Adaptive Curriculum Sequencing and Education Management System via Group-Theoretic Particle Swarm Optimization

Abstract

The Curriculum Sequencing (CS) problem is a challenging task to tackle in the field of online teaching and learning system development. The current methods of education management might still possess certain drawbacks that would cause ineffectiveness and incompatibility of the whole system. A solution for achieving better user satisfaction would be to treat users individually and to offer educational materials in a customized way. Adaptive Curriculum Sequencing (ACS) plays an important role in education management system, for it helps finding the optimal sequence of a curriculum among various possible solutions, which is a typical NP-hard combinatorial optimization problem. Therefore, this paper proposes a novel metaheuristic algorithm named Group-Theoretic Particle Swarm Optimization (GT-PSO) to tackle the ACS problem. GT-PSO would rebuild the search paradigm adaptively based on the solid mathematical foundation of symmetric group through encoding the solution candidates, decomposing the search space, guiding neighborhood movements, and updating the swarm topology. The objective function is the learning goal, with additional intrinsic and extrinsic information from those users. Experimental results show that GT-PSO has outperformed most other methods in real-life scenarios, and the insights provided by our proposed method further indicate the theoretical and practical value of an effective and robust education management system.

1. Introduction

With the rapid development of information technology and rising amounts of communication demand, the popularization of online education is growing and there have been great achievements in the last few decades. This social phenomenon has facilitated the advancement of e-education technologies, and distance education has become more and more widely recognized. Meanwhile, the relationship between teachers and students has been dispersed spatially and temporally, for the boundaries of the physical classrooms are expanded to the new space through e-education and e-classrooms [1]. The demands of online meeting and remote conference also come from various entities such as elementary schools, universities, governments, private companies, non-profit social organizations, etc. [2,3].

Especially since the beginning of 2020, because of the epidemic of coronavirus, social structures have been massively rebuilt, and more and more social interactions and communications quickly transferred from physical form to electronic form [4]. Furthermore, the traditional online education system is still imperfect and has many limitations such as lack of reusability and flexibility, as well as possible incompatibility with updating management systems over time, which could compromise the quality of education. Therefore, improving and perfecting online education systems is an imminent and crucial task [5]. One of the possible improvements could be developing the function of producing a curriculum with new ways of content delivery and innovative pedagogic strategies, customized to fit each user in the system. In this way, user satisfaction, teaching efficiency, and learning outcome could all be improved, and in the meantime avoid possible educational defects such as lack of commitment, cognitive overload, disorientation, distraction, etc.

However, because of the NP-hard nature of the Curriculum Sequencing (CS) problem, as well as the fact that nowadays the CS process is still mainly performed by teaching staff manually, customizing personal curricula for different students is a difficult and consuming process. Therefore, offering the optimal sequence of a curriculum with intelligent tutoring, personalized scenarios, and adaptive education according to each student’s skills and learning content, is known as the Adaptive Curriculum Sequencing (ACS) problem in the education management system. Specifically, ACS problems are divided into two separate categories: Individual Sequencing (IS) problem [6] and Social Sequencing (SS) problem [7]. The users’ own background information is the only item considered by IS, whereas SS treats the users’ both previous and current experiences as the impacting factors for final ruling. To tackle this task, a metaheuristic approach involving evolutionary computation and swarm intelligence could be considered and implemented to approximate optimal solutions [8]. These methods are endowed with a non-deterministic framework that can search the solution space based on the guidance of better candidates, trial-and-error interactions, and random components to some extent.

The challenge of the ACS problem mainly focuses on the automatic progress of reaching the most suitable sequence in terms of users’ profiles and goals, and different types of metaheuristic algorithms are dedicated to improving the approximation of optimal solutions. Previous research showed that the trending methods of metaheuristics such as Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Prey–Predator Algorithm (PPA) are increasingly being applied to IS and SS problems recently.

Vanitha et al. [9] propose the adaptation of ACO to select the optimal path for maximizing users’ acquisitions according to their pedagogical experiences. The improved ACO is unitary by integrating the individual profile, collective factors, and mutual interactions of the optimization approach. Birjali et al. [10] present e-learning paradigm optimization based on big data technologies to enhance the capacity of a large number of educational resources. The model consists of several levels using MapReduce-based GA, MapReduce-based ACO, and MapReduce-based Social Networks Analysis (SNA) for adequate e-assessment, adaptive learning path, and social productivity, respectively. Christudas et al. [11] introduce the Compatible Genetic Algorithm (CGA) to force the compatibility of unimplemented learning contents in the e-learning system, which further reduces the complexity of the search space of personalized learning contents. Peng et al. [12] use the Hybrid PSO by adding the components of a competitive-genetic crossover operator, adaptive polynomial mutation, and a curriculum scheduling model to ensure the practicability of their proposed model for the ACS problem under certain constraints. Govindarajan et al. [13] observe the competence and meta-competence values of students and utilize the Parallel PSO to predict the dynamic path for them, and it is able to auto-configure and auto-customize the proposed system simultaneously. Menai et al. [14] modify the swarm search with a random walk mechanism (SwarmRW) using incremental and random heuristic strategies, and the quantitative results show that the best trade-off between quality and time is obtained by SwarmRW_inc. Meanwhile, the maximum number of satisfied constraints are achieved by SwarmRW_rnd. Machado et al. [15] investigate the adaptive version of the Prey–Predator Algorithm (PPA) for the CS problem, and the performance is tested on both real-life and synthetic datasets, but it is not yet explored under the adaptive environment of e-learning. Martins et al. [16] conduct a series of metaheuristic algorithms including aforementioned GA, PSO, PPA, and DE. This comprehensive study analyzes the influence of multiple learning contents while matching the requirements of learning resources with learning paths, and it highlights the effort of DE compared to other algorithms as well.

