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

Abstract

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.

1. Introduction

Atanassov [1] defined intuitionistic fuzzy sets (IFSs), which are an extension of fuzzy sets (FSs) [2]. Atanassov and Gargov [3] and Atanassov [4] presented the definition of interval-valued IFSs (IVIFS); since then, FSs, IFSs, and IVIFSs have attracted more and more scholars’ and researchers’ attention [5,6,7,8,9,10,11,12,13,14,15,16]. Recently, Cuong and Kreinovich [17] developed the picture fuzzy set (PFS), and studied some basic operations, rules, and properties of PFS. Singh [18] explored the correlation coefficients of PFS. Son etc. [19,20] presented several novel fuzzy clustering algorithms based on PFSs. Thong and Son [21] defined a novel hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis and application to health care support systems. Wei [22] proposed the picture fuzzy cross-entropy method to deal with multiple attribute decision-making (MADM) problems. Thong and Son [23] gave Automatic Picture Fuzzy Clustering (AFC-PFS) for determining the most suitable number of clusters for FC-PFS. Wei [24] assigned some cosine similarity measures of PFSs for strategic decision-making on the basis of traditional similarity measures [25,26,27,28,29]. Wei [30] also defined some similarity measures for PFSs. Wei [31] defined some aggregation operators for MADM problems with respect to PFSs based on traditional aggregation operators [14,32,33,34,35,36,37]. Wei [38] proposed some picture fuzzy Hamacher aggregation operators with traditional Hamacher operations [39,40,41,42]. Zhang et al. [43] provided some relative projection models for PFSs. Wei [44] proposed the TODIM model for picture fuzzy MADM problems. Wang et al. [45] formulated a hybrid fuzzy MADM framework with PFSs. Wei et al. [46] designed PFN projection models to handle MADM problems. Wei [47] defined some picture 2-tuple linguistic Bonferroni mean operators in MADM. Wei et al. [48] proposed some picture 2-tuple linguistic operators. Wei [49] defined some picture uncertain linguistic Bonferroni mean operators. Wang and Li [50] combined PFSs and the hesitant fuzzy set [51,52,53,54,55,56] to propose the picture hesitant fuzzy set (PHFS) theory.

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.

2. Preliminaries

2.1. 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.

Definition 1

([17,65]). A PFS $A$ on the domain $X$ is an object which denotes as:

$$A=\left\{\langle x,{\mu }_A\left(x\right),{\eta }_A\left(x\right),{\nu }_A\left(x\right)\rangle |x\in X\right\}$$

(1)

where ${\mu }_A\left(x\right)\in \left[0,1\right]$ is known as the positive-membership degree function, ${\eta }_A\left(x\right)\in \left[0,1\right]$ is known as the neutral-membership degree function, and ${\nu }_A\left(x\right)\in \left[0,1\right]$ is known as the negative-membership degree function. At the same time, for all $\forall x\in X$, ${\mu }_A\left(x\right)$, ${\eta }_A\left(x\right)$ and ${\nu }_A\left(x\right)$ meet the following requirements: $0\le {\mu }_A\left(x\right)+{\eta }_A\left(x\right)+{\nu }_A\left(x\right)\le 1$. Furthermore, the refusal-membership degree function is presented as ${\pi }_A\left(x\right)=1-\left({\mu }_A\left(x\right)+{\eta }_A\left(x\right)+{\nu }_A\left(x\right)\right)$. For convenience, we call $\alpha =\left({\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }\right)$ a PFN, where ${\mu }_{\alpha }\in \left[0,1\right],{\eta }_{\alpha }\in \left[0,1\right],$ ${\nu }_{\alpha }\in \left[0,1\right]$, ${\mu }_{\alpha }+{\eta }_{\alpha }+{\nu }_{\alpha }\le 1.$

Definition 2

([31]). Assume that $\alpha =\left({\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }\right)$ and $\beta =\left({\mu }_{\beta },{\eta }_{\beta },{\nu }_{\beta }\right)$ are two PFNs. The score function of $\alpha$ $\beta$ can be denoted as $S\left(\alpha \right)={\mu }_{\alpha }-{\nu }_{\alpha }$ and $S\left(\beta \right)={\mu }_{\beta }-{\nu }_{\beta }$. Meanwhile, the accuracy functions of $\alpha$ and $\beta$ are presented as $H\left(\alpha \right)={\mu }_{\alpha }+{\eta }_{\alpha }+{\nu }_{\alpha }$ and $H\left(\beta \right)={\mu }_{\beta }+{\eta }_{\beta }+{\nu }_{\beta }$. Then, if $S\left(\alpha \right)<S\left(\beta \right)$, $\alpha <\beta$; if $S\left(\alpha \right)=S\left(\beta \right)$, then

(1) 

If$H\left(a\right)=H\left(\beta \right)$, $\alpha =\beta$;

(2) 

if$H\left(a\right)<H\left(\beta \right)$, $\alpha <\beta$.

Similar to the IFS [66,67], Wei [31] has proposed some operational rules for PFNs.

