Neutrosophic Cubic Power Muirhead Mean Operators with Uncertain Data for Multi-Attribute Decision-Making

The neutrosophic cubic set (NCS) is a hybrid structure, which consists of interval neutrosophic sets (INS) (associated with the undetermined part of information associated with entropy) and single-valued neutrosophic set (SVNS) (associated with the determined part of information). NCS is a better tool to handle complex decision-making (DM) problems with INS and SVNS. The main purpose of this article is to develop some new aggregation operators for cubic neutrosophic numbers (NCNs), which is a basic member of NCS. Taking the advantages of Muirhead mean (MM) operator and power average (PA) operator, the power Muirhead mean (PMM) operator is developed and is scrutinized under NC information. To manage the problems upstretched, some new NC aggregation operators, such as the NC power Muirhead mean (NCPMM) operator, weighted NC power Muirhead mean (WNCPMM) operator, NC power dual Muirhead mean (NCPMM) operator and weighted NC power dual Muirhead mean (WNCPDMM) operator are proposed and related properties of these proposed aggregation operators are conferred. The important advantage of the developed aggregation operator is that it can remove the effect of awkward data and it considers the interrelationship among aggregated values at the same time. Furthermore, a novel multi-attribute decision-making (MADM) method is established over the proposed new aggregation operators to confer the usefulness of these operators. Finally, a numerical example is given to show the effectiveness of the developed approach.


