The Decision-Making Trial and Evaluation Laboratory (DEMATEL) was initially developed by the Geneva Research Center of the Battelle Memorial Institute. To identify the correlation (causality) between alternatives, the DEMATEL method uses pairwise comparisons that capture the interrelationships between alternatives. A detailed review of the DEMATEL method can be found in Sorooshian et al. (2023). The procedure of the DEMATEL method is summarized as follows.

Step 1. Obtain the initial matrix M. For a pair of alternatives a_i and a_j, a decision maker (e.g., a student in one group) is asked how much a_i influences a_j. A scale of 0 to 100 can be used to make the recommendation, with 0 (resp. 100) representing no influence (resp. very high influence). The decision maker's response is denoted by xji, representing the degree of the influence of a_i on a_j. For the problem including n alternatives, the initial evaluation matrix has n rows and n columns, and its structure is displayed as

It is noted that xji=0 (i = j; i, j = 1, 2, …, n).

Step 2. Obtain the normalized evaluation matrix M¯. The normalized evaluation matrix is calculated by M̄=M/S where S=max{maxi1{∑i2xi1i2},maxi2{∑i1xi1i2}}.

Step 3. Calculate the total influence matrix T. The direct total influence matrix is calculated by T=M̄(I-M̄)-1 where I is an identity matrix. The total influence of a_i1 on a_i2 is represented by the element in column i₂ of row i₁ in the total influence matrix T, denoted by ti1i2.

Step 4. Construct the influential relation map. Based on T, the influence of a_i on the other alternatives is calculated by eir=∑i1tii1, which form the out influence vector Er=[eir]1×n. The impact of the other alternatives on a_i is reflected by eic=∑i1ti1i, which form the inner influence vector Ec=[eic]1×n. For instance, the total influence matrix is shown below.

The influence of a₁ on the other alternatives is calculated by e1r=∑it1i=2+3+3=8. The impact of the other alternatives on a₁ is calculated by e1c=∑iti1=2+2+2=6.

Step 5. Calculate the centrality and causality indices. The centrality index μi=eir+eic indicates the importance of alternative a_i to the entire decision problem. The causality index λi=eir-eic shows the total net influence of a_i on the other alternatives. When λ_i > 0, a_i has an impact on other alternatives; when λ_i < 0, other alternatives affect a_i. Continue to the above example, the centrality index of alternative a₁ is calculated by μ1=e1r+e1c=14. The causality index of alternative a₁ is calculated by λ1=e1r-e1c=2.

In the field of education, the DEMATEL has been widely applied to (1) identify the factors that promote or hinder student learning (Chen et al., 2023; Ma et al., 2020; Lu et al., 2024), (2) find the factors that make students stressful (Garg and Bhardwaj, 2024), and (3) analyze learning design (Pourhejazy and Isaksen, 2024). In other fields, it has been employed in business process evaluation (Ni et al., 2025), operational risk factor assessment (Zheng et al., 2024), and key success factors recognition (Zhang et al., 2022). Given the good performance of the DEMATEL method in the field of education, this paper uses the DEMATEL method to assess the level and quality of student engagement in group tasks.

As a generalization of t-norm and t-conorm, the uninorm aggregation operator (Yager and Rybalov, 1996; Fodor et al., 1997; De Baets, 1999) is a mapping R:[0, 1]² → [0, 1]. A popular uninorm aggregation operation is (Wang et al., 2024)

where g ∈ [0, 1] is the neutral element. The neutral element g divides the plane [0, 1]² into four areas: one reinforcement area, one weakening area, and two average areas. In Figure 1, when the neutral element g rises, the reinforcement area is reduced while the weakening area is expanded.

Corresponding to the reinforcement and weakening areas, the uninorm aggregation operator has two features: reinforcement and weakening. An example is given to illustrate these two features.

Example 1. (Reinforcement and weakening of the uninorm aggregation operator) Consider two evaluations x₁ and x₂. The neutral element g is set to 0.5. When both x₁ and x₂ are larger than 0.5, U(x₁, x₂) is larger than x₁ and x₂, referred to as the reinforcement feature. For instance, let x₁ = 0.6 and x₂ = 0.7. Referring to Equation 1, U(x₁, x₂) = 0.76, which is larger than 0.6 and 0.7. When both x₁ and x₂ are smaller than 0.5, U(x₁, x₂) is smaller than x₁ and x₂, referred to as the weakening feature. For instance, let x₁ = 0.3 and x₂ = 0.2. Referring to Equation 1, U(x₁, x₂) = 0.12, which is smaller than 0.3 and 0.2.

Since its introduction, the uninorm aggregation operator has been widely applied in multiple criteria decision-making (Wang et al., 2024), group decision-making (Jin et al., 2024; Wu et al., 2022; Gong et al., 2023), and recommended systems (Palomares et al., 2018). Given that the uninorm aggregation operator performs well in aggregating the information given by decision makers, this paper uses the uninorm aggregation operator to integrate the evaluations of teachers and students.

