Abstract
The concept of picture fuzzy soft sets (PFSSs) has been introduced as an efficacious extension of intuitionistic fuzzy soft sets to model uncertain information, especially when dealing with the responses “Yes”, “No”, and “Abstain”. Recently, Memiş [Another View on Picture Fuzzy Soft Sets and Their Product Operations with Soft Decision-Making. J New Theory 38(2022):1–13, 2022] has redefined the PFSSs and their basic operations to overcome some inconsistencies in the concept. The literature employs the aforesaid inconsistencies in some of Cuong’s definitions. Benefiting from these two inspiring studies, we herein first aim to introduce the picture fuzzy soft \(\sigma\)-algebra, picture fuzzy soft measure, and picture fuzzy soft exterior measure. We then present the famous “Dynkin \(\pi\)–\(\lambda\) theorem” under the picture fuzzy soft environment using the notions of \(\pi\)- and \(\lambda\)-systems. Afterwards, we provide a number of associated results accompanied by explanatory examples. Thereafter, we propose a multi-criteria decision-making method utilising the max-min-max composition on picture fuzzy soft mapping and show how to apply the method to a decision-making scenario about determining the best student semester-wise. Finally, we discuss the need for further research.
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References
Akram M, Bashir A, Edalatpanah SA (2021) A hybrid decision-making analysis under complex q-rung picture fuzzy Einstein averaging operators. Comput Appl Math 40:305. https://doi.org/10.1007/s40314-021-01651-y
Akram M, Ahmad U, Rukhsar A (2022a) Threshold graphs under picture Dombi fuzzy information. Granul Comput 7(3):691–707. https://doi.org/10.1007/s41066-021-00291-1
Akram M, Shahzadi G, Alcantud JCR (2022b) Extended MULTIMOORA method based on 2-tuple linguistic Pythagorean fuzzy sets for multi-attribute group decision-making. Granul Comput. https://doi.org/10.1007/s41066-022-00330-5
Akram M, Shahzadi G, Alcantud JCR (2022c) Multi-attribute decision-making with q-rung picture fuzzy information. Granul Comput 7(1):197–215. https://doi.org/10.1007/s41066-021-00260-8
Arikrishnan A, Sriram S (2020) Algebraic operations on picture fuzzy soft matrices. Adv Math Sci J 9(8):6349–6358. https://doi.org/10.37418/amsj.9.8.102
Arya V, Kumar S (2020) A new picture fuzzy information measure based on Shannon entropy with applications in opinion polls using extended VIKOR-TODIM approach. Comput Appl Math. https://doi.org/10.1007/s40314-020-01228-1
Arya V, Kumar S (2021) A picture fuzzy multiple criteria decision-making approach based on the combined TODIM-VIKOR and entropy weighted method. Cogn Comput 13:1172–1184. https://doi.org/10.1007/s12559-021-09892-z
Ashraf S, Abdullah S, Mahmood T et al. (2019) Cleaner production evaluation in gold mines using novel distance measure method with cubic picture fuzzy numbers. Int J Fuzzy Syst 21(8):2448–2461. https://doi.org/10.1007/s40815-019-00681-3
Atanassov KT (1984) Intuitionistic fuzzy sets. In: Sgurev V (ed) VII ITKRs Session, Sofia, June 1983. Central Sci Techn. Library, Bulg. Academy of Sciences, Sofia, Bulgaria
Atanassov KT (2010) Intuitionistic fuzzy sets: theory and application, Studies in Fuzziness and Soft Computing, XVIII Physica. Springer, Verlag, Heidelberg
Chellamani P, Ajay D, Broumi S et al. (2022) An approach to decision-making via picture fuzzy soft graphs. Granul Comput 7(3):527–548. https://doi.org/10.1007/s41066-021-00282-2
Cuong BC (2013a) Picture fuzzy sets-first results. Part 1. In: Seminar Neuro-Fuzzy Systems with Applications. Tech. rep., Institute of Mathematics, Hanoi, Vietnam
Cuong BC (2013b) Picture fuzzy sets-first results. Part 2. In: Seminar Neuro-Fuzzy Systems with Applications. Tech. rep., Institute of Mathematics, Hanoi, Vietnam
Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cybern 30(4):409–420. https://doi.org/10.15625/1813-9663/30/4/5032
Ganie AH, Singh S, Bhatia PK (2020) Some new correlation coefficients of picture fuzzy sets with applications. Neural Comput Appl 32(12):609–12. https://doi.org/10.1007/s00521-020-04715-y (625)
Garg H (2017) Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arab J Sci Eng 42:5275–5290. https://doi.org/10.1007/s13369-017-2625-9
