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N-valued neutrosophic trapezoidal numbers with similarity measures and application to multi-criteria decision-making problems

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Abstract

N-valued neutrosophic trapezoidal numbers (NVNT-numbers) are a special neutrosophic multi-sets on real number set based on neutrosophic sets. In the NVNT-numbers the occurrences are more than one with the possibility of the same or the different truth-membership function, indeterminacy-membership function and falsity-membership functions. In this paper, NVNT-numbers based on multi-criteria decision-making problems in which the ratings of alternatives are expressed with NVNT-numbers are defined. Firstly, some operational laws of NVNT-numbers by using t-norm and t-conorm are introduced. Then, some aggregation operators of NVNT-numbers including NVNT-numbers weighted geometric operator and NVNT-numbers weighted arithmetic operators are given. Also, a TOPSIS method for the ranking order of alternatives with NVNT-numbers is given according to the similarity of alternative with respect to the positive and negative ideal solution. Moreover, an illustrative example of multi-criteria decision-making in which the ratings of alternatives are expressed with NVNT-numbers is given to verify the developed TOPSIS method and to demonstrate its practicality and effectiveness. Finally, a comparative analysis is presented with illustrative example.

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References

  • Abdel-Basset MA, Mohamed M, Sangaiah AK (2018) Neutrosophic AHP-Delphi group decision making model based on trapezoidal neutrosophic numbers. J Ambient Intell Human Comput 9(5):1427–1443

    Article  Google Scholar 

  • Ashraf S, Abdullah S (2020) Decision support modeling for agriculture land selection based on sine trigonometric single valued neutrosophic information. Int J Neutrosophic Sci 9(2):60–73

    Google Scholar 

  • Ashraf S, Abdullah S, Mahmood T (2018) GRA method based on spherical linguistic fuzzy Choquet integral environment and its application in multi-attribute decision-making problems. Math Sci 12(4):263–275

    Article  MATH  Google Scholar 

  • Ashraf S, Abdullah S, Abdullah L (2019a) Child development influence environmental factors determined using spherical fuzzy distance measures. Mathematics 7(8):661–684

    Article  Google Scholar 

  • Ashraf S, Abdullah S, Smarandache F, Amin NU (2019b) Logarithmic hybrid aggregation operators based on single valued neutrosophic sets and their applications in decision support systems. Symmetry 11(3):364–386

    Article  Google Scholar 

  • Ashraf S, Abdullah S, Aslam M (2020a) Symmetric sum based aggregation operators for spherical fuzzy information: application in multi-attribute group decision making problem. J Intell Fuzzy Syst 38(4):5241–5255

    Article  Google Scholar 

  • Ashraf S, Abdullah S, Mahmood T (2020b) Spherical fuzzy Dombi aggregation operators and their application in group decision making problems. J Ambient Intell Humaniz Comput 11:2731–2749

    Article  Google Scholar 

  • Ashraf S, Abdullah S, Zeng S, Jin HH, Ghani F (2020c) Fuzzy decision support modeling for hydrogen power plant selection based on single valued neutrosophic sine trigonometric aggregation operators. Symmetry 12(2):298–326

    Article  Google Scholar 

  • Aslam M, Fahmi A (2020) New work of trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging operator and decision making. Soft Comput 24:3331–3354

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Batool B, Ahmad M, Abdullah S, Ashraf S, Chinram R (2018) Entropy based Pythagorean probabilistic hesitant fuzzy decision-making technique and its application for fog-haze factor assessment problem. Entropy 22(318):1–18

    MathSciNet  Google Scholar 

  • Brans JP, Vinvke P, Mareschal B (1986) How to select and how to rank projects: the PROMETHEE method. Eur J Oper Res 24:228–238

    Article  MathSciNet  MATH  Google Scholar 

  • Broumi S, Deli I (2015) Correlation measure for neutrosophic refined sets and its application in medical diagnosis. Palest J Math 5(1):135–143

    MathSciNet  MATH  Google Scholar 

  • Chatterjee R, Majumdar P, Samanta SK (2015) Single valued neutrosophic multisets. Ann Fuzzy Math Inform 10(3):499–514

    MathSciNet  MATH  Google Scholar 

  • Das S, Kar MB, Kar S (2013) Group multi-criteria decision making using intuitionistic multi-fuzzy sets. J Uncertain Anal Appl 10(1):1–16

