Methods for evaluating the development potential of digital media technology with fuzzy number intuitionistic fuzzy information

Abstract

In this paper, we investigate the multiple attribute decision making (MADM) problems for evaluating the development potential of digital media technology with fuzzy number intuitionistic fuzzy information. We first introduce some operations on the fuzzy number intuitionistic fuzzy sets. Then, we further develop the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator. Then, we apply the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator to deal with MADM problems with the fuzzy number intuitionistic fuzzy information. Finally, an illustrative example for evaluating the development potential of digital media technology with the fuzzy number intuitionistic fuzzy information is given to verify the developed approach.

Keywords

Multiple attribute decision making (MADM)fuzzy number intuitionistic fuzzy sets Hamacher geometric aggregation operators fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator development potential digital media technology

1. Introduction

Atanassov [1, 2] proposed the concept of intuitionistic fuzzy set (IFS) on the basis of the concept of fuzzy set [3]. Recently, many works have investigated the intuitionistic fuzzy sets [9, 10, 11, 12, 13, 14, 15]. Liu and Yuan [16] introduced the concept of fuzzy number intuitionistic fuzzy set (FNIFS). Wang [17] proposed some aggregation operators, including the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator, fuzzy number intuitionistic fuzzy ordered weighted averaging (FNIFOWA) operator and fuzzy number intuitionistic fuzzy hybrid aggregation (FNIFHA) operator. Wang [18] developed the fuzzy number intuitionistic fuzzy weighted geometric (FNIFWG) operator, fuzzy number intuitionistic fuzzy ordered weighted geometric (FNIFOWG) operator and fuzzy number intuitionistic fuzzy hybrid geometric (FNIFHG) operator.

With the development of era, the form of art has also been gradually diversified, the current is the booming of digital media art, digital media technology in today’s films, computer, video, Internet, and other advanced digital imaging fusion and integration of scientific and technological achievements and developed a kind of new style of art. Digital media technology based on digital transmission and digital compression processing technology is widely used in digital media network transport of streaming media technology, based on computer graphics technology is widely applied in the digital entertainment industry in the computer animation technology, and based on human-computer interaction, computer graphics and display technology and widely used in entertainment, broadcast, display and virtual reality technology in the field of education, etc, etc. Digital media technology in the application of film and television and animation game industry also plays a huge role. Such as within the scope of the animation game, many game maker, also gradually from the simple way of modeling, rendering and post-production, mostly using 3 dvia platform, the platform including modeling, rendering and VSL, concerns the background support of programming code, the language module, real-time dynamic virtual effect, immersive and realistic performance incisively and vividly. With digital media technology, can at any time using the simulation method, the concept of real-time, makes all the creativity in a timely manner to capture, in time, change in time, make good ideas better. Then again, in the film and television industry, many special effects or Hollywood, pixar, dream works blockbusters, is not the traditional producing method, but USES the digital media technology, to carry on the combination of virtual and reality, both can produce can’t realize the impact of traditional way, and can save a lot of money, time and manpower, therefore, digital media technology has become the core of the film and television production technology. Relatively speaking, although the digital media technology in the song and dance drama, stage performance, the party has many of the application, but its research is far below the film and the game industry. Domestic digital media technology combined with stage art is still in its infancy. Therefore, how to extend the Hamacher operations to aggregate the fuzzy number intuitionistic fuzzy information is a meaningful work. To do so, the remainder of this paper is set out as follows. In the next section, some basic concepts of fuzzy number intuitionistic fuzzy sets are introduced. In Section 3 we have introduced the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator. In Section 4, we have applied the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator to develop the model for MADM problems with fuzzy number intuitionistic fuzzy information. In Section 5, a practical example for evaluating the development potential of digital media technology is given. In Section 6, we conclude the paper and give some remarks.

2. Preliminaries

Liu and Yuan [16] introduced the concept of fuzzy number intuitionistic fuzzy set (FNIFS) as follows.

