1 Introduction

The emergence of virtual worlds and metaverse environments heralds a shift in the interplay of education, creativity, and technological advancements [1, 2]. As the digital revolution gains momentum, virtual worlds and the metaverse cater to vast populations, and an element that beckons scholarly attention is the incorporation of theater into these virtual realms. Theatrical performances have been praised for their effectiveness in improving various aspects of children's development for a considerable period [3]. The digital environment presents a new set of possibilities and challenges [4, 5]. One of the numerous challenges is to manage and form children's groups efficiently and effectively for theatrical engagements.

Theater's significance in the pedagogical domain cannot be underestimated. It has proven to be a necessary tool for promoting creativity, developing social skills, and enhancing communication and emotional intelligence among children [6,7,8]. As education moves into the digital world, it is essential to bring these benefits to the virtual realm as well. Digital theater in virtual worlds can be a way to keep the advantages of traditional theater while also going beyond its limitations [9].

However, the large and diverse population within virtual worlds brings about practical challenges that need to be addressed [10,11,12]. The importance of effective group formation for theatrical purposes within this context is crucial [13,14,15,16]. Children need to be grouped in a way that promotes the development of their creative abilities, encourages collaboration, and leads to successful preparation and execution of the theatrical performances.

As a result, the integration of Computational Intelligence (CI) techniques for the automated formation of student groups emerges as a promising and beneficial approach [17]. The particular techniques have the ability to use various factors like age, preferences, abilities, and schedules, for the automated grouping of students [18]. Furthermore, CI can simplify the process of forming groups, ensuring the use of the best teaching practices and enhancing group dynamics [19,20,21].

From a psychosocial perspective, the formation of these groups is also very important [22]. Groups consisting of members with similar psychosocial profiles help the collaborative nature of theatrical preparations. When groups are formed this way, they promote shared understanding and natural collaboration, which are essential for the complex process of preparing for a theatrical performance. CI can also examine psychosocial data to identify compatible profiles, forming the basis for cohesive and harmonious group formations [20].

Digital theater and drama in virtual environments represent an innovation, not only in their form but also in their essence [23,24,25,26,27]. They liberate participants from physical limitations and facilitate collaboration across geographical boundaries. Furthermore, the virtual environment offers a variety of tools and platforms that can be utilized for creative expression [28].

Even though automation in group formation is not only useful but also essential for digital theater and drama, creating optimal groups in virtual worlds remains a challenge. While CI has been employed in various educational settings, its application in digital drama education, particularly for group formation, is unexplored.

Flying Fox Optimizer (FFO) [29], is a recently developed metaheuristic, that distinguishes itself from other metaheuristic algorithms primarily through its unique inspiration from the survival strategies of flying foxes during heatwaves. A significant innovation in FFO is the use of Fuzzy Logic to autonomously determine individual parameters for each solution, creating a parameter-free optimizer, which contrasts with the manual parameter tuning required in most traditional optimizers. Additionally, FFO's hybrid algorithmic structure effectively combines operators from the existing algorithms, providing adaptability to the specific needs of various problems. Its movement mechanism, combined with the unique death and replacement strategy for solutions, ensures diversity and prevents premature convergence. As a result, FFO's global optimization capabilities mark a significant advancement over existing methods.

Given these factors, this study aims to present a CI technique based on FFO, for the automated formation of student groups in virtual worlds, with a focus on enhancing their creative and collaborative skills during theatrical performances. Specifically, this research aims to create homogeneous groups that foster collaborative learning and theatrical creativity, while ensuring manageability for educators. Thus, this research aims to significantly enhance education and theater in the digital era.

The research hypothesizes that the application of FFO can form student groups more systematically, ensuring homogeneity and compatibility. This approach aims to enhance collaborative learning, foster creativity, and ultimately optimize educational experiences in virtual theater settings. This research not only contributes to the field of educational technology but also bridges the gap between CI and the arts, offering novel perspectives and methods for educators and researchers in both disciplines. Additionally, it is hoped that this research will inspire future research and innovation in these important areas. To achieve this, a dataset encompassing a wide range of student attributes has been utilized. This dataset includes psychosocial data regarding social competence, academic competence, emotional competence, and potential behavior problems, offering a comprehensive perspective on each student's capabilities.

