Picture fuzzy heronian mean aggregation operators in multiple attribute decision making

Abstract

The Heronian mean and generalized Heronian mean provide two aggregation operators that consider the inter-dependent phenomena among the aggregated arguments. In this paper, the Heronian mean and generalized Heronian mean under picture fuzzy environments is studied. First, the generalized picture fuzzy Heronian mean (GPFHM) operator and generalized picture fuzzy weighted Heronian mean (GPFWHM) operator are proposed and some of their desirable properties and special cases are investigated in detail. Then, an approach to multiple attribute decision making (MADM) based on GPFWHM operator is proposed. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making (MADM)picture fuzzy set generalized picture fuzzy Heronian mean (GPFHM) operator generalized picture fuzzy weighted Heronian mean (GPFWHM) operator enterprise resource planning (ERP) system

1. Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the concept of fuzzy set [3]. Atanassov and Gargov [4] and Atanassov [5] proposed the concept of interval-valued intuitionistic fuzzy sets, which are characterized by a membership function, a non-membership function, and a hesitancy function whose values are intervals. The intuitionistic fuzzy set and interval-valued intuitionistic fuzzy sets have received more and more attention since its appearance [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Recently, Cuong [31] proposed picture fuzzy set (PFS) and investigated the some basic operations and properties of PFS. The picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of non-membership. The only constraint is that the sum of the three degrees must not exceed 1. Basically, PFS based models can be applied to situations requiring human opinions involving more answers of types: yes, abstain, no, refusal, which can’t be accurately expressed in the traditional FS and IFS. Until now, some progress has been made in the research of the PFS theory. Singh [32] investigated the correlation coefficients for picture fuzzy set and apply the correlation coefficient to clustering analysis with picture fuzzy information. Son [33] and Tong and Son [34] introduce several novel fuzzy clustering algorithms on the basis of picture fuzzy sets and applications to time series forecasting and weather forecasting. Thong [35] developed a novel hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis and application to health care support systems. Wei [36] proposed the picture fuzzy cross-entropy for multiple attribute decision making (MADM) problems. Wei et al. [37] developed the projection models for multiple attribute decision making with picture fuzzy information.

However, all the above approaches are unsuitable to aggregate these picture fuzzy numbers on the basis of the Heronian mean and generalized Heronian mean. Thus, how to aggregate these picture fuzzy numbers on the basis of the Heronian mean and generalized Heronian mean is an interesting topic. To solve this issue, in this paper, we shall develop some picture fuzzy aggregation operators on the basis of the traditional Heronian mean [38] and generalized Heronian mean [39, 40]. In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy set and picture fuzzy sets. In Section 3, we propose some picture fuzzy arithmetic aggregation operators on the basis of the Heronian mean and generalized Heronian mean. In Section 4, we present a model for multiple attribute decision making problems based on the generalized picture fuzzy weighted Heronian mean (GPFWHM) operator with picture fuzzy information. In Section 5, we present a numerical example for enterprise resource planning (ERP) system selection with picture fuzzy information in order to illustrate the method proposed in this paper. Section 6 concludes the paper with some remarks.

2. Preliminaries

In the following section, we introduce some basic concepts related to intuitionistic fuzzy sets.

Definition 1 [1, 2].An IFS $A$ in $X$ is given by

$\displaystyle A=\left\{\left\langle x,\mu_{A}(x),\nu_{A}(x)\right\rangle\left|% x\in X\right.\right\}$ (1)

where $\mu_{A}:X\to[0,1]$ and $\nu_{A}:X\to[0,1]$ , where $0\leqslant\mu_{A}(x)+\nu_{A}(x)\leqslant 1$ , $\forall x\in X$ . The number $\mu_{A}(x)$ and $\nu_{A}(x)$ represents, respectively, the membership degree and non-membership degree of the element $x$ to the set $A$ .

Picture fuzzy sets [31] are extension of Atanassov’s intuitionistic fuzzy sets.

Definition 2 [31]. A picture fuzzy set (PFS) A on the universe $X$ is an object of the form

$\displaystyle A=\left\{\left\langle x,\mu_{A}(x),\eta_{A}(x),\nu_{A}(x)\right% \rangle\left|x\in X\right.\right\}$ (2)

where $\mu_{A}(x)\in[0,1]$ is called the “degree of positive membership of $A$ ”, $\eta_{A}(x)\in[0,1]$ is called the “degree of neutral membership of $A$ ” and $\nu_{A}(x)\in[0,1]$ is called the “degree of negative membership of $A$ ”, and $\mu_{A}(x)$ , $\eta_{A}(x)$ , $\nu_{A}(x)$ satisfy the following condition: $0\leqslant\mu_{A}(x)+\eta_{A}(x)+\nu_{A}(x)\leqslant 1$ , $\forall x\in X$ . Then for $x\in X$ , $\pi_{A}(x)=1-(\mu_{A}(x)+\eta_{A}(x)+\nu_{A}(x))$ could be called the degree of refusal membership of $x$ in $A$ .

