An Extended EDAS Method for Multicriteria Decision-Making Based on Multivalued Neutrosophic Sets

Neutrosophic set (NS) is a generalization of intuitionistic fuzzy set (IFS). It depicts not only the incomplete information but also the indeterminate information and inconsistent information which exist commonly in belief systems. In this paper, the evaluation based on distance from average solution (EDAS) method is extended to handle multicriteria decision-making problems with multivalued neutrosophic numbers (MVNNs). The average solution under all the criteria is calculated by the proposed convex weighted average operator of MVNNs. Then, the positive distance and the negative distance from each solution to the average solution are calculated, and the comprehensive evaluations of alternatives are obtained by integrating two kinds of distance values to get the ranking result. Finally, the rationality and eﬃciency of the proposed method are shown by the parameter analysis and comparisons with some existing methods.


Introduction
Zadeh [1,2] proposed the fuzzy set (FS) theory, which describes the uncertain information in the form of membership function.In some cases, it is difficult for the membership function to give a complete description of the decision information alone.So, Atanassov [3,4] proposed the concept of intuitionistic fuzzy sets (IFSs) on this basis of FSs by adding a nonmembership function.Torra and Narukawa [5,6] proposed hesitant fuzzy sets (HFSs) to describe the situation in which the expert is hesitant when giving evaluations.IFSs and HFSs have attracted a lot of researchers' attention and obtained many research results, which have been widely used to solve problems of multicriteria decision-making (MCDM) in engineering and management [7][8][9].
However, in some decision-making environment, multiple different discrete values are needed to represent decision information, rather than using specific or interval values.Under these circumstances, Wang and Li [43] proposed the definition of multivalued neutrosophic sets (MVNSs) and developed the TODIM method to solve the multicriteria decision-making (MCDM) with MVNSs.Liu [44] proposed some aggregation operators of MVNSs and applied them in decision-making problems.Peng and Wang [45] introduced a comparison method and some aggregation operators of MVNSs.Peng et al. [46,47] also developed a multivalued neutrosophic qualitative flexible approach based on likelihood and an extended ELECTRE approach with multivalued neutrosophic information.Ji et al. [48] proposed a projection-based TODIM method under multivalued neutrosophic environments.
Ghorabaee et al. [49] proposed a new method called evaluation on distance from average solution (EDAS) for multicriteria inventory classification.is method ranks the alternatives by calculating the positive distance and negative distance between each alternative and the average alternative.e good performance of the EDAS method in solving the inventory classification problems and MCDM problems was also shown in [49].EDAS is a simple and effective method for MCDM, but it can only deal with decision information described by numerical values.
Motivated by these ideas, in this paper, we propose an extended EDAS method to solve the MCDM problem, which is described by MVNSs.We revise the aggregation method of the positive distance and negative distance in [49] and permit the decision makers to adjust their proportions according to their preferences in calculating the comprehensive evaluations of alternatives.So, the method is more flexible for decision makers in choosing the most appropriate solution.Moreover, compared with some existing methods, it needs fewer data preprocessing and calculation and is suitable to solve the decisionmaking problem with more criteria and alternatives.e proposed method also can be used to the situations in which the decision information is described by SVNSs, HFSs, or Dual HFSs.
is paper is constructed as follows.Section 2 introduces the basic concepts and operations of of MVNSs.In Section 3, the EDAS method for MCDM problem with MVNSs is proposed.Algorithm analysis and comparison analysis are made based on two numerical examples to illustrate the feasibility and effectiveness of the proposed method in Section 4. Finally, the conclusions are given in Section 5.

Preliminaries
2.1.Multivalued Neutrosophic Sets.In this section, we recall some definitions, operations, and properties regarding MVNSs, which will be used in the rest of the paper.Definition 1 (see [14]).Let X be a universal set and x be a generic element in X.A single-valued neutrosophic set (SVNS) in X is defined as where T A is a truth-membership function, I A is a indeterminacy-membership function, and F A is a falsity-membership function.For each point x ∈ X, we have T A , I A , F A ∈ [0, 1] and 0 ≤ T A + I A + F A ≤ 1.
For convenience, 〈 T A (x), MVNS is a generalized form of SVNS, hesitant fuzzy set, and dual hesitant fuzzy set.In fact, if T A (x), I A (x), and F A (x), for any x ∈ X, has only one value, respectively, then a MVNSs is reduced to a SVNS.If I A (x) � Φ for any x ∈ X, then a MVNS is reduced to a dual hesitant fuzzy set [50]; if I A (x) � F A (x) � Φ for any x ∈ X, a MVNS is reduced to a hesitant fuzzy set [5].
Example 1. Assume that X � {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } is the universal set, where x 1 is the functionality, x 2 is the reliability, x 3 is the ease of use, x 4 is the efficiency, x 5 is the maintainability, x 6 is the portability of software, and the evaluations of a software under x 1 , x 2 , x 3 , x 4 , x 5 , and x 6 are in [0, 1].Evaluation information about a software can be obtained from some experts in the field.
eir opinions about the degree of "good," a degree of indeterminacy, and a degree of "poor" can be described by the following MVNS in X: (3)

