Multiple attribute group decision-making method based on neutrosophic number generalized hybrid weighted averaging operator

Neutrosophic number is an important tool which is used to express indeterminate evaluation information. The purpose of the paper is to propose some aggregation operators based on neutrosophic number, which are used to handle multiple attribute group decision-making problems. Firstly, we introduce the definition, the properties and the operational laws of the neutrosophic numbers, and the possibility degree function is briefly introduced. Then, some neutrosophic number operators are proposed, such as the neutrosophic number weighted arithmetic averaging operator, the neutrosophic number ordered weighted arithmetic averaging operator, the neutrosophic number hybrid weighted arithmetic averaging operator, the neutrosophic number weighted geometric averaging operator, the neutrosophic number ordered weighted geometric averaging operator, the neutrosophic number hybrid weighted geometric averaging operator, the neutrosophic number generalized weighted averaging operator, the neutrosophic number generalized ordered weighted averaging operator, the neutrosophic number generalized hybrid weighted averaging (NNGHWA) operator. Furthermore, some properties of these operators are discussed. Moreover, a multiple attribute group decision-making method based on the NNGHWA operator is proposed. Finally, an illustrative example is proposed to demonstrate the practicality and effectiveness of the method.


Introduction
Multiple attribute group decision making (MAGDM) is an important branch of decision theory which has been widely applied in many fields. Because of the fuzziness of human thinking and the complexity of objective things, the attribute values expressed by the crisp numbers have difficulty in conveying people's thinking about objective things. Zadeh [1] firstly proposed the fuzzy set (FS) to deal with the fuzzy information. Because the fuzzy set only considered the membership degree and did not take the non-membership degree into account, Atanassov [2] further proposed the intuitionistic fuzzy set (IFS) which was used to overcome the shortcoming of the FS. In other words, the intuitionistic fuzzy set (IFS) consisted of membership degree and non-membership degree. Similar to the FS, IFS paid more attention to the membership degree and non-membership degree and did not consider the indeterminacy-membership degree. On the basis of the intuitionistic fuzzy set, Smarandache [3] further proposed the neutrosophic numbers (NNs), which can be divided into two Manuscript Click here to download Manuscript: neutrosophic numbers -3.pdf 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 parts: determinate part and indeterminate part. So the neutrosophic number (NN) was more practical to handle indeterminate information in real situations. Therefore, the neutrosophic number (NN) can be represented as the function bI a N   in which a is the determinate part and bI is the indeterminate part. Obviously, the indeterminate part related to the neutrosophic number (NN) is fewer, the information conveyed by NN is better. So, the worst scenario is bI N  , where the indeterminate part reach the maximum. Conversely, the best case is a N  where there is not indeterminacy related the neutrosophic number. Thus, it is more suitable to handle the indeterminate information in decision making problems. To this day, using neutrosophic numbers to handle indeterminate problems has made little progress in the fields of scientific and engineering techniques. Therefore, it is necessary to propose a new method based on the neutrosophic numbers to handle group decision making problems.
The information aggregation operators have attracted more and more attentions, and they have become a hot research topic. A variety of operators have been proposed to aggregate evaluation information in various environments [4][5][6][7][9][10][11][12][13] such as the arithmetic aggregation operator, the geometric aggregation operator and the generalized aggregation operator. Yager [8] firstly proposed the ordered weighted averaging (OWA) operator which was widely used in decision field. The OWA operator can weight the inputs according to the ranking position of all inputs. Many extension of the OWA operator have been proposed, Such as uncertain aggregation operators [12,14,15], the induced aggregation operators [16,17], the linguistic aggregation operators [18,19], the uncertain linguistic aggregation operators [7], the fuzzy aggregation operators [5,20], the fuzzy linguistic aggregation operators [21], the induced linguistic aggregation operators [22], the induced uncertain linguistic aggregation operators [23,24], the fuzzy induced aggregation operators [25] and the intuitionistic fuzzy aggregation operators [26]. Based on the operators mentioned above, Xu and Chen [27] proposed some interval-valued intuitionistic fuzzy arithmetic aggregation (IVIFAA) operators, such as the interval-valued intuitionistic fuzzy weighted aggregation(IVIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted aggregation (IVIFOWA) operator, and the interval-valued intuitionistic fuzzy hybrid aggregation (IVIFHA) operator. Zhao [28] proposed the generalized intuitionistic fuzzy weighted (GIFWA) operator, the generalized intuitionistic fuzzy ordered weighted (GIFOWA) operator, and the generalized intuitionistic fuzzy hybrid (GIFHA) operator.
To this day, there are not the researches on the combination between neutrosophic numbers and generalized aggregation operator. Thus, it is essential to do the research based on neutrosophic numbers aggregation operators. In this paper, we propose a new method, the generalized hybrid weighted averaging operator based on neutrosophic numbers, to handle multiple attribute group decision making problems. The new method not only can handle the indeterminacy of evaluation information but also can consider the relationship between the attributes.
The remainder of this paper is shown as follows. In section 2, we briefly introduce the basic concepts and the operational rules and the characteristics of NNs. In section 3, some aggregation operators based on neutrosophic numbers and these properties are proposed, such as the neutrosophic number weighted arithmetic averaging (NNWAA) operator, the neutrosophic number ordered weighted averaging (NNOWA) operator, the neutrosophic number hybrid weighted averaging (NNHWA) operator, the neutrosophic number weighted geometric averaging (NNWGA) operator, the neutrosophic number ordered weighted geometric averaging (NNOWGA) operator, the neutrosophic number hybrid weighted geometric averaging (NNHWGA) operator, the neutrosophic number generalized weighted averaging (NNGWA)operator, the neutrosophic number generalized ordered weighted averaging (NNGOWA) operator, the neutrosophic number generalized hybrid weighted 3 averaging (NNGHWA) operator. In section 4, we briefly introduce the procedure of multiple attribute group decision making method based on neutrosophic number generalized hybrid weighted averaging (NNGHWA) operator. In section 5, we give a numerical example to demonstrate the effective of the new proposed method. be two neutrosophic numbers, then the operational laws are defined as follows.
(3) for the left of the formula (11) for the right of the formula (11) So, we can get which completes the proof of the formula (11).
So, we can proof the formula (12) is right. (5) for the left of the formula (13) for the right of the formula (13) So, we can proof the formula (13) is right.
So, we can proof the formula (14) is right.