For convenience, we call $\alpha=(\mu_{\alpha},\eta_{\alpha},\nu_{\alpha})$ a picture fuzzy number (PFN), where $\mu_{\alpha}\in[0,1],\eta_{\alpha}\in[0,1],\nu_{\alpha}\in[0,1]$ , $\mu_{\alpha}+\eta_{\alpha}+\nu_{\alpha}\leqslant$ 1.

Based on the score function $S$ [41] and the accuracy function $H$ [42], in the following section, Wei [42] gave an order relation between two PFNs.

Definition 3 [42]. Let $\alpha=(\mu_{\alpha},\eta_{\alpha},\nu_{\alpha})$ and $\beta=(\mu_{\beta},\eta_{\beta},\nu_{\beta})$ be two picture fuzzy numbers, $S(a)=\mu_{\alpha}-\nu_{\alpha}$ and $S(\beta)=\mu_{\beta}-\nu_{\beta}$ be the scores of $\alpha$ and $\beta$ , respectively, and let $H(a)=\mu_{\alpha}+\eta_{\alpha}+\nu_{\alpha}$ and $H(\beta)=\mu_{\beta}+\eta_{\beta}+\nu_{\beta}$ be the accuracy degrees of $\alpha$ and $\beta$ , respectively, then if $S(a)<S(\beta)$ , then $\alpha$ is smaller than $\beta$ , denoted by $\alpha<\beta$ ; if $S(a)=S(\beta)$ , then (1) if $H(a)=H(\beta)$ , then $\alpha=\beta$ ; (2) if $H(a)<H(\beta)$ , then $\alpha<\beta$ .

Motivated by the operations of the intuitionistic fuzzy number [8, 9] and according to Definition 3, Wei [42] and Wei et al. [43]defined some operational laws of picture fuzzy number.

Definition 4. Let $\alpha=(\mu_{\alpha},\eta_{\alpha},\nu_{\alpha})$ and $\beta=(\mu_{\beta},\eta_{\beta},$ $\nu_{\beta})$ be two picture fuzzy numbers, then

$\displaystyle\bar{\alpha}=\alpha=(\nu_{\alpha},\eta_{\alpha},\mu_{\alpha})$ $\displaystyle\alpha\wedge\beta=(\min\{\mu_{\alpha},\mu_{\beta}\},\max\{\eta_{% \alpha},\eta_{\beta}\},\max\{\nu_{\alpha},\nu_{\beta}\})$ $\displaystyle\alpha\vee\beta=(\max\{\mu_{\alpha},\mu_{\beta}\},\min\{\eta_{% \alpha},\eta_{\beta}\},\min\{\nu_{\alpha},\nu_{\beta}\})$ $\displaystyle\alpha\oplus\beta=(\mu_{\alpha}+\mu_{\beta}-\mu_{\alpha}\mu_{% \beta},\eta_{\alpha}\eta_{\beta},\nu_{\alpha}\nu_{\beta});$ $\displaystyle\alpha\otimes\beta=(\mu_{\alpha}\mu_{\beta},\eta_{\alpha}+\eta_{% \beta}-\eta_{\alpha}\eta_{\beta},\nu_{\alpha}+\nu_{\beta}-\nu_{\alpha}\nu_{% \beta});$ $\displaystyle\lambda\alpha=\left(1-(1-\mu_{\alpha})^{\lambda},\eta_{\alpha}^{% \lambda},\nu_{\alpha}^{\lambda}\right)$ $\displaystyle\alpha^{\lambda}=\left(\mu_{\alpha}^{\lambda},1-(1-\eta_{\alpha})% ^{\lambda},1-(1-\nu_{\alpha})^{\lambda}\right)$

3. Generalized picture fuzzy Heronian mean (GPFHM) operators

In this Section, we first introduced the Heronian mean (HM) [38] and generalized Heronian mean (GHM) operator [39, 40]. Furthermore, considering that the PFSs are powerful to express the vaguess in real decision making process, we further extend the GHM to picture fuzzy situations and develop a generalized picture fuzzy Heronian mean (GPFHM) operator for aggregating PFNs, and discuss its desirable properties and a variety of special cases.

Definition 5 [38]. Let $a_{i}(i=1,2,\ldots,n)$ be a collection of nonnegative numbers. If

$\displaystyle\!\!\!\!\!\!\!\!\textit{HM}(a_{1},a_{2},\ldots,a_{n})=\frac{2}{n(% n+1)}\sum_{i=1}^{n}\sum_{j=i}^{n}\sqrt{a_{i}a_{j}}$ (3)

then HM is called the Heronian mean (HM).