e Comparison Method and Operational Rules.
In order to apply MVNNSs to solve practical problems, some comparison and operational rules are reviewed as follows.
For any two MVNNs, they can be compared by the method proposed by Peng et al. [45].(1) , and l T A , l I A , and l F A are the element numbers of T A , I A , and F A , respectively.
Definition 4 (see [45]).Let A and B be two MVNNs.en, their comparison rules are defined as follows: ( Definition 5 (see [45]).Let A � 〈 T A , I A , F A 〉 and B � 〈 T B , I B , F B 〉 be any two MVNNs.e operations on MVNNs are defined by be two MVNNs.en, the following properties are easily obtained from the abovementioned operations:

Aggregation Operators and EDAS
Method with MVNNSs 3.1.Aggregation Operators.In this section, we define the weighted average operators for MVNNS by using the idea of the convex combination [51].
MVNNs. e convex combination of A and B can be defined as follows: where A multivalued neutrosophic extended weighted average (MVNCWA) operator, where ω i is the weight vector of x i and We consider a special case of Definition 7. If then the MVNCWA operator reduces to the arithmetic average operator with the following form: 3.2.An Extended EDAS Method with MVNNS.In this section, we present an algorithm to handle MCDM problems with MVNNs based on the EDAS method proposed by Ghorabaee et al. [49].In this method, first, calculate the average alternative with the existing alternative.Next, calculate the positive distance and negative distance of each alternative relative to the average alternative.en, the comprehensive evaluations of alternatives are obtained by considering the influence of positive and negative distance on an average.Based on this idea, we propose an extended EDAS method to deal with decision information described by MVNNs.
e multiattribute decision-making problems based on MVNNs can be considered as follows.Let A � {A 1 , A 2 , . . ., A m } be the set of alternatives, C � {c 1 , c 2 , . . ., c n } be the set of criteria, B denote the index set of beneficial criteria, and N Complexity denote the index set of cost criteria.Let ω � {ω 1 , ω 2 , . . ., ω m } with ω i ≥ 0, m i�1 ω i � 1 be the weight vector of the creteria.Suppose that R � [R ij ] m×n is the decision matrix, where , and describes the evaluation information of the alternative A i with respect to the criterion c j .
Here, we give the basic idea of the extended EDAS decision-making method based on MVNNs.Firstly, the average alternative is calculated by the proposed weighted average operator given in Definition 7. Next, the score function of MVNN proposed in [45] is comonly used to calculate the dominance of the alternative, and the positive distance matrix and negative distance matrix are constructed by calculating the positive and negative distance between each alternative and the average alternative.Finally, the weighted average of the normalized positive and negative distance is used to obtain the comprehensive evaluation value of each alternative.
e concrete algorithm of EDAS for MCDM with MVNNs is described as follows.
Step 1: construct the decision matrix R � [R ij ] m×n , whose elements are described by the multivalued neutrosophic numbers.
Step 2: aggregate the evaluations of all the alternatives under each criterion by equation ( 6) and get the average alternative AV � (AV 1 , AV 2 , . . ., AV i , . . ., AV n ), where Step 3: calculate the positive distance matrix PDA � (PDA ij ) m×n and negative distance matrix NDA � (NDA ij ) m×n of each alternative from the average alternative according to equations ( 8) and ( 9): Remark 1.If s(AV j ) � s(R ij ) and a(AV j ) ≠ a(R ij ), then replace score function with accuracy function in equations ( 8) and (9); if s(AV j ) � s(R ij ) and a(AV j ) � a(R ij ), then replace score function with certainty function in equations ( 8) and (9).
Step 4: calculate the positive distance and the negative distance of each alternative A i (i � 1, . . ., m), denoted as PD i and ND i , respectively, by Step 5: the normalized values of PD and ND (NPD and NND) for all alternatives can be calculated by , Step 6: the comprehensive evaluation of each alternative AS i (i � 1, 2, . . ., m) can be calculated by where the decision makers can adjust the value of λ according to their preference for positive distance and negative distance.Especially, when the preference of the decision-maker is neutral and taken λ � (1/2), (12) is simplified to Step 7: rank the alternatives according to the comprehensive evaluation value AS i (i � 1, 2, . . ., m) and then choose the satisfactory scheme.

