Models for multiple attribute decision making with dual hesitant pythagorean fuzzy information

Abstract

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Muirhead Mean (MM) operators with dual Pythagorean hesitant fuzzy information. Then, motivated by the ideal of MM operators, we have developed some MM operators for aggregating dual hesitant Pythagorean fuzzy information. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the dual hesitant Pythagorean fuzzy multiple attribute decision making problems. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making (MADM)dual Pythagorean hesitant fuzzy sets Muirhead Mean (MM) operators supplier selection

1. Introduction

Atanassov [1] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [2] whose basic component is only a membership function. Xu [3] developed the intuitionistic fuzzy weighted averaging (IFWA) operator, intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. Xu and Yager [4] developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Li et al. [5] defined the Hamy mean (HM) operator and Dombi Hamy mean (DHM) operator [6, 7, 8, 9] with intuitionistic fuzzy information. Lu and Wei [10] defined the TODIM method for performance appraisal on social-integration-based rural reconstruction with interval-valued intuitionistic fuzzy information. Wu et al. [11] proposed the VIKOR method for financing risk assessment of rural tourism projects under interval-valued intuitionistic fuzzy environment. Wu et al. [12] gave some interval-valued intuitionistic fuzzy dombi heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas. Pythagorean fuzzy sets (PFSs) is put forward by Yager and Abbasov [13] compared with intuitionistic fuzzy sets [14], which expanded the membership and the non-membership degree to meet the conditions of the sum of the squares less than or equal to 1. Since the introduction of PFSs, many scholars have done a lot of research and expansion on it. Zhang and Xu [15] proposed a sort optimization method based on similar ideal solution for MADM problem with PFSs. Garg [16] first pointed out the shortcomings of calculating the correlation coefficients of intuitionistic fuzzy sets (IFSs) and designed a new (weighted) correlation coefficient formula to measure the relationship between two PFSs. In order to better understand PFSs, Peng and Yang [17] redefined the division and subtraction operation of PFS and discussed their properties in detail. Ma and Xu [18] defined a new class of operators based on the given algorithm and compared with the existing operators for PFSs. Liang et al. [19] proposed a concept based on interval Pythagorean fuzzy number (IVPFN) and innovated a method to deal with MAGDM problem. In order to enrich the PFS theory, Gou et al. [20] studied the basic properties of continuity, derivability and differentiability for PFSs. Garg [21] defined several operators of PFSs and studied the corresponding properties. Peng and Yang [22] studied the MAGDM problem of interval valued PFSs. Qin et al. [23] studied the new distance measurement method for PFSs, and innovatively introduced the generalized Pythagorean fuzzy weighted averaging distance (GPFWAD) operator and generalized Pythagorean fuzzy ordered weighted averaging distance (GPFOWAD) operator. Bolturk [24] extended the CODAS method to the PFSs and considered the decision maker’s hesitation. Wei et al. [25] defined the TODIM method based on the Pythagorean 2-tuple linguistic fuzzy sets (P2TLSs) by combining PFSs [26, 27] with 2-tuple linguistic set. Liu et al. [28] used the interval-valued Pythagorean uncertain linguistic sets to express the subjectivity and uncertainty of decision makers and used the QUALIFLEX technology to select the most suitable robot. Wei and Lu [29] proposed the concept and basic operations of the dual hesitant Pythagorean fuzzy sets (DHPFSs) based on the PFSs and dual hesitant fuzzy sets [30] and developed some Hamacher aggregation operators for aggregating dual hesitant Pythagorean fuzzy information. Tang and Wei [31] developed some dual hesitant Pythagorean fuzzy Bonferroni mean operators in multi-attribute decision making. Lu et al. [32] designed the Bidirectional project method for dual hesitant Pythagorean fuzzy multiple attribute decision-making with their application to performance assessment of new rural construction. Wei et al. [33] defined Dual hesitant Pythagorean fuzzy Hamy mean operators in multiple attribute decision making. Tang et al. [34] built dual hesitant pythagorean fuzzy Heronian mean operators in multiple attribute decision making.

Because DHPFSs can easily describe the fuzzy information, and the MM operator can capture interrelationships among any number of arguments assigned by a variable vector, it is necessary to extend the MM operator to deal with the DHPFSs. The purpose of this paper is to propose some MM operators under DHPFSs by extending MM and DMM operators to DHPFSs, then to study some properties of these operators, and applied them to solve the MADM problems in which the attributes take the form of DHPFSs.

In order to achieve this purpose, the rest of this paper is set out as follows. Section 2 reviews some basic concepts and theory of DHPFSs. In Section 3, we propose the dual hesitant Pythagorean fuzzy MM (DHPFMM) operator and dual hesitant Pythagorean fuzzy weighted MM (DHPFWMM) operator, and study some properties of these operators. In Section 4, we develop two MADM methods for DHPFSs based on the DHPFWMM operator. In Section 5, an illustrative example is given to verify the validity of the proposed methods and to show their advantages. In Section 6, we give some conclusions of this study.

2. Preliminaries

2.1 Pythagorean fuzzy set

The basic concepts of PFSs [35, 36] are briefly reviewed in this section. Afterwards, novel score and accuracy functions for PFSs are proposed. Furthermore, a new comparison method for PFSs is developed.

