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Neutrosophic fuzzy set and its application in decision making

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Abstract

Fuzzy set (FS) is used to tackle the uncertainty using the membership grade, whereas neutrosophic set (NS) is used to tackle uncertainty using the truth, indeterminacy and falsity membership grades which are considered as independent. In this paper, we introduce the notion of neutrosophic fuzzy set (NFS) by combining FS with NS, which gives rise to some new concepts. Since NFS finds some difficulties to deal with some real life problems due to the nonstandard interval of neutrosophic components, we introduce single valued NFS (SVNFS) which is considered as an instance of NFS. Some set theoretic operations of SVNFS are proposed and their properties are derived. We also propose the distance measures between two SVNFSs. Then a decision making approach is presented using similarity measures based on distance measures. Finally, we emonstrate the proposed approach using a numerical example.

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References

  • Abdel-Basset M, Mohamed M, Sangaiah AK (2018) Neutrosophic AHP-Delphi Group decision making model based on trapezoidal neutrosophic numbers. J Ambient Intell Hum Comput 9:1427–1443

    Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Biswas P, Pramanik S, Giri B (2014) Cosine similarity measure based multi-attribute decision-making with trapezoidal fuzzy neutrosophic numbers. Neutrosophic Sets Syst 8:46–56

    Google Scholar 

  • Brauers WKM, Zavadskas EK (2010) Project management by MULTIMOORA as an instrument for transition economies. Technol Econ Dev Econ 16(1):5–24

    Google Scholar 

  • Cabrerizo FJ, Moreno JM, Pérez IJ, Herrera-Viedma E (2010) Analyzing consensus approaches in fuzzy group decision making: advantages and drawbacks. Soft Comput 14:451–463

    Google Scholar 

  • Cabrerizo FJ, Al-Hmouz R, Morfeq A, Balamash AS, Martínez MA, Herrera-Viedma E (2017) Soft consensus measures in group decision making using unbalanced fuzzy linguistic information. Soft Comput 21:3037–3050

    MATH  Google Scholar 

  • Cereška A, Zavadskas EK, Cavallaro F, Podvezko V, Tetsman I, Grinbergien I (2016) Sustainable assessment of aerosol pollution decrease applying multiple attribute decision-making methods. Sustainability 8(7):586

    Google Scholar 

  • Cheng HD, Guo Y (2008) A new neutrosophic appraoch to image thresholding. New Math Nat Comput 4(3):291–308

    Google Scholar 

  • Das S, Kar S (2014) Group decision making in medical system: an intuitionistic fuzzy soft set approach. Appl Soft Comput 24:196–211

    Google Scholar 

  • Das S, Kar MB, Kar S (2013) Group multi criteria decision making using intuitionistic multi fuzzy sets. J Uncertain Anal Appl 1:10. https://doi.org/10.1186/2195-5468-1-10

    Article  Google Scholar 

  • Das S, Kar S, Pal T (2014a) Group decision making using interval-valued intuitionistic fuzzy soft matrix and confident weight of experts. Art Intel Soft Comput Res 4(1):57–77

    Google Scholar 

  • Das S, Kar MB, Pal T, Kar S (2014b) Multiple attribute group decision making using interval-valued intuitionistic fuzzy soft matrix. In: Proceedings of IEEE international conference on fuzzy systems (FUZZ-IEEE). IEEE, Beijing, pp 2222–2229. https://doi.org/10.1109/FUZZ-IEEE.2014.6891687

    Chapter  Google Scholar 

  • Das S, Kar S, Pal T (2017a) Robust decision making using intuitionistic fuzzy numbers. Granul Comput 2:41–54

    Google Scholar 

  • Das S, Kumar S, Kar S, Pal T (2017b) Group decision making using neutrosophic soft matrix: an algorithmic approach. J King Saud Univ-Comput Inf Sci. https://doi.org/10.1016/j.jksuci.2017.05.001

    Article  Google Scholar 

  • Das S, Malakar D, Kar S, Pal T (2018) A brief review and future outline on decision making using fuzzy soft set. Int J Fuzzy Syst Appl 7(2):1–43

    Google Scholar 

  • Deli I, Ali M, Smarandache F (2015) Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. In: Proceedings of the 2015 international conference on advanced mechatronic systems (ICAMechS). IEEE, Beijing, pp 249–254

    Google Scholar 

  • Garg H (2016a) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:989–999

    Google Scholar 

  • Garg H (2016b) Generalized Intuitionistic Fuzzy Multiplicative Interactive Geometric Operators and their application to multiple criteria decision making. Int J Mach Learn Cybern 7(6):1075–1092