    Google Scholar 

  • Das S, Roy BK, Kar MB, Kar S, Pamučar D (2020) Neutrosophic fuzzy set and its application in decision making. J Ambient Intell Humaniz Comput 11:5017–5029

    Article  Google Scholar 

  • Deli I (2016) Refined neutrosophic sets and refined neutrosophic soft sets: theory and applications. In: John SJ (ed) Handbook of research on generalized and hybrid set structures and applications for soft computing. IGI Global, pp 321–343

    Chapter  Google Scholar 

  • Deli I, Broumi S, Smarandache F (2015) On neutrosophic refined sets and their applications in medical diagnosis. J New Theory 6:88–98

    Google Scholar 

  • Ejegwa PA (2015) New operations on intuitionistic fuzzy multisets. J Math Inform 3:17–23

    Google Scholar 

  • Ejegwa PA, Awolola JA (2013) Some algebraic structures of intuitionistic fuzzy multisets. Int J Sci Technol 2(5):373–376

    Google Scholar 

  • Fan C, Hu K, Feng S, Ye J, Fan E (2019) Heronian mean operators of linguistic neutrosophic multisets and their multiple attribute decision-making methods. Int J Distrib Sens Netw 15(4):1–12

    Article  Google Scholar 

  • Garai T, Garg H, Roy TK (2020) A ranking method based on possibility mean for multi-attribute decision making with single valued neutrosophic numbers. J Ambient Intell Humaniz Comput 11:5245–5258

    Article  Google Scholar 

  • Hashim H, Abdullah L, Al-Quran A (2019) Algebraic operations on new interval neutrosophic vague sets. J Phys Conf Ser 1366(1):12–62

    Article  Google Scholar 

  • Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Jana C, Karaaslan F (2020) Dice and Jaccard similarity measures based on expected intervals of trapezoidal neutrosophic fuzzy numbers and their applications in multicriteria decision making. In: Smarandache F, Abdel-Basset M (eds) Optimization theory based on neutrosophic and polithogenic sets. Academic Press, Elsevier, pp 261–287

    Google Scholar 

  • Jana C, Pal M, Karaaslan F, Wang JQ (2018) Trapezoidal neutrosophic aggregation operators and its application in multiple attribute decision making process. Sci Iran 37(3):1655–1673

    Google Scholar 

  • Jin Y, Ashraf S, Abdullah S (2019) Spherical fuzzy logarithmic aggregation operators based on entropy and their application in decision support systems. Entropy 21(7):628–663

    Article  MathSciNet  Google Scholar 

  • Karaaslan F (2018) Gaussian single valued neutrosophic number and its application in multi-attribure decision making. Neutrosophic Sets Syst 22:101–117

    Google Scholar 

  • Karaaslan F, Hunu F (2020) Type-2 single-valued neutrosophic sets and their applications in multi-criteria group decision making based on TOPSIS method. J Ambient Intell Humaniz Comput 11:4113–4132

    Article  Google Scholar 

  • Khan AA, Ashraf S, Abdullah S, Qiyas M, Luo J, Khan SU (2019) Pythagorean fuzzy Dombi aggregation operators and their application in decision support system. Symmetry 11(3):383–401

    Article  MATH  Google Scholar 

  • Khatter K (2020) Interval valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements. J Ambient Intell Humaniz Comput 12:1039–1055

    Article  Google Scholar 

  • Miyamoto S (2001) Fuzzy multisets and their generalizations. Multiset processing. Springer, Berlin, pp 225–235

    Chapter  Google Scholar 

  • Miyamoto S (2004) Data structure and operations for fuzzy multisets. In: Transactions on Rough Sets II, Lecture Notes in Computer Science, Springer-Verlag, Berlin, vol 3135, pp 189–200

  • Mondal K, Pramanik S, Giri BC (2018a) Interval neutrosophic tangent similarity measure based MADM strategy and its application to MADM problems. Neutrosophic Sets Syst 19:47–58

    Google Scholar 

  • Mondal K, Pramanik S, Giri BC (2018b) Single valued neutrosophic hyperbolic sine similarity measure based MADM strategy. Neutrosophic Sets Syst 20:4–11