Definition 1 [35, 36]. Let $X$ be a fix set. A PFSs is an object having the form

$\displaystyle P=\{{\langle{x,({\mu_{P}(x),\nu_{P}(x)})}\rangle|{x\in X}}\}$ (1)

where the function $\mu_{P}:X\to[{0,1}]$ defines the degree of membership and the function $\nu_{P}:X\to[{0,1}]$ defines the degree of nonmembership of the element $x\in X$ to $P$ , respectively, and, for every $x\in X$ , it holds that

$\displaystyle({\mu_{p}(x)})^{2}+({\nu_{p}(x)})^{2}\leqslant 1.$ (2)

Definition 2 [15]. Let $\tilde{a}_{1}=({\mu_{1},\nu_{1}})$ , $\tilde{a}_{2}=({\mu_{2},\nu_{2}})$ , and $\tilde{a}=({\mu,\nu})$ be three Pythagorean fuzzy numbers, and some basic operations on them are defined as follows:

(1)

$\tilde{a}_{1}\oplus\tilde{a}_{2}=({\sqrt{({\mu_{1}})^{2}+({\mu_{2}})^{2}-({\mu% _{1}})^{2}({\mu_{2}})^{2}},\nu_{1}\nu_{2}})$ ;

(2)

$\tilde{a}_{1}\otimes\tilde{a}_{2}=({\mu_{1}\mu_{2},\sqrt{({\nu_{1}})^{2}+({\nu% _{2}})^{2}-({\nu_{1}})^{2}({\nu_{2}})^{2}}})$ ;

(3)

$\lambda\tilde{a}=({\sqrt{1-({1-\mu^{2}})^{\lambda}},\nu^{\lambda}}),\lambda>0$ ;

(4)

$(\tilde{a})^{\lambda}=({\mu^{\lambda},\sqrt{1-({1-\nu^{2}})^{\lambda}}}),% \lambda>0$ ;

(5)

$\tilde{a}^{c}=({\nu,\mu})$ .

2.2 Dual hesitant Pythagorean fuzzy set

In this section, we introduced dual hesitant Pythagorean fuzzy set (DHPFS) [29], which is a new extension of PFS and dual hesitant fuzzy set [30]. It’s clear that the DHPFSs consist of two parts, that is, the membership hesitancy function and the nonmembership hesitancy function, supporting a more exemplary and flexible access to assign values for each element in the domain, and we have to handle two kinds of hesitancy in this situation.

Definition 3 [29]. Let $X$ be a fixed set, then a dual hesitant Pythagorean fuzzy set (DHPFS) on $X$ is described as:

$\displaystyle D=({\langle{x,h_{P}(x),g_{P}(x)}\rangle|{x\in X}})$ (3)

in which $h_{P}(x)$ and $g_{P}(x)$ are two sets of some values in $[{0,1}]$ , denoting the possible membership degrees and non-membership degrees of the element $x\in X$ to the set $D$ respectively, with the conditions:

$\displaystyle\gamma^{2}+\eta^{2}\leqslant 1$ (4)

where $\gamma\in h_{P}(x)$ , $\eta\in g_{P}(x)$ , for all $x\in X$ . For convenience, the pair $d(x)=({h_{P}(x),g_{P}(x)})$ is called a dual hesitant Pythagorean fuzzy number (DHPFN) denoted by $d=({h,g})$ , with the conditions: $\gamma\in h$ , $\eta\in g$ , $0\leqslant\gamma$ , $\eta\leqslant 1$ , $0\leqslant\gamma^{2}+\eta^{2}\leqslant 1$ .

To compare the DHPFNs, in the following, [29] gave the following comparison laws:

Definition 4 [29]. Let $d=({h,g})$ be a DHPFNs, $s(d)=\frac{1}{2}\left({1+\frac{1}{\#h}\sum\nolimits_{\gamma\in h}{\gamma^{2}}-% \frac{1}{\#g}\sum\nolimits_{\eta\in g}{\eta^{2}}}\right)$ the score function of $d$ , and $p(d)=\frac{1}{\#h}\sum\nolimits_{\gamma\in h}{\gamma^{2}}+\frac{1}{\#g}\sum% \nolimits_{\eta\in g}{\eta^{2}}$ the accuracy function of $d$ , where $\#h$ and $\#g$ are the numbers of the elements in $h$ and $g$ respectively, then, Let $d_{i}=({h_{i},g_{i}})({i=1,2})$ be any two DHPFNs, we have the following comparison laws:

•

If $s({d_{1}})>s({d_{2}})$ , then $d_{1}$ is superior to $d_{2}$ , denoted by $d_{1}\succ d_{2}$ ;

•

If $s({d_{1}})=s({d_{2}})$ , then

(1)

If $p({d_{1}})=p({d_{2}})$ , then $d_{1}$ is equivalent to $d_{2}$ , denoted by $d_{1}\sim d_{2}$ ;

(2)

If $p({d_{1}})>p({d_{2}})$ , then $d_{1}$ is superior to $d_{2}$ , denoted by $d_{1}\succ d_{2}$ .

Then, Wei and Lu [29] defined some new operations on the DHPFNs $d$ , $d_{1}$ and $d_{2}$ :

(1)

$d^{\lambda}=\cup_{\gamma\in h,\eta\in g}\{{\{{\gamma^{\lambda}}\},\{{\sqrt{1-(% {1-\eta^{2}})^{\lambda}}}\}}\},\lambda>0$ ;

(2)

$\lambda d=\cup_{\gamma\in h,\eta\in g}\{{\{{\sqrt{1-({1-\gamma^{2}})^{\lambda}% }}\},\{{\eta^{\lambda}}\}}\},\lambda>0$ ;

(3)

$d_{1}\oplus d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}\in h_{2},\eta_{1}\in g_% {1},\eta_{2}\in g_{2}}\{{\{{\sqrt{({\gamma_{1}})^{2}+({\gamma_{2}})^{2}-({% \gamma_{1}})^{2}({\gamma_{2}})^{2}}}\},\{{\eta_{1}\eta_{2}}\}}\}$ ;

(4)

$d_{1}\otimes d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}\in h_{2},\eta_{1}\in g% _{1},\eta_{2}\in g_{2}}\{{\{{\gamma_{1}\gamma_{2}}\},\{{\sqrt{({\eta_{1}})^{2}% +({\eta_{2}})^{2}-({\eta_{1}})^{2}({\eta_{2}})^{2}}}\}}\}$ .

