1. Introduction
Multi-attribute decision making problems are very common in many research fields. In real life, many uncertain and inconsistent pieces of information cannot be described by specific values. In order to solve this problem, Zadeh [
1] firstly proposed the concept of fuzzy set theory. However, a fuzzy set cannot express the non-membership degree. Then, Atanassov [
2] presented the concept of intuitionistic fuzzy sets (IFSs), which involve degrees of membership and degrees of non-membership. The IFS includes the membership
and non-membership
, and
,
. Zwick et al. [
3] proposed triangular IFS. Zeng and Li [
4] defined trapezoidal IFS. Yager [
5] discuss the need for obtaining aggregation operations on ordinal-based intuitionistic fuzzy subsets (OBIFS) and applied fuzzy multi-criteria technology to mobile "apps." Verma [
6] defined some new operational-laws for LIFNs based on the linguistic scale function (LSF) and proposed a generalized linguistic intuitionistic fuzzy weighted average (GLIFWA) operator for aggregating LIFNs. However, IFSs cannot process indeterminate information and inconsistent information. Therefore, Smarandache [
7] introduced the neutrosophic set (NS), which is composed of truth membership
, falsity membership
, and indeterminacy membership
, and they are independent of each other. NS is an extension of IFS. Wang [
8] presented interval NS(INS), which can represent the function of truth membership, falsity membership, and indeterminacy membership by the interval value. Wang et al. [
9] extended some new sets of NSs and proposed a single-valued neutrosophic set (SVNS). Fan [
10] defined a new form of SVNULS and expressed the weight of each expert in the uncertain linguistic part. Ji [
11] established an MABAC–ELECTRE method under SVNUL environments and used it to solve the problem of outsourcing provider selection. Kamac [
12] proposed a Hamming distance-based formula to measure the distance between two SVNULNs and presented a game theory model based on the framework of LSVNS technique.
In daily life, in addition to multi-attribute decision making problems that can be processed by qualitative information, some uncertain information is difficult to be described by precise numbers. At this time, decision makers usually use some language terms (LTs), such as “good”, “very good”, “bad”, and “very bad”. Therefore, Herrera and Herrera-Viedma [
13] proposed the LTs to deal with this kind of non-qualitative information. However, linguistic variables can only express membership, and cannot express non-membership. On this basis, Wang [
14] presented the concept of the intuitionistic linguistic set, which combines the concepts of linguistic variables and intuitionistic fuzzy sets. After that, Ye [
15] proposed the SVNLNs and used LTs to describe the truth membership, indeterminacy membership, and falsity membership.
As an effective tool, aggregation operators are widely used in MADM problems [
16]. Yager [
17] proposed the ordered weighted average (OWA) operator [
18] for multi-attribute decision making. Bonferroni presented the Bonferroni mean (BM) [
19,
20,
21,
22] operator, which can capture the interrelationships among multiple input arguments. Furthermore, Beliakov [
23] proposed the Heronian mean (HM) [
24,
25,
26] operators. Like the BM operator, it can also capture the interrelationships among multiple input arguments. However, since they can only reflect the interrelationships between two arguments, they cannot effectively deal with multi-attribute value decision making problems. Therefore, in order to solve this problem, Maclaurin introduced the MSM operator, which can capture the interrelationships between multiple input arguments [
27]. Qin and Liu [
28] extended the intuitionistic fuzzy numbers to MSM operator. Ju et al. [
29] introduced some novel weighted intuitionistic linguistic MSM operators. Liu and Qin [
30] combined the MSM operator with linguistic intuitionistic fuzzy numbers to deal with the MADM problem. Zhong Y [
31] proposed hesitant fuzzy power Maclaurin symmetric mean operators. Furthermore, Qin [
32] proposed a new method with which to solve the MADM problem by combining MSM operators with uncertain linguistic variables. In addition, it is also an effective method for applying an MSM operator to hesitant fuzzy sets [
33,
34].
As we know, an MSM operator can capture the interrelationships between multiple input arguments, and language terms can describe non-qualitative information well. Meanwhile, a neutrosophic set has advantages in expressing incomplete, indeterminate, and inconsistent information. Therefore, the purpose of this paper is to apply the MSM operator to the neutrosophic uncertain linguictic numbers environment, and propose some Maclaurin symmetric mean operators based on single-valued neutrosophic uncertain linguistic numbers, including the weighted SVNULMSM operator and the weighted SVNULGMSM operator. Then, we give some definitions and properties and prove them. Finally, an investment case is used to verify the effectiveness of the proposed method, and compared it with other methods to prove its advantages.
The complete structure of the paper is as follows: In
Section 2, we give the definitions and properties of uncertain linguistic numbers, the single-valued neutrosophic set, the Maclaurin symmetric mean (MSM) operator, and the generalized Maclaurin symmetric mean (GMSM) operator. In
Section 3, we introduce the single-valued neutrosophic uncertain linguistic set composed of the uncertain linguistic numbers and the single-valued neutrosophic set, and give some operational rules. In
Section 4, we extend MSM operators to single-valued neutrosophic uncertain linguistic numbers, and propose the SVNULMSM operator, SVNULGMSM operator, WSVNULMSM operator, and WSVNULGMSM operator. In
Section 5, we introduce the decision making methods based on the WSVNULMSM and WSVNULGMSM operators and give the specific operation steps. In
Section 6, we demonstrate the effectiveness of the method with an investment example. In
Section 7, we give the conclusions.
