Public-Private Partnership Decision Making Based on Correlation Coefficients of Single-Valued Neutrosophic Hesitant Fuzzy Sets

(i)

μ ( i ) ⩾ 0 $\mu (i)\geqslant 0$

,

∀ i ∈ N $\forall i\in N$

;

(ii)

∑ { i , j } ⊆ N μ ( i , j ) − ( n − 2 ) ∑ ˍ i ∈ N μ ( i ) = 1 ${\textstyle\sum _{\{i,j\}\subseteq N}}\mu (i,j)-(n-2)\textstyle\sum \_ i\in N\mu (i)=1$

;

(iii)

∑ i ⊆ S ∖ j ( μ ( i , j ) − μ ( i ) ) ⩾ ( s − 2 ) μ ( j ) ${\textstyle\sum _{i\subseteq S\setminus j}}(\mu (i,j)-\mu (i))\geqslant (s-2)\mu (j)$

,

∀ j ∈ S ⊆ N $\forall j\in S\subseteq N$

and

s ⩾ 2 $s\geqslant 2$

.

Although 2-additive measure is powerful to reflect the interactions among the weights of criteria, it cannot be used as the weights directly. Considering this issue, the Shapley function (Shapley, 1953) is a useful tool. With respect to the 2-additive measure μ, the Shapley function can be expressed as in Meng and Tang (2013):

ϕ i ( μ , N ) = 3 − n 2 μ ( i ) + 1 2 ∑ j ∈ N ∖ i ( μ ( i , j ) − μ ( j ) ) \[ {\phi _{i}}(\mu ,N)=\frac{3-n}{2}\mu (i)+\frac{1}{2}\sum \limits_{j\in N\setminus i}\big(\mu (i,j)-\mu (j)\big)\]

(15)

for any

i ∈ N $i\in N$

.

Based on the 2-additive Shapley function and correlation coefficients offered in Section 4.1, single-valued neutrosophic hesitant fuzzy 2-additive Shapley weighted correlation coefficients (SVNHF-2ASWCCs) are defined as follows:

Definition 10.

Let

A = { x i , h A ( x i ) , ι A ( x i ) , g A ( x i ) : x ∈ X } $A=\{{x_{i}},{h_{A}}({x_{i}}),{\iota _{A}}({x_{i}}),{g_{A}}({x_{i}}):x\in X\}$

and

B = { x i , h B ( x i ) , ι B ( x i ) , g B ( x i ) : x ∈ X } $B=\{{x_{i}},{h_{B}}({x_{i}}),{\iota _{B}}({x_{i}}),{g_{B}}({x_{i}}):x\in X\}$

be any two SVNHFSs on the finite set

X = { x 1 , x 2 , … , x n } $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$

. Then, the geometric mean based SVNHF-2ASWCC between A and B is defined as follows:

C C 1 ϕ = C ϕ ( A , B ) E ϕ ( A ) E ϕ ( B A ) + E ϕ ( B ) E ϕ ( A B ) . \[ C{C_{1}^{\phi }}=\frac{{C_{\phi }}(A,B)}{\sqrt{{E_{\phi }}(A){E_{\phi }}({B^{A}})}+\sqrt{{E_{\phi }}(B){E_{\phi }}({A^{B}})}}.\]

(16)

The maximum based SVNHF-2ASWCC is defined as follows:

C C 2 ϕ = C ϕ ( A , B ) max { E ϕ ( A ) , E ϕ ( B A ) } + max { E ϕ ( B ) , E ϕ ( A B ) } , \[ C{C_{2}^{\phi }}=\frac{{C_{\phi }}(A,B)}{\max \{{E_{\phi }}(A),{E_{\phi }}({B^{A}})\}+\max \{{E_{\phi }}(B),{E_{\phi }}({A^{B}})\}},\]

