Single-Valued Neutrosophic Hybrid Arithmetic and Geometric Aggregation Operators and Their Decision-Making Method

Single-valued neutrosophic numbers (SVNNs) can express incomplete, indeterminate, and inconsistent information in the real world. Then, the common weighted aggregation operators of SVNNs may result in unreasonably aggregated results in some situations. Based on the hybrid weighted arithmetic and geometric aggregation and hybrid ordered weighted arithmetic and geometric aggregation ideas, this paper proposes SVNN hybrid weighted arithmetic and geometric aggregation (SVNNHWAGA) and SVNN hybrid ordered weighted arithmetic and geometric aggregation (SVNNHOWAGA) operators and investigates their rationality and effectiveness by numerical examples. Then, we establish a multiple-attribute decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator under a SVNN environment. Finally, the multiple-attribute decision-making problem about the design schemes of punching machine is presented as a case to show the application and rationality of the proposed decision-making method.

In the aforementioned aggregation operators, however, the interval neutrosophic number weighted arithmetic average and interval neutrosophic number weighted geometric average operators and the interval neutrosophic number ordered weighted arithmetic average and interval neutrosophic number ordered weighted geometric average operators are common aggregation operations in information fusion and decision-making areas.Especially when interval membership values in interval neutrosophic numbers are degenerated to any real numbers between 0 and 1, the interval neutrosophic number weighted arithmetic average (INNWAA) and interval neutrosophic number weighted geometric average (INNWGA) operators [13] can be reduced to the single-valued neutrosophic number weighted arithmetic average (SVNNWAA) and single-valued neutrosophic number weighted geometric average (SVNNWGA) operators, and the interval neutrosophic number ordered weighted arithmetic average (INNOWAA) and interval neutrosophic number ordered weighted geometric average (INNOWGA) operators [18] can be reduced to the single-valued neutrosophic number ordered weighted arithmetic average (SVNNOWAA) and single-valued neutrosophic number ordered weighted geometric average (SVNNOWGA) operators, respectively, as special cases of the existing interval neutrosophic number aggregation operators [13,18].However, they imply the drawbacks of their unreasonably aggregated results in some cases (see Section 2 in detail).For example, in the information aggregations of the SVNNWAA and SVNNOWAA operators, their aggregated results may result in tendency to the maximum value in some cases, while the aggregated results of the SVNNWGA and SVNNOWGA operators may result in tendency to the maximum weight value in some cases.Also, the SVNNWGA and SVNNOWGA operators emphasize personal major points [13,18] and the SVNNWAA and SVNNOWAA operators emphasize group's major points [13,18].Motivated by the hybrid arithmetic and geometric aggregation operators of intuitionistic fuzzy numbers [28], this paper proposes the single-valued neutrosophic number hybrid weighted arithmetic and geometric aggregation (SVNNHWAGA) and single-valued neutrosophic number hybrid ordered weighted arithmetic and geometric aggregation (SVNNHOWAGA) operators to realize more reasonable results in information aggregations of single-valued neutrosophic numbers, and then indicates some properties of the SVNNHWAGA and SVNNHOWAGA operators.Furthermore, a single-valued neutrosophic multiple-attribute decision-making method is established by using the SVNNHWAGA or SVNNHOWAGA operator, and then used for the decision-making problem of design schemes of punching machine under a single-valued neutrosophic environment.The main advantage of this study is that the proposed SVNNHWAGA and SVNNHOWAGA operators can overcome the drawbacks of the existing arithmetic/geometric average aggregation operators of single-valued neutrosophic numbers in some situations and reach the moderate aggregation values.
The remainder of this paper is structured as the following.In Section 2, we introduce some basic concepts and operations of single-valued neutrosophic numbers and investigate some drawbacks of the SVNNWAA, SVNNOWAA, SVNNWGA, and SVNNOWGA operators in some cases.In Section 3, we propose the SVNNHWAGA and SVNNHOWAGA operators and investigate their effectiveness and rationality based on numerical examples.Section 4 develops a single-valued neutrosophic multiple-attribute decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator.
Section 5 presents a multiple-attribute decision-making problem about the design schemes of punching machine as a case to illustrate the application and effectiveness of the presented decision-making method.Section 6 gives some conclusions and further research.