Definition 1 [16]. Given a fixed set $X=\{x_{1}$ , $x_{2}$ , …, $x_{n}\}$ , an FNIFS $\tilde{A}$ over $X$ is an object having the form:

$\displaystyle\tilde{A}=\left\{\left\langle x_{i},\tilde{t}_{A}(x_{i}),\tilde{f% }_{A}(x_{i})\right\rangle|x_{i}\in X\right\}$ (1)

where $\tilde{t}_{A}(x_{i})\subset[0,1]$ and $\tilde{f}_{A}(x_{i})\subset[0,1]$ are triangular fuzzy numbers, and

$\displaystyle\tilde{t}_{A}(x_{i})=(a(x_{i}),b(x_{i}),c(x_{i})),X\to[0,1]$ $\displaystyle\tilde{f}_{A}(x)=(l(x_{i}),m(x_{i}),p(x_{i})),X\to[0,1]$ $\displaystyle 0\leqslant c(x_{i})+p(x_{i})\leqslant 1,\forall x\in X.$

For convenience, let $\tilde{t}_{A}(x_{i})=(a(x_{i}),b(x_{i}),c(x_{i}))$ , $\tilde{f}_{A}(x)=(l(x_{i}),m(x_{i}),p(x_{i}))$ , so $\tilde{a}(x_{i})=$ $\langle(a(x_{i})$ , $b(x_{i})$ , $c(x_{i}))$ , $(l(x_{i}),m(x_{i}),p(x_{i}))\rangle$ and we call $\tilde{a}(x_{i})$ an fuzzy number intuitionistic fuzzy value (FNIFV).

Definition 2 [17]. Let $\tilde{a}(x_{i})=$ $\langle(a(x_{i})$ , $b(x_{i})$ , $c(x_{i}))$ , $(l(x_{i}),m(x_{i}),p(x_{i}))\rangle$ and $\tilde{a}(x_{j})=$ $\langle(a(x_{j})$ , $b(x_{j})$ , $c(x_{j}))$ , $(l(x_{j}),m(x_{j}),p(x_{j}))\rangle$ be two FNIFVs, then

(1)

$\tilde{a}(x_{i})\otimes\tilde{a}(x_{j})=$ $\langle(a(x_{i})a(x_{j})$ , $b(x_{i})b(x_{j})$ , $c(x_{i})c(x_{j}))$ , $(l(x_{i})+l(x_{j})-l(x_{i})l(x_{j})$ , $m(x_{i})+m(x_{j})-m(x_{i})m(x_{j})$ , $p(x_{i})+p(x_{j})-p(x_{i})p(x_{j}))\rangle$ ;

(2)

$\left(\tilde{a}(x_{i})\right)^{\lambda}=$ $\left\langle\left((a(x_{i}))^{\lambda}\right.\right.$ , $(b(x_{i}))^{\lambda}$ , $\left.(c(x_{i}))^{\lambda}\right)$ , $\left(1-(1-l(x_{i}))^{\lambda}\right.$ , $1-(1-m(x_{i}))^{\lambda}$ , $1-(1-$ $\left.\left.p(x_{i}))^{\lambda}\right)\right\rangle$ , $\lambda\geqslant 0$ ;

Definition 3 [17, 18]. Let $\tilde{a}(x_{i})=\langle(a(x_{i})$ , $b(x_{i})$ , $c(x_{i}))$ , $(l(x_{i})$ , $m(x_{i})$ , $p(x_{i}))\rangle$ be a FNIFV, a score function $S$ of a FNIFV $\tilde{a}(x_{i})$ can be represented as follows:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!S\left(\tilde{a}(x_{i})\right)=\frac{a(x_% {i})+2b(x_{i})+c(x_{i})}{4}-$ (2) $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\quad\frac{l(x_{i})+2m(x_{i})+p(x_{i})}{4% },\ \ S\left(\tilde{a}(x_{i})\right)\in[-1,1].$