The problem addressed in this research is of significant importance for several reasons. First, in the realm of digital drama education, the formation of student groups plays a crucial role in shaping the learning experience. Second, traditional methods of group formation often rely on subjective judgment or random assignments, which may not consistently result in the formation of the most effective groups. This research addresses this gap by offering a data-driven CI method for group formation. Such an approach is particularly crucial in the context of digital education, where managing student engagement and interaction dynamics can be challenging. By optimizing group formation, the study contributes to improved educational strategies, ensuring that students in digital drama settings are grouped in a way that maximizes their learning potential and fosters a more productive and engaging educational environment.

The results of the study demonstrate that the application of the FFO in group formation significantly enhances the homogeneity of student groups in digital drama education settings. This finding underscores the potential of CI techniques in revolutionizing group formation processes, offering a more effective, data-driven approach to fostering collaborative and creative learning environments in the arts, especially in digital drama education.

The rest of the paper is structured as follows: Sect. 2 presents a review of the literature on the automated formation of student groups. Section 3 provides a comprehensive demonstration of the FFO. The research objectives are outlined in Sect. 4, followed by the methodology in Sect. 5. Section 6 showcases the study's results, while Sect. 7 concludes with key findings, limitations, and suggestions for future research.

2 Computational Intelligence and the automated formation of student groups

In today's educational settings, properly forming student groups is crucial for promoting collaboration, enriching learning experiences, and developing a wide range of essential skills. As educational settings become more complex, and as student needs and profiles diversify, traditional methods of group formation may become inefficient or less effective. CI, an offshoot of Artificial Intelligence, offers promising solutions to address this challenge through the automated formation of student groups [18, 30, 31].

CI involves a set of methodologies, including neural networks, fuzzy systems, and optimization algorithms, and is capable of addressing complex real-world problems with large datasets [21, 32,33,34]. In educational environments, CI can be used to investigate various student attributes, such as learning styles, academic proficiency, interests, and social dynamics [35,36,37,38,39,40,41,42].

Automating student group formation using CI ensures not only a simple process but also an intelligent one. CI algorithms can process and evaluate a variety of factors, identifying patterns and relationships that might not be evident through simple statistical analyses. This allows for the creation of groups that are both diverse and harmonious in terms of interpersonal dynamics.

By incorporating CI into the educational framework, educators and institutions can optimize the grouping process, tailor learning experiences, and ultimately foster an environment conducive to academic excellence and personal growth for students [43]. In this context, Krouska, Troussas and Sgouropoulou [17], Sanchez et al. [44], Sukstrienwong [45], Miranda et al. [46], Krouska and Virvou [47], Ani et al. [48], Pinninghoff et al. [49], Zheng et al. [31], Moreno, Ovalle and Vicari [50], Wang, Lin, and Sun [51] use genetic algorithms for group formation, and Lin, Huang and Cheng [52] and Zervoudakis, Mastrothanasis and Tsafarakis [20] use Particle Swarm Optimization (PSO) to assign students to groups based on various characteristics, enhancing differentiated instruction.

Finally, in the recent research of Mastrothanasis et al. [21], it is demonstrated how CI can be applied in educational drama. The authors applied a Mayfly optimization algorithm [53] to a dataset of 774 student instances to categorize them based on their levels of performance anxiety as well as the emotions developed during a theatrical performance. The findings revealed the formation of distinct student groups with similar characteristics in terms of emotions and performance anxiety. According to this research, the proposed CI method enables drama educators to easily identify and support students at risk by understanding each group's unique characteristics and developing coping practices.

