3.1. Single Valued Neutrosophic Power Heronian Aggregation Operators
As an important aggregation operator, the PA operator is first proposed byYager [
36], which can overcome the influence of unreasonable arguments by considering the support degree between input arguments. The traditional PA operator is defined as below.
Definition 6. [
36].
Let , and be a group of nonnegative numbers. Ifwhere
. We denote as the support degree for from . satisfies the following axioms:(1) ;
(2) ;
(3) , if .
Then PA is called the power average (PA) operator.
For example, suppose , , are nonnegative numbers, the are calculated as follows:
Step 1. Calculate the Thereafter, we have
, , .
Step 2. Calculate the power weighting vector through Expression (19). Thereafter, we have
,
,
.
Step 3. Calculate the
using the equation 15 (suppose p = q = 1). Thereafter, we can have
Beliakov [
38] first proposed the Heronian mean operator, which can determine the interrelation of the input arguments [
46,
47,
48]. HM is defined as follows.
Definition 7. [
48].
Let , and be a group of nonnegative numbers. IfThen is called the Heronian mean (HM) operator.
For example, suppose
,
,
are nonnegative numbers, the
are calculated as follow (suppose
p =
q = 1):
Next, we shall develop the SVNPHA and SVNWPHA operators based on the operation laws of SVNNs.
Definition 8. Let , and be a collection of SVNNs. Ifwhere . We shall denote as the support degree for from . satisfies the following three properties: (1) ;
(2) ;
(3) , if in which is the distance between SVNNs and .
Then,
is called the single-valued neutrosophic power Heronian aggregation (SVNPHA) operator.
In order to simply this expression (18). We can define:
and call
as the power weighting vector with
Thereafter, Expression (18) can be expressed as follows:
Theorem 2. Let be a group of SVNNs and . Then, the result aggregated from SVNPHA operator is
still a SVNN, and even Proof. To prove Equation (21), we first prove that the following equation is right.
By the operational rules of SVNNs defined in (2–5), we have
By using Equation (2), we get
That is, when n = 2, the Equation (22) is right.
Assume
n =
m, Equation (22) is right:
Furthermore, when
n =
k + 1, we have
We shall prove Equation (27) on mathematical induction on k
For
k = 2, we have
Suppose
k =
a, the Equation (27) is right, that is
Then, when
k =
a + 1, we have
Therefore, when k = a + 1, the Equation (27) is true. Hence, Equation (27) is established for any k.
Similarly, we can prove the other parts of Equation (26).
So, Equation (26) becomes
Therefore, when n = k + 1, the Equation (22) is true. Hence, Equation (22) is established for any n. □
From Equation (22) and the operational rules of SVNNs defined in (2–5), we have
Therefore, Equation (21) is right and we complete the proof of the Theorem 2.
To compute the power weight vector
, the support degree between SVNNs should be calculate firstly. Under normal circumstances, we can use the similarity degree between SVNNs to replace the support degree and that is,
Example 1. Suppose three SVNNs exist: , , and . Accordingly, we can use SVNPHA to generate a comprehensive value. In the following, the steps are given.
Step 1. Calculate the by using Expressions (15) and (22). Thereafter, we have
= 0.97142 = 0.94070
= 0.97142 = 0.93958
= 0.94070 = 0.93958
Step 2. Calculate the power weighting vector through Expression (19). Thereafter, we have
=+= 1.91212
=+= 1.91100
=+= 1.88028
Step 3. Calculate the comprehensive value using the SVNPHA operator (suppose p = q = 1). Thereafter, we can haveandandThus, we can obtain the comprehensive value . Theorem 3. (Idempotency). Let be a collection of SVNNs and . Hence, Proof.
thereby completing the proof of Theorem 3. □
Theorem 4. (Commutativity). Let be any permutation of , then
Proof. Since
be any permutation of
, then
thereby completing the proof of Theorem 4.□
Theorem 5. (Boundedness). Let be a collection of SVNNs, , and . Hence, Proof. By the comparison method in Definition 3, we have
, then based on the theorem 2 and 3, we have
Similarly, we can obtain
thereby completing the proof of Theorem 5.□
In the follow, we can discuss some special cases about operator.
