Research on the Service Quality Index and Alternatives Evaluation and Ranking for Online Yue Kiln Celadon Art Education in Post COVID-19 Era

Abstract

Online education has been still a common way for teaching and learning in the post epidemic era. However, the related research on service quality for the online Yue kiln celadon art education industry is still a vital research gap during this period. Thus, a hybrid method of FANP and GRA is proposed in this study to analyse and evaluate the key factors for providing and maintaining high service quality of online Yue kiln celadon art education industry in the post coronavirus era. In this research, whether in the model of FANP and GRA, factors such as safety mechanism of transaction and education, personnel quality, and the ability of customer need handling are essential conditions for providing excellent service quality in the post-COVID-19 era. The main contribution of this study is to propose an integrated method of FANP and GRA to calculate and rank potential solutions of online Yue kiln celadon art education service quality in the post-COVID-19 era under fuzzy environment and discrete conditions. Finally, the research findings of this study have a guiding role, thereby becoming a guide for the industries related to online Yue kiln celadon art education to maintain good service quality in similar scenarios in the future.

1. Introduction

China’s porcelain industry flourished during the Tang (618–907 CE) and Five Dynasties (907–960 CE). Celadon from the Yue kiln was produced in huge quantities. It was famous for the quality of its production and established by the imperial court as a “tribute kiln” [1]. Literature from the Tang and Song Dynasties (960–1127 CE) indicates that Mi-se porcelain was produced by the Yue kiln as a tribute to the emperor [2,3]. According to archaeological results [4,5,6], the Yue kiln is closely related to the origin of Mi-se porcelain, which is a major discovery of Chinese archaeology. Zhao [7] argued that porcelain is still one of the representative objects of Chinese culture. Coupled with major archaeological discoveries, the education subjects related to Yue Kiln celadon began to develop gradually before the COVID-19 pandemic [7], especially among comprehensive higher education institutions in China.

Since the outbreak of COVID-19, many educational institutions have changed their teaching methods from traditional face-to-face teaching to online teaching. Sabol et al. [8] reported that art educators must learn to teach in a different way during this period. Therefore, it was necessary to explore the impact of the COVID-19 pandemic on activities of art education to understand how methods of art education could be adjusted to reflect the new art education landscape in the post coronavirus era. Tuttle et al. [9] mentioned that these methods of online teaching and hybrid teaching were helpful for art education industry in the face of such unprecedented disaster scenarios.

Meanwhile, the learning perception of higher educational students was studied in China [10]. These scholars [10] discovered that the degree to which learners interacted with course content has a significant impact on learning satisfaction. Pratama et al. [11] introduced that online classroom was popular and effective for students studying the performing arts. Martyniv et al. [12] considered that fundamental creativity skills are best developed through interdisciplinary collaboration in STEM-education method during the period of COVID-19 outbreak. Additionally, Gamage et al. [13] provided some suggestions of online lab teaching strategies in the coronavirus pandemic era. Moreover, Tolmach et al. [14] presented a case study of higher educational students that studying culture and art in Ukraine during the period of coronavirus pandemic. The research result of Tolmach et al. [14] mentioned that students should focus on developing digital technology skills in the online learning process.

Furthermore, we have noticed that some scholars [15,16,17] have begun to consider the development trend of art education and Yue kiln celadon in the post COVID-19 era. For example, Dik et al. [15] provided some suggestions of policies for art education industry in the post COVID-19 era. Li et al. [16] proposed a case study of the perspective for the hybrid performing art education in Hong Kong. Jin et al. [18] used scientific instruments to analyse Yue kiln celadon to find out the corresponding dynasty colour characteristics. This is the first research achievement to standardise the colour characteristics of Yue kiln celadon in the post pandemic era. Xie et al. [19] applied the flipped classroom to the field of art education to investigate students’ positive psychological performance on art education teaching in the post epidemic era.

In addition, much research [20,21,22,23,24,25,26,27,28,29] proposed some research results of service quality for online education industry using various methods in the era of COVID-19 pandemic and post coronavirus epidemic.

Despite the above research has provided many suggestions for the online education industry from many perspectives in the recent three years. However, the research related to online art education in the post coronavirus era is still insufficient, particularly in the field of online Yue kiln celadon art education.

Accordingly, this research will explore important factors that should be noticed and considered for the online Yue kiln celadon art education from the perspective of service quality, thereby providing decision-making basis and filling the research gap in the post COVID-19 era.

1.1. Fuzzy Analytic Network Process

The Analytic Network Process (ANP) was proposed by Saaty in 1996 [30]. Nowadays, it has been proven by many studies to be one of the effective ways to solve multi-criteria decision-making (MCDM) problems in many fields, particularly problems with special relational structures among sub-criteria, alternatives and identified nonlinear links [31,32,33,34,35,36,37,38].

However, ANP is not suitable for solving MCDM-related problems involving uncertain phenomena. It is necessary to introduce fuzzy theory. Fuzzy theory was proposed by Dr. Lotfi Zadeh in 1965 [39]. Since variables of human psychological perception are difficult to accurately assess. Therefore, fuzzy theory uses mathematical language to describe the fuzzy phenomenon of human psychology, which can make the statistical results closer to the state of human psychological perception [40,41].

Meanwhile, Kahraman et al. [42] further integrated fuzzy theory and ANP into a comprehensive approach, called Fuzzy ANP (FANP), thereby dealing with related decision-making problems caused by imprecise psychological perception variables.

In recent decades, FANP has been widely implemented in the research field of MCDM problems and proven to be a highly reliable and valid research method for MCDM problems [43,44,45,46,47,48,49,50,51].

