The quantitative evaluation on the advertisement design effects with fuzzy number intuitionistic fuzzy information

Abstract

With the rapid growth of the market economy, ranked according to the proportion of consumption, China has become the world’s second, a lot of foreign companies increasingly value the Chinese market, and constantly improve their design philosophy and business philosophy. Throughout today’s advertising industry, it can be seen in the many excellent advertising designs, non-mainstream appearance has a high rate of non-mainstream cultural elements is a common design approach in the design of modern advertising, and more and more people sought after. In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. Firstly, we analyze several operations on the fuzzy number intuitionistic fuzzy sets. Then, we use the induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator to solve multiple attribute decision making with the fuzzy number intuitionistic fuzzy information. Finally, an illustrative example for evaluating the advertisement design effects is given to verify the developed approach.

Keywords

Evaluation fuzzy number intuitionistic fuzzy sets Hamacher aggregation operators induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator advertisement design effects

1. Introduction

Green marketing [1, 2] has been a research topic for over thirty years [3, 4]. Focusing mainly on developing marketing strategies to approach the green consumer population, it was not meant or not able to attract the mainstream consumer. A study by the United Nations Environmental Program (UNEP) [5, p. 15] reports that only 4% of consumers actually buy sustainable products, this is in stark contrast with the 40% who stated that they were willing to buy more sustainable products. The Natural Marketing Institute published that, although 16% of the consumers indicate that they are willing to pay 20% more for a product that is produced in a sustainable and environmentally friendly way [6, p. 4], in reality even fewer consumers deliver on that promise [7]. The market for greener products is under-exploited by marketers [8]. There appears to be a potentially much larger market for sustainable products if the mainstream consumer could be reached. This would make higher sales volumes possible, which are necessary to cover the potential extra cost to produce in a more sustainable and a more environmentally friendly way. Although there is extensive qualitative research in green marketing publicized [3, 4], practical guidelines for the successful advertising of sustainable products substantiated by quantitative research are scarce, especially for marketing towards mainstream consumers. This makes effective advertising of sustainable products difficult [9, 10].

Internet has changed the ways that people receive and use information, which displays in business, education and daily life, etc., and Internet advertising suddenly becomes protagonist of network economy after the emergence of standardization software, scale operation, effective objective orientation. Owing to the different characteristics of communicative media, the essential difference between Internet advertising and traditional advertising lie in that communicative receivers can interact with Internet advertising or Internet advertising owners, or interact with themselves among communicative receivers. Internet advertising product supply chain is a kind of service supply chain that involves these activities related to Internet advertising product of R&D, design, manufacture, communication, release and reaction to communicative receivers. Firstly, each member of Internet advertising product supply chain separately belongs to different investors who own different interest and duty and limited information, which lead to conflict in interest and risk to various degree, while respective local interest and behavior of each member are often not identical with systematic goal of the whole supply chain, which result in reducing systematic performance of Internet advertising product supply chain and decreasing revenue of members. Secondly, natural attribute of Internet and Internet advertising product make it difficult in constituting acknowledged standard of measuring the number of visiting Internet advertising, scaling the reactive degree to Internet advertising by receivers, and optimizing Internet advertising pricing model, which causes great difficulty in adjusting the efficiency of Internet advertising, and designing the compensation mechanism for the manufacturing and communicating work of advertising agency and media owner. Therefore, it is necessary to analyze the behavior strategy between the members of Internet advertising product supply chain, design and use coordinating mechanism of supply chain in order to share useful information, reconcile decision behavior of individual member, and make the individual behavior and objective of members identify with systematic behavior and objective of supply chain. The problem of evaluating the advertisement design effects with fuzzy number intuitionistic fuzzy information is the multiple attribute decision making (MADM) problems [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. Firstly, we analyze several operations on the fuzzy number intuitionistic fuzzy sets. Then, we use the induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator to solve multiple attribute decision making with the fuzzy number intuitionistic fuzzy information. Finally, an illustrative example for evaluating the advertisement design effects is given to verify the developed approach.

