Abstract
The paper aims to give some new kinds of operational laws named as neutrality addition and scalar multiplication for the pairs of single-valued neutrosophic numbers. The main idea behind these operations is to include the neutral characters of the decision-maker towards the preferences of the objects when it shows the equal degrees to membership functions. Some salient features of them are investigated also. Further based on these laws, some new aggregation operators are developed to aggregate the different preferences of the decision-makers. Desirable relations and properties are investigated in detail. Finally, a multiattribute group decision-making approach based on the proposed operators is presented and investigated with numerous numerical examples. The superiors, as well as the advantages of the operators, are also discussed in it.
Similar content being viewed by others
References
Arora R, Garg H (2019) Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties. Comput Appl Math 38(2):1–36. https://doi.org/10.1007/s40314-019-0764-1
Arqub OA (2017) Adaptation of reproducing kernel algorithm for solving fuzzy fredholm-volterra integrodifferential equations. Neural Comput Appl 28(7):1591–1610
Arqub OA, Abo-Hammour Z (2014) Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf Sci 279:396–415
Arqub OA, Mohammed AS, Momani S, Hayat T (2016) Numerical solutions of fuzzy differential equations using reproducing kernel hilbert space method. Soft Comput 20(8):3283–3302
Arqub OA, Al-Smadi M, Momani S, Hayat T (2017) Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput 21(23):7191–7206
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Chen SM, Cheng SH, Lan TC (2016) A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition. Inf Sci 343–344:15–40
Garg H (2016) 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
Garg H (2019) Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems. Iran J Sci Technol Trans Electr Eng 43(3):597–613
Garg H, Kumar K (2018) An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making. Soft Comput 22(15):4959–4970
Garg H, Kumar K (2019) A novel possibility measure to interval-valued intuitionistic fuzzy set using connection number of set pair analysis and their applications. Neural Comput Appl. https://doi.org/10.1007/s00521-019-04291-w
Garg H, Nancy (2018a) Multi-criteria decision-making method based on prioritized muirhead mean aggregation operator under neutrosophic set environment. Symmetry 10(7):280. https://doi.org/10.3390/sym10070280
Garg H, Nancy (2018b) New logarithmic operational laws and their applications to multiattribute decision making for single-valued neutrosophic numbers. Cogn Syst Res 52:931–946
Garg H, Nancy (2018c) Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment. Appl Intell 48(8):2199–2213
Garg H, Nancy (2018d) Some hybrid weighted aggregation operators under neutrosophic set environment and their applications to multicriteria decision-making. Appl Intell 48(12):4871–4888
Huang JY (2014) Intuitionistic fuzzy Hamacher aggregation operator and their application to multiple attribute decision making. J Intell Fuzzy Syst 27:505–513
Kaur G, Garg H (2019) Generalized cubic intuitionistic fuzzy aggregation operators using t-norm operations and their applications to group decision-making process. Arab J Sci Eng 44(3):2775–2794
Kumar K, Garg H (2018) Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making. Appl Intell 48(8):2112–2119
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
Liu P, Chu Y, Li Y, Chen Y (2014) Some generalized neutrosophic number hamacher aggregation operators and their application to group decision making. Int J Fuzzy Syst 16(2):242–255
