Some Interval Neutrosophic Linguistic Maclaurin Symmetric Mean Operators and Their Application in Multiple Attribute Decision Making

: There are many practical decision-making problems in people’s lives, but the information given by decision makers (DMs) is often unclear and how to describe this information is of critical importance. Therefore, we introduce interval neutrosophic linguistic numbers (INLNs) to represent the less clear and uncertain information and give their operational rules and comparison methods. In addition, since the Maclaurin symmetric mean ( MSM ) operator has the special characteristic of capturing the interrelationships among multi-input arguments, we further propose an MSM operator for INLNs ( INLMSM ). Furthermore, considering the weights of attributes are the important parameters and they can inﬂuence the decision results, we also propose a weighted INLMSM ( WINLMSM ) operator. Based on the WINLMSM operator, we develop a multiple attribute decision making (MADM) method with INLNs and some examples are used to show the procedure and effectiveness of the proposed method. Compared with the existing methods, the proposed method is more convenient to express the complex and unclear information. At the same time, it is more scientiﬁc and ﬂexible in solving the MADM problems by considering the interrelationships among multi-attributes.


Introduction
The unclear set (FS) theory was put forward by Zadeh [1] in 1965.In this theory, the membership degree (MD) T(x) is used to describe fuzzy information and it has also been widely used in practice.However, the inadequacies of FS are evident.For example, it is difficult to express the non-membership degree (NMD) F(x).In order to fix this problem, Intuitionistic FS (IFS) was proposed by Atanassov [2] in 1986.It is made up of two parts: MD and NMD.IFS is an extension and development of Zadeh' FS and Zadeh' FS is a special case of IFS [3].IFS needs to meet two conditions: (1) T(x), F(x) ∈ [0, 1]; (2) 0 ≤ T(x) + F(x) ≤ 1 [2].Subsequently, the IFS theory was further extended such as Zadeh [4] proposed interval IFS (IIFS).Zwick et al. [5] put forward the triangular IFS while Zeng and Li [6] defined trapezoidal IFS.However, under some circumstances due to the limited cognitive ability of the DMs, they may hesitate in the two choices for accuracy and uncertainty.Since they choose both of them at the same time, this can produce an imprecise or contradictory evaluation result.Therefore, Smarandache [7,8] introduced a concept called neutrosophic set (NS), which included MD, NMD, and indeterminacy membership degree (IMD) in a non-standard unit interval [9].Clearly, the NS is the generalization of FS and IFS.Furthermore, Wang [10] proposed the definition of interval NS (INS) which uses the standard interval to express the function of MD, IMD, and NMD.Broumi and Smarandache [11] presented the correlation coefficient of INS.
When dealing with the MADM problems with qualitative information, it is difficult for DMs to describe their own ideas with precise values.Generally, DMs ordinarily uses some linguistic terms (LTs) like "excellent", "good", "bad", "very bad", or "general" to indicate their evaluations.For example, when we look at a person's height, we usually describe him as "high" or "very high" by visual inspection, but we will not give the exact value.In order to easily process the qualitative information, Herrera and Herrera-Viedma [12] proposed the LTs to deal with this kind of information instead of numerical computation.However, because LT such as "high" is not with MD, or we can think its MD is 1, which means LTs cannot describe the MD and NMD.Therefore, in order to facilitate DMs to describe the MD and NMD for one LT, Liu and Chen [13] defined the linguistic intuitionistic fuzzy number (LIFN), which combined the advantages of intuitionistic fuzzy numbers (IFNs) and linguistic variables (LVs).Therefore, LIFN can fully express the complex fuzzy information and there is a good prospect in MADM.After that, Ye [14] came up with the single-valued neutrosophic linguistic number (SVNLN).The most striking feature of the SVNLN is that it used LTs to describe the MD, IMD, and NMD.Sometimes, the three degrees are not expressed in a single real number, but is expressed in intervals [15].And then, Ye [16] defined an interval neutrosophic linguistic set (INLS) and INLNs.INLNs is used to represent three values of MD, IMD, and NMD in the form of intervals.Clearly, INLS is a generalization of FS, IFS, NS, INS, LIFN, and SVNLN.It is general and beneficial for describing practical problems.
The aggregation operators (AOs) are an efficient way to handle MADM problems [17,18].Many AOs are proposed for achieving some special functions.Yager [19] employed the ordered weighted average (OWA) operator for MADM.Bonferroni [20] proposed the Bonferroni mean (BM) operator, which can capture the correlation between input variables very well.Then BM operators have been extended to process different uncertain information such as IFS [21,22], interval-valued IFS [23], q-Rung Orthopai Fuzzy set [24], and Multi-valued Ns [25].In addition, Beliakov [26] presented the Heronian mean (HM) operators, which have the same feature as the BM (i.e., they can capture the interrelationship between input parameters).Some HM operators have been proposed [27][28][29][30].Furthermore, Yu [31] gave the comparison of BM with HM.However, since the BM operator and the HM operator can only reflect the relationship between any two parameters, they cannot process the MADM problems, which require the relationship for multiple inputs.In order to solve this shortcoming, Maclaurin [32] proposed the MSM operator, which has prominent features of capturing the relationship among multiple input parameters.Afterward, Qin and Liu [33] developed some MSM operators for uncertain LVs.Liu and Qin [34] developed some MSM for LIFNs.Liu and Zhang [35] proposed some MSM operators for single valued trapezoidal neutrosophic numbers.
Since the INLNs are superior to other ways of expressing complex uncertain information [16] and the MSM has good flexibility and adaptability, it can capture the relationship among multiple input parameters.However, now the MSM cannot deal with INLNs.Therefore, the objectives of this paper are to extend the MSM and weighted MSM (WMSM) operators to INLNs and to propose the INLMSM operator and the WINLMSM operator, to prove some properties of them and discuss some special cases, to propose a MADM approach with INLNs, and show the advantages of the proposed approach by comparing with other studies.
In Section 2 of this paper, we introduce some basic concepts about NS, INS, INLS, and MSM.In Section 3, we introduce the INLN and its operations including a new scoring function and a comparison method of INLN.In Section 4, we introduce an operator of INLMSM.Additionally, in order to improve flexibility, we propose the INLGMSM operator based on the GMSM operator.Furthermore, we develop the WINLMSM operator and the WINLGMSM operator to compare with operators that lack weight.Afterwards, we use examples to prove our theories.In Section 5, we give a MADM method for INLNs.In Section 6, we provide an example to demonstrate the effectiveness of the proposed method.Lastly, we provide the conclusions.