Based on Definition 4, Yu [39] and Yu and Wu [40] proposed the generalized Heronian mean as follows.

Definition 6 [39, 40]. Let $p,q\geqslant 0$ , and $a_{i}(i=1,2,\ldots,n)$ be a collection of nonnegative numbers. If

$\displaystyle\textit{GHM}^{p,q}(a_{1},a_{2},\ldots,a_{n})=\left(\frac{2}{n(n+1% )}\sum_{i=1}^{n}\sum_{j=i}^{n}a_{i}^{p}a_{j}^{q}\right)^{\frac{1}{p+q}}$ (4)

then GHM is called the generalized Heronian mean (GHM).

It should noted that, when $p=q=\frac{1}{2}$ , the GHM reduces to the HM [38]. Base on the Definitions 4, the generalized picture fuzzy Heronian mean (GPFHM) operator is defined as follows:

Definition 7. Let $\alpha_{j}(j=1,2,\ldots,n)$ be a collection of PFNs. The generalized picture fuzzy Heronian mean (GPFHM) operator is a mapping $P^{n}\to P$ such that

$\displaystyle\textit{GPFHM}^{p,q}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})=$ (5) $\displaystyle\qquad\left(\frac{2}{n(n+1)}\mathop{\oplus}_{i=1}^{n}\mathop{% \oplus}_{j=i}^{n}\left(\alpha_{i}^{p}\otimes\alpha_{j}^{q}\right)\right)^{% \frac{1}{p+q}}$

where $p,q\geqslant 0$ .

Based on the Definition 4, we can get the following result:

Theorem 1. The aggregated value by using GPFHM operator is also a PFN, where

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\textit{GPFHM}^{p,q}(\alpha_{1},\alpha_{2% },\ldots,\alpha_{n})$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!=\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_% {i=1}^{n}\mathop{\oplus}_{j=i}^{n}\left(\!\alpha_{i}^{p}\otimes\alpha_{j}^{q}% \!\right)\!\right)^{\frac{1}{p+q}}$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!=\left(\!\begin{array}[]{l}\bigg{(}\!1\!-% \!\prod\limits_{i=1,j=i}^{n}\left(\!1\!-\!(\mu_{\alpha_{i}})^{p}(\mu_{\alpha_{% j}})^{q}\!\right)^{\frac{2}{n(n+1)}}\!\bigg{)}^{\frac{1}{p+q}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}\!1\!-\!(1\!-\!\eta_{% \alpha_{i}})^{p}\\ \qquad(1\!-\!\eta_{\alpha_{j}})^{q}\!\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{% \frac{1}{p+q}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}\!1\!-\!(1\!-\!\nu_{% \alpha_{i}})^{p}\\ \qquad(1\!-\!\nu_{\alpha_{j}})^{q}\!\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{% \frac{1}{p+q}}\end{array}\!\right)$ (6)

where $p,q\geqslant 0$ .

Proof: By the operational laws for PFNs, we have

$\displaystyle\alpha_{i}^{p}=\left(\!(\mu_{\alpha_{i}})^{p},1-(1-\eta_{\alpha_{% i}})^{p},1-(1-\nu_{\alpha_{i}})^{p}\!\right)$ $\displaystyle\alpha_{j}^{p}=\left(\!(\mu_{\alpha_{j}})^{p},1-(1-\eta_{\alpha_{% j}})^{p},1-(1-\nu_{\alpha_{j}})^{p}\!\right)$

and

$\displaystyle\!\!\!\!\alpha_{i}^{p}\otimes\alpha_{j}^{q}=\Big{(}\!(\mu_{\alpha% _{i}})^{p}(\mu_{\alpha_{j}})^{q},1-(1-\eta_{\alpha_{i}})^{p}\!\!\!\!(1-\eta_{% \alpha_{j}})^{q},1-(1-\nu_{\alpha_{i}})^{p}(1-\nu_{\alpha_{j}})^{q}\!\Big{)}$

then

$\displaystyle\mathop{\oplus}_{i=1}^{n}\mathop{\oplus}_{j=i}^{n}\left(\!\alpha_% {i}^{p}\otimes\alpha_{j}^{q}\!\right)=\left(\!\begin{array}[]{l}1-\prod\limits% _{i=1,j=i}^{n}\left(\!1-(\mu_{\alpha_{i}})^{p}(\mu_{\alpha_{j}})^{q}\!\right),% \\ \prod\limits_{i=1,j=i}^{n}\left(\!1-(1-\eta_{\alpha_{i}})^{p}(1-\eta_{\alpha_{% j}})^{q}\!\right),\\ \prod\limits_{i=1,j=i}^{n}\left(\!1-(1-\nu_{\alpha_{i}})^{p}(1-\nu_{\alpha_{j}% })^{q}\!\right)\end{array}\!\right)$