2. Preliminaries

On the basis of intuitionistic fuzzy set [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], Liu et al. [48] discussed the fuzzy number intuitionistic fuzzy set (FNIFS). Next, Wang et al. [49] presented the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator, and then they showed how to exploit this operator. Wang et al. [50] proposed several aggregation operators, such as the FNIFWG operator, the FNIFOWG operator and fuzzy number intuitionistic fuzzy hybrid geometric (FNIFHG) operator.

Definition 1 [48]. Given a fixed set $X=\{x_{1},x_{2},\ldots,$ $x_{n}\}$ , an FNIFS $\tilde{A}$ over $X$ denotes an object which is represented as follows:

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

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

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

Definition 2 [49]. Suppose that $\tilde{a}({x_{i}})=\langle(a({x_{i}}),b({x_{i}}),$ $c({x_{i}})),({l({x_{i}}),m({x_{i}}),p({x_{i}})})\rangle$ 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

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

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

Definition 3 [49, 50]. 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 $\tilde{a}({x_{i}})$ is defined as follows:

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

Definition 4 [49, 50]. 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 $\tilde{a}({x_{i}})$ is defined as follows:

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

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 $H({\tilde{a}({x_{i}})})\in[0,1]$ . The value of $H({\tilde{a}(x_{i})})$ increases when the accuracy of the FNIFV $\tilde{a}({x_{i}})$ increases as well.

3. IFNIFHCA operators

In the following, we introduced some fuzzy number intuitionistic fuzzy Aggregation operator based on the operations of HIVFEs and Hamacher sum [51, 52, 53, 54, 55, 56].

Definition 5 [57]. Assume that $\tilde{a}({x_{j}})=\langle(a({x_{j}}),b({x_{j}}),$ $c({x_{j}})),\!{({l(x_{j}),\!m({x_{j}}),\!p({x_{j}})})}\rangle$ refers to a set of FNIFEs, then the fuzzy number intuitionistic fuzzy Hamacher ordered weighted average (FNIFHOWA) operator is defined as follows:

$\displaystyle\text{FNIFHOWA}({\tilde{a}({x_{1}}),\tilde{a}(x_{2}),\ldots,% \tilde{a}({x_{n}})})=\mathop{\oplus}\limits_{j=1}^{n}({w_{j}\tilde{a}({x_{% \sigma(j)}})})=$ $\displaystyle\scalebox{1.5}{\Bigg{\langle}}\left({\frac{\prod\limits_{j=1}^{n}% {({1+(\gamma-1)a({x_{\sigma(j)}})})^{w_{j}}-\prod\limits_{j=1}^{n}{({1-a({x_{% \sigma(j)}})})^{w_{j}}}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)a({x_{\sigma(j)% }})})^{w_{j}}+(\gamma-1)\prod\limits_{j=1}^{n}{(1-a({x_{\sigma(j)})})^{w_{j}}}% }},}\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}{({1+(\gamma-1)b(x_{\sigma(j)})})^{w_% {j}}-\prod\limits_{j=1}^{n}{({1-b({x_{\sigma(j)}})})^{w_{j}}}}}{\prod\limits_{% j=1}^{n}{({1+(\gamma-1)b({x_{\sigma(j)}})})^{w_{j}}+(\gamma-1)\prod\limits_{j=% 1}^{n}{(1-b({x_{\sigma(j)})})^{w_{j}}}}},$ $\displaystyle\left.\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})c({x_{\sigma(j% )}})})^{w_{j}}-\prod\limits_{j=1}^{n}{({1-c({x_{\sigma(j)}})})^{w_{j}}}}}{% \prod\limits_{j=1}^{n}{({1+(\gamma-1)c({x_{\sigma(j)}})})^{w_{j}}+(\gamma-1)% \prod\limits_{j=1}^{n}{(1-c({x_{\sigma(j)})})^{w_{j}}}}}\right),$ $\displaystyle\left({\frac{\gamma\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{w_{% j}}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)({1-l({x_{\sigma(j)}})})})^{w_{j}}}% +(\gamma-1)\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{w_{j}}}}},\right.$ $\displaystyle\frac{\gamma\prod\limits_{j=1}^{n}{m({x_{\sigma(j)}})^{w_{j}}}}{% \prod\limits_{j=1}^{n}{({1+({\gamma-1})({1-m({x_{\sigma(j)}})})})^{w_{j}}}+(% \gamma-1)\prod\limits_{j=1}^{n}{m(x_{\sigma(j)})^{w_{j}}}},$ $\displaystyle\left.{\frac{\gamma\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{w_{% j}}}}{\prod\limits_{j=1}^{n}{(1+(\gamma-1)({1-p({x_{\sigma(j)})})})^{w_{j}}}+(% {\gamma-1})\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{w_{j}}}}}\right)% \scalebox{1.5}{\Bigg{\rangle}}$ (4)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,$ $\ldots,n)$ , where $\tilde{a}({x_{\sigma(j-1)}})\geqslant\tilde{a}({x_{\sigma(j)}})$ for all $j=2,\ldots,n$ , and $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is the aggregation-associated weight vector, and $w_{j}\in[0,1]$ , $\sum_{j=1}^{n}{w_{j}}$ $=1$ should be satisfied.