Garg H, Ullah K, Mahmood T et al. (2021) T-spherical fuzzy power aggregation operators and their applications in multi-attribute decision making. J Ambient Intell Humaniz Comput 12:9067–9080. https://doi.org/10.1007/s12652-020-02600-z
Guiwu W (2017) Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33(2):713–724. https://doi.org/10.3233/JIFS-161798
Guleria A, Bajaj RK (2019) T-sphericalfuzzy soft sets and its aggregation operators with application in decision making. Scientia Iranica 28(2):1014–1029. https://doi.org/10.24200/SCI.2019.53027.3018
Gupta R, Kumar S (2022) A new similarity measure between picture fuzzy sets with applications to pattern recognition and clustering problems. Granul Comput 7(3):561–576. https://doi.org/10.1007/s41066-021-00283-1
Jana C, Senapat T, Pal M et al. (2019) Picture fuzzy Dombi aggregation operators: application to MADM process. Appl Soft Comput 74:99–109. https://doi.org/10.1016/j.asoc.2018.10.021
Jayaraman P, Vetrikkani K, Selvakumar A et al. (2021) Application of picture fuzzy soft relations in multi-altribute decision making. J Univ Shanghai Sci Technol 23(9):911–928
Jin J, Zhao P, You T (2021) Picture fuzzy TOPSIS method based on CPFRS model: an application to risk management problems. Sci Prog. https://doi.org/10.1155/2021/6628745
Joshi R (2020) A novel decision-making method using R-norm concept and VIKOR approach under picture fuzzy environment. Expert Syst Appl 147(1):1–12. https://doi.org/10.1016/j.eswa.2020.113228
Joshi R, Kumar S (2022) A novel VIKOR approach based on weighted correlation coefficients and picture fuzzy information for multicriteria decision making. Granul Comput 7(2):323–336. https://doi.org/10.1007/s41066-021-00267-1
Karaaslan F (2015) Neutrosophic soft sets with applications in decision making. Int J Inform Sci Intell Syst 4(2):1–20. https://doi.org/10.5281/zenodo.23151
Khameneh AZ, Kilicman A (2013) On soft -algebras. Malays J Math Sci 7(1):17–29
Khan MJ, Kumam P, Kumam W et al. (2021) Picture fuzzy soft robust VIKOR method and its applications in decision-making. Fuzzy Inform Eng 13(3):296–322. https://doi.org/10.1080/16168658.2021.1939632
Kumari PN, Kumar DVSRA (2018) Soft sets and sigma algebras. Int J Math Trends Technol 55(1):10–13. https://doi.org/10.14445/22315373/IJMTT-V55P502
Liu P, Zang X (2018) A novel picture fuzzy linguistic aggregation operators and its application to group decision-making. Cogn Comput 10:242–259. https://doi.org/10.1007/s12559-017-9523-z
Liu P, Akram M, Bashir A (2018) Extensions of power aggregation operators for decision making based on complex picture fuzzy knowledge. J Intell Fuzzy Syst 40(1):1107–1128. https://doi.org/10.3233/JIFS-201385
Maji PK (2013) Neutrosophic soft set. Ann Fuzzy Math Inform 5(1):157–168
Maji PK, Biswas R, Roy AR (2001) Intuitionistic fuzzy soft sets. J Fuzzy Math 9(3):677–692
Meksavang P, Shi H, Lin SM et al. (2019) An extended picture fuzzy VIKOR approach for sustainable supplier management and its application in the beef industry. Symmetry 11(4):468. https://doi.org/10.3390/sym11040468
Memiş S (2021) A study on picture fuzzy sets. In: Çuvalcıoğlu G (ed) 7th IFS and Contemporary Mathematics Conference. Mersin University, Mersin, Turkey, pp 125–132
Memiş S (2022) Another view on picture fuzzy soft sets and their product operations with soft decision-making. J New Theory 2022(38):1–13. https://doi.org/10.53570/jnt.1037280
Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37(4–5):19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
Munir M, Kalsoom H, Ullah K et al. (2020) T-spherical fuzzy Einstein hybrid aggregation operators and their applications in multi-attribute decision making problems. Symmetry 12(3):365. https://doi.org/10.3390/sym12030365
Naeem K, Riaz M, Afzal D (2020) Fuzzy neutrosophic soft -algebra and fuzzy neutrosophic soft measure with applications. J Intell Fuzzy Syst 39(1):277–287. https://doi.org/10.3233/JIFS-191062
Peng X, Dai J (2020) A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017. Artif Intell Rev 53:199–255. https://doi.org/10.1007/s10462-018-9652-0
Riaz M, Naeem K (2016) Measurable soft mappings. Punjab Univ J Math 48(2):19–34