    Google Scholar 

  • Mondal K, Pramanik S, Giri BC (2018c) Multi-criteria group decision making based on linguistic refined neutrosophic strategy. In: Smarandache F, Pramanik S (eds) New trends in neutrosophic theory and applications, vol 2. Pons Editions, Brussels, pp 125–139

    Google Scholar 

  • Muhammad R, Shahzaib A, Saleem A, Tahir M, Shakoor M (2019) The cosine similarity measures of spherical fuzzy sets and their applications in decision making. J Intell Fuzzy Syst 36(6):6059–6073

    Article  Google Scholar 

  • Opricovic S, Tzeng GH (2004) Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur J Oper Res 156:445–455

    Article  MATH  Google Scholar 

  • Peng X (2019) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev 52(3):1873–1927

    Article  Google Scholar 

  • Pramanik S, Banerjee D, Giri BC (2016) TOPSIS approach for multi attribute group decision making in refined neutrosophic environment. In: Smarandache F, Pramanik S (eds) New trends in neutrosophic theory and applications, vol 1. Pons Editions, Brussels, pp 79–91

    Google Scholar 

  • Rajarajeswari P, Uma N (2013a) On distance and similarity measures of intuitionistic fuzzy multi set. IOSR J Math 5(4):19–23

    Article  Google Scholar 

  • Rajarajeswari P, Uma N (2013b) Intuitionistic fuzzy multi similarity measure based on cotangent function. Int J Eng Res Technol 2(11):1323–1329

    Google Scholar 

  • Rajarajeswari P, Uma N (2014) Correlation measure for intuitionistic fuzzy multi sets. Int J Res Eng Technol 3(1):611–617

    Article  Google Scholar 

  • Roy B (1991) The outranking approach and the foundations of ELECTRE methods. Theory Decis 31:49–73

    Article  MathSciNet  Google Scholar 

  • Sahin M, Olgun N, Ulucay V, Kargın A, Smarandache F (2017) A new similarity measure based on falsity value between single valued neutrosophic sets based on the centroid points of transformed single valued neutrosophic numbers with applications to pattern recognition. Neutrosophic Sets Syst 15:31–48

    Google Scholar 

  • Sahin M, Deli I, Ulucay V (2018) Extension principle based on neutrosophic multi-sets and algebraic operations. J Math Ext 12(1):69–90

    Google Scholar 

  • Sahin M, Ulucay V, Yılmaz FS (2019a) Improved hybrid vector similarity measures and their applications on trapezoidal fuzzy multi numbers. In: Smarandache F, Şahin M (eds) Neutrosophic triplet structures, vol 1. Pons Editions, Brussels, Belgium, pp 158–184

    Google Scholar 

  • Sahin M, Ulucay V, Yılmaz FS (2019b) Dice vector similarity measure of trapezoidal fuzzy multi-numbers based on multi-criteria decision making. In: Smarandache F, Şahin M (eds) Neutrosophic triplet structures, vol 1. Pons Editions, Brussels, Belgium, pp 185–196

    Google Scholar 

  • Sebastian S, Ramakrishnan TV (2010) Multi-fuzzy sets. Int Math Forum 5(50):2471–2476

    MathSciNet  MATH  Google Scholar 

  • Sebastian S, Ramakrishnan TV (2011a) Multi-fuzzy extension of crisp functions using bridge functions. Ann Fuzzy Math Inform 2(1):1–8

    MathSciNet  MATH  Google Scholar 

  • Sebastian S, Ramakrishnan TV (2011b) Multi-fuzzy subgroups. Int J Contemp Math Sci 6(8):365–372

    MathSciNet  MATH  Google Scholar 

  • Shahzadi G, Akram M, Saeid AB (2017) An application of single-valued neutrosophic sets in medical diagnosis. Neutrosophic Sets Syst 18:80–88

    Google Scholar 

  • Shahzaib A, Saleem A, Tahir M, Fazal G, Tariq M (2019) Spherical fuzzy sets and their applications in multi-attribute decision making problems. J Intell Fuzzy Syst 36(3):2829–2844

    Article  Google Scholar 

  • Shinoj TK, John SJ (2012) Intuitionistic fuzzy multisets and its application in medical diagnosis. World Acad Sci Eng Technol 6(1):1418–1421

    Google Scholar 

  • Shinoj TK, John SJ (2013) Intuitionistic fuzzy multisets. Int J Eng Sci Innov Technol IJESIT 2(6):1–24

