Abstract
In this paper, we extend the Bonferroni mean (BM) operator, generalized Bonferroni mean (GBM) operator, dual generalized Bonferroni mean (DGBM) operator and dual generalized geometric Bonferroni mean (DGGBM) operator with 2-tuple linguistic neutrosophic numbers (2TLNNs) to propose 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (2TLNNWBM) operator, 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (2TLNNWGBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (G2TLNNWBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (G2TLNNWGBM) operator, dual generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (DG2TLNNWBM) operator, and dual generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (DG2TLNNWGBM) operator. Then, the MADM methods are proposed with these operators. In the end, we utilize an applicable example for green supplier selection in green supply chain management to prove the proposed methods.
Keywords:
multiple attribute decision making (MADM); neutrosophic numbers; 2-tuple linguistic neutrosophic numbers set (2TLNNSs); Bonferroni mean (BM) operator; generalized Bonferroni mean (GBM) operator; dual generalized Bonferroni mean (DGBM) operator; dual generalized geometric Bonferroni mean (DGGBM) operator; green supplier selection; green supply chain management 1. Introduction
Zadeh [] introduced a membership function between 0 and 1 instead of traditional crisp value of 0 and 1, and defined the fuzzy set (FS). In order to overcome the insufficiency of FS, Atanassov [] proposed the concept of an intuitionistic fuzzy set (IFS), which is characterized by its membership function and non-membership function between 0 and 1. Furthermore, Atanassov and Gargov [] introduced the concept of an interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by its interval membership function and interval non-membership function in the unit interval [0,1]. Because IFSs and IVIFSs cannot depict indeterminate and inconsistent information, Smarandache [] introduced a neutrosophic set (NS) from a philosophical point of view to express indeterminate and inconsistent information. A NS has more potential power than other modeling mathematical tools, such as fuzzy set [], IFS [], and IVIFS []. But, it is difficult to apply NSs in solving of real life problems. Therefore, Smarandache [] and Wang et al. [,] defined a single valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), which are characterized by a truth-membership, an indeterminacy membership, and a falsity membership. Ye [] introduced a simplified neutrosophic set (SNS), including the concepts of SVNS and INS, which are the extension of IFS and IVIFS. Obviously, SNS is a subclass of NS, while SVNS and INS are subclasses of SNS. Ye [] proposed the correlation and correlation coefficient of single-valued neutrosophic sets (SVNSs) that are based on the extension of the correlation of intuitionistic fuzzy sets and demonstrates that the cosine similarity measure is a special case of the correlation coefficient in SVNS. Broumi and Smarandache [] extended the correlation coefficient to INSs. Biswas et al. [] developed a new approach for multi-attribute group decision-making problems by extending the technique for order preference by similarity to ideal solution to single-valued neutrosophic environment. Liu et al. [] combined Hamacher operations and generalized aggregation operators to NSs, and proposed the generalized neutrosophic number Hamacher weighted averaging (GNNHWA) operator, generalized neutrosophic number Hamacher ordered weighted averaging (GNNHOWA) operator, and generalized neutrosophic number Hamacher hybrid averaging (GNNHHA) operator, and explored some properties of these operators and analyzed some special cases of them. Sahin and Liu [] developed a maximizing deviation method for solving the multiple attribute decision-making problems with the single-valued neutrosophic information or interval neutrosophic information. Ye [] defined the Hamming and Euclidean distances between the interval neutrosophic sets (INSs) and proposed the similarity measures between INSs based on the relationship between similarity measures and distances. Zhang et al. [] defined the operations for INSs and put forward a comparison approach that was based on the related research of interval valued intuitionistic fuzzy sets (IVIFSs) and developed two interval neutrosophic number aggregation operators. Peng et al. [] developed a new outranking approach for multi-criteria decision-making (MCDM) problems in the context of a simplified neutrosophic environment, where the truth-membership degree, indeterminacy-membership degree, and falsity-membership degree for each element are singleton subsets in [0,1] and defined some outranking relations for simplified neutrosophic number (SNNs) based on ELECTRE (ELimination and Choice Expressing REality), and the properties within the outranking relations are further discussed in detail. Zhang et al. [] proposed a novel outranking approach for multi-criteria decision-making (MCDM) problems to address situations where there is a set of numbers in the real unit interval and not just a specific number with a neutrosophic set. Liu and Liu [] proposed 