Liu P, Liu J, Chen SM (2018) Some intuitionistic fuzzy dombi bonferroni mean operators and their application to multi-attribute group decision making. J Oper Res Soc 69(1):1–24
Nancy, Garg H (2016a) An improved score function for ranking neutrosophic sets and its application to decision-making process. Int J Uncertain Quantif 6(5):377–385
Nancy, Garg H (2016b) Novel single-valued neutrosophic decision making operators under Frank norm operations and its application. Int J Uncertain Quantif 6(4):361–375
Nancy, Garg H (2019) A novel divergence measure and its based TOPSIS method for multi criteria decision—making under single—valued neutrosophic environment. J Intell Fuzzy Syst 36(1):101–115
Peng JJ, Wang JQ, Wang J, Zhang HY, Chen ZH (2016) Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int J Syst Sci 47(10):2342–2358
Peng XD, Dai JG (2018) A bibliometric analysis of neutrosophic set: two decades review from 1998-2017. Artif Intell Rev, pp 1 – 57. https://doi.org/10.1007/s10462-018-9652-0
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
Rani D, Garg H (2019) Some modified results of the subtraction and division operations on interval neutrosophic sets. J Exp Theor Artif Intell 31(4):677–698
Sahin R, Liu P (2016) Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput Appl 27(7):2017–2029
Smarandache F (1998) Neutrosophy. Neutrosophic probability, set, and logic. ProQuest Information & Learning. Ann Arbor, Michigan, USA
Wang H, Smarandache F, Zhang YQ, Smarandache R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix, AZ
Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistruct 4:410–413
Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187
Yager RR (1988) On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans Syst Man Cybern 18(1):183–190
Yang L, Li B (2016) A multi-criteria decision-making method using power aggregation operators for single-valued neutrosophic sets. Int J Database Theory Appl 9(2):23–32
Ye J (2014) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26(5):2459–2466
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Proofs
Appendix: Proofs
1.1 Proof of Proposition 1
We proof the result by using principle of mathematical induction (PMI) on \(\lambda \). For SVNN \({\mathcal {N}}=( \varrho _{\mathcal {N}}, \theta _{\mathcal {N}},\eta _{\mathcal {N}})\), the following steps of the induction are executed.
- Step 1:
For \(\lambda = 2\) and by using Eq. (8), we have
$$\begin{aligned}&\text {PS}\left( \lambda (\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}})\right) \\&= \text {PS}\left( (\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}), \text {PS}(\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}})\right) \\&= \text {PS}\left( \varrho _{\mathcal {N}} +\theta _{\mathcal {N}}+\eta _{\mathcal {N}}, \varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}} \right) \\&= 3\left( 1-\pi _{\mathcal {N}}^{2}\right) \end{aligned}$$Thus, result is true for \(\lambda =2\).
- Step 2:
Assume that result holds for \(\lambda =n-1\), then for \(\lambda =n\), we have
$$\begin{aligned}&\text {PS}(n(\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}))\\&=\text {PS}\left( \left( \varrho _{\mathcal {N}}{+}\theta _{\mathcal {N}}{+}\eta _{\mathcal {N}}\right) , \text {PS}((n{-}1)(\varrho _{\mathcal {N}}{+}\theta _{\mathcal {N}}{+}\eta _{\mathcal {N}})) \right) \\&= \text {PS}\left( \varrho _{\mathcal {N}} +\theta _{\mathcal {N}}+\eta _{\mathcal {N}}, 3\left( 1-\pi _{\mathcal {N}}^{n-1}\right) \right) \\&= 3\left( 1-\pi _{\mathcal {N}}^{n}\right) \end{aligned}$$which is true for \(\lambda =n\). Hence, by the PMI, Eq. (9) true for all \(\lambda \).
1.2 Proof of Theorem 5
For SVNNs \({\mathcal {N}}\), \({\mathcal {M}}\) and real numbers \(\lambda ,\lambda _1,\lambda _2>0\), we have
- (i)
Easily follows from Eq. (10).