Some Concepts and Operations of Single-Valued Neutrosophic Numbers
Definition 1 [1].Let X be a universal of discourse.A single-valued neutrosophic set N in X is characterized by truth, indeterminacy, and falsity membership functions T N (x), U N (x), and V N (x), respectively, where the values of the three functions T N (x), U N (x), and V N (x) are real numbers between 0 and 1, satisfying T N (x), U N (x), V N (x) ∈ [0,1] and 0 ≤ T N (x) + U N (x) + V N (x) ≤ 3 for x ∈ X.Thus, the single-valued neutrosophic set N is denoted as the following form: For convenience, a basic element <x, T N (x), U N (x), V N (x)> in the single-valued neutrosophic set N is denoted by z = <T, U, V> for short, which is called a single-valued neutrosophic number.
Since a single-valued neutrosophic number is a special case of an interval neutrosophic number, the concepts and operations of interval neutrosophic numbers can be introduced to single-valued neutrosophic numbers.
Let z j = <T j , U j , V j > (j = 1, 2, . . ., n) be a collection of single-valued neutrosophic numbers.Then, the SVNNWAA and SVNNWGA operators [13] are introduced, respectively, as follows: where w j (j = 1, 2, . . ., n) is the weight of z j (j = 1, 2, . . ., n), satisfying w j ∈ [0,1] and ∑ n j=1 w j = 1.When the orders of all the arguments are considered by important positions in the aggregation process of single-valued neutrosophic numbers, the SVNNOWAA and SVNNOWGA operators [18] are introduced, respectively, as follows: where (p(1), Then, the SVNNOWAA and SVNNOWGA operators can reflect the important degrees of the ordered positions of arguments.
Although the above four aggregation operators are common aggregation operations in information fusion and decision-making areas, they imply some drawbacks, which result in tendency to the maximum arguments or weight values of their aggregated values.For example, some drawbacks are shown by the following two numerical examples.
From the aggregated results of the two examples, it is obvious that the aggregated values of the SVNNWAA and SVNNOWAA operators indicate tendency to the maximum value, and then the aggregated values of the SVNNWGA and SVNNOWGA operators indicate tendency to the maximum weight value.Therefore, the SVNNWAA, SVNNOWAA, SVNNWGA and SVNNOWGA operators may result in unreasonably aggregated results of single-valued neutrosophic numbers in some cases.To overcome these drawbacks, we need to improve these aggregation operators and to propose hybrid arithmetic and geometric aggregation operators of single-valued neutrosophic numbers as the extension of hybrid arithmetic and geometric aggregation operators of intuitionistic fuzzy numbers in [28].
Proof.Corresponding to the operational laws of single-valued neutrosophic numbers in Section 2 and the SVNNWAA and SVNNWGA operators, we have the following result: .
Therefore, this completes the proof of Equation ( 8).

Numerical Examples
We still consider the aforementioned two numerical examples to illustrate the effectiveness and rationality of the aggregated values of the SVNNHWAGA and SVNNHOWAGA operators.Generally taking α = 0.5, we apply the SVNNHWAGA and SVNNHOWAGA operators to calculate the two numerical examples in Section 2.
From the above aggregated results of the two numerical examples, the SVNNHWAGA and SVNNHOWAGA operators indicate their moderate values.Obviously, the SVNNHWAGA and SVNNHOWAGA operators demonstrate their effectiveness and rationality in the information aggregations.

