The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making

Neutrosophic number (NN) is a useful tool which is used to overcome the difficulty of describing indeterminate evaluation information. The purpose of the study is to propose some power aggregation operators based on neutrosophic number which is used to deal with multiple attributes group decision making problems more effectively. Firstly, the basic concepts and the operational rules and the characteristics of NNs are introduced. Then, some aggregation operators based on neutrosophic numbers are developed, included the neutrosophic number weighted power averaging (NNWPA) operator, the neutrosophic number weighted geometric power averaging (NNWGPA) operator, the generalized neutrosophic number weighted power averaging (GNNWPA) operator. At the same time, the properties of above operators are studied such as idempotency, monotonicity, boundedness and so on. Then, the generalized neutrosophic number weighted power averaging (GNNWPA) operator is applied to solve multiple attribute group decision making (MAGDM) problems. Afterwards, a numerical example is given to demonstrate the effective of the new developed method, and some comparison are conducted to verify the influence of different parameters or to reveal the difference with another method. In the end, the main conclusion of this paper is summarized.


Introduction
In real decision making, since the fuzziness and complexity of decision making problems, sometimes it is different to express the people's judgments by crisp numbers in conveying their opinions thoroughly.Zadeh [1] innovatively proposed the fuzzy set (FS) to cope with the fuzzy information.Since the fuzzy set has only the membership degree and has not the non-membership degree, Atanassov [2] made an improvement to overcome this shortcoming, and proposed the intuitionistic fuzzy set (IFS) which is made up with membership degree and non-membership degree.However IFS did not consider the indeterminacymembership degree.Further, Smarandache [3] proposed the neutrosophic set (NS) which added the independent indeterminacy-membership function to the IFS.Obviously, NS is easier to express uncertain information and there are some research results for NS [4][5][6][7][8][9][10][11][12][13][14].In addition, Smarandache [3,15] further proposed the neutrosophic numbers (NNs), and it can be divided into determinate part and indeterminate part.The neutrosophic number (NN) is in the form of N ¼ a þ bI.As we can see that a is the determinate part and bI represents the indeterminate part.Obviously, about the indeterminate part, the fewer it is, the better it is.So, the worst scenario is N = bI.Conversely, the best case is N = a.Ye [16] developed a de-neutrosophication method and a ranking method for neutrosophic numbers based on the possibility degree.To this day, there is the little progress to cope with indeterminate problems by neutrosophic numbers in fields of scientific and engineering techniques.Therefore, it is necessary to propose a new method based on neutrosophic numbers (NNs) to handle group decision making problems.
The information aggregation operators have a wide range of applications in the MAGDM fields, and a lot of them are proposed, such as uncertain aggregation operators [17,18], the induced aggregation operators [19,20], the linguistic aggregation operators or the uncertain linguistic aggregation operators [21][22][23], and the intuitionistic fuzzy or interval-valued intuitionistic fuzzy aggregation operators [24][25][26].However, these operators didn't consider the relationship between the attributes.Yager [27] developed a power average (PA) operator which can consider the relationship between the attributes, then a large amount of operators based on PA have been developed to aggregate evaluation information in order to adapt to various environments.Zhou et al. [28] proposed some generalized power aggregation operators; Xu and Wang [29], Zhou and Chen [30] proposed some power aggregation operators for linguistic information; Peng et al. [31] developed multivalued neutrosophic power aggregation operators; Liu and Yu [32] proposed 2-dimension uncertain linguistic power generalized aggregation operator; Liu and Wang [33] developed intuitionistic linguistic power generalized aggregation operators; Liu and Liu [34] proposed intuitionistic trapezoidal fuzzy power generalized aggregation operator; Liu and Shi [35] proposed intuitionistic uncertain linguistic powered einstein aggregation operators; Zhang et al. [36] proposed intuitionistic fuzzy frank power aggregation operators; Li et al. [37] developed trapezoidal fuzzy two-dimension linguistic power generalized aggregation operators; Liu and Teng [38] proposed normal neutrosophic generalized weighted power averaging operator; Liu and Tang [39] developed some power generalized aggregation operators.
To this day, there is not the research on the combination the neutrosophic numbers with power aggregation operator.Thus, it is very necessary to do the research based on neutrosophic numbers aggregation operators.In this study, we will propose the generalized hybrid weighted power averaging operator under neutrosophic numbers environment, and then propose a new method for the multiple attribute group decision problems, which has two advantages, one is that it can cope with the indeterminacy of evaluation information precisely; another is that it can take the relationship between the attributes into consideration.
This paper is written as follows: The Sect. 2 is about basic concepts, the operational rules and the characteristics of NNs.In Sect.3, some aggregation operators based on neutrosophic numbers are developed, such as the neutrosophic number weighted power averaging (NNWPA) operator, the neutrosophic number weighted geometric power averaging (NNWGPA) operator, the generalized neutrosophic number weighted power averaging (GNNWPA) operator, and then their properties are proved.In Sect.4, we propose a MAGDM method based on the GNNWPA operator, and introduce the decision steps.In Sect.5, a numerical example is given to demonstrate the effective of the new developed method.In Sect.6, the conclusion is made.

