In this study, the Gray Delphi method^{40} And gray DEMATEL technique^{41,42} It was used to identify factors that influence students' academic burnout. Matlab version R2020 software and Excel version 2021 software were used for data analysis. DEMATEL is used to derive correlations between criteria. A gray DEMATEL method was developed for group decision making and causal analysis in gray environments. The gray DEMATEL is applied to identify the causality of the criteria. This allows decision makers to focus on what has the most impact and act more efficiently. Study participants were selected through a onestage cluster sampling method. For this reason, we randomly selected four faculties and four departments from each of the 12 faculties. In the Delphi stage, 86 graduate students were randomly selected, and in the DEMATEL stage, 37 students (with a probability drop rate) were randomly selected from each department.
Identifying the cause
In order to explore the factors that influence students' academic burnout, we randomly selected 86 graduate students (master's and doctoral programs) studying at Islamic Azad University's scientific research department, conducted a public questionnaire, and identified 43 factors. was extracted.
Gray Delphi method (GDM)
Dalkey and Helmer established the Delphi method in 1963^{43}. The Delphi method is a structured process for screening and ranking factors, which is implemented by gathering the opinions of decision makers through questionnaires. The three basic characteristics of the Delphi method are anonymous response, repeated and controlled feedback, and statistical group response.^{44,45}. Although the traditional Delphi method has been used in many studies, it has been criticized for its long process, poor convergence, and the loss of some valuable expert information.^{46}. Additionally, traditional processes for quantifying people's perspectives are subject to ambiguity, inaccuracy, and poor compatibility with linguistic and sometimes ambiguous human explanations, judgments, and priorities, and therefore do not fully capture human thinking styles. It is not reflected in^{45,47}. Gray system theory was started by:^{48}. The goal of the Gray System is to bridge the gap that exists between the social and natural sciences.^{48}. In the gray system, all messages can be classified into three categories: white, gray, and black. The white area ultimately indicates a clear message in the system, the black area has unknown characteristics, and the gray area occurs between the known and unknown messages and covers both.This theory includes four parts^{49,50}.
Using Gray theory and Delphi method simultaneously is the proposed solution^{40,51}.
step 1: A Delphi questionnaire was distributed to 86 graduate students who were asked to judge the importance of each criterion using linguistic variables.Linguistic variables and their corresponding gray scales according to criterion importance weighting.^{40} Shown in Table 1.
Step 2: Based on the method proposed by^{40},j (\(j = 1,…,5\)) gray class is considered and the selection range of the ith criterion, i.e.\(\left[ {a_{i}^{1} ,b_{i}^{5} } \right]\) I was divided into 5Gy classes.
Step 3: Equations (1) and (2) show the halftrapezoidal whitening weight function applied to j = 1 and 5.
$$f_{i}^{1} (x) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & {x \le a_{i}^{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{b_{i}^{1} – x}}{ {b_{i}^{1} – a_{i}^{1} }}} & {a_{i}^{1} < x \le b_{i}^{1} } \\ \end{配列} } \\ {\begin{array}{*{20}c} 0 & {x > b_{i}^{1} } \\ \end{array} } \\ \end{array} } \right. $$
(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. The triangular whitening weight function is: (3) has been 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: We calculated the synthetic clustering coefficient using equation (4) (\(\rho_{i}^{j}\)).
$$\rho_{i}^{j} = \sum\limits_{k = 1}^{m} {f_{i}^{j} (x) \cdot \eta_{i}^{k} }$ $
(Four)
Step 5: where \(f_{i}^{j} (x)\) The whitening weight function for the jth gray class criterion i. m is the number of categories of student opinions. \(\eta_{i}^{k}\) is the weight of criterion i in the composite cluster.
Step 6: The decision vector for the evaluation criteria has been identified.standard \(\max_{1 \le j \le 5} (\rho_{i}^{j} ) = \rho_{i}^{{j^{*} }}\) was used to determine whether criterion j belongs to the class \(j^{*}\).
The selection criteria are:

(1)
For classes \(j^{*}\) It belongs to classes 4 and 5. That is, the criterion is accepted if the value of the important or very important class is the maximum value of the vector decision.

(2)
If the class ratio \(j^{*}\) Comes with class 4 and class 5 \(j^{*}\) This criterion is accepted if the ratings associated with classes 1 and 2 are greater than 1, that is, if the important and very important classes, excluding the undetermined classes, account for more than 50 percent.