1.2. Grey Rational Analysis

Grey Rational Analysis was proposed by Deng [52]. This method is mainly aimed at problems with uncertainty and incomplete information for solving problems with uncertain or discrete data through steps such as correlation analysis, model building, prediction and decision-making [53]. Such characteristics make the GRA a suitable method for solving MCDM problems of multi-attribute and multi-scenario [54,55,56,57].

Meanwhile, many scholars [58,59,60,61] integrated the GRA with other MCDM research methods, such as: AHP and ANP, for solving MCDM problems in many fields within these decades. Additionally, some studies [62,63] proposed some hybrid approaches of FAHP, FANP, and GRA for energy storage industry and ERP packages evaluation.

The above research results provided significant inspiration for the development of research process and methodology for this study.

1.3. Research Objectives

In the view of this, we will construct the structure for the measurement of online Yue kiln celadon art education service quality through expert questionnaires and literature review from other’s research. Afterwards fuzzy analytic network process (FANP) will be implemented to calculate weights of evaluation dimensions and indicators. Finally, grey rational analysis (GRA) will be applied to rank all alternatives, thereby achieving the following research objectives:

• To construct the structure of service quality evaluation for the online Yue kiln celadon art education industry.
• To integrate the perspective of experts for analysing weights of evaluation dimensions and indicators for the service quality of online Yue kiln celadon art education using FANP.
• To evaluate and rank alternatives of online Yue kiln celadon art education service quality by applying GRA.
• To fill in the research gap of online Yue kiln celadon art education service quality in the post COVID-19 era, thereby providing relevant decision-making suggestions for online Yue kiln celadon art education industry.

2. Materials and Methods

In this study, a hybrid approach of FANP and GRA was proposed for the service quality evaluation of online Yue kiln celadon art education in the post COVID-19 era. The research process is shown in Figure 1.

2.1. The Identification of Evaluation Criteria

Firstly, evaluation criteria were identified and selected based on expert questionnaires and literature review. As for the literature review, extensive searches and discussions using keywords such as “Yue Kiln Celadon”, “Art Education”, “Multi-Criteria Decision Analysis”, “Fuzzy ANP”, and “Gray Rational Analysis” were conducted using SCI and SSCI databases in this study. Then, the study aggregated the search results. Afterwards experts were asked to provide suggestions. Finally, this study revised the statement of evaluation criteria based on expert suggestions.

As for the number of experts, Darko et al. [64] argued that a large number of experts will easily produce “cold-called” experts, which could seriously affect the results of consistency assessment. Meanwhile, we found that much research [65,66,67,68,69,70,71,72,73,74] utilised small sample size of four to nine experts to obtain valuable and reliable decision-making basis. In the view of this, this study selected ten experts in online Yue kiln celadon art education to avoid the influence on consistency check. Accordingly, the evaluation criteria were identified and selected by ten experts. Afterwards the result of evaluation criteria identification was obtained. It consisted of four levels; the first level is the goal of service quality evaluation for online Yue kiln celadon art education in the post COVID-19 era. The second level consists of five main criteria, followed by the 15 sub-criteria and five alternatives, as shown in Figure 2.

Additionally, this study obtained the interdependence of sub-criteria and alternatives based on suggestions of above-mentioned 10 experts, thereby establishing the hierarchy and network structure of service quality evaluation for online Yue kiln celadon art education industry in the post coronavirus era, as shown in Figure 3.

2.2. Questionnaire Establishment and Measuring

After the hierarchy and network structure was obtained. We inputted the associated evaluation indicators in the hierarchy and network structure into the Super Decision software to create a pairwise comparison questionnaire of a nine-point evaluation scale. The result of pairwise comparison questionnaires were integrated by geometric mean method and analysed by Fuzzy ANP. Meanwhile, this study established a direct rating scale questionnaire of five alternatives. The results of direct rating scale questionnaires were analysed using GRA.

Then, in this study, a total of 30 expert questionnaires, including 15 pairwise comparison questionnaires and 15 direct scoring questionnaires, were sent to the online Yue Kiln Celadon art education experts from 10 April 2022 to 15 June 2022. Subsequently, a total of 20 valid questionnaires were recovered, including 10 valid pairwise comparison questionnaires and 10 valid direct rating scale questionnaires.

2.3. Fuzzy Logic and Linguistic Variables

Scholars [75,76] considered that linguistic variables utilised measuring importance, such as “Very important”, “Relatively important”, and “Unimportant”, are often ambiguous. Therefore, it is necessary to introduce fuzzy logic to clarify the state of human psychological perception.

Fuzzy logic was proposed by Zadeh in 1975 [77]. It is an algorithm involving fuzzy numbers, which introduces a concept like reasoning, thereby finding the result that is closest to the variable of human psychological perception.

Therefore, we studied fuzzy numbers and found that fuzzy numbers are generally expressed in mathematical way [78,79,80,81,82,83,84,85]. For example, triangular fuzzy number $A$ ($L$, $M$, $U$) given by the following equation is shown in Figure 4.

$${\mu }_{\tilde{A}}\left(x\right)=\left\{\begin{array}{ccc}\frac{x-L}{M-L}& ,& L\le x\le M\\ \frac{x-U}{M-U}& ,& M\le x\le U\\ 0& ,& \mathrm{otherwise}\end{array}\right.$$

(1)

Meanwhile, much research [78,79,80,81,82,83,84,85] mentioned that the most likely evaluation value of triangular fuzzy numbers is the crisp value. The crisp value of triangular fuzzy numbers is given by the following equation.

$$A_a=\left[L^a,M^a\right]=\left[\left(M-L\right)a-\left(U-M\right)a+U\right]$$

(2)

Additionally, Buckley [86] reported that the characteristics of triangular fuzzy numbers is helpful to accurately present human fuzzy psychological perception variables through converting fuzzy numbers into clear and practical numbers. Moreover, triangular fuzzy numbers have been proven by Pedrycz [87] to be very suitable for expressing the degree of relative psychological perception and judgment of each criteria and alternative in the hierarchy and network structure. Accordingly, triangular fuzzy numbers are utilised to represent linguistic variable scales in this research.

