1. Introduction
The goal of the multiple attribute group decision-making (MAGDM) method is to select the optimal scheme from finite alternatives. First of all, decision makers (DMs) evaluate each alternative under the different attributes. Then, based on the DMs’ evaluation information, the alternatives are ranked in a certain way. As a research hotspot in recent decades, the MAGDM theory and methods have widely been used in all walks of life, such as supplier selection [
1,
2,
3], medical diagnosis, clustering analysis, pattern recognition, and so on [
4,
5,
6,
7,
8,
9,
10,
11]. When evaluating alternatives, DMs used to evaluate alternatives by crisp numbers, but sometimes it is hard to use precise numbers because the surrounding environment has too much redundant data or interfering information. As a result, DMs have difficulty fully understanding the object of evaluation and exploiting exact information. As an example, when we evaluate people’s morality or vehicle performance, we can easily use linguistic term such as good, fair, or poor, or fuzzy concepts such as slightly, obviously, or mightily, to give evaluation results. For this reason, Zadeh [
12] put forward the concept of linguistic variables (LVs) in 1975. Later, Herrera and Herrera-Viedma [
5,
6] proposed a linguistic assessments consensus model and further developed the steps of linguistic decision analysis. Subsequently, it has become an area of wide concern, and resulted in several in-depth studies, especially in MAGDM [
8,
11,
13,
14,
15]. In addition, for the reason of fuzziness, Atanassov [
16] introduced the intuitionistic fuzzy set (IFS) on the basis of the fuzzy set developed by Zadeh [
17]. IFS can embody the degrees of satisfaction and dissatisfaction to judge alternatives, synchronously, and has been studied by large numbers of scholars in many fields [
1,
2,
9,
10,
18,
19,
20,
21,
22,
23]. However, intuitionistic fuzzy numbers (IFNs) use the two real numbers of the interval [0,1] to represent membership degree and non-membership degree, which is not adequate or sufficient to quantify DMs’ opinions. Hence, Chen et al. [
24] used LVs to express the degrees of satisfaction and dissatisfaction instead of the real numbers of the interval [0,1], and proposed the linguistic intuitionistic fuzzy number (LIFN). LIFNs contain the advantages of both linguistic term sets and IFNs, so that it can address vague or imprecise information more accurately than LVs and IFNs. Since the birth of LIFNs, some scholars have proposed some improved aggregation operators and have applied them to MAGDM problems [
10,
25,
26,
27,
28].
With the further development of fuzzy theory, Fang and Ye [
29] noted while LIFNs can deal with vague or imprecise information more accurately than LVs and IFNs, it can only express incomplete information rather than indeterminate or inconsistent information. Since the indeterminacy of LIFN
is reckoned by
in default, evaluating the indeterminate or inconsistent information, i.e.,
or
, is beyond the scope of the LIFN. Hence, a new form of information expression needs to be found. Fortunately, the neutrosophic sets (NSs) developed by Smarandache [
30] are able to quantify the indeterminacy clearly, which is independent of truth-membership and false-membership, but NSs are not easy to apply to the MAGDM. So, some stretched form of NS was proposed for solving MAGDM, such as single-valued neutrosophic sets (SVNSs) [
31], interval neutrosophic sets (INSs) [
32], simplified neutrosophic sets (SNSs) [
33], and so on. Meanwhile, they have attracted a lot of research, especially related to MAGDM [
34,
35,
36,
37,
38,
39,
40,
41]. Due to the characteristic of SNSs that use three crisp numbers of the interval [0,1] to depict truth-membership, indeterminacy-membership, and false-membership, motivated by the narrow scope of the LIFN, Fang and Ye [
29] put forward the concept of linguistic neutrosophic numbers (LNNs) by combining linguistic terms and a simplified neutrosophic number (SNN). LNNs use LVs in the predefined linguistic term set to express the truth-membership, indeterminacy-membership, and falsity-membership of SNNs. So, LNNs are more appropriate to depict qualitative information than SNNs, and are also an extension of the LIFNs, obviously. Therefore, in this paper, we tend to study the MAGDM problems with LNNs.