3 Flying Foxes Optimization (FFO)

The optimizer called Flying Foxes Optimization algorithm (FFO) emerges as an attractive alternative optimization algorithm [29]. Its biological inspiration, combined with dynamic parameter adjustment through fuzzy logic, gives it a unique blend of characteristics conducive to efficient and effective optimization. Due to its effectiveness in addressing real-world problems, researchers have already embraced FFO as a valuable optimization approach [54, 55].

FFO is an innovative global optimization approach that takes its inspiration from the survival strategies of flying foxes during heatwaves. The particular species of bat, known for being one of the largest species of megabats, exhibits specific behavior under heat stress. Typically, these bats spend the night foraging and return to their trees at dawn. However, during heatwaves, they face rapidly rising temperatures and, to avoid overheating, they fan their wings. As temperatures continue to escalate, and the flying foxes become exhausted, they desperately search for cooler trees to rest. They often follow each other in this quest and, most of the time, end up overcrowding and suffocating one another, leading to fatalities.

The FFO algorithm starts by randomly generating a set of potential solutions, each represented by the position of a flying fox in a search space. Just as flying foxes instinctively head toward cooler areas during a heatwave, the virtual flying foxes in the algorithm do the same by seeking the most optimum solutions. When a flying fox’s position is not an optimal one, it moves toward the optimal solution. The algorithm also accounts for the possibility of a flying fox's death, mirroring the real-life situation in which a flying fox could die due to extreme heat or suffocation. This is simulated by the flying foxes either moving to a very poor solution or becoming overcrowded at the optimal solution. To counterbalance this, the algorithm has mechanisms for the replacement of the fallen flying foxes, ensuring the continuity and robustness of the search process.

A notable feature of the FFO algorithm is the incorporation of a Fuzzy Rule-Based System (FRBS) consisting of six fuzzy rules. These rules allow the algorithm to dynamically adjust its parameter values in an automatic way, overcoming the constraints that fixed parameter values could impose. This mechanism makes the algorithm capable of addressing a range of problems without the necessity for manual calibration of parameters.

According to the literature, the FFO algorithm has undergone extensive benchmark testing against a variety of other metaheuristic optimization algorithms on numerous test functions and real-world engineering optimization problems. The outcomes were promising, as FFO exhibited superior local and global search capabilities. Moreover, it has demonstrated a tendency to reach optimal solutions more quickly than several competing algorithms. This is partly attributed to its solution replacement technique, which enables the algorithm to effectively disregard unproductive regions of the search space.

According to the literature, the FFO algorithm offers an approach to global optimization, showcasing significant benefits over other metaheuristic methods. Its movement and replacement mechanisms enhance exploration capabilities, aiding in escaping local optima points and avoiding unfavorable regions of the search space. Combined with an FRBS, FFO dynamically determines parameter values, unlike other metaheuristics that require manual tuning, which can be very time-consuming. Finally, FFO demonstrates rapid convergence, competitive performance, and promising results in real-world engineering problems, establishing it as a powerful optimization tool [29].

In FFO, a set of potential solutions to the problem is initially generated randomly, with each one corresponding to the position of a flying fox. A d-dimensional vector \({{\varvec{x}}} = (x_1 , \ldots ,x_d )\) represents these potential solutions, which are evaluated using the problem’s objective function f(x). During the iterative process of the algorithm, flying foxes desperately search for cooler spots to stay alive during the heatwave.

3.1 Movement of flying foxes

When not resting in the coolest position, the flying foxes fly toward a cooler spot to avoid heat. This behavior can be described in the following formulation:

$$ x_{i,j}^{t + 1} = x_{i,j}^t + a\cdot rand\left( {cool_j - x_{ij}^t } \right), $$
(1)

with xi0 ~ U(xmin, xmax), \(x_{ij}^t\) is the jth element of flying fox i, at iteration t, a is a positive attraction constant, rand ~ U(0,1), and cool is the position of the best solution ever found. Equation (1) is selected for use when a flying fox is considered to be far from the coolest spot, as mentioned in the research of Zervoudakis and Tsafarakis [29].