In this section, the uninorm DEMATEL method is introduced to evaluate the level and quality of student engagement based on the evaluations given by students within the group.

Consider a group of n students represented as A = {a_i|i = 1, 2, …, n}. Each student a_i is asked to rate how positively the other students influence him/her during group tasks. This recommendation is given on a scale from 0 to 100, where 0 (resp. 100) indicates no positive impact (resp. a very strong positive impact). The influence of student a_j on student a_i is represented as xji. The evaluations from all students can be organized into an evaluation matrix, expressed as

In the evaluation matrix M, xii (i = 1, 2, …, n) are set to 0.

Once the evaluation matrix has been established, following the procedure outlined in Section 2.1, the DEMATEL method is used to compute the centrality and causality indices (Ni et al., 2025). For student a_i, the centrality index u_i indicates the total influence exerted by a_i as well as the influence received by a_i. A high centrality index u_i suggests that student a_i is actively engaged in group tasks, reflecting their level of participation. The causality index λ_i signifies the positive impact of student a_i on other students. A large causality index λ_i implies that student a_i offers constructive feedback to others in group tasks, which highlights the participation quality of student a_i.

When a student has high centrality and causality indices, he/she actively engages in group tasks, leading to a high score. Conversely, if the centrality and causality indices of a student are low, he/she may be free-riding and not participating actively in the group work, which results in a low score.

In the above discussion, the first requirement is adapted to the reinforcement feature of the uninorm aggregation operator, and the second requirement is adapted to the weakening feature. Since the features of reinforcement and weakening of the uninorm aggregation operator satisfy the two aforementioned requirements, the uninorm aggregation operator is used to aggregate the centrality and causality indices. The participation index Ω_i for a_i is determined by

where λī and μī are the normalized centrality and causality indices, calculated by

where Δλ=maxi1=1,2,…,n{λi1}-mini2=1,2,…,n{λi2} and Δμ=maxi1=1,2,…,n{μi1}-mini2=1,2,…,n{μi2}. If a student has a small composite index, then the likelihood of the student free-riding is high.

When the neutral element g is large, the weakening area is large (referring to Figure 1). In such a situation, students must have large centrality and causality indices (i.e., high level and quality of engagement) to obtain a large participation index. In such a context, the neutral element g can reflect the rigor degree of the assessment of group tasks.

In this section, we calculate the unfairness index of each group and generate the discount parameter for every student through the uninorm DEMATEL method. The neutral element g is set to 0.5. The teacher gives the largest discount rate ψ of 0.6.

We take the first group as an example to show how the unfairness index and the discounted scores are calculated. For the first group, referring to Step 2 in Section 2.1, the normalized evaluation matrix is calculated by M̄1=M1/S1 where S₁ = max{max{99, 90}, max{99, 90}} = 99, shown as

Referring to Step 3 in Section 2.1, the total influence matrix for the first group is calculated as

For student a₁, the centrality index μ₁ (resp. the causality index λ₁) is calculated as μ₁ = (10 + 11) + (10 + 10) = 41 (resp. λ₁ = (10 + 11)−(10 + 10) = 1). For student a₂, the centrality index μ₂ (resp. the causality index λ₂) is μ₂ = 41 (resp. λ₂ = −1).

Referring to Equations 3, 4, the normalized centrality and causality indices of a₁ and a₂ are calculated as μ̄1=1, λ̄1=1, μ̄2=1, and λ̄2=0. According to Equation 2, the participation indices of a₁ and a₂ are 1 and 0.5, respectively. Referring to Equation 7, the discounted scores of a₁ and a₂ are calculated as 95 and 76. The unfairness index of the first group is calculated as F1=[(1-0.75)2 + (0.5-0.75)2]÷2=0.25.

The unfairness index for the other groups is calculated in the same way, and the final scores for each of the 14 groups are F₁ = 0.25, F₂ = 0.44, F₃ = 0.41, F₄ = 0.31, F₅ = 0.47, F₆ = 0.37, F₇ = 0.4, F₈ = 0.41, F₉ = 0.38, F₁₀ = 0.45, F₁₁ = 0.37, F₁₂ = 0.43, F₁₃ = 0.25, and F₁₄ = 0.39, as shown in Figure 3.

Further analysis reveals that the number of students does not exhibit a direct correlation with unfairness indices, as statistically confirmed by the nonsignificant p-value (p = 0.1239) in Figure 4. It can be known that when the number of students in the group is equal to 2, the unfairness index (i.e., the F score) is low (< 0.26). When the number of students in the group is equal to 3, the unfairness index varies from a low unfairness index of about 0.3 to a high unfairness index of about 0.5. For the medium-to-large-sized groups (4 − 7 members), the unfairness index is at least a medium level (>0.36).