In MAGDM, the key step is how to select the optimal alternative according to the existing information. Usually, we adopt the traditional evaluation methods or the information aggregation operators. The common traditional evaluation methods include TOPSIS [
7,
9], VIKOR [
19], ELECTRE [
42], TODIM [
20,
43], PROMETHE [
18], etc., and they can only give the priorities in order regarding alternatives. However, the information aggregation operators first integrate DMs’ evaluation information into a comprehensive value, and then rank the alternatives. In other words, they not only give the prioritization orders of alternatives, they also give each alternative an integrated assessment value, so that the information aggregation operators are more workable than the traditional evaluation approaches in solving MAGDM problems. Hence, our study is concentrated on how to use information aggregation operators to solve the MAGDM problems with LNNs. In addition, in real MAGDM problems, there are often homogeneous connections among the attributes. Using a common example, quality is related to customer satisfaction when picking goods on the Internet. In order to solve this MAGDM problems where the attributes are interrelated, many related results have been achieved as a result, especially information aggregation operators such as the Bonferroni mean (BM) operator [
23,
44], the Maclaurin symmetric mean (MSM) operator [
45], the Hamy mean operator [
46], the generalized MSM operator [
47], and so forth. However, the heterogeneous connections among the attributes may also exist in real MAGDM problems. For instance, in order to choose a car, we may consider the following attributes: the basic requirements (
G1), the physical property (
G2), the brand influence (
G3), and the user appraisal (
G4), where the attribute
G1 is associated with the attribute
G2, and the attribute
G3 is associated with the attribute
G4, but the attributes
G1 and
G2 are independent of the attributes
G3 and
G4. So, the four attributes can be sorted into two clusters,
P1 and
P2, namely
P1 = {
G1,
G2} and
P2 = {
G3,
G4} meeting the condition where
P1 and
P2 have no relationship. To solve this issue, Dutta and Guha [
48] proposed the partition Bonferroni mean (
PBM) operator, where all attributes are sorted into several clusters, and the members have an inherent connection in the same clusters, but independence in different clusters. Subsequently, Banerjee et al. [
4] extended the
PBM operator to the general form that was called the generalized partitioned Bonferroni mean (GPBM) operator, which further clarified the heterogeneous relationship and individually processed the elements that did not belong to any cluster of correlated elements, so the GPBM operator can model the average of the respective satisfaction of the independent and dependent input arguments. Besides, the GPBM operator can be translated into the BM operator, arithmetic mean operator, and
PBM operator, so the GPBM operator is a wider range of applications for solving MAGDM problems with related attributes. Therefore, in this paper, we are further focused on how to combine the GPBM operator with LNNs to address the MAGDM problems with heterogeneous relationships among attributes. Inspired by the aforementioned ideas, we aim at:
- (1)
establishing a linguistic neutrosophic GPBM (LNGPBM) operator and the weighted form of the LNGPBM operator (the form of shorthand is LNGWPBM).
- (2)
discussing their properties and particular cases.
- (3)
proposing a novel MAGDM method in light of the proposed LNGWPBM operator to address the MAGDM problems with LNNs and the heterogeneous relationships among its attributes.
- (4)
showing the validity and merit of the developed method.
The arrangement of this paper is as follows. In
Section 2, we briefly retrospect some elementary knowledge, including the definitions, operational rules, and comparison method of the LNNs. We also review some definitions and characteristics of the
PBM operator and GPBM operator. In
Section 3, we construct the
LNGPBM operator and
LNGWPBM operator for LNNs, including their characteristics and some special cases. In
Section 4, we propose a novel MAGDM method based on the proposed
LNGWPBM operator to address the MAGDM problems where heterogeneous connections exist among the attributes. In
Section 5, we give a practical application related to the selection of green suppliers to show the validity and the generality of the MAGDM method, and compare the experimental results of the proposed MAGDM method with the ones of Fang and Ye’s MAGDM method [
29] and Liang et al.’s MAGDM method [
7].
Section 6 presents the conclusions.
3. The Linguistic Neutrosophic GPBM Operators
In this section, we will construct the LNGPBM operator from the GPBM operator and LNNs. Moreover, with respect to the different weights of different attributes in real life, we will propose the corresponding weighted operators, and call it the LNGWPBM operator. They are defined as follows.
3.1. The LNGPBM Operator
Definition 8. Letandbe LNNs, which are sorted into two groups: F1 and F2. In F1, the elements are divided intoclusters, which satisfies, x ≠ y and; in F2, the elements are irrelevant to any element. The LNGPBM operator of the LNNsandis defined as follows:whereand;and; |F2| denotes the number of elements in F2,indicates the number of elements in clusterand. Theorem 1. Letandbe LNNs, whereand. The synthesized result of the LNGPBM operator of the LNNsandis still a LNN, which is shown as follows:
where
,
and
.
