Abstract
In this paper, we propose the concepts of hesitant bipolar-valued fuzzy sets (HBVFSs) and bipolar-valued hesitant fuzzy sets (BVHFSs). The HBVFSs encompass bipolar-valued fuzzy sets (BVFSs) and hesitant fuzzy sets (HFSs) as special cases. The BVHFS is the generalization of the concept of HFS in which the membership degrees of an element to a given set are not exactly defined, but denoted by several possible bipolar-valued fuzzy values. Then, we investigate the basic operations, properties, and comparison laws of HBVFSs and BVHFSs. Based on these basic operations, properties, and comparison laws, we also develop some aggregation operators for solving the multi-attribute group decision-making problems with numerical examples.
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References
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349
Beg I, Rashid T (2014) Group decision making using intuitionistic hesitant fuzzy sets. Int J Fuzzy Logic Intell Syst 14(3):181–187
Bustince H, Barrenechea E, Pagola M, Fernandez J, Xu Z, Bedregal B, Montero J, Hagras H, Herrera F, Baets B (2016) A historical account of types of fuzzy sets and their relationships. IEEE Trans Fuzzy Syst 24:179–194
Chen SM, Chen CD (2011) Handling forecasting problems based on high-order fuzzy logical relationships. Expert Syst Appl 38(4):3857–3864
Chen SM, Hong JA (2014) Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation. Inf Sci 286:63–74
Chen SM, Wang NY, Pan JS (2009) Forecasting enrollments using automatic clustering techniques and fuzzy logical relationships. Expert Syst Appl 36(8):11,070–11,076
Chen SM, Munif A, Chen GS, Liu HC, Kuo BC (2012) Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights. Expert Syst Appl 39(7):6320–6334
Chen N, Xu ZS, Xia MM (2013) Interval-valued hesitant preference relations and their applications to group decision making. Knowl-Based Syst 37:528–540
Chen SM, Cheng SH, Lin TE (2015) Group decision making systems using group recommendations based on interval fuzzy preference relations and consistency matrices. Inf Sci 298:555–567
Deschrijver G, Kerre E (2003) On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst 133:227–235
Dubois D, Prade H (2008) An introduction to bipolar representations of information and preference. Int J Intell Syst 23:866–877
Farhadinia B (2014) Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int J Intell Syst 29:184–205
Han Y, Shi P, Chen S (2015) Bipolar-valued rough fuzzy set and its applications to decision information system. IEEE Trans Fuzzy Syst 33(6):2358–2370
Kaplan RS, Norton D (1996) The balanced scorecard: translating strategy into action. Harvard Business School Press, Boston
Khan MSA, Abdullah S, Ali A, Siddiqui N, Amin F (2017) Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information. J Intell Fuzzy Syst 33(6):3971–3985
Khan MSA, Abdullah S, Ali A, Amin F (2018) An extension of vikor method for multi-attribute decision-making under pythagorean hesitant fuzzy setting. Granul Comput. https://doi.org/10.1007/s41066-018-0102-9
Lee KM (2004) Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets and bipolar-valued fuzzy sets. J Fuzzy Logic Intell Syst 14(2):125–129
Lee LW, Chen SM (2015a) Fuzzy decision making and fuzzy group decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets. J Intell Fuzzy Syst 29(3):1119–1137
Lee LW, Chen SM (2015b) Fuzzy decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Inf Sci 294:513–529
Liu N, Meng N (2018) Approaches to the selection of cold chain logistics enterprises under hesitant fuzzy environment based on decision distance measures. Granul Comput 3(1):27–38
Mandal P, Ranadive AS (2017) Multi-granulation bipolar-valued fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes. Soft Comput. https://doi.org/10.1007/s00500-017-2765-6
Mandal P, Ranadive AS (2018a) Decisiontheoretic rough sets under pythagorean fuzzy information. Int J Intell Syst 33(4):818–835
Mandal P, Ranadive AS (2018b) Multi-granulation fuzzy decision-theoretic rough sets and bipolar-valued fuzzy decision-theoretic rough sets and their applications. Granul Comput. https://doi.org/10.1007/s41066-018-0111-8
Mandal P, Ranadive AS (2018c) Multi-granulation interval-valued fuzzy probabilistic rough sets and their corresponding three-way decisions based on interval-valued fuzzy preference relations. Granul Comput. https://doi.org/10.1007/s41066-018-0090-9
Mishra AR, Rani P, Pardasani KR (2018) Multiple-criteria decision-making for service quality selection based on shapley COPRAS method under hesitant fuzzy sets. Granul Comput. https://doi.org/10.1007/s41066-018-0103-8
Miyamoto S (2005) Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets Syst 156(3):427–432
Parreiras RO, Ekel PY, Martini JSC, Palhares RM (2010) A flexible consensus scheme for multicriteria group decision making under linguistic assessments. Inf Sci 180:1075–1089
Pedrycz W, Chen SM (2011) Granular computing and intelligent systems: design with information granules of high order and high type. Springer, Heidelberg
Pedrycz W, Chen SM (2015a) Granular computing and decision-making: interactive and iterative approaches. Springer, Heidelberg
Pedrycz W, Chen SM (2015b) Information granularity, big data, and computational intelligence. Springer, Heidelberg
Peng XD (2017) Hesitant trapezoidal fuzzy aggregation operators based on archimedean t-norm and t-conorm and their application in madm with completely unknown weight information. Int J Uncertain Quantif 7(6):475–510
Peng XD, Dai JG (2017) Hesitant fuzzy soft decision making methods based on WASPAS, MABAC and COPRAS with combined weights. J Intell Fuzzy Syst 33(2):1313–1325
Peng XD, Liu C (2017) Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set. J Intell Fuzzy Syst 32(1):955–968
Peng XD, Yang Y (2017) Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight. Appl Soft Comput 54:415–430
Qian G, Wang H, Feng XQ (2013) Generalized hesitant fuzzy sets and their application in decision support system. Knowl-Based Syst 37:357–365
Rodríguez LMRM, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20(1):109–119
Rodríguez RM, Bedregal B, Bustince H, Dong YC, Kahraman C, Martínez L, Torra V, Xu YJ, Xu ZS, Herrera F (2016) A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. towards high quality progress. Inf Fusion 29:89–97
Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539
Wang H (2015) Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making. Int J Computat Intell Syst 8(1):14–33
Wang HY, Chen SM (2008) Evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence. IEEE Trans Fuzzy Syst 16(2):403–415
Xia M, Xu Z (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407
Yager RR (2013) Pythagorean fuzzy subsets. In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada pp 57–61
Yu DJ (2013) Triangular hesitant fuzzy set and its application to teaching quality evaluation. J Inf Comput Sci 10(7):1925–1934
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–358
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning–I. Inf Sci 8(3):199–249
Zhang WR (1998) Bipolar fuzzy sets. Proc FUZZIEEE 1:835–840
Zhang WR (1994) Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp 305–309
Zhu B, Xu Z, Xia M (2012) Dual hesitant fuzzy sets. J Appl Math 2012:1–13
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Mandal, P., Ranadive, A.S. Hesitant bipolar-valued fuzzy sets and bipolar-valued hesitant fuzzy sets and their applications in multi-attribute group decision making. Granul. Comput. 4, 559–583 (2019). https://doi.org/10.1007/s41066-018-0118-1
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DOI: https://doi.org/10.1007/s41066-018-0118-1