Dual Generalized Nonnegative Normal Neutrosophic Bonferroni Mean Operators and Their Application in Multiple Attribute Decision Making

For multiple attribute decision making, ranking and information aggregation problems are increasingly receiving attention. In a normal neutrosophic number, the ranking method does not satisfy the ranking principle. Moreover, the proposed operators do not take into account the correlation between any aggregation arguments. In order to overcome the deficiencies of the existing ranking method, based on the nonnegative normal neutrosophic number, this paper redefines the score function, the accuracy function, and partial operational laws. Considering the correlation between any aggregation arguments, the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean operator and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean operator were investigated, and their properties are presented. Here, these two operators are applied to deal with a multiple attribute decision making problem. Example results show that the proposed method is effective and superior.


Introduction
During the decision making process, the evaluation information given by decision makers is often incomplete, indeterminate, and inconsistent. To deal with this uncertain information, fuzzy set (FS) was proposed by Zadeh [1] in 1965. On the basis of FS, intuitionistic fuzzy set (IFS) was introduced by Atanassov [2] in 1986. However, IFS can not deal with all types of indeterminate and inconsistent information. Hence, considering the indeterminacy-membership based on IFS, Smarandache [3] developed the neutrosophic set (NS) in 1995. In NS, the truth-membership function, indeterminacy-membership function, and false-membership function are independent of each other. In real life, normal distribution is widely applied. Nevertheless, FS, IFS, and NS do not take the normal distribution into account. Therefore, the normal fuzzy number (NFN) was firstly introduced by Yang and Ko [4] in 1996, and NFN can deal with normal fuzzy information. Based on IFS and NFN, normal intuitionistic fuzzy number (NIFN) was defined by Wang and Li [5] in 2002. Further, combining NFN with NS, Liu [6] proposed the normal neutrosophic number (NNN).
With the development of society, many achievements have been made in the research of multiple attribute decision making (MADM) [7][8][9][10]. Chatterjee et al. [7] proposed a novel hybrid method encompassing factor relationship (FARE) and multi-attributive border approximation area comparison (MABAC) methods. Petković et al. [8] introduced the performance selection index (PSI) method for solving machining MADM problems. Roy et al. [9] developed a rough strength relational-decision making and trial evaluation laboratory model. Badi et al. [10] used a new combinative distance-based assessment (CODAS) method to handle MADM problems. Lee et al. [11] developed fuzzy entropy, Definition 4. [6] Let X be a universe of discourse, with a generic element in X denoted by x, and (a, σ) ∈Ñ; then, an NNN A in X is expressed as: A(x) = x|(a, σ), (T A (x), I A (x), F A (x)) , x ∈ X, where the truth-membership function T A (x) satisfies: T A (x) = T A e −( x−a σ ) 2 , x ∈ X, where the indeterminacy-membership function I A (x) satisfies: x ∈ X, where the falsity-membership function F A (x) satisfies: 1], and 0 ≤ T A (x) + I A (x) + F A (x) ≤ 3. Then, we denoteã = (a, σ), (T, I, F) as an NNN.

Ranking of Nonnegative Normal Neutrosophic Number
Liu and Li (2017) [6] introduced the concept of the score function s 1 and s 2 , and the accuracy function h 1 and h 2 , as shown in Definition 6. We found some deficiencies with the ranking of these functions, as shown below.
Letã k = (a k , σ k ), (T k , I k , F k ) (k = 1, 2) be any two NNNs. When T 1 < T 2 , I 1 > I 2 , F 1 > F 2 , a 1 ≤ a 2 , and σ 1 ≥ σ 2 : (1) If a k > 0 or a k < 0, then ranking results may be completely opposite; (2) When s 1 can determine the ranking result of a k , the influence of σ k is not considered; (3) Neither the score function nor the accuracy function satisfy the monotonicity.
We use the following example to illustrate problems (1) and (3) mentioned above. Example 1. Letã 1 andã 2 be two NNNs, where the specific values are as shown in Table 1. According to we can get its score function and accuracy function from Table 1. For number 1, 28, by −2.1 > −2.28, we haveã 1 >ã 2 for the numerical results, which are shown in Table 2.
In order to avoid the disadvantages of the ranking, we propose the nonnegative normal neutrosophic number (NNNN). Additionally, we take σ into account and introduce the score function and accuracy function of the NNNN. According to the score function and accuracy function, the following propositions are derived. Proposition 1. Letã k = (a k , σ k ), (T k , I k , F k ) (k = 1, 2) be any two NNNNs, then the following conclusions are obtained.
Therefore, we have the following ranking principles.
Definition 11. Letã k = (a k , σ k ), (T k , I k , F k ) (k = 1, 2) be any two NNNNs, then we have the following method for ranking an NNNN: We introduce some operational laws as follows: Definition 12. Letã k = (a k , σ k ), (T k , I k , F k ) (k = 1, 2) be any two NNNNs, then the operational rules are defined as follows: Moreover, the relations of the operational laws are given as below, and these properties are obvious.