Definition 3

([31]). Assume that $\alpha =\left({\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }\right)$ and $\beta =\left({\mu }_{\beta },{\eta }_{\beta },{\nu }_{\beta }\right)$ are two PFNs. Then,

$$\begin{array}{l}\overline{\alpha }=\alpha =\left({\nu }_{\alpha },{\eta }_{\alpha },{\mu }_{\alpha }\right)\\ \alpha \wedge \beta =\left(\min \left\{{\mu }_{\alpha },{\mu }_{\beta }\right\},\max \left\{{\eta }_{\alpha },{\eta }_{\beta }\right\},\max \left\{{\nu }_{\alpha },{\nu }_{\beta }\right\}\right)\\ \alpha \vee \beta =\left(\max \left\{{\mu }_{\alpha },{\mu }_{\beta }\right\},\min \left\{{\eta }_{\alpha },{\eta }_{\beta }\right\},\min \left\{{\nu }_{\alpha },{\nu }_{\beta }\right\}\right)\\ \alpha \oplus \beta =\left({\mu }_{\alpha }+{\mu }_{\beta }-{\mu }_{\alpha }{\mu }_{\beta },{\eta }_{\alpha }{\eta }_{\beta },{\nu }_{\alpha }{\nu }_{\beta }\right);\\ \alpha \otimes \beta =\left({\mu }_{\alpha }{\mu }_{\beta },{\eta }_{\alpha }+{\eta }_{\beta }-{\eta }_{\alpha }{\eta }_{\beta },{\nu }_{\alpha }+{\nu }_{\beta }-{\nu }_{\alpha }{\nu }_{\beta }\right);\\ \lambda \alpha =\left(1-{\left(1-{\mu }_{\alpha }\right)}^{\lambda },{\eta }_{\alpha }^{\lambda },{\nu }_{\alpha }^{\lambda }\right)\\ {\alpha }^{\lambda }=\left({\mu }_{\alpha }^{\lambda },1-{\left(1-{\eta }_{\alpha }\right)}^{\lambda },1-{\left(1-{\nu }_{\alpha }\right)}^{\lambda }\right)\end{array}$$

According to Definition 3, Wei [31] obtained the following properties.

Theorem 1.

Assume that $\alpha =\left({\mu }_{\alpha },{\eta }_{\alpha },{\nu }_{\alpha }\right)$ and$\beta =\left({\mu }_{\beta },{\eta }_{\beta },{\nu }_{\beta }\right)$are two PFNs, $\lambda ,{\lambda }_1,{\lambda }_2>0$, then

$\begin{array}{l}(1)\alpha \oplus \beta =\beta \oplus \alpha ;\\ (2)\alpha \otimes \beta =\beta \otimes \alpha ;\\ (3)\lambda \left(\alpha \oplus \beta \right)=\lambda \alpha \oplus \lambda \beta ;\\ (4){\left(\alpha \otimes \beta \right)}^{\lambda }={\alpha }^{\lambda }\otimes {\beta }^{\lambda };\\ (5){\lambda }_1\alpha \oplus {\lambda }_2\alpha =\left({\lambda }_1+{\lambda }_2\right)\alpha ;\\ (6){\alpha }^{{\lambda }_1}\otimes {\alpha }^{{\lambda }_2}={\alpha }^{\left({\lambda }_1+{\lambda }_2\right)};\\ (7){\left({\alpha }^{{\lambda }_1}\right)}^{{\lambda }_2}={\alpha }^{{\lambda }_1{\lambda }_2}.\end{array}$

2.2. MM Operators

Muirhead [57] proposed the Muirhead mean (MM) operator.

Definition 4

([57]). Assume that $a_j(j=1,2,\dots ,n)$ is a set of non-negative real numbers, and let $P=(p_1,p_2,\dots ,p_n)\in R^n$ be a vector of parameters. Then

$${\mathrm{MM}}^{\mathrm{P}}(a_1,a_2,\dots ,a_n)={\left(\frac{1}{n!}{\displaystyle \sum _{\sigma \in S_n}{\displaystyle \prod _{j=1}^n{a}_{\sigma \left(j\right)}^{p_j}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}.$$

(2)

Then, we call MM^P the Muirhead mean (MM) operator, where $\sigma (j)(j=1,2,\dots ,n)$ is any permutation of $\left\{1,2,\dots ,n\right\}$, and $S_n$ is the set of all permutations of $\left\{1,2,\dots ,n\right\}$.

3. 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.

3.1. The PFMM Operator

Definition 5.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a list of PFNs. The definition of the PFMM operator is expressed as:

$${\mathrm{PFMM}}^{\mathrm{P}}({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}.$$

(3)

Theorem 2.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a set of PFNs. We can fuse all the PFN information by utilizing the PFMM operator, and the fused results are shown as:

$$\begin{array}{l}{\mathrm{PFMM}}^{\mathrm{P}}({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\\ =\left\{\begin{array}{l}{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(4)

Proof. 

$${\alpha }_{\sigma \left(j\right)}^{p_j}=\left\{{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j},1-{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j},1-{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}\right\}.$$

(5)

Thus,

$$\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}=\left\{{\displaystyle \prod _{j=1}^n{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j}},1-{\displaystyle \prod _{j=1}^n{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}},1-{\displaystyle \prod _{j=1}^n{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right\}.$$