(17)

where

C ϕ ( A , B ) = ∑ k = 1 n ϕ x k ( μ , X ) ( ( 1 l ( h A ( x k ) ) ∑ γ i ( x k ) ∈ h A ( x k ) γ i ( x k ) γ j i ( x k ) + 1 l ( h B ( x k ) ) ∑ γ j ( x k ) ∈ h B ( x k ) γ j ( x k ) γ i j ( x k ) ) + ( 1 l ( ι A ( x k ) ) ∑ δ i ( x k ) ∈ ι A ( x k ) δ i ( x k ) δ j i ( x k ) + 1 l ( ι B ( x k ) ) ∑ δ j ( x k ) ∈ ι B ( x k ) δ j ( x k ) δ i j ( x k ) ) + ( 1 l ( g A ( x k ) ) ∑ η i ( x k ) ∈ g A ( x k ) η i ( x k ) η j i ( x k ) + 1 l ( g B ( x k ) ) ∑ η j ( x k ) ∈ g B ( x k ) η j ( x k ) η i j ( x k ) ) ) , E ϕ ( A ) = ∑ k = 1 n ϕ x k ( μ , X ) ( 1 l ( h A ( x k ) ) ∑ γ ( x k ) ∈ h A ( x k ) γ ( x k ) 2 + 1 l ( ι A ( x k ) ) ∑ δ ( x k ) ∈ ι A ( x k ) δ ( x k ) 2 + 1 l ( g A ( x k ) ) ∑ η ( x k ) ∈ g A ( x k ) η ( x k ) 2 ) , E ϕ ( B ) = ∑ k = 1 n ϕ x k ( μ , X ) ( 1 l ( h B ( x k ) ) ∑ γ ( x k ) ∈ h B ( x k ) γ ( x k ) 2 + 1 l ( ι B ( x k ) ) ∑ δ ( x k ) ∈ ι B ( x k ) δ ( x k ) 2 + 1 l ( g B ( x k ) ) ∑ η ( x k ) ∈ g B ( x k ) η ( x k ) 2 ) , E ϕ ( B A ) = ∑ k = 1 n ϕ x k ( μ , X ) ( 1 l ( h A ( x k ) ) ∑ γ j i ( x k ) ∈ h B ( x k ) : γ i ( x k ) ∈ h A ( x k ) γ j i ( x k ) 2 + 1 l ( ι A ( x k ) ) ∑ δ j i ( x k ) ∈ ι B ( x k ) : δ i ( x k ) ∈ ι A ( x k ) δ j i ( x k ) 2 + 1 l ( g A ( x k ) ) ∑ η j i ( x k ) ∈ g B ( x k ) : η i ( x k ) ∈ g A ( x k ) η j i ( x k ) 2 ) , E ϕ ( A B ) = ∑ k = 1 n ϕ x k ( μ , X ) ( 1 l ( h B ( x k ) ) ∑ γ i j ( x k ) ∈ h A ( x k ) : γ j ( x k ) ∈ h B ( x k ) γ i j ( x k ) 2 + 1 l ( ι B ( x k ) ) ∑ δ i j ( x k ) ∈ ι A ( x k ) : δ j ( x k ) ∈ ι B ( x k ) δ i j ( x k ) 2 + 1 l ( g B ( x k ) ) ∑ η i j ( x k ) ∈ g A ( x k ) : η j ( x k ) ∈ g B ( x k ) η i j ( x k ) 2 ) , \[\begin{aligned}{}{C_{\phi }}(A,B)& ={\sum \limits_{k=1}^{n}}{\phi _{{x_{k}}}}(\mu ,X)\bigg(\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{{\gamma _{i}}({x_{k}})\in {h_{A}}({x_{k}})}{\gamma _{i}}({x_{k}}){\gamma _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{{\gamma _{j}}({x_{k}})\in {h_{B}}({x_{k}})}{\gamma _{j}}({x_{k}}){\gamma _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{{\delta _{i}}({x_{k}})\in {\iota _{A}}({x_{k}})}{\delta _{i}}({x_{k}}){\delta _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{{\delta _{j}}({x_{k}})\in {\iota _{B}}({x_{k}})}{\delta _{j}}({x_{k}}){\delta _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{{\eta _{i}}({x_{k}})\in {g_{A}}({x_{k}})}{\eta _{i}}({x_{k}}){\eta _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{{\eta _{j}}({x_{k}})\in {g_{B}}({x_{k}})}{\eta _{j}}({x_{k}}){\eta _{i}^{j}}({x_{k}})\bigg)\bigg),\\ {} {E_{\phi }}(A)& ={\sum \limits_{k=1}^{n}}{\phi _{{x_{k}}}}(\mu ,X)\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\hspace{-0.1667em}\hspace{-0.1667em}\sum \limits_{\gamma ({x_{k}})\in {h_{A}}({x_{k}})}\gamma {({x_{k}})^{2}}+\frac{1}{l({\iota _{A}}({x_{k}}))}\hspace{-0.1667em}\hspace{-0.1667em}\sum \limits_{\delta ({x_{k}})\in {\iota _{A}}({x_{k}})}\delta {({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{\eta ({x_{k}})\in {g_{A}}({x_{k}})}\eta {({x_{k}})^{2}}\bigg),\\ {} {E_{\phi }}(B)& ={\sum \limits_{k=1}^{n}}{\phi _{{x_{k}}}}(\mu ,X)\bigg(\frac{1}{l({h_{B}}({x_{k}}))}\hspace{-0.1667em}\hspace{-0.1667em}\sum \limits_{\gamma ({x_{k}})\in {h_{B}}({x_{k}})}\gamma {({x_{k}})^{2}}+\frac{1}{l({\iota _{B}}({x_{k}}))}\hspace{-0.1667em}\hspace{-0.1667em}\sum \limits_{\delta ({x_{k}})\in {\iota _{B}}({x_{k}})}\delta {({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{\eta ({x_{k}})\in {g_{B}}({x_{k}})}\eta {({x_{k}})^{2}}\bigg),\\ {} {E_{\phi }}\big({B^{A}}\big)& ={\sum \limits_{k=1}^{n}}{\phi _{{x_{k}}}}(\mu ,X)\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{{\gamma _{j}^{i}}({x_{k}})\in {h_{B}}({x_{k}}):{\gamma _{i}}({x_{k}})\in {h_{A}}({x_{k}})}{\gamma _{j}^{i}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{{\delta _{j}^{i}}({x_{k}})\in {\iota _{B}}({x_{k}}):{\delta _{i}}({x_{k}})\in {\iota _{A}}({x_{k}})}{\delta _{j}^{i}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{{\eta _{j}^{i}}({x_{k}})\in {g_{B}}({x_{k}}):{\eta _{i}}({x_{k}})\in {g_{A}}({x_{k}})}{\eta _{j}^{i}}{({x_{k}})^{2}}\bigg),\\ {} {E_{\phi }}({A^{B}})& ={\sum \limits_{k=1}^{n}}{\phi _{{x_{k}}}}(\mu ,X)(\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{{\gamma _{i}^{j}}({x_{k}})\in {h_{A}}({x_{k}}):{\gamma _{j}}({x_{k}})\in {h_{B}}({x_{k}})}{\gamma _{i}^{j}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{{\delta _{i}^{j}}({x_{k}})\in {\iota _{A}}({x_{k}}):{\delta _{j}}({x_{k}})\in {\iota _{B}}({x_{k}})}{\delta _{i}^{j}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{{\eta _{i}^{j}}({x_{k}})\in {g_{A}}({x_{k}}):{\eta _{j}}({x_{k}})\in {g_{B}}({x_{k}})}{\eta _{i}^{j}}{({x_{k}})^{2}}),\end{aligned}\]

