Abstract
As a subclass of neutrosophic sets, single-valued neutrosophic sets (SVNS) can be used to represent uncertainty and inconsistent information that exist in real-world situations. Information measures play an important role in SVNS theory, which has received more and more attention in recent years. In this paper, we develop a multi-attribute decision-making (MADM) method based on single-valued neutrosophic information measures. To this end, three axiomatic definitions of information measures are first introduced. These include entropy, similarity measure and cross-entropy. Then, we construct information measure formulas on the basis of the cosine function. The relationship among entropy, similarity measure and cross-entropy as well as their mutual transformations are further discussed. Moreover, an approach to single-valued neutrosophic MADM based on these information measure formulas is presented. Finally, a numerical example of city pollution evaluation is provided. The comparative analysis demonstrates the applicability and effectiveness of the proposed method.
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Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Atanassov K (1989) More on intuitionistic fuzzy sets. Fuzzy Sets Syst 33:37–46
Atanassov K (2000) Two theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst 110:267–269
Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349
Beliakov G, Pagola M, Wilkin T (2014) Vector valued similarity measures for Atanassov’s intuitionistic fuzzy sets. Inf Sci 280:352–367
Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148:319–328
Hu JH, Xiao KL, Chen XH, Liu YM (2015) Interval type-2 hesitant fuzzy set and its application in multi-criteria decision-making. Comput Ind Eng 87:91–103
Hung WL, Yang MS (2007) Similarity measures of intuitionistic fuzzy sets based on Lp metric. Int J Approx Reason 46(1):120–136
Jin FF, Pei LD, Chen HY, Zhou LG (2014) Interval-valued intuitionistic fuzzy continuous weighted entropy and its application to multicriteria fuzzy group decision-making. Knowl Based Syst 59:132–141
Jin FF, Ni ZW, Chen HY, Li YP (2016a) Approaches to group decision-making with intuitionistic fuzzy preference relations based on multiplicative consistency. Knowl Based Syst 97:48–59
Jin FF, Ni ZW, Chen HY, Li YP, Zhou LG (2016b) Multiple attribute group decision-making based on interval-valued hesitant fuzzy information measures. Comput Ind Eng 101:103–115
Liu XC (1992) Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Sets Syst 52:305–318
Luca DA, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy sets theory. Inf Control 20:301–312
Majumdar P, Samanta SK (2014) On similarity and entropy of neutrosophic sets. J Intell Fuzzy Syst 26(3):1245–1252
Meng F, Chen X (2015) Interval-valued intuitionistic fuzzy multi-criteria group decision-making based on cross entropy and 2-additive measures. Soft Comput 19(7):2071–2082
Onar SC, Oztaysi B, Otay İ, Kahraman C (2015) Multi-expert wind energy technology selection using interval-valued intuitionistic fuzzy sets. Energy 90:274–285
Peng JJ, Wang JQ, Zhang HY, Chen XH (2014) An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl Soft Comput 25:336–346
Smarandache F (1999) A unifying field in logics neutrosophic logic. Neutrosophic probability, set and logic. American Research Press, Rehoboth
Smarandache F (2003) A unifying field in logics neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability. American Research Press, Rehoboth
Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118:467–477
Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information-applications to pattern recognition. Pattern Recognit Lett 28:197–206
Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single-valued neutrosophic sets. Multisp Multistruct 4:410–413
Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf Sci 181:4273–4286
Wu J, Chiclana F (2014) A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions. Appl Soft Comput 22:272–286
Xu ZS, Xia MM (2012) Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. Int J Intell Syst 27:799–822
Ye J (2010) Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Appl Math Model 34:3864–3870
Ye J (2014a) Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision-making. Int J Fuzzy Syst 16(2):204–215
Ye J (2014b) Single-valued neutrosophic cross-entropy for multicriteria decision-making problems. Appl Math Model 38(3):1170–1175
Ye J (2015) Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med 63(3):171–179
Ye J (2017) Single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Comput 21(3):817–825
Ye J, Fu J (2016) Multi-period medical diagnosis method using a single-valued neutrosophic similarity measure based on tangent function. Comput Methods Programs Biomed 123:142–149
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427
Zeng WY, Li HX (2006) Relationship between similarity measure and entropy of interval-valued fuzzy sets. Fuzzy Sets Syst 157:1477–1484
Zhang HY, Zhang WX, Mei CL (2009) Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure. Knowl Based Syst 22:449–454
Zhou LG, Chen HY (2013) The induced linguistic continuous ordered weighted geometric operator and its application to group decision-making. Comput Ind Eng 66:222–232
Zhou LG, Tao ZF, Chen HY, Liu JP (2014a) Continuous interval-valued intuitionistic fuzzy aggregation operators and their applications to group decision-making. Appl Math Model 38:2190–2205
Zhou LG, Tao ZF, Chen HY, Liu JP (2014b) Intuitionistic fuzzy ordered weighted cosine similarity measure. Group Decis Negot 23:879–900
Zhou LG, Jin FF, Chen HY, Liu JP (2016) Continuous intuitionistic fuzzy ordered weighted distance measure and its application to group decision-making. Technol Econ Dev Econ 22(1):75–99
Acknowledgements
The work was supported by the Natural Science Foundation of Jiangsu Province (BK20170546). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper.
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Communicated by A. Genovese, G. Bruno.
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Wu, H., Yuan, Y., Wei, L. et al. On entropy, similarity measure and cross-entropy of single-valued neutrosophic sets and their application in multi-attribute decision making. Soft Comput 22, 7367–7376 (2018). https://doi.org/10.1007/s00500-018-3073-5
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DOI: https://doi.org/10.1007/s00500-018-3073-5