(6)

Thereafter,

$$\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}\right)=\left\{\begin{array}{l}1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j}}\right)},{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)},\\ {\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}\end{array}\right\}.$$

(7)

Furthermore,

$$\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}\right)\right)=\left\{\begin{array}{l}\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j}}\right)}}^{\frac{1}{n!}}\right),{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}},\\ {{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\end{array}\right\}.$$

(8)

Therefore,

$$\begin{array}{l}{\mathrm{PFMM}}^{\mathrm{P}}({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\alpha }_{\sigma \left(j\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\\ =\left\{\begin{array}{l}{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\mu }_{{\alpha }_{\sigma \left(j\right)}}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(9)

Hence, (4) is kept. □

Example 1.

Let$\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)$be three PFNs, and$\mathrm{P}=(0.2,0.3,0.5),{\displaystyle \sum _{j=1}^3{p}_j}=1$. Then, according to (4), we have

$$\begin{array}{l}{\mathrm{PFMM}}^{(0.2,0.3,0.5)}\left(\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)\right)\\ =\left\{\begin{array}{l}{\left(1-{\left(\begin{array}{l}\left(1-{0.43}^{0.2}\times {0.79}^{0.3}\times {0.44}^{0.5}\right)\times \left(1-{0.43}^{0.2}\times {0.44}^{0.3}\times {0.79}^{0.5}\right)\\ \times \left(1-{0.79}^{0.2}\times {0.43}^{0.3}\times {0.44}^{0.5}\right)\times \left(1-{0.79}^{0.2}\times {0.44}^{0.3}\times {0.43}^{0.5}\right)\\ \times \left(1-{0.44}^{0.2}\times {0.43}^{0.3}\times {0.79}^{0.5}\right)\times \left(1-{0.44}^{0.2}\times {0.79}^{0.3}\times {0.43}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ 1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-0.36\right)}^{0.2}\times {\left(1-0.02\right)}^{0.3}\times {\left(1-0.45\right)}^{0.5}\right)\times \left(1-{\left(1-0.36\right)}^{0.2}\times {\left(1-0.45\right)}^{0.3}\times {\left(1-0.02\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.02\right)}^{0.2}\times {\left(1-0.36\right)}^{0.3}\times {\left(1-0.45\right)}^{0.5}\right)\times \left(1-{\left(1-0.02\right)}^{0.2}\times {\left(1-0.45\right)}^{0.3}\times {\left(1-0.36\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.45\right)}^{0.2}\times {\left(1-0.36\right)}^{0.3}\times {\left(1-0.02\right)}^{0.5}\right)\times \left(1-{\left(1-0.45\right)}^{0.2}\times {\left(1-0.02\right)}^{0.3}\times {\left(1-0.36\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ 1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-0.19\right)}^{0.2}\times {\left(1-0.10\right)}^{0.3}\times {\left(1-0.08\right)}^{0.5}\right)\times \left(1-{\left(1-0.19\right)}^{0.2}\times {\left(1-0.08\right)}^{0.3}\times {\left(1-0.10\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.10\right)}^{0.2}\times {\left(1-0.19\right)}^{0.3}\times {\left(1-0.08\right)}^{0.5}\right)\times \left(1-{\left(1-0.10\right)}^{0.2}\times {\left(1-0.08\right)}^{0.3}\times {\left(1-0.19\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.08\right)}^{0.2}\times {\left(1-0.19\right)}^{0.3}\times {\left(1-0.10\right)}^{0.5}\right)\times \left(1-{\left(1-0.08\right)}^{0.2}\times {\left(1-0.10\right)}^{0.3}\times {\left(1-0.19\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}}\end{array}\right\}\\ =\left(0.5340,0.2935,0.0942\right)\end{array}$$

It is clear that the PFMM operator satisfies the following three properties.

Theorem 3 (Idempotency).

If all${\alpha }_j\left(j=1,2,\cdots ,n\right)$are equal, i.e., ${\alpha }_j=\alpha$for all$j$, then

$${\mathrm{PFMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)=\alpha$$

(10)

Theorem 4 (Boundedness).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$is a group of PFNs. If

$${\alpha }^{-}=\underset{j}{\min }{\alpha }_j,{\alpha }^{+}=\underset{j}{\max }{\alpha }_j$$

Then

$${\alpha }^{-}\le {\mathrm{PFMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\alpha }^{+}$$

(11)

Theorem 5 (Monotonicity).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$and${{\alpha }^{\prime }}_j\left(j=1,2,\cdots ,n\right)$are two lists of PFNs. Let${\alpha }_j\le {{\alpha }^{\prime }}_j$for all$j$. Then,

$${\mathrm{PFMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\mathrm{PFMM}}^{\mathrm{P}}\left({{\alpha }^{\prime }}_1,{{\alpha }^{\prime }}_2,\cdots ,{{\alpha }^{\prime }}_n\right).$$

(12)

3.2. 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${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a list of PFNs. The PFWMM operator can be defined as:

$${\mathrm{PFWMM}}_{nw}^P({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}$$