2.3 Muirhead Mean (MM) operator

The MM was firstly introduced by Muirhead [37], the advantage of the MM operator is that it can capture the overall interrelationships among the multiple input arguments and it is a generalization of some existing aggregation operators. It was defined as follows:

Definition 5 [37]. Let $a_{j}({j=1,2,\cdots,n})$ be a set of crisp numbers and $[\lambda]=(\lambda_{1},\lambda{}_{2},\cdots,\lambda_{n})\in R$ , then the Muirhead mean (MM) operator is defined as

$\displaystyle\mbox{MM}^{[\lambda]}({a_{1},a_{2},\cdots,a_{n}})=\left({\frac{1}% {n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{a_{\vartheta(j)}^{% \lambda_{j}}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ (5)

Where $\vartheta(j)({j=1,2,\cdots,n})$ is any permutation of $({1,2,\cdots,n})$ and $S_{n}$ is the set of all permutation of $({1,2,\cdots,n})$ .

By assigning some special vectors to $\lambda$ , we can obtain some special cases of the MM:

If $\lambda=(1,0,0,\cdots,0)$ the MM is reduced to

$\displaystyle\mbox{MM}^{({1,0,0,\cdots,0})}({a_{1},a_{2},\cdots,a_{n}})=\frac{% 1}{n}\sum\limits_{j=1}^{n}{a_{j}}$ (6)

Which is the arithmetic averaging operator.

If $\lambda=(1,1,0,0,\cdots,0)$ the MM is reduced to

$\displaystyle\mbox{MM}^{({1,1,0,0,\cdots,0})}({a_{1},a_{2},\cdots,a_{n}})=% \frac{1}{n({n+1})}\sum\limits_{\begin{array}[]{c}i,j=1\\ i\neq j\\ \end{array}}^{n}{a_{i}a_{j}}$ (7)

Which is the BM operator [41].

If $\lambda=(\overbrace{1,1,\cdots,1}^{k},\overbrace{0,0,\cdots,0}^{n-k})$ the MM is reduced to

$\displaystyle\mbox{MM}^{(\overbrace{1,1,\cdots,1}^{k},\overbrace{0,0,\cdots,0}% ^{n-k})}({a_{1},a_{2},\cdots,a_{n}})=\left({\frac{\sum\limits_{1\leqslant i_{1% }\leqslant\cdots\leqslant i_{i}\leqslant n}{\prod\limits_{j=1}^{n}{a_{i_{j}}}}% }{C_{n}^{k}}}\right)^{1\mathord{/{\vphantom{1k}}.\kern-1.2pt}k}$ (8)

which is the Maclaurin symmetric mean (MSM) operator [54].

If $\lambda=(1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n,1\mathord{/{\vphantom{1n}}.% \kern-1.2pt}n,\cdots,1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n)$ the MM is reduced to

$\displaystyle\mbox{MM}^{({1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n,1\mathord{/% {\vphantom{1n}}.\kern-1.2pt}n,\cdots,1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n}% )}({a_{1},a_{2},\cdots,a_{n}})=\prod\limits_{j=1}^{n}{a_{j}^{1\mathord{/{% \vphantom{1n}}.\kern-1.2pt}n}}$ (9)

which is the arithmetic averaging operator.

3. Dual Hesitant Pythagorean Fuzzy operators

In this section, we’ll apply MM into Dual Hesitant Pythagorean Fuzzy environment to deduce some new operators which will be used to solve multiple attributions decision making problems later.

3.1 Dual Hesitant Pythagorean Fuzzy Muirhead Mean operator

Definition 6. Let $p_{j}=({h_{p_{j}},g_{p_{j}}})$ be a set of DHPFNs and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ be a vector of parameters, then the Dual Hesitant Pythagorean Fuzzy MM (DHPFMM) operator is defined as

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})=\left({% \frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{p_{% \vartheta(j)}^{\lambda_{j}}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}% {\lambda_{j}}}}}$

(10) $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\left({% \prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}\right)^{2}}\right% )}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{% \lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{\left({1-\eta_{\vartheta(j)}^{2}}\right)^{\lambda_{j}}}% }\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^% {n}{\lambda_{j}}}}}}}\right\}\\ \end{array}}\right\}$

Where $\vartheta_{j}({j=1,2,\cdots,n})$ is any permutation of $({1,2,\cdots,n})$ , and $S_{n}$ is the collection of all permutations of $({1,2,\cdots,n})$ .

Based on the operations of the DHPFNs described, we can derive the Theorem 1.

Theorem 1. Let $p_{j}=({h_{p_{j}},g_{p_{j}}})$ be a set of DHPFNs and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ be a vector of parameters, then its aggregated value by using the DHPFMM operator is also a DHPFN, and

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})=\left({% \frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{p_{% \vartheta(j)}^{\lambda_{j}}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}% {\lambda_{j}}}}}$ (11) $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\left({% \prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}\right)^{2}}\right% )}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{% \lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}}}\right)}}% \right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{% \lambda_{j}}}}}}}\right\}\\ \end{array}}\right\}$

Proof:

$\displaystyle p_{\vartheta(j)}^{\lambda_{j}}=\cup_{\gamma\in h,\eta\in g}\{{\{% {\gamma_{\vartheta(j)}^{\lambda_{j}}}\},\{{\sqrt{1-({1-\eta_{\vartheta(j)}^{2}% })^{\lambda_{j}}}}\}}\}$ $\displaystyle\prod\limits_{j=1}^{n}{p_{\vartheta(j)}^{\lambda_{j}}}=\cup_{% \gamma\in h,\eta\in g}\left\{{\left\{{\prod\limits_{j=1}^{n}{\gamma_{\vartheta% (j)}^{\lambda_{j}}}}\right\},\left\{{\sqrt{1-\prod\limits_{j=1}^{n}{({1-\eta_{% \vartheta(j)}^{2}})^{\lambda_{j}}}}}\right\}}\right\}$