Basic concepts of neutrosophic numbers and their operators
The concept of neutrosophic number is firstly proposed by Smarandache in neutrosophic probability.It includes two parts: determinate part and indeterminate part.
Definition 1 [15,16] Let I 2 ½b À ; b þ be an indeterminate part, a neutrosophic number N is denoted as: where a and b are both real numbers, and I is the indeterminate part, such that I 2 = I, 0ÁI = 0, and ð Þbe two neutrosophic numbers, then, operational relations of neutrosophic numbers are shown as follows: ð2Þ ð3Þ 0, the operational laws have the following characteristics: ð2Þ Definition 3 [16] Suppose that N i ¼ a i þ b i Á I with I 2 ½b À ; b þ ði ¼ 1; 2; . ..; nÞ is any neutrosophic number for , where R is the set of real numbers.To normalize N i , we get Definition 4 [16] Suppose that N i ¼ a i þ b i Á I with I 2 ½b À ; b þ ði ¼ 1; 2; . ..; nÞ is any neutrosophic number for , where R is the set of real numbers.We can give the distance between N i and N j as follow: which meets the following criteria: Definition 5 [16,36] Let . ..; nÞ be a set of neutrosophic numbers, I 2 ½b À ; b þ ði ¼ 1; 2; . ..; nÞ, a i ; b i , b À ; b þ 2 R, where R is the set of real numbers, the neutrosophic number so the possibility degree is where, P ij !0, P ij þ P ji ¼ 1, and P ii ¼ 0:5.Then, the value of N i ði ¼ 1; 2; . ..; nÞ can be used for ranking order as follows: Therefore, if the value of q i ði ¼ 1; 2; . ..; nÞ is bigger, information that neutrosophic numbers represent is more precise.In consequence, we rank the neutrosophic numbers of q i ði ¼ 1; 2; . ..; nÞ in an ascending order in order to get the best N i ði ¼ 1; 2; . ..; nÞ.