Furthermore, ANP used a nine-point evaluation scale to indicate the importance of each evaluation criterion and alternative. Therefore, we integrate the triangular fuzzy number and ANP evaluation scale, thereby assessing and measuring human psychological true preferences for specific options. The corresponding fuzzy numbers are provided in

Table 1. Fuzzy numbers and scales.

Triangular Fuzzy Number	Linguistic Variables
$\tilde{1}=\left(1,1,1\right)$	Equally Preferred
$\tilde{2}=\left(1,2,3\right)$	Intermediate
$\tilde{3}=\left(2,3,4\right)$	Moderately Preferred
$\tilde{4}=\left(3,4,5\right)$	Intermediate
$\tilde{5}=\left(4,5,6\right)$	Strongly Preferred
$\tilde{6}=\left(5,6,7\right)$	Intermediate
$\tilde{7}=\left(6,7,8\right)$	Very Strongly Preferred
$\tilde{8}=\left(7,8,9\right)$	Intermediate
$\tilde{9}=\left(9,9,9\right)$	Extremely Preferred

.

2.4. Fuzzy Analytic Network Process

2.4.1. Consolidate Opinions of All Experts

In ANP model, the method of geometric mean is suggested by Saaty [88] to integrate perspectives of all experts and calculated as follows:

$${\left({\displaystyle \prod }_{i=1}^n{x}_i\right)}^{\frac{1}{n}}=\sqrt[n]{x_1{x}_2\cdots x_n},$$

(3)

where

$n$ is the number of experts.

2.4.2. The Construction of Fuzzy Pairwise Comparison Matrix

In this step, a fuzzy pairwise comparison matrix is performed and presented as follows:

$$\widetilde{A^k}=\left[\begin{array}{cccc}\widetilde{a_{11}^k}& \widetilde{a_{12}^k}& \cdots & \widetilde{a_{1n}^k}\\ \widetilde{a_{21}^k}& \widetilde{a_{22}^k}& \cdots & \widetilde{a_{2n}^k}\\ ⋮& ⋮& \ddots & ⋮\\ \widetilde{a_{n1}^k}& \widetilde{a_{n2}^k}& \cdots & \widetilde{a_{nn}^k}\end{array}\right],$$

(4)

where

$\widetilde{A^k}$ represents the fuzzy pairwise comparison matrix.

$\widetilde{a_{nn}^k}$ is triangular fuzzy mean value for comparing priority pairs among elements.

2.4.3. Fuzzy Decomposition

As for fuzzy decomposition, the process of defuzzification is presented as follows [89,90,91]:

$$t_{\alpha ,\beta }\left({\overline{a}}_{ij}\right)=\left[\beta f_a\left(L_{ij}\right)+\left(1-\beta \right)f_a\left(U_{ij}\right)\right], \mathsf{\alpha }\in \left[0,1\right], \mathsf{\beta }\in \left[0,1\right],$$

(5)

where

$$f_a\left(L_{ij}\right)=\left(M_{ij}-L_{ij}\right)\alpha +L_{ij},$$

(6)

$$f_a\left(U_{ij}\right)=U_{ij}-\left(M_{ij}-L_{ij}\right)\alpha ,$$

(7)

where

$L_{ij}$ is the lower bound value of the triangular fuzzy number.

$M_{ij}$ represents the median value of the triangular fuzzy number.

$U_{ij}$ is the upper bound value of the triangular fuzzy number.

When the diagonal matrix is matching, we have

$$t_{\alpha ,\beta }\left({\overline{a}}_{ij}\right)=\frac{1}{t_{\alpha ,\beta }\left({\overline{a}}_{ij}\right)}, \hspace{1em}\mathsf{\alpha }\in \left[0,1\right], \mathsf{\beta }\in \left[0,1\right],i>j$$

(8)

2.4.4. Set up the De-Fuzzified Comparison Matrix

The de-fuzzified pairwise comparison matrix can be built using $\alpha =\left[0,1\right]$ and $\beta =\left[0,1\right]$ in Equation (5) for calculating the weight of each dimension and indicator. The de-fuzzified pairwise comparison matrix is expressed as follows:

$$A={\left(a_{ij}\right)}_{n\times n}=\left[\begin{array}{cccc}1& a_{12}& \cdots & a_{1n}\\ a_{21}& 1& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & 1\end{array}\right]$$

(9)

2.4.5. The Calculation of Priority Vector

The local priority vector is used in ANP model to estimate the relative importance associated with elements or components being compared. The local priority vector is calculated as follows:

$$A·\omega ={\lambda }_{max}·\omega ,$$

(10)

where

$A$ is the de-fuzzified pairwise comparison matrix.

${\lambda }_{max}$ represents the maximum value of the matrix.

$\omega$ is the eigenvector.