Thereafter,

$\displaystyle\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{p_{% \vartheta(j)}^{\lambda_{j}}}}=\cup_{\gamma\in h,\eta\in g}\left\{{\left\{{% \sqrt{1-\prod\limits_{\vartheta\in S_{n}}{\left({1-\left({\prod\limits_{j=1}^{% n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}\right)^{2}}\right)}}}\right\},\left\{% {\prod\limits_{\vartheta\in S_{n}}{\sqrt{1-\prod\limits_{j=1}^{n}{({1-\eta_{% \vartheta(j)}^{2}})^{\lambda_{j}}}}}}\right\}}\right\}$ $\displaystyle\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{% n}{p_{\vartheta(j)}^{\lambda_{j}}}}=\cup_{\gamma\in h,\eta\in g}\left\{{\begin% {array}[]{l}\left\{{\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}\right)^{2}% }\right)}}\right)^{\frac{1}{n!}}}}\right\},\\ \left\{{\left({\prod\limits_{\vartheta\in S_{n}}{\sqrt{1-\prod\limits_{j=1}^{n% }{({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}}}}}\right)^{\frac{1}{n!}}}\right% \}\\ \end{array}}\right\}$

Thus,

$\displaystyle\left({\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_% {j=1}^{n}{p_{\vartheta(j)}^{\lambda_{j}}}}}\right)^{\textstyle{1\over{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}}=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{% array}[]{l}\left\{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{% \left({1-\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}% \right)^{2}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}}}\right)}}% \right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{% \lambda_{j}}}}}}}\right\}\\ \end{array}}\right\}$

Considering $\gamma^{2}+\eta^{2}\leqslant 1$ , we can easily know that

$\displaystyle\left({\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{({% 1-\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}\right)^{% 2}})}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}% {\lambda_{j}}}}}}\right)^{2}$ $\displaystyle\quad{}+\left({\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S% _{n}}{\left({1-\prod\limits_{j=1}^{n}{({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{% j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j% =1}^{n}{\lambda_{j}}}}}}}\right)^{2}$ $\displaystyle\quad\leqslant\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{% \left({1-\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta(j)}^{\lambda_{j}}}}% \right)^{2}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad{}+1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left% ({1-\prod\limits_{j=1}^{n}{({\gamma_{\vartheta(j)}^{2}})^{\lambda_{j}}}}\right% )}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{n}{% \lambda_{j}}}}}=1$

So, we complete the proof that elucidate the result calculated by DHPFMM also a DHPFN.

Property 1. (Idempotency) let $p_{j}=({h_{p_{j}},g_{p_{j}}})=p({j=1,2,3,\cdots,n})$ , then

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})=p$ (12)

Proof:

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})$ $\displaystyle\quad=\left({\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod% \limits_{j=1}^{n}{p^{\lambda_{j}}}}}\right)^{\textstyle{1\over{\sum\limits_{j=% 1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad=\left({p^{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\right)^{% \textstyle{1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad=p$

Property 2. (Monotonicity) let $p_{j}=({h_{p_{j}},g_{p_{j}}})$ and $q_{j}=({h_{q_{j}},g_{q_{j}}})({j=1,2,3,\cdots,n})$ be two sets of DHPFNs, If, $\gamma_{p_{j}}\leqslant\gamma_{q_{j}}$ and $\eta_{p_{j}}\geqslant\eta_{q_{j}}$ , $\gamma_{p_{j}}\in h_{p_{j}},\eta_{p_{j}}\in g_{p_{j}},\gamma_{q_{j}}\in h_{q_{% j}},\eta_{q_{j}}\in g_{q_{j}}$ then

$\displaystyle\mbox{DHPFMM}^{\lambda}({p_{1},p_{2},\cdots,p_{n}})\leqslant\mbox% {DHPFMM}^{\lambda}({q_{1},q_{2},\cdots,q_{n}})$ (13)

Proof:

$\displaystyle\gamma_{\vartheta({p_{j}})}^{\lambda_{j}}\leqslant\gamma_{% \vartheta({q_{j}})}^{\lambda_{j}}$ $\displaystyle\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta({p_{j}})}^{% \lambda_{j}}}}\right)^{2}\leqslant\left({\prod\limits_{j=1}^{n}{\gamma_{% \vartheta({q_{j}})}^{\lambda_{j}}}}\right)^{2}$

Thereafter,

$\displaystyle\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\left({\prod% \limits_{j=1}^{n}{\gamma_{\vartheta({p_{j}})}^{\lambda_{j}}}}\right)^{2}}% \right)}}\right)^{\frac{1}{n!}}\geqslant\left({\prod\limits_{\vartheta\in S_{n% }}{\left({1-\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta({q_{j}})}^{\lambda% _{j}}}}\right)^{2}}\right)}}\right)^{\frac{1}{n!}}$

Thus,

$\displaystyle\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta({p_{j}})}^{\lambda_{j}}}}% \right)^{2}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad\leqslant\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_% {n}}{\left({1-\left({\prod\limits_{j=1}^{n}{\gamma_{\vartheta({q_{j}})}^{% \lambda_{j}}}}\right)^{2}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle% {1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$