- (ii)
For SVNNs \({\mathcal {N}}\) and \({\mathcal {M}}\), real number \(\lambda >0\), we have
$$\begin{aligned} \lambda {\mathcal {N}} = \left( \begin{aligned}&\frac{\varrho _{\mathcal {N}}}{\varrho _{\mathcal {N}} + \theta _{\mathcal {N}} + \eta _{\mathcal {N}} } \cdot 3 (1-\pi _{\mathcal {N}}^\lambda ), \frac{\theta _{\mathcal {N}}}{\varrho _{\mathcal {N}} + \theta _{\mathcal {N}} + \eta _{\mathcal {N}} } \cdot 3 (1-\pi _{\mathcal {N}}^\lambda ), \\&\frac{\eta _{\mathcal {N}}}{\varrho _{\mathcal {N}} + \theta _{\mathcal {N}}+\eta _{\mathcal {N}}} \cdot 3 (1-\pi _{\mathcal {N}}^\lambda ) \end{aligned}\right) \end{aligned}$$and
$$\begin{aligned} \lambda {\mathcal {M}} = \left( \begin{aligned}&\frac{\varrho _{\mathcal {M}}}{\varrho _{\mathcal {M}} + \theta _{\mathcal {M}} + \eta _{\mathcal {M}} } \cdot 3 (1-\pi _{\mathcal {M}}^\lambda ), \frac{\theta _{\mathcal {M}} }{\varrho _{\mathcal {M}} + \theta _{\mathcal {M}} + \eta _{\mathcal {M}} } \cdot 3 (1-\pi _{\mathcal {M}}^\lambda ), \\&\frac{\eta _{\mathcal {M}}}{\varrho _{\mathcal {M}} + \theta _{\mathcal {M}}+\eta _{\mathcal {M}} } \cdot 3 (1-\pi _{\mathcal {M}}^\lambda ) \end{aligned}\right) \end{aligned}$$
Therefore, by Eq. (7), we get
- (iii)
For two positive real numbers \(\lambda _1\) and \(\lambda _2\), and by Eq. (7), we get
$$\begin{aligned}&\lambda _1 {\mathcal {N}} \varTheta \lambda _2 {\mathcal {N}} \\&= \left( \begin{aligned} \frac{\text {MCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})}{\text {MCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {ICS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {NCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \frac{\text {ICS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})}{\text {MCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {ICS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {NCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \frac{\text {NCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})}{\text {MCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {ICS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}}) + \text {NCS}(\lambda _1 {\mathcal {N}}, \lambda _2 {\mathcal {N}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \end{aligned} \right) \\&= \left( \begin{aligned} \frac{(\lambda _1+\lambda _2) \varrho _{\mathcal {N}}}{(\lambda _1+\lambda _2) \varrho _{\mathcal {N}} + (\lambda _1+\lambda _2) \theta _{\mathcal {N}} + (\lambda _1+\lambda _2) \eta _{\mathcal {N}}} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \frac{(\lambda _1+\lambda _2) \theta _{\mathcal {N}}}{(\lambda _1+\lambda _2) \varrho _{\mathcal {N}} + (\lambda _1+\lambda _2) \theta _{\mathcal {N}} + (\lambda _1+\lambda _2) \eta _{\mathcal {N}}} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \frac{(\lambda _1+\lambda _2) \eta _{\mathcal {N}}}{(\lambda _1+\lambda _2) \varrho _{\mathcal {N}} + (\lambda _1+\lambda _2) \theta _{\mathcal {N}}+ (\lambda _1+\lambda _2) \eta _{\mathcal {N}}} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^{\lambda _1}), 3(1-\pi _{\mathcal {N}}^{\lambda _2})\right) , \\ \end{aligned} \right) \\&= \left( \begin{aligned}&\frac{\varrho _{\mathcal {N}}}{ \varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}} \cdot 3(1-\pi _{\mathcal {N}}^{\lambda _1+\lambda _2}), \frac{\theta _{\mathcal {N}}}{ \varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}} \cdot 3(1-\pi _{\mathcal {N}}^{\lambda _1+\lambda _2}), \\&\frac{\eta _{\mathcal {N}}}{ \varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}} \cdot 3(1-\pi _{\mathcal {N}}^{\lambda _1+\lambda _2}) \end{aligned} \right) \\&= (\lambda _1+\lambda _2) {\mathcal {N}} \end{aligned}$$
1.3 Proof of Theorem 6
For SVNNs \({\mathcal {N}}_i(i=1,2,\ldots ,n)\) and real numbers \(\kappa _i>0\), the first result holds immediately from Theorem 4. Now, in order to show Eq. (13) holds, we follow the steps of PMI on n which are summarized as follows:
Step 1: For \(n=1\), we have \({\mathcal {N}}_1=( \varrho _1, \theta _1, \eta _1)\) and \(\kappa _i=1\). Thus, we can write as
Thus, Eq.(13) holds.
