Novel neutrality aggregation operator-based multiattribute group decision-making method for single-valued neutrosophic numbers

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.

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

$$\begin{aligned}&\lambda {\mathcal {N}} \varTheta \lambda {\mathcal {M}} \nonumber \\&= \left( \begin{aligned} \frac{\text {MCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})}{\text {MCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {ICS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {NCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})} \cdot \text {PS}\left( \text {PS}\left( \lambda 3(\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}})\right) , \text {PS}\left( \lambda 3(\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}})\right) \right) , \\ \frac{\text {ICS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})}{\text {MCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {ICS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {NCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})} \cdot \text {PS}\left( \text {PS}\left( \lambda 3(\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}})\right) , \text {PS}\left( \lambda 3(\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}})\right) \right) , \\ \frac{\text {NCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})}{\text {MCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {ICS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}}) + \text {NCS}(\lambda {\mathcal {N}}, \lambda {\mathcal {M}})} \cdot \text {PS}\left( \text {PS}\left( \lambda 3(\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}})\right) , \text {PS}\left( \lambda 3(\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}})\right) \right) \\ \end{aligned}\right) \nonumber \\&= \left( \begin{aligned} \frac{\lambda (\varrho _{\mathcal {N}}+\varrho _{\mathcal {M}})}{\lambda (\varrho _{\mathcal {N}}+\varrho _{\mathcal {M}}) + \lambda (\theta _{\mathcal {N}}+\theta _{\mathcal {M}})+\lambda (\eta _{\mathcal {N}}+\eta _{\mathcal {M}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^\lambda ), 3(1-\pi _{\mathcal {M}}^\lambda )\right) , \\ \frac{\lambda (\theta _{\mathcal {N}}+\theta _{\mathcal {M}})}{\lambda (\varrho _{\mathcal {N}}+\varrho _{\mathcal {M}}) + \lambda (\theta _{\mathcal {N}}+\theta _{\mathcal {M}})+\lambda (\eta _{\mathcal {N}}+\eta _{\mathcal {M}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^\lambda ), 3(1-\pi _{\mathcal {M}}^\lambda )\right) , \\ \frac{\lambda (\eta _{\mathcal {N}}+\eta _{\mathcal {M}})}{\lambda (\varrho _{\mathcal {N}}+\varrho _{\mathcal {M}}) + \lambda (\theta _{\mathcal {N}}+\theta _{\mathcal {M}})+\lambda (\eta _{\mathcal {N}}+\eta _{\mathcal {M}})} \cdot \text {PS}\left( 3(1-\pi _{\mathcal {N}}^\lambda ), 3(1-\pi _{\mathcal {M}}^\lambda )\right) \\ \end{aligned} \right) \nonumber \\&= \left( \begin{aligned} \frac{\varrho _{\mathcal {N}} + \varrho _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot 3\left( 1-\pi _{\mathcal {N}}^{\lambda }\pi _{\mathcal {M}}^{\lambda }\right) \\ \frac{\theta _{\mathcal {N}} + \theta _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot 3\left( 1-\pi _{\mathcal {N}}^{\lambda }\pi _{\mathcal {M}}^{\lambda }\right) , \\ \frac{\eta _{\mathcal {N}} + \eta _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot 3\left( 1-\pi _{\mathcal {N}}^{\lambda }\pi _{\mathcal {M}}^{\lambda }\right) \\ \end{aligned} \right) \nonumber \\&= \left( \begin{aligned} \frac{\varrho _{\mathcal {N}} + \varrho _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot \text {PS}\left( \lambda \left( 3(1-\pi _{\mathcal {N}}\pi _{\mathcal {M}})\right) \right) , \\ \frac{\theta _{\mathcal {N}} + \theta _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot \text {PS}\left( \lambda \left( 3(1-\pi _{\mathcal {N}}\pi _{\mathcal {M}})\right) \right) , \\ \frac{\eta _{\mathcal {N}} + \eta _{\mathcal {M}}}{\varrho _{\mathcal {N}}+\theta _{\mathcal {N}}+\eta _{\mathcal {N}}+\varrho _{\mathcal {M}}+\theta _{\mathcal {M}}+\eta _{\mathcal {M}}} \cdot \text {PS}\left( \lambda \left( 3(1-\pi _{\mathcal {N}}\pi _{\mathcal {M}})\right) \right) \\ \end{aligned}\right) \nonumber \\&= \lambda (A \varTheta B) \end{aligned}$$

