Abstract
Soft set theory initiated by Molodtsov in 1999 has been emerging as a generic mathematical tool for dealing with uncertainty. A noticeable progress is found concerning the practical use of soft set in decision-making problems. The purpose of this manuscript is to explore the novel of multi-valued picture fuzzy set (MPFS) and multi-valued picture fuzzy soft set (MPFSS) which are the generalizations of the notions of picture fuzzy soft set (PFSS) and multi-fuzzy soft set (MFSS). This notion can be used to express fuzzy information in more general and effective way. In particular, some basic operations such as union, intersection, complement and product of the proposed MPFSS are developed, and their properties are investigated. Furthermore, some aggregation operators corresponding to the proposed MPFSSs are called multi-picture fuzzy soft weighted averaging, multi-picture fuzzy soft ordered weighted averaging and multi-picture soft hybrid weighted averaging operators for a collections of MPFSSs are also developed. Moreover, based on these operators, we presented a new method to deal with the multi‐attribute group decision-making problems under the multi-valued picture fuzzy soft environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods. The graphical interpretation of the explored approaches is also utilized with future directions.
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References
Alcantud JCR, Khameneh AZ, Kilicman A (2020) Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information. Inf Sci 514:106–117
Ali Z, Mahmood T (2020) Complex neutrosophic generalized dice similarity measures and their application to decision making. CAAI Trans Intell Technol 5:1–20
Ali MI, Feng F, Liu X, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57(9):1547–1553
Atanassov KT (1994) Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 64(2):159–174
Atanassov KT (1999a) Intuitionistic fuzzy sets. In: Intuitionistic fuzzy sets. Physica, Heidelberg, pp 1–137
Atanassov KT (1999b) Interval valued intuitionistic fuzzy sets. In: Intuitionistic fuzzy sets. Physica, Heidelberg, pp 139–177
Boran FE, Genç S, Kurt M, Akay D (2009) A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Exp Syst Appl 36(8):11363–11368
Bustince H, Burillo P (1995) Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 74(2):237–244
Çağman N, Enginoğlu S (2010) Soft set theory and uni–int decision making. Eur J Oper Res 207(2):848–855
Cagman N, Enginoglu S, Citak F (2011) Fuzzy soft set theory and its applications. Iran J Fuzzy Syst 8(3):137–147
Chen SM, Cheng SH, Chiou CH (2016) Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inf Fus 27:215–227
Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cybern 30(4):409–420
Cuong BC, Kreinovich V (2013) Picture fuzzy sets-a new concept for computational intelligence problems. In: 2013 Third world congress on information and communication technologies (WICT 2013). IEEE, pp 1–6
Das S, Kar S (2013) Intuitionistic multi fuzzy soft set and its application in decision making. In: International conference on pattern recognition and machine intelligence. Springer, Berlin, Heidelberg, pp 587–592
Das S, Kar S (2014) Group decision making in medical system: an intuitionistic fuzzy soft set approach. Appl Soft Comput 24:196–211
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17(2–3):191–209
Feng Q, Guo X (2018) A novel approach to fuzzy soft set-based group decision-making. Complexity 2018
Garg H (2016a) A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst 31(12):1234–1252
Garg H (2016b) A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. Int J Intell Syst 31(9):886–920
Garg H (2017a) Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arab J Sci Eng 42(12):5275–5290
Garg H (2017b) Generalized pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process. Int J Intell Syst 32(6):597–630
Garg H, Kaur G (2020) Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment. Sci Iran 27(1):396–410
Garg H, Kumar K (2020) A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory. Artif Intell Rev 53(1):595–624
Jaisankar N, Ganapathy S, Yogesh P, Kannan A, Anand K (2012) An intelligent agent based intrusion detection system using fuzzy rough set based outlier detection. In: Soft computing techniques in vision science. Springer, Berlin, Heidelberg, pp 147–153
Jan N, Zedam L, Mahmood T, Ullah K, Ali Z (2019a) Multiple attribute decision making method under linguistic cubic information. J Intell Fuzzy Syst 36:253–269