DGNNNWBM Operator and DGNNNWGBM Operator
This section extends the DGWBM and DGWGBM to NNNN, and proposes the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean (DGNNNWBM) operator and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean (DGNNNWGBM) operator.
The DGNNNWBM operator can consider the relationship between any elements. Here are some special cases of it. Remark 1. If R = (λ, 0, 0, ..., 0) T , that is, consider the relationship of a single element, then the DGNNNWBM reduces to: which is called a generalized nonnegative normal neutrosophic weighted averaging (GNNNWA) operator.
If R = (s, t, 0, 0, ..., 0) T , that is, consider the relationship between any two elements, then the DGNNNWBM reduces to: which is the nonnegative normal neutrosophic weighted Bonferroni mean(NNNWBM) operator. If R = (s, t, r, 0, 0, ..., 0) T , that is, consider the relationship between any three elements, then the DGNNNWBM reduces to: which is called a generalized nonnegative normal neutrosophic weighted Bonferroni mean (GNNNWBM) operator.
.., n} be a set of NNNNs, then the aggregated result of the DGNNNWBM is also an NNNN and r j , Proof. By Definition 5 and 12, we havẽ r j , Thereafter The following example is used to explain the calculation of the DGNNNWBM operator.
hold for any i, Theorem 3 is still holds.
Likewise, an example is used to explain the calculation of the DGNNNWGBM operator.

A Multiple Attribute Decision-Making Method on the Basis of the DGNNNWBM Operator and DGNNNWGBM Operator
In this section, based on the NNNN, we utilize the DGNNNWBM operator or DGNNNWGBM operator to solve the MADM problem.
Let A = {A 1 , A 2 , ..., A m } be a set of the alternatives, and C = {C 1 , C 2 , ..., C n } be a set of the attributes; the weight vector of the attribute is ω = (ω 1 , ω 2 , ..., ω n ) T , where ω j ∈ [0, 1] and ∑ n j=1 ω j = 1. Let D = (ã ij ) m×n be the decision matrix, andã ij = (a ij , σ ij ), (T ij , I ij , F ij ) be the evaluation value of the alternative A i with respect to attribute C j , denoted by the form of NNNN.
The DGNNNWBM operator or DGNNNWGBM operator can be used to handle the MADM problem, and the steps are shown as follows: Step 1. Standardize the decision matrix.
If all the attributes C i are of the same type, then the attribute values do not need standardization. If there is a different type, the attributes should be converted so they are of the same type. Suppose the decision matrix D = (ã ij ) m×n transforms to the standardized matrixD = (ã ij ) m×n .
According to [6], we have the following standardization method. For the benefit attribute: For the cost attribute: Step 2. Utilize the DGNNNWBM operator Step 3. According to rank principles, which are shown in Definitions 10 and 11, rank the alternatives A 1 , A 2 , ..., A m and choose the best one.

Numerical Example and Comparative Analysis
In this section, the effectiveness of the proposed MADM method is illustrated, demonstrating the effect of different parameter values on the final ranking results. Finally, the advantages of the proposed method are illustrated by comparison.

The Numerical Example
In the following, the application of the proposed method is illustrated by a numerical example.

Example 5.
Patients choose a hospital according to their own needs. There are five alternatives hospitals to choose from: (1) A 1 is a people's hospital; (2) A 2 is a city hospital; (3) A 3 is a second city hospital; (4) A 4 is the first affiliated hospital; and (5) A 5 is the second affiliated hospital. There are four evaluation attributes: (1) C 1 is the hardware and software facilities; (2) C 2 is the physician team; (3) C 3 is the consumption index; and (4) C 4 is the service quality. We know the attributes C 1 ,C 2 , and C 4 are benefit criteria, and C 3 is cost. The weight vector of the attributes is ω = (0.2, 0.4, 0.3, 0.1) T . The final evaluation outcomes are expressed by the NNNN, which is shown in Table 3. Table 3. The nonnegative normal neutrosophic decision matrix D. Step 1. Since C 1 ,C 2 , and C 4 are benefit attributes, we have a 11 = . The normalized decision matrix is shown in Table 4. Table 4. Normalized decision matrixD. Step 2. Calculate the comprehensive evaluation value of each alternative by using the DGNNNWBM (DGNNNWGBM) operator (suppose R = (1, 1, 1, 1) T ), which is shown in Table 5. (There are 256 cases in this example, which are not listed here. MATLAB can be used for the calculations.) Table 5. Utilization of the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean (DGNNNWBM) operator and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean (DGNNNWGBM) operator R = (1,1,1,1). By the ranking principle of Definition 11, we obtain A 2 > A 5 > A 1 > A 4 > A 3 , which is shown in Table 6. The best alternative is A 2 . Table 6. The score of the alternatives.