and

$\displaystyle\!\!\!\!\frac{2}{n(n+1)}\mathop{\oplus}_{i=1}^{n}\mathop{\oplus}_% {j=i}^{n}\left(\!\alpha_{i}^{p}\otimes\alpha_{j}^{q}\!\right)=$ $\displaystyle\left(\!\begin{array}[]{l}1-\prod\limits_{i=1,j=i}^{n}\left(\!1-(% \mu_{\alpha_{i}})^{p}(\mu_{\alpha_{j}})^{q}\!\right)^{\frac{2}{n(n+1)}},\\ \prod\limits_{i=1,j=i}^{n}\left(\!1-(1-\eta_{\alpha_{i}})^{p}(1-\eta_{\alpha_{% j}})^{q}\!\right)^{\frac{2}{n(n+1)}},\\ \prod\limits_{i=1,j=i}^{n}\left(\!1-(1-\nu_{\alpha_{i}})^{p}(1-\nu_{\alpha_{j}% })^{q}\!\right)^{\frac{2}{n(n+1)}}\end{array}\!\right)$

Therefore, we have

The proof is completed.

It can be easily proved that the GPFHM operator has the following properties.

Property 1. (Idempotency) If all $\alpha_{j}$ ( $j=$ 1, 2, …, $n$ ) are equal, i.e. $\alpha_{j}=\alpha$ for all $j$ , then

$\displaystyle\textit{GPFHM}^{p,q}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})=\alpha$ (7)

Property 2. (Boundedness) Let $\alpha_{j}(j=1,2,\ldots,n)$ be a collection of PFNs, and let $\alpha^{-}=\mathop{\min}_{j}\alpha_{j}$ , $\alpha^{+}=\mathop{\max}_{j}\alpha_{j}$ ,

Then

$\displaystyle\alpha^{-}\leqslant\textit{GPFHM}^{p,q}(\alpha_{1},\alpha_{2},% \ldots,\alpha_{n})\leqslant\alpha^{+}$ (8)

Property 3. (Monotonicity) Let $\alpha_{j}(j=1,2,\ldots,n)$ and $\alpha^{\prime}_{j}(j=1,2,\ldots,n)$ be two set of PFNs, if $\alpha_{j}\leqslant\alpha^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\textit{GPFHM}^{p,q}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\leqslant$ (9) $\displaystyle\quad\textit{GPFHM}^{p,q}(\alpha^{\prime}_{1},\alpha^{\prime}_{2}% ,\ldots,\alpha^{\prime}_{n})$

By assigning different values to the parameters $p$ and $q$ , some special cases of the GPFHM can be obtained as follows:

Case 1. If $q=q=\frac{1}{2}$ , then the GPFHM reduces to

$\displaystyle\textit{GPFBM}^{\frac{1}{2},\frac{1}{2}}(\alpha_{1},\alpha_{2},% \ldots,\alpha_{n})$ $\displaystyle=\left(\!\begin{array}[]{l}1\!-\!\prod\limits_{i=1,j=i}^{n}\left(% 1\!-\!\sqrt{\mu_{\alpha_{i}}\mu_{\alpha_{j}}}\right)^{\frac{2}{n(n+1)}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\\ \Big{(}1\!-\!\sqrt{(1\!-\!\eta_{\alpha_{i}})(1\!-\!\eta_{\alpha_{j}})}\Big{)}^% {\frac{2}{n(n+1)}}\!\bigg{)},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\\ \Big{(}1\!-\!\sqrt{(1\!-\!\nu_{\alpha_{i}})(1\!-\!\nu_{\alpha_{j}})}\Big{)}^{% \frac{2}{n(n+1)}}\!\bigg{)}\end{array}\!\right)$ (10)

which we call the picture fuzzy Heronian mean.

Case 2. If $q\to 0$ then the GPFHM reduces to

$\displaystyle\!\!\!\!\!\mathop{\lim}_{q\to 0}\textit{GPFHM}^{p,q}(\alpha_{1},% \alpha_{2},\ldots,\alpha_{n})$ $\displaystyle\!\!\!\!\!=\left(\!\frac{1}{n}\mathop{\oplus}_{i=1}^{n}\alpha_{i}% ^{p}\!\right)^{\frac{1}{p}}$ $\displaystyle\!\!\!\!\!=\left(\!\begin{array}[]{l}\left(\!1\!-\!\prod\limits_{% i=1}^{n}\left(\!1-(\mu_{\alpha_{i}})^{p}\!\right)^{\frac{1}{n}}\!\right)^{% \frac{1}{p}},\\ 1-\left(\!1-\prod\limits_{i=1}^{n}\left(\!1-(1-\eta_{\alpha_{i}})^{p}\!\right)% ^{\frac{1}{n}}\!\right)^{\frac{1}{p}},\\ 1-\left(\!1-\prod\limits_{i=1}^{n}\left(\!1-(1-\nu_{\alpha_{i}})^{p}\!\right)^% {\frac{1}{n}}\!\right)^{\frac{1}{p}}\end{array}\!\right)$ (11)

which is the generalized picture fuzzy mean.