• (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

$$\begin{aligned}&\text {SVNWNA}({\mathcal {N}}_1) = \kappa _1{\mathcal {N}}_1 = \big ( \varrho _{1},\theta _{1}, \eta _{1}\big )\\&= \left( \begin{aligned}&\frac{\kappa _1 \varrho _{1}}{ \kappa _1(\varrho _{1}+\theta _{1}+\eta _{1})} 3\left( 1-\pi _1^{\kappa _1}\right) , \frac{\kappa _1 \theta _{1}}{ \kappa _1(\varrho _{1}+\theta _{1}+\eta _{1})} 3\left( 1-\pi _1^{\kappa _1}\right) , \\&\frac{\kappa _1 \eta _{1}}{ \kappa _1(\varrho _{1}+\theta _{1}+\eta _{1})} 3\left( 1-\pi _1^{\kappa _1}\right) \end{aligned} \right) \end{aligned}$$

Thus, Eq.(13) holds.

Step 2: Assume that Eq.(13) holds for $n=k$, that is

$$\begin{aligned}&\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_k) \\&= \left( \begin{aligned}&\frac{\sum \nolimits _{i=1}^k \kappa _i\varrho _{i}}{\sum \nolimits _{i=1}^k \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^k \pi _{i}^{\kappa _i}\right) , \frac{\sum \nolimits _{i=1}^k \kappa _i\theta _{i}}{\sum \nolimits _{i=1}^k \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^k \pi _{i}^{\kappa _i}\right) , \\&\frac{\sum \nolimits _{i=1}^k \kappa _i \eta _{i}}{\sum \nolimits _{i=1}^k \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^k\pi _{i}^{\kappa _i}\right) \end{aligned}\right) \end{aligned}$$

Now, for $n=k+1$, we have

$$\begin{aligned}&\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_{k+1}) \\&= \text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_k) \varTheta (\kappa _{k+1}{\mathcal {N}}_{k+1})\\&= \left( \begin{aligned} \frac{\text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1})}{ \left( \begin{aligned} \text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ + \text {ICS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ +\text {NCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \end{aligned} \right) } \cdot \text {PS}\left( 3\left( 1-\prod \nolimits _{i=1}^k(\pi _{i})^{\kappa _i}\right) , 3\left( 1-(\pi _{k+1})^{\kappa _{k+1}}\right) \right) , \\ \frac{\text {ICS}(\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1})}{ \left( \begin{aligned} \text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ + \text {ICS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ + \text {NCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \end{aligned} \right) } \cdot \text {PS}\left( 3\left( 1-\prod \nolimits _{i=1}^k(\pi _{i})^{\kappa _i}\right) , 3\left( 1-(\pi _{k+1})^{\kappa _{k+1}}\right) \right) , \\ \frac{\text {NCS}(\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1})}{ \left( \begin{aligned} \text {MCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ + \text {ICS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \\ + \text {NCS}(\text {SVNWNA}({\mathcal {N}}_1,\ldots ,{\mathcal {N}}_k),\kappa _{k+1}{\mathcal {N}}_{k+1}) \end{aligned} \right) } \cdot \text {PS}\left( 3\left( 1-\prod \nolimits _{i=1}^k(\pi _{i})^{\kappa _i}\right) , 3\left( 1-(\pi _{k+1})^{\kappa _{k+1}}\right) \right) , \\ \end{aligned} \right) \end{aligned}$$

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

$$\begin{aligned}&\text {PS}\left( 3\left( 1-\prod \nolimits _{i=1}^k (\pi _i)^{\kappa _i}\right) , 3\left( 1-(\pi _{k+1})^{\kappa _{k+1}}\right) \right) \\&= 3\left( 1-\left( 1-1+\prod \nolimits _{i=1}^k (\pi _i)^{\kappa _i}\right) \left( 1-1+(\pi _{k+1})^{\kappa _{k+1}}\right) \right) \\&= 3\left( 1-\prod \nolimits _{i=1}^{k+1} (\pi _i)^{\kappa _i}\right) \end{aligned}$$

Thus,

$$\begin{aligned}&\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_{k+1}) \\&= \left( \begin{aligned}&\frac{\sum \nolimits _{i=1}^{k+1} \kappa _i \varrho _{i} }{\sum \nolimits _{i=1}^{k+1} \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{k+1}(\pi _{i})^{\kappa _i}\right) , \frac{\sum \nolimits _{i=1}^{k+1} \kappa _i \theta _{i} }{\sum \nolimits _{i=1}^{k+1} \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{k+1}(\pi _{i})^{\kappa _i}\right) , \\&\frac{\sum \nolimits _{i=1}^{k+1} \kappa _i \eta _{i} }{\sum \nolimits _{i=1}^{k+1} \kappa _i(\varrho _{i}+\theta _{i}+\eta _{i})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{k+1}(\pi _{i})^{\kappa _i}\right) , \end{aligned} \right) \end{aligned}$$