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 studied the properties of above operators are studied, such as idempotency, monotonicity, boundedness, and so on. Peng et al. [] introduced the multi-valued neutrosophic sets (MVNSs), which allow for the truth-membership, indeterminacy- membership, and falsity-membership degree to have a set of crisp values between zero and one, respectively, and the multi-valued neutrosophic power weighted average (MVNPWA) operator and proposed the multi-valued neutrosophic power weighted geometric (MVNPWG) operator. Zhang et al. [] presented a new correlation coefficient measure, which satisfies the requirement of this measure equaling one if and only if two interval neutrosophic sets (INSs) are the same and used the proposed weighted correlation coefficient measure of INSs to solve decision-making problems, which take into account the influence of the evaluations’ uncertainty and both the objective and subjective weights. Chen and Ye [] presented the Dombi operations of single-valued neutrosophic numbers (SVNNs) based on the operations of the Dombi T-norm and T-conorm, and then proposed the single-valued neutrosophic Dombi weighted arithmetic average (SVNDWAA) operator and the single-valued neutrosophic Dombi weighted geometric average (SVNDWGA) operator to deal with the aggregation of SVNNs and investigated their properties. Liu and Wang [] proposed the single-valued neutrosophic normalized weighted Bonferroni mean (SVNNWBM) operator on the basis of Bonferroni mean, the weighted Bonferroni mean (WBM), and the normalized WBM, and developed an approach to solve the multiple attribute decision-making problems with SVNNs that were based on the SVNNWBM operator. Wu et al. [] defined the prioritized weighted average operator and the prioritized weighted geometric operator for simplified neutrosophic numbers (SNNs) and proposed two novel effective cross-entropy measures for SNSs. Li et al. [] proposed the improved generalized weighted Heronian mean (IGWHM) operator and the improved generalized weighted geometric Heronian mean (IGWGHM) operator, the single valued neutrosophic number improved generalized weighted Heronian mean (NNIGWHM) operator, and single valued the neutrosophic number improved generalized weighted geometric Heronian mean (NNIGWGHM) operator for multiple attribute group decision making (MAGDM) problems, in which attribute values take the form of SVNNs. Wang et al. [] combined the generalized weighted BM (GWBM) operator and generalized weighted geometric Bonferroni mean (GWGBM) operator with single valued neutrosophic numbers (SVNNs) to propose the generalized single-valued neutrosophic number weight BM (GSVNNWBM) operator and the generalized single-valued neutrosophic numbers weighted GBM (GSVNNWGBM) operator and developed the MADM methods based on these operators. Wei & Zhang [] utilized power aggregation operators and the Bonferroni mean to develop some single-valued neutrosophic Bonferroni power aggregation operators and single-valued neutrosophic geometric Bonferroni power aggregation operators. Peng & Dai [] initiated a new axiomatic definition of single-valued neutrosophic distance measure and similarity measure, which is expressed by a single-valued neutrosophic number that will reduce the information loss and retain more original information.
Although SVNS theory has been successfully applied in some areas, the SVNS is also characterized by the truth-membership degree, indeterminacy-membership degree, and falsity-membership degree information. However, all of the above approaches are unsuitable to describe the truth-membership degree, indeterminacy-membership degree, and falsity-membership degree information of an element to a set by linguistic variables on the basis of the given linguistic term sets, which can reflect the decision maker’s confidence level when they are making an evaluation. In order to overcome this limit, we shall propose the concept of 2-tuple linguistic neutrosophic numbers set (2TLNNSs) to solve this problem based on the SVNS [,,] and the 2-tuple linguistic information processing model [,,,,,,,,,,,,,,,]. Thus, how to aggregate these 2-tuple linguistic neutrosophic numbers is an interesting topic. To solve this issue, in this paper, we shall develop some 2-tuple linguistic neutrosophic information aggregation operators that are based on the traditional Bonferroni mean (BM) operations [,,,,,,,]. In order to do so, the remainder of this paper is set out as follows. In the next section, we shall propose the concept of 2TLNNSs. In Section 3, we shall propose some Bonferroni mean (BM) operators with 2TLNNs. In Section 4, we shall propose some generalized Bonferroni mean (GBM) operators with 2TLNNs. In Section 5, we shall propose some dual generalized Bonferroni mean (DGBM) operators with 2TLNNs. In Section 6, we shall present a numerical example for green supplier selection in order to illustrate the method that is proposed in this paper. Section 7 concludes the paper with some remarks.
2. Preliminaries
In this section, we shall propose the concept of 2-tuple linguistic neutrosophic number sets (2TLNNSs) to solve this problem based on the SVNSs [,] and 2-tuple linguistic sets (2TLSs) [,].