Definition 4 [17, 18]. Let $\tilde{a}(x_{i})=\langle(a(x_{i})$ , $b(x_{i})$ , $c(x_{i}))$ , $(l(x_{i}),m(x_{i}),p(x_{i}))\rangle$ be a FNIFV, an accuracy function $H$ of a FNIFV $\tilde{a}(x_{i})$ can be represented as follows:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!H\left(\tilde{a}(x_{i})\right)=(a(x_{i}% )+2b(x_{i})+c(x_{i}))$ (3) $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\>\>+(l(x_{i})+2m(x_{i})+p(x_{i}))/4\quad H% \left(\tilde{a}(x_{i})\right)\in[0,1].$

to evaluate the degree of accuracy of the FNIFV $\tilde{a}(x_{i})=\langle(a(x_{i}),b(x_{i}),c(x_{i})),(l(x_{i}),m(x_{i}),p(x_{i% }))\rangle$ , where $H\left(\tilde{a}(x_{i})\right)\in[0,1]$ . The larger the value of $H\left(\tilde{a}(x_{i})\right)$ , the more the degree of accuracy of the FNIFV $\tilde{a}(x_{i})$ is.

Based on the score function S and the accuracy function $H$ , in the following, we shall give an order relation between two fuzzy number intuitionistic fuzzy values, which is defined as follows.

Definition 5. Let $\tilde{a}(x_{i})$ and $\tilde{a}(x_{j})$ be two FNIFVs, $s(\tilde{a}(x_{i}))$ and $s(\tilde{a}(x_{j}))$ be the scores of $\tilde{a}(x_{i})$ and $\tilde{a}(x_{j})$ , respectively, and let $H(\tilde{a}(x_{i}))$ and $H(\tilde{a}(x_{j}))$ be the accuracy degrees of $\tilde{a}(x_{i})$ and $\tilde{a}(x_{j})$ , respectively, then if $s(\tilde{a}(x_{i}))<s(\tilde{a}(x_{j}))$ , then $\tilde{a}(x_{i})$ is smaller than $\tilde{a}(x_{j})$ , denoted by $\tilde{a}(x_{i})<\tilde{a}(x_{j})$ ; if $s(\tilde{a}(x_{i}))=s(\tilde{a}(x_{j}))$ , then

(1)

if $H(\tilde{a}(x_{i}))=H(\tilde{a}(x_{j}))$ , then $\tilde{a}(x_{i})$ and $\tilde{a}(x_{j})$ represent the same information, denoted by $\tilde{a}(x_{i})$ $=\tilde{a}(x_{j})$ ;

(2)

if $H(\tilde{a}(x_{i}))<H(\tilde{a}(x_{j}))$ , $\tilde{a}(x_{i})$ is smaller than $\tilde{a}(x_{j})$ , denoted by $\tilde{a}(x_{i})<\tilde{a}(x_{j})$ .

3. Fuzzy number intuitionistic fuzzy Hamacher geometric aggregation operators

T-norm and t-conorm are used to define a generalized union and intersection of fuzzy sets [19]. Roychowdhury and Wang [20] gave the definition and conditions of t-norm and t-conorm. A generalized union and a generalized intersection of intuitionistic fuzzy sets were introduced by Deschrijver and Kerre [21]. Further, Hamacher [22] proposed the Hamacher product and Hamacher sum.

Hamacher product $\otimes$ is a t-norm and Hamacher sum $\oplus$ is a t-conorm, where

$\displaystyle\!\!\!T(a,b)=a\otimes b=\frac{ab}{\gamma+(1-\gamma)(a+b-ab)}$ (4) $\displaystyle\!\!\!T*(a,b)=a\oplus b=\frac{a+b-ab-(1-\gamma)ab}{1-(1-\gamma)ab}$ (5)

Especially, when $\gamma=$ 1, then Hamacher t-norm and t-conorm will reduce to

$\displaystyle T(a,b)=a\otimes b=ab$ (6) $\displaystyle T*(a,b)=a\oplus b=a+b-ab$ (7)

which are the algebraic t-norm and t-conorm respectively; when $\gamma=$ 2, then Hamacher t-norm and t-conorm will reduce to

$\displaystyle T(a,b)=a\otimes b=\frac{ab}{1+(1-a)(1-b)}$ (8) $\displaystyle T*(a,b)=a\oplus b=\frac{a+b}{1+ab}$ (9)

which are called the Einstein t-norm and t-conorm respectively [21].