Introduction
One of the drawbacks of real MADM problems is expressing attribute values in fuzzy and indeterminate DM environments.Fuzzy sets (FSs) developed by Zadeh [1] emerged as a tool for describing and communicating uncertainties and vagueness.Since its beginning, FS has gained a significant focus from researchers all over the world who studied its practical and theoretical aspects.Several extensions of FSs have been developed, such as interval-valued FS (IVFS) [2], which explained the truth membership degree (TMD) on a closed interval value in the interval [0, 1], and intuitionistic FS (IFS) [3], which explained the TMD and falsity-membership degree (FMD).Therefore, IFS defines fuzziness and uncertainty more comprehensively than FS.However, neither FS nor IFS are capable to handle indeterminate and inconsistent information.For example, when we take a student opinion about the teaching skills of a professor with about 0.6 being the possibility that the teaching skills Symmetry 2018, 10, 444 2 of 23 of the professor are good, 0.5 being the possibility that the teaching skills of the professor are bad and 0.3 is the possibility that he/she may not be sure about the teaching skills of the professor whether bad or good.To handle such type of information, Smarandache [4] added a new component "indeterminacy membership degree" (IMD) to the TMD and FMD, all being independent elements lying in ]0 − , 1 + [.The resulting set is now familiarly known as neutrosophic set (NS).To use NS in practical and engineering problems, some scholars developed simplified forms of NS, such as SVNS [5], INS [6,7], simplified neutrosophic sets [8,9], multi-valued NS [10], Q-neutrosophic soft set [11], complex neutrosophic soft expert set [12] and others.
In the real world, sometimes it is difficult to express the TMD in some fuzzy problems completely by an exact value or interval value.Therefore, Jun et al. [13] developed the concept of cubic set (CS) by combining FS and IVFS.CS defined uncertainty and vagueness by an interval value and a fuzzy value concurrently.In recent years, some researchers established some extended forms of CS.Garg et al. [14,15] combined IFS and interval-valued intuitionistic FS (IVIFS) to form cubic IFS (CIFS), while Ali et al. [16] and Jun et al. [17] combined INS and SVNS to develop the cubic NS (CNS), consisting of internal and external NCSs.Jun et al. [18] further investigated P-union and P-intersection of NCS and discussed their related properties.Since then, various studies to solve MADM problems based on NCSs are developed.Zhang et al. [19] and Ye [20] developed some aggregation operators such as weighted averaging operators and weighted geometric operators on NCSs and applied these to MADM.Shi et al. [21], developed some aggregation operator for NCNs based on Dombi T-norm and T-conorm and applied these to MADM.To solve MADM problems under NC information, various similarity measures are developed for NCSs [22,23].Pramanik et al. [24] introduced the NC-TODIM method to solve multiple-attribute group decision-making (MAGDM) problem.
Aggregation operator (AO) plays a dominant role in DM.Consequently, many scholars proposed different aggregation operators and their generalizations, such as Bonferroni mean (BM) operator [25,26], Heronian mean (HM) operator [27], Muirhead mean (MM) operator [28], Maclaurin symmetric mean (MSM) operator [29,30] and others.Certainly, different AOs have different functions.Some can remove the effect of awkward data given by prejudiced DMs, such as power average (PA) operator [31,32] developed by Yager [31] which can aggregate the input information by giving the weighted vector based on support degree among the input arguments.Some aggregation operators are capable to consider the interrelationship among two or more input arguments such as BM operator, HM operators, MSM operator and MM operator.
Due to the enhanced complexity in real decision-making problems, it is necessary to look over the following questions when selecting the best alternative.Firstly, the values of the attributes provided by the decision makers may be too low or too high, thus giving a negative impact on the final ranking results.The PA operator, however, permits the evaluated values to be mutually supported and enhanced.Therefore, we may use the PA operator to diminish such awful impact by designating distinct weights produced by the support measure.Secondly, the values of attributes are required to be dependent.Hence, the interrelationship among the values of the attributes should be examined.Some advantages of MM operator over BM and HM are discussed by Liu et al. [33,34].Some existing aggregation operator such as the BM and MSM operators are special cases of the MM operator.The MM operator consists of the parameter vector, which enlarges the flexibility in the aggregation process.Recently, Li et al. [35] developed the concept of power Muirhead mean operator under Pythagorean fuzzy environment.From the existing literature, the PA operator and MM operator have not been yet combined to deal with NC information.To handle the issues raised, a few new aggregation operators will be proposed by incorporating both the PA and MM operators.These new aggregation operators are NC power MM operator (NCPMM), weighted NC power MM operator, NC power dual MM operator (NCPDMM) and weighted NC power dual MM (WNCPDMM) operator.Discussions on some basic properties and related cases with respect to the parameter vector will be dealt at length.The advantages of these proposed aggregation operators are to capture the interrelationship among input arguments by the MM operator, and simultaneously eliminate the effect of awkward data.Finally, a novel approach to solve MADM problems based on these proposed aggregation operators will be developed.
The rest of the article is organized as follows.In Section 2, some basic definitions and properties of NCSs, MM and PA operators are recalled.In Section 3, the PA and MM operators in the construction of new operators, namely NCPMM, WNCPMM, NCPDMM and WNCPDMM operators are incorporated followed by discussions on their related properties.In Section 4, a novel method to MADM is established based on the developed aggregation operators.In Section 5, a numerical example is illustrated to show the effectiveness of the proposed method to solve a MADM problem.In Section 6, a comparison with the existing methods is given followed by the conclusion.

Preliminaries
In this part, some basic concepts about SVNSs, INSs, NCSs, PA and MM operators are briefly overviewed.

The NCSs and Their Operations
Definition 1 ([4]).Let Γ be a space of points (objects), with a generic element in Γ denoted by n.A neutrosophic set N in Γ is defined as N = { n; T N (n), I N (n), F N (n) n ∈ Γ} where T N (n), I N (n) and F N (n) are the truth membership function, the indeterminacy membership function and the falsity-membership function respectively, such that T; F; I : Smarandache [4] developed the concept of NS as a generalization of FS, IFS and IVIFS.To apply NS to real and engineering problems easily, its parameters should be specified.Hence, Wang et al. [5] provided the following definition.Definition 2 ([5]).Let Γ be a space of points (objects), with a generic element in Γ denoted by n.A single-valued neutrosophic set S in Γ is defined as: when Γ is continuous, and when Γ is discrete, where T S (n), I S (n) and F S (n) are the truth membership function, the indeterminacy membership function and the falsity-membership function respectively, such that T; F; I : Γ → [0, 1] and 0 ≤ T S (n) + I S (n) + F S (n) ≤ 3.