Definition 6 [58]. Suppose that $\tilde{a}({x_{j}})\!=\!\langle(a({x_{j}}),b({x_{j}}),$ $c({x_{j}})),{({l(x_{j}),m({x_{j}}),p({x_{j}})})}\rangle$ denotes FNIFEs, and $\mu$ be a fuzzy measure on $X$ , next, the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator is defined as follows.

$\displaystyle\text{FNIFHCA}({\tilde{a}({x_{1}}),\tilde{a}(x_{2}),\ldots,\tilde% {a}({x_{n}})})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A% _{\sigma(j-1)}})})\tilde{a}({x_{\sigma(j)}})})=$ $\displaystyle\scalebox{1.5}{\Bigg{\langle}}\left({\frac{\prod\limits_{j=1}^{n}% {({1+(\gamma-1)a({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}% })}-\prod\limits_{j=1}^{n}{({1-a({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu(% {A_{\sigma(j-1)}})}}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)a(x_{\sigma(j)})})% ^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n% }{({1-a(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}},}\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}{({1+(\gamma-1)b(x_{\sigma(j)})})^{% \mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}-\prod\limits_{j=1}^{n}{({1-b({x_{% \sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}{\prod\limits_{% j=1}^{n}{({1+(\gamma-1)b(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma% (j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n}{({1-b(x_{\sigma(j)})})^{\mu({A_{% \sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}},$ $\displaystyle\left.{\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})c({x_{\sigma(% j)}})})^{\mu(A_{\sigma(j)})-\mu({A_{\sigma({j-1})}})}-\prod\limits_{j=1}^{n}{(% {1-c({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}{\prod% \limits_{j=1}^{n}{({1+(\gamma-1)c(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n}{({1-c(x_{\sigma(j)})})^{% \mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}}\right),$ $\displaystyle\left({\frac{\gamma\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{\mu% ({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma% -1)({1-l({x_{\sigma(j)}})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(% \gamma-1)\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}}}},\right.$ $\displaystyle\frac{\gamma\prod\limits_{j=1}^{n}{m({x_{\sigma(j)}})^{\mu({A_{% \sigma(j)}})-\mu(A_{\sigma(j-1)})}}}{\prod\limits_{j=1}^{n}{(1+(\gamma-1)({1-m% ({x_{\sigma(j)})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(\gamma-1)% \prod\limits_{j=1}^{n}{m({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma% (j-1)}})}}},$ $\displaystyle\left.{\frac{\gamma\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{\mu% ({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma% -1)(1-p({x_{\sigma(j)})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(% \gamma-1)\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}}}}\right)\scalebox{1.5}{\Bigg{\rangle}}$ (5)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ refers to a arrange of $({1,2,\ldots,n})$ , where $\tilde{a}({x_{\sigma(j-1)}})\geqslant\tilde{a}({x_{\sigma(j)}})$ for all $j=2,\ldots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

In the following, Zeng [59] developed the induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator based on the I-COA operator [60] and fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator.