(13)

Theorem 6.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a group of PFNs. We can fuse all the PFN information by utilizing the PFWMM operator, and the fused results are shown as:

$$\begin{array}{l}{\mathrm{PFWMM}}_{nw}^P={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\\ =\left\{\begin{array}{l}{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(14)

Proof. 

$$n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}=\left\{1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}},{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}},{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right\}$$

(15)

Thus,

$${\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}=\left\{{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j},1-{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j},1-{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}\right\}.$$

(16)

Therefore,

$$\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}=\left\{\begin{array}{l}{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}},1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}},\\ 1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\end{array}\right\}.$$

(17)

Thereafter,

$$\begin{array}{l}\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}\right)\\ =\left\{\begin{array}{l}1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)},{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)},\\ {\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}\end{array}\right\}\end{array}$$

(18)

Furthermore,

$$\begin{array}{l}\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}\right)\right)\\ =\left\{\begin{array}{l}1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}},{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}},\\ {{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\end{array}\right\}\end{array}$$

(19)

Therefore,

$$\begin{array}{l}{\mathrm{PFWMM}}_{nw}^P={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\alpha }_{\sigma \left(j\right)}\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\\ =\left\{\begin{array}{l}{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{\sigma \left(j\right)}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(20)

Hence, (14) is kept. □

Example 2.

Let$\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)$be three PFNs, and$\mathrm{P}=(0.2,0.3,0.5),{\displaystyle \sum _{j=1}^3{p}_j}=1$and$w=(0.4,0.3,0.3)$. Then, according to (4), we have

$$\begin{array}{l}{\mathrm{PFWMM}}_{(0.4,0.3,0.3)}^{(0.2,0.3,0.5)}\left(\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)\right)\\ =\left\{\begin{array}{l}{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.44\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.79\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.43\right)}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ 1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{0.36}^{1.2}\right)}^{0.2}\times {\left(1-{0.02}^{0.9}\right)}^{0.3}\times {\left(1-{0.45}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.36}^{1.2}\right)}^{0.2}\times {\left(1-{0.45}^{0.9}\right)}^{0.3}\times {\left(1-{0.02}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.02}^{0.9}\right)}^{0.2}\times {\left(1-{0.36}^{1.2}\right)}^{0.3}\times {\left(1-{0.45}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.02}^{0.9}\right)}^{0.2}\times {\left(1-{0.45}^{0.9}\right)}^{0.3}\times {\left(1-{0.36}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.45}^{0.9}\right)}^{0.2}\times {\left(1-{0.36}^{1.2}\right)}^{0.3}\times {\left(1-{0.02}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.45}^{0.9}\right)}^{0.2}\times {\left(1-{0.02}^{0.9}\right)}^{0.3}\times {\left(1-{0.36}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ 1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{0.19}^{1.2}\right)}^{0.2}\times {\left(1-{0.10}^{0.9}\right)}^{0.3}\times {\left(1-{0.08}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.19}^{1.2}\right)}^{0.2}\times {\left(1-{0.08}^{0.9}\right)}^{0.3}\times {\left(1-{0.10}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.10}^{0.9}\right)}^{0.2}\times {\left(1-{0.19}^{1.2}\right)}^{0.3}\times {\left(1-{0.08}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.10}^{0.9}\right)}^{0.2}\times {\left(1-{0.08}^{0.9}\right)}^{0.3}\times {\left(1-{0.19}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.08}^{0.9}\right)}^{0.2}\times {\left(1-{0.19}^{1.2}\right)}^{0.3}\times {\left(1-{0.10}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.08}^{0.9}\right)}^{0.2}\times {\left(1-{0.10}^{0.9}\right)}^{0.3}\times {\left(1-{0.19}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}}\end{array}\right\}\\ =\left(0.5347,0.2885,0.0853\right)\end{array}$$

It is clear that the PFWMM operator satisfies the following two properties.

Theorem 7 (Boundedness).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$is a group of PFNs. If

$${\alpha }^{-}=\underset{j}{\min }{\alpha }_j,{\alpha }^{+}=\underset{j}{\max }{\alpha }_j$$

Then

$${\alpha }^{-}\le {\mathrm{PFWMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\alpha }^{+}.$$

(21)

Theorem 8 (Monotonicity).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$and${{\alpha }^{\prime }}_j\left(j=1,2,\cdots ,n\right)$are two groups of PFNs. Let${\alpha }_j\le {{\alpha }^{\prime }}_j$for all$j$. Then,

$${\mathrm{PFWMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\mathrm{PFWMM}}_{nw}^P\left({{\alpha }^{\prime }}_1,{{\alpha }^{\prime }}_2,\cdots ,{{\alpha }^{\prime }}_n\right).$$

(22)

3.3. The PFDMM Operator

Qin and Liu [68] proposed the dual MM (DMM) as follows.