Definition 3 ([6]
).Let Γ be a space of points (objects), with a generic element in Γ denoted by n.An interval neutrosophic set A in Γ is defined as: when Γ is continuous, and when Γ is discrete, where T A (n), I A (n) and F A (n) are the truth membership function, the indeterminacy membership function and the falsity-membership function respectively.For each element n in Γ, we have Symmetry 2018, 10, 444 Definition 4 ([16,17]).Let Γ be a non-empty set.A neutrosophic cubic set (NCS) in Γ is a pair For simplicity, a basic element {n, T(n), I(n), F(n) , λ T (n), λ I (n), λ F (n) } in a NCS can be expressed by z = ( T, I, F , λ T , λ I , λ F ), which is called neutrosophic cubic number (NCN), where

Power Average (PA) Operator
The PA operator was first introduced by Yager [31] for classical number.The dominant edge of PA operator is its capacity to diminish the inadequate effect of unreasonably too high and too low arguments on the inconclusive results.Definition 7 ([31]).Let g (g = 1, 2, . . ., a) be a group of classical numbers.The PA operator is then represented as follows: where, T( z ) = a ∑ x=1 g =x Supp g , x and Supp( z , x ) is the support degree for g and x .The support degree must satisfy the following axioms: (1) Supp g , x ∈ [0, 1]; (2) Supp g , x = Supp x , g ; (3) If D g , x < D( l , m ), then Supp g , x > Supp( l , m ), where D g , x is the distance measure among g and x .

Muirhead Mean (MM) Operator
The MM operator was first introduced by Muirhead [28] for classical numbers.MM operator has the advantage of considering the interrelationship among all aggregated arguments.Definition 8 ([28]).Let g (g = 1, 2, . . ., a) be a group of classical numbers and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the MM operator is described as: where, S a is the group of permutation of (1, 2, . . . ,a) and θ(g) is any permutation of (1, 2, . . . ,a).Now we can give some special cases with respect to the parameter vector Q of the MM operator, which are shown as follows: (1) If Q = (1, 0, 0, . . . , 0),then the MM operator degenerates to the following form: That is, the MM operator degenerates into arithmetic averaging operator. ( a , then the MM operator degenerates to the following form: That is, the MM operator degenerates into geometric averaging operator.(3) If Q = (1, 1, 0, . . . , 0),then the MM operator degenerates to the following form: Symmetry 2018, 10, 444 That is, the MM operator degenerates into BM operator. ( a−c 0, . . ., 0   , then the MM operator degenerates to the following form: That is, the MM operator degenerates into MSM operator.

Some Power Muirhead Mean Operator for NCNs
In this part, we first give the definitions of PMM operator and propose the concept of power dual Muirhead mean (PDMM) operator.Then, we extended both the aggregation operator to NCN environment.Definition 9 ([35]).Let g (g = 1, 2, . . ., a) be a group of classical numbers and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the PMM operator is explained as, where, T g = a ∑ x=1,x =g Supp g , x and Supp g , x is the support degree for g and x , satisfying the above conditions.Definition 10.Let g (g = 1, 2, . . ., a) be a group of classical numbers and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the PDMM operator is described as, where, T Supp g , x and Supp g , x is the support degree for g and x , satisfying the above conditions.