ϕ x k ( μ , X ) ${\phi _{{x_{k}}}}(\mu ,X)$

is the Shapley value of

x k ${x_{k}}$

with respect to the 2-additive measure μ on the set

X = { x 1 , x 2 , … , x n } $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$

,

k = 1 , 2 , … , n $k=1,2,\dots ,n$

.

Note that correlation coefficients in Definition 10 have the properties for SVNHFCCs offered in Section 4.1.

Proposition 4.

Let

A = { x i , h A ( x i ) , ι A ( x i ) , g A ( x i ) : x ∈ X } $A=\{{x_{i}},{h_{A}}({x_{i}}),{\iota _{A}}({x_{i}}),{g_{A}}({x_{i}}):x\in X\}$

and

B = { x i , h B ( x i ) , ι B ( x i ) , g B ( x i ) : x ∈ X } $B=\{{x_{i}},{h_{B}}({x_{i}}),{\iota _{B}}({x_{i}}),{g_{B}}({x_{i}}):x\in X\}$

be any two SVNHFSs on the finite set

X = { x 1 , x 2 , … , x n } $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$

. Then, correlation coefficients between A and B satisfy: (i)

C C q ϕ ( A , A ) = 1 $C{C_{q}^{\phi }}(A,A)=1$

;

(ii)

C C q ϕ ( A , B ) = C C q ϕ ( B , A ) $C{C_{q}^{\phi }}(A,B)=C{C_{q}^{\phi }}(B,A)$

;

(iii)

0 ⩽ C C q ϕ ( A , B ) ⩽ 1 $0\leqslant C{C_{q}^{\phi }}(A,B)\leqslant 1$

,

where

q = 1 , 2 $q=1,2$

.

Remark 1.

When there are no interactions between the SVNHFSs A and B, correlation coefficients in Definition 10 degenerate to the corresponding weighted correlation coefficients as follows: (i)

The geometric mean based weighted SVNHFCC:

C C 1 ω = C ω ( A , B ) E ω ( A ) E ω ( B A ) + E ω ( B ) E ω ( A B ) ; \[ C{C_{1}^{\omega }}=\frac{{C_{\omega }}(A,B)}{\sqrt{{E_{\omega }}(A){E_{\omega }}({B^{A}})}+\sqrt{{E_{\omega }}(B){E_{\omega }}({A^{B}})}};\]

(18)

(ii)

The maximum based weighted SVNHFCC:

C C 2 ω = C ω ( A , B ) max { E ω ( A ) , E ω ( B A ) } + max { E ω ( B ) , E ω ( A B ) } , \[ C{C_{2}^{\omega }}=\frac{{C_{\omega }}(A,B)}{\max \{{E_{\omega }}(A),{E_{\omega }}({B^{A}})\}+\max \{{E_{\omega }}(B),{E_{\omega }}({A^{B}})\}},\]

(19)