Illustrative Example
4.1.An Illustrative Example of the EDAS Method.Based on the multicriteria decision-making problem used in [22,43], the feasibility and effectiveness of the proposed method are analyzed.e multiattribute decision-making problem is described as follows.An investment company needs to choose appropriate investment projects so that to get rich reward.ere are four possible investment options: (1) A 1 : automobile company; (2) A 2 : food company; (3) A 3 : computer company; (4) A 4 : military industrial enterprise.In order to select appropriate investment project and obtain satisfied investment returns, the investment company should consider the enterprise performance of the investment project.
e enterprise performance is a comprehensive reflection of its profitability, operation, risk management, debt repayment, and social responsibility in the company's production and operation process.e ability of enterprises to fulfill their social responsibilities is the basis of their long-term existence and development.e social 4 Complexity responsibility ability of enterprises is manifested in many aspects.We choose relatively important environmental factors.After discussion by the company's development committee, the following three criteria are considered in the selection of investment options: (1) C 1 : risk control capability; (2) C 2 : growth factors; (3) C 3 : environmental impact factors, where C 1 and C 2 are benefit criteria and C 3 is a cost criterion.In this problem, the weight vector of criteria is given by w � (0.2, 0.25, 0.55).
Next, we apply the algorithm proposed in Section 3.2 to solve this decision-making problem.
Step 1: the decision matrix described by the multivalued neutrosophic sets is given by the decision maker, which is shown in Table 1: Step 2: calculate the average solution AV � (AV 1 , AV 2 , AV 3 ) according to equation (6), where So, by Definition 3, the score function value of AV 1 can be obtained as In a similar way, we can get s(AV 2 ) � 0.7127, s(AV 3 ) � 0.6489.
Step 3: calculate the positive distance matrix PDA and negative distance matrix NDA by equations ( 8) and ( 9 0.1014 0.1347 0 0 0 0.1044 0.1475 0.0646 0.0530 Step 4: according to equation ( 10), the positive distance PD i and the negative distance ND i (i � 1, 2, 3, 4) of each alternative are calculated as follows: Step 6: when λ � 0.5 is taken in equation ( 12), the comprehensive evaluations AS i (i � 1, 2, 3, 4) of alternatives can be obtained: Step 7: the ranking of the four projects is Obviously, A 4 is the best alternative.

Algorithmic Analysis. In
Step 6 of the proposed algorithm, decision makers can adjust the value of λ according to their preference for positive distance and negative distance.In order to study the influence of different parameter values on ranking results, we take different values of λ to calculate the ranking results of the alternatives, which are shown in Table 2.
From Table 2, it can be seen that the ranking results are consistent for λ ≤ 0.5 and λ > 0.5, respectively.On the other hand, the ranking for λ ≤ 0.5 is slightly different from that for λ > 0.5.When λ ≤ 0.5, the best scheme is A 4 and the worst Complexity scheme is A 3 ; when λ > 0.5, the best scheme is A 1 and the worst scheme is A 3 .e best one is A 4 or A 1 depending on the value of λ, while the worst one is always A 3 for different values of λ.So, the ranking results are stable with the different values of λ.
Compared with the original EDAS in [49] that fixed the value of λ to 0.5, the proposed method allows parameter λ from 0 to 1 and gives full consideration of the preference of decision makers for loss and gain to the average solution by adjusting the values of λ.When λ < 0.5, the loss of the alternative relative to the average solution is magnified.Otherwise, more attention is paid to the gain of the scheme over the average solution.e decision makers can adjust the value of λ according to their preference so as to solve the actual decision-making problem with a perfect choice.ere are other methods for the multicriteria decision problem with MVNSs, for example, the MVNWA and MVNGA operators proposed by Peng and Wang [45] are used to solve MCDM with MVNSs.

Comparison Analysis
e TODIM method was put forward by Wang and Li [43] to deal with decision information described with MVNSs.In order to verify the effectiveness of the proposed algorithm, we made a comparative analysis with these methods by using the case in Section 4.1.e calculation results of the three methods are shown in Table 3.
From Table 3, it can be seen that the best alternative is A 4 for all the methods and the worst one is A 2 for the method using the MVNGA operator in [45], while the worst one is A 3 for all other methods.e ranking result of our proposed method taking λ ≤ 0.5 is the same as that of the method in [43] and the method using the MVNWA operator in [45].
e TODIM method [43] can take into account the decision makers' psychological behavior and obtain more decision information by calculating the dominance of any two alternatives under each criterion.But when there are more alternatives in the MCDM problems, we need to deal with the dominance degree matrices with larger order.By the methods in [45], different aggregation operators can be used to deal with the different relationships of the aggregated arguments.But the number of operations and the sizes of the results will exponentially increase if more MVNNs are involved in the operations.erefore, it is not suitable to deal with the MCDM problems with large number of criteria.Compared with these methods, the proposed method has the advantages of the simplicity and less calculation process and is more suitable to deal with MVNNs under large number of criteria.