Step 2: Assume that Eq.(13) holds for \(n=k\), that is
Now, for \(n=k+1\), we have
By definition of MCS and NCS, we have \(\text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k), \kappa _{k+1}{\mathcal {N}}_{k+1}) = \text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k)) + \text {MCS}(\kappa _{k+1}{\mathcal {N}}_{k+1})\)\(=\sum _{i=1}^k \kappa _i \varrho _i + \kappa _{k+1}\varrho _{k+1}\)\(=\sum _{i=1}^{k+1} \kappa _i \varrho _i\). Similarly, we get \(\text {ICS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k)\), \(\kappa _{k+1}{\mathcal {N}}_{k+1})\) = \(\sum _{i=1}^{k+1} \kappa _i \theta _i\) and \(\text {NCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k)\), \(\kappa _{k+1}{\mathcal {N}}_{k+1})\) = \(\sum _{i=1}^{k+1} \kappa _i \eta _i\). Further, by definition of PS, we have
Thus,
i.e. Eq. (13) holds for \(n=k+1\). Therefore, by PMI, Eq. (13) holds for all n, which completes the theorem.
1.4 Proof of Theorem 7
For a collection of SVNNs \({\mathcal {N}}_i=( \varrho _i, \theta _i, \eta _i)\) and \({\mathcal {N}}_0=( \varrho _0, \theta _0, \eta _0)\) such that \({\mathcal {N}}_i = {\mathcal {N}}_0\) we have \(\varrho _i = \varrho _0\), \(\theta _i = \theta _0\) and \(\eta _i = \eta _0\) for all i. Then, by Eq. (13) and by weight vector \(\kappa _i>0\) with \(\sum \nolimits _{i=1}^n \kappa _i=1\), we have
1.5 Proof of Theorem 8
For a collection of SVNNs \({\mathcal {N}}_i=( \varrho _i, \theta _i,\eta _i) (i=1,2,\ldots ,n)\), we have
- (i)$$\begin{aligned}&\min \nolimits _i\left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i} }{3} \right\} \\&\quad = 1-\left( 1-\min \nolimits _i \left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3} \right\} \right) ^{\sum \nolimits _{i=1}^n \kappa _i} \\&\quad = 1-\prod \nolimits _{i=1}^n \left( 1-\min \nolimits _i\left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3}\right\} \right) ^{\kappa _i} \\&\quad \le 1 - \prod \nolimits _{i=1}^n \left( 1-\frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3}\right) ^{\kappa _i} \\&\quad \le 1 - \prod \nolimits _{i=1}^n \left( 1-\max \nolimits _i\left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3}\right\} \right) ^{\kappa _i} \\&\quad = 1 - \left( 1-\max \nolimits _i\left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3} \right\} \right) ^{\sum \nolimits _{i=1}^n \kappa _i}\\&\quad = \max \nolimits _i\left\{ \frac{\varrho _{i} + \theta _{i}+\eta _{i}}{3}\right\} \end{aligned}$$
Thus, we have
$$\begin{aligned} \min \nolimits _i\left\{ \varrho _{i} + \theta _{i}+\eta _{i} \right\} \le 3\left( 1 - \prod \nolimits _{i=1}^n \pi _i^{\kappa _i} \right) \le \max \nolimits _i\left\{ \varrho _{i} + \theta _{i}+\eta _{i} \right\} \end{aligned}$$Now by Eq. (13), we get