If $A$ was the consistency matrix, the eigenvector $X$ would be calculated as follows:

$$\left|A-{\lambda }_{max}·X\right|=0$$

(11)

2.4.6. Consistency Check

Saaty [92] proposed adopting the consistency index (C.I.) and consistency ratio (C.R.) to verify the consistency of the comparison matrix. The C.I. and C.R. are defined as follows:

$$C.I.=\frac{\lambda max-n}{n-1},$$

(12)

where

$n$ is the number of criteria.

$$C.R.=\frac{C.I.}{R.I.},$$

(13)

Random index (R.I.) is a consistency index that produced by positive reciprocal matrices of different orders.

Table 2. Random indexes (R.I.).

The Order of Matrix	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
R.I.	-	-	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49	1.51	1.53	1.56	1.57	1.59

shows values of random index.

When $C.I.\le 0.1$, it refers to the best acceptable error. When $C.R.\le 0.1$, it means that the consistency of the matrix is satisfactory [92].

2.4.7. The Construction of Super Matrix

After completing the above steps, the super matrix is formed as follows:

$$W_N=\left[\begin{array}{ccc}0& 0& 0\\ w_1& W_3& 0\\ 0& W_2& W_4\end{array}\right],$$

(14)

where

$W_N$ represents the weight of indicators in the super matrix.

$w_1$ is the vector of the feature.

$W_2$ means the vector of the criterion.

$W_3$ represents the dependency of dimensions.

$W_4$ is the dependency of criteria.

Finally, the weight of indicators in the super matrix ($W_N$) is calculated as follows:

$$\begin{array}{c}{W}_3\times w_1=W_c\\ W_4\times W_2=W_e\end{array}\rangle \Rightarrow W_e\times W_c=W_N,$$

(15)

where

$W_c$ is the weight matrix of main criteria considering the interdependence degree.

$W_e$ is the evaluation weight matrix of sub-criteria considering the interdependence degree.

2.5. Grey Rational Analysis

2.5.1. The Definition of Evaluation Indicators and Data Treatment

The five alternatives in this study correspond to the 15 sub-criteria. Meanwhile, the direct evaluation (with a rating from 1 to 9, with a higher value indicating better the ability) is utilized to measure these 15 sub-criteria. In this study, experts are asked to score 15 sub-criteria. Then, the average scores of experts are taken as the score of the indicator corresponding to the five alternatives.

2.5.2. The Calculation of Referential Series and Compared Series

The referential series ($x_0$) with the number of indicators ($n$) is defined as follows:

$$x_0=\left(x_0\left(1\right),x_0\left(2\right),\dots ,x_0\left(n\right)\right)$$

(16)

Then, the compared series ($x_i$) is defined as follows:

$$x_i=\left(x_i\left(1\right),x_i\left(2\right),\dots ,x_i\left(n\right)\right), i=1,2,\dots ,m$$

(17)

2.5.3. Normalisation

Afterwards the data of referential series and compared series should be normalised, thereby making them comparable.

In this research, scores of all criteria are larger-the-better. Thus, the process of normalisation for referential series and compared series is expressed as follows [93]:

$$x_i^{*}\left(k\right)=\frac{x_i\left(k\right)-\underset{k}{\min }{x}_i\left(k\right)}{\underset{k}{\max }{x}_i\left(k\right)-\underset{k}{\min }{x}_i\left(k\right)},$$

(18)

where

$\underset{k}{\max }{x}_i\left(k\right)$ is the maximum value of $k$ indicator.

$\underset{k}{\min }{x}_i\left(k\right)$ represents the minimum value of $k$ indicator.

2.5.4. Calculate the Difference between Referential Series and Compared Series

The series difference is calculated as follows:

$${\Delta }_{0i}\left(k\right)=\left|x_0\left(k\right)-x_i\left(k\right)\right|, k=1,2,\dots ,15,$$

(19)

where

$x_0\left(k\right)$ is the referential series of 15 sub-criteria.

$x_i\left(k\right)$ represents the compared series of 15 sub-criteria.

2.5.5. Calculate the Grey Rational Coefficient

The grey relational coefficient between the compared series ($x_i$) and the referential series ($x_0$) at the $j\mathrm{th}$ indicator is defined as follows:

$${\gamma }_{0i}\left(k\right)=\frac{\Delta \min +\zeta \Delta \max }{{\Delta }_{0i}\left(k\right)+\zeta \Delta \max }$$

(20)

2.5.6. The Calculation the Grey Rational Grade

The grey rational grade (GRG) of a series ($x_i$) is calculated as follows:

$${\mathsf{\Gamma }}_{0i}={\displaystyle \sum }_{k=1}^n{\omega }_k{\gamma }_{0i}\left(k\right)$$

(21)

Finally, the alternatives are prioritised based on the magnitude of GRG values (${\mathsf{\Gamma }}_{0i}$). The alternative with the largest GRG value represents the best alternative and so on.

3. Results

3.1. Numerical Analysis

3.1.1. Fuzzy Analytic Network Process

The expert questionnaires were utilized in FANP model for gathering opinions from experts. Then, Equation (3) were used to integrate experts’ opinions. Afterwards the fuzzy pairwise comparison matrix for all criteria from FANP model was established.

Table 3. The fuzzy pairwise comparison matrix for five main criteria from FANP model.