Definition 3[32-33]. Let
, where R is all real numbers, the neutrosophic number i N is equivalent to , then the possibility degree is

Neutrosophic Number Aggregation Operators
A neutrosophic number includes two parts, determinate part a and indeterminate part bI .
Therefore, the neutrosophic number has an advantage in expressing indeterminate and incomplete information in real decision making. On the basis of neutrosophic numbers, it is necessary to propose some aggregation operators and apply them to the MAGDM problems in which the attribute values take the form of NNs. Here, some neutrosophic number aggregation operators are proposed firstly.

The neutrosophic number hybrid weight arithmetic averaging operator
be a set of neutrosophic numbers (NNs), and NNWAA : Then NNWAA is called neutrosophic number weighted arithmetic averaging operator.
Specially, when 1 1 1 , ,..., n n n         , the NNWAA operator will degenerate into neutrosophic number arithmetic averaging (NNAA) operator: be a set of NNs, and ) ,..., , ( Then the result obtained by Eq. (17) is still an NN and The Eq. (19) can be proved by Mathematical induction on n as follows:
According to (i) and (ii), we can get when the Eq. (19) is right for all n .
which completes the proof of theorem 3.
be a set of NNs, and NNOWAA : NNS n → NNS. If is the weight vector correlative with the NNOWAA operator . Then NNOWAA operator is called neutrosophic number ordered weighted arithmetic averaging operator.
. Then the result obtained using Eq.

(20) is still an NN and
The proof is similar with theorem 2, it is omitted here. Similar to Theorems 3-5, it is easy to prove the NNOWAA operator has the following 7-9 properties.

The neutrosophic number hybrid weighted geometric averaging operator
be a set of NNs, and ) ,..., , ( The proof of this theorem is similar with theorem 2, it's omitted here.
The proof of this theorem is similar with theorem 2, it's omitted here. can be expressed as The proof is similar with the theorem 2, it is omitted here.
The proof is similar with the theorem 2, it is omitted here.
Obviously, there are some properties for the NNGWA operator as follows.
.Then the result obtained using Eq.
(33) is still an NN and The proof is similar with the theorem 2, it is omitted here.
Obviously, there are some properties for the NNGOWA operator as follows.
So, the NNGOWA operator is reduced to the NNOWGA operator.
be a collection of NNs, then the result obtained using Eq.
(35) can be expressed as The proof is similar with the theorem 2, it is omitted here.
It is easy to prove that when , the NNGHWA operator reduce to the NNGWA operator.
Obviously, there are some properties for the NNGHWA operator as follows.
So, the NNGHWA operator is reduced to the NNHWGA operator.
So, the NNGHWA operator is reduced to the NNHWAA operator. Therefore, the NNHWGA operator and the NNHWAA operator are two particular cases of the NNGHWA operator, and the NNGHWA operator is the generalized form of the NNHWGA operator and NNHWAA operator.

Multiple Attribute Group decision-making method based on Neutrosophic Number Generalized Aggregation Operator
As we all known, the objective things are complex in real decision making, so it is difficult to express people's judgments to some objective things by the crisp numbers. The neutrosophic number is a more suitable and effective tool which is used to express the indeterminate information in decision making problems. The decision makers can evaluate the alternatives with respect to every attribute and give the final evaluation results by the neutrosophic number. Therefore, we show a method for processing group decision making problems with neutrosophic numbers, including a de-neutrosophication process and a possibility degree ranking method for neutrosophic numbers.
In a multiple attribute group decision making problem with neutrosophic numbers, let Then, the steps of the decision making method are described as follows: Step 1: Utilized the NNGHWA operator ) ,..., , ( of each decision maker.
Step 2: Utilized the NNGHWA operator ) ,..., , ( to derive the collective overall values ) ,..., Step 3: Calculate the possibility degree ) ( can be given by the Eq.(16) So, the matrix of possibility degrees is structured as

A numerical example
In this section, we give a numerical example to demonstrate the multiple attribute group decision making method based on neutrosophic number generalized hybrid weighted averaging operator (which is cited from [34]). An investment company wants to choose a best investment project. There are four possible alternatives : (1) 1 A is a car company; (2) 2 A is a food company; (3) 3 A is a computer company; (4) 4 A is an arms company. The investment company makes a choice according to the following three attributes: (1) 1 C is the risk factor; (2) 2 C is the growth factor; (3) 3 C is the environmental factor.

The influence of the parameter  and the indeterminate range for I on the ordering of the alternatives
We use the values of parameter  to express the mentality of the decision makers. The bigger  is, the more optimistic decision makers are. In this part, in order to verify the influence of the parameter  on decision making results, the different values  are used to compute the ordering results.

Conclusions
In this paper, we propose a new multiple attribute group decision making method based on neutrosophic number generalized hybrid weighted averaging (NNGHWA) operator, which is a widely practical tool used to handle indeterminate evaluation information in decision making problems.