Definition 7

([68]). Assume that $a_i(i=1,2,\dots ,n)$ is a set of non-negative real numbers, and let $P=(p_1,p_2,\dots ,p_n)\in R^n$ be a vector of parameters. Then,

$${\mathrm{DMM}}^{\mathrm{P}}(a_1,a_2,\dots ,a_n)=\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}{\left({\displaystyle \prod _{\sigma \in S_n}{\displaystyle \sum _{j=1}^n{p}_j{a}_{\sigma \left(j\right)}}}\right)}^{\frac{1}{n!}}.$$

(23)

Then, we call ${\mathrm{DMM}}^{\mathrm{P}}$ the dual MM (DMM) operator, where $\sigma (j)(j=1,2,\dots ,n)$ is any permutation of $\left\{1,2,\dots ,n\right\}$, and $S_n$ is the set of all permutations of $\left\{1,2,\dots ,n\right\}$.

Combining the PFN information and the DMM operator, the definition of the PFDMM operator can be developed as follows.

Definition 8.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a group of PFNs, and let$P=(p_1,p_2,\dots ,p_n)\in R^n$be a vector of parameters. Then,

$${\mathrm{PFDMM}}^P({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_j)=\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}{\left(\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)\right)\right)}^{\frac{1}{n!}}.$$

(24)

Then, we call ${\mathrm{PFDMM}}^P$ the picture fuzzy DMM (PFDMM) operator, where $\sigma (j)(j=1,2,\dots ,n)$ is any permutation of $\left\{1,2,\dots ,n\right\}$, and $S_n$ is the set of all permutations of $\left\{1,2,\dots ,n\right\}$.

Theorem 9.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a list of PFNs. We can fuse all the PFN information by utilizing the PFDMM operator, and the fused results are shown as:

$$\begin{array}{l}{\mathrm{PFDMM}}^P({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}{\left(\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},{\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\eta }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\nu }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(25)

Proof. 

$$p_j{\alpha }_{\sigma (j)}=\left\{1-{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j},{\left({\eta }_{{\alpha }_j}\right)}^{p_j},{\left({\nu }_{{\alpha }_j}\right)}^{p_j}\right\}$$

(26)

Thus,

$$\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)=\left\{1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j}},{\displaystyle \prod _{j=1}^n{\left({\eta }_{{\alpha }_j}\right)}^{p_j}},{\displaystyle \prod _{j=1}^n{\left({\nu }_{{\alpha }_j}\right)}^{p_j}}\right\}.$$

(27)

Therefore,

$$\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)\right)=\left\{\begin{array}{l}{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j}}\right)},1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left({\eta }_{{\alpha }_j}\right)}^{p_j}}\right)},\\ 1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left({\nu }_{{\alpha }_j}\right)}^{p_j}}\right)}\end{array}\right\}.$$

(28)

Furthermore,

$$\begin{array}{l}{\left(\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j}}\right)}}^{\frac{1}{n!}},1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\eta }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}},\\ 1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\nu }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\end{array}\right\}\end{array}$$

(29)

Therefore,

$$\begin{array}{l}{\mathrm{PFDMM}}^P({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)=\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}{\left(\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma (j)}\right)\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\mu }_{{\alpha }_j}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},{\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\eta }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left({\nu }_{{\alpha }_j}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(30)

Hence, (25) is kept. □

Example 3.

Let$\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)$be three PFNs, and$\mathrm{P}=(0.2,0.3,0.5),{\displaystyle \sum _{j=1}^3{p}_j}=1$. Then, according to (25), we have

$$\begin{array}{l}{\mathrm{PFDMM}}^{(0.2,0.3,0.5)}\left(\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)\right)\\ =\left\{\begin{array}{l}1-\left(1-{\left(\begin{array}{l}\left(1-{\left(1-0.43\right)}^{0.2}\times {\left(1-0.79\right)}^{0.3}\times {\left(1-0.44\right)}^{0.5}\right)\times \left(1-{\left(1-0.43\right)}^{0.2}\times {\left(1-0.44\right)}^{0.3}\times {\left(1-0.79\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.02\right)}^{0.2}\times {\left(1-0.43\right)}^{0.3}\times {\left(1-0.44\right)}^{0.5}\right)\times \left(1-{\left(1-0.79\right)}^{0.2}\times {\left(1-0.44\right)}^{0.3}\times {\left(1-0.43\right)}^{0.5}\right)\\ \times \left(1-{\left(1-0.44\right)}^{0.2}\times {\left(1-0.43\right)}^{0.3}\times {\left(1-0.79\right)}^{0.5}\right)\times \left(1-{\left(1-0.44\right)}^{0.2}\times {\left(1-0.79\right)}^{0.3}\times {\left(1-0.43\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)\\ {\left(1-{\left(\begin{array}{l}\left(1-{0.36}^{0.2}\times {0.02}^{0.3}\times {0.45}^{0.5}\right)\times \left(1-{0.36}^{0.2}\times {0.45}^{0.3}\times {0.02}^{0.5}\right)\\ \times \left(1-{0.02}^{0.2}\times {0.36}^{0.3}\times {0.45}^{0.5}\right)\times \left(1-{0.02}^{0.2}\times {0.45}^{0.3}\times {0.36}^{0.5}\right)\\ \times \left(1-{0.45}^{0.2}\times {0.36}^{0.3}\times {0.02}^{0.5}\right)\times \left(1-{0.45}^{0.2}\times {0.02}^{0.3}\times {0.36}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ {\left(1-{\left(\begin{array}{l}\left(1-{0.19}^{0.2}\times {0.10}^{0.3}\times {0.08}^{0.5}\right)\times \left(1-{0.19}^{0.2}\times {0.08}^{0.3}\times {0.10}^{0.5}\right)\\ \times \left(1-{0.10}^{0.2}\times {0.19}^{0.3}\times {0.08}^{0.5}\right)\times \left(1-{0.10}^{0.2}\times {0.08}^{0.3}\times {0.19}^{0.5}\right)\\ \times \left(1-{0.08}^{0.2}\times {0.19}^{0.3}\times {0.10}^{0.5}\right)\times \left(1-{0.45}^{0.2}\times {0.10}^{0.3}\times {0.19}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}}\end{array}\right\}\\ =\left(0.5887,0.1598,0.0563\right)\end{array}$$