Definition 7 [59]. Let $\langle{u_{j},\tilde{a}(x_{j})}\rangle({j=1,2,\ldots,n})$ be a collection of 2-tuples on $X$ , and $\mu$ be a fuzzy measure on $X$ , the induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator is defined as follows.

$\displaystyle\text{IFNIFHCA}_{\mu}({\langle{u_{1},\tilde{a}({x_{1}})}\rangle,% \langle{u_{2},\tilde{a}({x_{2}})}\rangle,\ldots,\langle{u_{n},\tilde{a}(x_{n})% }\rangle})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu(A_{% \sigma(j-1)})})\tilde{a}(x_{\sigma(j)})})=$ $\displaystyle\scalebox{1.5}{\Bigg{\langle}}\left({\frac{\prod\limits_{j=1}^{n}% {({1+(\gamma-1)a({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}% })}-\prod\limits_{j=1}^{n}{({1-a({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu(% {A_{\sigma(j-1)}})}}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)a(x_{\sigma(j)})})% ^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n% }{({1-a(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}},}\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}{({1+(\gamma-1)b(x_{\sigma(j)})})^{% \mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}-\prod\limits_{j=1}^{n}{({1-b({x_{% \sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}{\prod\limits_{% j=1}^{n}{({1+(\gamma-1)b(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma% (j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n}{({1-b(x_{\sigma(j)})})^{\mu({A_{% \sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}},$ $\displaystyle\left.{\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})c({x_{\sigma(% j)}})})^{\mu(A_{\sigma(j)})-\mu({A_{\sigma({j-1})}})}-\prod\limits_{j=1}^{n}{(% {1-c({x_{\sigma(j)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}{\prod% \limits_{j=1}^{n}{({1+(\gamma-1)c(x_{\sigma(j)})})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}+(\gamma-1)\prod\limits_{j=1}^{n}{({1-c(x_{\sigma(j)})})^{% \mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}}}\right),$ $\displaystyle\left({\frac{\gamma\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{\mu% ({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma% -1)({1-l({x_{\sigma(j)}})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(% \gamma-1)\prod\limits_{j=1}^{n}{l({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}}}},\right.$ $\displaystyle\frac{\gamma\prod\limits_{j=1}^{n}{m({x_{\sigma(j)}})^{\mu({A_{% \sigma(j)}})-\mu(A_{\sigma(j-1)})}}}{\prod\limits_{j=1}^{n}{(1+(\gamma-1)({1-m% ({x_{\sigma(j)})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(\gamma-1)% \prod\limits_{j=1}^{n}{m({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma% (j-1)}})}}},$ $\displaystyle\left.{\frac{\gamma\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{\mu% ({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma% -1)(1-p({x_{\sigma(j)})})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(% \gamma-1)\prod\limits_{j=1}^{n}{p({x_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})}}}}\right)\scalebox{1.5}{\Bigg{\rangle}}$ (6)

where $\tilde{a}({x_{\sigma(j)}})$ is the $\tilde{a}({x_{j}})$ value of the IFNIFHCA pair $\langle{u_{i},\tilde{a}({x_{i}})}\rangle$ having the $j$ th largest $u_{i}({u_{i}\in[0,1]})$ , and $u_{i}$ in $\langle{u_{i},\tilde{a}({x_{i}})}\rangle$ is referred to as the order inducing variable and $\tilde{a}({x_{i}})$ as the fuzzy number intuitionistic fuzzy arguments, $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

4. 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, $i=1,2,\ldots,m$ ,.

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

Step 1.
Use the decision information which are provided from the matrix $\tilde{R}$ , and the IFNIFHCA operator.