Criteria	Environment (A)	Personnel (B)	Feedback (C)	Safety (D)	Flexibility (E)
Environment (A)	(1,1,1)	(4,5,6)	(3,4,5)	(6,7,8)	(3,4,5)
Personnel (B)	(1/6,1/5,1/4)	(1,1,1)	(1/3,1/2,1)	(2,3,4)	(1/3,1/2,1)
Feedback (C)	(1/5,1/4,1/3)	(1,2,3)	(1,1,1)	(2,3,4)	(1,2,3)
Safety (D)	(1/8,1/7,1/6)	(1/4,1/3,1/2)	(1/4,1/3,1/2)	(1,1,1)	(1/4,1/3,1/2)
Flexibility (E)	(1/5,1/4,1/3)	(1,2,3)	(1/3,1/2,1)	(2,3,4)	(1,1,1)

reveals the fuzzy pairwise comparison matrix for five main criteria.

The process of fuzzy decomposition using $\alpha =0.5$ and $\beta =0.5$ for main criteria between Environment (A) and Personnel (B) is as follows [94,95,96]:

$$t_{0.5,0.5}\left(\overline{a_{A,B}}\right)=\left[0.5\times 4.5+\left(1-0.5\right)\times 5.5\right]=5$$

$$f_a\left(L_{A,B}\right)=\left(5-4\right)\times 0.5+4=4.5$$

$$f_a\left(U_{A,B}\right)=6-\left(5-4\right)\times 0.5=5.5$$

$$t_{0.5,0.5}\left(\overline{a_{B,A}}\right)=\frac{1}{5}$$

The processes of fuzzy decomposition for remaining main criteria are like the above calculation. Afterwards the de-fuzzified pairwise comparison matrix for five main criteria from FANP model are shown in

Table 4. The de-fuzzified pairwise comparison matrix for five main criteria from FANP model.

Criteria	Environment (A)	Personnel (B)	Feedback (C)	Safety (D)	Flexibility (E)
Environment (A)	1	5	4	7	4
Personnel (B)	1/5	1	1/2	3	1/2
Feedback (C)	1/4	2	1	3	2
Safety (D)	1/7	1/3	1/3	1	1/3
Flexibility (E)	1/4	2	1/2	3	1

.

The calculation process of maximum individual value for each main criteria is shown in

Table 5. The maximum individual value calculation process.

Dimensions	The Calculation Process	Maximum Individual Value
Environment (A)	${\left(1\times 5\times 4\times 7\times 4\right)}^{\frac{1}{5}}$	3.5452
Personnel (B)	${\left(\frac{1}{5}\times 1\times \frac{1}{2}\times 3\times \frac{1}{2}\right)}^{\frac{1}{5}}$	0.6843
Feedback (C)	${\left(\frac{1}{4}\times 2\times 1\times 3\times 2\right)}^{\frac{1}{5}}$	1.2457
Safety (D)	${\left(\frac{1}{7}\times \frac{1}{3}\times \frac{1}{3}\times 1\times \frac{1}{3}\right)}^{\frac{1}{5}}$	0.3505
Flexibility (E)	${\left(\frac{1}{4}\times 2\times \frac{1}{2}\times 3\times 1\right)}^{\frac{1}{5}}$	0.9441
	Total	6.7698

.

The calculation process of weight for each dimension is shown in

Table 6. The calculation of weight for five dimensions.

Dimensions	The Calculation Process	Weight
Environment (A)	$\frac{3.5452}{6.7698}$	0.5237
Personnel (B)	$\frac{0.6843}{6.7698}$	0.1011
Feedback (C)	$\frac{1.2457}{6.7698}$	0.184
Safety (D)	$\frac{0.3505}{6.7698}$	0.0518
Flexibility (E)	$\frac{0.9441}{6.7698}$	0.1395
	Total	1

.

The calculation of normalised matrix is shown in

Table 7. Normalised matrix calculation.

Criteria	Environment (A)	Personnel (B)	Feedback (C)	Safety (D)	Flexibility (E)
Environment (A)	$1\times 0.5237$	$5\times 0.1011$	$4\times 0.184$	$7\times 0.0518$	$4\times 0.1395$
Personnel (B)	$1/5\times 0.5237$	$1\times 0.1011$	$1/2\times 0.184$	$3\times 0.0518$	$1/2\times 0.1395$
Feedback (C)	$1/4\times 0.5237$	$2\times 0.1011$	$1\times 0.184$	$3\times 0.0518$	$2\times 0.1395$
Safety (D)	$1/7\times 0.5237$	$1/3\times 0.1011$	$1/3\times 0.184$	$1\times 0.0518$	$1/3\times 0.1395$
Flexibility (E)	$1/4\times 0.5237$	$2\times 0.1011$	$1/2\times 0.184$	$3\times 0.0518$	$1\times 0.1395$

.

The calculation of maximum eigenvector ($W_1$) is shown in

Table 8. The calculation of maximum eigenvector for five main criteria.

Dimensions	A	B	C	D	E	Total	$\mathit{\omega }$	${\mathit{W}}_1$
A	0.5237	0.5054	0.7361	0.3624	0.5578	2.6854	0.5237	$2.6854/0.5237=5.1279$
B	0.1047	0.1011	0.0920	0.1553	0.0697	0.5229	0.1011	$0.5229/0.1011=5.1731$
C	0.1309	0.2022	0.1840	0.1553	0.2789	0.9513	0.184	$0.9513/0.184=5.1699$
D	0.0748	0.0337	0.0613	0.0518	0.0465	0.2681	0.0518	$0.2681/0.0518=5.1781$
E	0.1309	0.2022	0.0920	0.1553	0.1395	0.7199	0.1395	$0.7199/0.1395=5.1619$

.

