Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment

: The aim of this paper is to introduce some new operators for aggregating single-valued neutrosophic (SVN) information and to apply them to solve the multi-criteria decision-making (MCDM) problems. Single-valued neutrosophic set, as an extension and generalization of an intuitionistic fuzzy set, is a powerful tool to describe the fuzziness and uncertainty, and Muirhead mean (MM) is a well-known aggregation operator which can consider interrelationships among any number of arguments assigned by a variable vector. In order to make full use of the advantages of both, we introduce two new prioritized MM aggregation operators, such as the SVN prioritized MM (SVNPMM) and SVN prioritized dual MM (SVNPDMM) under SVN set environment. In addition, some properties of these new aggregation operators are investigated and some special cases are discussed. Furthermore, we propose a new method based on these operators for solving the MCDM problems. Finally, an illustrative example is presented to testify the efﬁciency and superiority of the proposed method by comparing it with the existing method.


Introduction
Multicriteria decision-making (MCDM) is one of the hot topics in the decision-making field to choose the best alternative to the set of the feasible one.In this process, the rating values of each alternative include both precise data and experts' subjective information [1,2].However, traditionally, it is assumed that the information provided by them are crisp in nature.However, due to the complexity of the system day-by-day, the real-life contains many MCDM problems where the information is either vague, imprecise or uncertain in nature [3].To deal with it, the theory of fuzzy set (FS) [4] or extended fuzzy sets such as intuitionistic fuzzy set (IFS) [5], interval-valued IFS (IVIFS) [6] are the most successful ones, which characterize the criterion values in terms of membership degrees.Since their existence, numerous researchers were paying more attention to these theories and developed several approaches using different aggregation operators [7][8][9][10] and ranking methods [11][12][13] in the processing of the information values.
It is remarked that neither the FS nor the IFS theory are able to deal with indeterminate and inconsistent data.For instance, consider an expert which gives their opinion about a certain object in such a way that 0.5 being the possibility that the statement is true, 0.7 being the possibility that the statement is false and 0.2 being the possibility that he or she is not sure.Such type of data is not handled with FS, IFS or IVIFS.To resolve this, Smarandache [14] introduced the concept neutrosophic sets (NSs).In NS, each element in the universe of discourse set has degrees of truth membership, indeterminacy-membership and falsity membership, which takes values in the non-standard unit interval (0 − , 1 + ).Due to this non-standard unit interval, NS theory is hard to implement on the practical problems.So in order to use NSs in engineering problems more easily, some classes of NSs and their theories were proposed [15,16].Wang et al. [16] presented the class of NS named as interval NS while in Wang et al. [15], a class of single-valued NS (SVNS) is presented.Due to its importance, several researchers have made their efforts to enrich the concept of NSs in the decision-making process and some theories such as distance measures [17], score functions [18], aggregation operators [19][20][21][22][23] and so on.
Generally, aggregation operators (AOs) play an important role in the process of MCDM problems whose main target is to aggregate a collection of the input to a single number.In that direction, Ye [21] presented the operational laws of SVNSs and proposed the single-valued neutrosophic (SVN) weighted averaging (SVNWA) and SVN weighted geometric average (SVNWGA) operators.Peng et al. [22] defined the improved operations of SVN numbers (SVNNs) and developed their corresponding ordered weighted average/geometric aggregation operator.Nancy and Garg [24] developed the weighted average and geometric average operators by using the Frank norm operations.Liu et al. [25] developed some generalized neutrosophic aggregation operators based on Hamacher operations.Zhang et al. [26] presented the aggregation operators under interval neutrosophic set (INS) environment and Aiwu et al. [27] proposed some of its generalized operators.Garg and Nancy [19] developed a nonlinear optimization model to solve the MCDM problem under the INS environment.