$$\begin{aligned} \varrho _P&= \frac{\sum \nolimits _{i=1}^n \kappa _i \varrho _{i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^n \pi _i^{\kappa _i}\right) ; \\ \theta _P&= \frac{\sum \nolimits _{i=1}^n \kappa _i \theta _{i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^n \pi _i^{\kappa _i}\right) \\ \text {and } \eta _P&= \frac{\sum \nolimits _{i=1}^n \kappa _i \eta _{i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^n\pi _i^{\kappa _i}\right) \end{aligned}$$Therefore, \(\varrho _P + \theta _P + \eta _P = 3\left( 1 - \prod \nolimits _{i=1}^n \pi _i^{\kappa _i} \right) \). Hence, we get
$$\begin{aligned} \min \nolimits _i\big \{\varrho _{i}+\theta _{i}+\eta _{i}\big \} \le \varrho _{P}+ \theta _{P}+\eta _{P} \le \max \nolimits _i\big \{\varrho _{i} + \theta _{i}+\eta _{i} \big \}. \end{aligned}$$ - (ii)
Since \(\varrho _i\ge \min \nolimits _i \{\varrho _i\}\), so by expression of \(\varrho _P\), we have
$$\begin{aligned} \varrho _P&\ge \frac{\sum \nolimits _{i=1}^n \kappa _i(\min \nolimits _i \{\varrho _{i}\})}{\sum \nolimits _{i=1}^n \kappa _i(\max \nolimits _i \{\varrho _{i}+\theta _{i}+\eta _{i}\})} \\&\quad 3\left[ 1-\prod \nolimits _{i=1}^n \left( 1-\min \nolimits _i \left\{ \frac{\varrho _i+\theta _i+\eta _i}{3}\right\} \right) ^{\kappa _i} \right] \\&= \frac{\min \nolimits _i \{\varrho _{i}\}}{\max \nolimits _i \{\varrho _{i}+\theta _{i}+\eta _{i}\}} \\&\quad 3\left[ 1-\left( 1-\min \nolimits _i \left\{ \frac{\varrho _i+\theta _i+\eta _i}{3}\right\} \right) ^{\sum \nolimits _{i=1}^n \kappa _i}\right] \\&= \frac{\min \nolimits _i \{\varrho _i+\theta _i+\eta _i\} \min \nolimits _i \{\varrho _{i}\}}{\max \nolimits _i \{\varrho _{i}+\theta _{i}+\eta _{i}\}} \end{aligned}$$Moreover,
$$\begin{aligned} \varrho _P&\le \frac{\sum \nolimits _{i=1}^n \kappa _i(\max \nolimits _i \{\varrho _{i}\})}{\sum \nolimits _{i=1}^n \kappa _i(\min \nolimits _i \{\varrho _{i}+\theta _{i}+\eta _{i}\})} \\&\quad 3\left[ 1-\prod \nolimits _{i=1}^n \left( 1-\max \nolimits _i \left\{ \frac{\varrho _i+\theta _i+\eta _i}{3}\right\} \right) ^{\kappa _i} \right] \\&= \frac{\max \nolimits _i \{\varrho _{i}\}}{\min \nolimits _i \{\varrho _{i}+\theta _i+\eta _{i}\}} \\&\quad 3\left[ 1-\left( 1-\max \nolimits _i \left\{ \frac{\varrho _i+\theta _i+\eta _i}{3}\right\} \right) ^{\sum \nolimits _{i=1}^n \kappa _i}\right] \\&=\frac{\max \nolimits _i \{\varrho _i+\theta _i+\eta _i\} \max \nolimits _i \{\varrho _{i}\}}{\min \nolimits _i \{\varrho _{i}+\theta _{i}+\eta _{i}\}} \end{aligned}$$Also, by definition of SVNN and Theorem 6, we get \(\varrho _p\le 1\). Thus, we have
$$\begin{aligned}&\frac{\min \nolimits _i\big \{\varrho _{i}+\theta _i+\eta _{i}\big \}\cdot \min \nolimits _i\big \{\varrho _{i}\big \}}{\max \nolimits _i \big \{\varrho _{i}+\theta _i+\eta _{i}\big \}} \\&\quad \le \varrho _P \le \min \nolimits _i\Bigg \{\frac{\max \nolimits _i \big \{\varrho _{i}+\theta _i+\eta _{i}\big \}\cdot \max \nolimits _i\big \{\varrho _{i}\big \}}{\min \nolimits _i\big \{\varrho _{i} +\theta _i+ \eta _{i}\big \}},1\Bigg \} \end{aligned}$$ - (iii)
As similar to part (ii), we can obtain it. So, we omit here.