where

C ω ( A , B ) = ∑ k = 1 n ω x k ( ( 1 l ( h A ( x k ) ) ∑ γ i ( x k ) ∈ h A ( x k ) γ i ( x k ) γ j i ( x k ) + 1 l ( h B ( x k ) ) ∑ γ j ( x k ) ∈ h B ( x k ) γ j ( x k ) γ i j ( x k ) ) + ( 1 l ( ι A ( x k ) ) ∑ δ i ( x k ) ∈ ι A ( x k ) δ i ( x k ) δ j i ( x k ) + 1 l ( ι B ( x k ) ) ∑ δ j ( x k ) ∈ ι B ( x k ) δ j ( x k ) δ i j ( x k ) ) + ( 1 l ( g A ( x k ) ) ∑ η i ( x k ) ∈ g A ( x k ) η i ( x k ) η j i ( x k ) + 1 l ( g B ( x k ) ) ∑ η j ( x k ) ∈ g B ( x k ) η j ( x k ) η i j ( x k ) ) ) , E ω ( A ) = ∑ k = 1 n ω x k ( 1 l ( h A ( x k ) ) ∑ γ ( x k ) ∈ h A ( x k ) γ ( x k ) 2 + 1 l ( ι A ( x k ) ) ∑ δ ( x k ) ∈ ι A ( x k ) δ ( x k ) 2 + 1 l ( g A ( x k ) ) ∑ η ( x k ) ∈ g A ( x k ) η ( x k ) 2 ) , E ω ( B ) = ∑ k = 1 n ω x k ( 1 l ( h B ( x k ) ) ∑ γ ( x k ) ∈ h B ( x k ) γ ( x k ) 2 + 1 l ( ι B ( x k ) ) ∑ δ ( x k ) ∈ ι B ( x k ) δ ( x k ) 2 + 1 l ( g B ( x k ) ) ∑ η ( x k ) ∈ g B ( x k ) η ( x k ) 2 ) , E ω ( B A ) = ∑ k = 1 n ω x k ( 1 l ( h A ( x k ) ) ∑ γ j i ( x k ) ∈ h B ( x k ) : γ i ( x k ) ∈ h A ( x k ) γ j i ( x k ) 2 + 1 l ( ι A ( x k ) ) ∑ δ j i ( x k ) ∈ ι B ( x k ) : δ i ( x k ) ∈ ι A ( x k ) δ j i ( x k ) 2 + 1 l ( g A ( x k ) ) ∑ η j i ( x k ) ∈ g B ( x k ) : η i ( x k ) ∈ g A ( x k ) η j i ( x k ) 2 ) , E ω ( A B ) = ∑ k = 1 n ω x k ( 1 l ( h B ( x k ) ) ∑ γ i j ( x k ) ∈ h A ( x k ) : γ j ( x k ) ∈ h B ( x k ) γ i j ( x k ) 2 + 1 l ( ι B ( x k ) ) ∑ δ i j ( x k ) ∈ ι A ( x k ) : δ j ( x k ) ∈ ι B ( x k ) δ i j ( x k ) 2 + 1 l ( g B ( x k ) ) ∑ η i j ( x k ) ∈ g A ( x k ) : η j ( x k ) ∈ g B ( x k ) η i j ( x k ) 2 ) \[\begin{aligned}{}{C_{\omega }}(A,B)& ={\sum \limits_{k=1}^{n}}{\omega _{{x_{k}}}}\bigg(\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{{\gamma _{i}}({x_{k}})\in {h_{A}}({x_{k}})}{\gamma _{i}}({x_{k}}){\gamma _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{{\gamma _{j}}({x_{k}})\in {h_{B}}({x_{k}})}{\gamma _{j}}({x_{k}}){\gamma _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{{\delta _{i}}({x_{k}})\in {\iota _{A}}({x_{k}})}{\delta _{i}}({x_{k}}){\delta _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{{\delta _{j}}({x_{k}})\in {\iota _{B}}({x_{k}})}{\delta _{j}}({x_{k}}){\delta _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{{\eta _{i}}({x_{k}})\in {g_{A}}({x_{k}})}{\eta _{i}}({x_{k}}){\eta _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{{\eta _{j}}({x_{k}})\in {g_{B}}({x_{k}})}{\eta _{j}}({x_{k}}){\eta _{i}^{j}}({x_{k}})\bigg)\bigg),\\ {} {E_{\omega }}(A)& ={\sum \limits_{k=1}^{n}}{\omega _{{x_{k}}}}\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{\gamma ({x_{k}})\in {h_{A}}({x_{k}})}\gamma {({x_{k}})^{2}}+\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{\delta ({x_{k}})\in {\iota _{A}}({x_{k}})}\delta {({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{\eta ({x_{k}})\in {g_{A}}({x_{k}})}\eta {({x_{k}})^{2}}\bigg),\\ {} {E_{\omega }}(B)& ={\sum \limits_{k=1}^{n}}{\omega _{{x_{k}}}}\bigg(\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{\gamma ({x_{k}})\in {h_{B}}({x_{k}})}\gamma {({x_{k}})^{2}}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{\delta ({x_{k}})\in {\iota _{B}}({x_{k}})}\delta {({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{\eta ({x_{k}})\in {g_{B}}({x_{k}})}\eta {({x_{k}})^{2}}\bigg),\\ {} {E_{\omega }}\big({B^{A}}\big)& ={\sum \limits_{k=1}^{n}}{\omega _{{x_{k}}}}\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{{\gamma _{j}^{i}}({x_{k}})\in {h_{B}}({x_{k}}):{\gamma _{i}}({x_{k}})\in {h_{A}}({x_{k}})}{\gamma _{j}^{i}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{{\delta _{j}^{i}}({x_{k}})\in {\iota _{B}}({x_{k}}):{\delta _{i}}({x_{k}})\in {\iota _{A}}({x_{k}})}{\delta _{j}^{i}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{{\eta _{j}^{i}}({x_{k}})\in {g_{B}}({x_{k}}):{\eta _{i}}({x_{k}})\in {g_{A}}({x_{k}})}{\eta _{j}^{i}}{({x_{k}})^{2}}\bigg),\\ {} {E_{\omega }}\big({A^{B}}\big)& ={\sum \limits_{k=1}^{n}}{\omega _{{x_{k}}}}\bigg(\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{{\gamma _{i}^{j}}({x_{k}})\in {h_{A}}({x_{k}}):{\gamma _{j}}({x_{k}})\in {h_{B}}({x_{k}})}{\gamma _{i}^{j}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{{\delta _{i}^{j}}({x_{k}})\in {\iota _{A}}({x_{k}}):{\delta _{j}}({x_{k}})\in {\iota _{B}}({x_{k}})}{\delta _{i}^{j}}{({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{{\eta _{i}^{j}}({x_{k}})\in {g_{A}}({x_{k}}):{\eta _{j}}({x_{k}})\in {g_{B}}({x_{k}})}{\eta _{i}^{j}}{({x_{k}})^{2}}\bigg)\end{aligned}\]