From the above mentioned AOs, it is analyzed that all these studies assume that all the input arguments used during aggregation are independent of each other and hence there is no interrelationship between the argument values.However, in real-world problems, there always occurs a proper relationship between them.For instance, if a person wants to purchase a house then there is a certain relationship between its cost and the locality.Clearly, both the factors are mutually dependent and interacting.In order to consider the interrelationship of the input arguments, Bonferroni mean (BM) [28], Maclaurin symmetric mean (MSM) [29], Heronian mean (HM) [30] etc., are the useful aggregation functions.Yager [31] proposed the concept of BM whose main characteristic is its capability to capture the interrelationship between the input arguments.Garg and Arora [32] presented BM aggregation operators under the intuitionistic fuzzy soft set environment.In these functions, BM can capture the interrelationship between two arguments while others can capture more than two relationships.Taking the advantages of these functions in a neutrosophic domain, Liu and Wang [33] applied the BM to a neutrosophic environment and introduce the SVN normalized weighted Bonferroni mean (SVNNWBM) operator.Wang et al. [34] proposed the MSM aggregation operators to capture the correlation between the aggregated arguments.Li et al. [20] presented HM operators to solve the MCDM problems under SVNS environment.Garg and Nancy [35] presented prioritized AOs under the linguistic SVNS environment to solve the decision-making problems.Wu et al. [36] developed some prioritized weighted averaging and geometric aggregation operators for SVNNs.Ji et al. [37] established the single-valued prioritized BM operator by using the Frank operations.An alternative to these aggregations, the Muirhead mean (MM) [38] is a powerful and useful aggregation technique.The prominent advantage of the MM is that it can consider the interrelationships among all arguments, which makes it more powerful and comprehensive than BM, MSM and HM.In addition, MM has a parameter vector which can make the aggregation process more flexible.
Based on the above analysis, we know the decision-making problems are becoming more and more complex in the real world.In order to select the best alternative(s) for the MCDM problems, it is necessary to express the uncertain information in a more profitable way.In addition, it is important to deal with how to consider the relationship between input arguments.Keeping all these features in mind, and by taking the advantages of the SVNS, we combine the prioritized aggregation and MM and propose prioritized MM (PMM) operator by considering the advantages of both.These considerations have led us to consider the following main objectives for this paper: 1.
to handle the impact of the some unduly high or unduly low values provided by the decision makers on to the final ranking; 2.
to present some new aggregation operators to aggregate the preferences of experts element; 3.
to develop an algorithm to solve the decision-making problems based on proposed operators; 4.
to present some example in which relevance of the preferences in SVN decision problems is made explicit.
Since in our real decision-making problems, we always encounter a problem of some attributes' values, provided by the decision makers, whose impact on the decision-making process are unduly high or unduly low; this consequently results in a bad impression on the final results.To handle it, in the first objective we utilize prioritized averaging (PA) as an aggregation function which can handle such a problem very well.To achieve the second objective, we develop two new AOs, named as SVN prioritized MM (SVNPMM) and SVN prioritized dual MM (SVNPDMM) operators, by extending the operations of SVNNs by using MM and PA operators.MM operator is a powerful and useful aggregation technique with the feature that it considers the interrelationships among all arguments which makes it more powerful and comprehensive than BM [28], MSM [29] and HM [30].Moreover, the MM has a parameter vector which can make the aggregation process more flexible.Several properties and some special cases from the proposed operators are investigated.To achieve the third objective, we establish an MCDM method based on these proposed operators under the SVNS environment where preferences related to each alternative is expressed in terms of SVNNs.An illustrative example is presented to testify the efficiency and superiority of the proposed method by comparative analysis with the other existing methods for fulfilling the fourth objective.Further, apart from these, we verify that the methods proposed in this paper have advantages with respect to existing operators as follows: (1) some of the existing AOs can be taken as a special case of the proposed operators under NSs environment, (2) they consider the interrelationship among all arguments, (3) they are more adaptable and feasible than the existing AOs based on the parameter vector, (4) the presented approach considers the preferences of the decision maker in terms of risk preference as well as risk aversion.
The rest of the manuscript is organized as follows.In Section 2, we briefly review the concepts of SVNS and the aggregation operators.In Section 3, two new AOs based on PA and MM operations are developed under SVNS environment and their desirable properties are investigated.In addition, some special cases of the operators by varying the parametric value are discussed.In Section 4, we explore the applications of SVNN to MCDM problems with the aid of the proposed decision-making method and demonstrate with a numerical example.Finally, Section 5 gives the concluding remarks.