1.6 Proof of Theorem 9
For a collection of SVNNs \({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_n\) and \({\mathcal {M}}_1,{\mathcal {M}}_2,\ldots ,{\mathcal {M}}_n\) and by using Theorem 6, we get \(\text {SVNWNA}\)\(({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_n) = ( \varrho _{P_{\mathcal {N}}}, \theta _{P_{\mathcal {N}}}, \eta _{P_{\mathcal {N}}})\) and \(\text {SVNWNA}({\mathcal {M}}_1,{\mathcal {M}}_2,\ldots ,{\mathcal {M}}_n) = ( \varrho _{p_{\mathcal {M}}}, \theta _{P_{\mathcal {M}}}, \eta _{P_{\mathcal {M}}})\) where
Based on these information, we have
- (i)
If \(\varrho _{{\mathcal {N}}_i} + \theta _{{\mathcal {N}}_i} + \eta _{{\mathcal {N}}_i} \le \varrho _{{\mathcal {M}}_i} + \theta _{{\mathcal {M}}_i} + \eta _{{\mathcal {M}}_i}\), then we have \(\varrho _{P_{\mathcal {N}}} + \theta _{P_{\mathcal {N}}}+ \eta _{P_{\mathcal {N}}} = 3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] \)\(\le \)\(3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {M}}_i}^{\kappa _i}\right] \)\(=\varrho _{P_{\mathcal {M}}} + \theta _{P_{\mathcal {M}}}+\eta _{P_{\mathcal {M}}}\).
- (ii)
If \(\varrho _{{\mathcal {N}}_i} + \theta _{{\mathcal {N}}_i}+\eta _{{\mathcal {N}}_i} = \varrho _{{\mathcal {M}}_i} + \theta _{{\mathcal {M}}_i}+ \eta _{{\mathcal {M}}_i}\), and \(\varrho _{{\mathcal {N}}_i}\le \varrho _{{\mathcal {M}}_i}\), then we have
$$\begin{aligned} \varrho _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \varrho _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i} + \eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] \\&\le \frac{\sum \nolimits _{i=1}^n \kappa _i \varrho _{{\mathcal {M}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i}+\theta _{{\mathcal {M}}_i} + \eta _{{\mathcal {M}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {M}}_i}^{\kappa _i}\right] = \varrho _{P_{\mathcal {M}}} \end{aligned}$$Similarly, for \(\eta _{{\mathcal {N}}_i}\ge \eta _{{\mathcal {M}}_i}\) and \(\theta _{{\mathcal {N}}_i}\ge \theta _{{\mathcal {M}}_i}\), we have
$$\begin{aligned} \theta _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \theta _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i}+\eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] \\&\ge \frac{\sum \nolimits _{i=1}^n \kappa _i \theta _{{\mathcal {M}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i}+\theta _{{\mathcal {M}}_i}+\eta _{{\mathcal {M}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n\pi _{{\mathcal {M}}_i}^{\kappa _i}\right] = \theta _{P_{\mathcal {M}}} \end{aligned}$$and
$$\begin{aligned} \eta _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \eta _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i}+\eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] \\&\ge \frac{\sum \nolimits _{i=1}^n \kappa _i \eta _{{\mathcal {M}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i}+\theta _{{\mathcal {M}}_i}+\eta _{{\mathcal {M}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n\pi _{{\mathcal {M}}_i}^{\kappa _i}\right] = \eta _{P_{\mathcal {M}}} \end{aligned}$$Hence, the result.
- (iii)
From part (ii), we obtain that \(\varrho _{P_{\mathcal {N}}} \le \varrho _{P_{\mathcal {M}}}\), \(\theta _{P_{\mathcal {N}}} \ge \theta _{P_{\mathcal {M}}}\) and \(\eta _{P_{\mathcal {N}}} \ge \eta _{P_{\mathcal {M}}}\). Therefore, by definition of score function given in Definition 4, we get \(\varrho _{P_{\mathcal {N}}} - \theta _{P_{\mathcal {N}}} -\eta _{P_{\mathcal {N}}} \le \varrho _{P_{\mathcal {M}}}-\theta _{P_{\mathcal {M}}} - \eta _{P_{\mathcal {M}}}\). Hence, based on an order relation between SVNNs, we have \(\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_n) \le \text {SVNWNA}({\mathcal {M}}_1,{\mathcal {M}}_2,\ldots ,{\mathcal {M}}_n)\).
Rights and permissions
About this article
Cite this article
Garg, H. Novel neutrality aggregation operator-based multiattribute group decision-making method for single-valued neutrosophic numbers. Soft Comput 24, 10327–10349 (2020). https://doi.org/10.1007/s00500-019-04535-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-019-04535-w