Preliminaries
In this section, we will introduce some existing definitions and basic concepts in order to understand this study.

The NS and INS
Definition 1 [7][8][9].Let X be a space of points (objects) with a generic element in X denoted by x.A NS A in X is expressed by a MD T A (x), an IMD I(x), and a NMD F A (x).
Then a NS A is denoted below.
T A (x), I(x), and F A (x) are real standard or non-standard subsets of ] − 0, 1 + [.That is Definition 2 [10,11].Let X be a space of points (objects) with a generic element in X denoted by x.
For convenience, the lower and upper ends of T, I, F are expressed as For each point x in X, we have that T Definition 3 [10,11].
Because the LTS is convenient and efficient, it is widely used by DMs in decision making.For instance, when we evaluate the production quality, we can set l = 9, then S is given below.S = {s 0 = extremely bad, s 1 = very bad, s 2 = bad, s 3 = slightly bad, s 4 = f air, s 5 = slightly good, s 6 = good, s 7 = very good, s 8 = extremely good} To relieve the loss of linguistic information in operations, Xu [38,39] extended LTS S to continuous LTS S = { s θ |0 ≤ θ ≤ l}.About the characteristics of LTS, please refer to References [38][39][40].
Definition 5 [13].Let s α and s β be any two LVs in S. The related operations can be defined below.

MSM Operator
Definition 6 [15,32].Let x i (i = 1, 2, . . ., n) be the set of the non-negative real number.An MSM operator of dimension n is a mapping MSM (m) : (R + ) n → R + and it can be defined below.
where (i 1 , i 2 , . . ., i m ) traverses all the m-tuple combination of (1, 2, . . ., n) and C m n = n! m!(n−m)! is the binomial coefficient.In addition, x i j refers to i j th element in a particular arrangement.
There are some properties of the MSM (m) operator, which are defined below.
(1) Idempotency.If x i = x for each i, and then MSM (m) (x, x, . . ., x) = x; (2) Monotonicity.If Furthermore, the MSM (m) operator would degrade some particular forms when m takes some special values, which are shown as follows.

1.
When m = 1, the MSM (m) operator would become the average operator.
When m = 2, the MSM (m) operator would become the following BM operator (p = q = 1). ) When m = n, the MSM (m) operator would become the geometric mean.
A generalized MSM operator of dimension n is a mapping GMSM (m,p 1 ,p 2 ,...,p m ) : (R + ) n → R + and it is defined below.
In addition, the GMSM (m,P 1 ,P 2 ,...,P m ) operator would degrade to some particular forms when m takes some special values, which are shown below.

