Identification of factors affecting student academic burnout in online education during the COVID19 pandemic using grey Delphi and greyDEMATEL techniques
In this study, the Gray Delphi method^{40} and, the Gray DEMATEL technique^{41,42} was used to identify the factors affecting student academic burnout. Matlab version R2020 software and Excel version 2021 software were employed for data analysis. DEMATEL is used for deriving the interrelationships between criteria. A grey DEMATEL method for group decisionmaking and analysis of the causeeffect relationship in grey environments was developed. The grey DEMATEL is applied to discriminate the causeandeffect relationships of criteria. This help decisionmakers to take more efficient actions by focusing on the ones with the greatest influence. The participants of the study were selected by onestage cluster sampling method. For this purpose, out of 12 faculties, four faculties, and four departments from each faculty were randomly selected. In the Delphi stage, 86 graduate students and in the DEMATEL stage, 37 students (with the probability drop rate) were randomly selected from each department.
Identification of factors
To explore factors affecting student academic burnout, an open questionnaire was administered to a randomly selected sample of 86 graduate students (Master courses and, Ph.D. courses) studying at Islamic Azad University, Science and Research Branch, and 43 factors were extracted.
Gray Delphi method (GDM)
Dalkey and Helmer established Delphi method in 1963^{43}. The Delphi method is a structured process for screening and ranking factors, which is implemented by gathering decisionmakers’ opinions through questionnaires. Three basic characteristics of the Delphi method are anonymous response, iteration and controlled feedback, and statistical group response^{44,45}. Although the traditional Delphi method has been used in many studies, however, it has been criticized for its lengthy process, low convergence, and loss of some valuable expert information^{46}. In addition, the traditional process of quantifying people’s perspectives does not fully reflect the human thinking style due to the fuzziness, imprecision, and low compatible with linguistic and sometimes ambiguous human explanations, judgment, and priorities^{45,47}. Grey system theory was initiated by^{48}. The goal of the Grey System is to bridge the gap existing between social science and natural science^{48}. In the grey system, all messages can be divided into three categories: white, grey, and black. The white part shows clear messages in a system ultimately, the black part has unknown characteristics, and the grey part happens between and covers both known and unknown messages. This theory includes four parts^{49,50}.
Simultaneous use of the gray theory and the Delphi method is a proposed solution^{40,51}.
Step 1: The Delphi questionnaire was distributed among 86 graduate students and asked them to determine the importance of each Criteria using linguistic variables. Linguistic variables and their corresponding gray scales for the importance weight of criteria according to^{40} are shown in Table 1.
Step 2: Based on the method proposed by^{40}, j (\(j = 1,…,5\)) gray classes were considered and, the selection range of ith criteria i.e.,\(\left[ {a_{i}^{1} ,b_{i}^{5} } \right]\) was divided into 5 Gy classes.
Step 3: Eqs. (1) and (2) show the half trapezoidal whitening weight function applied for j = 1 and 5.
$$f_{i}^{1} (x) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & {x \le a_{i}^{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{b_{i}^{1} – x}}{{b_{i}^{1} – a_{i}^{1} }}} & {a_{i}^{1} < x \le b_{i}^{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & {x > b_{i}^{1} } \\ \end{array} } \\ \end{array} } \right.$$
(1)
$$f_{i}^{5} (x) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {x \le a_{i}^{5} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{x – a_{i}^{5} }}{{b_{i}^{5} – a_{i}^{5} }}} & {a_{i}^{5} \le x < b_{i}^{5} } \\ \end{array} } \\ {\begin{array}{*{20}c} 1 & {x \ge b_{i}^{5} } \\ \end{array} } \\ \end{array} } \right.$$
(2)
For J = 2,3,4. triangular whitening weight function as Eqs. (3) was applied.