Similarly, we can easily prove that

$\displaystyle\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({% 1-\prod\limits_{j=1}^{n}{({1-\eta_{\vartheta({p_{j}})}^{2}})^{\lambda_{j}}}}% \right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum\limits_{j=1}^{% n}{\lambda_{j}}}}}}$ $\displaystyle\quad\geqslant\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S% _{n}}{\left({1-\prod\limits_{j=1}^{n}{({1-\eta_{\vartheta({q_{j}})}^{2}})^{% \lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}}}$

then, the proof is completed. And

If $\gamma_{p_{j}}<\gamma_{q_{j}}$ and $\eta_{p_{j}}>\eta_{q_{j}}$ then

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})<\mbox{% DHPFMM}^{[\lambda]}({q_{1},q_{2},\cdots,q_{n}});$

If $\gamma_{p_{j}}<\gamma_{q_{j}}$ and $\eta_{p_{j}}\mbox{ = }\eta_{q_{j}}$ then

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})<\mbox{% DHPFMM}^{[\lambda]}({q_{1},q_{2},\cdots,q_{n}})$

If $\gamma_{p_{j}}=\gamma_{q_{j}}$ and $\eta_{p_{j}}=\eta_{q_{j}}$ then

$\displaystyle\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})\mbox{ = % DHPFMM}^{[\lambda]}({q_{1},q_{2},\cdots,q_{n}})$

Property 3. (Boundedness) Let $p_{j}=({h_{p_{j}},g_{p_{j}}}){\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}({j=1,2,% \cdots,n})$ be a set of DHPFNs. If $p^{+}=((\max_{j}({h_{j}}),\min_{j}\linebreak({g_{j}})))$ and $p^{-}=({({\min_{j}({h_{j}}),\max_{j}({g_{j}})})})$ , According the properties of Monotonicity and Idempotency, it is derived easily that

$\displaystyle p^{-}\leqslant\mbox{DHPFMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n% }})\leqslant p^{+}$ (14)
3.2 Dual Hesitant Pythagorean Fuzzy Weighted Muirhead mean operator

In this section, we shall propose the dual Hesitant Pythagorean Fuzzy Weighted MM (DHPFWMM) operator.

Definition 7. Let $p_{j}=({h_{p_{j}},g_{p_{j}}})$ be a set of DHPFNs and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ be a vector of parameters, then the dual hesitant Pythagorean fuzzy weighted MM (DHPFWMM) operator is defined as:

Where $\vartheta_{j}({j=1,2,\cdots,n})$ is any permutation of $({1,2,\cdots,n})$ , and $S_{n}$ is the collection of all permutations of $({1,2,\cdots,n})$ .

Based on the operations of the DHPFNs described, we can derive the Theorem 2.

Theorem 2. Let $p_{j}=({h_{p_{j}},g_{p_{j}}})({j=1,2,\cdots,n})$ be a set of DHPNs and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ be a vector of parameters, then its aggregated value by using the DHPFWMM operator is also a DHPFN, and

$\displaystyle\text{DHPFWMM}_{\text{W}}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}})% =\left({\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{({% nw_{\vartheta(j)}p_{\vartheta(j)}^{\lambda_{j}}})}}}\right)^{\textstyle{1\over% {\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ (16) $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\prod% \limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{% \vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-({1-({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}})^{% nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1% \over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}}}\right\}\\ \end{array}}\right\}$

Proof:

$\displaystyle p_{\vartheta(j)}^{\lambda_{j}}\mbox{ = }\cup_{\gamma\in h,\eta% \in g}\{{\{{\gamma_{\vartheta(j)}^{\lambda_{j}}}\},\{{\sqrt{1-({1-\eta_{% \vartheta(j)}^{2}})^{\lambda_{j}}}}\}}\}$ $\displaystyle nw_{\vartheta(j)}p_{\vartheta(j)}^{\lambda_{j}}=\cup_{\gamma\in h% ,\eta\in g}\{{\{{\sqrt{1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{% \vartheta(j)}}}}\},\{{({\sqrt{1-({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}}})% ^{nw_{\vartheta(j)}}}\}}\}$

Thereafter,

$\displaystyle\prod\limits_{j=1}^{n}{({nw_{\vartheta(j)}p_{\vartheta(j)}^{% \lambda_{j}}})}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\left\{{\prod\limits_{j% =1}^{n}{\sqrt{1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{\vartheta(j)}}% }}}\right\},\left\{{\sqrt{1-\prod\limits_{j=1}^{n}{({1-({1-({1-\eta_{\vartheta% (j)}^{2}})^{\lambda_{j}}})^{nw_{\vartheta(j)}}})}}}\right\}}\right\}$ $\displaystyle\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{({nw_{% \vartheta(j)}p_{\vartheta(j)}^{\lambda_{j}}})}}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\sqrt{1-\prod\limits_{\vartheta\in S_{n}}{\left({1-\prod\limits_{j=1}^{n}{(% {1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{\vartheta(j)}}})}}\right)}}% }\right\},\\ \left\{{\prod\limits_{\vartheta\in S_{n}}{\sqrt{1-\prod\limits_{j=1}^{n}{\left% ({1-({1-({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}})^{nw_{\vartheta(j)}}}% \right)}}}}\right\}\\ \end{array}}\right\}$ $\displaystyle\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{% n}{({nw_{\vartheta(j)}p_{\vartheta(j)}^{\lambda_{j}}})}}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\prod\limits_{j=1% }^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{\vartheta(j)}}})}}% \right)}}\right)^{\frac{1}{n!}}}}\right\},\\ \left\{{\left({\prod\limits_{\vartheta\in S_{n}}{\sqrt{1-\prod\limits_{j=1}^{n% }{({1-({1-({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}})^{nw_{\vartheta(j)}}})}% }}}\right)^{\frac{1}{n!}}}\right\}\\ \end{array}}\right\}$

