. International Journal of Intelligent Systems. 2023; 5540085, 14 pages. DOI: https://doi.org/10.1155/2023/5540085
[8] Ejegwa PA, Kausar N, Agba JA, Ugwuh F, Özbilge E, Ozbilge E. Determination of medical emergency via new intuitionistic fuzzy correlation measures based on Spearman’s correlation coefficient. AIMS Mathematics. 2024; 9(6): 15639-15670. DOI: https://doi.org/10.3934/math.2024755
[9] Mahmood T, Idrees A, Albaity M, ur Rehman U. Selection of artificial intelligence provider via multiattribute decision-making technique under the model of complex intuitionistic fuzzy rough sets. AIMS Mathematics. 2024; 9(11): 33087-33138. DOI: https://doi.org/10.3934/math.20241581
[10] Ejegwa PA, Anum MT, Isife KI. A new method of distance measure between intuitionistic fuzzy sets and its application in admission procedure. Journal of Uncertain Systems. 2024; 17(2): 2440005. DOI: https://doi.org/10.1142/S1752890924400051
[11] Ahemen S, Ujah R, Ejegwa PA. New cosine similarity measures for intuitionistic fuzzy sets with application in decision-making. In: I. Ali, UM Modibbo, A. L. Bolaji, H. Garg (eds.) Optimization and Computing using Intelligent Data-Driven Approaches for Decision-Making: Optimization Applications. CRC Press; 2024. p.55-69. DOI: https://doi.org/10.1201/9781003536796
[12] Boran FE, Akay D. A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Information Sciences. 2014; 255(10): 45-57. DOI: https://doi.org/10.1016/j.ins.2013.08.013
[13] Ejegwa PA, Ahemen S. Enhanced intuitionistic fuzzy similarity operator with applications in emergency management and pattern recognition. Granular Computing. 2023; 8: 361-372. DOI: https://doi.org/10.1007/s41066-022-00334-1
[14] Zhou Y, Ejegwa PA, Johnny SE. Generalized similarity operator for intuitionistic fuzzy sets and its applications based on recognition principle and multiple criteria decision making technique. International Journal of Computational Intelligence Systems. 2023; 16: 85. DOI:
https://doi.org/10.1007/s44196-023- 00245-2
[15] Atanassov KT. Geometrical interpretation of the elements of the intuitionistic fuzzy objects. Preprint IM-MFAIS-1-89, Sofia. 1989. Reprinted: Int J Bioautomation, 2016, 20(S1), S27-S42.
[16] Yager RR, Abbasov AM. Pythagorean membership grades, complex numbers and decision making. International Journal of Intelligent Systems. 2013; 28(5): 436-452. DOI: https://doi.org/10.1002/int.21584
[17] Garg H. A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. International Journal of Intelligent Systems. 2016; 31(9): 886-920. DOI: https://doi.org/10.1002/int.21809
[18] Garg H. Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and tconorm for multicriteria decision making process. International Journal of Intelligent Systems. 2017; 32(6): 597-630. DOI: https://doi.org/10.1002/int.21860
[19] Memis S, Enginoglu S, Erkan U. A new classification method using soft decision-making based on an aggregation operator of fuzzy parameterized fuzzy soft matrices. Turkish Journal of Electrical Engineering and Computer Science. 2022; 30(3): 871-890. DOI: https://doi.org/10.55730/1300-0632.3816
[20] Ejegwa PA, Wen S, Feng Y, Zhang W, Chen J. Some new Pythagorean fuzzy correlation techniques via statistical viewpoint with applications to decision-making problems. Journal of Intelligent and Fuzzy Systems. 2021; 40(5): 9873-9886. DOI: https://doi.org/10.3233/JIFS-202469
[21] Ejegwa PA, Wen S, Feng Y, Zhang W. Determination of pattern recognition problems based on a Pythagorean fuzzy correlation measure from statistical viewpoint. In: Proceeding of the 13th International Conference on Advanced Computational Intelligence, Wanzhou, China. 2021. p.132-139. DOI:
https://doi.org/10.1109/ICACI52617.2021.9435895
[22] Ejegwa PA, Wen S, Feng Y, Zhang W, Liu J. A three-way Pythagorean fuzzy correlation coefficient approach and its applications in deciding some real-life problems. Applied Intelligence. 2023; 53(1): 226- 237. DOI: https://doi.org/10.1007/s10489-022-03415-5
[23] Yan D, Wu K, Ejegwa PA, Xie X, Feng Y. Pythagorean fuzzy partial correlation measure and its application. Symmetry. 2023; 15: 216. DOI: https://doi.org/10.3390/sym15010216
[24] Ejegwa PA, Feng Y, Tang S, Agbetayo JM, Dai X. New Pythagorean fuzzy-based distance operators and their applications in pattern classification and disease diagnostic analysis. Neural Computing and Application. 2023; 35(14): 10083-10095. DOI: https://doi.org/10.1007/s00521-022-07679-3
[25] Li DQ, Zeng WY. Distance measure of Pythagorean fuzzy sets. International Journal of Intelligent Systems. 2018; 33: 348-361. DOI: https://doi.org/10.1002/int.21934