Case 3. If $p=$ 2, $q\to 0$ , then the GPFHM is transformed as:

$\displaystyle\textit{GPFHM}^{2,0}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\ldots% ,\tilde{\alpha}_{n})$ $\displaystyle=\left(\!\frac{1}{n}\left(\!\mathop{\oplus}_{i=1}^{n}\alpha_{i}^{% 2}\!\right)\!\right)^{\frac{1}{2}}$ $\displaystyle=\left(\!\begin{array}[]{l}\left(\!1-\prod\limits_{i=1}^{n}\left(% \!1-(\mu_{\alpha_{i}})^{2}\!\right)^{\frac{1}{n}}\!\right)^{\frac{1}{2}},\\ 1-\left(\!1-\prod\limits_{i=1}^{n}\left(\!1-(1-\eta_{\alpha_{i}})^{2}\!\right)% ^{\frac{1}{n}}\!\right)^{\frac{1}{2}},\\ 1-\left(\!1-\prod\limits_{i=1}^{n}\left(\!1-(1-\nu_{\alpha_{i}})^{2}\!\right)^% {\frac{1}{n}}\!\right)^{\frac{1}{2}}\end{array}\!\right)$ (12)

which is the picture fuzzy square mean.

Case 4. If $p=$ 1, $q\to 0$ , then the GPFHM reduces to the picture fuzzy average:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\textit{GPFHM}^{1,0}(\tilde{\alpha}_{1}% ,\tilde{\alpha}_{2},\ldots,\tilde{\alpha}_{n})=\frac{1}{n}\left(\!\mathop{% \oplus}_{i=1}^{n}\alpha_{i}\!\right)$ (13) $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!=\left(\!\left(\!1\!-\!\prod\limits_{i=% 1}^{n}\left(\!1\!-\!\mu_{\alpha_{i}}\!\right)^{\frac{1}{n}}\!\right),\prod% \limits_{i=1}^{n}\left(\!\eta_{\alpha_{i}}\!\right)^{\frac{1}{n}},\prod\limits% _{i=1}^{n}\left(\!\nu_{\alpha_{i}}\!\right)^{\frac{1}{n}}\!\right)$

Case 5. If $p=q=$ 1, then the GPFHM reduces to the following:

$\displaystyle\!\!\!\!\!\!\textit{GPFHM}^{1,1}(\alpha_{1},\alpha_{2},\ldots,% \alpha_{n})$ $\displaystyle\!\!\!\!\!\!=\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_{i=1}^{n}% \mathop{\oplus}_{j=i}^{n}\left(\!\alpha_{i}\otimes\alpha_{j}\!\right)\!\right)% ^{\frac{1}{2}}$ $\displaystyle\!\!\!\!\!\!=\left(\!\begin{array}[]{l}\bigg{(}\!1-\prod\limits_{% i=1,j=i}^{n}\left(1-\mu_{\alpha_{i}}\mu_{\alpha_{j}}\right)^{\frac{2}{n(n+1)}}% \!\bigg{)}^{\frac{1}{2}},\\ 1-\bigg{(}\!1-\prod\limits_{i=1,j=i}^{n}\\ \quad\left(1-(1-\eta_{\alpha_{i}})(1-\eta_{\alpha_{j}})\right)^{\frac{2}{n(n+1% )}}\!\bigg{)}^{\frac{1}{2}},\\ 1-\bigg{(}\!1-\prod\limits_{i=1,j=i}^{n}\\ \quad\left(1-(1-\nu_{\alpha_{i}})(1-\nu_{\alpha_{j}})\right)^{\frac{2}{n(n+1)}% }\!\bigg{)}^{\frac{1}{2}}\end{array}\!\right)$ (14)

which we call an generalized picture fuzzy interrelated square mean.