    Google Scholar 

  • Shinoj TK, John SJ (2015) Intuitionistic fuzzy multigroups. Ann Pure Appl Math 9(1):131–143

    Google Scholar 

  • Smarandache F (2013) n-Valued refined neutrosophic logic and its applications in physics. Prog Phys 4:143–146

    Google Scholar 

  • Syropoulos A (2010) On nonsymmetric multi-fuzzy sets. Crit Rev 4:35–41

    Google Scholar 

  • Syropoulos A (2012) On generalized fuzzy multisets and their use in computation. Iran J Fuzzy Syst 9(2):113–125

    MathSciNet  MATH  Google Scholar 

  • Ulucay V (2020) Some concepts on interval-valued refined neutrosophic sets and their applications. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-020-02512-y

    Article  Google Scholar 

  • Ulucay V, Deli I, Şahin M (2018) Trapezoidal fuzzy multi-number and its application to multi-criteria decision making problems. Neural Comput Appl 30(5):1469–1478

    Article  Google Scholar 

  • Ulucay V, Kılıç A, Sahin M, Deniz H (2019a) A new hybrid distance-based similarity measure for refined neutrosophic sets and its application in medical diagnosis. Matematika 35(1):83–96

    Article  Google Scholar 

  • Ulucay V, Deli I, Şahin M (2019b) Intuitionistic trapezoidal fuzzy multi-numbers and its application to multi-criteria decision-making problems. Complex Intell Syst 5(1):65–78

    Article  Google Scholar 

  • Wenyu YW, Zhang Z, Zhong Q (2019) Consensus reaching for MAGDM with multi-granular hesitant fuzzy linguistic term sets: a minimum adjustment-based approach. Ann Oper Res. https://doi.org/10.1007/s10479-019-03432-7

    Article  MATH  Google Scholar 

  • Yager RR (1986) On the theory of bags (multi sets). Int J Gen Syst 13:23–37

    Article  Google Scholar 

  • Yager RR (2014) Pythagorean membership grades in multicriteria decision-making. IEEE Trans Fuzzy Syst 22:958–965

    Article  Google Scholar 

  • Ye J (2015) Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment. J Intell Syst 24(1):23–36

    Google Scholar 

  • Ye J, Smarandache F (2016) Similarity measure of refined single-valued neutrosophic sets and its multicriteria decision making method. Neutrosophic Sets Syst 12:41–44

    Google Scholar 

  • Ye S, Ye J (2014) Dice similarity measure between single valued neutrosophic multisets and its application in medical diagnosis. Neutrosophic Sets Syst 6:48–52

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Zhang Z, Yu W, Martinez L, Gao Y (2019) Managing multigranular unbalanced hesitant fuzzy linguistic information in multiattribute large-scale group decision making: a linguistic distribution-based approach. IEEE Trans Fuzzy Syst 28(11):2875–2889

    Article  Google Scholar 

  • Zhang Z, Kou X, Yu W, Gao Y (2020a) Consistency improvement for fuzzy preference relations with self-confidence: an application in two-sided matching decision making. J Oper Res Soc 2020:1–14. https://doi.org/10.1080/01605682.2020.1748529

    Article  Google Scholar 

  • Zhang Z, Gao Y, Li Z (2020b) Consensus reaching for social network group decision making by considering leadership and bounded confidence. Knowl Based Syst 204(27):1–12

    Google Scholar 

  • Zimmermann HJ (1993) Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Zindani D, Maity SR, Bhowmik S (2021) Complex interval-valued intuitionistic fuzzy TODIM approach and its application to group decision making. J Ambient Intell Humaniz Comput 12:2079–2102

    Article  Google Scholar 

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  1. Kilis 7 Aralık University, 79000, Kilis, Turkey

    İrfan Deli, Vakkas Uluçay & Yadigar Polat

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  1. İrfan Deli
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  2. Vakkas Uluçay
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  3. Yadigar Polat
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Correspondence to İrfan Deli.

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Deli, İ., Uluçay, V. & Polat, Y. N-valued neutrosophic trapezoidal numbers with similarity measures and application to multi-criteria decision-making problems. J Ambient Intell Human Comput 13, 4493–4518 (2022). https://doi.org/10.1007/s12652-021-03294-7

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