Should a flying fox be near the coolest spot ever found, it explores the search space to avoid suffocation. This can be formulated as follows:

$$ nx_{i,j}^{t + 1} = x_{i,j}^t + rand_{1,j} \cdot \left( {cool_j - x_{i,j}^t } \right) + rand_{2,j} \cdot \left( {x_{R_1 j}^t - x_{R_2 j}^t } \right) $$
(2)
$$ x_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}l} {nx_{i,j}^{t + 1} ,} \hfill & {{\text{if}}\; j = k\;{\text{or}}\; rnd_j \ge pa} \hfill \\ {x_{i,j}^t ,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right., $$
(3)

where rand ~ U(0,1), \(rnd_j\) is a randomly generated number between \(\left( {0,1} \right)\), pa is a probability constant, and \({{\varvec{x}}}_{{{\varvec{R}}}_1 }^{{\varvec{t}}}\) and \({{\varvec{x}}}_{{{\varvec{R}}}_2 }^{{\varvec{t}}}\) are two different uniformly distributed random members selected from the current population. Finally, k, which is randomly selected from {1,2,…,d}, ensures that at least one component from \(nx_{i,j}^{t + 1}\) is chosen by \(x_{i,j}^{t + 1}\), to prevent the new solution from being an exact copy of the existing ones.

The new solutions are assessed, and if a flying fox discovers a better spot (an improved solution), it stays in the new one. If the flying fox does not find a better spot, it returns to its previous position.

3.2 Death of flying foxes

One possible reason for a flying fox to die is by flying into an extremely hot region of the search space, which is a distant area from the optimum solution ever found. In such situations, flying foxes find themselves in extremely hot regions from which they are unable to return, resulting in their demise. To replace the deceased ones, Zervoudakis and Tsafarakis [29] proposed a list called the Survival List (SL) with the NL unique best solutions ever found. An integer n ∈ [2, NL] is then randomly generated, and the position of a new solution is generated as follows:

$$ x_{i,j}^{t + 1} = \frac{{\sum_{k = 1}^n SL_{k,j}^t }}{n}, $$
(4)

where \({{\varvec{SL}}}_{{\varvec{k}}}^{{\varvec{t}}}\) is the kth flying fox on the SL during t iteration. The purpose of Eq. (4) is to increase the probability of generating a new solution in a better region of the search space.

Another reason a flying fox may die is when it becomes enclosed, surrounded, and suffocated by others. Hence, before the end of each iteration, a probability is computed based on the number of flying foxes in the coolest spot ever found (those having the same objective function value as the best solution ever found), calculated as

$$ pD = \frac{nc - 1}{{population\; size}}, $$
(5)

where nc is the number of flying foxes having the same objective function value as the best solution ever found.

As nc rises, and therefore pD rises as well, the flying foxes start suffocating and dying. For every two flying foxes in the coolest spot, a random number between \(\left( {0,1} \right)\) is generated. If this random number is less than the calculated pD probability, the two flying foxes die, and two new flying foxes are introduced to replace them in the population. The new flying foxes are either the offspring of two different random flying foxes or are generated using Eq. (4), with each scenario having a probability of 0.5.

3.2.1 Mating of flying foxes

The process of mating between two flying foxes involves using the genetic crossover operator. First, two different flying foxes are randomly selected from the population. The outcomes of this crossover technique are two offspring, calculated as [56]

$$ \begin{array}{*{20}c} {{{\varvec{offspring}}}1 = c \cdot {{\varvec{FF}}}_1 + \left( {1 - c} \right) \cdot {{\varvec{FF}}}_2 } \\ {{{\varvec{offspring}}}2 = c \cdot {{\varvec{FF}}}_2 + \left( {1 - c} \right) \cdot {{\varvec{FF}}}_1 ,} \\ \end{array} $$
(6)

where \({{\varvec{FF}}}_1\) and \({{\varvec{FF}}}_2\) are two different randomly selected population members, and \(c\) is a randomly generated value between \(\left( {0,1} \right)\).