$$f_{i}^{j} (x) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {x \notin \left[ {a_{i}^{j} ,b_{i}^{j} } \right]} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{2(x – a_{i}^{j} )}}{{b_{i}^{j} – a_{i}^{j} }}} & {x \in \left[ {a_{i}^{j} ,\frac{{a_{i}^{j} + b_{i}^{j} }}{2}} \right]} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{2(b_{i}^{j} – x)}}{{b_{i}^{j} – a_{i}^{j} }}} & {x \notin \left[ {\frac{{a_{i}^{j} + b_{i}^{j} }}{2},b_{i}^{j} } \right]} \\ \end{array} } \\ \end{array} } \right.$$
(3)
Step 4: Eq. (4) was employed to calculate the synthetic clustering coefficient (\(\rho_{i}^{j}\)).
$$\rho_{i}^{j} = \sum\limits_{k = 1}^{m} {f_{i}^{j} (x) \cdot \eta_{i}^{k} }$$
(4)
Step 5: where \(f_{i}^{j} (x)\) is the whitening weight function of jth’s grey class criteria i; m is the number of categories of students’ opinions; \(\eta_{i}^{k}\) is the weight of criteria i in the synthetic cluster.
Step 6: The decision vectors of evaluation criteria were identified. The criterion \(\max_{1 \le j \le 5} (\rho_{i}^{j} ) = \rho_{i}^{{j^{*} }}\) was used to judge whether criterion j belongs to the class \(j^{*}\).
The selection criteria are as follows:

(1)
If class \(j^{*}\) belong to classes 4 and 5, namely, the value of classes of important or very important is the maximum in the vector decision, the criterion is accepted.

(2)
If the ratio of class \(j^{*}\) attached to class 4 and 5 to class \(j^{*}\) attached to class 1 and 2 is more than 1, namely, classes of important and very important account for an over 50 percent degree except for the class of undecided, the criterion is accepted.
Gray DEMATEL
Fontela used decisionMaking Trial and Evaluation Laboratory (DEMATEL) technique at first and, Gabus in 1976^{50}. DEMATEL is helpful in analyzing the causeandeffect relationships among the components of a system. DEMATEL can demonstrate the existence of a relationship/interdependence among criteria or explain the relative level of relationships within them^{50}. The DEMATEL does not depend on a large data sample and simplifies the correlation analysis of factors^{52,53,54}. However, the traditional DEMATEL doesn’t consider the fuzziness and uncertainties in the reallife^{54,55,56}. Tseng (2009) expanded fuzzy triangular numbers to establish hierarchical grey DEMATEL to analyze criteria and alternatives in incomplete information^{41}.
So, in this study, the gray DEMATEL technique was applied to develop the causal relationships model of factors affecting student academic burnout.
Step 1: A questionnaire with two square matrices (a square matrix of order 7 for the main criteria and, seven square matrices of order n, where n is a number of subcriteria for each main criteria) was designed for pairwise comparison of the Criteria.
Step 2: A randomly selected sample of 37 graduate students (Master courses and, Ph.D. courses) evaluated interrelations among the Criteria by pairwise comparisons.
Step 3: The students used ten linguistic variables to illustrate the degree of causality between the Criteria. Linguistic variables and their corresponding grayfuzzy numbers, according to^{41,57,58}, to define the degree of influence of factors affecting student academic burnout are shown in Table 2.
Step 4: Assume \(\otimes X\) an interval grey number is defined as \(\otimes X = \left[ {\underline{X} ,\overline{X} } \right]\), and X’s lower and upper bound is limited.