The Neutrosophic Cubic Power Muirhead Mean (NCPMM) Operator
In this subsection, we extend the PMM operator to neutrosophic cubic environment and discuss some basic properties, and special cases of these developed aggregation operators with respect to the parameter Q. Definition 11.Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.If, then, we call NCPMM Q the neutrosophic cubic power Muirhead mean operator, where S a is the group of all permutation, θ(g) is any permutation of (1, 2, . . . ,a) and T(z x ) = a ∑ x=1,x =g Supp z g , z x , Supp z g , z x is the support degree for z g and z x , satisfying the following axioms: (1) Supp z g , z x ∈ [0, 1]; (2) Supp z g , z x = Supp(z x , z z ); , where D(z g , z x ) is the distance among z g and z x .
To write Equation ( 20) in a simple form, we can specify it as: For suitability, we can call (Θ 1 , Θ 2 , . . . ,Θ a ) T the power weight vector (PMV), such that Θ g ∈ [0, 1]   and From the use of Equation (20), Equation ( 19) can be expressed as: Based on the operational rules given in Definition 3 for NCNs, and Definition 11, we can have the following Theorem 2. Theorem 2. Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the aggregated value obtained by using Equation ( 21) is still an NCN and, Proof.According to the operational laws for NCNs, we have Therefore, Symmetry 2018, 10, 444 Therefore, and Furthermore, Hence, This is the required proof of Theorem 2.
In the above equations, we calculate the PWV Θ, after calculating the support degree Supp z g , z x .First, we determined the Supp z g , z x using where, Therefore, we use the equation to obtain the values of T z g (g = 1, 2, . . ., a).Then using Equation (20) we can get the PWV.
In a similar way we can show that NCPMM Q (z 1 , z 2 , . . . ,z a ) ≤ n.
The NCPMM operator does not have the property of monotonicity.One of the leading advantages of NCPMM is its capacity to represent the interrelationship among NCNs.Furthermore, the NCPMM operator is more flexible in aggregation process due to parameter vector.Now, we discuss some special cases of NCPMM operators by assigning different values to the parameter vector.
z−i 0, 0, . . ., 0   , then the NCPMM operator degenerates into the following form: This is the NC power Maclaurin symmetric mean operator.

Weighted Neutrosophic Cubic Power Muirhead Mean (WNCPMM) Operator
The NCPMM operator does not consider the weight of the aggregated NCNs.In this subsection, we develop the WNCPMM operator, which has the capacity of taking the weights of NCNs.Definition 12. Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.If, then, we W NCPMM Q the weighted neutrosophic cubic power Muirhead mean operator, where Ξ = (Ξ 1 , Ξ 2 , . . . ,Ξ a ) T is the weight vector of z g (g = 1, 2, . . ., a) such that Ξ z ∈ [0, 1], a ∑ z=1 Ξ z = 1, S a is the group of all permutation, θ(z) is any permutation of (1, 2, . . . ,a) and Θ g is power weight vector (PWV) satisfying Supp z g , z x , Supp z g , z x is the support degree for z g and z x , satisfying the following axioms: (1) Supp z g , z x ∈ [0, 1]; (2) Supp z g , z x = Supp z x , z g ; (3) If D(z g , z x ) < D(z u , z v ), then Supp(z g , z x ) > Supp(z u , z v ), where D(z g , z x ) is distance among z g and z x .From Definition 12, we have the following Theorem 5.
Theorem 5. Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the aggregated value obtained by using Equation ( 32) is still an NCN and Proof.Proof of Theorem 5 is same as Theorem 2.

The Neutrosophic Cubic Power Dual Muirhead Mean (NCPDMM) Operator
In this subsection, we develop the NCPDMM operator and discuss some related properties.Definition 13.Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.If, then, we call NCPDMM Q the neutrosophic cubic power dual Muirhead mean operator, where S a is the group of all permutation, θ(g) is any permutation of (1, 2, . . . ,a) and is the support degree for z g and z x , satisfying the following axioms: (1) Supp z g , z x ∈ [0, 1]; (2) Supp z g , z x = Supp z x , z g ; , where D(z g , z x ) is distance among z g and z x .
To write Equation ( 34) in a simple form, we can specify it as: For suitability, we can call (Θ 1 , Θ 2 , . . . ,Θ a ) T the power weight vector (PMV), such that Θ g ∈ [0, 1]   and From, the use of Equation (35), Equation ( 34) can be expressed as, Theorem 6.Let z g (g = 1, 2, . . ., a) be a group of SVNNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the aggregated value obtained by using Equation ( 36) is still an NCN and, Proof.Proof of Theorem 6 is similar to that of Theorem 2.

