Methods for MADM with Picture Fuzzy Muirhead Mean Operators and Their Application for Evaluating the Financial Investment Risk

: In this article, we study multiple attribute decision-making (MADM) problems with picture fuzzy numbers (PFNs) information. Afterwards, we adopt a Muirhead mean (MM) operator, a weighted MM (WMM) operator, a dual MM (DMM) operator, and a weighted DMM (WDMM) operator to deﬁne some picture fuzzy aggregation operators, including the picture fuzzy MM (PFMM) operator, the picture fuzzy WMM (PFWMM) operator, the picture fuzzy DMM (PFDMM) operator, and the picture fuzzy WDMM (PFWDMM) operator. Of course, the precious merits of these deﬁned operators are investigated. Moreover, we have adopted the PFWMM and PFWDMM operators to build a decision-making model to handle picture fuzzy MADM problems. In the end, we take a concrete instance of appraising a ﬁnancial investment risk to demonstrate our deﬁned model and to verify its accuracy and scientiﬁc merit.

The Muirhead mean (MM) [57] is a useful decision-making tool that can identify the inter-relationships among any number of information fusions, and some existing operators, such as arithmetic and geometric operators (not considering the inter-relationships).Both the Bonferroni mean (BM) operator [58][59][60][61][62][63] and the Maclaurin symmetric mean (MSM) operator [8,64] are special issues in the MM operator.So, the MM can provide a flexible and robust mechanism to process information fusion problems and more effectively solve MADM problems.However, in order to make the original MM operator process PFSs, it needs to be constrained to take only numeric arguments.
Although the IFSs theory has been applied in different fields, there are some real-life cases where IFSs are inappropriate.Voting can be a good example of this, because human voters can be divided into four groups: those who vote for, those who vote against, those who abstain, and those who refuse to vote.On the whole, PFSs [17] can handle human opinions that involve more answers, such as: yes, abstain, no, and refusal.However, none of the above methods is suitable for fusing picture fuzzy numbers (PFNs).Thus, the question of how to fuse PFN information is an interesting topic.In order to handle this case, in this article, we will present some picture fuzzy aggregation operators based on the traditional MM operators [57].
This research has four main purposes.The first is to develop a comprehensive MADM method for appraising financial investment risk with PFNs.The second lies in exploring several picture fuzzy aggregation operators based on the traditional MM operators.The third is to establish an integrated outranking decision-making method by the PFWMM (PFWDMM) operators.The final purpose is to demonstrate the application, practicality, and effectiveness of the proposed MADM method using a case study about financial investment risk.
For the sake of clarity, the rest of this research is organized as follows.Some basic definitions, operation rules, and score and accuracy functions of PFSs are introduced in the next section.Section 3 presents some picture fuzzy Muirhead mean aggregation operators, such as the PFMM operator; the picture fuzzy weighted MM (PFWMM) operator; the picture fuzzy dual MM (PFDMM) operator; and the picture fuzzy weighted dual (PFWDMM) operator.In Section 4, based on our defined aggregation operators and the PFN information, we build decision-making models to solve MADM problems.Section 5 gives a numerical example for evaluating a financial investment risk with picture fuzzy information in order to verify the method proposed in this article.Finally, some remarks are given to conclude this article.

Picture Fuzzy Sets
Picture fuzzy sets (PFSs) [17], as the extension of intuitionistic fuzzy sets (IFSs) [1], have been considered to be an effective tool to depict uncertain information in the application of MADM problems.The basic definition and fundamental theory of PFSs are introduced as follows.

Picture Fuzzy Muirhead Mean Aggregation Operators
In this part, based on PFN information and the MM operator, we are going to propose some new aggregation operators, including the PFMM operator and the PFWMM operator.

The PFMM Operator
Definition 5. Assume that α j = µ α j , η α j , ν α j (j = 1, 2, • • • , n) is a list of PFNs.The definition of the PFMM operator is expressed as: We can fuse all the PFN information by utilizing the PFMM operator, and the fused results are shown as: Proof. Thus, Thereafter, Furthermore, Therefore, Hence, ( 4) is kept.

The PFWMM Operator
To take an attribute's weight into account, the picture fuzzy weighted MM (PFWMM) operator can be defined as follows.Definition 6. Assume that α j = µ α j , η α j , ν α j (j = 1, 2, • • • , n) is a list of PFNs.The PFWMM operator can be defined as: Theorem 6. Assume that We can fuse all the PFN information by utilizing the PFWMM operator, and the fused results are shown as: Proof.