Step 5: Using Eq. (5) to aggregate students’ opinions and a directrelation matrix (n × n) (\(i,j = 1, \ldots ,n\)) was achieved to show that criteria i affects the criteria j.
$$\otimes X_{ij} = \frac{1}{h}\left( { \otimes X_{ij}^{1} + \otimes X_{ij}^{2} + \cdots + \otimes X_{ij}^{h} } \right)$$
(5)
$$X = \left[ {\begin{array}{*{20}c} { \otimes X_{11} } & \ldots & { \otimes X_{1n} } \\ \vdots & \vdots & \vdots \\ { \otimes X_{n1} } & \ldots & { \otimes X_{nn} } \\ \end{array} } \right]$$
(6)
Step 6: Normalized the grey relation decision matrix (\(X^{\prime}\))
$$X{\prime} = \left[ {\begin{array}{*{20}c} { \otimes X_{11}{\prime} } & \ldots & { \otimes X_{1n}{\prime} } \\ \vdots & \vdots & \vdots \\ { \otimes X_{n1}{\prime} } & \ldots & { \otimes X_{nn}{\prime} } \\ \end{array} } \right]$$
(7)
Step 7: The relation normalized greyDEMATEL decision matrix (M*).
$$M^{*} = \left[ {\begin{array}{*{20}c} { \otimes M_{11} } & \ldots & { \otimes M_{1n} } \\ \vdots & \vdots & \vdots \\ { \otimes M_{n1} } & \ldots & { \otimes M_{nn} } \\ \end{array} } \right]$$
(8)
Where
$$\otimes M_{ij} = \frac{{ \otimes X_{ij}{\prime} }}{{\max_{1 \le i \le n} \sum\limits_{j = 1}^{n} {M_{ij} } }}$$
(9)
Step 8: The total relation matrix (T)
$$T = M^{*} (I – M^{*} )^{ – 1}$$
(10)
where matrix I is the identity matrix of order n.
for transforming the grey weights into the crisp weights applies the average method, which is a simple and practical method to calculate the best nongrey performance (BNP) value of the grey weights of each aspect.
Step 9: The sum of each row and column of the total directrelation matrix was stamped as two vectors \(\overrightarrow{D}={\left[{d}_{i}\right]}_{n\times 1}\), \(\vec{R} = \mathop {\left[ {r_{j} } \right]_{1 \times n} }\limits^{\prime }\), \(\overrightarrow{D}\) + \(\overrightarrow{R}\) and, \(\overrightarrow{D}\)—\(\overrightarrow{R}\) vectors. \(When i=j , if {d}_{i}>{r}_{j}\to {d}_{i}{r}_{j}>0\), then the criterion is a net cause; \(When i=j , if {d}_{i}<{r}_{j}\to {d}_{i}{r}_{j}<0\), then criterion is a net effect. \({d}_{i}\) indicates the sum of direct and indirect effects of criterion i on other criteria. \({r}_{j}\) indicates the sum of direct and indirect effects on criterion j.
Step 10: A Cartesian coordinate system consisting of a horizontal axis (\(\overrightarrow{D}\)+\(\overrightarrow{R}\)) and a vertical axis (\(\overrightarrow{D}\) –\(\overrightarrow{R}\)) was drawn in which the coordinates of each criterion are displayed in ordered pairs (\({d}_{i}+{r}_{j}\) ,\({d}_{i}{r}_{j}\)).
Step 11: Also determine the influential weights of criteria. The relative importance of the criteria is calculated by using the following equation.
$$W_{i} = \left[ {\left( {d_{i} + r_{i} } \right)^{2} + \left( {d_{i} – r_{i} } \right)^{2} } \right]^{\frac{1}{2}} \,\,\,\,\,\,\,\,\begin{array}{*{20}c} {\forall i} & {i = 1, \ldots ,n} \\ \end{array}$$
(11)
The normalized weight of any criterion was measured as follows:
$$\overline{W}_{i} = \frac{{W_{i} }}{{\sum\limits_{i = 1}^{n} {W_{i} } }}\,\,\,\,\,\,\,\,\,\begin{array}{*{20}c} {\forall i} & {i = 1, \ldots ,n} \\ \end{array}$$
(12)
where \(\overline{W}_{i}\) shows the total criteria weights that would be required in the decisionmaking process. Therefore, therefore, the influential weight for each criterion (i.e., global influential weight) by applying the modified 2tuple DEMATEL approach was calculated.