Gray DEMATEL
Fonterra initially used the Decision Making Test and Evaluation Laboratory (DEMATEL) method and in 1976 used Gabus.^{50}. DEMATEL is useful for analyzing causeandeffect relationships between components of a system. DEMATEL can demonstrate the existence of relationships/interdependencies between criteria or describe the relative levels of relationships within criteria.^{50}. DEMATEL does not rely on large data samples and simplifies factor correlation analysis.^{52,53,54}. However, his traditional DEMATEL does not take into account the ambiguity and uncertainty of reality.^{54,55,56}. Tseng (2009) extended fuzzy triangular numbers to establish his hierarchical gray DEMATEL for analyzing incomplete information criteria and alternatives.^{41}.
Therefore, in this study, we applied the Gray DEMATEL method to develop a causal model of factors influencing students' academic burnout.
Step 1: A questionnaire containing two square matrices (a square matrix of order 7 of the main criterion and a square matrix of order n of order 7, where n is the number of subcriteria of each main criterion) is created for the following pairwise comparisons. It was designed to. standard.
Step 2: A randomly selected sample of 37 graduate students (master's and doctoral students) assessed the interrelationships between the criteria through pairwise comparisons.
Step 3: Students used 10 linguistic variables to explain the degree of causal relationship between the criteria. Linguistic variables and their corresponding grayfuzzy numbers (as per below)^{41,57,58}Indicators for defining the influence of factors that influence students' academic burnout are shown in Table 2.
Step 4: assume \(\otimes\) The interval gray number is defined as: \(\times X = \left[ {\underline{X} ,\overline{X} } \right]\)the lower and upper bounds of X are restricted.
Step 5: Using the formula, (5) Summarize the students' opinions and the direct relationship matrix (n × n) (\(i,j = 1, \ldots ,n\)) was achieved to show that criterion i influences criterion j.
$$\otimes X_{ij} = \frac{1}{h}\left( { \otimes X_{ij}^{1} + \otimes X_{ij}^{2} + \cdots + \otimes X_{ ij}^{h} } \right)$$
(Five)
$$X = \left[ {\begin{array}{*{20}c} { \otimes X_{11} } & \ldots & { \otimes X_{1n} } \\ \vdots & \vdots & \vdots \\ { \otimes X_{n1} } & \ldots & { \otimes X_{nn} } \\ \end{array} } \right]$$
(6)
Step 6: We normalized the gray relational 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 relationship is a normalized gray DEMATEL 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: Overall relationship matrix (T)
$$T = M^{*} (I – M^{*} )^{ – 1}$$
(Ten)
Here, matrix I is an identity matrix of order n.
To convert gray weights to crisp weights, apply the averaging method. This is a simple and practical method to calculate the best nongray performance (BNP) value for each side's gray weight.
Step 9: The sum of each row and column of the entire direct relationship 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, \(\overrightarrow{D}\)—\(\overrightarrow{R}\) vector. \(If i=j, {d}_{i}>{r}_{j}\to {d}_{i}{r}_{j}>0\)If , the criterion is the true cause. \(If i=j, {d}_{i}<{r}_{j}\to {d}_{i}{r}_{j}<0\)If , the criterion is the net effect. \({d}_{i}\) Indicates the total direct and indirect influence of criterion i on other criteria. \({r}_{j}\) Denotes the sum of direct and indirect effects on criterion j.
Step 10: Horizontal axis (\(\overrightarrow{D}\)+\(\overrightarrow{R}\)) and the vertical axis (\(\overrightarrow{D}\) –\(\overrightarrow{R}\)) is an ordered pair of coordinates for each criterion (\({d}_{i}+{r}_{j}\) ,\({d}_{i}{r}_{j}\)).
Step 11: It also determines the influence weight of the criterion. The relative importance of criteria is calculated using the following formula:.
$$W_{i} = \left[ {\left( {d_{i} + r_{i} } \right)^{2} + \left( {d_{i} – r_{i} } \right)^{2} } \right]^{\frac{1}{2}} \,\,\,\,\,\,\,\,\begin{array}{*{20}c} {\forall i} & {i = 1, \ldots ,n} \\ \end{array}$$
(11)
The normalized weight of any standard was determined 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 \(\overline{W}_{i}\) indicates the total weight of the criteria required in the decisionmaking process. Therefore, the influence weight for each criterion (i.e., the overall influence weight) by applying a modified 2tuple DEMATEL approach was calculated.