Proof. According to Formula (16), first of all, we can part two steps: the processing of F1 and F2, and then combine them to prove.
(i) The processing of F1:
Based on the operational rules of LNNs, we can get and .
Then, we can calculate the average satisfaction of the elements in
except
:
and the conjunction of the satisfaction of element
with the average satisfaction of the rest of elements in
:
Then, the satisfaction of the interrelated elements of
is:
So, the average satisfaction of all of the elements of the
clusters is:
We suppose
,
and
, then the upper formula can be rewritten as:
(ii) The processing of F2:
The average satisfaction of all the elements that are irrelevant to any element is:
Finally, we can compute the average satisfaction of the elements
and
:
That proves that Formula (17) is kept. Then, we prove that the aggregated result of Formula (17) is a LNN. It is easy to prove the following inequalities:
and:
Firstly, we prove , and .
Since
and
, we can get
,
and
. Owing to
, the following inequalities are established:
According to , it is easy to obtain the below inequality: , and .
In addition, because
,
, and
, we can get the following inequalities:
Besides, on the basis of the upper inequalities, we can get:
which can derive directly:
Therefore, Theorem 1 is kept if some of the partitions only contain one element. ☐
In the following, we will demonstrate the desired properties of the proposed LNGPBM operator:
- (1)
Idempotency: If and are LNNs meeting the condition ; then, .
Proof. Since
, we can get:
In the same way, we can obtain and .
According to Theorem 1, we can obtain:
☐
- (2)
Monotonicity: If and are any two sets of LNNs; they satisfy the condition , and , then .
Proof. Suppose that
and
, then:
In order to prove this property, we need to compute their score function values and , and their accuracy values and to compare their synthesized result, i.e., . Firstly, on the basis of the condition , , and , we can get the compared result of their truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees, respectively.
- (i)
The comparison of the truth-membership degrees:
Based on
, we can get:
In accordance with the upper two inequalities, we have:
That is, .
- (ii)
The comparision of indeterminacy-membership degrees and falsity-membership degrees, respectively:
Based on and , we can also obtain and ; this process is similar to the process of the truth-membership degrees.
Thus, it can be obtained that . In the following, we discuss two cases.
- (i)
If , then , according to Definition 2.
- (ii)
If , then . Since in the light of and , now we assume , then , which is in contradiction with the previous proof . So, we can conclude that . That is, , which testifies .
In conclusion, the synthesized result
, which explains:
☐
- (3)
Boundedness: Let
be an arbitrary set of LNNs, then:
Proof. Since
, according to the monotonicity and idempotency of the proposed
LNGPBM operator, we can obtain the following result:
Similarly, we can obtain the corresponding result for
:
Therefore, . ☐
Based on the character of F2, some special cases are discussed about the LNGPBM operator, and shown in the following.
- (1)
When |
F2| = 0, all arguments belong to the group
F1, and are divided into
clusters; then, the proposed
LNGPBM operator is simplified as the following form:
The LNPBM is called the linguistic neutrosophic PBM operator.
- (2)
When |
F2| = 0 and
, all arguments belong to the same cluster, i.e.,
hr =
n; then, the proposed
LNGPBM operator becomes the following form:
The LNBM is called the linguistic neutrosophic BM operator.
- (3)
When |
F2| =
n, there is no element in group
F1 and all elements are independent; then, the proposed
LNGPBM operator reduces to the following form:
The LNPRAM is called the linguistic neutrosophic power root arithmetic mean operator.
Moreover, we can also get some special cases by distributing different values to the parameters p and q.
- (1)
When , the proposed LNGPBM operator becomes the LNPRAM operator, which was described in the previous discussion. Since there is no inner connection in group F1, all of the elements are independent.
- (2)
When
and
, the proposed
LNGPBM operator reduces to the linguistic neutrosophic arithmetic mean (
LNAM) operator, which is shown as follows:
- (3)
When
and
, the proposed
LNGPBM operator is transformed into the linguistic neutrosophic square root arithmetic mean (
LNSRAM) operator, which is shown as follows:
- (4)
When
, the proposed
LNGPBM operator is simplified as the simplest form of the
LNGPBM operator, which is shown as follows:
It is often used to simplify the calculation in a problem.
3.2. The LNGWPBM Operator
In Definitions 8, we assume that all the input arguments have the same position. However, in many realistic decision-makings, every input argument may have different importance. Accordingly, we give different values to the weights of input arguments, and propose the weighted form of the LNGPBM operator. Let the weight of input argument be , where and . The weighted form of the LNGPBM operator is shown in the following.