Thus,

$\displaystyle\left({\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_% {j=1}^{n}{({nw_{\vartheta(j)}p_{\vartheta(j)}^{\lambda_{j}}})}}}\right)^{% \textstyle{1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\prod% \limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{% \vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-({1-({1-\eta_{\vartheta(j)}^{2}})^{\lambda_{j}}})^{% nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1% \over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}}}\right\}\\ \end{array}}\right\}$

Considering $\gamma^{2}+\eta^{2}\leqslant 1$ , we can easily know that

$\displaystyle\left({\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{% \left({1-\prod\limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})% ^{nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1% \over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}}\right)^{2}$ $\displaystyle\quad{}+\left({\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S% _{n}}{\left({1-\prod\limits_{j=1}^{n}{({1-({1-({1-\eta_{\vartheta(j)}^{2}})^{% \lambda_{j}}})^{nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^% {\textstyle{1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}}}\right)^{2}$ $\displaystyle\quad\leqslant\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{% \left({1-\prod\limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})% ^{nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1% \over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}+1$ $\displaystyle\quad-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta(j)}^{2\lambda_{j}}})^{nw_{% \vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1\over{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad=1$

DHPFWMM has the property of boundedness and monotonicity, but it does not satisfy the property of idempotency. In the following, we omitted the process of prove, because it is similar with the DHPFMM monotonicity property.

Property 4. (Monotonicity) let $p_{j}=({h_{p_{j}},g_{p_{j}}})$ and $q_{j}=({h_{q_{j}},g_{q_{j}}})({j=1,2,3,\cdots,n})$ be two sets of DHPFNs, If, $\gamma_{p_{j}}\leqslant\gamma_{q_{j}}$ and $\eta_{p_{j}}\geqslant\eta_{q_{j}},\gamma_{p_{j}}\in h_{p_{j}},\eta_{p_{j}}\in g% _{p_{j}},\gamma_{q_{j}}\in h_{q_{j}},\eta_{q_{j}}\in g_{q_{j}}$ then

$\displaystyle\mbox{DHPFWMM}^{\lambda}({p_{1},p_{2},\cdots,p_{n}})\leqslant% \mbox{DHPFWMM}^{\lambda}({q_{1},q_{2},\cdots,q_{n}})$ (17)

Property 5. (Boundedness) Let $p_{j}=({h_{p_{j}},g_{p_{j}}}){\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}({j=1,2,% \cdots,n})$ be a set of DHPFNs. If $p^{+}=((\max_{j}({h_{j}}),\min_{j}\linebreak({g_{j}})))$ and $p^{-}=({({\min_{j}({h_{j}}),\max_{j}({g_{j}})})})$ , According the properties of Monotonicity, it is derived easily that

$\displaystyle p^{-}\leqslant\mbox{DHPFWMM}^{[\lambda]}({p_{1},p_{2},\cdots,p_{% n}})\leqslant p^{+}$ (18)

4. Models for MADM with DHPFNs

Based the DHPFWMM and DHPFWDMM operators, in this section, we shall propose the model for MADM with DHPFNs. Let $A=\{{A_{1},A_{2},\cdots,A_{m}}\}$ be a discrete set of alternatives, and $G=\{{G_{1},G_{2},\cdots,G_{n}}\}$ be the set of attributes, $\omega=({\omega_{1},\omega_{2},\cdots,\omega_{n}})$ is the weighting vector of the attribute $G_{j}({j=1,2,\cdots,n})$ , where $\omega_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{\omega_{j}}=1$ . Suppose that $P=({p_{ij}})_{m\times n}=({\mu_{ij},\nu_{ij}})_{m\times n}$ is the Dual hesitant Pythagorean fuzzy decision matrix, where $\mu_{ij}$ indicates the degree that the alternative $A_{i}$ satisfies the attribute $G_{j}$ given by the decision maker, $\nu_{ij}$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ given by the decision maker, $\mu_{ij}\subset[{0,1}]$ , $\nu_{ij}\subset[{0,1}]$ , $({\mu_{ij}})^{2}+({\nu_{ij}})^{2}\leqslant 1$ , $i=1,2,\cdots,m$ , $j=1,2,\cdots,n$ .

In the following, we apply the DHPFWMM (DHPFWDMM) operator to the MADM problems with DHPFNs.

Step 1. We utilize the DHPFNs given in matrix $\tilde{R}$ , and the DHPFWMM operator

$\displaystyle p_{i}=\text{DHPFWMM}_{\text{W}}^{[\lambda]}({p_{1},p_{2},\cdots,% p_{n}})=\left({\frac{1}{n!}\sum\limits_{\vartheta\in S_{n}}{\prod\limits_{j=1}% ^{n}{({nw_{\vartheta(j)}p_{\vartheta({ij})}^{\lambda_{j}}})}}}\right)^{% \textstyle{1\over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\left({\sqrt{1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-\prod% \limits_{j=1}^{n}{({1-({1-\gamma_{\vartheta({ij})}^{2\lambda_{j}}})^{nw_{% \vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\textstyle{1\over{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}}}\right\},\\ \left\{{\sqrt{1-\left({1-\left({\prod\limits_{\vartheta\in S_{n}}{\left({1-% \prod\limits_{j=1}^{n}{({1-({1-({1-\eta_{\vartheta({ij})}^{2}})^{\lambda_{j}}}% )^{nw_{\vartheta(j)}}})}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\textstyle{1% \over{\sum\limits_{j=1}^{n}{\lambda_{j}}}}}}}\right\}\\ \end{array}}\right\},$ $\displaystyle\quad i=1,2,\cdots,m.$ (19)