It is clear that the PFDMM operator satisfies the following three properties.

Theorem 10 (Idempotency).

Let${\alpha }_j\left(j=1,2,\cdots ,n\right)$be equal, i.e., ${\alpha }_j=\alpha$for all$j$. Then,

$${\mathrm{PFDMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)=\alpha .$$

(31)

Theorem 11 (Boundedness).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$is a list of PFNs. If

$${\alpha }^{-}=\underset{j}{\min }{\alpha }_j,{\alpha }^{+}=\underset{j}{\max }{\alpha }_j$$

Then

$${\alpha }^{-}\le {\mathrm{PFDMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\alpha }^{+}.$$

(32)

Theorem 12 (Monotonicity).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$and${{\alpha }^{\prime }}_j\left(j=1,2,\cdots ,n\right)$are two groups of PFNs, and let${\alpha }_j\le {{\alpha }^{\prime }}_j$for all$j$. Then,

$${\mathrm{PFDMM}}^{\mathrm{P}}\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\mathrm{PFDMM}}^{\mathrm{P}}\left({{\alpha }^{\prime }}_1,{{\alpha }^{\prime }}_2,\cdots ,{{\alpha }^{\prime }}_n\right).$$

(33)

3.4. The PFWDMM Operator

To take an attribute’s weight into account, the picture fuzzy weighted DMM (PFWDMM) operator can be defined as follows.

Definition 9.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a set of PFNs. The definition of the PFWDMM operator can be expressed as:

$${\mathrm{PFWDMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)=\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}{\left(\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)\right)}^{\frac{1}{n!}}.$$

(34)

Theorem 13.

Assume that${\alpha }_j=\left({\mu }_{{\alpha }_j},{\eta }_{{\alpha }_j},{\nu }_{{\alpha }_j}\right)\left(j=1,2,\cdots ,n\right)$is a group of PFNs. We can fuse all the PFN information by utilizing the PFWDMM operator, and the fused results are shown as:

$$\begin{array}{l}{\mathrm{PFWDMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\\ =\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}\underset{\sigma \in S_n}{\otimes }{\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(35)

Proof. 

$${\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}=\left\{{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}},1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}},1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right\}$$

(36)

Then,

$$p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}=\left\{1-{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j},{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j},{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}\right\}.$$

(37)

Thus,

$$\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)=\left\{\begin{array}{l}1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}},{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}},\\ {\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\end{array}\right\}.$$

(38)

Therefore,

$$\begin{array}{l}\underset{\sigma \in S_n}{\otimes }\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)\\ =\left\{\begin{array}{l}{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)},1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)},\\ 1-{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}\end{array}\right\}\end{array}$$

(39)

Furthermore,

$$\begin{array}{l}\underset{\sigma \in S_n}{\otimes }{\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}},1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}},\\ 1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\end{array}\right\}\end{array}$$

(40)

Therefore,

$$\begin{array}{l}{\mathrm{PFWDMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\\ =\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}\underset{\sigma \in S_n}{\otimes }{\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\}\end{array}$$

(41)

Hence, (35) is kept. □

Example 4.

Let$\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)$be three PFNs, and$\mathrm{P}=(0.2,0.3,0.5),{\displaystyle \sum _{j=1}^3{p}_j}=1$and$w=(0.4,0.3,0.3)$. Then, according to (35), we have