Case 6. If $p\to\infty$ , $q=$ 0, then the GPFHM reduces to

$\displaystyle\mathop{\lim}_{p\to\infty}\textit{GPFHM}^{p,0}(\alpha_{1},\alpha_% {2},\ldots,\alpha_{n})=$ (15) $\displaystyle\mathop{\lim}_{p\to\infty}\left(\!\frac{1}{n}\mathop{\oplus}_{i=1% }^{n}(\alpha_{i})^{p}\!\right)^{1/p}=\mathop{\max}_{i}(\alpha_{i})$

Case 7. If $p\to 0$ , $q=$ 0 then the GPFHM reduces to

$\displaystyle\!\!\!\!\!\!\!\!\mathop{\lim}_{p\to 0}\textit{GPFHM}^{p,0}(\alpha% _{1},\alpha_{2},\ldots,\alpha_{n})\!=$ (16) $\displaystyle\!\!\!\!\!\!\!\!\mathop{\lim}_{p\to 0}\left(\!\frac{1}{n}\mathop{% \oplus}_{i=1}^{n}(\alpha_{i})^{p}\!\right)^{1/p}\!=\!\mathop{\otimes}_{i=1}^{n% }(\alpha_{i})^{\frac{1}{n}}\!=\!\left(\!\mathop{\otimes}_{i=1}^{n}\alpha_{i}\!% \right)^{\frac{1}{n}}$

which is the picture fuzzy geometric mean operator.

In most cases, the aggregated arguments have their weights, therefore it is necessary to consider the weighted form of the GPFHM operator. In this section we shall propose the generalized picture fuzzy weighted Heronian mean (GPFWHM) operator.

Definition 8. Let $\alpha_{j}(j=1,2,\ldots,n)$ be a collection of PFNs, $w=(w_{1},w_{2},\ldots,w_{n})^{T}$ is the weight vector of $\alpha_{j}(j=1,2,\ldots,n)$ , where $w_{j}$ indicates the importance degree of $\alpha_{j}$ , satisfying $w_{j}>$ 0, $i=1,2,\ldots,n$ and $\sum_{j=1}^{n}w_{j}=1$ . If

$\displaystyle\!\!\!\!\!\!\!\!\!\!\textit{GPFWHM}_{w}^{p,q}(\alpha_{1},\alpha_{% 2},\ldots,\alpha_{n})$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\!\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_{i% =1}^{n}\mathop{\oplus}_{j=i}^{n}\left(\!\left(\!(\alpha_{i})^{w_{i}}\!\right)^% {p}\otimes\left(\!(\alpha_{j})^{w_{j}}\!\right)^{q}\!\right)\!\right)^{\frac{1% }{p+q}}$ (17) $\displaystyle\!\!\!\!\!\!\!\!\!\!=\!\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_{i% =1}^{n}\mathop{\oplus}_{j=i}^{n}\left(\!(\alpha_{i})^{pw_{i}}\otimes(\alpha_{j% })^{qw_{j}}\!\right)\!\right)^{\frac{1}{p+q}}$

then GPFWHM is called the generalized picture fuzzy weighted Heronian mean (GPFWHM) operator.

Similar to Theorem 1, the Theorem 2 can be derived easily.

Theorem 2. Let $p,q>$ 0, and $\alpha_{j}(j=1,2,\ldots,n)$ be a collection of PFNs, whose weight vector is $w=(w_{1},w_{2},\ldots,w_{n})^{T}$ , which satisfies $w_{j}>$ 0, $i=1,2,\ldots,n$ and $\sum_{j=1}^{n}w_{j}=$ 1. Then the aggregated value by using the GPFWHM is also an PFN, and

$\displaystyle\!\!\!\!\!\!\!\!\!\!\textit{GPFWHM}_{w}^{p,q}(\alpha_{1},\alpha_{% 2},\ldots,\alpha_{n})$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\bigg{(}\!\frac{2}{n(n+1)}\mathop{\oplus}_{i% =1}^{n}\mathop{\oplus}_{j=i}^{n}\left(\!\left(\!(\alpha_{i})^{w_{i}}\!\right)^% {p}\otimes\left(\!\left(\!\alpha_{j}\!\right)^{w_{j}}\!\right)^{q}\!\right)\!% \bigg{)}^{\frac{1}{p+q}}$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_{i=1% }^{n}\mathop{\oplus}_{j=i}^{n}\left(\!(\alpha_{i})^{pw_{i}}\otimes\left(\!% \alpha_{j}\!\right)^{qw_{j}}\!\right)\!\right)^{\frac{1}{p+q}}$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\left(\!\begin{array}[]{l}\bigg{(}\!1\!-\!% \prod\limits_{i=1,j=i}^{n}\Big{(}1\!-\!(\mu_{\alpha_{i}})^{pw_{i}}\\ \qquad(\mu_{\alpha_{j}})^{qw_{j}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{\frac{1% }{p+q}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}1\!-\!(1\!-\!\eta_{% \alpha_{i}})^{pw_{i}}\\ \qquad(1\!-\!\eta_{\alpha_{j}})^{qw_{j}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{% \frac{1}{p+q}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}1\!-\!(1\!-\!\nu_{% \alpha_{i}})^{pw_{i}}\\ \qquad(1\!-\!\nu_{\alpha_{j}})^{qw_{j}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{% \frac{1}{p+q}}\end{array}\!\right)$ (18)