Case 2. If
a , 1 a , . . .., 1 a , then NCPMM operators degenerate into the following form: This is NC power arithmetic averaging operator.
(43) This is the NC power dual Maclaurin symmetric mean operator.

Weighted Neutrosophic Cubic Power Dual Muirhead Mean (WNCPDMM) Operator
The NCPDMM operator does not consider the weight of the aggregated NCNs.In this subsection, we develop the WNCPDMM operator, which has the capacity of taking the weights of NCNs.Definition 14.Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.If, Symmetry 2018, 10, 444 16 of 23 then, we call W NCPDMM Q the weighted neutrosophic cubic power dual Muirhead mean operator, where Ξ = (Ξ 1 , Ξ 2 , . . . ,Ξ a ) T is the weight vector of z g (g = 1, 2, . . ., a) such that the group of all permutation, θ(g) is any permutation of (1, 2, . . . ,a) and Θ g is PVW satisfying Θ g = (1+T(zg)) Supp z g , z x , and Supp z g , z x is the support degree for z g and z x , satisfying the following axioms: (1) Supp z g , z x ∈ [0, 1]; (2) Supp z g , z x = Supp z x , z g ; , where D(z g , z x ) is distance among z g and z x .
From Definition 14, we have the following Theorem 9.
Theorem 9. Let z g (g = 1, 2, . . ., a) be a group of NCNs and Q = (q 1 , q 2 , . . . ,q a ) ∈ R a be a vector of parameters.Then, the aggregated value obtained by using Equation ( 44) is still an NCN and Proof.Proof of Theorem 9 is similar to that of Theorem 2.

The MADM Approach Based on WNCPMM Operator and WNCPDMM Operator
In this section, we give a novel method to MADM with NCNs, in which the attributes values gain the form of NCNs.For a MADM problem, let the series of alternatives is represented by = { 1 , 2 , . . . ,a }, and the series of attributes is represented by permutation, ( ) g θ is any permutation of ( ) 1,2,..., a and g Θ is PVW satisfying , and ( ) Supp z z is the support degree for g z and , x z satisfying the following axioms: (1) Supp z z Supp z z > where ( , )   g x D z z is distance among g z and x z .
From Definition 14, we have the following Theorem 9. , ,..., be a vector of parameters.Then, the aggregated value obtained by using Equation ( 44) is still an NCN and ( ) .
Proof.Proof of Theorem 9 is similar to that of Theorem 2. □

The MADM Approach Based on WNCPMM Operator and WNCPDMM Operator
In this section, we give a novel method to MADM with NCNs, in which the attributes values gain the form of NCNs.For a MADM problem, let the series of alternatives is represented by and the series of attributes is represented by The weight vector of the attributes is denoted by , ,..., is the assessment values of the alternatives g  on the attribute h  , which is expressed by the form of NCN.Then, the main aim is to rank the alternatives.The following decision steps are to be followed.

= {
Hence, the decision matrix Step 2. Determine the supports ( ..., , 1 Step 5. Use the WNCPMM or WNCPDMM operators ( . Step 6. Determine the score values of the collective NCNs ( ) , usin Step 7. Rank all the alternatives according to their score values, and the sele Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative from [19,21] with NC information.Step 2. Determine the supports ( , where, ( ) , gh g h D δ δ is the distance measure among two NCNs gh δ and gl δ defined in Equation ( 2Step 3. Determine ( ) ..., , 1 Step 5. Use the WNCPMM or WNCPDMM operators .
Step 6. Determine the score values of the collective NCNs ( ) , using Definition 6.
Step 7. Rank all the alternatives according to their score values, and the select the best one Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is emb from [19,21] with NC information.