Preliminaries
In this section, some basic concepts related to SVNSs have been defined over the universal set X with a generic element x ∈ X.
By assigning some special vectors to P, we can obtain some special cases of the MM: 1.
which is the MSM operator [29].
The working of the proposed operator is demonstrated through a numerical example, which is illustrated as follow.
Further, we have .
Hence, we get .
Proof.Since min j {µ j } ≤ µ j , therefore min In the same manner, we get which completes the proof.
This property is called commutativity.
Proof.The proof of this theorem can be easily followed from Equation ( 12), so we omit it here.
Theorem 5.If the priority level of all the SVNNs is taken to be the same then SVNPMM operator reduces to single-valued neutrosophic Muirhead mean (SVNMM) operator.This property is called reducibility.
Proof.Take ξ j = n for all j denotes the prioritized level.As ξ j is same for all j, so, we have (nξ j )α σ(j) = α σ(j) , which implies However, apart from these, the following particular cases are observed from the proposed SVNPMM operator by assigning different values to P = (p 1 , p 2 , . . ., p n ).

Single-Valued Neutrosophic Prioritized Dual Muirhead Mean Operator
In this section, we propose prioritized dual aggregation operator based on the MM under the SVNS environment.Definition 8.A SVNPDMM operator is a mapping SVNPDMM : Ω n → Ω given by SVNPDMM(α 1 , α 2 , . . . , Theorem 6.The collective value by using Equation ( 13) is still a SVNN and is given as Proof.The proof follows from Theorem 1.
In order to illustrate the working of this operator, we demonstrate it through an illustrative example as follows.
Example 2. If we have taken the data as considered in Example 1 to illustrate the aggregation operator as defined in Theorem 6 then, we have Similarly, we have , 1 − (0.0268) Similar to SVNPMM operator, it is observed that this SVNPDMM operator also satisfies same properties for a collection of SVNNs α j (j = 1, 2, . . ., n) which are stated without proof as below.

Multi-Criteria Decision-Making Approach Based on Proposed Operators
In this section, we present an MCDM approach for solving the decision-making problem under the SVNS environment by using the proposed operators.A practical example from a field of decision-making has been taken to illustrate it.

Proposed Decision-Making Approach
Consider an MCDM problem which consists of m alternatives A 1 , A 2 , . . ., A m which are evaluated under the n criteria C 1 , C 2 , . . ., C n .For this, an expert was invited to evaluate these alternatives under the SVN environment such that their rating values were given in the form of SVNNs.For instance, corresponding to alternative A i under criterion C j , when we ask the opinion of an expert about the alternative A i with respect to the criterion C j , he or she may observe that the possibility degree in which the statement is good is µ ij , the statement is false is ν ij and the degree in which he or she is unsure is ρ ij .In this case, the evaluation of these alternatives are represented as SVNN This collective information is represented in the form of the neutrosophic decision-matrix D which is represented as Based on this information, the procedure to find the best alternative(s) is summarized as follows: Step 1: If in the considered decision-making problem, there exist two kinds of criteria, namely the benefit and the cost types, then all the cost type criteria should be normalized into the benefit type by using the following equation Step 2: Compute H ij (i = 1, 2, . . ., m) as Step 3: For a given parameter P = (p 1 , p 2 , . . ., p n ), utilize either SVNPMM or SVNPDMM operator to get the collective values r i = (µ i , ρ i , ν i )(i = 1, 2, . . ., m) for each alternative as or Step 4: Calculate score values of the overall aggregated values r i = (µ i , ρ i , ν i ) (i = 1, 2, . . ., m) by using equation Step 5: Rank all the feasible alternatives A i (i = 1, 2, . . ., m) according to Definition 3 and hence select the most desirable alternative(s).
The above mentioned approach has been illustrated with a numerical example discussed in Section 4.2.