3.3 Fuzzy self-tuning method of FFO

Metaheuristic algorithms, by their nature, rely on a set of parameters to guide their search process in finding optimal solutions. These parameters often include elements such as population size, the number of iterations, and specific values that control the behavior of the algorithm. The need for these parameters arises from the requirement to balance exploration (searching for new areas) and exploitation (refining known good areas) within the solution space. Metaheuristic algorithms require these parameters to be fine-tuned to ensure efficiency and effectiveness in solving complex optimization problems.

To make the FFO independent of any parameter setting, Zervoudakis and Tsafarakis [29] introduced an automatic parameter tuning method, where its parameter settings are automatically determined using Fuzzy Logic (FL), similar to the method used by Nobile et al. [57] and Tsafarakis et al. [58] in their research for PSO and DE, respectively. More details about how the fuzzy self-tuning process works can be found in the work of Zervoudakis and Tsafarakis [29]. As a result, FFO does not require the user to specify exact parameter values at the outset. Instead, it intelligently adapts its parameters based on the ongoing performance and the characteristics of the problem it is solving. This not only simplifies the process for users, especially those who may not be experts in metaheuristics but also enhances the robustness and adaptability of the algorithm, ensuring optimal performance across a range of different scenarios and datasets.

3.4 FFO pseudo-code

The pseudo-code of FFO is presented below.

figure a

4 Purpose of the research

The purpose of the present study is to design and develop an innovative mechanism that utilizes CI to create psychosocially homogeneous groups of students in theater education. In the realm of virtual theater, pedagogical efficacy often relies on the synergistic collaboration among the participants. This necessitates the formation of groups that share compatible psychosocial characteristics, as such shared traits can foster a conducive environment for collaborative learning and theatrical creativity. Additionally, the study acknowledges the realistic considerations of a theater educator's capacity to effectively manage a group; therefore, it aims to ensure that the numerical size of these groups is manageable. Through the application of CI techniques, the mechanism aspires to analyze arrays of psychosocial attributes to form student groups that are not only homogeneous but also optimally composed, thus providing theater educators with a powerful tool to enhance the learning experiences and creative outcomes of their students.

Based on this context, this study addresses the following research questions:

  1. 1.

    Can an FFO-based CI methodology effectively form homogeneous student groups in virtual theater education that enhance collaborative learning and theatrical creativity?

  2. 2.

    Does the FFO methodology offer advantages in terms of ease of implementation and efficiency when compared to other metaheuristic algorithms for forming homogeneous student groups?

5 Method

5.1 Dataset

To collect the psychosocial data necessary for conducting tests and developing the mechanism under study, we utilized the standardized measurement instrument known as the “Psychosocial Adaptation Test” by Chatzichristou et al. [59]. This instrument was used for data collection purposes. The test consists of 112 questions and is completed by teachers, who assess the frequency of specific behaviors using a five-point Likert-type scale.

The statements within the teacher-completed scale are categorized into four subscales: (a) "social competence subscale" (27 items), (b) "academic competence subscale" (29 items), (c) "emotional competence subscale" (25 items), and (d) "behavior problems subscale" (31 items).

The social competence subscale evaluates three dimensions: (a) assertiveness or leadership skills (DH), which measures the extent to which children display initiative and take on leadership roles (6 statements); (b) interpersonal communication (DE), which assesses the specific behaviors that lead to desirable social outcomes for the individual employing them (15 statements); and (c) peer cooperation (SO), which gauges the level of interaction children engage in with their peers to accomplish shared goals (6 statements).

The academic competence subscale assesses four dimensions: (a) motivation (KI), which evaluates the student's drive to achieve a goal (4 statements); (b) organization or planning (OS), which examines behaviors that aid the child in implementing strategies, starting and completing activities within given timeframes (13 statements); (c) school effectiveness (SA), which focuses on behaviors that contribute to the students’ overall effectiveness in school (6 statements); and (d) school adjustment (SX), which considers behaviors related to the child's successful adaptation within the school environment (6 statements).