Example 1.
A passenger wants to travel and select the best vans (alternatives) ( 1,2,3,4) , and ( ) Supp z z is the support degree for g z and , x z satisfying the following axioms: ( , , ; Supp z z Supp z z > where ( , )   g x D z z is distance among g z and x z .
From Definition 14, we have the following Theorem 9.
Theorem 9. Let ( 1,2,..., ) g z g a = be a group of NCNs and , ,..., be a vector of parameters.Then, the aggregated value obtained by using Equation ( 44) is still an NCN and ( ) , , Proof.Proof of Theorem 9 is similar to that of Theorem 2. □

The MADM Approach Based on WNCPMM Operator and WNCPDMM Operator
In this section, we give a novel method to MADM with NCNs, in which the attributes values gain the form of NCNs.For a MADM problem, let the series of alternatives is represented by { } , ,..., is the assessment values of the alternatives g  on the attribute h  , which is expressed by the form of NCN.Then, the main aim is to rank the alternatives.The following decision steps are to be followed.

}. The weight vector of the attributes is denoted by
is the assessment values of the alternatives g on the attribute l h , which is expressed by the form of NCN.Then, the main aim is to rank the alternatives.The following decision steps are to be followed.
Step 1. Standardize the decision matrix.Generally, there are two types of attributes, one is of cost type and the other is of benefit type.We need to convert the cost type of attributes into benefit types of attributes by using the following formula: where, D δ gh , δ gh is the distance measure among two NCNs δ gh and δ gl defined in Equation (25).
Step 7. Rank all the alternatives according to their score values, and the select the best one using Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is embraced from [19,21] with NC information.
Example 1.A passenger wants to travel and select the best vans (alternatives) g (g = 1, 2, 3, 4) among the possible four vans.The customer takes the following four attributes into account to evaluate the possible four alternatives: (1) the facility type and the other is of benefit type.We need to convert the cost type of attributes into benefit types of attributes by using the following formula: , for cost attribute .Step 5. Use the WNCPMM or WNCPDMM operators to calculate the overall NCNs, ( ) , using Definition 6.
Step 7. Rank all the alternatives according to their score values, and the select the best one using Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is embraced from [19,21] with NC information.


The importance degree ; (2) saving rent type and the other is of benefit type.We need to convert the cost type of attributes into benefit types of attributes by using the following formula: Hence, the decision matrix Step 4. Determine the weights related with the NCN ( )   Step 5. Use the WNCPMM or WNCPDMM operators to calculate the overall NCNs, ( ) , using Definition 6.
Step 7. Rank all the alternatives according to their score values, and the select the best one using Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is embraced from [19,21] with NC information.


The importance degree ; (3) comfort type and the other is of benefit type.We need to convert the cost type of attributes into benefit types of attributes by using the following formula: Hence, the decision matrix Step 4. Determine the weights related with the NCN ( )   Step 5. Use the WNCPMM or WNCPDMM operators to calculate the overall NCNs, ( ) , using Definition 6.
Step 7. Rank all the alternatives according to their score values, and the select the best one using Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is embraced from [19,21] with NC information.


The importance degree ; (4) safety Step 1. Standardize the decision matrix.Generally, there are two types of attributes, one is of cost type and the other is of benefit type.We need to convert the cost type of attributes into benefit types of attributes by using the following formula: Hence, the decision matrix Step 4. Determine the weights related with the NCN ( )   Step 5. Use the WNCPMM or WNCPDMM operators , using Definition 6.
Step 7. Rank all the alternatives according to their score values, and the select the best one using Theorem 1.