Illustrative Example
A travel agency named Marricot Tripmate has excelled in providing travel related services to domestic and inbound tourists.The agency wants to provide more facilities like detailed information, online booking capabilities, the ability to book and sell airline tickets, and other travel related services to their customers.For this purpose, the agency intends to find an appropriate information technology (IT) software company that delivers affordable solutions through software development.To complete this motive, the agency forms a set of five companies (alternatives), namely, Zensar Tech (A 1 ), NIIT Tech (A 2 ), HCL Tech(A 3 ), Hexaware Tech(A 4 ), and Tech Mahindra (A 5 ) and the selection is held on the basis of the different criteria, namely, technology expertise (C 1 ), service quality (C 2 ), project management (C 3 ) and industry experience (C 4 ).The prioritization relationship for the criterion is In order to access these alternatives, an expert was invited and he gives their preferences toward each alternative in the form of SVNN.Their complete preferences of the expert are summarized in Table 1.Then, the following steps of the proposed approach have been executed as below Step 1: As all the criteria values are of the same types, the original decision matrix need not be normalized.
Step 5: Since s(r 3 ) > s(r 2 ) > s(r 5 ) > s(r 4 ) > s(r 1 ) and thus ranking order of their corresponding alternatives is Here refers "preferred to".Therefore, A 3 is the best one according to the requirement of the travel agency.
Contrary to this, if we utilize SVNPDMM operator then the following steps are executed as: Step 1: Similar to above Step 1.
Step 2: Similar to above Step 2.
Step 5: The ranking order of the alternatives, based on the score values, is A 5 A 3 A 1 A 2 A 4 and hence A 5 as the best alternative among the others.
The approaches in [21,22,25] are based on a simple weighted averaging operator.However, in these approaches, some of the weakness are (1) they assume that all the input arguments are independent, which is somewhat inconsistent with reality; (2) they cannot consider the interrelationship among input arguments.However, on the contrary, the proposed method can capture the interrelationship among input arguments.In addition to that, the proposed operator has an additional parameter P which provide a feasible aggregation process.In addition, some of the existing operators are deduced from the proposed operators.Thus, the proposed method is more powerful and flexible than the methods in [21,22,25].
In [33,37], authors presented an approach based on the BM aggregation operator where they considered the interrelationship between the arguments.However, the main flaws of these approaches are that they consider only two arguments during the interrelationship.On the other hand, in [34] authors have presented an aggregation operator based on MSM by considering two or more arguments during the interrelationship; however, these methods [33,34,37] fail to reflect the interrelationship among all input arguments.Finally, in [20] authors used the Heronian mean AOs without considering any interrelationship between the arguments.
As compared with these existing approaches, the merits of the proposed approach are that it can reflect the interrelationships among all the input arguments.In addition, the proposed operators have an additional parameter P which makes the proposed approach more flexible and feasible.

Conclusions
Muirhead mean aggregation operator is more flexible by using a variable and considering the multiple interrelationships between the pairs of the input arguments.On the other hand, SVNS is more of a generalization of the fuzzy set, intuitionistic fuzzy set to describe the uncertainties in the data.In order to combine their advantages, in the present paper, we develop some new MM aggregation operators for the SVNSs including the SVNPMM and the SVNPDMM.The desirable properties of these proposed operators and some special cases are discussed in detail.Moreover, we presented two new methods to solve the MCDM problem based on the proposed operators.The proposed method is more general and flexible, not only by considering the parametric vector P but also by taking into account the multiple interrelationships between the input argument.Apart from this, the remarkable characteristic of the proposed operator is to reflect the correlations of the aggregated arguments by considering the fact that those different criteria having different priority levels.The mentioned approach has been demonstrated through a numerical example and compares their corresponding proposed results with some of the results of existing approaches.From the computed results, it has been observed that the proposed approach can be efficiently utilized to solve decision-making problems where uncertainties and vagueness in the data occur concurrently.Moreover, by changing the values of the parameter P, an analysis has been done which concludes that the proposed operators provide more choices to the decision makers according to their preferences.In addition, it is also regarded as considering the risk preference of decision makers by the parameter P. So, the proposed approach is more suitable and flexible to solve the practical and complex MCDM problems.
In future works, we will apply our proposed method for more practical decision-making problems.In addition, considering the superiority of MM operator, we can extend it to some new fuzzy sets, such as Pythagorean fuzzy sets [39][40][41], applications to MCDM [42][43][44], multiplicative sets [45,46] and so on.

Table 3 .
Comparison of different approaches and aggregation operators.