$$\begin{array}{l}{\mathrm{PFWDMM}}_{(0.4,0.3,0.3)}^{(0.2,0.3,0.5)}\left(\left(0.43,0.36,0.19\right),\left(0.79,0.02,0.10\right),\left(0.44,0.45,0.08\right)\right)\\ =\left\{\begin{array}{l}1-\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{0.43}^{1.2}\right)}^{0.2}\times {\left(1-{0.79}^{0.9}\right)}^{0.3}\times {\left(1-{0.44}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.43}^{1.2}\right)}^{0.2}\times {\left(1-{0.44}^{0.9}\right)}^{0.3}\times {\left(1-{0.79}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.79}^{0.9}\right)}^{0.2}\times {\left(1-{0.43}^{1.2}\right)}^{0.3}\times {\left(1-{0.44}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.79}^{0.9}\right)}^{0.2}\times {\left(1-{0.44}^{0.9}\right)}^{0.3}\times {\left(1-{0.43}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.44}^{0.9}\right)}^{0.2}\times {\left(1-{0.43}^{1.2}\right)}^{0.3}\times {\left(1-{0.79}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{0.44}^{0.9}\right)}^{0.2}\times {\left(1-{0.79}^{0.9}\right)}^{0.3}\times {\left(1-{0.43}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right),\\ {\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.45\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.02\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.36\right)}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}},\\ {\left(1-{\left(\begin{array}{l}\left(1-{\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.2}\times {\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.3}\times {\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.5}\right)\\ \times \left(1-{\left(1-{\left(1-0.08\right)}^{0.9}\right)}^{0.2}\times {\left(1-{\left(1-0.10\right)}^{0.9}\right)}^{0.3}\times {\left(1-{\left(1-0.19\right)}^{1.2}\right)}^{0.5}\right)\end{array}\right)}^{\frac{1}{3!}}\right)}^{\frac{1}{1}}\end{array}\right\}\\ =\left(0.5946,0.1586,0.0560\right)\end{array}$$

It is clear that the PFWDMM operator satisfies the following two properties.

Theorem 14 (Boundedness).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$is a group of PFNs. If

$${\alpha }^{-}=\underset{j}{\min }{\alpha }_j,{\alpha }^{+}=\underset{j}{\max }{\alpha }_j$$

Then

$${\alpha }^{-}\le {\mathrm{PFDWMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\alpha }^{+}.$$

(42)

Theorem 15 (Monotonicity).

Assume that${\alpha }_j\left(j=1,2,\cdots ,n\right)$and${{\alpha }^{\prime }}_j\left(j=1,2,\cdots ,n\right)$are two groups of PFNs, and let${\alpha }_j\le {{\alpha }^{\prime }}_j$for all$j$. Then,

$${\mathrm{PFWDMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\le {\mathrm{PFWDMM}}_{nw}^P\left({{\alpha }^{\prime }}_1,{{\alpha }^{\prime }}_2,\cdots ,{{\alpha }^{\prime }}_n\right).$$

(43)

4. Models for MADM with PFNs

According to the PFWMM (PFWDMM) operators, an MADM model with PFNs is briefly introduced in this part. Assume there are $m$ alternatives $A=\left\{A_1,A_2,\cdots ,A_m\right\}$, and $n$ attributes $G=\left\{G_1,G_2,\cdots ,G_n\right\}$ with a weighting vector $\omega =\left({\omega }_1,{\omega }_2,\cdots ,{\omega }_n\right)$, where ${\omega }_j\in \left[0,1\right]$, ${\displaystyle \sum _{j=1}^n{\omega }_j}=1$. Construct the picture fuzzy decision matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}={\left({\mu }_{ij},{\eta }_{ij},{\nu }_{ij}\right)}_{m\times n}$, where ${\mu }_{ij}$ means the positive-membership degree function that the alternative $A_i$ meets the attribute $G_j$, ${\eta }_{ij}$ is the neutral-membership degree function that the alternative $A_i$ does not meet the attribute $G_j$, ${\nu }_{ij}$ denotes the negative-membership degree function that the alternative $A_i$ does not meet the attribute $G_j$, ${\mu }_{ij}\in \left[0,1\right]$, ${\eta }_{ij}\in \left[0,1\right]$ ${\nu }_{ij}\in \left[0,1\right]$, ${\mu }_{ij}+{\eta }_{ij}+{\nu }_{ij}\le 1$, ${\pi }_{ij}=1-\left({\mu }_{ij}+{\eta }_{ij}+{\nu }_{ij}\right)$ $i=1,2,\cdots ,m$, $j=1,2,\cdots ,n$.

Afterwards, we utilize the PFWMM (PFDWMM) operator to solve MADM problems with PFN information.

Step 1.

We fuse the PFN information given in matrix $\tilde{R}$ by using the PFWMM operator

$$\begin{array}{l}{\alpha }_i={\mathrm{PFWMM}}_{nw}^P\left({\tilde{r}}_{i1},{\tilde{r}}_{i2},\cdots ,{\tilde{r}}_{in}\right)\\ ={\left(\frac{1}{n!}\left(\underset{\sigma \in S_n}{\oplus }\left(\underset{j=1}{\overset{n}{\otimes }}{\left(n{w}_{\sigma \left(j\right)}{\tilde{r}}_{ij}\right)}^{p_j}\right)\right)\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\\ =\left\{\begin{array}{l}{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\mu }_{{\alpha }_{ij}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\eta }_{{\alpha }_{ij}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ 1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\nu }_{{\alpha }_{ij}}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\},i=1,2,\cdots ,m.\end{array}$$

(44)

Or the PFWDMM operator

$$\begin{array}{l}{\mathrm{PFWDMM}}_{nw}^P\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\\ =\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}\underset{\sigma \in S_n}{\otimes }{\left(\underset{j=1}{\overset{n}{\oplus }}\left(p_j{\alpha }_{\sigma \left(j\right)}^{n{w}_{\sigma \left(j\right)}}\right)\right)}^{\frac{1}{n!}}\\ =\left\{\begin{array}{l}1-{\left(1-{{\displaystyle \prod _{\sigma \in S_n}\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\mu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\eta }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}},\\ {\left(1-{\displaystyle \prod _{\sigma \in S_n}{\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left(1-{\nu }_{{\alpha }_j}\right)}^{n{w}_{\sigma \left(j\right)}}\right)}^{p_j}}\right)}^{\frac{1}{n!}}}\right)}^{\frac{1}{{\displaystyle {\sum }_{j=1}^n{p}_j}}}\end{array}\right\},i=1,2,\cdots ,m.\end{array}$$

(45)

to obtain the overall preference results ${\alpha }_i\left(i=1,2,\cdots ,m\right)$ of the alternative $A_i$.