2.1. 2-Tuple Fuzzy Linguistic Representation Model
Definition 1
([,]). Let be a linguistic term set with odd cardinality. Any label, represents a possible value for a linguistic variable, and can be defined as:
Herrera and Martinez [,] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple, whereis a linguistic label for predefined linguistic term setandis the value of symbolic translation, and
2.2. SVNSs
Let be a space of points (objects) with a generic element in fix set , denoted by . A single-valued neutrosophic sets (SVNSs) in is characterized as following [,,]:
where the truth-membership function , indeterminacy-membership , and falsity-membership function are single subintervals/subsets in the real standard , that is, and . The sum of , , and satisfies the condition . Then, a simplification of is denoted by , which is a SVNS.
For a SVNS , the ordered triple components , are described as a single-valued neutrosophic number (SVNN), and each SVNN can be expressed as , where , and .
2.3. 2TLNNSs
Definition 2.
Assume thatis a 2TLSs with odd cardinality. Ifis defined forwhereexpress independently the truth degree, indeterminacy degree, and falsity degree by 2TLSs, then 2TLNNSs is defined as follows:
where, and
Definition 3.
Letbe a 2TLNN in. Then the score and accuracy functions ofare defined as follows:
Definition 4.
Letandbe two 2TLNNs, then
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- ,
- (5)
- ,
Definition 5.
Letandbe two 2TLNNs, then
- (1)
- (2)
- (3)
- (4)
- .
2.4. BM Operators
Definition 6
([]). Let and be a collection of nonnegative crisp numbers then the Bonferroni mean (BM) is defined as follows:
Thenis called Bonferroni mean (BM) operator.
3. 2TLNNWBM and 2TLNNWGBM Operators
3.1. 2TLNNWBM Operator
To consider the attribute weights, the weighted Bonferroni mean (WBM) is defined, as follows.
Definition 7
([]). Let and be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . The weighted Bonferroni mean (WBM) is defined as follows:
Then we extend WBM to fuse the 2TLNNs and propose 2-tuple linguistic neutrosophic number weighted Bonferroni mean (2TLNNWBM) aggregation operator.
Definition 8.
Letbe a set of 2TLNNs. The 2-tuple linguistic neutrosophic number weighted Bonferroni mean (2TLNNWBM) operator is:
Theorem 1.
Letbe a set of 2TLNNs. The aggregated value by using 2TLNNWBM operators is also a 2TLNN where
Proof.
According to Definition 5, we can obtain
Thus,
Thereafter,
Furthermore,
Therefore,
Hence, (8) is kept. □
Then, we need to prove that (8) is a 2TLNN. We need to prove two conditions, as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Example 1.
Letbe two 2TLNNs,according to (8), we have
Then, we will discuss some properties of 2TLNNWBM operator.
Property 1.
(Idempotency) Ifare equal, then
Proof.
Since , then
□
Property 2.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Proof.
Let and , given that , we can obtain
Thereafter,
Furthermore,
That means . Similarly, we can obtain and .
If and
If and
If and
So, Property 2 is right. □
Property 3.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
From Property 1,
From Property 2,
3.2. 2TLNNWGBM Operator
Similarly to WBM, to consider the attribute weights, the weighted geometric Bonferroni mean (WGBM) is defined, as follows:
Definition 9
([]). Let and be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . If
Then we extend WGBM to fuse the 2TLNNs and propose 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (2TLNNWGBM) operator.
Definition 10.
Letbe a set of 2TLNNs with their weight vector be, thereby satisfyingand. If
Then we calledthe 2-tuple linguistic neutrosophic number weighted geometric BM.
Theorem 2.
Letbe a set of 2TLNNs. The aggregated value by using 2TLNNWGBM operators is also a 2TLNN where
Proof.
From Definition 5, we can obtain,
Thus,
Therefore,
Thereafter,
Furthermore,
Hence, (27) is kept. □
Then, we need to prove that (27) is a 2TLNN. We need to prove two conditions, as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Example 2.
Letbe two 2TLNNs,according to (27), we have
Similar to 2TLNNWBM, the 2TLNNWGBM has the same properties, as follows. The proof are omitted here to save space.
Property 4.
(Idempotency) Ifare equal, then
Property 5.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Property 6.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
4. G2TLNNWBM and G2TLNNWGBM Operators
4.1. G2TLNNWBM Operator
The primary advantage of BM is that it can determine the interrelationship between arguments. However, the traditional BM can only consider the correlations of any two aggregated arguments. Thereafter, Beliakov et al. [] extended the BM and introduced the generalized BM(GBM) operator. Zhu et al. [] introduced the generalized weighted BM(GWBM) operator, as follows.
Definition 11
([]). Let and be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . The generalized weighted Bonferroni mean (GWBM) is defined as follows:
Then we extend GWBM to fuse the 2TLNNs and propose generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (G2TLNNWBM) aggregation operator.
