Fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing

Abstract

As a constituent element of tourism system, tourist attraction is the spatial carrier of tourism activity. Those tourist attractions that can provide the high-quality service for the tourists will obtain the satisfactory tourists, drum for more tourists and achieve the good performance. Therefore service quality is the crucial factor to build the market image, acquire the steady tourist flow and increase the profit. It is the life-line of tourist attractions. The crux of service quality management in tourist attractions is how to evaluate and proceed to improve the service quality of tourist attractions. In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the triangular fuzzy comprehensive evaluation problems for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing. Finally, a practical example for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing is given to verify the developed approach.

Keywords

Fuzzy comprehensive evaluation triangular fuzzy information Einstein aggregation operators triangular fuzzy Einstein hybrid average (TFEHA) operator tourist scenic spot service quality perspective of Wechat marketing

1. Introduction

The modern tourism is a new industry full of vigor and vitality, and is called the sunrise industry because of its strong requirement, fast development and broad prospect. As the core component of tourism, scenic regions will become the source of power to push the deep developments of tourism and the increase pole to the tourism industry. The continuous development of economy in our country and the improvement of the economical globalization and the people’s living standard have brought unprecedented opportunities to scenic regions. At the same time, the competitions in the scenic regions are becoming intense, superheated and global day by day. The service quality comes to be the key point to the competition. This situation makes the scenic regions aim at the service quality in order to win the reputation, markets and tourists, So that they will develop healthily and continuously. So the control of the service quality has become one of the core problems to the development of the scenic regions. The most important thing to improve the service quality is quality standardization, which also the most urgent matter to the development of the scenic regions in our country. The service quality will bring about valuable promotion only after the management of scenic regions is strengthened and the standard service is implemented well.

A multiple attribute decision making problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Xu [20] developed some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator, fuzzy hybrid harmonic mean (FHHM) operator. Wei [21] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making. Wei et al. [22] proposed the generalized triangular fuzzy correlated averaging operator for multiple attribute decision making. Wei et al. [23] proposed the fuzzy power aggregating operators for multiple attribute group decision making. Zhao et al. [24] proposed the fuzzy prioritized operators for multiple attribute group decision making.

The fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing is the multiple attribute decision making problems. In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing. Finally, a practical example for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing is given to verify the developed approach.

2. Preliminaries

In the following, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers.

Definition 1 [25]. A triangular fuzzy numbers $\tilde{{a}}$ can be defined by a triplet $({a^{L},a^{M},a^{R}})$ . The membership function $\mu_{\tilde{{a}}}(x)$ is defined as:

$\mu_{\tilde{{a}}}(x)=\begin{cases}0,&x<a^{L},\\ {\displaystyle\frac{x-a^{L}}{a^{M}-a^{L}}},&a^{L}\leqslant x\leqslant a^{M},\\ {\displaystyle\frac{x-a^{R}}{a^{M}-a^{R}}},&a^{M}\leqslant x\leqslant a^{R},\\ 0,&x\geqslant a^{R}.\\ \end{cases}$ (1)

where $0<a^{L}\leqslant a^{M}\leqslant a^{R}$ , $a^{L}$ and $a^{R}$ stand for the lower and upper values of the support of $\tilde{{a}}$ , respectively, and $a^{M}$ for the modal value.

Definition 2 [26]. If $A$ is a triangular fuzzy variable $[{a^{L},a^{M},a^{R}}]$ , according to the Eq. (2), then the expected value of $A$ is

$E(A)=\frac{1}{3}({a^{L}+a^{M}+a^{R}})$ (2)

Motivated by the Definition of the Einstein operations, let a t-norm $T$ be the Einstein product $\otimes_{\varepsilon}$ and t-conorm $S$ be the Einstein sum $\oplus_{\varepsilon}$ , then the generalized intersection and union on two fuzzy numbers $\tilde{{a}}_{1}=[{a_{1}^{L},a_{1}^{M},a_{1}^{R}}]$ and $\tilde{{a}}_{2}=[{a_{2}^{L},a_{2}^{M},a_{2}^{R}}]$ become the Einstein product (denoted by $\tilde{{a}}_{1}\otimes_{\varepsilon}\tilde{{a}}_{2}$ ) and Einstein sum (denoted by $\tilde{{a}}_{1}\oplus_{\varepsilon}\tilde{{a}}_{2}$ ) of two fuzzy numbers $\tilde{{a}}_{1}=[{a_{1}^{L},a_{1}^{M},a_{1}^{R}}]$ and $\tilde{{a}}_{2}=[{a_{2}^{L},a_{2}^{M},a_{2}^{R}}]$ , respectively, as follows:

$\tilde{{a}}_{1}\oplus_{\varepsilon}\tilde{{a}}_{2}=\left({{\displaystyle\frac{% a_{1}^{L}+a_{2}^{L}}{1{+}a_{1}^{L}a_{2}^{L}}},{\displaystyle\frac{a_{1}^{M}+a_% {2}^{M}}{1{+}a_{1}^{M}a_{2}^{M}}},{\displaystyle\frac{a_{1}^{R}+a_{2}^{R}}{1{+% }a_{1}^{R}a_{2}^{R}}}}\right);$

$\tilde{{a}}_{1}\otimes_{\varepsilon}\tilde{{a}}_{2}=\left({\displaystyle\frac{% a_{1}^{L}a_{2}^{L}}{1+(1-a_{1}^{L})(1-a_{2}^{L})}},\right.$ $\left.{\displaystyle\frac{a_{1}^{M}a_{2}^{M}}{1{+}(1{-}a_{1}^{M})(1{-}a_{2}^{M% })}},{\displaystyle\frac{a_{1}^{R}a_{2}^{R}}{1{+}(1{-}a_{1}^{R})(1{-}a_{2}^{R}% )}}\right);$

$\lambda._{\varepsilon}\tilde{{a}}_{1}=\left({\displaystyle\frac{({1+a_{1}^{L}}% )^{\lambda}-({1-a_{1}^{L}})^{\lambda}}{({1+a_{1}^{L}})^{\lambda}+({1-a_{1}^{L}% })^{\lambda}}},\right.$ $\left.{\displaystyle\frac{({1{+}a_{1}^{M}})^{\lambda}{-}({1{-}a_{1}^{M}})^{% \lambda}}{({1{+}a_{1}^{M}})^{\lambda}{+}({1{-}a_{1}^{M}})^{\lambda}}},{% \displaystyle\frac{({1{+}a_{1}^{R}})^{\lambda}{-}({1{-}a_{1}^{R}})^{\lambda}}{% ({1{+}a_{1}^{R}})^{\lambda}{+}({1{-}a_{1}^{R}})^{\lambda}}}\right),$ $\lambda>0;$

$\tilde{{a}}_{1}^{\wedge_{\varepsilon}\lambda}=\left({\displaystyle\frac{2({a_{% 1}^{L}})^{\lambda}}{(2-a_{1}^{L})^{\lambda}+({a_{1}^{L}})^{\lambda}}},\right.$ $\left.{\displaystyle\frac{2({a_{1}^{M}})^{\lambda}}{(2{-}a_{1}^{M})^{\lambda}{% +}({a_{1}^{M}})^{\lambda}}},{\displaystyle\frac{2({a_{1}^{R}})^{\lambda}}{(2{-% }a_{1}^{R})^{\lambda}{+}({a_{1}^{R}})^{\lambda}}}\right),\lambda>0$ .

3. Triangular fuzzy Einstein aggregation operators

In the following, we shall develop some fuzzy Einstein arithmetic aggregation operator based on the operations of fuzzy numbers and Einstein sum.

Definition 3. Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, then we define the triangular fuzzy Einstein weighted average (TFEWA) operator as follows:

${\rm TFEWA}_{\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{a}}_{n}})% =\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}\omega_{j}\tilde{{a}}_{j}$ (3)

where $\omega=({\omega_{1},\omega_{2},\ldots,\omega_{n}})^{T}$ be the weight vector of $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ , and $\omega_{j}>0$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ .

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive the Theorem 1.