(1) If
, then the Expression (21) operator is reduced to the single-valued neutrosophic power generalized linear descending weight operator as follows:
(2) If
then the Expression (21) operator is reduced to the single-valued neutrosophic power generalized linear ascending weight operator as follows:
(3) If
, then the Expression (21) operator is reduced to the single-valued neutrosophic power basic Heronian operator as follows:
(4) If
, then the operator of Equation (21) is reduced to the single-valued neutrosophic number power line Heronian operator as follows:
In the SVNPHA operators, we only take into account the power weight vector and interrelationship among SVNNs but not the weight of every SVNN. However, in many realistic decision-making, the weights of attributes are also an important parameter. Thus, we propose the single-valued neutrosophic weight PHA (SVNWPHA) operator as follows.
Definition 9. Let be a group of SVNNs and , is the weight vector of , satisfying and . If:where and . we shall denote as the support degree for from . satisfies the following three properties: (1) ;
(2) ;
(3) , if in which is the distance between SVNNs and .
Then, is called the single-valued neutrosophic weight power Heronian aggregation (SVNPHA) operator.
Theorem 6. Let and be a collection of SVNNs, , satisfying and . Then, the result aggregated from SVNWPHA is still a SVNNs, and even As with the proof of Theorem 2, it is omitted from this paper.
Obviously, when , the SVNWPHA operator is reduced to the SVNPHA operator.
Similar to the above SVNPHA operator, the SVNWPHA operator also has the same properties.
Theorem 7. (Idempotency). Let be a collection of SVNNs, , satisfying and , and . Hence, Theorem 8. (Commutativity). Let be any permutation of , , satisfying and . Hence,.
Theorem 9. (Boundedness). Let be a collection of SVNNs , satisfying and , , and . Hence, 3.2. Single Valued Neutrosophic Geometric Power Heronian Aggregation Operators
Based on the PA operator [
36] and geometric mean [
49], Xu [
37] further defined a power geometric (PG) operator:
Definition 10. [
37].
Let , and be a collection of nonnegative numbers. if:where . We denote as the support degree for from . satisfies the following axioms:(1) ;
(2)
(3) , if
Definition 11. [
39].
Let , and be a collection of nonnegative numbers. If:Then is called the geometric Heronian mean (GHM) operator.
Next, we shall develop the SVNGPHA and SVNWGPHA operators based on the operation laws of SVNNs.
Definition 12. Let , and be a collection of single-valued neutrosophic numbers. if:where . we shall denote as the support degree for from . satisfies the following three properties: (1) ;
(2)
(3) , if in which is the distance between SVNNs and
Then, is called the single-valued neutrosophic geometric power Heronian aggregation (SVNGPHA) operator.
In order to simply this expression X. We can define
and call
as the power weighting vector with
.
Then, Expression (18) can be shown as follows:
Theorem 10. Let , and be a collection of SVNNs. Then, the result aggregated from SVNGPHA is still a SVNN, and even Similar to the above SVNPHA operator, the SVNGPHA operator also has the same properties.
Theorem 11. (Idempotency). Let be a collection of SVNNs, and , then Theorem 12. (Commutativity). Let be any permutation of , then Theorem 13. (Boundedness). Let be a collection of SVNNs, and , then In the SVNGPHA operators, we only take into account the power weight vector and interrelationship among SVNNs but not the weight of every SVNN. However, in many realistic decision-making, the weights of attributes are also an important parameter. Thus, we propose the single-valued neutrosophic numbers weight geometric power Heronian aggregation (SVNWGPHA) operator as follows.
Definition 13. Let , and be a collection of single-valued neutrosophic numbers, is the weight vector of , satisfying and . If:where and . . We shall denote as the support degree for from . satisfies the following three properties: (1) ;
(2) ;
(3) , if in which is the distance between SVNNs and .
Then, is called the single-valued neutrosophic weight geometric power Heronian aggregation (SVNWGPHA) operator.
Theorem 14. Let , and be a collection of SVNNs, , satisfying and . Then, the result aggregated from SVNWGPHA is still a SVNNs, and even Similar to the proof of Theorem 2 it is omitted in this study.
Obviously, when , the SVNWGPHA operator is reduced to the SVNGPHA operator.
Similar to the SVNWPHA operator, the SVNWGPHA operator has the same properties.
Theorem 15. (Idempotency). Let be a collection of SVNNs, , satisfying and , and , then .
Theorem 16. (Commutativity). Let be any permutation of , satisfying and then Theorem 17. (Boundedness). Let be a collection of SVNNs, , satisfying And and , then