$\displaystyle\tilde{r}_{i}=\text{IFNIFHCA}({\langle{u_{1},\tilde{r}_{i1}}% \rangle,\langle{u_{2},\tilde{r}_{i2}}\rangle,\ldots,\langle{u_{n},\tilde{r}_{% \rm in}}\rangle})=\mathop{\oplus}\limits_{j=1}^{n}({\mu({A_{\sigma(j)}})-\mu({% A_{\sigma(j-1)}})\tilde{r}_{\sigma(ij)}})=$ $\displaystyle\scalebox{1.5}{\Bigg{\langle}}\left({\frac{\prod\limits_{j=1}^{n}% {({1+(\gamma-1)a_{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}-% \prod\limits_{j=1}^{n}{({1-a_{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{% \sigma(j-1)})}}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)a_{\sigma(ij)}})^{\mu({% A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}+({\gamma-1})\prod\limits_{j=1}^{n}{({1-a% _{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}}}},}\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}{({1+(\gamma-1)b_{\sigma(ij)}})^{\mu(% {A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}-\prod\limits_{j=1}^{n}{({1-b_{\sigma(% ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}}}}{\prod\limits_{j=1}^{n}{(% {1+(\gamma-1)b_{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}+({% \gamma-1})\prod\limits_{j=1}^{n}{({1-b_{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-% \mu(A_{\sigma(j-1)})}}}},$ $\displaystyle\left.{\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})c_{\sigma(ij)% }})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}-\prod\limits_{j=1}^{n}{({1-c% _{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}}}}{\prod\limits_{j% =1}^{n}{({1+(\gamma-1)c_{\sigma(ij)}})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1% )})}+({\gamma-1})\prod\limits_{j=1}^{n}{({1-c_{\sigma(ij)}})^{\mu({A_{\sigma(j% )}})-\mu(A_{\sigma(j-1)})}}}}}\right),$ $\displaystyle\left({\frac{\gamma\prod\limits_{j=1}^{n}{l_{\sigma(ij)}^{\mu({A_% {\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)(% {1-l_{\sigma(ij)}})})^{\mu({A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}}+({\gamma-1}% )\prod\limits_{j=1}^{n}{l_{\sigma(ij)}^{\mu(A_{\sigma(j)})-\mu({A_{\sigma({j-1% })}})}}}},\right.$ $\displaystyle\frac{\gamma\prod\limits_{j=1}^{n}{m_{\sigma(ij)}^{\mu({A_{\sigma% (j)}})-\mu({A_{\sigma(j-1)}})}}}{\prod\limits_{j=1}^{n}{({1+(\gamma-1)({1-m_{% \sigma(ij)}})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+(\gamma-1)\prod% \limits_{j=1}^{n}{m_{\sigma(ij)}^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}% }},$ $\displaystyle\left.{\frac{\gamma\prod\limits_{j=1}^{n}{p_{\sigma({ij})}^{\mu({% A_{\sigma(j)}})-\mu(A_{\sigma(j-1)})}}}{\prod\limits_{j=1}^{n}{(1+(\gamma-1)({% 1-p_{\sigma(ij)})})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma(j-1)}})}}+({\gamma-1}% )\prod\limits_{j=1}^{n}{p_{\sigma(ij)}^{\mu(A_{\sigma(j)})-\mu({A_{\sigma({j-1% })}})}}}}\right)\scalebox{1.5}{\Bigg{\rangle}}$ (7)

to calculate $\tilde{r}_{i}({i=1,2,\ldots,m})$ of the alternative $A_{i}$ .
Step 2.
Compute the scores $S({\tilde{r}_{i}})({i=1,2,\ldots,m})$ of the fuzzy number intuitionistic fuzzy values $\tilde{r}_{i}(i=1,$ $2,\ldots,m)$ to sort all the alternatives $A_{i}(i\!=\!1,2,\ldots,m)$ and choose the optimal one.
Step 3.
Sort all the alternatives $A_{i}(i=1,2,\ldots,m)$ and choose the optimal one according to $S({\tilde{r}_{i}})$ and $H({\tilde{r}_{i}})({i=1,2,\ldots,m})$ .