Step 2.

Compute the score values $S\left({\alpha }_i\right)\left(i=1,2,\cdots ,m\right)$ of the overall PFNs ${\alpha }_i\left(i=1,2,\cdots ,m\right)$ to order all the alternatives $A_i\left(i=1,2,\cdots ,m\right)$. If two scores $S\left({\alpha }_i\right)$ and $S\left({\alpha }_j\right)$ are equal, we can compute the accuracy values $H\left({\alpha }_i\right)$ of the overall PFNs ${\alpha }_i$, and then order the all the alternatives $A_i$.

Step 3.

Order all the alternatives $A_i\left(i=1,2,\cdots ,m\right)$ and select the best choice by $S\left({\alpha }_i\right)\left(i=1,2,\cdots ,m\right)$.

Step 4.

End.

5. Numerical Example and Comparative Analysis

5.1. 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 $A_i\left(i=1,2,3,4,5\right)$ are evaluated. The team of experts must make a decision according to the following four attributes: ① G₁ is the market risk; ② G₂ is the enterprise’s operation and management risk; ③ G₃ is the enterprise’s assets structure risk; and ④ G₄ is the environmental risk. The experts use the above attributes to evaluate the five possible financial investment alternatives $A_i\left(i=1,2,\cdots ,5\right)$ by using the PFNs by the decision-makers under the above four attributes (whose weighting vector is $\omega =\left(0.3,0.2,0.4,0.1\right)$), and construct the following matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{5\times 4}$ as shown in

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

	A1	A2	A3	A4	A5
G1	(0.43,0.36,0.19)	(0.43,0.32,0.18)	(0.71,0.23,0.01)	(0.25,0.49,0.15)	(0.50,0.45,0.03)
G2	(0.79,0.02,0.01)	(0.73,0.04,0.11)	(0.87,0.02,0.03)	(0.64,0.12,0.13)	(0.78,0.03,0.11)
G3	(0.43,0.45,0.08)	(0.03,0.62,0.33)	(0.04,0.55,0.30)	(0.01,0.69,0.25)	(0.03,0.57,0.26)
G4	(0.18,0.39,0.04)	(0.53,0.25,0.18)	(0.48,0.26,0.16)	(0.02,0.54,0.26)	(0.13,0.65,0.19)

.

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.

• Step1. Based on Table 1, fuse all PFNs ${\tilde{r}}_{ij}\left(j=1,2,\cdots ,n\right)$ by utilizing the PFWMM (PFWDMM) operator to obtain the overall PFNs ${\alpha }_i\left(i=1,2,3,4,5\right)$ of the financial investment alternative $A_i$. The fused values are listed in

  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.

  	PFWMM	PFWDMM
  A1	(0.4056,0.3604,0.1213)	(0.5234,0.2365,0.0555)
  A2	(0.3036,0.3700,0.2605)	(0.5571,0.2222,0.1703)
  A3	(0.3719,0.3239,0.1844)	(0.6637,0.1880,0.0666)
  A4	(0.0965,0.5274,0.2694)	(0.2934,0.3709,0.1701)
  A5	(0.2166,0.4819,0.2210)	(0.4738,0.3379,0.1087)

  .
• Step2. Based on the fused values shown in Table 2, the score values of the financial investment alternatives are given in

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

  	PFWMM	PFWDMM
  A1	0.2843	0.4679
  A2	0.0431	0.3867
  A3	0.1874	0.5971
  A4	−0.1730	0.1233
  A5	−0.0044	0.3651

  .
• Step 3. Based on the score values of the overall alternatives (

  Table 4. Ordering of the financial investment alternatives.

  	Ordering
  PFWMM	A1 > A3 > A2 > A5 > A4
  PFWDMM	A3 > A1 > A2 > A5 > A4

  ), we can rank all the alternatives, and the ranking of the financial investment alternatives is slightly different.

5.2. 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. Ranking of the financial investment alternatives.

	Ordering
PFWA	A3 > A1 > A5 > A2 > A4
PFWG	A1 > A3 > A5 > A2 > A4

.

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.

6. 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 $\lambda$. 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 PFWMM and PFWDMM operators to build some decision-making models 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. In the future, we can apply our defined PFN aggregation operators into other decision-making fields, such as the decision-making, risk analysis, and other fields that operate in an uncertain environment [54,69,70,71,72,73,74,75,76,77,78,79,80].