Theorem 1. Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEWA operator is also a triangular fuzzy number, and

$\displaystyle{\rm TFEWA}_{\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}\omega_{j}% \tilde{{a}}_{j}$ $\displaystyle=\left({\frac{\prod\limits_{j=1}^{n}{({1+a_{j}^{L}})^{\omega_{j}}% -\prod\limits_{j=1}^{n}{({1-a_{j}^{L}})^{\omega_{j}}}}}{\prod\limits_{j=1}^{n}% {({1+a_{j}^{L}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-a_{j}^{L}})^{\omega_{% j}}}}},}\right.$ $\displaystyle\mspace{40.0mu }\frac{\prod\limits_{j=1}^{n}{({1+a_{j}^{M}})^{% \omega_{j}}-\prod\limits_{j=1}^{n}{({1-a_{j}^{M}})^{\omega_{j}}}}}{\prod% \limits_{j=1}^{n}{({1+a_{j}^{M}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-a_{j% }^{M}})^{\omega_{j}}}}},$ $\displaystyle\mspace{35.0mu }\left.{\frac{\prod\limits_{j=1}^{n}{({1+a_{j}^{R}% })^{\omega_{j}}-\prod\limits_{j=1}^{n}{({1-a_{j}^{R}})^{\omega_{j}}}}}{\prod% \limits_{j=1}^{n}{({1+a_{j}^{R}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-a_{j% }^{R}})^{\omega_{j}}}}}}\right)$ (4)

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

Theorem 2. (Idempotency) If all $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{a}}_{j}=\tilde{{a}}$ for all $j$ , then

${\rm TFEWA}_{\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{a}}_{n}})% =\tilde{{a}}$ (5)

Theorem 3. (Boundedness) Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, and let

$\displaystyle\tilde{{a}}^{-}=\min\limits_{j}\tilde{{a}}_{j}=\min\limits_{j}({a% _{j}^{L},a_{j}^{M},a_{j}^{R}}),$ $\displaystyle\tilde{{a}}^{+}=\max\limits_{j}\tilde{{a}}_{j}=\max\limits_{j}({a% _{j}^{L},a_{j}^{M},a_{j}^{R}})$

Then

$\tilde{{a}}^{-}\leqslant{\rm TFEWA}_{\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},% \ldots,\tilde{{a}}_{n}})\leqslant\tilde{{a}}^{+}$ (6)

Theorem 4. (Monotonicity) Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ and $\tilde{{a}}^{\prime}_{j}=(a^{\prime L}_{j},a_{j}^{\prime M},a^{\prime U}_{j})(% j=1,2,$ $\ldots,$ $n)$ be two set of triangular fuzzy numbers, if $\tilde{{a}}_{j}\leqslant{\tilde{{a}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle{\rm TFEWA}_{\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}})$ $\displaystyle\leqslant{\rm TFEWA}_{\omega}({{\tilde{{a}}}^{\prime}_{1},{\tilde% {{a}}}^{\prime}_{2},\ldots,{\tilde{{a}}}^{\prime}_{n}})$ (7)

Definition 4. Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, then we define the triangular fuzzy Einstein ordered weighted average (TFEOWA) operator as follows:

$\displaystyle{\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{% a}}_{n}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}w_{j}\tilde{{a}}_{% \sigma(j)}$ (8)

where $(\sigma(1),\sigma(2),\ldots,\sigma(n))$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $\tilde{{a}}_{\sigma({j-1})}\geqslant\tilde{{a}}_{\sigma(j)}$ for all $j=2,\ldots,n$ , and $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is the aggregation-associated weight vector such that $w_{j}\in[{0,1}]$ and $\sum\nolimits_{j=1}^{n}{w_{j}}=1$ .

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive the Theorem 5.