Afterwards the numbers of main dimensions are 5, we get $n=5$. Therefore, ${\lambda }_{max}$ and $C.I.$ are calculated as follows:

$${\lambda }_{max}=\frac{\left(5.1279+5.1731+5.1699+5.1781+5.1619\right)}{5}=5.1622$$

$$C.I.=\frac{{\lambda }_{max}-n}{n-1}=\frac{5.1622-5}{5-1}=0.0405$$

For $C.R.$, with $n=5$, we have $R.I.=1.12$.

$$C.R.=\frac{C.I.}{R.I.}=\frac{0.0405}{1.12}=0.0362$$

The calculation result of de-fuzzified pairwise comparison matrix between five main criteria is shown in

Table 9. The pairwise comparison matrix for five main criteria from FANP model.

Dimensions	Environment (A)	Personnel (B)	Feedback (C)	Safety (D)	Flexibility (E)	Weight
Environment (A)	1	5	4	7	4	0.5237
Personnel (B)	1/5	1	1/2	3	1/2	0.1011
Feedback (C)	1/4	2	1	3	2	0.184
Safety (D)	1/7	1/3	1/3	1	1/3	0.0518
Flexibility (E)	1/4	2	1/2	3	1	0.1395
	Total					1
	$C.I.=0.0405, C.R.=0.0362$

.

Additionally, the calculation processes of defuzzification, maximum eigenvalue (${\lambda }_{max}$), consistency index ($C.I.$) and consistency ratio ($C.R.$) for all sub-criteria are like above calculation. Afterwards the de-fuzzified pairwise comparison matrix for all sub-criteria is shown in

Table 10. The de-fuzzified pairwise comparison matrix for all sub-criteria.

Environment (A)					Personnel (B)
	A1	A2	A3	Weight		B1	B2	B3	Weight
A1	1	1/4	2	0.187	B1	1	1/2	1/4	0.1429
A2	4	1	7	0.7153	B2	2	1	1/2	0.2857
A3	1/2	1/7	1	0.0977	B3	4	2	1	0.5714
${\lambda }_{max}=3.002$, $C.I.=0.001$, $C.R.=0.0017$					${\lambda }_{max}=3$, $C.I.=0.0$, $C.R.=0.0$
Feedback (C)					Safety (D)
	C1	C2	C3	Weight		D1	D2	D3	Weight
C1	1	2	3	0.5396	D1	1	1/2	5	0.3643
C2	1/2	1	2	0.297	D2	2	1	4	0.5368
C3	1/3	1/2	1	0.1634	D3	1/5	1/4	1	0.0989
${\lambda }_{max}=3.0092$, $C.I.=0.0046$, $C.R.=0.0079$					${\lambda }_{max}=3.094$, $C.I.=0.047$, $C.R.=0.081$
Flexibility (E)
	E1	E2	E3	Weight
E1	1	1	2	0.3764
E2	1	1	4	0.4742
E3	1/2	1/4	1	0.1494
${\lambda }_{max}=3.0536$, $C.I.=0.0268$, $C.R.=0.0462$

.

Moreover,

Table 11. The de-fuzzified pairwise comparison matrix for each criterion under specific criterion.

Personnel (B)					Safety (D)
	C	D		Weight		C	D		Weight
C	1	2		0.6667	C	1	3		0.25
D	1/2	1		0.3333	D	1/3	1		0.75
${\lambda }_{max}=2$, $C.I.=0.00$, $C.R.=0.0$					${\lambda }_{max}=2$, $C.I.=0.0$, $C.R.=0.0$
Well-trained employees (B1)					Well-qualified instructors (B2)
	D1	D2	D3	Weight		D1	D2	D3	Weight
D1	1	2	1/2	0.2764	D1	1	1/5	1/3	0.1047
D2	1/2	1	1/5	0.1283	D2	5	1	3	0.637
D3	2	5	1	0.5954	D3	3	1/3	1	0.2583
${\lambda }_{max}=3.0055$, $C.I.=0.0028$, $C.R.=0.0048$					${\lambda }_{max}=3.0385$, $C.I.=0.0193$, $C.R.=0.0332$
Customer needs are effectively handled (C1)					Safety transaction mechanism (D1)
	E1	E2	E3	Weight		C1	C2		Weight
E1	1	2	4	0.5584	C1	1	3		0.75
E2	1/2	1	3	0.3196	C2	1/3	1		0.25
E3	1/4	1/3	1	0.122
${\lambda }_{max}=3.0183$, $C.I.=0.0091$, $C.R.=0.0158$					${\lambda }_{max}=2$, $C.I.=0.0$, $C.R.=0.0$

demonstrates the de-fuzzified pairwise comparison matrix for each criterion under specific criterion.

As shown in Table 9, Table 10 and Table 11, Values of consistency index ($C.I.$) and consistency ratio ($C.R.$) for main criteria and sub-criteria are all less than 0.1. It means that the data in the pairwise comparison matrix is consistent.

Finally, the super matrix was calculated using Super Decision software. The value of each column in the limit super matrix is the weight of each indicator, as shown in

Table 12. The super matrix.

Main Criteria		Environment			Personnel			Feedback			Safety			Flexibility
Sub-criteria		A1	A2	A3	B1	B2	B3	C1	C2	C3	D1	D2	D3	E1	E2	E3
Environment	A1	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	A2	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	A3	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
Personnel	B1	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	B2	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	B3	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
Feedback	C1	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	C2	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	C3	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
Safety	D1	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	D2	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	D3	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
Flexibility	E1	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	E2	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116
	E3	0.007	0.002	0.011	0.051	0.029	0.015	0.026	0.054	0.045	0.141	0.189	0.220	0.031	0.063	0.116

.