Jan N, Zedam L, Mahmood T, Ullah K, Ali Z (2019b) Multiple attribute decision making method under linguistic cubic information. J Intell Fuzzy Syst 36(1):253–269
Khan MJ, Kumam P, Ashraf S, Kumam W (2019) Generalized picture fuzzy soft sets and their application in decision support systems. Symmetry 11(3):415
Kumar PS (2020) Intuitionistic fuzzy zero point method for solving type-2 intuitionistic fuzzy transportation problem. Int J Oper Res 37(3):418–451
Liu P, Wang P (2018) Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int J Intell Syst 33(2):259–280
Liu Z, Qin K, Pei Z (2017) A method for fuzzy soft sets in decision-making based on an ideal solution. Symmetry 9(10):246
Liu P, Ali Z, Mahmood T (2019) A method to multi-attribute group decision-making problem with complex q-Rung Orthopair linguistic information based on Heronian mean operators. Int J Comput Intell Syst 12(2):1465–1496
Molodtsov D (1999) Soft set theory—first results. Comput Math Appl 37(4–5):19–31
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356
Roy AR, Maji PK (2007) A fuzzy soft set theoretic approach to decision making problems. J Comput Appl Math 203(2):412–418
Sebastian S, Ramakrishnan TV (2011) Multi-fuzzy sets: an extension of fuzzy sets. Fuzzy Inf Eng 3(1):35–43
Singh P (2015) Correlation coefficients for picture fuzzy sets. J Intell Fuzzy Syst 28(2):591–604
Son LH (2016) Generalized picture distance measure and applications to picture fuzzy clustering. Appl Soft Comput 46(C):284–295
Tan C, Chen X (2010) Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making. Exp Syst Appl 37(1):149–157
Ullah K, Mahmood T, Jan N (2018a) Similarity measures for t-spherical fuzzy sets with applications in pattern recognition. Symmetry 10(6):193
Ullah K, Mahmood T, Jan N, Ali Z (2018b) A note on geometric aggregation operators in T-spherical fuzzy environment and their applications in multi-attribute decision making. J Eng Appl Sci 37(2):75–86
Ullah K, Hassan N, Mahmood T, Jan N, Hassan M (2019a) Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry 11(3):357
Ullah K, Mahmood T, Ali Z, Jan N (2019b) On some distance measures of complex pythagorean fuzzy sets and their applications in pattern recognition. Complex Intell Syst, pp 1–13
Ullah K, Garg H, Mahmood T, Jan N, Ali Z (2020a) Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making. Soft Comput 24:1–13
Ullah K, Garg H, Mahmood T, Jan N, Ali Z (2020b) Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making. Soft Comput 24(3):1647–1659
Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information–applications to pattern recognition. Pattern Recognit Lett 28(2):197–206
Wei G (2016) Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag 17(4):491–502
Wei G (2017) Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33(2):713–724
Wei G (2018) Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fundam Inform 157(3):271–320
Wei G, Gao H, Wei Y (2018a) Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. Int J Intell Syst 33(7):1426–1458
Wei G, Alsaadi FE, Hayat T, Alsaedi A (2018b) Projection models for multiple attribute decision making with picture fuzzy information. Int J Mach Learn Cybern 9(4):713–719
Xu ZS (2007) Models for multiple attribute decision making with intuitionistic fuzzy information. Int J Uncertain Fuzziness Knowl Based Syst 15(03):285–297
Xu Z, Hu H (2010) Projection models for intuitionistic fuzzy multiple attribute decision making. Int J Inf Technol Decis Mak 9(02):267–280
Xu Z, Yager RR (2008) Dynamic intuitionistic fuzzy multi-attribute decision making. Int J Approx Reason 48(1):246–262
Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision-making. IEEE Trans Syst Man Cybern 18(1):183–190
Yang Y, Tan X, Meng C (2013) The multi-fuzzy soft set and its application in decision making. Appl Math Model 37(7):4915–4923
Yang Y, Liang C, Ji S, Liu T (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
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zadeh LA (1988) Fuzzy logic. Computer 21(4):83–93
Zhang Z, Zhang S (2012) Type-2 fuzzy soft sets and their applications in decision making. J Appl Math 2012
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Jan, N., Mahmood, T., Zedam, L. et al. Multi-valued picture fuzzy soft sets and their applications in group decision-making problems. Soft Comput 24, 18857–18879 (2020). https://doi.org/10.1007/s00500-020-05116-y
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DOI: https://doi.org/10.1007/s00500-020-05116-y