4. Models for multiple attribute decision making with picture fuzzy information

Based the GPFWHM operator, in this section, we shall propose the model for multiple attribute decision making with picture fuzzy information. Let $A=\{A_{1},A_{2},\ldots,A_{m}\}$ be a discrete set of alternatives, and $G=\{G_{1},G_{2},\ldots,G_{n}\}$ be the set of attributes, $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})$ is the weighting vector of the attribute $G_{j}(j=1,2,\ldots,n)$ , where $\omega_{j}\in[0,1]$ , $\sum_{j=1}^{n}\omega_{j}=$ 1. Suppose that $\tilde{R}=\left(\!\tilde{r}_{ij}\!\right)_{m\times n}=(\mu_{ij},\eta_{ij},\nu_% {ij})_{m\times n}$ is the picture fuzzy decision matrix, where $\mu_{ij}$ indicates the degree of positive membership that the alternative $A_{i}$ satisfies the attribute $G_{j}$ given by the decision maker, $\eta_{ij}$ indicates the degree of neutral membership that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ , $\nu_{ij}$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ given by the decision maker, $\mu_{ij}\in[0,1]$ , $\eta_{ij}\in[0,1]$ , $\nu_{ij}\in[0,1]$ , $\mu_{ij}+\eta_{ij}+\nu_{ij}\leqslant$ 1, $\pi_{ij}=1-(\mu_{ij}+\eta_{ij}+\nu_{ij})$ , $i=1,2,\ldots,m$ , $j=1,2,\ldots,n$ .

In the following section, we apply the GPFWHM operator to the MADM problems with picture fuzzy information.

Table 1
The picture fuzzy decision matrix

	$A_{1}$	$A_{2}$	$A_{3}$	$A_{4}$
$G_{1}$	(0.53,0.33,0.09)	(0.73,0.12,0.08)	(0.91,0.03,0.02)	(0.85,0.09,0.05)
$G_{2}$	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.07,0.09,0.05)	(0.74,0.16,0.10)
$G_{3}$	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)
$G_{4}$	(0.08,0.89,0.02)	(0.73,0.15,0.08)	(0.68,0.26,0.06)	(0.08,0.84,0.06)

Table 2

The aggregating results of the ERP systems by the GPFWHM operators

	$A_{1}$	$A_{2}$	$A_{3}$	$A_{4}$
$p=$ 0, $q=$ 1	(0.822,0.127,0.010)	(0.798,0.109,0.024)	(0.800,0.076,0.013)	(0.739,0.185,0.013)
$p=$ 0.5, $q=$ 0.5	(0.827,0.121,0.009)	(0.795,0.108,0.020)	(0.810,0.065,0.010)	(0.779,0.151,0.010)
$p=q=$ 1	(0.838,0.114,0.009)	(0.805,0.105,0.019)	(0.820,0.062,0.010)	(0.804,0.139,0.010)
$p=$ 5, $q=$ 0	(0.933,0.046,0.005)	(0.878,0.064,0.009)	(0.924,0.018,0.003)	(0.942,0.039,0.005)
$p=q=$ 5	(0.891,0.081,0.009)	(0.852,0.087,0.019)	(0.875,0.045,0.009)	(0.895,0.082,0.010)

Step 1. We utilize the decision information given in matrix $\tilde{R}$ , and the GPFWHM operator

$\displaystyle\!\!\!\!\!\!\!\!\!\!\alpha_{i}=\textit{GPFWHM}_{w}^{p,q}\left(\!% \tilde{r}_{i1},{\tilde{r}}_{i2},\ldots,{\tilde{r}}_{in}\!\right)$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\left(\!\frac{2}{n(n+1)}\mathop{\oplus}_{j=1% }^{n}\mathop{\oplus}_{k=j}^{n}\left(\!(\alpha_{ij})^{pw_{j}}\otimes(\alpha_{ik% })^{qw_{k}}\!\right)\!\right)^{\frac{1}{p+q}}$ $\displaystyle\!\!\!\!\!\!\!\!\!\!=\left(\!\begin{array}[]{l}\bigg{(}\!1\!-\!% \prod\limits_{j=1,k=j}^{n}\Big{(}1\!-\!(\mu_{ij})^{pw_{j}}\\ \quad(\mu_{ik})^{qw_{k}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{\frac{1}{p+q}},% \\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}1\!-\!(1\!-\!\eta_{ij})% ^{pw_{j}}\\ \quad(1\!-\!\eta_{ik})^{qw_{k}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{\frac{1}{% p+q}},\\ 1\!-\!\bigg{(}\!1\!-\!\prod\limits_{i=1,j=i}^{n}\Big{(}1\!-\!(1\!-\!\nu_{ij})^% {pw_{j}}\\ \quad(1\!-\!\nu_{ik})^{qw_{k}}\Big{)}^{\frac{2}{n(n+1)}}\!\bigg{)}^{\frac{1}{p% +q}}\end{array}\!\right),$ (19) $\displaystyle\!\!\!\!\!\!\!\!\!\!i=1,2,\ldots,m.$

to derive the overall preference values $\alpha_{i}$ ( $i=$ 1, 2, …, $m$ ) of the alternative $A_{i}$ .