    Google Scholar 

  • Garg H (2016c) Generalized intuitionistic fuzzy interactive geometric interaction operators using einstein t-norm and t-conorm and their application to decision making. Comput Ind Eng 101:53–69

    Google Scholar 

  • Garg H (2016d) A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. Intell Syst 31(9):886–920

    Google Scholar 

  • Garg H (2017) Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application. Eng Appl Art Int 60:164–174

    Google Scholar 

  • Garg H, Nancy (2018) Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making. J Ambient Intell Humaniz Comput 9:1975–1997. https://doi.org/10.1007/s12652-018-0723-5

    Article  Google Scholar 

  • Grattan-Guiness I (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z Math Logik Grundladen Math 22:149–160

    MathSciNet  Google Scholar 

  • Guo Y, Cheng HD (2009) New neutrosophic approach to image segmentation. Pattern Recogn 42:587–595

    MATH  Google Scholar 

  • Guo Z-L, Liu Y-L, Yang H-L (2017) A novel rough set model in generalized single valued neutrosophic approximation spaces and its application. Symmetry 9:119. https://doi.org/10.3390/sym9070119

    Article  Google Scholar 

  • Jahn KU (1975) Intervall-wertige Mengen. Math Nachr 68:115–132

    MathSciNet  MATH  Google Scholar 

  • Jana C, Muhiuddin G, Pal M (2019) Multiple-attribute decision making problems based on SVTNH methods. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-019-01568-9

    Article  Google Scholar 

  • Kahraman C, Ghorabaee MK, Zavadskas EK, Onar SC, Yazdani M, Oztaysi B (2017) Intuitionistic fuzzy EDAS method: an application to solid waste disposal site selection. J Environ Eng Landsc Manag 25(1):1–12

    Google Scholar 

  • Kamal K, Garg H (2018) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. J Comput Appl Math 37(2):1319–1329. https://doi.org/10.1007/s40314-016-0402-0

    Article  MathSciNet  MATH  Google Scholar 

  • Karaaslan F, Hunu F (2020) Type-2 single-valued neutrosophic sets and their applications in multicriteria group decision making based on TOPSIS method. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-020-01686-9

    Article  Google Scholar 

  • Li Y, Liu P, Chen Y (2016) Some single valued neutrosophic number heronian mean operators and their application in multiple attribute group decision making. Informatica 27(1):85–110

    MATH  Google Scholar 

  • Liu F, Aiwu G, Lukovac V, Vukic M (2018) A multicriteria model for the selection of the transport service provider: a single valued neutrosophic DEMATEL multicriteria model. Decis Mak Appl Manag Eng 1(2):121–130

    Google Scholar 

  • Maiti I, Mandal T, Pramanik S (2019) Neutrosophic goal programming strategy for multi-level multi-objective linear programming problem. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-019-01482-0

    Article  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. Adapt Learn Optimiz 19:97–115. https://doi.org/10.1007/978-3-319-16598-1_4

  • Mardani A, Jusoh A, Zavadskas EK (2015) Fuzzy multiple criteria decision-making techniques and applications—two decades review from 1994 to 2014. Expert Syst Appl 42:4126–4148

    Google Scholar 

  • Meng F, Wang N, Xu Y (2019) Interval neutrosophic preference relations and their application in virtual enterprise partner selection. J Ambient Intell Humaniz Comput 10:5007–5036. https://doi.org/10.1007/s12652-019-01178-5

    Article  Google Scholar 

  • Morente-Molinera JA, Pérez IJ, Ureña MR, Herrera-Viedma E (2015) On multi-granular fuzzy linguistic modelling in group decision making problems: a systematic review and future trends. Knowl-Based Syst 74:49–60

    Google Scholar 

  • Nancy Garg H (2016a) Single-valued neutrosophic Entropy of order alpha. Neutrosophic Sets Syst 14:21–28

    Google Scholar 

  • Nancy Garg H (2016b) An improved score function for ranking neutrosophic sets and its application to decision—making process. Int J Uncertain Quan 6(5):377–385

    Google Scholar 

  • Nancy Garg H (2016c) Novel single-valued neutrosophic decision making operators under Frank norm operations and its application. Int J Uncertain Quan 6(4):361–375

    Google Scholar 

  • Nie R-X, Wang J-Q, Zhang H-Y (2017) Solving solar-wind power station location problem using an extended weighted aggregated sum product assessment (WASPAS) technique with interval neutrosophic sets. Symmetry 9:106. https://doi.org/10.3390/sym9070106