$\displaystyle p_{i}=\mbox{DHPFWDMM}_{w}^{[\lambda]}({p_{1},p_{2},\cdots,p_{n}}% )=\frac{1}{\sum\nolimits_{j=1}^{n}{\lambda_{j}}}\left({\prod\limits_{\sigma\in S% _{n}}{\sum\limits_{j=1}^{n}{\lambda_{j}p_{\sigma({ij})}^{nw_{\sigma(j)}}}}}% \right)^{\frac{1}{n!}}$ $\displaystyle\quad=\cup_{\gamma\in h,\eta\in g}\left\{{\begin{array}[]{l}\left% \{{\sqrt{1-\left({1-\left({\prod\limits_{\sigma\in S_{n}}{\left({1-\prod% \limits_{j=1}^{n}{({1-({\gamma_{\vartheta({ij})}^{nw_{\sigma(j)}}})^{2}})^{% \lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{\frac{1}{\sum\nolimits_% {j=1}^{n}{\lambda_{j}}}}}}\right\},\\ \left\{{\left({\sqrt{1-\left({\prod\limits_{\sigma\in S_{n}}{\left({1-\prod% \limits_{j=1}^{n}{({1-({1-\eta_{\vartheta({ij})}^{2}})^{nw_{\sigma(j)}}})^{% \lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}}\right)^{\frac{1}{\sum\nolimits% _{j=1}^{n}{\lambda_{j}}}}}\right\}\\ \end{array}}\right\},$ $\displaystyle\quad i=1,2,\cdots,m.$ (20)

to derive the $p_{i}({i=1,2,\cdots,m})$ of the alternative $A_{i}$ .

Step 2. Calculate the scores $S({p_{i}})({i=1,2,\cdots,m})$ of the overall DHPFNs $p_{i}({i=1,2,\cdots,m})$ to rank all the alternatives $A_{i}({i=1,2,\cdots,m})$ and then to select the best one(s). If there is no difference between two scores $S({p_{i}})$ and $S({p_{j}})$ , then we need to calculate the accuracy degrees $H({p_{i}})$ and $H({p_{j}})$ of the overall DHPFNs $p_{i}$ and $p_{j}$ , respectively, and then rank the alternatives $A_{i}$ and $A_{j}$ in accordance with the accuracy degrees $H({p_{i}})$ and $H({p_{j}})$ .

Step 3. Rank all the alternatives $A_{i}({i=1,2,\cdots,m})$ and select the best one(s) in accordance with $S({p_{i}})({i=1,2,\cdots,m})$ .

Step 4. End.

5. Numerical example and comparative analysis

5.1 Numerical example

Nowadays, the ecological environment around the world is deteriorating and human survival is threatened. As a result of it, the development of circular economy has become a hot topic. In 2015, the government proposed to actively promote economic restructuring and upgrading, and the manufacturing sector is the focus of rectification and reform. Supplier is the “Source” of the whole supply chain, and the green supplier selection is the foundation of green supply chain management [38, 39, 40, 41]. The quality of suppliers will directly affect the environmental performance of enterprises. First, the green supply chain management and the traditional supply chain management were compared, then the characteristics of green supplier partnerships were analyzed from various aspects [42, 43, 44]. Then, we shall give the application to select green suppliers in green supply chain management with DHPFNs. There are five possible green suppliers in green supply chain management $O_{i}({i=1,2,3,4,5})$ to select. The experts select four attributes to assess the five possible green suppliers: ⟀ C ${}_{1}$ is the product quality factor; ⟁ C ${}_{2}$ is environmental factors; ⟂ C ${}_{3}$ is delivery factor; ⟃ C ${}_{4}$ is price factor. Five green suppliers $O_{i}({i=1,2,3,4,5})$ are to be assessed with DHPFNs according to four attributes (whose weighting vector $W=({0.2,0.3,0.3,0.2})$ and $\lambda=({2,2,2,2})$ , as shown in Table 1.

Table 1
DHPFN decision matrix

	C ${}_{1}$	C ${}_{2}$	C ${}_{3}$	C ${}_{4}$
O ${}_{1}$	{(0.5, 0.8), (0.4, 0.7)}	{(0.6, 0.5)}	{(0.3, 0.6)}	{(0.6, 0.7), (0.6, 0.4)}
O ${}_{2}$	{(0.7, 0.5)}	{(0.6, 0.5), (0.6, 0.7)}	(0.6, 0.20, (0.7, 0.4)	{(0.4, 0.6)}
O ${}_{3}$	{(0.7, 0.5) (0.5, 0.3)}	{(0.5, 0.7)}	{(0.4, 0.5), (0.5, 0.2), (0.6, 0.4)}	{(0.6, 0.2)}
O ${}_{4}$	{(0.8, 0.2)}	{(0.6, 0.3) (0.6, 0.6)}	{(0.4, 0.5) (0.6, 0.5)}	{(0.5, 0.2), (0.6, 0.4), (0.7, 0.2)}
O ${}_{5}$	{(0.5, 0.2)}	{(0.6, 0.3), (0.6, 0.5)}	{(0.7, 0.5)}	{(0.6, 0.8)}

In the following, in order to select the most desirable green suppliers in GSCM, we utilize the DHPFWMM and DHPFWDMM operators to solve MADM problem with DHPFNs, which concludes the following calculating steps:

Step 1. We utilize the DHPFNs given in matrix $\tilde{R}$ , and the DHPFWMM operator to get aggregation results, we illustrate one of alternative for save space.

$\displaystyle O_{1}=\mbox{DHPFWMM}_{\omega}^{t,r}({G_{1},G_{2},G_{3},G_{4}})=% \{{{\{(0.5,0.8),(0.4,0.7)\},\{(0.6,0.5)\},\{(0.3,0.6)\},\{(0.6,0.70),(0.60,0.4% 0)\}}}\}=\{{({{0.3933,0.8969}}),({{0.3933,0.8773}}),({{0.3719,0.8860}}),({{0.3% 719,0.8641}})}\}$

Step 2. According to the aggregating results and the score functions of the green suppliers are shown in Table 2.