Definition 12.
Letbe a set of 2TLNNs. The generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (G2TLNNWBM) operator is:
Theorem 3.
Letbe a set of 2TLNNs. The aggregated value by using G2TLNNWBM operators is also a 2TLNN where
Proof.
According to Definition 5, we can obtain
Thus,
Thereafter,
Furthermore,
Therefore,
Hence, (42) is kept. □
Then, we need to prove that (42) is a 2TLNN. We need to prove two conditions as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Then we will discuss some properties of G2TLNNWBM operator.
Property 7.
(Idempotency) Ifare equal, then
Proof.
Since , then
□
Property 8.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Proof.
Let and , given that , we can obtain
Thereafter,
Furthermore,
That means . Similarly, we can obtain and .
If and
If and
If and
So Property 8 is right. □
Property 9.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
From Property 7,
From Property 8,
4.2. G2TLNNWGBM Operator
Similarly to GWBM, to consider the attribute weights, the generalized weighted geometric Bonferroni mean (GWGBM) is defined, as follows.
Definition 13
([]). Let and be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . If
Then we extend GWGBM to fuse the 2TLNNs and propose generalized 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (G2TLNNWGBM) aggregation operator.
Definition 14.
Letbe a set of 2TLNNs with their weight vector be, thereby satisfyingand. If
Then we calledthe generalized 2-tuple linguistic neutrosophic number weighted geometric BM.
Theorem 4.
Letbe a set of 2TLNNs. The aggregated value by using G2TLNNWGBM operators is also a 2TLNN where
Proof.
From Definition 5, we can obtain,
Thus,
Therefore,
Thereafter,
Furthermore,
Hence, (62) is kept. □
Then, we need to prove that (62) is a 2TLNN. We need to prove two conditions, as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Similar to G2TLNNWBM, the G2TLNNWGBM has the same properties as follows. The proof are omitted here to save space.
Property 10.
(Idempotency) Ifare equal, then
Property 11.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Property 12.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
5. DG2TLNNWBM and DG2TLNNWGBM Operators
5.1. DG2TLNNWBM Operator
However, the GBM still has some drawbacks, GBWM and GWGBM can only consider the interrelationship between any three aggregated arguments. So, Zhang et al. [] introduced a new generalization of the traditional BM because the correlations are ubiquitous among all of the arguments. The new generalization of the traditional BM is called the dual GWBM (DGBM). The DGWBM is defined, as follows.
Definition 15
([]). Let be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . The dual generalized weighted Bonferroni mean (DGWBM) is defined as follows:
where is the parameter vector with
Then we extend DGWBM to fuse the 2TLNNs and propose dual generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (DG2TLNNWBM) operator.
Definition 16.
Letbe a set of 2TLNNs. The dual generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (DG2TLNNWBM) operator is:
Theorem 5.
Letbe a set of 2TLNNs. The aggregated value by using DG2TLNNWBM operators is also a 2TLNN where
Proof.
According to Definition 5, we can obtain
Thus,
Thereafter,
Furthermore,
Therefore,
Hence, (78) is kept. □
Then, we need to prove that (78) is a 2TLNN. We need to prove two conditions as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
Furthermore,
Therefore,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Then, we will discuss some properties of the DG2TLNNWBM operator.
Property 13.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Proof.
Let and , given that , we can obtain
Thereafter,
Furthermore,
That means . Similarly, we can obtain and .
If and
If and
If and
So Property 13 is right. □
Property 14.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
From Theorem 5, we can obtain
From Property 13,
5.2. DG2TLNNWGBM Operator
Similarly to DGWBM, to consider the attribute weights, the dual generalized weighted geometric Bonferroni mean (DGWGBM) is defined, as follows.
Definition 17
([]). Let and be a collection of nonnegative crisp numbers with the weights vector being , thereby satisfying and . If
where is the parameter vector with
Then we extend DGWGBM to fuse the 2TLNNs and propose dual generalized 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (DG2TLNNWGBM) aggregation operator.
Definition 18.
Letbe a set of 2TLNNs with their weight vector be, thereby satisfyingand. If
Then we calledthe dual generalized 2-tuple linguistic neutrosophic number weighted geometric BM.
Theorem 6.
Letbe a set of 2TLNNs. The aggregated value by using DG2TLNNWGBM operators is also a 2TLNN where
Proof.
From Definition 5, we can obtain,
Thus,
Therefore,
Thereafter,
Furthermore,
Hence, (96) is kept. □
Then, we need to prove that (96) is a 2TLNN. We need to prove two conditions, as follows:
- ①
- ②
Let
Proof.