5. Numerical example

Chinese traditional painting elements in modern plane advertisement design to join to the innovation of the work provides a wider space, composed of elements of Chinese traditional painting a variety of advertising design, elegance, of primitive simplicity, or clever, more can enhance the cultural connotation of advertisement design works. Modern plane advertisement design art as a mass media combined with traditional Chinese painting elements can form the beauty of the imposing manner, implicit beauty, the beauty of artistic conception, aesthetic characteristics, highly symbolic, in such a combination can transmission under the condition of the deep cultural inside information of the visual arts. Is known as the quintessence of Chinese traditional painting is an important part of the Chinese nation traditional culture, it has been influencing the Chinese nation and even to the modern aesthetic taste of the Orient. Will be skilful use of Chinese traditional painting elements in modern plane advertisement design, Chinese traditional national characteristics and cultural connotation will increase in the design work, improve the design level and quality, make our cultural products enter the international market, to promote the modernization construction in our country, has the extremely important practical significance. is the important content of our research, therefore, injection of traditional Chinese painting elements in the design of print ads, has a very important cultural significance. This section presents a numerical example to evaluate the advertisement design effects with uncertain linguistic variables to illustrate the method proposed. To test five possible advertisement enterprises, the expert group should make a decision according to four attributes include innovation resources input ability $({G_{1}})$ , research and development ability $({G_{2}})$ , manufacturing capacity and marketing ability $({G_{3}})$ and innovation output capacity $(G_{4})$ , respectively. The five possible advertisement enterprises $A_{i}({i=1,2,\ldots,5})$ should be calculated by the fuzzy number intuitionistic fuzzy numbers by the decision maker using the given attributes, the decision matrices as illustrated in $\tilde{R}=(\tilde{r}_{ij})_{5\times 4}$ :

$\displaystyle\tilde{R}=\left[\begin{array}[]{ll}{\langle{(0.2,0.3,0.4),(0.3,0.% 4,0.4)}\rangle}&{\langle{(0.5,0.6,0.7),(0.1,0.1,0.1)}\rangle}\\ {\langle{(0.3,0.4,0.5),(0.1,0.2,0.3)}\rangle}&{\langle{(0.4,0.5,0.5),(0.1,0.1,% 0.2)}\rangle}\\ {\langle{(0.1,0.2,0.3),(0.3,0.4,0.5)}\rangle}&{\langle{(0.3,0.4,0.5),(0.2,0.2,% 0.3)}\rangle}\\ {\langle{(0.4,0.5,0.6),(0.1,0.1,0.1)}\rangle}&{\langle{(0.7,0.7,0.7),(0.1,0.1,% 0.1)}\rangle}\\ {\langle{(0.6,0.6,0.7),(0.1,0.1,0.1)}\rangle}&{\langle{(0.4,0.4,0.4),(0.1,0.2,% 0.3)}\rangle}\end{array}\right.\left.\begin{array}[]{ll}{\langle{(0.5,0.5,0.6)% ,(0.1,0.1,0.2)}\rangle}&{\langle{(0.4,0.5,0.6),(0.1,0.1,0.1)}\rangle}\\ {\langle{(0.3,0.4,0.5),(0.1,0.2,0.3)}\rangle}&{\langle{(0.1,0.2,0.3),(0.3,0.4,% 0.5)}\rangle}\\ {\langle{(0.6,0.7,0.8),(0.1,0.1,0.1)}\rangle}&{\langle{(0.1,0.1,0.2),(0.4,0.5,% 0.6)}\rangle}\\ {\langle{(0.4,0.5,0.5),(0.1,0.2,0.3)}\rangle}&{\langle{(0.3,0.4,0.5),(0.2,0.3,% 0.3)}\rangle}\\ {\langle{(0.6,0.6,0.6),(0.1,0.1,0.1)}\rangle}&{\langle{(0.2,0.3,0.3),(0.3,0.4,% 0.5)}\rangle}\end{array}\right]$

In the following, we use the IFNIFHCA operator to MADM problems for evaluating the advertisement design effects with fuzzy number intuitionistic fuzzy information.