Emotional competence is evaluated across four sub-dimensions: (a) self-control (AE), (b) emotion management (DS), (c) empathy (E), and (d) stress management (Dt). Self-control refers to the child's ability to regulate their emotions (8 statements). Emotion management assesses the child's awareness of their emotional state and their ability to control their emotions (5 statements). Empathy examines the child's capacity to recognize and understand the feelings of others (5 statements). Finally, stress management focuses on the child's utilization of appropriate self-regulation strategies (7 statements).

The subscale concerning behavior problems is evaluated along three dimensions: (a) interpersonal adjustment (DP), (b) intrapersonal adjustment (EP), and (c) hyperactivity, impulsivity, and attention difficulties (YD). Interpersonal adjustment refers to the extent to which the child displays aggressive and reactive behaviors (9 statements). Intrapersonal adjustment assesses the child's level of emotional difficulties (9 statements). The third-dimension measures impulsive behaviors and hyperactivity (13 statements). It should be noted that for the "behavior problems" subscale and its sub-dimensions, a high score indicates a lower level of adjustment compared to the other subscales of the instrument.

The descriptive data utilized in this study were derived from the fourteen subscales described above, encompassing data from 1077 fourth-, fifth-, and sixth-grade primary school children attending various schools in Greece. The descriptive statistics are demonstrated in Table 1.

Table 1 Descriptive statistics of datasets
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5.2 Application design

The purpose of this research is to minimize the differences in characteristics among individuals within each group as much as possible. Therefore, the objective function of the problem is defined as the sum of the sum of the distances between characteristics of all students within each group. For this purpose, the pairwise distances between students are calculated.

In this research, a range of 4–9 people per group was considered sufficient for the preparation of a theatrical performance. As a result, if a group has a number of people that falls out of the specified bounds (less than 4 or greater than 9), the solution is rejected.

To achieve the optimal division of individuals into homogeneous groups of 4–9 individuals, the FFO algorithm was adapted to the problem as follows:

Initially, the position of each flying fox is randomly generated as a two-dimensional table with dimensions (number of students × [number of students/4]). The number 4 is derived from the fact that each group must have at least 4 people. Each row represents a student (1077 students), and the columns represent the (rounded) maximum number of groups that can be constructed ([1077/4] = 269).

Given that this problem is a discrete optimization problem, and the FFO algorithm is primarily designed for continuous optimization, the Smallest Position Value (SPV) technique was employed. This technique has proven effective in the previous studies for optimizing discrete problems using continuous algorithms [58]. Hence, each student is assigned to the group whose corresponding column contains the lowest number.

For instance, consider a solution table with three individuals, as shown in Fig. 1. The first student is assigned to the second group, because the minimum value of the first row corresponds to the second column. The second student is assigned to the first group, because the minimum value of the second row belongs to the first column. Finally, the third student is assigned to the first group, because the minimum value of the third row belongs to the first column.

Fig. 1
figure 1

Solution example

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To improve the convergence speed of the algorithm, a modification was implemented in the initialization of solutions, as supported by previous research [60]. In this way, a student is randomly selected to form a group. Immediately afterward, the n students who are closest to the selected student are chosen based on their characteristics and the pairwise distances between them, where n is a random number between 3 and 8. The selected students are assigned a value of 1 for all groups (columns) except for the group they are currently enrolled in, where they are assigned a value of 0. The process is repeated to form the next group, and this continues until all students are allocated to a single group.

Another modification made to the algorithm concerns the movement of the flying fox that is in the optimal position. A student is randomly selected from a random cluster and is then moved to the cluster of a student closest to him, based on their characteristics and the pairwise distances between them. If the student who is to be moved is in a cluster of 4 students, then he exchanges position with a student of the cluster to be moved. This student is selected based on the distance of their characteristics, and therefore, the student with the greatest distance is chosen. Similarly, if the student who is to be moved is about to be assigned to a cluster consisting of 9 students, he exchanges position with another student from this cluster based on their characteristics, as before (the student with the greatest distance).