① Since we get
Then,
That means , so ① is maintained, similarly, we can get . ② Since , we get . □
Similar to DG2TLNNWBM, the DG2TLNNWGBM has the same properties, as follows. The proof are omitted here to save space.
Property 15.
(Monotonicity) Letandbe two sets of 2TLNNs. Ifhold for all, then
Property 16.
(Boundedness) Letbe a set of 2TLNNs. Ifandthen
6. Numerical Example and Comparative Analysis
6.1. Numerical Example
Given the rise in environmental and resource conservation importance, green supply chain management has seen growth within industry. In addition, balancing economic development and environmental development is one of the critical issues faced by managers to help organizations maintain a strategically competitive position. Green supplier management as one of important part of green supply chain is also critical issue for effective green supply chain management. Thus, in this section, we shall present a numerical example to select green suppliers in green supply chain management with 2TLNNs in order to illustrate the method that is proposed in this paper. There is a panel with five possible green suppliers in green supply chain management to select. The experts selects four attribute to evaluate the five possible green suppliers: ① G1 is the product quality factor; ② G2 is the environmental factors; ③ G3 is the delivery factor; and, ④ G4 is the price factor. The five possible green suppliers are to be evaluated using the 2TLNNs by the three decision maker under the above four attributes (whose weighting vector , expert weighting vector, which are listed in Table 1, Table 2 and Table 3.
Table 1.
2-tuple linguistic neutrosophic number (2TLNN) decision matrix .
Table 2.
2TLNN decision matrix .
Table 3.
2TLNN decision matrix .
In the following, we utilize the approach that was developed to select green suppliers in green supply chain management.
Definition 19.
Letbe a set of 2TLNNs with their weight vector be, thereby satisfyingand, then we can obtain
Step 1. According to 2TLNNs , we can aggregate all of the 2TLNNs by using the 2TLNNWAA (2TLNNWGA) operator to get the overall 2TLNNs of the green suppliers . Then the aggregating results are shown in Table 4.
Table 4.
The aggregating results by the 2TLNNWAA operator.
Step 2. According to Table 4, we can aggregate all of the 2TLNNs by using the DG2TLNNWBM (DG2TLNNWGBM) operator to get the overall 2TLNNs of the green suppliers . Suppose that , then the aggregating results are shown in Table 5.
Table 5.
The aggregating results of the green suppliers by the DG2TLNNWBM (DG2TLNNWGBM) operator.
Step 3. According to the aggregating results shown in Table 5 and the score functions of the green suppliers are shown in Table 6.
Table 6.
The score functions of the green suppliers.
Step 4. According to the score functions shown in Table 6 and the comparison formula of score functions, the ordering of the green suppliers is shown in Table 7. Note that “>” means “preferred to”. As we can see, depending on the aggregation operators that were used, the best green supplier is A1.
Table 7.
Ordering of the green suppliers.
6.2. Influence of the Parameter on the Final Result
In order to show the effects on the ranking results by changing parameters of in the DG2TLNNWBM (DG2TLNNWGBM) operators, all of the results are shown in Table 8 and Table 9.
Table 8.
Ranking results for different operational parameters of the DG2TLNNWBM operator.
Table 9.
Ranking results for different operational parameters of the DG2TLNNWGBM operator.
6.3. Comparative Analysis
Then, we compare our proposed method with other existing methods, including the LNNWAA operator and the LNNWGA operator proposed by Fang & Ye [] and cosine measures of linguistic neutrosophic numbers []. The comparative results are shown in Table 10.
Table 10.
Ordering of the green suppliers.
From above, we can that we get the same results to show the practicality and effectiveness of the proposed approaches. However, the existing aggregation operators, such as the LNNWAA operator and the LNNWGA operator, do not consider the information about the relationship between the arguments being aggregated, and thus cannot eliminate the influence of unfair arguments on the decision result. Our proposed DG2TLNNWBM and DG2TLNNWGBM operators consider the information about the relationship among arguments being aggregated.
7. Conclusions
In this paper, we investigate the MADM problems with 2TLNNs. Then, we utilize the Bonferroni mean (BM) operator, generalized Bonferroni mean (GBM) operator, and dual generalized Bonferroni mean (DGBM) operator to develop some Bonferroni mean aggregation operators with 2TLNNs: 2-tuple linguistic neutrosophic number weighted Bonferroni mean (2TLNNWBM) operator, 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (2TLNNWGBM) operator, generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (G2TLNNWBM) operator, generalized 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (G2TLNNWGBM) operator, dual generalized 2-tuple linguistic neutrosophic number weighted Bonferroni mean (DG2TLNNWBM) operator, and dual generalized 2-tuple linguistic neutrosophic number weighted geometric Bonferroni mean (DG2TLNNWGBM) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the MADM problems with 2TLNNs. Finally, a practical example for green supplier selection in green supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, the application of the proposed aggregating operators of 2TLNNs needs to be explored in the decision making, risk analysis, and many other fields under uncertain environments [,,,,,,,,,,,,,,,,,,,,,,].