3.1.2. Grey Rational Analysis

In this study, scores of all sub-criteria given by experts are larger-the-better. Therefore, the largest value of each sub-criteria is considered as referential series ($x_0$), the value of each indicator is considered as compared series ($x_i$).

Table 13. Referential series and compared series.

Indicators	Referential Series (${\mathit{x}}_0$)	Compared Series (${\mathit{x}}_{\mathit{i}}$)
		Alt 1	Alt 2	Alt 3	Alt 4	Alt 5
A1	7.09	1.00	5.93	4.28	1.00	7.09
A2	8.36	1.74	5.67	4.04	8.36	7.56
A3	8.36	1.74	5.43	3.37	8.36	7.16
B1	8.14	1.52	5.14	3.18	8.14	6.79
B2	8.14	1.00	5.43	3.18	8.14	6.88
B3	7.71	1.74	4.64	3.73	7.71	7.00
C1	8.11	1.74	5.67	3.17	8.11	6.69
C2	8.36	1.52	5.93	3.44	8.36	7.38
C3	8.36	1.32	5.14	3.17	8.36	7.16
D1	7.89	1.52	5.43	3.73	7.89	7.33
D2	8.11	1.32	5.67	3.95	8.11	7.33
D3	8.11	1.52	4.74	3.95	8.11	7.04
E1	8.11	1.52	4.74	3.73	8.11	7.07
E2	8.30	1.32	4.54	3.64	8.30	6.90
E3	8.30	1.32	4.74	3.64	8.30	7.16

reveals the referential series ($x_0$) and compared series ($x_i$).

Then, the normalised data is calculated using Equation (18). After that, the calculation of deviation sequences using Equation (19). Afterwards the calculation of grey rational coefficient using Equation (20). Finally, the calculation results of normalised data, deviation sequences and grey rational coefficient are shown in

Table 14. Normalised data.

Indicators	Alt 1	Alt 2	Alt 3	Alt 4	Alt 5
A1	0.0000	0.8095	0.5386	0.0000	1.0000
A2	0.0000	0.5937	0.3474	1.0000	0.8792
A3	0.0000	0.5574	0.2462	1.0000	0.8187
B1	0.0000	0.5468	0.2508	1.0000	0.7961
B2	0.0000	0.6204	0.3053	1.0000	0.8235
B3	0.0000	0.4858	0.3333	1.0000	0.8811
C1	0.0000	0.6170	0.2245	1.0000	0.7771
C2	0.0000	0.6447	0.2807	1.0000	0.8567
C3	0.0000	0.5426	0.2628	1.0000	0.8295
D1	0.0000	0.6138	0.3469	1.0000	0.9121
D2	0.0000	0.6406	0.3873	1.0000	0.8851
D3	0.0000	0.4886	0.3687	1.0000	0.8376
E1	0.0000	0.4886	0.3354	1.0000	0.8422
E2	0.0000	0.4613	0.3324	1.0000	0.7994
E3	0.0000	0.4900	0.3324	1.0000	0.8367

,

Table 15. Deviation sequences.

Indicators	Alt 1	Alt 2	Alt 3	Alt 4	Alt 5
A1	1.0000	0.1905	0.4614	1.0000	0.0000
A2	1.0000	0.4063	0.6526	0.0000	0.1208
A3	1.0000	0.4426	0.7538	0.0000	0.1813
B1	1.0000	0.4532	0.7492	0.0000	0.2039
B2	1.0000	0.3796	0.6947	0.0000	0.1765
B3	1.0000	0.5142	0.6667	0.0000	0.1189
C1	1.0000	0.3830	0.7755	0.0000	0.2229
C2	1.0000	0.3553	0.7193	0.0000	0.1433
C3	1.0000	0.4574	0.7372	0.0000	0.1705
D1	1.0000	0.3862	0.6531	0.0000	0.0879
D2	1.0000	0.3594	0.6127	0.0000	0.1149
D3	1.0000	0.5114	0.6313	0.0000	0.1624
E1	1.0000	0.5114	0.6646	0.0000	0.1578
E2	1.0000	0.5387	0.6676	0.0000	0.2006
E3	1.0000	0.5100	0.6676	0.0000	0.1633

and

Table 16. Grey rational coefficient.

Indicators	Alt 1	Alt 2	Alt 3	Alt 4	Alt 5
A1	0.3333	0.7241	0.5201	0.3333	1.0000
A2	0.3333	0.5517	0.4338	1.0000	0.8054
A3	0.3333	0.5304	0.3988	1.0000	0.7339
B1	0.3333	0.5246	0.4002	1.0000	0.7103
B2	0.3333	0.5685	0.4185	1.0000	0.7391
B3	0.3333	0.4930	0.4286	1.0000	0.8078
C1	0.3333	0.5662	0.3920	1.0000	0.6916
C2	0.3333	0.5846	0.4101	1.0000	0.7773
C3	0.3333	0.5223	0.4041	1.0000	0.7458
D1	0.3333	0.5642	0.4336	1.0000	0.8505
D2	0.3333	0.5818	0.4494	1.0000	0.8132
D3	0.3333	0.4944	0.4420	1.0000	0.7549
E1	0.3333	0.4944	0.4293	1.0000	0.7601
E2	0.3333	0.4814	0.4282	1.0000	0.7137
E3	0.3333	0.4950	0.4282	1.0000	0.7538

.