1.
When m = 1, we have the formula below.
When m = n, the MSM (m) operator would become the following formula.

INLNs and Operations
Definition 8 [16,41].Let X be a finite universal set.An INLS in X is defined by the equation below.
represent the MD, the IMD, and the NMD of the element x in X to the LV s θ(x) , respectively, with the condition 0 ≤ T Then the seven tuple Definition 9 [16,37,42].Let a 1 = s θ(a 1 ) , ( T L (a 1 ), T U (a 1 ) , I L (a 1 ), I U (a 1 ) , F L (a 1 ), F U (a 1 ) ) and a 2 = s θ(a 2 ) , ( T L (a 2 ), T U (a 2 ) , I L (a 2 ), I U (a 2 ) , F L (a 2 ), F U (a 2 ) ) be two INLNs and λ ≥ 0. Then the operation of the INLNs can be expressed by the equation below.
As seen from the above examples, these results are not reasonable because they exceed the range of LTS.In order to overcome these limitations, we will improve these operations by Definition 10.
) ) be two INLNs and λ ≥ 0. Then the operations of the INLNs can be defined by the equations below. ) Based on the operational rules above, the above example is recalculated as follow.
From the above example, the results are more reasonable than the previous ones.
In the following definitions, a new scoring function and a comparison method of INLN are described.
Then the score function of a can be expressed by the equation below.
where the values of α ∈ [0, 1] reflect the attitudes of the decision makers.

Some Interval Neutrosophic Linguistic MSM Operators
In this section, we will propose INLMSM operators and INLGMSM operators.
According to the operational laws of INLNs in Definition 10, we can get the expression of the INLMSM operator shown below.
where k = 1, 2, ...C m n , a i j (k) is the i j th element of k th permutation.

Proof.
Because Therefore, Theorem 1 is kept.
.., n) be sets of INLNs.There are four properties of I NLMSM (m) operator, which is shown below.
(1) Idempotency.If the INLNs If x i is a permutation of y i for all i (i = 1, 2, ..., n) and then

1
If each a i = x, then we get the equation below.
This property is clear and it is now omitted. .
Therefore, we can get the following conclusion.
According to the monotonicity, if x a ≤ x i and x b ≥ x i for all i, then we have x a = I NLMSM (m) (x a , x a , ..., x a ) ≤ I NLMSM (m) (x 1 , x 2 , ..., x n ) and Therefore, we can get the conclusion below.
Furthermore, the I NLMSM (m) operator would degrade to some particular forms when m takes some special values.
(1) When m = 1, we have the formula below.
(2) When m = 2, we have the formula below. )) (3) When m = n, the I NLMSM (m) operator would reduce to the following form.
According to the operational laws of INLNs in Definition 10, we can get the expression of the INLMSM operator shown below.I NLGMSM (m,p 1 ,p 2 ,...,pm ) (a 1 , . . ., where k = 1, 2, ...C m n , a i j (k) is the i j th element of k j th permutation.Therefore, Theorem 2 is kept.The process of proof is similar to Theorem 1 and is now omitted.

Property 2. Let x
There are four properties of I NLGMSM (m,p 1 ,p 2 ,...,p m ) operator shown as follows.
The proofs are similar to Property 1, which are now omitted.Furthermore, the I NLGMSM (m,p 1 ,p 2 ,...,p m ) operator would degrade to some particular forms when m takes some special values.
(1) When m = 1, we have the following formula.
(2) When m = 2, we have the following formula.
(3) When m = n, the I NLMSM (m) operator would reduce to the form below.