An Illustrative Example
To show the application of the developed MADM method, an illustrative example is embraced from [19,21] with NC information.Example 1.A passenger wants to travel and select the best vans (alternatives) ( 1,2,3,4)


The importance degree .The importance degree of the attributes is expressed by = (0.5, 0.25, 0.125, 0.125) T .Therefore, the following decision matrix M = z gh 4×4 can be obtained in the form of NCNs shown in Table 1.Then, we apply the WNCPMM operator or WNCPDMM operator to solve the MADM problem.Now, we use the WNCPMM operator for this decision-making problem as follows: Step 3. Use Equation (48), to get ( )( , 1 Step 3. Use Equation (48), to get T δ gh (g, h = 1 to 4).We denote T δ gh by T gh .
Step 7. According to the score values, ranking order of the alternative is 3 > 4 > 1 > 2 .
Hence using Theorem 1, the best alternative is 3 and the worst is 2 .Similarly, by using WNCPDMM operator for this decision-making problem, we will have, the Steps 1 to 4 are similar to that of weighted neutrosophic cubic power Muirhead mean operator.
Step 7. According to the score values, ranking order of the alternative is 3 > 4 > 1 > 2 .
Hence using Theorem 1, the best alternatives is 3 , while the worst is 2 .
From the above obtained results, we can see that by using WNCPMM operator or WNCPDMM operator, the best alternative obtained is 3 , while the worst is 2 .

Effect of the Parameter Q on the Decision Result
In this subsection, different values to the parameter vector and the results obtained from these values are shown in Tables 2 and 3. From Tables 2 and 3, it can be seen that, when the value of the parameter vector Q is (1, 0, 0, 0), that is, when the interrelationship among the attributes is not considered, then according to the score values the best alternative is 4 while the worst is 2 .Similarly, when the value of the parameter vector Q is (1, 1, 0, 0), that is, when WCNPMM operator and WNCPDMM operator degenerate into neutrosophic cubic power Bonferroni mean operator and neutrosophic cubic power geometric Bonferroni mean operator respectively, the alternative is 3 and 4 while the worst both cases is 2 .When the value of the parameter vector Q is (1, 1, 1, 0), the best alternative is 3 and the worst is 2 .When the value of the parameter vector Q is (1, 1, 1, 1), the best alternative is 3 and the worst is 2 .Similarly, for other values of the parameter vector the score values and ranking order vary.Thus, one can select the value of the parameter vector according to the needs of the situations.

Step 3 .
is the distance measure among two NCNs gh δ and gl δ define Determine ( ) gh T δ by,

Step 4 .
Determine the weights related with the NCN ( )

Example 1 .
A passenger wants to travel and select the best vans (alternatives)  possible four vans.The customer takes the following four attributes into account to ev alternatives: (1) the facility 1 ;

Step 4 .
Determine the weights related with the NCN ( ) g g =  amon possible four vans.The customer takes the following four attributes into account to evaluate the possibl alternatives: (1) the facility 1 ; ..., a and g Θ is PVW satisfying weight vector of the attributes is denoted by

Step 3 .Step 4 .
is the distance measure among two NCNs gh δ and gl δ defined in Equation(25).Determine ( ) Determine the weights related with the NCN ( ) four vans.The customer takes the following four attributes into account to evaluate the possible four alternatives: (1) the facility 1 ;

Step 3 .
is the distance measure among two NCNs gh δ and gl δ defined in Equation(25).Determine ( ) four vans.The customer takes the following four attributes into account to evaluate the possible four alternatives: (1) the facility 1 ;

Step 3 .
is the distance measure among two NCNs gh δ and gl δ defined in Equation(25).Determine ( ) four vans.The customer takes the following four attributes into account to evaluate the possible four alternatives: (1) the facility 1 ;

Step 3 .
is the distance measure among two NCNs gh δ and gl δ defined in Equation(25).Determine ( )

Table 1 .
The decision matrix M = CN gh 4×4 .canbeobtained in the form of NCNs shown in Table1.

Table 1 .
The decision matrix