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

$$\begin{aligned}&\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_n) \\&= \left( \begin{aligned}&\frac{\sum \nolimits _{i=1}^{n} \kappa _i \varrho _{0} }{\sum \nolimits _{i=1}^{n} \kappa _i(\varrho _{0}+\theta _0+\eta _{0})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{n}\left( \pi _{0}\right) ^{\kappa _i}\right) , \frac{\sum \nolimits _{i=1}^{n} \kappa _i \theta _{0} }{\sum \nolimits _{i=1}^{n} \kappa _i(\varrho _{0}+\theta _0+\eta _{0})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{n}\left( \pi _{0}\right) ^{\kappa _i}\right) ,\\&\qquad \frac{\sum \nolimits _{i=1}^{n} \kappa _i \eta _{0} }{\sum \nolimits _{i=1}^{n} \kappa _i(\varrho _{0}+\theta _0+\eta _{0})} \cdot 3\left( 1-\prod \nolimits _{i=1}^{n}\left( \pi _{0}\right) ^{\kappa _i}\right) \end{aligned} \right) \\&= \left( \begin{aligned}&\frac{\varrho _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot 3\left( 1-(\pi _{0})^{\sum \nolimits _{i=1}^{n}\kappa _i}\right) , \frac{\theta _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot 3\left( 1-(\pi _{0})^{\sum \nolimits _{i=1}^{n}\kappa _i}\right) , \\&\frac{\eta _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot 3\left( 1-(\pi _{0})^{\sum \nolimits _{i=1}^{n}\kappa _i}\right) \end{aligned} \right) \\&= \left( \begin{aligned}&\frac{\varrho _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot (\varrho _0+\theta _0+\eta _0), \frac{\theta _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot (\varrho _0+\theta _0+\eta _0), \\&\frac{\eta _{0}}{\varrho _{0}+\theta _0+\eta _{0}} \cdot (\varrho _0+\theta _0+\eta _0) \end{aligned} \right) \\&= \left( \varrho _0, \theta _0, \eta _0\right) \\&= {\mathcal {N}}_0 \end{aligned}$$

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

$$\begin{aligned} \varrho _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \varrho _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i} + \eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] ;\\ \theta _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \theta _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i} + \eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] ; \\ \eta _{P_{\mathcal {N}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \eta _{{\mathcal {N}}_i}}{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {N}}_i}+\theta _{{\mathcal {N}}_i} + \eta _{{\mathcal {N}}_i})}3\left[ 1-\prod \nolimits _{i=1}^n \pi _{{\mathcal {N}}_i}^{\kappa _i}\right] ; \\ \varrho _{P_{\mathcal {M}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \varrho _{{\mathcal {M}}_i} }{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i} + \theta _{{\mathcal {M}}_i} + \eta _{{\mathcal {M}}_i} )} 3\left[ 1-\prod \nolimits _{i=1}^n\pi _{{\mathcal {M}}_i}^{\kappa _i}\right] ;\\ \theta _{P_{\mathcal {M}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \theta _{{\mathcal {M}}_i} }{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i} + \theta _{{\mathcal {M}}_i} + \eta _{{\mathcal {M}}_i} )} 3\left[ 1-\prod \nolimits _{i=1}^n\pi _{{\mathcal {M}}_i}^{\kappa _i}\right] ;\\ \eta _{P_{\mathcal {M}}}&= \frac{\sum \nolimits _{i=1}^n \kappa _i \eta _{{\mathcal {M}}_i} }{\sum \nolimits _{i=1}^n \kappa _i(\varrho _{{\mathcal {M}}_i} + \theta _{{\mathcal {M}}_i} + \eta _{{\mathcal {M}}_i} )} 3\left[ 1-\prod \nolimits _{i=1}^n\pi _{{\mathcal {M}}_i}^{\kappa _i}\right] ; \end{aligned}$$

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 $\text {SVNWNA}({\mathcal {N}}_1,{\mathcal {N}}_2,\ldots ,{\mathcal {N}}_n) \le \text {SVNWNA}({\mathcal {M}}_1,{\mathcal {M}}_2,\ldots ,{\mathcal {M}}_n)$.