The power aggregation (PA) operator
Definition 6 [27] For real numbers a i ði ¼ 1; 2; . ..; nÞ, the power average operator is defined as PAða 1 ; a 2 ; . ..; a n Þ ¼ where and sup a i ; a j À Á means the degree to which a j supports a i .It satisfies the following rules.
3 Neutrosophic number aggregation operators A neutrosophic number includes two parts: determinate part and indeterminate part.Thus, it is a good tool to express the indeterminate and incomplete information.At the same time, the Power aggregation can take the relationship between the attributes into consideration.
For this reason, we combine them together, and develop some kinds of neutrosophic number aggregation operators.
3.1 The neutrosophic number weighted power averaging operator . ..; nÞ be a set of neutrosophic numbers, then we define NNPA (neutrosophic number powered aggregation) operator as follows: where , and supðN i ; N j Þ means the support for a i from a j , supðN i ; Obviously, it satisfies the following rules: . ..; nÞ be a set of neutrosophic numbers and NNPA: NNS n ?NNS.then, the result aggregated from Definition 7 is still a NN, and even We use Mathematical induction on n to testify the Eq.(32) as follows.
Proof (i) When n = 1, it's clear that the Eq. ( 32) is right.
(ii) Suppose when n = k, the Eq. ( 32) is right, i.e., Then when n = k?1, we have Thus, when n = k?1, the Eq. ( 32) is right too.Accordingly, we can get that the Eq. ( 32) is right for all n.
Theorem 3 If Sup ãk ; ãj À Á ¼ c, then the NNPA operator will reduce to the arithmetic averaging operator of NNs shown as follows.
NNPA N 1 ; N 2 ; . ..; N n ð Þ¼ In the following, we can prove the NNPA operator has some desirable characteristics, such as idempotency, boundedness and commutativity.
which completes the proof of this Theorem 4.
Theorem 5 (Boundedness) Proof Since a min a i a max , b min b i b max , in the case of all i, we can obtain Similarly, we have So, we can get So, Theorem 6 is right.However, the NNPA operator doesn't have the monotonicity.
According to (i) and (ii), we can get that the Eq. ( 34) is right for all n.
½0; 1; k 6 ¼ j, then the weighted power averaging operator of NNs will reduce to the weighted arithmetic averaging operator of NNs(NNWAA) as follows: Similar to the NNPA operator, we can also prove the NNWPA operator has the idempotency, Boundedness and commutativity, it is omitted here.

The neutrosophic number weighted geometric
power averaging operator Definition 9 Let N i ðN i ¼ a i þ b i iÞð1; 2; . ..; nÞ be a set of NNs, and NNGPA:NNS n ?NNS.The neutrosophic number geometric power averaging operator is defined as: . Obviously, the NNGPA operator is a nonlinear weighted-geometric aggregation operator.
Similarly, the NNGPA operator has the characteristics, such as idempotency, boundedness and commutativity.
Theorem 12 Let N i ðN i ¼ a i þ b i iÞð1; 2; . ..; nÞ be a set of NNs, and Then the result obtained using Eq. ( 36) is still a NN and NNWGPAðN 1 ; N 2 ; . ..; The proof process is similar to Theorem 2, so we can omit it here.Let then the equation turns into: Similarly, the NNWGPA operator has the characteristics, such as commutativity, idempotency, and boundedness.
Similarly, the GNNPA operator also has the commutativity, idempotency and boundedness.
Theorem 13 Let N i ðN i ¼ a i þ b i iÞð1; 2; . ..; nÞ be a set of NNs, and k 2 ð0; þ1Þ.Then the result obtained by Eq. ( 41) is still an NN and GNNWPAðN 1 ; N 2 ; . ..; The proof is similar to the Theorem 2, it is omitted here.
Obviously, there are some properties for the GNNWPA operator as follows. (1) When k !0, So, the GNNWPA operator is reduced to the NNWGPA operator. ( So, the GNNWPA operator is reduced to the NNWPA operator.
Similarly, the GNNWPA operator has the characteristics, such as commutativity, idempotency, and boundedness.
Step 5 Calculate the possibility degree P ij ¼ PðN i !N j Þ, we have Step 6 Calculate the values q i ði ¼ 1; 2; . ..; mÞ for ranking the orders, we have Step 7 Rank the values of q i ði ¼ 1; 2; . ..; mÞ in descending order according, and then the best alternative is obtained.
We use the generalized neutrosophic number weighted power averaging operator to deal with multiple attribute group decision making problems.An investment company wants to choose a best investment project from four possible alternatives: (1) A 1 is a car company; (2) A 2 is a food company; (3) A 3 is a computer company; (4) A 4 is an arms company.There are three attributes that the investment company wants to take into consideration: (1) C 1 is the risk factor; (2) C 2 is the growth factor; (3) C 3 is the environmental factor.The weighting vector of the attributes is x ¼ ð0:35; 0:25; 0:4Þ.The company invites three experts D 1 ; D 2 ; D 3 f gto evaluate the four alternatives.The expert weight vector is w ¼ ð0:37; 0:33; 0:3Þ.The kth ðk ¼ 1; 2; 3Þ expert evaluates these four potential alternatives in terms of these three attributes by the form of neutrosophic number ð1; 2; 3Þ i ¼ ð1; 2; 3; 4Þj ¼ ð1; 2; 3Þ and the evaluation values are shown in Tables 1, 2, 3.
Then we can make the best alternative for this investment.
Since q 2 1 q 4 1 q 1 1 q 3 , the ranking order of the four alternatives is A 2 1 A 4 1 A 1 1 A 3 .So the best choice is A 2 .