[26] Liu Z. Hellinger distance measures on Pythagorean fuzzy environment via their applications. International Journal of Knowledge-based and Intelligent Engineering Systems. 2024; 28(2): 211-229. DOI: https://doi.org/10.3233/KES-230150
[27] Mahanta J, Panda S. Distance measure for Pythagorean fuzzy sets with varied applications. Neural Computing and Application. 2021; 33: 17161-17171. DOI: https://doi.org/10.1007/s00521-021-06308-9
[28] Peng X. New similarity measure and distance measure for Pythagorean fuzzy set. Complex and Intelligent Systems. 2019; 5(2): 101-111. DOI: https://doi.org/10.1007/s40747-018-0084-x
[29] Wu K, Ejegwa PA, Feng Y, Onyeke IC, Johnny SE, Ahemen S. Some enhanced distance measuring approaches based on Pythagorean fuzzy information with applications in decision making. Symmetry. 2022; 14: 2669. DOI: https://doi.org/10.3390/sym14122669
[30] Xiao F, Ding W. Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis. Applied Soft Computing. 2019; 79: 254-267. DOI: https://doi.org/10.1016/j.asoc.2019.03.043
[31] Zeng W, Li D, Yin Q. Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. International Journal of Intelligent Systems. 2018; 33(11): 2236-2254. DOI: https://doi.org/10.1002/int.22027
[32] Hemalatha K, Venkateswarlu B. Pythagorean fuzzy transportation problem: New way of ranking for Pythagorean fuzzy sets and mean square approach. Heliyon. 2023; 9(10): e20775. DOI: https://doi:10.1016/j.heliyon.2023.e20775
[33] Li R, Ejegwa PA, Li K, Agaji I, Feng Y, Onyeke IC. A new similarity function for Pythagorean fuzzy sets with application in football analysis. AIMS Mathematics. 2024; 9(2): 4990-5014. DOI: https://doi.org/10.3934/math.2024242
[34] Akram M, Sultan M, Adeel A, Al-Shamiri MMA. Pythagorean fuzzy N-soft PROMETHEE approach: a new framework for group decision making. AIMS Mathematics. 2023; 8(8): 17354-17380. DOI: https://doi.org/10.3934/math.2023887
[35] Bozyigit MC, Olgun M, Unver M. Circular Pythagorean fuzzy sets and applications to multi-criteria decision making. Informatica. 2023; 34(4): 713-742. DOI: https://doi.org/10.15388/23-INFOR529
[36] Thakur P, Kaczyrska A, Gandotra N, Saini N, Salabun W. The application of the new Pythagorean fuzzy entropy to decision-making using linguistic terms. Procedia Computer Science. 2022; 207: 4525-4534. DOI: https://doi.org/10.1016/j.procs.2022.09.516
[37] Thakur P, Paradowski B, Gandotra N, Thakur P, Saini N, Salabun WA. Study and application analysis exploring Pythagorean fuzzy set distance metrics in decision making. Information. 2024; 15: 28. DOI: https://doi.org/10.3390/info15010028
[38] Peng XD, Yuan HY, Yang Y. Pythagorean fuzzy information measures and their applications. International Journal of Intelligent Systems. 2017; 32(10): 991-1029. DOI: https://doi.org/10.1002/int.21880
[39] Zhang XL. A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. International Journal of Intelligent Systems. 2016; 31(6): 593-611. DOI: https://doi.org/10.1002/int.21796
[40] Wei G, Wei Y. Similarity measures of Pythagorean fuzzy sets based on cosine function and their applications. International Journal of Intelligent Systems. 2018; 33(3): 634-652. DOI: https://doi.org/10.1002/int.21965
[41] Zhang Q, Hu J, Feng J, Liu A, Li Y. New similarity measures of Pythagorean fuzzy sets and their applications. IEEE Access. 2019; 7: 38192-138202. DOI: https://doi.org/10.1109/ACCESS.2019.2942766
[42] Ejegwa JO, Mngutyo J. An evaluation of the effects of virtual library on the academic performance of undergraduate students in the Department of Medicine and Surgery, Benue State University, Makurdi, Nigeria. Library Philosophy and Practice (e-journal). 2023; 7624. https://digitalcommons.unl.edu/libphilprac/7624
[43] Yamane T. Statistics: an introductory analysis. 2 nd Ed. New York: Harper and Row; 1967.
[44] Cuong BC. Picture fuzzy sets. Journal of Computer and Cybernetics. 2014; 30(4): 409-420. DOI: https://doi.org/10.15625/1813-9663/30/4/5032
[45] Hassan K, Shaheen T, Ali W, Haq IU, Bibi N, Adak AK. Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application. Decision Making and Analysis. 2024; 2(1): 58-72. DOI:
https://doi.org/10.55976/dma.22024127458-72
[46] Ejegwa PA, Sarkar A. Fermatean fuzzy approach of diseases diagnosis based on new correlation coefficient operators. In: H. Garg, J. M. Chatterjee (eds.) Deep Learning in Personalized Healthcare and Decision Support. Academic Press; 2023. p. 23-38. DOI:
https://doi.org/10.1016/B978-0-443-19413-9.00021-7