Influence Analysis
To show the effects on the ranking results by altering the parameters of the DGNNNWBM and DGNNNWGBM operators, according to Definition 10 and 11, we can get the results by using MATLAB, which is shown in Tables 7 and 8. Table 7. Ranking for different parameters of DGNNNWBM.

R S(A 1 ) S(A 2 ) S(A 3 ) S(A 4 ) S(A 5 ) Ranking
(  (16,17,18,19) 31.59 53.67 31.74 28.52 37.37 (20,20,20,20) 31.52 74.73 31.64 28.30 37.27 As shown in Table 7, when the parameter values are small, the ranking of the alternatives may be of little influence. When the parameter values are large, the ordering of A 1 and A 5 changes. However, the best alternative is the same, i.e., A 2 . As shown in Table 8, when the parameter values are small, the ranking of the alternatives may be of little influence, but when the parameter values are large, it has a great impact on the ranking results. Although the ranking changes greatly, the best alternative is still A 2 . In practical applications, we usually take R = (1, 1, .., 1) T , which is not only intuitive but also takes into account the effect of multiple parameters.

Comparison Analysis
In this section, we compare the DGNNNWBM and DGNNNWGBM operators proposed in this paper with the normal neutrosophic weighted Bonferroni mean (NNWBM) operator and normal neutrosophic weighted geometric Bonferroni mean (NNWGBM) operator proposed by Liu P and Li H [6] for dealing with Example 5.1. The results are shown in Tables 8-14, where we take the first two values of the parameter R in the DGNNNWBM and DGNNNWGBM operators as the parameter values p, q in the NNWBM and NNWGBM operators.
According to the result, we conclude the following: (1) From Tables 7 and 9, when p, q take different values and the values are small, the NNWBM operator has three different ranking results, while the DGNNNWBM operator has only one. It shows that the stability of the DGNNNWBM operator is better than that of the NNWBM operator.
(2) From Tables 8 and 10, there is only one ranking result of the NNWGBM operaotr. However,  show that when the parameter values p, q are taken as (10, 10) and (14,15), the result of the NNWBM operator is T = 0, I = 1, F = 1, and the NNWGBM operator result is T = 1, I = 0, F = 0. Regardless of whether the parameters p, q change, the values of a and δ in the NNWGBM operator are invariant. However, in this case, the DGNNNWBM and DGNNNWGBM operators consider more parameters, so they can overcome these problems that arise in the NNWBM and NNWGBM operators.
From this, we know that the NNWBM and NNWGBM operators lack stability and sensitivity. Compared to the NNWBM and NNWGBM operators, the DGNNNWBM and DGNNNWGBM are not only more general, but they are also more flexible. Table 9. Liu and Li's method [6] (ranking for different parameters of the normal neutrosophic weighted Bonferroni mean (NNWBM)).  Table 10. Liu and Li's method [6] (ranking for different parameters of the normal neutrosophic weighted geometric Bonferroni mean (NNWGBM)).

Conclusions
The multiple attribute decision-making method has a wide range of applications in many domains. The nonnegative normal neutrosophic number is more suitable for dealing with uncertain information, and the dual generalized weighted Bonferroni mean operator and dual generalized weighted geometric Bonferroni mean operator take into account the relationship between arbitrary aggregation arguments. Therefore, in this paper, the definition of nonnegative normal neutrosophic number has been proposed. The score function and accuracy function have been developed to overcome the deficiency, i.e., that the original function does not satisfy the ranking principle. Considering the connections between any two or more than two aggregation arguments, the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean operator and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean operator were discussed. Meanwhile, some properties were investigated, such as idempotency, monotonicity, boundedness, and commutativity. Based on the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean operator and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean operator, a method was developed to deal with a multiple attribute decision-making problem with nonnegative normal neutrosophic number. Further, we used the dual generalized nonnegative normal neutrosophic weighted Bonferroni mean and dual generalized nonnegative normal neutrosophic weighted geometric Bonferroni mean operators for aggregative information. Decision making obtain the satisfactory alternative according to actual need and preference by changing the values of R, which makes our proposed multiple attribute decision-making method more flexible and reliable. Further, compared with the method in Liu [6], our method shows that when the relationship between more aggregation arguments are considered, the aggregation result is more stable; when the parameter value is larger, the aggregation result is more sensitive.