The influence of the parameter k
and the indeterminate range I on the ordering of the alternatives Different parameter values k are used to express different level of the mentality of decision makers, because the bigger k is, more optimistic decision makers are.In this sub-section, we will check the influences of different parameter k on decision making results.The ranking results are shown in Table 10 (suppose I 2 ½0:02; 0:04).
From Table 10, we can get the different values k may lead to different sequence in GNNWPA operator.
(1) When 0\k\1, the order of the alternatives is A 2 1 A 4 1 A 1 1 A 3 , and the best choice is A 2 .From Table 11, we can get the different values I may lead to different sequence in GNNWPA operator.
(1) When I = 0, the order of the alternatives is ) When I 2 ½0; 0:2, the order of the alternatives is ) When I 2 ½0; 0:4; I 2 ½0; 0:8, the order of the alternatives is A 2 1 A 4 1 A 1 1 A 3 , so the best choice is A 2 .(4) When I 2 0; 1 ½ , the order of the alternatives is A 2 1 A 4 1 A 3 1 A 1 and the best alternative is A 2 .
In order to demonstrate the effectiveness of the new method in this paper, we compare the ordering results of the new method with the method proposed by Ye [16].From the Tables 11 and 12, we can find that the two methods produce different ranking results.When I 2 ½0; 0:4, the ranking is the same, the other is not.The reason can be analyzed as follows.
The method proposed by Ye [16] is based on de-neutrosophication process, it doesn't realize the importance of the rules of powering operation.The new method proposed in this paper is based on the neutrosophic number generalized weighted power averaging operators which considered the effects of unreasonable attributes.Even the value I is same, when we change the value k, the result is different.The example identifies the validity of the proposed method, and it provides the more general and flexible features as I and k are assigned different values.

Conclusions
In this paper, we firstly use neutrosophic numbers to express uncertain or inaccurate evaluation information.
Then we propose generalized neutrosophic number weighted power averaging (GNNWPA) operator as a new method to deal with MAGDM problems, which can take the relationship between the decision arguments and the mentality of the decision makers into consideration.Since the decision makers have their interest and the actual need, they can assign the different value k, which makes the result more flexible and reliable.Finally, we use the possibility degree ranking method to choose the best choice.Afterward, we give a numerical example to reveal the practicability of the new method.Especially, we use the different values k and different indeterminate ranges I to analyze the effectiveness.The significance of the paper is that we combine neutrosophic number with power aggregation operators to cope with multiple attribute group decision making problems.For further research, we will extend GPA operator to refined neutrosophic numbers [40] or n-valued refined numbers [41], or apply them to solve some actual problems, such as fuzzy reasoning [42], fuzzy classifier [43,44], big data applications [45], and so on.

Table 4
The evaluation values of four alternatives with respect to the three attributes by experts D 1

Table 5
The evaluation values of four alternatives with respect to the three attributes by experts D 2

Table 6
The evaluation values of four alternatives with respect to the three attributes by experts D 3

Table 7
Results from calculating dðN1

Table 8
Results from calculating dðN 2

Table 9
Results from calculating dðN3 When 1 k 2, the order of the alternatives isA 2 1 A 1 1 A 4 1 A 3 ,and the best choice is A 2 .(3)When2:5k 30, the order of the alternatives isA 2 1 A 4 1 A 3 1 A 1 ,and the best choice is A 2 .Similarly, we can check the influences of different parameter I on decision making results, the d the ordering results are shown in Table 11 (suppose k ¼ 1).

Table 10
Ordering of the alternatives by utilizing the different k in GNNWPA operator

Table 11
Ordering of the alternatives by different indeterminate ranges for I in NNGWPA operator