Definition 9. Letandbe LNNs that are sorted into two groups: F1 and F2. In F1, the elements are divided intoclusters, which satisfy, x ≠ y and; in F2, the elements are irrelevant to any element. The weighted form of the LNGPBM operator of the LNNsandis defined as follows:whereand;is the weight of input argumentmeetingand;and; |F2| denotes the number of elements in F2;indicates the number of elements in partition; and. Then, we call it a linguistic neutrosophic generalized weighted PBM (LNGWPBM) operator. Theorem 2. Letandbe LNNs, whereand, and let the weight of input argumentbe, whereand. Then, the synthesized result of the LNGWPBM operator of the LNNsandis still a LNN, which is shown as follows:
where
,
Proof. Along the lines of Theorem 1, we also process the groups F1 and F2 separately, and then combine them to prove.
(i) The processing of F1:
Firstly, we successively use Formulas (7), (6), and (4) to get the following formula:
Then, we have:
Since
, we can get:
Hence, the following equation is established in the light of the upper:
Since the expression
is too long, we suppose:
Then, the expression
can be written as
. Next, we can get the below expression:
(ii) The processing of F2:
Based on the operational laws of LNNs, it is easy to obtain:
Finally, we compute the synthesized result of the
LNGWPBM operator:
That proves that Formula (19) is kept. Then, we prove that the aggregated result of Formula (19) is an LNN. It is easy to prove the following inequalities:
Firstly, we prove , , and .
Based on the previous conditions such as
,
,
,
, and so on, we can get
and
, which can deduce the following inequality:
So, we can easily obtain , i.e., .
Similarly, we also have and .
Next, we put the first to prove .
According to
,
and
, we can illustrate
and
, which can deduce the following inequality:
Then, we find that .
Likewise, we can illustrate and .
Therefore, Theorem 2 is kept. ☐
In the following, we demonstrate the desired properties of the proposed LNGWPBM operator:
- (1)
Monotonicity: If
and
are any two sets of LNNs, they satisfy the conditions
,
and
, then:
Proof. Similar to the monotonicity property of the LNGPBM operator, we also suppose that and .
In order to prove this property, we need to compute their score function values and , and their accuracy values and to compare their synthesized result, i.e., . Firstly, on the basis of the condition , and , we can get the compared result of their truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees, respectively.
- (i)
The comparison of the truth-membership degrees:
Based on
, we can get:
In accordance with the upper two inequalities, we have:
That is, .
- (ii)
The comparison of indeterminacy-membership degrees and falsity-membership degrees, respectively:
Based on and , we can also obtain and , which is similar to the process of the truth-membership degrees.
Thus, it can be obtained that . In the following, we discuss two cases.
- (i)
If , then according to Definition 2.
- (ii)
If , then . Since in the light of and , now we assume , then , which is in contradiction with the previous proof . So, we can conclude that . That is , which testifies .
In conclusion, the synthesized result is , which explains . ☐
- (2)
Boundedness: Let
be any set of LNNs, then:
Based on the monotonicity property of the LNGWPBM operator, it is easy to prove, and the detailed process is omitted here.
Based on the character of F2, some special cases are discussed about the LNGPBM operator, as shown in the following.
- (1)
When |
F2| = 0, all of the arguments belong to the group
F1, and are divided into
partitions; then, the proposed
LNGWPBM operator is simplified as the linguistic neutrosophic weighted
PBM (
LNWPBM) operator:
- (2)
When |
F2| = 0 and
, all of the arguments belong to the same partition, i.e.,
hr= n; then, the proposed
LNGWPBM operator is translated into the linguistic neutrosophic normalized weighted
BM (
LNNWBM) operator:
- (3)
When |
F2| =
n, there is no element in group
F1 and all of the elements are independent; then, the proposed
LNGWPBM operator reduces to the linguistic neutrosophic power root weighted mean (
LNPRWM) operator:
Moreover, we can also get some special cases by distributing different values to the parameters p and q.
- (1)
When , the LNGWPBM operator is translated into the LNPRWM operator, as described in the previous discussion. Since there are no inner connections in group F1, all of the elements are unrelated.
- (2)
When
and
, the
LNGWPBM operator becomes the
LNNWAA operator, as defined by Fang and Ye [
29]:
- (3)
When
and
, the
LNGWPBM operator is transformed into the linguistic neutrosophic square root weighted mean (
LNSRWM) operator, which is shown as follows:
- (4)
When
, the
LNGWPBM operator is simplified as the simplest form of the
LNGWPBM operator, which is shown as follows:
It is often used to simplify the calculation in a problem with different weights.