The PFDMM Operator
Qin and Liu [68] proposed the dual MM (DMM) as follows.
Combining the PFN information and the DMM operator, the definition of the PFDMM operator can be developed as follows.

The PFWDMM Operator
To take an attribute's weight into account, the picture fuzzy weighted DMM (PFWDMM) operator can be defined as follows.
The definition of the PFWDMM operator can be expressed as: Theorem 13.Assume that α j = µ α j , η α j , ν α j (j = 1, 2, • • • , n) is a group of PFNs.We can fuse all the PFN information by utilizing the PFWDMM operator, and the fused results are shown as: Proof.
Step 2. Compute the score values S(α i If two scores S(α i ) and S α j are equal, we can compute the accuracy values H(α i ) of the overall PFNs α i , and then order the all the alternatives A i .
Step 3. Order all the alternatives A i (i = 1, 2, • • • , m) and select the best choice by S(α i ) Step 4. End.

A Numerical Example
As a transitional state, China has implemented reform and an opening-up policy for more than 30 years.During this period, China's economy has made marvelous achievements, and so did reform in financial circles.However, people still worry about the accumulation of financial risks and other factors that make a financial system unstable.China did successfully bear the impact of the global financial crisis in 2008; however, this does not mean that our financial system has the ability to resist any risk.In fact, there are many potential factors that can make our financial system unstable.Thus, in this section, we shall present a numerical example for evaluating financial investment risk with IVPULNs in order to illustrate the method proposed in this paper.The project's aim is to evaluate the best financial investment alternatives from the different financial investment alternatives in an enterprise financial risk environment.In order to select most desirable enterprise, the desirability levels of five possible financial investment alternatives A i (i = 1, 2, 3, 4, 5) are evaluated.The team of experts must make a decision according to the following four attributes: 1 G 1 is the market risk; 2 G 2 is the enterprise's operation and management risk; 3 G 3 is the enterprise's assets structure risk; and 4 G 4 is the environmental risk.The experts use the above attributes to evaluate the five possible financial investment alternatives A i (i = 1, 2, • • • , 5) by using the PFNs by the decision-makers under the above four attributes (whose weighting vector is ω = (0.3, 0.2, 0.4, 0.1)), and construct the following matrix R = r ij 5×4 as shown in Table 1.To select the most desirable financial investment alternative, we use the PFWMM (PFWDMM) operator to solve the MADM model with PFNs.The computing steps are listed as follows.

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Step 2. Based on the fused values shown in Table 2, the score values of the financial investment alternatives are given in Table 3.

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Step 3. Based on the score values of the overall alternatives (Table 4), we can rank all the alternatives, and the ranking of the financial investment alternatives is slightly different.

Comparative Analysis
In addition, a comparative analysis was made between the PFWMM(PFWDMM) operator and the PFWA and PFWG operators defined by Wei [31].The comparative results are given in Table 5.From above, we can see that the fused values are slightly different in the ordering of the alternatives to show the accuracy and scientific merit of the proposed approaches.However, the PFWA and PFWG operators have the limitation of not considering the relationships between the attributes in the fused information.Our defined PFWMM and PFWDMM operators have the advantage of taking the interaction relationships among any number of attributes into account, and can be more effective and accurate.

Conclusions
Aggregation operators have become a hot issue and an important tool in the decision-making fields in recent years.However, they still have some limitations in practical applications.For example, some aggregation operators suppose that the attributes are independent of each other.However, the MM operator and the dual MM operator have a prominent characteristic: they can consider the interaction relationships among any number of attributes by a parameter vector λ.According to the MM operator and the dual MM operator, in this article, we defined some new MM and DMM aggregation operators to deal with MADM problems under a PFN environment, including the PFMM operator, the PFWMM operator, the PFDMM operator and the PFWDMM operator.Of course, the precious merits of these defined operators are investigated.Moreover, we have adopted

Table 1 .
The picture fuzzy number (PFN) information decision matrix.

Table 2 .
The fused values of the financial investment alternatives by the picture fuzzy weighted Muirhead mean (PFWMM) operator and the picture fuzzy weighted dual Muirhead mean (PFWDMM) operator.

Table 3 .
The score functions of the financial investment alternatives.

Table 4 .
Ordering of the financial investment alternatives.

Table 5 .
Ranking of the financial investment alternatives.