Theorem 5. Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEOWA operator is also a triangular fuzzy numbers, and

$\displaystyle{\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{% a}}_{n}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}w_{j}\tilde{{a}}_{% \sigma(j)}$ $\displaystyle=\left({\frac{\prod\limits_{j=1}^{n}{({1+a_{\sigma(j)}^{L}})^{w_{% j}}-\prod\limits_{j=1}^{n}{({1-a_{\sigma(j)}^{L}})^{w_{j}}}}}{\prod\limits_{j=% 1}^{n}{({1+a_{\sigma(j)}^{L}})^{w_{j}}+\prod\limits_{j=1}^{n}{({1-a_{\sigma(j)% }^{L}})^{w_{j}}}}},}\right.$ $\displaystyle\mspace{40.0mu }\frac{\prod\limits_{j=1}^{n}{({1+a_{\sigma(j)}^{M% }})^{w_{j}}-\prod\limits_{j=1}^{n}{({1-a_{\sigma(j)}^{M}})^{w_{j}}}}}{\prod% \limits_{j=1}^{n}{({1+a_{\sigma(j)}^{M}})^{w_{j}}+\prod\limits_{j=1}^{n}{({1-a% _{\sigma(j)}^{M}})^{w_{j}}}}},$ $\displaystyle\mspace{34.0mu }\left.{\frac{\prod\limits_{j=1}^{n}{({1{+}a_{% \sigma(j)}^{R}})^{w_{j}}{-}\prod\limits_{j=1}^{n}{({1{-}a_{\sigma(j)}^{R}})^{w% _{j}}}}}{\prod\limits_{j=1}^{n}{({1{+}a_{\sigma(j)}^{R}})^{w_{j}}{+}\prod% \limits_{j=1}^{n}{({1{-}a_{\sigma(j)}^{R}})^{w_{j}}}}}}\right)$ (9)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $\tilde{{a}}_{\sigma({j-1})}\geqslant\tilde{{a}}_{\sigma(j)}$ for all $j=2,\ldots,n$ , and $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is the aggregation-associated weight vector such that $w_{j}\in[{0,1}]$ and $\sum\nolimits_{j=1}^{n}{w_{j}}=1$ .

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

Theorem 6. (Idempotency) If all $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{a}}_{j}=\tilde{{a}}$ for all $j$ , then

${\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{a}}_{n}})=% \tilde{{a}}$ (10)

Theorem 7. (Boundedness) Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, and let

Then

$\tilde{{a}}^{-}\leqslant{\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},% \ldots,\tilde{{a}}_{n}})\leqslant\tilde{{a}}^{+}$ (11)

Theorem 8. (Monotonicity) Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ and ${\tilde{{a}}}^{\prime}_{j}=({{a^{\prime}}_{j}^{L},{a^{\prime}}_{j}^{M},{a^{% \prime}}_{j}^{R}})(j=1,2,$ $\ldots,n)$ be two set of triangular fuzzy numbers, if $\tilde{{a}}_{j}\leqslant{\tilde{{a}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle{\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{% a}}_{n}})$ $\displaystyle\leqslant{\rm TFEOWA}_{w}({{\tilde{{a}}}^{\prime}_{1},{\tilde{{a}% }}^{\prime}_{2},\ldots,{\tilde{{a}}}^{\prime}_{n}})$ (12)

Theorem 9. (Commutativity) Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})$ $({j=1,2,\ldots,n})$ and ${\tilde{{a}}}^{\prime}_{j}=({{a^{\prime}}_{j}^{L},{a^{\prime}}_{j}^{M},{a^{% \prime}}_{j}^{R}})(j=1,2,$ $\ldots,n)$ be two set of triangular fuzzy numbers, then

$\displaystyle{\rm TFEOWA}_{w}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{% a}}_{n}})$ $\displaystyle={\rm TFEOWA}_{w}({{\tilde{{a}}}^{\prime}_{1},{\tilde{{a}}}^{% \prime}_{2},\ldots,{\tilde{{a}}}^{\prime}_{n}})$ (13)

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

In the following we shall propose the triangular fuzzy Einstein hybrid average (TFEHA) operator.