The discount rate of the 14 groups revealed that there were significant differences in the fairness levels of the groups (The value range: 0.25 − 0.47), reflecting the dynamic characteristics of the recognition of contribution distribution in group collaboration. Among them, the smallest 2-member group (Group 1 and Group 13) had the best fairness (F₁ = F₁₃ = 0.25), indicating that small-scale collaboration is more likely to achieve transparent division of labor and consensus; while Group 5, consisting of 3 members, had the worst fairness (F₅ = 0.47), probably due to the large divergence in members' evaluations or the existence of free-riding behaviors. Further analysis showed that group size did not show a simple linear relationship with fairness: the unfairness indices (i.e., F) of medium-to-large-sized groups (4–7 members) were mostly concentrated in the range of 0.37–0.45, indicating that although more members increase the complexity of collaboration, moderate fairness can still be maintained through effective management (e.g., F₉ = 0.38 for the group of 7 members).

The analysis reveals that group scores do not exhibit direct correlation with unfairness indices, as statistically confirmed by the nonsignificant p-value (p = 0.3525) in Figure 5. More specifically, high Group 14 (Score = 87) still has moderate inequity (F₁₄ = 0.39), while the high unfairness index of low Group 12 (Score = 70) (F₁₂ = 0.43) suggests that its underperformance may be superimposed on the imbalance in the distribution of contributions. The results of this paper emphasize that the inequity index can provide a quantitative basis for identifying collaboration problems, and teachers need to focus on groups with F>0.4 (e.g., Groups 2, 5, 10, and 12) to optimize the team division of labor mechanism and dynamic monitoring strategies, thus enhancing the fairness of course assessment and the achievement of educational goals.

The discounted scores presented in Table 3 demonstrate the practical implications of integrating the unfairness index into academic assessments. Significant score variations within high-unfairness-index groups (e.g., Group 5, F₅ = 0.47) highlight pronounced disparities in perceived contributions, as evidenced by contrasting outcomes such as Student a₁₄ (Score = 51) vs. a₁₃ and a₁₅ (Score = 85), suggesting potential free-riding or evaluation conflicts. Conversely, low-unfairness-index groups (e.g., Group 1, F₁ = 0.25) exhibit minimal score adjustments (e.g., a₁ = 95, a₂ = 76), reinforcing that smaller team sizes enhance transparency and consensus. Notably, even high-performing groups (e.g., Group 14, score = 87) show moderate unfairness (F₁₄ = 0.39), with substantial discounts for some students (e.g., a₅₄ = 52.2 versus a₅₅ = 85.608), underscoring that academic excellence does not inherently ensure equitable contribution distribution. These findings validate the unfairness index as a robust tool for identifying collaboration inefficiencies, particularly in groups with F>0.4 (e.g., Groups 5 and 12), where extreme discounts (e.g., a₅₀ = 42) signal urgent needs for pedagogical interventions. This approach equips educators with actionable insights to refine team dynamics and align assessment outcomes with educational equity goals.

This section performs the sensitivity analysis to discover the influence of the neutral element on unfairness indices and discounted scores. The neutral element is set to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. The unfairness indices with different neutral elements are shown in Figure 6.

The means and standard variances of the unfairness indices of the 14 group are shown in Table 4. The sensitivity analysis results show that the variation of the neutral element (g) has a limited overall impact on the unfairness index, which verifies the advantages of the uninorm DEMATEL method in terms of parameter robustness. The mean of unfairness indices fluctuations of the 14 groups under different neutral elements are narrower (0.25–0.4633), and the standard deviation is generally lower (e.g., the standard deviation of Groups 1 and 13 is 0, and the others are mostly lower than 0.05). This indicates that although the adjustment of the neutral element slightly affects the quantitative results of fairness, it does not trigger a significant systematic bias, suggesting that the method is insensitive to parameter selection.

Then, the discounted scores of all students with different neutral elements are calculated, which are shown in Figure 7. Of all the students, those whose scores changed with the neutral element are shown in Figure 8. The means and standard variances of the discounted scores of the 57 students are given in Table 5, and the following findings can be drawn:

(1) The proposed model is robust to changes in the neutral element (g). Many of the students' discounted scores in the table remain stable under different g, and both the mean and standard deviation show low volatility. For example, the discounted scores of students a₁ and a₂ have mean values of 95 and 76, respectively, and a standard deviation of 0, indicating that their discounted scores are not affected at all regardless of changes in g. Also, the results from 22 other students further show that the model is stable.

(2) The standard deviations are generally low, indicating that the model outputs are not sensitive to the choice of parameters. Most of the students have standard deviations lower than 5. For example, Student a₃₇ has a discounted score mean of 81.9 and a standard deviation of 2.87, indicating that the range of fluctuation of his discounted score across different g is small. The mean value of the discounted scores of Student a₄₉ is 69.5, and the standard deviation is only 0.37, which further indicates the stability of the model.

(3) Even if there are individual students whose discounted scores fluctuate greatly, their impact is still within a reasonable range. For example, the discounted score of student a₂₅ has a mean value of 70.44 and a standard deviation of 4.43. Although the fluctuation is a bit large, the mean value is still within a reasonable range, which shows that the model can effectively deal with the effects of parameter changes.

In summary, the sensitivity analysis proves the robustness of the uninorm DEMATEL method in parameter selection, which provides a reliable guarantee for the practical application of the model.