5. MCDM Approach Based on WSVNULMSM Operator and WSVNULGMSM Operator
In this section, we apply the proposed SVNULMSM operator and SVNULGMSM operator to cope with a MADM issue. Suppose is a set of alternatives, and let be a collection of attributes. The weight vector of the attribute is while satisfying , and each represents the importance of . Let be the set of decision makers, and be the weight vector of decision makers , and . is the decision matrix, where is attribute value given by the decision maker for alternative with respect to attribute . The main steps as follows:
Step 1 Utilize the WSVNULMSM operator:
Utilize the WSVNULGMSM operator:
We use Definition 12 and Definition 13 to aggregate the attribute values of each alternative for decision maker , and obtain the overall preference value corresponding to alternative .
Step 2 Utilize the WSVNULMSM operator:
Utilize the WSVNULGMSM operator:
According to the aggregation value from the above, we also use Definition 12 and Definition 13 to aggregate the evaluation values of the decision maker.
Step 3 Calculate the value of according to Definition 7.
Step 4 According to Definition 9, rank the alternatives.
6. Illustrative Example
In this section, we provide an investment example (adapted from [
24]) to illustrate the application of WSVNULMSM and WSVNUGMSM operators.
There are four alternatives, including a car company (A1), a food company (A2), a computer company (A3), and an arms company (A4); and these companies evaluated by three decision makers,
,
, and
. The weight vector of the decision makers is
. We use the following attributes: C1 (the risk index), C2 (the growth index), and C3 (the social-political impact index). Suppose the attribute weight vector is
. Decision makers use linguistic term set
to express their evaluation results. The decision matrices
as listed in
Table 1,
Table 2 and
Table 3.
6.1. The Decision Making Method Based on the WSVNULMSM Operator
We can give , so m = 1 or m = 2.
(1) When , the steps are shown below.
Step 1 Get the aggregate values of each alternative for decision maker with the WSVNULMSM operator.
Step 2 Get the aggregate values for each alternative with the WSVNULMSM operator.
Step 3 Calculate the value of .
, , ,
Step 4 According to Definition 7, rank the alternatives:
.
(2) When , the steps are shown below.
Step 1 Get the aggregate values of each alternative for decision maker with the WSVNULMSM operator.
Step 2 Get the aggregate values for each alternative with the WSVNULMSM operator.
Step 3 Calculate the value of .
, , ,
Step 4 According to Definition 7, rank the alternatives:
.
6.2. The Method Based on the WSVNULGMSM Operator
When , p = 1, the operator is the same as the operator. The steps are omitted here. When , the steps are below.
Step 1 Get the aggregate values of each alternative for decision maker with the WSVNULGMSM operator.
Step 2 Get the aggregate values for each alternative with the WSVNULGMSM operator.
Step 3 Calculate the value of .
, , ,
Step 4 According to Definition 7, rank the alternatives:
.
6.3. Comparative Analysis and Discussion
Based on the experiment in
Section 6.1 and
Section 6.2, we acquired the ranking results shown in
Table 4. According to
Table 4, we can achieve the same ranking results when
. This is because the relationship between the attributes is not considered. When
, we can see that the ranking results are slightly different. It can be stated that we should consider the relationship between the attributes.
From
Table 5, we see the comparisons for when
and
have different values. When
, we do not need to consider the relationship between multiple attributes, so the ranking results are the same:
. When
, the relationship between the attributes also does not need to be considered, and the ranking results are
. When
and
are not equal to zero, we should consider the interrelationship between the attributes, and we find the best alternative to be the arms company (
). According to the sorting results, we can know that the number of parameters has a great influence on the results. Therefore, sometimes the best alternative was the food company (
). Meanwhile, the worst alternative was the computer company (
).
In order to prove the effectiveness of the method proposed in this paper, we have performed a comparison with Liu’s [
24] method, and the results are shown in
Table 6.
As shown in
Table 6, the best choice found by the other method was
, which is the same as in our former results. Obtaining the same ranking results shows that our method is effective and reasonable. When
, our method had the same values as Liu’s method [
24]. This is because neither method considers the relationship between attributes. When
, the ranking results are different from those of Liu’s method [
24]. This is because the method proposed by Liu [
24] only considers the relationship between two input parameters, whereas our method takes the relationship between multiple parameters into account, which effectively solves the problem of multiple input parameters. Therefore, our method has a wider range of applications.
7. Conclusions
SVNULNs can well represent incomplete, indeterminate, and inconsistent information. Meanwhile, as an effective aggregation tool, the MSM operator can consider the relationship between multiple input parameters. In this paper, we combined the neutrosophic uncertain linguistic numbers with MSM operators, and proposed some MSM operators based on a neutrosophic uncertain linguistic environment, including the SVNULMSM operator, SVNULGMSM operator, weighted SVNULMSM operator, and weighted SVNULGMSM operator. Then, we proposed a method for solving the MADM problem by using a WSVNULMSM operator and a WSVNULGMSM operator.
Finally, the effectiveness of the proposed method was proved by comparing it with other methods. According to the same ranking results, the rationality of the method was proved. The research on the determination of parameter values is not thorough, so we will further study and explore the area. In the future, we will further extend the aggregation operator and apply it to a wider range of fields, such as fault diagnosis, financial analysis, and algorithm selection. In future research, we will extend the MSM operator to other fuzzy environments, such as an intuitionistic fuzzy set, and further validate the effectiveness of MSM operator.