Definition 5. The triangular fuzzy Einstein hybrid average (TFEHA) operator is defined as follows:

${\rm TFEHA}_{w,\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,\tilde{{a}}_{n}% })=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}w_{j}\dot{{\tilde{{a}}}}_{% \sigma(j)}$ (14)

where $w=({w_{1},w_{2},\ldots,w_{n}})$ is the associated weighting vector, with $w_{j}\in[{0,1}]$ , $\sum\nolimits_{j=1}^{n}{w_{j}}=1$ , and $\dot{{\tilde{{a}}}}_{\sigma(j)}$ is the $j$ -th largest element of the triangular fuzzy arguments $\dot{{\tilde{{a}}}}_{j}({\dot{{\tilde{{a}}}}_{j}=n\omega_{j}._{\varepsilon}% \tilde{{a}}_{j},j=1,2,\ldots,n})$ , $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})$ is the weighting vector of triangular fuzzy arguments $\tilde{{a}}_{j}({j=1,2,\ldots,n})$ , with $\omega_{i}\in[{0,1}]$ , $\sum\nolimits_{i=1}^{n}{\omega_{i}}=1$ , and $n$ is the balancing coefficient. Especially, if $w=({1/n,1/n,\ldots,1/n})^{T}$ , then TFEHA is reduced to the triangular fuzzy Einstein weighted average (TFEWA) operator; if $\omega=({1/n,1/n,\ldots,1/n})$ , then TFEHA is reduced to the triangular fuzzy Einstein ordered weighted average (TFEOWA) operator.

Based on Einstein sum operations of the triangular fuzzy numbers described, we can drive the Theorem 10.

Theorem 10. Let $\tilde{{a}}_{j}=({a_{j}^{L},a_{j}^{M},a_{j}^{R}})({j=1,2,\ldots,n})$ be a set of triangular fuzzy numbers, then their aggregated value by using the TFEHA operator is also a triangular fuzzy number, and

$\displaystyle{\rm TFEHA}_{w,\omega}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}w_{j}\dot{{% \tilde{{a}}}}_{\sigma(j)}$ $\displaystyle=\left({\frac{\prod\limits_{j=1}^{n}{({1+\dot{{a}}_{\sigma(j)}^{L% }})^{w_{j}}-\prod\limits_{j=1}^{n}{({1-\dot{{a}}_{\sigma(j)}^{L}})^{w_{j}}}}}{% \prod\limits_{j=1}^{n}{({1+\dot{{a}}_{\sigma(j)}^{L}})^{w_{j}}+\prod\limits_{j% =1}^{n}{({1-\dot{{a}}_{\sigma(j)}^{L}})^{w_{j}}}}},}\right.$ $\displaystyle\mspace{40.0mu }\frac{\prod\limits_{j=1}^{n}{({1+\dot{{a}}_{% \sigma(j)}^{M}})^{w_{j}}-\prod\limits_{j=1}^{n}{({1-\dot{{a}}_{\sigma(j)}^{M}}% )^{w_{j}}}}}{\prod\limits_{j=1}^{n}{({1+\dot{{a}}_{\sigma(j)}^{M}})^{w_{j}}+% \prod\limits_{j=1}^{n}{({1-\dot{{a}}_{\sigma(j)}^{M}})^{w_{j}}}}},$ $\displaystyle\mspace{34.0mu }\left.\frac{\prod\limits_{j=1}^{n}{({1{+}\dot{{a}% }_{\sigma(j)}^{R}})^{w_{j}}{-}\prod\limits_{j=1}^{n}{({1{-}\dot{{a}}_{\sigma(j% )}^{R}})^{w_{j}}}}}{\prod\limits_{j=1}^{n}{({1{+}\dot{{a}}_{\sigma(j)}^{R}})^{% w_{j}}{+}\prod\limits_{j=1}^{n}{({1{-}\dot{{a}}_{\sigma(j)}^{R}})^{w_{j}}}}}\right)$ (15)

where $w=({w_{1},w_{2},\ldots,w_{n}})$ is the associated weighting vector, with $w_{j}\in[{0,1}]$ , $\sum\nolimits_{j=1}^{n}{w_{j}}=1$ , and $\dot{{\tilde{{a}}}}_{\sigma(j)}$ is the $j$ -th largest element of the triangular fuzzy arguments $\dot{{\tilde{{a}}}}_{\sigma(j)}({\dot{{\tilde{{a}}}}_{\sigma(j)}=n\omega_{j}._% {\varepsilon}\tilde{{a}}_{j},j=1,2,\ldots,n})$ , $\omega=(\omega_{1},$ $\omega_{2},\ldots,\omega_{n})$ is the weighting vector of triangular fuzzy arguments $\tilde{{a}}_{j}({j=1,2,\ldots,n})$ , with $\omega_{i}\in[{0,1}]$ , $\sum\nolimits_{i=1}^{n}{\omega_{i}}=1$ , and $n$ is the balancing coefficient.