5.3 Evaluation of the application

After the formation of the groups, the solutions were assessed for their intragroup homogeneity using Coefficient of Variation (CV). Values close to one denote heterogeneity, whereas values close to zero indicate homogeneity with respect to the characteristic. The average degrees of homogeneity of the resulting groups are classified according to the values of the CV index as follows: (a) satisfactory level of homogeneity (0.00 < CV ≤ 0.40), and (b) dissatisfactory level of homogeneity (CV > 0.40) [20]. Then, the final clustering solution is evaluated through a Multivariate Analysis of Covariance test (MANCOVA) for the test of the discriminant validity of the proposed method.

6 Results

FFO is executed 50 times until 200,000 function evaluations are reached. The performance of FFO was compared to that of state-of-the-art metaheuristics like PSO and DE. Since FFO has the ability to calculate the parameter values automatically according to its needs, the fuzzy self-tuning PSO and the fuzzy self-tuning DE algorithms were used in this comparison, as proposed by Nobile et al. [57] and Tsafarakis et al. [58], respectively, in order for a fair comparison (Table 2).

Table 2 Comparison of optimizers
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The Shapiro–Wilk test, which was conducted to assess the normality of the results, revealed that the results significantly deviated from a normal distribution, W = 0.15, p < 0.001. As a result, a Kruskal–Wallis test was performed, to reveal that the optimizer selection significantly affects the objective function value, H(2) = 134.23, p < 0.001, ε2 = 0.87. The p values of the post hoc Dwass–Steel–Critchlow–Fligner pairwise comparisons across all group pairs test are presented in Table 3.

Table 3 p values of the post hoc Dwass–Steel–Critchlow–Fligner pairwise comparisons
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The convergence characteristic curves of all algorithms are depicted in Fig. 2.

Fig. 2
figure 2

Convergence characteristic curves of FFO, PSO, and DE

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Based on the presented results, it is evident that FFO demonstrates superior optimization capabilities compared to fuzzy self-tuning PSO and DE. Figure 2 further illustrates that FFO converges more rapidly to optimal points when compared to both PSO and DE variants.

As a result, it is evident that FFO exhibited superior performance in group formation within digital drama education settings. This is particularly highlighted by its effectiveness in achieving lower values of the objective function, which is a crucial metric for assessing the quality of groupings. Lower values of this objective are indicative of optimal group formations, characterized by high internal cohesion within groups. The results, as illustrated in Table 2 and Fig. 2, demonstrate the FFO's rapid convergence to minimal objective function values, compared to the rest of the optimizers.

To further demonstrate the performance of FFO, its results were evaluated. FFO created 173 distinct groups of students with similar psychosocial profiles within each group, as described by the 14 subscales. Table 4 provides information about the distribution of groups based on the number of members they have. The range of 4–9 people per group (Mean = 6.23; s.d. = 0.78), was considered suitable for preparing a theatrical performance and, therefore, for the performance of the algorithm.

Table 4 Frequencies and percentages of individuals across groups
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The mean degree of homogeneity of the generated groups ranged from 0.07 to 0.36 across the 14 psychosocial characteristics. This value can be considered satisfactory according to the evaluation standards for homogeneity. Furthermore, the results of the multivariate tests of MANCOVA for the dependent variables are presented in Table 5, revealing the discriminant validity of the proposed method.

Table 5 Multivariate tests
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The multivariate analysis of covariance revealed significant differences among the groups. Using Pillai's Trace as the criterion, the combined dependent variables were significantly affected by the independent variables, V = 6.95, F(2408, 12,656) = 5.19, p < 0.001. Similarly, significant results were found using Hotelling's Trace, T2 = 53.63, F(2408, 12,448) = 53.63, p < 0.001, and Roy's Largest Root, Θ = 30.74, F(172, 904) = 30.74, p < 0.001. The results of the univariate tests of MANCOVA test for the dependent variables, are presented in Table 6, which reveals the discriminant validity of FFO.