and

ω x k ${\omega _{{x_{k}}}}$

is the weight of the element

x k ${x_{k}}$

,

k = 1 , 2 , … , n $k=1,2,\dots ,n$

.

Furthermore, when we have

ω x k = 1 / n ${\omega _{{x_{k}}}}=1/n$

for all

k = 1 , 2 , … , n $k=1,2,\dots ,n$

, correlation coefficients in Definition 10 reduce to those in Definition 7.

Similar to correlation coefficients offered in Section 4.1, we obtain the following property:

Proposition 5.

Let

A = { x i , h A ( x i ) , ι A ( x i ) , g A ( x i ) : x ∈ X } $A=\{{x_{i}},{h_{A}}({x_{i}}),{\iota _{A}}({x_{i}}),{g_{A}}({x_{i}}):x\in X\}$

and

B = { x i , h B ( x i ) , ι B ( x i ) , g B ( x i ) : x ∈ X } $B=\{{x_{i}},{h_{B}}({x_{i}}),{\iota _{B}}({x_{i}}),{g_{B}}({x_{i}}):x\in X\}$

be any two SVNHFSs on the finite set

X = { x 1 , x 2 , … , x n } $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$

. Then, correlation coefficients between A and B satisfy: (i)

C C 1 ϕ ( A , B ) ⩾ C C 2 ϕ ( A , B ) $C{C_{1}^{\phi }}(A,B)\geqslant C{C_{2}^{\phi }}(A,B)$

;

(ii)

C C 1 ω ( A , B ) ⩾ C C 2 ω ( A , B ) $C{C_{1}^{\omega }}(A,B)\geqslant C{C_{2}^{\omega }}(A,B)$

.

Noticeably, the pessimists can choose (12) or (14), while the optimists could adopt (11) and (13).

5 An Approach to Evaluating PPP Models