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

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

Step 4. End.

5. Numerical example

In this section, we utilize a practical multiple attribute decision making problems to illustrate the application of the developed approaches. Suppose an organization plans to implement enterprise resource planning (ERP) system (adapted from [44]). The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all possible information about ERP vendors and systems, project term choose four potential ERP systems $A_{i}(i=1,2,3,4)$ as candidates. The company employs some external professional organizations (or experts) to aid this decision-making. The project team selects four attributes to evaluate the alternatives: (1) function and technology G ${}_{1}$ , (2) strategic fitness G ${}_{2}$ , (3) vendor’s ability G ${}_{3}$ ; (4) vendor’s reputation G ${}_{4}$ . The four possible ERP systems $A_{i}(i=1,2,3,4)$ are to be evaluated using the picture fuzzy numbers by the decision makers under the above four attributes (whose weighting vector is $\omega=(0.2,0.1,0.3,0.4)$ ), and construct the following matrix $\tilde{R}=(\tilde{r}_{ij})_{4\times 4}$ is shown in Table 1.

In the following Section, in order to select the most desirable ERP systems, we utilize the GPFWHM operator to develop an approach to MADM problem with picture fuzzy information, which can be described as following.

Step 1. According to Table 1, aggregate all picture fuzzy numbers ${\tilde{r}}_{ij}(j=1,2,\ldots,n)$ by using the GPFWHM operator to derive the overall picture fuzzy numbers $\alpha_{i}(i=1,2,3,4)$ of the alternative $A_{i}$ . The aggregating results are shown in Table 2.

Step 2. According to the aggregating results shown in Table 2 and the score functions of the ERP systems are shown in Table 3.

Table 3
The score functions of the ERP systems

	$A_{1}$	$A_{2}$	$A_{3}$	$A_{4}$
$p=$ 0, $q=$ 1	0.812	0.774	0.788	0.726
$p=$ 0.5, $q=$ 0.5	0.817	0.775	0.800	0.769
$p=q=$ 1	0.828	0.786	0.811	0.794
$p=$ 5, $q=$ 0	0.928	0.868	0.921	0.937
$p=q=$ 5	0.882	0.833	0.865	0.885

Step 3. According to the score functions shown in Table 3 and the comparison formula of score functions, the ordering of the ERP systems are shown in Table 4. Note that “ $>$ ” means “preferred to”. As we can see, depending on the aggregation operators used, the ordering of the ERP systems is different.

Table 4

Ordering of the ERP systems

	Ordering
$p=$ 0, $q=$ 1	$A_{1}>A_{3}>A_{2}>A_{4}$
$p=$ 0.5, $q=$ 0.5	$A_{1}>A_{3}>A_{2}>A_{4}$
$p=q=$ 1	$A_{1}>A_{3}>A_{4}>A_{2}$
$p=$ 5, $q=$ 0	$A_{4}>A_{1}>A_{3}>A_{2}$
$p=q=$ 5	$A_{4}>A_{1}>A_{3}>A_{2}$

From the above analysis, we can easily find that our method is very flexible because it can provide the decision makers more choices as the parameters are assigned different values.

6. Conclusion

In this paper, we have given an interesting research on picture fuzzy information aggregation operators and their application in multiple attribute decision making. In order to aggregate the picture fuzzy information, the generalized picture fuzzy Heronian mean (GPFHM) and the generalized picture fuzzy weighted Heronian mean (GPFWHM) operators have been developed. Moreover, we have applied the GPFWHM operator to solve the multiple attribute decision making problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. By the illustrated example, we have roughly shown that the parameters of the aggregation operators indeed have an impact on the ranks of alternatives. The GPFHM and GPFWHM operators were distinguished from other existed operators not only due to the fact that the operators accommodate the picture fuzzy environment, but also due to the consideration of the inter-dependent phenomena among the arguments, which allows our operators to have more wide practical application potentials. In the future, the application of the proposed aggregating operators of PFSs needs to be explored in the decision making [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], risk analysis and many other fields under uncertain environments [58, 59, 60, 61, 62, 63, 64, 65, 66].

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China (grant no. 71571128), the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (grant nos 16YJA630033 and 17XJA630003), and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (grant no. 15TD0004).

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Picture fuzzy heronian mean aggregation operators in multiple attribute decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

Table 1 The picture fuzzy decision matrix

Table 3 The score functions of the ERP systems

Footnotes

Acknowledgments

References

Table 1
The picture fuzzy decision matrix

Table 3
The score functions of the ERP systems