Some Weighted INLMSM Operators
We will introduce two operators, which are the weighted forms of the INLMSM operator and INLGMSM operator.
According to the operational laws of INLNs in Definition 10, we can get the expression of the WINLMSM operator, which is shown below.
.., n) be a set of INLNs and m = 1, 2, ..., n, then the value aggregated from Definition 15 is still a WINLMSM operator.
where k = 1, 2, ..., C m n , a i j (k) is the i j th element of kth permutation.The process of proof is similar to Theorem 1.Now it is omitted. Proof. 2 The proofs of Monotonicity and Boundedness are similar to Property 1, which are now omitted.
Furthermore, the WINLMSM (m) operator would degrade a particular form when m takes some special values.
(1) When m = 1, we have the formula below.
(2) When m = 2, we have the formula below.
(3) When m = n, we have the formula below.
.., ω n ) T is the weight vector and it satisfies ∑ n i=1 ω i = 1 with ω i > 0 (i = 1, 2, ..., n).Each ω i represents the importance of a i .Then the WINLGMSM operator: Ω is a set of INLNs and m = 1, 2, ..., n.According to the operational laws of INLNs in Definition 10, we can get the expression of WINLMSM operator shown below.W I NLGMSM (m,p 1 ,p 2 ,...,pm ) (a 1 , . . ., where k = 1, 2, ..., C m n , a i j (k) is the i j th element of kth permutation.The process of proof is similar to Theorem 1.It is now omitted.
There are some properties of the W I NLGMSM (m,p 1 ,p 2 ,...,p m ) operator shown below.
The process of proof is similar to Property 3 and is now omitted.Furthermore, the W I NLGMSM (m,p 1 ,p 2 ,...,p m ) operator would degrade some particular forms when m takes some special values.
(3) When m = n, we have the formula below.

MADM Method Based on INLMSM Operator
In this section, we introduce the MADM method based on the WINLMSM and WINLGMSM operators.Let d = {d 1 , d 2 , ..., d m } be a collection of alternatives and c = {c 1 , c 2 , ..., c n } is a collection of n criteria.The weight vector is ω = (ω 1 , ω 2 , ..., ω n ) T with satisfying ∑ n i=1 ω i = 1(ω i ≥ 0, i = 1, 2, ..., n), and each ω i represents the importance of c j .The performance of alternative d j in criteria c j is surveyed by INLNs and the decision matrix is A = (a ij ) m×n , where ) .The objective is to rank the alternatives.
The detailed steps are shown below.
Step 1 Normalize the decision matrix.
We should normalize the decision-making information in the matrix.The benefit (the bigger the better) and the cost (the smaller the better) are the two possible types.In order to keep the consistency of the types, it is necessary to convert the decision matrix A into a standardized matrix R = (r ij )m × n.
Step 2 Aggregate the criterion values of each alternative.We would use Definition 15 and Definition 16 to aggregate r ij (j = 1, 2, ..., n) of the ith alternative and get the overall value r i .
Step 3 Calculate the score values of r i (i =

Illustrative Example
There are many decision-making problems to be solved in the current society, which requires some decision-making methods.
In this section, we investigate an example (adapted from Ref [43]) about the MADM.In a MADM problem, there are four possible alternatives for an investment company including a car company (A 1 ), a food company (A 2 ), a computer company (A 3 ), and an arms company (A 4 ).The following three attributes can be used to evaluate alternatives by the investment company: the risk (C 1 ), the growth (C 2 ), and the environmental impact (C 3 ) where C 1 and C 2 are benefit types and C 3 is cost type.Then the evaluation values of alternatives are shown in Table 1 where the LTS is S = {s 0 = extremely poor(EP), s 1 = very poor(VP), s 2 = poor(P), s 3 = medium(M), s 4 = good(G), s 5 = very good(VG), s 6 = extremely good(EG)}, and the weight vector of criteria is ω = (0.35, 0.25, 0.4) T .Now we will use the method proposed in this paper, according to the above LTs and three criteria.Then we evaluate and sort the four options in Table 1.Generally, we can give m = n 2 , so m = 1 and m = 2.Then, according to Section 5, we have the statements below.

Step 1
Normalize the decision matrix.
From the example, the risk (C 1 ) and the growth (C 2 ) are benefit types while the environmental impact (C 3 ) is cost type.We set up the decision matrix as shown below.
Step 2 Aggregate all attribute values of each alternative and get the overall value of each alternative a i denoted as r i (i = 1, 2, 3, 4).Step 3 According to Definition 11, we assume α = 0.7 and calculate the score values of r i (i = 1, 2, 3, 4) below.
S (r 1 ) = s 0.6228 , S (r 2 ) = s 0.8306 , S (r 3 ) = s 0.7462 , S (r 4 ) = s 0.7778 Step 4 According to Step 3 and Definition 12, we would get the ranking of the alternatives, which are (2) When m = 2, the steps are shown below.
Step 1 Normalize the decision matrix.
From the example, the risk (C 1 ) and the growth (C 2 ) are benefit types while the environmental impact (C 3 ) is cost type.We set up the decision matrix as shown below.Step 2 Aggregate all attribute values of each alternative and get the overall value of each alternative a i denoted as r i (i = 1, 2, 3, 4).Step 3 According to Definition 11, we assume α = 0.7 and calculate the score values of r i (i = 1, 2, 3, 4).
We get the values below.
Step 4 According to Step 3 and Definition 12, we get the ranking of the alternatives below.