    Article  Google Scholar 

  • Pamucar D, Bozanic D (2019) Selection of a location for the development of multimodal logistics center: application of single-valued neutrosophic MABAC model. Oper Res Eng Sci Theory Appl 2(2):55–71

    Google Scholar 

  • Pamucar D, Badi I, Korica S, Obradovic R (2018) A novel approach for the selection of power generation technology using an linguistic neutrosophic combinative distance-based assessment (CODAS) method: a case study in Libya. Energies 11(9):2489. https://doi.org/10.3390/en11092489

    Article  Google Scholar 

  • Pamucar D, Sremac S, Stevic Z, Cirović G, Tomic D (2019) New multi-criteria LNN WASPAS model for evaluating the work of advisors in the transport of hazardous goods. Neural Comput Appl. https://doi.org/10.1007/s00521-018-03997-7

    Article  Google Scholar 

  • Peng J-J, Wang J-Q, Zhang H-Y, Chen X-H (2014) An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl Soft Comput 25:336–346

    Google Scholar 

  • Peng J-J, Wang J-Q, Yang W-E (2017) A multi-valued neutrosophic qualitative flexible approach based on likelihood for multi-criteria decision-making problems. Int J Syst Sci 48(2):425–435

    MathSciNet  MATH  Google Scholar 

  • Pouresmaeil H, Shivanian E, Khorram E, Fathabadi HS (2017) An extended method using TOPSIS and VIKOR for multiple attribute decision making with multiple decision makers and single valued neutrosophic numbers. Adv Appl Stat 50(4):261–292

    MATH  Google Scholar 

  • Rivieccio U (2008) Neutrosophic logics: prospects and problems. Fuzzy Sets Syst 159:1860–1868

    MathSciNet  MATH  Google Scholar 

  • Sambuc R (1975) Fonctions φ-floues: Application l’aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis Univ. Marseille, France

  • Smarandache F (1999) A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic. American Research Press, Rehoboth

    MATH  Google Scholar 

  • Smarandache F (2005) Neutrosophic set—a generalization of the intuitionistic fuzzy set. Int J Pure Appl Math 24(3):287–297

    MathSciNet  MATH  Google Scholar 

  • Stanujkic D, Zavadskas EK, Smarandache F, Brauers WKM, Karabasevic DA (2017) Neutrosophic extension of the MULTIMOORA method. Informatica 28(1):181–192

    Google Scholar 

  • Tan R, Zhang W (2017) Multiple attribute group decision making methods based on trapezoidal fuzzy neutrosophic numbers. J Intell Fuzzy Syst 33:2547–2564

    MATH  Google Scholar 

  • Ureña MR, Chiclana F, Morente-Molinera JA, Herrera-Viedma E (2015) Managing incomplete preference relations in decision making: a review and future trends. Inf Sci 302:14–32

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Wei G, Zhang Z (2019) Some single-valued neutrosophic Bonferroni power aggregation operators in multiple attribute decision making. J Ambient Intell Humaniz Comput 10:863–882. https://doi.org/10.1007/s12652-018-0738-y

    Article  Google Scholar 

  • Ye J (2013) Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int J Gen Syst 42(4):386–394

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Ye J (2014b) Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making. J Intell Fuzzy Syst 26(1):165–172

    MathSciNet  MATH  Google Scholar 

  • Ye J (2014c) Single valued neutrosophic cross-entropy for multicriteria decision making problems. Appl Math Model 38(3):1170–1175

    MathSciNet  Google Scholar 

  • Ye J (2015) Trapezoidal neutrosophic set and its application to multiple attribute decision-making. Neural Comput Appl 26(5):1157–1166

    Google Scholar 

  • Ye J (2016) Correlation coefficients of interval neutrosophic hesitant fuzzy sets and its application in a multiple attribute decision making method. Informatica 27(1):179–202

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249

    MathSciNet  MATH  Google Scholar 

  • Zhang H-Y, Wang J-Q, Chen X-H (2014) Interval neutrosophic sets and their application in multicriteria decision making problems. Sci World J Artic ID. https://doi.org/10.1155/2014/645953

    Article  Google Scholar 

  • Zhang H, Wang J, Chen X (2016) An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput Appl 27(3):615–627

    Google Scholar 

  • Zhang H, Dong Y, Herrera-Viedma E (2018) Consensus building for the heterogeneous large-scale GDM with the individual concerns and satisfactions. IEEE T Fuzzy Syst 26(2):884–898. https://doi.org/10.1109/TFUZZ.2017.2697403