4.3.2.
Comparison with the Methods for SVNSs.In this section, the proposed method is compared with the two methods of Ye [21] and Chu and Liu [17] for simplified neutrosophic information.
e proposed method is also suitable for MCDM with SVNSs.In this situation, Ye [21] proposed the crossentropy of SVNSs and use it to solve MCDM problem, and Chi and Liu [17] proposed an expanded TOPSIS method to deal with interval neutrosophic information, which is suitable for simplified neutrosophic information since a numerical number can be seen an interval.In order to facilitate the comparative analysis, we cite the same illustrative example in [21] to compare the three methods.e problem to be solved is to choose the best supplier based on the suppliers' core competitiveness.Suppose that there are four suppliers A 1 , A 2 , A 3 , and A 4 whose core competencies are evaluated by the following four criteria C 1 , C 2 , C 3 , and C 4 : (1) the level of technology innovation C 1 , (2) the control ability of flow C 2 , (3) the ability of management C 3 , and (4) the level of service C 4 .Suppose the weight vector of the four criteria is w � (0.3, 0.25, 0.25, 0.2).
e decision matrix described by the single-valued

The ranking
The best alternative The worst alternative λ � 0.1 6 Complexity neutrosophic sets which are given by the decision maker is shown in Table 4. e above three methods are used to solve the decisionmaking problem, respectively, and the results of them are shown in Table 5.
From Table 5, it can be seen that the worst alternative is A 4 for all methods, the best one is A 3 for the proposed method and the crossentropy method in [21], while the best one is A 2 for the TOPSIS method in [17].e ranking results are identical using the crossentropy method in [21] and the proposed method with λ � 0.5.e result of the TOPSIS method differs from the other two methods.
e three methods have the different ideas to solve the MCDM with SVNSs.e crossentropy method in [21] calculates the crossentropy of each alternative to the positive ideal solution, which is used to rank alternatives.e TOPSIS method in [17] used the relative closeness coefficients to rank all alternative.e relative closeness coefficient is determined by the distances of an alternative from the positive ideal solution and the negative ideal solution.On the other hand, the two methods are not suitable for solving the MCDM problems with multivalue neutrosophic information, while the proposed method can be used to deal with single-value or multivalue neutrosophic information.

Conclusion
MVNSs are very useful in modeling the problems with uncertain, imprecise, incomplete, and inconsistent information, which widely exist in scientific and engineering cases.In this paper, based on the EDAS method in [49] and the proposed multivalued neutrosophic extended weighted average (MVNCWA) operator, we propose an extended EDAS method for decision-making problem described by MVNSs.In this method, the average alternative is calculated by aggregating the MVNNs under each criterion using the MVNCWA operator.Next, the positive distance and negative distance between each alternative and the average alternative under all the criteria are calculated.And then we revise the aggregation method of the positive distance and negative distance in [49] and permit the decision makers to adjust their proportions in calculating the comprehensive evaluations of alternatives according to their preferences.Two cases study are analyzed.
e first one is to illustrate that the proposed method is stable for the adjusted parameter and is feasible and effective for solving MCDM problems with MVNSs by comparing with some existing methods.And the other is to compare the proposed method with two methods for SVNSs.
From the above analysis, it is obvious that the proposed method needs fewer data preprocessing and less calculation and is suitable to solve the decision-making problem with more criteria and alternatives.It can also be used to the situations in which the decision information is described by SVNSs, HFSs, or Dual HFSs, which are the special forms of MVNSs.Moreover, the decision makers' preference for the loss or gain of an alternative is taken into consideration in the proposed method, so as to get the most favourite solution.
In the future, we will investigate the decision-making method for MVNSs under criteria with different relationships and intend to study the social network-based methods for different formats of NSs.

Data Availability
All of the data used to support the findings of this study are included within the article.

The method
The ranking The best The worst Crossentropy The proposed method (λ � 0.5) Table 3: Ranking results by different methods.

The method
The ranking The best The worst MVNWA The proposed method (λ ≤ 0.5)

2 ComplexityDefinition 3 .
Let A � 〈 T A , I A , F A 〉 be a MVNN.en, its score function s(A) , accuracy function a(A), and certainty function c(A) are defined as follows: ), where B � {C 1 , C 2 } and N � {C 3 }:

( 17 )NPD 1 � 1
Step 5: the normalized values NPD i and NND i for all alternatives are shown as follows:

Table 2 :
Compared results utilizing different values of λ.

Table 5 :
Ranking results by different methods.