3.2. Research Results

3.2.1. Fuzzy Analytic Network Process

As for the ranking of all criteria in FANP model, they are ordered by overall weights. Therefore, we inputted all data in the pairwise comparison matrix into Super Decision software. Afterwards overall weights of main criteria and sub-criteria were calculated. The overall weights and ranking of main criteria and sub-criteria are shown in Figure 5 and Figure 6.

As shown in Figure 5, the main criteria that were ordered by overall weights as safety (0.44), personnel (0.215), flexibility (0.17), feedback (0.13), and environment (0.046).

As shown in Figure 6, the top 3 sub-criteria that were ordered by overall weights as “Safe online teaching and learning mechanism” (0.22), “Legally and reliably services” (0.189), and “Safe transaction mechanism” (0.141).

The overall weights of sub-criteria ranked fourth to fifth are “Individual customer demand” (0.116) and “Compatibility and timely adjustments” (0.063).

The sixth to eighth important sub-criteria are “The willingness to help customers” (0.054), “Well-trained employees” (0.051) and “Customer requests are responded instantly” (0.045).

3.2.2. Grey Rational Analysis

The priority of all alternatives is based on the grey rational grade (${\mathsf{\Gamma }}_{0i}$). The calculation of grey rational grade using Equation (21) is shown in

Table 17. Grey rational grade.

Alternatives	Description	Grey Rational Grade (${\mathbf{\Gamma }}_{0\mathit{i}}$)	Rank
Alt 1	Appealing facility and environment	0.3333	5
Alt 2	Personnel quality	0.5451	3
Alt 3	Customer demand processing ability	0.4278	4
Alt 4	Safe environment and legal service content	0.9556	1
Alt 5	Potential customer need	0.7772	2

.

The larger value of grey rational grade (${\mathsf{\Gamma }}_{0i}$) represents that the alternative is closer to optimal solution. Accordingly, rankings of all alternatives based on grey rational grade are “Safe environment and legal service content” (0.9556), “Potential customer need” (0.7772), “Personnel quality” (0.5451), “Customer demand processing ability” (0.4278) and “Appealing facility and environment” (0.3333).

4. Discussion

In this study, the overall weight of all criteria is calculated and obtained using Super Decision software in FANP model. Afterwards all criteria are ranked based on the overall weight in FANP model. Meanwhile, the score of each alternative with respect to each sub-criterion is obtained based on the judgement of experts on each alternative under each sub-criterion. Afterwards all alternatives are ranked according to the grey rational grade (GRG) in the GRA model.

As for the ranking of main criteria in FANP model, the most vital main criterion ordered by overall weight is safety. It means that service providers of online Yue kiln celadon art education should give top priority to the security of service content in the post COVID-19 era.

The second to fourth important main criteria are personnel, flexibility, and feedback. Therefore, personnel factors such as employees and teachers, the flexibility of service content and the ability to respond to customer needs are important factors for maintaining high service quality of online Yue kiln celadon art education industry in the post coronavirus era.

As for ranking of sub-criteria in FANP model, the most three vital sub-criteria are “Safe online teaching and learning mechanism”, “Legally and reliably services”, and “Safe transaction mechanism”. Accordingly, the service providers of online Yue kiln celadon art education should pay attention to the safety and legality of the education mechanism, service content and transactions. In the view of this, this study suggests that the service providers of online Yue kiln celadon art education can ensure that the service content in the post COVID-19 era is safe and legal through employee training and the establishment of a safe transactional mechanism.

The fourth and fifth important sub-criteria are “Individual customer demand” and “Compatibility and timely adjustment”. Thus, we suggest that service providers of online Yue kiln celadon art education should care about the individual needs of consumers and establish highly compatible and flexible service capabilities, thereby maintaining high service quality in the post coronavirus era.

The sixth to eighth important sub-criteria are “The willingness to help customers”, “Well-trained employees” and “Customer requests are responded instantly”. It means that factors such as the willingness of customer service, the level of staff training and the respond speed to customer need are crucial condition to provide good service quality for online Yue kiln celadon art education industry in the post coronavirus era.

As for the ranking of all alternatives in the GRA model, the top alternative ordered by grey rational grade is “Safe environment and legal service content”, followed by “Potential customer need”, “Personnel quality” and “Customer demand processing ability”. In the view of this, this research suggests that the safe environment and legal service content should be the primary goals of online Yue kiln celadon art education industry. Meanwhile, personnel quality and abilities of customer needs discovery and processing are also the key factors for maintaining excellent service quality of online Yue kiln celadon art education industry in the post epidemic era.

Finally, whether in the model of FANP and GRA, the main criterion and alternative ranked last is environmental factors. This means that the environment of online Yue kiln celadon art education industry is relatively less important in the post coronavirus era.

5. Conclusions

This research established the hierarchy and network structure of service quality evaluation for the online Yue kiln celadon art education industry based on suggestions of experts. Then, the overall weight of all criteria was calculated and obtained in FANP model. Afterwards all alternatives were analysed and ranked using GRA.

The main contribution of this research is to propose a hybrid approach of FANP and GRA for the service quality evaluation of online Yue kiln celadon art education in the post COVID-19 era under fuzzy environment and discrete condition. Meanwhile, whether in the model of FANP and GRA, factors such as safety mechanism of transaction and education, personnel quality and the ability of potential customer need discovery and handling are crucial conditions for providing and maintaining excellent service quality in the post COVID-19 era.

Additionally, this study has an indicative role for the online Yue kiln celadon art education industry to provide good service quality in the post COVID-19 era. Finally, the research findings of this study provide the guidance for the online English teaching industry to maintain good service quality in future related scenarios.