Acknowledgments
The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (16XJA630005) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).
Author Contributions
Jie Wang, Guiwu Wei and Yu Wei conceived and worked together to achieve this work, Jie Wang compiled the computing program by Matlab and analyzed the data, Jie Wang and Guiwu Wei wrote the paper. Finally, all the authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflicts of interest.
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Atanassov, K.; Gargov, G. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
- Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis; American Research Press: Rehoboth, DE, USA, 1998. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistructure 2010, 4, 410–413. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. [Google Scholar]
- Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 2459–2466. [Google Scholar]
- Ye, J. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int. J. Gen. Syst. 2013, 42, 386–394. [Google Scholar] [CrossRef]
- Broumi, S.; Smarandache, F. Correlation coefficient of interval neutrosophic set. Appl. Mech. Mater. 2013, 436, 511–517. [Google Scholar] [CrossRef]
- Biswas, P.; Pramanik, S.; Giri, B.C. TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 2016, 27, 727–737. [Google Scholar] [CrossRef]
- Liu, P.D.; Chu, Y.C.; Li, Y.W.; Chen, Y.B. Some generalized neutrosophic number Hamacher aggregation operators and their application to Group Decision Making. Int. J. Fuzzy Syst. 2014, 16, 242–255. [Google Scholar]
- Şahin, R.; Liu, P.D. Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput. Appl. 2016, 27, 2017–2029. [Google Scholar] [CrossRef]
- Ye, J. Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making. J. Intell. Fuzzy Syst. 2014, 26, 165–172. [Google Scholar]
- Zhang, H.Y.; Wang, J.Q.; Chen, X.H. Interval neutrosophic sets and their application in multicriteria decision making problems. Sci. Word J. 2014, 2014, 1–15. [Google Scholar] [CrossRef] [PubMed]
- Peng, J.J.; Wang, J.Q.; Zhang, H.Y.; Chen, X.H. An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl. Soft Comput. 2014, 25, 336–346. [Google Scholar] [CrossRef]
- Zhang, H.; Wang, J.Q.; Chen, X.H. An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput. Appl. 2016, 27, 615–627. [Google Scholar] [CrossRef]
- Liu, P.D.; Xi, L. The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making. Int. J. Mach. Learn. Cybernet. 2016. [Google Scholar] [CrossRef]
- Peng, J.J.; Wang, J.Q.; Wu, X.H.; Wang, J.; Chen, X.H. Multi-valued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems. Int. J. Comput. Intell. Syst. 2015, 8, 345–363. [Google Scholar] [CrossRef]
- Zhang, H.Y.; Ji, P.; Wang, J.Q.; Chen, X.H. An improved weighted correlation coefficient based on integrated weight for interval neutrosophic sets and its application in multi-criteria decision-making problems. Int. J. Comput. Intell. Syst. 2015, 8, 1027–1043. [Google Scholar] [CrossRef]
- Chen, J.Q.; Ye, J. Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making. Symmetry 2017, 9, 82. [Google Scholar] [CrossRef]
- Liu, P.D.; Wang, Y.M. Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 2014, 25, 2001–2010. [Google Scholar] [CrossRef]
- Wu, X.H.; Wang, J.Q.; Peng, J.J.; Chen, X.H. Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. J. Intell. Fuzzy Syst. 2016, 18, 1104–1116. [Google Scholar] [CrossRef]
- Li, Y.; Liu, P.; Chen, Y. Some Single Valued Neutrosophic Number Heronian Mean Operators and Their Application in Multiple Attribute Group Decision Making. Informatica 2016, 27, 85–110. [Google Scholar] [CrossRef]
- Wang, J.; Tang, X.; Wei, G. Models for Multiple Attribute Decision-Making with Dual Generalized Single-Valued Neutrosophic Bonferroni Mean Operators. Algorithms 2018, 11, 2. [Google Scholar] [CrossRef]
- Wei, G.W.; Zhang, Z.P. Some Single-Valued Neutrosophic Bonferroni Power Aggregation Operators in Multiple Attribute Decision Making. J. Ambient Intell. Humaniz. Comput. 2018. [Google Scholar] [CrossRef]
- Peng, X.D.; Dai, J.G. Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function. Neural Comput. Appl. 2018, 29, 939–954. [Google Scholar] [CrossRef]