Decision-Making Method Using the SVNNHWAGA or SVNNHOWAGA Operator
This section develops a multiple-attribute decision-making method by using the SVNNHWAGA or SVNNHOWAGA operator.
In a multiple-attribute decision-making problem, suppose that Z = {Z 1 , Z 2 , . . ., Z m } is a set of alternatives and A = {A 1 , A 2 , . . ., A n } is a set of attributes.By decision-makers' suitability evaluation for each attribute A j over each alternative Z i , all the evaluation values are expressed by single-valued neutrosophic numbers In the single-valued neutrosophic number z ij , T ij indicates the degree that the alternative Z i is suitable for the attribute A j , U ij indicates the degree that the alternative Z i is unsure/indeterminate for the attribute A j , and V ij indicates the degree that the alternative Z i is unsuitable for the attribute A j .Thus, all the evaluation values can be constructed as a single-valued neutrosophic decision matrix D = (z ij ) m×n .
Hence, we can apply the proposed decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator to the multiple-attribute decision-making problem and give the following decision procedures: Step 1. Suppose that the weight vector of attributes is w = (w 1 , w 2 , . . ., w n ) and satisfies ∑ n j=1 w j = 1 for w j ∈ [0,1].Then, the aggregated value of z i (i = 1, 2, . . ., m) for each alternative Z i (i = 1, 2, . . ., m) is calculated by the following SVNNHWAGA operator: On the other hand, suppose that the ordered important positions of all the arguments are given by the associated weight vector . Thus, the aggregated value of z i (i = 1, 2, . . ., m) for each alternative Z i (i = 1, 2, . . ., m) is calculated by the following SVNNHOWAGA operator: Step 2. By Equation (1) (Equation ( 2) if necessary), we calculate the score values of E(z i ) (accuracy degrees of H(z i ) if necessary) (i = 1, 2, . . ., m).Step 3. Corresponding to the score values (accuracy degrees), we rank all the alternatives in a descending order and determine the best choice based on the alternative with the largest value.Step 4. End.

MADM Problem of the Design Schemes of Punching Machine
In this section, an applied example about the multiple-attribute decision-making (MADM) problem of the design schemes (alternatives) of punching machine is introduced from [29] to illustrate the application and rationality of the proposed decision-making method by the actual case.
If the weight vector of the five attributes is considered as w = (0.25, 0.2, 0.25, 0.15, 0.15) in the multiple-attribute decision-making problem, then the decision steps are presented as follows: Step 1.By Equation (11) (generally take α = 0.5), we calculate the aggregated values of z i (i = 1, 2, 3, 4) for each alternative Z i (i = 1, 2, 3, 4) as the following results: According to E(z 4 ) > E(z 1 ) > E(z 3 ) > E(z 2 ), the ranking of the four design schemes is Z 4 Z 1 Z 3 Z 2 .So, the best design scheme is Z 4 .These results are the same as in [29].
Although the above two ranking orders show little difference, the best scheme Z 4 is identical.
For comparative convenience, all the results of the proposed decision-making approach and the related decision-making methods based on the SVNNWAA, SVNNWGA, SVNNOWAA, and SVNNOWGA operators are summarized in Table 2.
However, the presented decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator demonstrates its suitability and effectiveness in some decision-making situations since it can overcome the drawbacks of the unreasonably aggregated results in some cases (as mentioned in Section 2).Since some value of α is specified by the preference and actual requirements of decision makers, the decision-making method proposed in this study appears to be more flexible and more reasonable than the decision-making method based on one of the SVNNWAA, SVNNOWAA, SVNNWGA, and SVNNOWGA operators.

Conclusions
This paper presented the SVNNHWAGA and SVNNHOWAGA operators to overcome some drawbacks of the SVNNWAA, SVNNOWAA, SVNNWGA and SVNNOWGA operators to aggregate single-valued neutrosophic numbers in some cases, and then investigated their some properties and rationality.Furthermore, we established a multiple-attribute decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator.Finally, a multiple-attribute decision-making problem about design schemes of punching machine is presented as a case to show the application and rationality of the proposed decision-making method.However, the proposed decision-making method based on the SVNNHWAGA or SVNNHOWAGA operator provides an effective and reasonable way for multiple-attribute decision-making problems since the SVNNWAA, SVNNOWAA, SVNNWGA and SVNNOWGA operators are special cases of the SVNNHWAGA and SVNNHOWAGA operators under a single-valued neutrosophic environment.Similarly, this study will be also further extended to interval neutrosophic sets and applications, like group decision making, pattern recognition, fault diagnosis, and medical diagnosis, and so on.