Riaz M, Naeem K, Ahmad MO (2017) Novel concepts of soft sets with applications. Ann Fuzzy Math Inform 13(2):239–251. https://doi.org/10.30948/afmi.2017.13.2.239
Singh P (2015) Correlation coefficients for picture fuzzy sets. J Intell Fuzzy Syst 28(2):591–604. https://doi.org/10.3233/IFS-141338
Singh A, Kumar S (2017) Picture fuzzy Choquet integral-based VIKOR for multicriteria group decision-making problems. Granul Comput 6(1):587–601. https://doi.org/10.1007/s41066-020-00218-2
Siraj A, Fatima T, Afzal D et al. (2022) Pythagorean m-polar fuzzy neutrosophic topology with applications. Neutrosophic Sets Syst 48:251–290. https://doi.org/10.5281/zenodo.6041514
Smarandache F (1998) Neutrosophy: neutrosophic probability. Set and logic. American Research Press, Rehoboth
Smarandache F (2006) A unifying field in logics: meutrosophic logic, neutrosophy, neutrosophic set, neutrosophic probability and statistics. American Research Press, Rehoboth
Smarandache F (2016) Neutrosophic overset, neutrosophic underset, and neutrosophic offset. Similarly for neutrosophic over-/under-/off- logic, probability, and statistics. Pons Editions, Bruxelles
Son LH (2016) Generalized picture distance measure and applications to picture fuzzy clustering. Appl Soft Comput 46:284–295. https://doi.org/10.1016/j.asoc.2016.05.009
Son LH, Thong PH (2017) Some novel hybrid forecast methods based on picture fuzzy clustering for weather nowcasting from satellite image sequences. Appl Intell 46(1):1–15. https://doi.org/10.1007/s10489-016-0811-1
Son LH, Viet PV, Hai PV (2017) Picture inference system: a new fuzzy inference system on picture fuzzy set. Appl Intell 46(3):652–669. https://doi.org/10.1007/s10489-016-0856-1
Thong NT, Son LH (2015) Hifcf: an effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis. Expert Syst Appl 42(7):3682–3701. https://doi.org/10.1016/j.eswa.2014.12.042
Thong PH, Son LH (2016) Picture fuzzy clustering for complex data. Eng Appl Artif Intell 56:121–130. https://doi.org/10.1016/j.engappai.2016.08.009
Tian C, Peng JJ, Zhang S et al. (2019) Weighted picture fuzzy aggregation operators and their applications to multi-criteria decision-making problems. Comput Ind Eng 137(1):106037. https://doi.org/10.1016/j.cie.2019.106037
Wang R, Li Y (2018) Picture hesitant fuzzy sets and its application to multiple criteria decision-making. Symmetry 10(7):295. https://doi.org/10.3390/sym10070295
Wang L, Zhang HY, Wang JQ et al. (2018) Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project. Appl Soft Comput 64:216–226. https://doi.org/10.1016/j.asoc.2017.12.014
Wei GW (2016) Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag 17(4):491–502. https://doi.org/10.3846/16111699.2016.1197147
Wei GW (2018) Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fund Inform 157(3):271–320. https://doi.org/10.3233/FI-2018-1628
Yager RR (2013) Pythagorean fuzzy subsets. In: Krol M, Mitra S, Lee DJ (eds) 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS. IEEE, Edmonton, Canada, pp 57–61, https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
Yang Y, Liang C, Ji S et al. (2015) Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making. J Intell Fuzzy Syst 29(4):1711–1722. https://doi.org/10.3233/IFS-151648
Yue C (2020) Picture fuzzy normalized projection and extended VIKOR approach to software reliability assessment. Appl Soft Comput 88(3):106056. https://doi.org/10.1016/j.asoc.2019.106056
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
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KN devised the main conceptual ideas and developed the theoretical framework. SM validated the results. KN and SM wrote the manuscript. SM reviewed and edited the manuscript. Both authors discussed the results and contributed to the final manuscript.
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Naeem, K., Memiş, S. Picture fuzzy soft \(\sigma\)-algebra and picture fuzzy soft measure and their applications to multi-criteria decision-making. Granul. Comput. 8, 397–410 (2023). https://doi.org/10.1007/s41066-022-00333-2
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DOI: https://doi.org/10.1007/s41066-022-00333-2