In the following, Wang et al. [23] introduced some fuzzy number intuitionistic fuzzy Aggregation operator based on the Hamacher operations [24, 25, 26, 27, 28].

Definition 6 [23]. Let $\tilde{a}(x_{j})=\langle(a(x_{j})$ , $b(x_{j})$ , $c(x_{j}))$ , $(l(x_{j})$ , $m(x_{j})$ , $p(x_{j}))\rangle$ be a collection of FNIFEs, then we define the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator as follows:

$\displaystyle\text{FNIFHWG}(\tilde{a}(x_{1}),\tilde{a}(x_{2}),\ldots,\tilde{a}% (x_{n}))$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}\left(\tilde{a}(x_{j})\right)^{% \omega_{j}}$ $\displaystyle=\left[\left(,\right.\right.$ $\displaystyle\quad\frac{\gamma\prod\limits_{j=1}^{n}b(x_{j})^{\omega_{j}}}{% \makecell{\prod\limits_{j=1}^{n}(1+(\gamma-1)(1-b(x_{j})))^{\omega_{j}}+(% \gamma-1)},}$ $\displaystyle\quad\left.\right),$ $\displaystyle\quad\left(,\right.$ $\displaystyle\frac{\makecell{\prod\limits_{j=1}^{n}(1+(\gamma-1)m(x_{j}))^{% \omega_{j}}-}{\makecell{\prod\limits_{j=1}^{n}(1+(\gamma-1)m(x_{j}))^{\omega_{% j}}+(\gamma-1)}}}$ $\displaystyle\left.\left.\right)\right]$ (10)

where $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})^{T}$ be the weight vector of $\tilde{a}(x_{j})$ $(j=1,2,\ldots,n)$ , and $\omega_{j}>$ 0, $\sum_{j=1}^{n}\omega_{j}=$ 1.

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

Theorem 1. (Idempotency) If all $\tilde{a}(x_{j})$ $(j=1,2,\ldots$ , $n)$ are equal, i.e. $\tilde{a}(x_{j})=\tilde{a}(x)$ for all $j$ , then

$\displaystyle\text{FNIFHWG}(\tilde{a}(x_{1}),\tilde{a}(x_{2}),\ldots,\tilde{a}% (x_{n}))=\tilde{a}(x)$ (11)

Theorem 2. (Boundedness) Let $\tilde{a}(x_{j})$ $(j=1,2,\ldots$ , $n)$ be a collection of FNIFEs, and let

$\displaystyle\tilde{a}{(x_{j})^{-}}=\mathop{\min}\limits_{j}\tilde{a}(x_{j}),% \tilde{a}{(x_{j})^{+}}=\mathop{\max}\limits_{j}\tilde{a}(x_{j})$ (12)

Then

$\displaystyle\@setsize{\footnotesize}{9.5pt}{\viiipt}{\@viiipt}\!\!\!\!\!\!\!% \!\!\!\!\!\tilde{a}{(x)^{-}}\!\leqslant\!\text{FNIFHWG}(\tilde{a}(x_{1}),% \tilde{a}(x_{2}),\ldots,\tilde{a}(x_{n}))\!\leqslant\!\tilde{a}{(x)^{+}}$

Theorem 3. (Monotonicity) Let $\tilde{a}(x_{j})$ $(j=1,2,\ldots$ , $n)$ and $\tilde{a}^{\prime}(x_{j})$ $(j=1,2,\ldots,n)$ be two set of FNIFEs, if $\tilde{a}(x_{j})\leqslant\tilde{a}^{\prime}(x_{j})$ , for all $j$ , then