Table 6 Multivariate analysis of covariance summary table
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The results indicate that there are significant differences in the means of all dependent variables (e.g., DHI, DE, SO, etc.) when accounting for the covariates and comparing across the different levels of the "Group" independent variable. Residuals were calculated for each dependent variable, indicating the unexplained variability after accounting for the effects of the "Group".

7 Discussion

The primary objective of this study was to utilize a CI technique to create homogeneous student groups in virtual theater education. The results revealed that the FFO-based CI method is highly effective at forming these groups of students.

The study revealed that the proposed method is capable of forming homogeneous student groups based on various characteristics, including social, cognitive, emotional, and behavioral ones. This finding aligns with previous research that has successfully applied CI techniques to automatically group students based on their characteristics [20, 21, 47]. Such homogeneity helps in creating a friendly environment among students for working together, which is very important for being creative in theater and learning together in online theater classes. As a result, the proposed innovative methodology has the potential to significantly change the landscape of digital drama and theater education, since using CI for pedagogical purposes establishes a more effective and efficient approach to help students' development in digital theater education.

Another finding relates to how the FFO method works in practice. Evidently, FFO avoids the requirement for initial parameter settings and parameter fine-tuning, thanks to its integrated fuzzy self-tuning method. The particular technique not only makes the application of FFO simpler but also has the potential to increase its effectiveness. This finding is in line with the existing literature. Specifically, studies [29, 57, 58] have highlighted that metaheuristics combined with a fuzzy self-tuning method have superior performance compared to other metaheuristic algorithms that require parameter tuning by the user.

An additional merit of the proposed methodology lies in its scalability and speed. The methodology's exceptional adaptability and versatility stem from its capacity to accommodate a wide range of students, without limitations on quantitative data or the variety of characteristics assessed holistically. The ability to rapidly generate solutions further emphasizes the practicality and viability of the methodology within the practical limitations of educational settings [20]. Unlike other metaheuristic algorithms, FFO does not need parameter initialization, due to its fuzzy self-tuning method, which makes it quite easy to implement.

Regarding the limitations of this study, while the research strongly highlights the uniformity of student groups in theater, it does not address the potential dynamics in groups with varied characteristics. Such diversity often introduces numerous perspectives, enriching creative fields. Another limitation is that the proposed methodology relies on quantitative data, and the quality and accuracy of these data can significantly influence the outcomes.

8 Conclusion

The internal homogeneity of student groups concerning psychosocial aspects is of paramount importance in the context of theater. This homogeneity fosters an environment conducive to collaboration, which is a cornerstone of theatrical preparation and performance. Such an environment is likely to enhance communication, facilitate decision-making, and instill a sense of creative harmony among the constituents of the group. These elements are indispensable for the effective realization of theatrical objectives and contribute to a more polished and cogent final output.

The CI FFO-based methodology proposed in this research equips theater educators with an enhanced ability to manage groups effectively. Possessing an awareness of the distinctive characteristics of each group enables educators to customize their instructional approaches and resources to suit the needs and attributes of the students. This informed approach to management is vital for optimizing educational outcomes and ensuring that the engagements are both relevant and impactful.

Considering future directions and the potential for further development, this research holds significant promise. As the digital landscape continues to evolve, the diverse nature and requirements of educational engagements within it will undoubtedly expand. The current methodology serves as a foundation for additional research, allowing for continuous refinement and adaptation to address emerging demands and seize new opportunities.

One of the most promising directions is the investigation of its applicability in diverse educational settings. While its efficacy in theater is evident, it would be interesting to examine its impact in other artistic domains such as music or dance. Each of these disciplines could benefit from a methodology that emphasizes group homogeneity and effective collaboration.

Finally, the long-term effects of this methodology need to be studied. It is important to evaluate how this method affects the improvement of students' skills, when it comes to both theater and education. Feedback from both students and teachers is crucial to evaluate the method's effectiveness in grouping and to understand its advantages and any potential disadvantages.