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, National Institute of Technology Warangal, Warangal, 506004, India

    Sujit Das

  2. Department of Mathematics, National Institute of Technology, Durgapur, 713209, India

    Bikash Koli Roy & Samarjit Kar

  3. Department of CSE, Heritage Institute of Technology, Kolkata, 700107, India

    Mohuya B. Kar

  4. Department of Logistics, Military Academy, University of Defence in Belgrade, Pavla Jurisica Sturma 33, 11000, Belgrade, Serbia

    Dragan Pamučar

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  1. Sujit Das
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  2. Bikash Koli Roy
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Correspondence to Dragan Pamučar.

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Appendix A

Appendix A

In this section we only give the proof of \( \left( {D4} \right) \). For this let \( P,Q,R \) be three SVNFSs over the universe \( Y = \left\{ {y_{1} ,y_{2} , \ldots ,y_{n} } \right\}. \)

Let \( P \subseteq Q \subseteq R \). Then

$$ T_{P} \left( {y_{i} ,u} \right) \le T_{Q} \left( {y_{i} ,u^{\prime}} \right) \le T_{R} \left( {y_{i} ,u^{\prime\prime}} \right),I_{P} \left( {y_{i} ,u} \right) \ge I_{Q} \left( {y_{i} ,u^{\prime}} \right) \ge I_{R} \left( {y_{i} ,u ^{\prime\prime}} \right),F_{P} \left( {y_{i} ,u} \right) \ge F_{Q} \left( {y_{i} ,u ^{\prime}} \right) \ge F_{R} \left( {y_{i} ,u^{\prime\prime}} \right). $$

Therefore for every \( y_{i} \in Y \), for \( m = 1,2 \) we have

$$ \left| {T_{P} \left( {y_{i} ,u} \right) - T_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|T_{P} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} $$
$$ \left| {T_{Q} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u ^{\prime}} \right)|^{m} \le } \right|T_{P} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} $$
$$ \left| {I_{P} \left( {y_{i} ,u} \right) - I_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|I_{P} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} $$
$$ \left| {I_{Q} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|I_{P} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} $$
$$ \left| {F_{P} \left( {y_{i} ,u} \right) - {\text{F}}_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|F_{P} \left( {y_{i} ,u} \right) - F_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} $$
$$ \left| {F_{Q} \left( {y_{i} ,u} \right) - F_{R} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|F_{P} \left( {y_{i} ,u} \right) - F_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} $$

Hence

$$ \left| {T_{P} \left( {y_{i} ,u} \right) - T_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} + } \right|I_{P} \left( {y_{i} ,u} \right) - I_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} + \left| {F_{P} \left( {y_{i} ,u} \right) - F_{Q} \left( {y_{i} ,u^{\prime}} \right)|^{m} \le } \right|T_{P} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} + \left| {I_{P} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} + } \right|F_{P} \left( {y_{i} ,u} \right) - F_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} $$

and

$$ \left| {T_{Q} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u ^{\prime}} \right)|^{m} + } \right|I_{Q} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u^{\prime}} \right)|^{m} + \left| {F_{Q} \left( {y_{i} ,u} \right) - F_{R} \left( {y_{i} ,u ^{\prime}} \right)|^{m} \le } \right|T_{P} \left( {y_{i} ,u} \right) - T_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} + \left| {I_{P} \left( {y_{i} ,u} \right) - I_{R} \left( {y_{i} ,u^{\prime\prime}} \right)|^{m} + } \right|F_{P} \left( {y_{i} ,{\text{u}}} \right) - F_{R} \left( {y_{i} ,u ^{\prime\prime}} \right)|^{m} . $$

Using the above inequalities with the above mentioned distance eypressions of \( d_{j} \left( {P,Q} \right)\left( {j = 1,2,3,4} \right) \) we obtain \( d_{j} \left( {P,Q} \right) \le d_{j} \left( {Q,R} \right)\;{\text{and}}\; d_{j} \left( {Q,R} \right) \le d_{j} \left( {P,R} \right)\;{\text{for}}\; j = 1,2,3,4. \)

Hence the property \( \left( {D4} \right) \) is proved for \( d_{j} \left( {P,Q} \right)\left( {j = 1, 2,3,4} \right). \)

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Das, S., Roy, B.K., Kar, M.B. et al. Neutrosophic fuzzy set and its application in decision making. J Ambient Intell Human Comput 11, 5017–5029 (2020). https://doi.org/10.1007/s12652-020-01808-3

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