4. MADM models for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing with triangular fuzzy information

The following assumptions or notations are used to represent the MADM problems for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing with triangular fuzzy information. Let $A=\{{A_{1},A_{2},\ldots,A_{m}}\}$ be a discrete set of alternatives, $G=\{{G_{1},G_{2},\ldots,G_{n}}\}$ be the set of attributes, $D=\{{D_{1},D_{2},\ldots,D_{s}}\}$ be the set of decision makers. Suppose that $R=(\tilde{{r}}_{ij})_{m\times n}$ $=(r_{ij}^{L},r_{ij}^{M},r_{ij}^{R})_{m\times n}$ is the decision making matrix, where $\tilde{{r}}_{ij}$ is a preference values, which take the form of triangular fuzzy numbers, given by the decision maker $D_{k}\in D$ , for the alternative $A_{i}\in A$ with respect to the attribute $G_{j}\in G$ , $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})$ is the weighting vector of the attributes $G_{j}({j=1,2,\ldots,n})$ , where $\omega_{j}\in[{0,1}]$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ .

In the following, we apply the TFEWA (or TFEWG) operator to the MADM problems with triangular fuzzy information.

Table 1
Evaluating matrix $A$

	$G_{1}$	$G_{2}$	$G_{3}$	$G_{4}$
$A_{1}$	(0.73, 0.74, 0.75)	(0.73, 0.75, 0.76)	(0.56, 0.59, 0.69)	(0.72, 0.73, 0.76)
$A_{2}$	(0.57, 0.59, 0.62)	(0.69, 0.73, 0.76)	(0.56, 0.58, 0.64)	(0.53, 0.56, 0.57)
$A_{3}$	(0.65, 0.67, 0.70)	(0.38, 0.45, 0.49)	(0.63, 0.68, 0.73)	(0.48, 0.56, 0.59)
$A_{4}$	(0.35, 0.42, 0.46)	(0.43, 0.47, 0.53)	(0.63, 0.66, 0.68)	(0.43, 0.55, 0.59)
$A_{5}$	(0.51, 0.55, 0.57)	(0.69, 0.75, 0.79)	(0.68, 0.72, 0.74)	(0.67, 0.68, 0.78)

Table 2

The overall preference values of the tourist scenic spots

	TFEWA
$A_{1}$	(0.152, 0.174, 0.196)
$A_{2}$	(0.191, 0.232, 0.257)
$A_{3}$	(0.204, 0.226, 0.248)
$A_{4}$	(0.163, 0.189, 0.236)
$A_{5}$	(0.185, 0.193, 0.214)

Step 1.

We utilize the decision information given in matrix $R=({\tilde{{r}}_{ij}})_{m\times n}=({r_{ij}^{L},r_{ij}^{M},r_{ij}^{R}})_{m% \times n}$ , and the TFEWA operator

$\displaystyle\tilde{{r}}_{i}=({r_{i}^{L},r_{i}^{M},r_{i}^{R}})$ $\displaystyle={\rm TFEWA}_{\omega}({\tilde{{r}}_{i1},\tilde{{r}}_{i2},\ldots,% \tilde{{r}}_{in}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}\omega_{j}% \tilde{{r}}_{ij}$ $\displaystyle=\left({\frac{\prod\limits_{j=1}^{n}{({1+r_{ij}^{L}})^{\omega_{j}% }-\prod\limits_{j=1}^{n}{({1-r_{ij}^{L}})^{\omega_{j}}}}}{\prod\limits_{j=1}^{% n}{({1+r_{ij}^{L}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-r_{ij}^{L}})^{% \omega_{j}}}}},}\right.$ $\displaystyle\frac{\prod\limits_{j=1}^{n}{({1+r_{ij}^{M}})^{\omega_{j}}-\prod% \limits_{j=1}^{n}{({1-r_{ij}^{M}})^{\omega_{j}}}}}{\prod\limits_{j=1}^{n}{({1+% r_{ij}^{M}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-r_{ij}^{M}})^{\omega_{j}}% }}},$ $\displaystyle\left.{\frac{\prod\limits_{j=1}^{n}{({1+r_{ij}^{R}})^{\omega_{j}}% -\prod\limits_{j=1}^{n}{({1-r_{ij}^{R}})^{\omega_{j}}}}}{\prod\limits_{j=1}^{n% }{({1+r_{ij}^{R}})^{\omega_{j}}+\prod\limits_{j=1}^{n}{({1-r_{ij}^{R}})^{% \omega_{j}}}}}}\right)$ (16)