$\displaystyle\text{FNIFHWG}(\tilde{a}(x_{1}),\tilde{a}(x_{2}),\ldots,\tilde{a}% (x_{n}))\leqslant$ (13) $\displaystyle\text{FNIFHWG}(\tilde{a}^{\prime}(x_{1}),\tilde{a}^{\prime}(x_{2}% ),\ldots,\tilde{a}^{\prime}(x_{n}))$

Theorem 4. (Commutativity) Let $\tilde{a}(x_{j})$ $(j=1,2,\ldots$ , $n)$ and $\tilde{a}^{\prime}(x_{j})$ $(j=1,2,\ldots,n)$ be two set of FNIFEs, then

$\displaystyle\text{FNIFHWG}(\tilde{a}(x_{1}),\tilde{a}(x_{2}),\ldots,\tilde{a}% (x_{n}))=$ (14) $\displaystyle\text{FNIFHWG}(\tilde{a}^{\prime}(x_{1}),\tilde{a}^{\prime}(x_{2}% ),\ldots,\tilde{a}^{\prime}(x_{n}))$

where $\tilde{a}^{\prime}(x_{j})$ $(j=1,2,\ldots,n)$ is any permutation of $\tilde{a}(x_{j})$ $(j=1,2,\ldots,n)$ .

4. An approach to multiple attribute decision making with fuzzy number intuitionistic 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 $\tilde{R}=(\tilde{r}_{ij})_{m\times n}=\langle(a_{ij},b_{ij},c_{ij}),(l_{ij},m% _{ij},p_{ij})\rangle_{m\times n}$ is the fuzzy number intuitionistic fuzzy decision matrix, where ( $a_{ij},b_{ij},c_{ij}$ ) indicates the degree that the alternative $A_{i}$ satisfies the attribute $G_{j}$ , $(l_{ij},m_{ij},p_{ij})$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ $(a_{ij},b_{ij},c_{ij})\subset[0,1]$ , $(l_{ij},m_{ij},p_{ij})\subset[0,1]$ , $c_{ij}+p_{ij}\leqslant$ 1, $i=1,2,\ldots,m$ , $j=1,2,\ldots,n$ .

In the following, we apply the FNIFHWG operator to MADM problems for with fuzzy number intuitionistic fuzzy information.

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the FNIFHWG operator.

$\displaystyle\!\!\!\tilde{r}_{i}=\text{FNIFHWG}_{\omega}\left(\tilde{r}_{i1},% \tilde{r}_{i2},\ldots,\tilde{r}_{in}\right)=\mathop{\otimes}\limits_{j=1}^{n}% \left(\tilde{r}_{ij}\right)^{\omega_{j}}$ $\displaystyle=\left[\left(,\right.\right.$ $\displaystyle\frac{\gamma\prod\limits_{j=1}^{n}b_{ij}^{\omega_{j}}}{\makecell{% \prod\limits_{j=1}^{n}\left(1+(\gamma-1)\left(1-b_{ij}^{\omega_{j}}\right)% \right)^{\omega_{j}}+(\gamma-1)},}$ $\displaystyle\left.\right),$ $\displaystyle\left(,\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}(1+(\gamma-1)m_{ij})^{\omega_{j}}-% \prod\limits_{j=1}^{n}(1-m_{ij})^{\omega_{j}}}{\makecell{\prod\limits_{j=1}^{n% }(1+(\gamma-1)m_{ij})^{\omega_{j}}+(\gamma-1)\prod\limits_{j=1}^{n}},}$ $\displaystyle\left.\left.\right)\right]$ (15)

to derive the values $\tilde{r}_{i}$ $(i=1,2,\ldots,m)$ of the alternative $A_{i}$ .