The Method Based on the WINLGMSM Operator
When m = 1, p = 1, the W I NLGMSM (1) operator is the same as the W I NLMSM (1) operator.The steps are omitted here.When m = 2, the steps are below.
Step 1 Normalize the decision matrix.
From the example, the risk (C 1 ), the growth (C 2 ) are benefit types and the environmental impact (C 3 ) is cost type, so we set up the matrix as step 1 of Section 6.1.
Step 2 Aggregate all attribute values of each alternative by the W I NLMSM (2) operator and get the overall value of each alternative a i denoted as r i (i = Step 3 According to Definition 11, we assume α = 0.7, calculate the score values of r i (i = 1, 2, 3, 4), and get the values shown below.

Comparative Analysis and Discussion
(1) Based on the results in Sections 6.1 and 6.2, we can show them by using Table 2. From Table 2, we know that there are the same ranking results in two methods when m = 1 or m = 2.However, the result when m = 1 is different from the one when m = 2.It can be explained that, when m = 1, the interrelationship between the attributes doesn't need to be considered when m = 2.We can consider the interrelationship between two attributes.(2) Furthermore, we get the comparisons for different values of P 1 and P 2 when m = 2, which are shown in Table 3. From Table 3, we know when m = 2 and P 1 and P 2 are not equal to zero, we can get the same ranking results, i.e., A 2 A 4 A 3 A 1 .However, when P 1 = 0 or P 2 = 0, the ranking results are different from the ones when P 1 and P 2 are not equal to zero.When P 1 = 0 or P 2 = 0, the interrelationship between the attributes doesn't need to be considered, so it can get the same ranking results as the ones when m = 1.
From Table 4, we know that the best choice is A 2 for all methods, which is the same as the results produced above.However, the ranking results are different.Compared with the approach proposed by Ye [16], when m = 1, our ranking results have the same values as that of Ye [16], but when m = 2, our ranking results are different from the Ye method [16].When m = 1, all methods don't consider the interrelationship.They produce the same results, however, when m = 2. Our methods in this paper can take into account the interrelationship while the method by Ye [16] doesn't consider the interrelationship.Therefore, there are different ranking results.Therefore, our methods are more suitable for the different applications.

Operator Ranking
Methods in this paper From the above comparison results, we can obtain that the methods proposed by this paper are feasible and adaptable for the MADM problems.Additionally, they have better reliability and wider application space than other existing methods.

Conclusions
In this study, we propose the concept of INLMSM, which can not only adapt to the cognitive situation of decision maker, but also provide convenience for decision making.We introduce the basic concept of INLMSM and its generalized form, give some operators based on INLMSM, and introduce the theory of weight to investigate WINLMSM and WINLGMSM.Afterwards, we put forward the INLMSM operator, the INLGMSM operator, the WINLMSM operator, and the WINLGMSM operator.In addition, we proved these operators.In addition, we introduce the MADM methods with INLMSM in detail and illustrate their usefulness and effectiveness by showing examples.Finally, we compare other methods to demonstrate our approach.From this paper, we can see that WINLGMSM is more practical and flexible in application and INLMSM can express fuzzy information more conveniently.In further study, we can use the INLMSM operator to solve practical problems and pattern recognition.We should develop other aggregation operators for future research.
For convenience, an INLN can be represented as a = s θ(a) , ( T L (a), T U (a) , I L (a), I U (a) , F L (a), F U (a) ) .Then we introduced the operational rules of operators of INLNs.

Definition 12 .
[37].Let a and b be two INLNs.Then the INLN comparison method can be expressed by the statements below.If S(a) > S(b), then a b; (26) If S(a) = S(b), then a ∼ b; (27) If S(a)<(b), then a ≺ b; a set of INLNs and m = 1, 2, ..., n.Then the value aggregated from Definition 13 is still an INLN.
1, 2, ..., m) according to Definition 11.If two score values are equal, then calculate the accuracy values and certainty values.Step 4 According to Step 3 and Definition 12, rank the alternatives.

Table 1 .
Evaluation values of alternatives.

Table 2 .
Comparison of different operator.

Table 4 .
Comparison of different methods.