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

Step 2.

Calculate the expected values as follows:

$E({\tilde{{r}}_{i}})=\frac{r_{i}^{L}+r_{i}^{M}+r_{i}^{R}}{{\rm 3}}.$ (17)

Step 3.

Rank all the alternatives $X_{i}(i=1,2,\ldots,$ $m)$ and select the best one(s) in accordance with the expected values $E({\tilde{{r}}_{i}})(i=1,2,\ldots,m)$ .

Step 4.

End.

5. Numerical example

Thus, in this section we shall present a numerical example for fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing with triangular fuzzy information in order to illustrate the method proposed in this paper. Suppose an organization plans to evaluate the tourist scenic spot service quality in the perspective of Wechat marketing. Project term choose five potential tourist scenic spots $A_{i}({i=1,2,\ldots,5})$ as candidates. The Project team selects four attributes to evaluate the tourist scenic spot service quality: (1) tourist satisfaction degree to the scenic spot- $G_{1}$ , (2)scenic spot transportation service- $G_{2}$ , (3) scenic spot information service- $G_{3}$ , (4) scenic spot ticket service- $G_{4}$ . The five possible tourist scenic spots $A_{i}({i=1,2,\ldots,5})$ are to be evaluated using the triangular fuzzy numbers under the above four attributes, and construct the following matrix $A=\left({\tilde{{r}}_{ij}}\right)_{5\times 4}$ is shown in Table 1.

The information about the attribute weights is known as follows: $\omega=({0.3,0.2,0.4,0.1})$ .

Then, we utilize the approach developed to get the most desirable tourist scenic spot(s).

Step 1.
We utilize the evaluating information given in matrix $\tilde{{R}}=({\tilde{{r}}_{ij}})_{5\times 4}$ , and the TFEWA operator to derive the overall preference values $\tilde{{r}}_{i}=({r_{i}^{L},r_{i}^{M},r_{i}^{U}})({i=1,2,3,4,5})$ of the enterprises $A_{i}$ . The results are shown in Table 2.

Table 3
Ordering of the tourist scenic spots by utilizing the TFEWA operator

Ordering

TFEWA $A_{2}>A_{3}>A_{4}>A_{5}>A_{1}$

Step 2.
According to the aggregating results shown in Table 2 and the expected values Eq. (17), the ordering of the tourist scenic spots are shown in Table 3. Note that $>$ means “preferred to”.

6. Conclusion

Einstein product is a t-norm and Einstein sum is a t-conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing triangular fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Einstein operations to develop some triangular fuzzy aggregation operators: triangular fuzzy Einstein weighted average (TFEWA) operator, triangular fuzzy Einstein ordered weighted average (TFEOWA) operator and triangular fuzzy Einstein hybrid average (TFEHA) operator. Then, we have utilized these operators to develop some approaches to solve the fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing. Finally, a practical example for evaluating the tourist scenic spot service quality in the perspective of Wechat marketing is given to verify the developed approach. In the future, we shall extend the proposed methods to other domains [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

Footnotes

Acknowledgments

The work was supported by the Culture and Arts Scientific Planned Project of Jiangxi Province in 2016 under Grant No. YG2016137.

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Fuzzy quantitative evaluation on tourist scenic spot service quality in the perspective of Wechat marketing

Abstract

Keywords

1. Introduction

2. Preliminaries

Table 1 Evaluating matrix A

Footnotes

Acknowledgments

References

Table 1
Evaluating matrix $A$