Table 2

The rank and score of green suppliers by using DHPF operators

	O ${}_{1}$	O ${}_{2}$	O ${}_{3}$	O ${}_{4}$	O ${}_{5}$	Order
DHPFWMM	0.1851	0.2490	0.2579	0.3073	0.2539	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
DHPFWDMM	0.6411	0.7149	0.7302	0.7800	0.7308	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$

According the result of green suppliers order, we can know that the best choice is supplier 4, we get same result by different aggregation, that proved the effectiveness of result.

5.2 Influence of the parameter on the final result

The aggregation method of extend DHPFS with MM has two advantages, one is that it can reduce the bad effects of the unduly high and low assessments on the final result, the other is that it can capture the interrelationship between dual hesitate Pythagorean fuzzy numbers. These aggregation operators have a parameter vector, which make extended operator more flexible, so the different vector lead to different aggregation results, different scores and ranking results. In order to illustrate the influence of the parameter vector $l_{i}$ on the ranking result, we discuss the influence with several parameter vectors, the result you can find in Tables 3 and 4.

Table 3
Ranking results by utilizing different parameter vector $l_{i}$ in the DHPFWMM operator

$l_{i},i=1,\cdots,6$	Scores					Order
	O ${}_{1}$	O ${}_{2}$	O ${}_{3}$	O ${}_{4}$	O ${}_{5}$
(1,1,1,1)	0.0966	0.1357	0.1423	0.1749	0.1405	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(2,2,2,2)	0.1851	0.2490	0.2579	0.3073	0.2539	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(3,3,3,3)	0.2362	0.3117	0.3207	0.3773	0.3155	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(4,4,4,4)	0.2689	0.3511	0.3598	0.4204	0.3536	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(5,5,5,5)	0.2916	0.3782	0.3864	0.4497	0.3795	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(6,6,6,6)	0.2121	0.3980	0.4060	0.4710	0.3982	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$

Table 4

Ranking results by utilizing different parameter vector $l_{i}$ in the DHPFWDMM operator

$l_{i},i=1,\cdots,6$	Scores					Order
	O ${}_{1}$	O ${}_{2}$	O ${}_{3}$	O ${}_{4}$	O ${}_{5}$
(1,1,1,1)	0.7533	0.8057	0.8150	0.8507	0.8164	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$
(2,2,2,2)	0.6411	0.7149	0.7302	0.7800	0.7308	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$
(3,3,3,3)	0.5913	0.6748	0.6926	0.7489	0.6929	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$
(4,4,4,4)	0.5636	0.6526	0.6857	0.7638	0.6720	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(5,5,5,5)	0.5459	0.6568	0.7027	0.7649	0.6916	O ${}_{4}>$ O ${}_{3}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{1}$
(6,6,6,6)	0.5337	0.6690	0.7084	0.7586	0.7265	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$

We can see that the different parameters lead to different result and different ranking order. More attributes we consider more bigger the scores, more bigger the attribute value more lower the scores. Therefore, the parameter vector can be considered as decision maker’s risk preference.

5.3 Comparative analysis

In this section, we get aggregation result by some existing method, such as DHPFWHM operator, DHPFWDHM operator and DHPFWA operator, and we can find in the Table 5.

Table 5
The aggregation result of exist method

Method name	Scores					Order
	O ${}_{1}$	O ${}_{2}$	O ${}_{3}$	O ${}_{4}$	O ${}_{5}$
DHPFWHM	0.7831	0.8433	0.8296	0.8631	0.8466	O ${}_{4}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{3}>$ O ${}_{1}$
DHPFWDHM	0.1552	0.2160	0.2302	0.2670	0.2312	O ${}_{4}>$ O ${}_{5}>$ O ${}_{3}>$ O ${}_{2}>$ O ${}_{1}$
DHPFWA	0.4600	0.5792	0.5729	0.6423	0.6007	O ${}_{4}>$ O ${}_{5}>$ O ${}_{2}>$ O ${}_{3}>$ O ${}_{1}$

From the scores and order result overall, we can see that these results different with the novel method we proposed, it is because that these method in Table 5 is not consider the interrelationship of DHPFNs.

6. Conclusion

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 the attributes are independent of each other. However, the MM operator has a prominent characteristic that it can consider the interaction relationships among any number of attributes by a parameter vector $\lambda$ . Motivated by the studies about MM operator, in this paper, we proposed some new MM aggregation operators to deal with MADM problems under dual hesitant Pythagorean fuzzy environment, included dual hesitant Pythagorean fuzzy MM (DHPFMM) operator and dual hesitant Pythagorean fuzzy weighted MM (DHPFWMM) operator. Then, the desirable properties were proved. Moreover, these proposed operators are utilized to solve the MADM problems with DHPFSs. Finally, we used an illustrative example for green suppliers selections in green supply chain management to show the feasibility and validity of the proposed operators by comparing with the other existing methods. In the future, we shall extend the proposed operators to the different fields [45, 46, 47, 48, 49, 50, 51] as well as to propose some new aggregation operators under the uncertain environment [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65].

Footnotes

Acknowledgments

The work was supported by the Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 14XJCZH002).

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Models for multiple attribute decision making with dual hesitant pythagorean fuzzy information

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Pythagorean fuzzy set

3.1 Dual Hesitant Pythagorean Fuzzy Muirhead Mean operator

5.1 Numerical example

Table 1 DHPFN decision matrix

Table 3 Ranking results by utilizing different parameter vector l i in the DHPFWMM operator

Table 5 The aggregation result of exist method

Footnotes

Acknowledgments

References

Table 1
DHPFN decision matrix

Table 3
Ranking results by utilizing different parameter vector $l_{i}$ in the DHPFWMM operator

Table 5
The aggregation result of exist method