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Similarity measures based on the generalized distance of neutrosophic Z-number sets and their multi-attribute decision making method

  • Fuzzy systems and their mathematics
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Abstract

Zadeh firstly proposed a Z-number with help of both a fuzzy value and a reliability measure in uncertain environment. Its highlight is that the fuzzy value is closely related to the reliability measure. However, the Z-number lacks the indeterminacy and falsity fuzzy values and reliability measures. Although a single-valued neutrosophic set is described by the truth, indeterminacy and falsity membership degrees, it lacks the reliability measures of the truth, indeterminacy and falsity membership degrees. By the hybrid information of the truth, indeterminacy and falsity Z-numbers, a neutrosophic Z-number (NZN) can be introduced to express the inconsistent, incomplete and indeterminate knowledge and judgments of human cognitions in the real world. Then, the similarity measure is an important mathematical tool for decision making (DM) and pattern recognition, but there is no similarity measure of any NZN set in the existing literature. Motivated by this idea, this study firstly proposes NZN sets and generalized distance and similarity measures between NZN sets and then further introduces cosine and cotangent similarity measures based on the weighted generalized distance of NZN sets. Next, a multi-attribute DM method using the proposed similarity measures is established in the setting of NZN sets. An illustrative example is provided to demonstrate the feasibility and rationality of the proposed method, and then the comparative analysis indicates the two highlighting advantages: (a) NZN enriches the neutrosophic information expression by means of the truth, indeterminacy and falsity Z-numbers and (b) the established DM method based on the proposed similarity measures not only enhances the reliability of DM problems, but also shows its superiority over existing neutrosophic DM methods.

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References

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96

    Article  Google Scholar 

  • Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349

    Article  MathSciNet  Google Scholar 

  • Broumi S, Smarandache F (2013a) Several similarity measures of neutrosophic sets. Neutrosophic Sets Syst 1:54–62

    Google Scholar 

  • Broumi S, Smarandache F (2013b) Cosine similarity measures of interval valued neutrosophic sets. Neutrosophic Sets Syst 5:15–20

    Google Scholar 

  • Chai JS, Selvachandran G, Smarandache F et al (2021) New similarity measures for single-valued neutrosophic sets with applications in pattern recognition and medical diagnosis problems. Complex Intell Syst 7:703–723

    Article  Google Scholar 

  • Du S, Ye J, Yong R, Zhang F (2021) Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method. Complex Intell Syst 7:429–438

    Article  Google Scholar 

  • Jiang W, Shou Y (2017) A novel single-valued neutrosophic set similarity measure and its application in multicriteria decision-making. Symmetry 9:127. https://doi.org/10.3390/sym9080127

    Article  Google Scholar 

  • Jiang W, Xie C, Zhuang M, Shou Y, Tang Y (2016) Sensor data fusion with z-numbers and its application in fault diagnosis. Sensors 16(9):1509

    Article  Google Scholar 

  • Kang B, Hu Y, Deng Y, Zhou D (2016) A new methodology of multicriteria decision-making in supplier selection based on z-numbers. Math Probl Eng 2016, Article ID 8475987. https://doi.org/10.1155/2016/8475987

  • Kang B, Chhipi-Shrestha G, Deng Y, Hewage K, Sadiq R (2018) Stable strategies analysis based on the utility of z-number in the evolutionary games. Appl Math Comput 324:202–217

    MathSciNet  MATH  Google Scholar 

  • Liu C (2018) New similarity measures of simplified neutrosophic sets and their applications. J Inf Process Syst 14(3):790–800

    Google Scholar 

  • Majumdar P (2014) On similarity and entropy of neutrosophic sets. J Intell Fuzzy Syst 26:1245–1252

    Article  MathSciNet  Google Scholar 

  • Majumdar P (2015) Neutrosophic sets and its applications to decision making. In: Acharjya D, Dehuri S, Sanyal S (eds) Computational intelligence for big data analysis. Adaptation, learning, and optimization, vol 19. Springer, Cham, pp 97–115

    Google Scholar 

  • Smarandache F (1998) Neutrosophy: neutrosophic probability, set, and logic. American Research Press, Rehoboth

    MATH  Google Scholar 

  • Saeed M, Saqlain M, Mehmood A (2020) Application of similarity measure on m-polar interval-valued neutrosophic set in decision making in sports. Neutrosophic Sets Syst 38:317–334

    Google Scholar 

  • Wang H, Smarandache F, Zhang YQ, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix

  • Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistruct 4:410–413

    MATH  Google Scholar 

  • Xian S, Yang Z, Guo H (2019) Double parameters TOPSIS for multi-attribute linguistic group decision making based on the intuitionistic Z-linguistic variables. Appl Soft Comput J 85:105835

    Article  Google Scholar 

  • Ye J (2014a) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26:2459–2466

    Article  MathSciNet  Google Scholar 

  • Ye J (2014b) Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J Intell Fuzzy Syst 27(6):2927–2935

    Article  MathSciNet  Google Scholar 

  • Ye J (2014c) Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. Int J Fuzzy Syst 16(2):204–211

    Google Scholar 

  • Ye J (2014d) Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. J Intell Syst 23(4):379–389

    Article  Google Scholar 

  • Ye J (2015) Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med 63(3):171–179

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  Google Scholar 

  • Zadeh LA (2011) A note on z-numbers. Inf Sci 181(14):2923–2932

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Ningbo University, Ningbo, 315211, People’s Republic of China

    Jun Ye

  2. Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, 312000, People’s Republic of China

    Jun Ye

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  1. Jun Ye
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Correspondence to Jun Ye.

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Ye, J. Similarity measures based on the generalized distance of neutrosophic Z-number sets and their multi-attribute decision making method. Soft Comput 25, 13975–13985 (2021). https://doi.org/10.1007/s00500-021-06199-x

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