- Herrera, F.; Martinez, L. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 2000, 8, 746–752. [Google Scholar]
- Herrera, F.; Martinez, L. An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making. Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 2000, 8, 539–562. [Google Scholar] [CrossRef]
- Merigó, J.M.; Casanovas, M.; Martínez, L. Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of Evidence. Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 2010, 18, 287–304. [Google Scholar] [CrossRef]
- Wei, G.W. A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information. Expert Syst. Appl. 2010, 37, 7895–7900. [Google Scholar] [CrossRef]
- Wei, G.W. Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Knowl. Inf. Syst. 2010, 25, 623–634. [Google Scholar] [CrossRef]
- Merigo, J.M.; Casanovas, M. Decision Making with Distance Measures and Linguistic Aggregation Operators. Int. J. Fuzzy Syst. 2010, 12, 190–198. [Google Scholar]
- Merigo, J.M.; Gil-Lafuente, A.M.; Zhou, L.G.; Chen, H.Y. Generalization of the linguistic aggregation operator and its application in decision making. J. Syst. Eng. Electron. 2011, 22, 593–603. [Google Scholar] [CrossRef]
- Wei, G.W. Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Expert Syst. Appl. 2011, 38, 4824–4828. [Google Scholar] [CrossRef]
- Wei, G.W. Some harmonic averaging operators with 2-tuple linguistic assessment information and their application to multiple attribute group decision making. Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 2011, 19, 977–998. [Google Scholar] [CrossRef]
- Wei, G.W.; Zhao, X.F. Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making. Expert Syst. Appl. 2012, 39, 5881–5886. [Google Scholar] [CrossRef]
- Xu, Y.J.; Wang, H.M. Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment. Appl. Soft Comput. 2011, 11, 3988–3997. [Google Scholar] [CrossRef]
- Liao, X.W.; Li, Y.; Lu, B. A model for selecting an ERP system based on linguistic information processing. Inf. Syst. 2007, 32, 1005–1017. [Google Scholar] [CrossRef]
- Wang, W.P. Evaluating new product development performance by fuzzy linguistic computing. Expert Syst. Appl. 2009, 36, 9759–9766. [Google Scholar] [CrossRef]
- Tai, W.S.; Chen, C.T. A new evaluation model for intellectual capital based on computing with linguistic variable. Expert Syst. Appl. 2009, 36, 3483–3488. [Google Scholar] [CrossRef]
- Zhao, X.F.; Li, Q.; Wei, G.W. Some prioritized aggregating operators with linguistic information and their application to multiple attribute group decision making. J. Intell. Fuzzy Syst. 2014, 26, 1619–1630. [Google Scholar]
- Jiang, X.P.; Wei, G.W. Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2014, 27, 2153–2162. [Google Scholar]
- Beliakov, G.; James, S.; Mordelová, J.; Rückschlossová, T.; Yager, R.R. Generalized Bonferroni mean operators in multicriteria aggregation. Fuzzy Sets Syst. 2010, 161, 2227–2242. [Google Scholar] [CrossRef]
- Bonferroni, C. Sulle medie multiple di potenze. Boll. Unione Mat. Ital. 1950, 5, 267–270. [Google Scholar]
- Xia, M.M.; Xu, Z.; Zhu, B. Generalized intuitionistic fuzzy Bonferroni means. Int. J. Intell. Syst. 2012, 27, 23–47. [Google Scholar] [CrossRef]
- Xu, Z.S.; Yager, R.R. Intuitionistic fuzzy Bonferroni means. IEEE Trans. Syst. Man Cybern. 2011, 41, 568–578. [Google Scholar]
- Yager, R.R. On generalized Bonferroni mean operators for multi-criteria aggregation. Int. J. Approx. Reason. 2009, 50, 1279–1286. [Google Scholar] [CrossRef]
- Wei, G.W. Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 19, 997–1010. [Google Scholar] [CrossRef]
- Tang, X.Y.; Wei, G.W. Models for green supplier selection in green supply chain management with Pythagorean 2-tuple linguistic information. IEEE Access 2018, 6, 18042–18060. [Google Scholar] [CrossRef]
- Wei, G.W.; Zhao, X.F.; Lin, R.; Wang, H.J. Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Appl. Math. Model. 2013, 37, 5277–5285. [Google Scholar] [CrossRef]
- Zhu, B.; Xu, Z.S.; Xia, M.M. Hesitant fuzzy geometric Bonferroni means. Inform. Sci. 2012, 205, 72–85. [Google Scholar] [CrossRef]
- Zhang, R.; Wang, J.; Zhu, X.; Xia, M.; Yu, M. Some Generalized Pythagorean Fuzzy Bonferroni Mean Aggregation Operators with Their Application to Multiattribute Group Decision-Making. Complexity 2017, 2017, 5937376. [Google Scholar] [CrossRef]
