Abstract
In this paper, the idea of the bipolar Pythagorean fuzzy sets (BPFSs) and its activities, which is a generalization of fuzzy sets, bipolar fuzzy sets (BFSs), intuitionistic fuzzy sets and bipolar intuitionistic fuzzy sets is proposed, with the goal that it can deal with dubious data all the more flexibly during the process of decision making. The key objective of this research paper has presented another variant of the Pythagorean fuzzy sets so called BPFSs. In bipolar Pythagorean fuzzy sets, membership degrees are satisfying the condition \(0 \le \left( {\mu_{p}^{ + } \left( x \right)} \right)^{2}\) + \(\left( {v_{p}^{ + } \left( x \right)} \right)^{2} \le 1\) and \(0 \le \left( {\mu_{p}^{ - } \left( x \right)} \right)^{2}\) + \(\left( {v_{p}^{ - } \left( x \right)} \right)^{2} \le 1\) instead of \(0 \le \left( {\mu_{p} \left( x \right)} \right)^{2}\) + \(\left( {v_{p} \left( x \right)} \right)^{2} \le 1\) as is in Pythagorean fuzzy sets and \(0 \le \mu_{p} \left( x \right)\) + \(v_{p} \left( x \right) \le 1\) as is in the intuitionistic fuzzy sets. Here, negative membership degree means the certain counter-property comparing to a bipolar Pythagorean fuzzy set. Also, the BPFSs weighted average operator and the BPFSs weighted geometric operator to aggregate the BPFSs is developed here. Further a multi attribute decision making technique is developed and the proposed aggregation operators are used. Finally, a numerical methodology for execution of the proposed system is introduced.
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References
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Atanassov, (1999) Intuitionistic fuzzy sets. Springer, Heidelberg
Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22:958–965
Chen SJ, Chen SM (2003) A new method for handling multicriteria fuzzy decision-making problems using FN-IOWA operators. Cybern Syst 34:109–137
Chen SJ, Hwang CL (1992) Fuzzy multiple attribute decision making. Springer, Berlin
Hong DH, Choi CH (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114:103–113
Bustine H, Burillo P (1996) Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 79:403–405
Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(343):349
Yager RR, Kacprzyk J (1997) On ordered weighted averaging operator: theory and application. Kluwer, Norwell
Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18:183–190
Yager RR, Filev DP (1999) Induced ordered weighted averaging operators. IEEE Trans Syst Man Cybern Part B Cybern 29:141–150
Yager RR (2002) The induced fuzzy integral aggregation operator. Int J Intell Syst 17:1049–1065
Yager RR (2003) Induced aggregation operators. Fuzzy Sets Syst 137:59–69
Xu ZS, Chen J (2007) On geometric aggregation over interval-valued intuitionistic fuzzy information. In: Fourth international conference on fuzzy systems and knowledge discovery, vol 2. pp 466–471
Chen SM, Tan JM (1994) Handling multicriteria fuzzy decision making problems based on vague set theory. Fuzzy Sets Syst 67:163–172
Chiclana F, Herrera-Viedma E, Herrera F, Alonso S (2004) Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations. Int J Intell Syst 19:233–255
Chiclana F, Herrera-Viedma E, Herrera F, Alonso S (2007) Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations. Eur Jo Oper Res 182:383–399
Herrera-Viedma E, Pasi G, Lopez-Herrera A, Porcel C (2006) Evaluating the information quality of web sites—a methodology based on fuzzy computing with words. J Am Soc Inf Sci Technol 57:538–549
Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433
Xu ZS (2006a) Induced uncertain linguistic OWA operators applied to group decision making. Inf Fusion 7:231–238
Tan CQ, Chen XH (2010) Induced choquet ordered averaging operator and its application to group decision making. Int J Intell Syst 25:59–82
Xu ZS (2006b) An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis Support Syst 41:488–499
Xu ZS, Yager RR (2008) Dynamic intuitionistic fuzzy multiple attribute decision making. Int J Approx Reason 48:246–262
Wei GW (2010) Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Appl Soft Comput 10:423–431
Zhao H, Xu ZS, Ni MF, Liu SS (2010) Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 25:1–30
Xu ZS (2007a) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187
Xu ZS (2007b) Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Deci 22:215–219 ((in Chinese))
Peng X, Yang Y (2015) Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators. Int J Intell Syst 31:1–44
Garai T, Chakraborty D, Roy TK (2019) Multi-objective inventory model with both stock-dependent demand rate and holding cost rate under fuzzy random environment. Ann Data Sci 6:61–81
Wang P (2015a) Oriental thinking and fuzzy logic (2015 celebration of the 50th anniversary of fuzzy sets). Ann Data Sci 2:243–244
Bera AK, Jana DK, Banerjee D et al (2020) A two-phase multi-criteria fuzzy group decision making approach for supplier evaluation and order allocation considering multi-objective, multi-product and multi-period. Ann Data Sci. https://doi.org/10.1007/s40745-020-00255-3
Shi YM (2001) Multiple criteria multiple constraint-level linear programming: concepts, techniques and applications. World Scientific Publishing, Singapore
Xu Z, Shi Y (2015) Exploring big data analysis: fundamental scientific problems. Ann Data Sci 2:363–372
Shi Y, Tian YJ, Kou G, Peng Y, Li JP (2011) Optimization based data mining: theory and applications. Springer, Berlin
Shi Winter Y (2014) Big data: history, current status, and challenges going forward. Bridge US Natl Acad Eng 44(4):6–11
Olson D, Shi Y (2007) Introduction to business data mining. McGraw-Hill/Irwin, New York
Lakovic V (2020) Crisis management of android botnet detection using adaptive neuro-fuzzy inference system. Ann Data Sci 7:347–355
Wang P (2015b) Oriental Thinking and Fuzzy Logic, Celebration of the 50th Anniversary of Fuzzy Sets. Ann Data Sci 2:243–244
Zhang WR, Zhang JH, Shi Y, Chen SS (2009) Bipolar linear algebra and YinYang-N-element cellular networks for equilibrium-based bio-system simulation and regulation. J Biol Syst 17(4):547–576
Lee KM (2000) Bipolar-valued fuzzy sets and their operations. In: Proceedings of the international conference on intelligent technologies Bangkok Thailand. pp 307–312
Zhang XL, Xu ZS (2014) Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078
Hang XL, Xu ZS (2014) Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078
Ezhilmaran D, Sankar K (2015) Morphism of bipolar intuitionistic fuzzy graphs. J Discrete Math Sci Cryptogr 18:605–621
Yang Y, Ding H, Chen ZS, Li YL (2015) A note on extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 31:1–5
Hwang CL, Yoon K (1981) Multiple attribute decision making methods and applications. Springer, New York
Chen J, Li S, Ma S, Wang X (2014) m-Polar fuzzy sets: an extension of bipolar fuzzy sets. Sci World J. https://doi.org/10.1155/2014/416530
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Mandal, W.A. Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-attribute Decision Making Problems. Ann. Data. Sci. 10, 555–587 (2023). https://doi.org/10.1007/s40745-020-00315-8
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DOI: https://doi.org/10.1007/s40745-020-00315-8