Step 1.
Suppose the fuzzy measure of attribute of $G_{j}(j=1,2,\ldots,n)$ and attribute sets of $G$ as follows:

$\displaystyle\mu({G_{1}})=0.25,\mu({G_{2}})=0.30,\mu({G_{3}})=0.35,\mu({G_{4}}% )=0.30,\mu({G_{1},G_{2}})=0.60,\mu({G_{1},G_{3}})=0.70$ $\displaystyle\mu({G_{1},G_{4}})=0.60,\mu({G_{2},G_{3}})=0.65,\mu({G_{2},G_{4}}% )=0.50,\mu({G_{3},G_{4}})=0.55$ $\displaystyle\mu({G_{1},G_{2},G_{3}})=0.70,\mu({G_{1},G_{2},G_{4}})=0.75,\mu({% G_{1},G_{3},G_{4}})=0.80,\mu({G_{2},G_{3},G_{4}})=0.85$ $\displaystyle\mu({G_{1},G_{2},G_{3},G_{4}})=1.00$
Step 2.
The experts use order-inducing variables to represent the complex attitudinal character involving the opinion of different members of the board of directors. The results are shown in Table 1.

Table 1
Inducing variables

$G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$

$A_{1}$ 21 15 18 13

$A_{2}$ 15 13 12 9

$A_{3}$ 11 20 19 22

$A_{4}$ 19 12 17 13

$A_{5}$ 20 13 23 24

Step 3.
We utilize the decision information given in matrix $\tilde{R}$ , and the IFNIFHCA operator to obtain the overall values $\tilde{r}_{i}$ of the advertisement enterprises $A_{i}(i=1,2,\ldots,5)$ .

$\displaystyle\tilde{r}_{1}=\langle{(0.385,0.471,0.575),(0.155,0.174,0.193)}% \rangle,\tilde{r}_{2}=\langle{(0.306,0.407,0.469),(0.122,0.178,0.285)}\rangle$ $\displaystyle\tilde{r}_{3}=\langle{(0.329,0.425,0.540),(0.210,0.239,0.288)}% \rangle,\tilde{r}_{4}=\langle{(0.516,0.568,0.613),(0.113,0.132,0.139)}\rangle$ $\displaystyle\tilde{r}_{5}=\langle{(0.505,0.516,0.556),(0.122,0.149,0.170)}\rangle$
Step 4.
Compute the scores $S({\tilde{r}_{i}})(i=1,2,\ldots,5)$ of the fuzzy number intuitionistic fuzzy values $\tilde{r}_{i}(i=1,2,$ $\ldots,5)$

$\displaystyle S({\tilde{r}_{1}})=0.603,∼{}S({\tilde{r}_{2}})=0.413,∼{}S({% \tilde{r}_{3}})=0.372,∼{}S({\tilde{r}_{4}})=0.874,∼{}S({\tilde{r}_{5}})=0.751$
Step 5.
Rank all the advertisement enterprises $A_{i}(i=1,2,3,4,5)$ according to the scores $S({\tilde{r}_{i}})(i=$ $1,2,\ldots,5)$ of values $\tilde{r}_{i}({i=1,2,\ldots,5})$ : $A_{4}\succ A_{5}\succ A_{1}\succ A_{2}\succ A_{3}$ , and thus the most desirable advertisement enterprise is $A_{4}$ .

6. Conclusion

	$G_{1}$	$G_{2}$	$G_{3}$	$G_{4}$
$A_{1}$	21	15	18	13
$A_{2}$	15	13	12	9
$A_{3}$	11	20	19	22
$A_{4}$	19	12	17	13
$A_{5}$	20	13	23	24

Print ad design development in the 20th century, the dominant role of commerce and art of these two attributes are constantly undergoing conversion and integration of this conversion and integration can be said the print ads evolution and change in the nature of law, not only to promote the print ads design their own rapid development, but also to further clarify the nature of advertising. Print ad design concepts and forms of development is an evolving process of change, in a specific period of time, tend to appear in the commercial and artistic Game one party dominates, to form the trend of conversion, but they never completely separated. New forms of these new ideas developed before the concept and design patterns are not completely negate, but there is some degree of blending. In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. Firstly, we analyze several operations on the fuzzy number intuitionistic fuzzy sets. Then, we use the induced fuzzy number intuitionistic fuzzy Hamacher correlated average (IFNIFHCA) operator to solve multiple attribute decision making with the fuzzy number intuitionistic fuzzy information. Finally, an illustrative example for evaluating the advertisement design effects is given to verify the developed approach. In the future, we will extend the proposed models to some other fields [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79].

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