- Fang, Z.; Ye, J. Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers. Symmetry 2017, 9, 111. [Google Scholar] [CrossRef]
- Shi, L.; Ye, J. Cosine Measures of Linguistic Neutrosophic Numbers and Their Application in Multiple Attribute Group Decision-Making. Information 2017, 8, 117. [Google Scholar]
- Chen, T. The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making. Appl. Soft Comput. 2015, 26, 57–73. [Google Scholar] [CrossRef]
- Wei, G.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Projection models for multiple attribute decision making with picture fuzzy information. Int. J. Mach. Learn. Cybern. 2018, 9, 713–719. [Google Scholar] [CrossRef]
- Merigo, J.M.; Gil-Lafuente, A.M. Fuzzy induced generalized aggregation operators and its application in multi-person decision making. Expert Syst. Appl. 2011, 38, 9761–9772. [Google Scholar] [CrossRef]
- Wei, G.W.; Gao, H. The generalized Dice similarity measures for picture fuzzy sets and their applications. Informatica 2018, 29, 1–18. [Google Scholar] [CrossRef]
- Gao, H.; Wei, G.W.; Huang, Y.H. Dual hesitant bipolar fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making. IEEE Access 2018, 6, 11508–11522. [Google Scholar] [CrossRef]
- Merigó, J.M.; Gil-Lafuente, A.M. Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making. Inform. Sci. 2013, 236, 1–16. [Google Scholar] [CrossRef]
- Yu, D.J.; Wu, Y.Y.; Lu, T. Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl.-Based Syst. 2012, 30, 57–66. [Google Scholar] [CrossRef]
- Gao, H.; Lu, M.; Wei, G.W.; Wei, Y. Some novel Pythagorean fuzzy interaction aggregation operators in multiple attribute decision making. Fundam. Inform. 2018, 159, 385–428. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2018, 20, 1–12. [Google Scholar] [CrossRef]
- Wei, G.W. Some similarity measures for picture fuzzy sets and their applications. Iran. J. Fuzzy Syst. 2018, 15, 77–89. [Google Scholar]
- Wei, G.W.; Wei, Y. Similarity measures of Pythagorean fuzzy sets based on cosine function and their applications. Int. J. Intell. Syst. 2018, 33, 634–652. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M. Pythagorean fuzzy power aggregation operators in multiple attribute decision making. Int. J. Intell. Syst. 2018, 33, 169–186. [Google Scholar] [CrossRef]
- Ma, Z.M.; Xu, Z.S. Symmetric Pythagorean Fuzzy Weighted Geometric_Averaging Operators and Their Application in Multicriteria Decision-Making Problems. Int. J. Intell. Syst. 2016, 31, 1198–1219. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M. Pythagorean Fuzzy Maclaurin Symmetric Mean Operators in multiple attribute decision making. Int. J. Intell. Syst. 2018, 33, 1043–1070. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft Comput. 2018, 22, 989–1002. [Google Scholar] [CrossRef]
- Wei, G.W. Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fundam. Inform. 2018, 157, 271–320. [Google Scholar] [CrossRef]
- Merigo, J.M.; Casanovas, M. Induced aggregation operators in decision making with the Dempster-Shafer belief structure. Int. J. Intell. Syst. 2009, 24, 934–954. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M.; Tang, X.Y.; Wei, Y. Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making. Int. J. Intell. Syst. 2018, 1–37. [Google Scholar] [CrossRef]
- Wei, G.W.; Gao, H.; Wei, Y. Some q-Rung Orthopair Fuzzy Heronian Mean Operators in Multiple Attribute Decision Making. Int. J. Intell. Syst. 2018. [Google Scholar] [CrossRef]
- Xu, Z.; Gou, X. An overview of interval-valued intuitionistic fuzzy information aggregations and applications. Granul. Comput. 2017, 2, 13–39. [Google Scholar] [CrossRef]
- Meng, S.; Liu, N.; He, Y. GIFIHIA operator and its application to the selection of cold chain logistics enterprises. Granul. Comput. 2017, 2, 187–197. [Google Scholar] [CrossRef]
- Wang, C.; Fu, X.; Meng, S.; He, Y. Multi-attribute decision making based on the SPIFGIA operators. Granul. Comput. 2017, 2, 321–331. [Google Scholar] [CrossRef]
- Xu, Z.; Wang, H. Managing multi-granularity linguistic information in qualitative group decision making: An overview. Granul. Comput. 2016, 1, 21–35. [Google Scholar] [CrossRef]
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).