Step 2. Calculate the scores $S\left(\tilde{r}_{i}\right)$ $(i=1,2,\ldots,m)$ of the overall FNIFVs $\tilde{r}_{i}$ $(i=1,2,\ldots,m)$ to rank all the alternatives $A_{i}$ $(i=1,2,\ldots,m)$ and then to select the best one(s).

Step 3. Rank all the alternatives $A_{i}$ $(i=1,2,\ldots$ , $m)$ and select the best one(s) in accordance with $S\left(\tilde{r}_{i}\right)$ .

Step 4. End.

5. Numerical example

The rapid development of digital technology, makes the graphic object of visual communication changes, from the previous print static, to the current dynamic multiple media convey, various senses such as vision and hearing expression has already replaced the pure visual communication; And digital media, which attracts the audience recently, also comes in a variety of art forms, and it directly affects the audience’s needs, social experience and psychological activities. Then the digital media art has been closely connected with people’s daily life. Obviously, the demand of the flash dynamic video promoted more and more, which also shows the current market demand. Web page, digital image processing and other dynamic graphic visual symbol appeared in the 1990s, when the people describe digital media as multiple-amphibian animals. And in order to develop the new vision progress, visual symbol developer needs to constantly summarize in the study to explore the visual form, philosophy and psychology to subjects to accumulate the experience and theory. Under the action of these new factors, visual symbol works toward a rich and diversified development, the digital media discussed hereby is mainly about the new forms of digital media communication under the high-tech, and also some further study on the dynamic visual symbol characteristics in the new digital media. Visual symbols can be said to be the spiritual civilization and material civilization span and the bridge of communication. During the past more than a century, the unique characteristics of the technique of expression, rich image make visual communication field of visual symbols expanded the people’s experience, it also played a important role for the understanding of people’s knowledge to the objective world, and it also has extremely profound influence for the spiritual wealth and material life and even the development of all kinds of ways of thinking And digital media is also called as pioneer media since the pioneer art form appeared earliest. Once the technology of the digital media can be transformed into practical application, and popularized effectively, then the digital media will be a new focus after the Internet media. Digital media has spread on the Internet and multimedia technology and digital technology is relatively mature in the field of visual communication, and then the related research of the visual symbols in the form of visual communication and information dissemination has commenced. The development potential evaluation of digital media technology with fuzzy number intuitionistic fuzzy information is a MADM problem [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Thus, in this section we shall present a numerical example for evaluating the development potential of digital media technology with the fuzzy number intuitionistic fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible digital media technology enterprises $A_{i}$ $(i=1,2,\ldots,5)$ to select. The experts selects four attribute to evaluate the five possible digital media technology enterprises: ⟀ $G_{1}$ is the performance of digital media technology; ⟁ $G_{2}$ is the availability of the digital media technology; ⟂ $G_{3}$ is the reliability of digital media technology; ⟃ $G_{4}$ is the security of digital media technology. The five possible digital media technology enterprises $A_{i}$ $(i=1,2,\ldots,5)$ are to be evaluated using the FNIFs, and construct, respectively, the decision matrices as listed as follows:

$\displaystyle\tilde{R}=\left[\begin{array}[]{cc}\langle(0.5,0.5,0.6),(0.1,0.1,% 0.2)\rangle\\ \langle(0.4,0.5,0.6),(0.1,0.1,0.1)\rangle\\ \langle(0.3,0.4,0.5),(0.1,0.2,0.3)\rangle\\ \langle(0.1,0.2,0.3),(0.3,0.4,0.5)\rangle\\ \langle(0.6,0.7,0.8),(0.1,0.1,0.1)\rangle\\ \langle(0.1,0.1,0.2),(0.4,0.5,0.6)\rangle\\ \langle(0.4,0.5,0.5),(0.1,0.2,0.3)\rangle\\ \langle(0.3,0.4,0.5),(0.2,0.3,0.3)\rangle\\ \langle(0.6,0.6,0.6),(0.1,0.1,0.1)\rangle\\ \langle(0.2,0.3,0.3),(0.3,0.4,0.5)\rangle\\ \langle(0.2,0.3,0.4),(0.3,0.4,0.4)\rangle\\ \langle(0.5,0.6,0.7),(0.1,0.1,0.1)\rangle\\ \langle(0.3,0.4,0.5),(0.1,0.2,0.3)\rangle\\ \langle(0.4,0.5,0.5),(0.1,0.1,0.2)\rangle\\ \langle(0.1,0.2,0.3),(0.3,0.4,0.5)\rangle\\ \langle(0.3,0.4,0.5),(0.2,0.2,0.3)\rangle\\ \langle(0.4,0.5,0.6),(0.1,0.1,0.1)\rangle\\ \langle(0.7,0.7,0.7),(0.1,0.1,0.1)\rangle\\ \langle(0.6,0.6,0.7),(0.1,0.1,0.1)\rangle\\ \langle(0.4,0.4,0.4),(0.1,0.2,0.3)\rangle\\ \end{array}\right]$

In the following, we apply the FNIFHWG operator to MADM problems for evaluating the development potential of digital media technology of digital media technology enterprises with fuzzy number intuitionistic fuzzy information.

Step 1. We utilize the information given in matrix $\tilde{R}$ , and the FNIFHWG operator to obtain the values $\tilde{r}_{i}$ of the digital media technology enterprises $A_{i}$ $(i=1,2,\ldots,5)$ .

$\displaystyle\tilde{r}_{1}=\langle(0.326,0.429,0.543),(0.216,0.245,0.286)\rangle$ $\displaystyle\tilde{r}_{2}=\langle(0.519,0.561,0.617),(0.111,0.143,0.154)\rangle$ $\displaystyle\tilde{r}_{3}=\langle(0.516,0.525,0.553),(0.126,0.142,0.173)\rangle$ $\displaystyle\tilde{r}_{4}=\langle(0.384,0.478,0.579),(0.152,0.179,0.187)\rangle$ $\displaystyle\tilde{r}_{5}=\langle(0.302,0.403,0.476),(0.126,0.173,0.281)\rangle$

Step 2. Calculate the scores $S\left(\tilde{r}_{i}\right)$ $(i=1,2,\ldots,5)$ of the overall FNIFVs $\tilde{r}_{i}$ $(i=1,2,\ldots,5)$

$\displaystyle S\left(\tilde{r}_{1}\right)=0.606,S\left(\tilde{r}_{2}\right)=0.% 421,S\left(\tilde{r}_{3}\right)=0.365,$ $\displaystyle S\left(\tilde{r}_{4}\right)=0.871,S\left(\tilde{r}_{5}\right)=0.% 746$

Step 3. Rank all the digital media technology enterprises $A_{i}$ $(i=$ 1,2,3,4,5) in accordance with the scores $S\left(\tilde{r}_{i}\right)$ $(i=$ 1, 2, …, 5) of the FNIFVs $\tilde{r}_{i}$ $(i=$ 1, 2, …, 5): $A_{4}\succ A_{5}\succ A_{1}\succ A_{2}\succ A_{3}$ , and thus the most desirable digital media technology enterprises is $A_{4}$ .

6. Conclusion

In this paper, we investigate the multiple attribute decision making problems for evaluating the development potential of digital media technology with the fuzzy number intuitionistic fuzzy information. We first introduce some operations on the FNIFV set. Then, we further develop some new Hamacher geometric aggregation operators with FNIFV information, such as the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator. Then, we apply the fuzzy number intuitionistic fuzzy Hamacher weighted geometric (FNIFHWG) operator to deal with MADM problems. Finally, an illustrative example for evaluating the development potential of digital media technology is given to verify the developed approach. In the future, we shall continue the application of the developed operators to other uncertain [41, 42, 43, 44, 45, 46, 47, 48] and fuzzy environments [49, 50, 51, 52, 53, 54, 55, 56, 57].

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