4. A Novel MAGDM Method by the Introduced LNGWPBM Operator
In this section, we develop a novel MAGDM method based on the proposed LNGWPBM operator to address the kind of problems where the attributes are sorted into two groups: one group contains several clusters where the attributes are relevant in same cluster, but independent in different clusters, and another contains the attributes that are irrelevant to any other attribute. Firstly, we put this kind of problem in a nutshell. Then, we detail the procedures of the proposed method to solve the above problems.
Suppose is a set of alternatives, and is a set of attributes, is the weight of the attribute , where , . Experts can use the LNNs to judge the alternative for attribute and denote it as in a linguistic term set , which meets , and is an even number. The experts’ weight vector is satisfying with , . Thus, we form the evaluation values given by expert into a decision matrix .
We further hypothesize that the set of attributes is sorted into two groups: F1 and F2. In F1, the attributes are divided into clusters , which satisfies , x ≠ y and . It means that the group F1 contains several clusters, where the attributes are relevant in same cluster, but independent in different clusters; in F2, the attributes are irrelevant to any attribute. Afterwards, we decide the priority of alternatives according to the information provided above.
The procedures of the proposed method are designed as follows.
- Step 1.
Normalize the LNNs.
Since the attributes generally fall into two types, the corresponding attribute values have the two types. In order to achieve normalization, we generally transform the cost attribute values into benefit attribute values. First of all, we assume that
is the normalized matrix of
, where
,
,
, and
. Then, the standardizing method is described in the following [
7]:
- (1)
For benefit attribute values:
- (2)
For cost attribute values:
- Step 2.
Calculate the collective decision information by the
LNGWPBM operator fixed with |
F2| = 0 and
(i.e., the
LNNWBM operator discussed in
Section 3.2), because there is no need to divide the experts into different clusters. Then, we can get the unfolding form:
where
,
,
, and
.
- Step 3.
Compute the comprehensive value of each alternative based on the
LNGWPBM operator; the unfolding form is detailed in the following:
where
,
, and
; |
F2| denotes the number of attributes in
F2,
indicates the number of attributes in cluster
, and
.
- Step 4.
Calculate the score value and the accuracy value of the synthesized evaluation value in the light of Definitions 2 and 3, where .
- Step 5.
Compare the obtained score values , , …, and based on Definition 4. The larger the value of , the more front the order of alternative , where . If the value of is the same, then compare the obtained accuracy values , , …, and to determine the ranking orders of alternatives.
- Step 6.
Ends.
6. Conclusions
The GPBM operator can model the average of the respective satisfaction of the independent and dependent inputs, and is an extended form of the
PBM operator, the arithmetic mean operator, and the BM operator. Its merit is to capture the heterogeneous relationship among attributes where all of the attributes are sorted into two groups:
F1 and
F2. In
F1, the elements are divided into several clusters, and the members have inherent connections in the same cluster, but independence in different clusters; in
F2, the elements do not belong to any cluster of the correlated input arguments in
F1. Besides, LNNs can depict the qualitative information more appropriately than the SNNs, and are also an extension of the LIFNs. However, now, based on LNNs, we yet have not seen any studies addressing the MAGDM problems with the heterogeneous relationships among attributes. Therefore, in order to fill this gap, we have expanded the GPBM operator to adapt the linguistic neutrosophic environment, and have proposed the
LNGPBM operator in this paper. At the same time, its desired properties and special cases have been discussed. Moreover, aiming at the condition where different attributes have different weights in practical applications, we also have introduced its weighted version, namely the
LNGWPBM operator, including discussing its desired properties and special cases. Then, based on the developed
LNGWPBM operator, we have developed a novel MAGDM method with LNNs to solve the MAGDM problems with the heterogeneous relationship among attributes. By comparing with Fang and Ye’s MAGDM method [
29] and Liang et al.’s MAGDM method [
7], we find that the developed MAGDM method is more valid and general for solving the MAGDM problems with co-dependent attributes. This is because the developed MAGDM method can intuitively and realistically depict qualitative information and reflect the heterogeneous relationship among attributes. In further research, our developed operators can be improved by considering the unknown weights, objective data, or other forms of information, such as unbalanced linguistic information [
50]. Besides, we can apply our developed operators to the other practices such as medical diagnosis, clustering analysis, pattern recognition, discordance analysis, and so on.