Generally speaking, the abovementioned methods are proven to be effective in respect of their convergence and the performance of objective functions during the optimization process. The experimental results illustrated the contributions of their valuable works but there is still room for improvement. Many of the pedagogical works take only into account the intrinsic characteristics of users and evaluate the optimal path mainly based on synthetic datasets. As for the search behavior, the trade-off between diversification (exploration) and intensification (exploitation) is significant to the solution quality and the dynamically balanced relationship should be investigated for ACS optimization.

Group theory is the study about algebraic structures known as groups [17]. Groups are special sets equipped with an operation (such as multiplication, addition, composition, etc.) that satisfies certain basic rules of properties. In practice, many combinatorial applications from multiple disciplines can be modeled by the framework of a symmetric group, especially for those with symmetric characteristics of the whole system. The theory of the symmetric group usually embodies the internal symmetry of some structures in the form of permutation and combination representations. It is also the case that the symmetric relationship resides in the path of curriculum sequences for the ACS problem. The internal symmetry of the underlying structure often exists as an invariant property of the system, which is a great effort in solving the combinatorial optimization problem simultaneously.

The objective of this study is to propose a novel metaheuristic mechanism named Group-Theoretic Particle Swarm Optimization (GT-PSO) for the ACS problem in the education management system. GT-PSO is the extension of traditional PSO with the solid mathematical foundation of a symmetric group to rebuild the search paradigm from four aspects, which are, respectively, particle representation, landscape decomposition, moving neighborhood, and swarm topology. GT-PSO is originally designed for the combinatorial optimization problem and trends of searching characteristics are implemented via the utilization of group theory concept. The ACS problem is formulated as the combination of users’ preferences, learning subjects, and knowledge concepts, and this sort of NP-hard problem is addressed in the form of the systematic procedure of GT-PSO. The main contributions of this paper are summarized as follows:

• A new search paradigm combining symmetric group theory with metaheuristics is proposed for tackling an NP-hard problem;
• Both intrinsic and extrinsic characteristics of users are involved in the learning process as the objective function;
• The balanced relationship between diversification and intensification is maintained during the optimization procedure.

The remainder of this paper is organized as follows: Section 2 describes the materials and methodology of our proposed work; Section 3 presents the results generated from experiments on real-life datasets; Section 4 discusses the performance and analyzes the behavior of the proposed method; Section 5 concludes the paper.

2. Materials and Methodology

The proposed method fully uses group theory as the unifying mathematical framework to design an optimization pattern in the heuristic mode. Specifically, GT-PSO is composed of six blocks. Particle representation, landscape decomposition, moving neighborhood, and swarm topology are the main body of the method. Problem formulation and the objective function are the partial components referring to different application scenarios. Important concepts regarding the symmetric group in the group theory domain such as permutation, group action, conjugation, conjugacy class, cycle, cyclic form, orbit, and orbital plane are involved and play important roles in our proposed GT-PSO method. Please refer to [18,19,20] for related research and detail information.

2.1. Problem Formulation

The ACS problem can be formulated as the function that receives the tuple of users’ preferences $lp$, learning contents $lo$, and knowledge concepts $kc$ as model variables and returns a sequence $s\in S$, which represents the suitable approximation of the final solution of the model among multiple possible candidates in $S$.

$$f\left(lp,lo,kc\right)\to S$$

(1)

An example of ACS problem formulation is illustrated in Figure 1. As three levels of preferences $l{p}_1$, $l{p}_2$, and $l{p}_3$ are concerned, if users are interested in knowledge concepts $k{c}_1$, $k{c}_2$, $k{c}_5$, and $k{c}_6$ according to their own preferences, then the associated learning contents might be $l{o}_2$, $l{o}_7$, or l_o8, which generate one of the optimal sequences of a curriculum $s=(s_2,s_7,s_8)$ to cover the maximal targets with the minimal paths. Note that each user’s preference is represented by a level, each learning object is represented by a rectangle, and each knowledge concept is represented by a circle. The solid arrow connecting two concepts is the prerequisite relationship and the dash arrow stands for the dependent relationship between knowledge concepts and learning contents.

2.2. Particle Representation

The permutation is used as the representation of solutions encoded in particles of GT-PSO. The degree of a solution or a particle is equal to the dimension of the objective function or the problem. The degree of the ACS problem is the size of a possible curriculum path. The solution is represented by the mapping function in permutation form and can be seen as a potential candidate of the problem.

$$sol=\left(p\left(s_1\right) p\left(s_2\right) \cdots \right)\cdots \left(\cdots p\left(s_{n-1}\right) p\left(s_n\right)\right)$$

(2)

2.3. Landscape Decomposition

Four layers of hierarchical decomposition are adopted to describe the systematic view of the landscape of the solution space. The hierarchical partitions of different levels are conjugacy class, cyclic form, orbital plane, and orbit, respectively. The demonstration of the hierarchical decomposition of the solution space is shown in Figure 2. The layered partition strategy would lead to the complete and exclusive results of the original space.

Conjugacy class is the first-order partition of hierarchy, and it is made up of various cyclic factors, with the structure denoted by the factor form. Suppose the degree of the ACS problem is $n$.

$$cnj=1^{k_1}{2}^{k_2}\cdots i^{k_i}\cdots n^{k_n}, \forall i\in \left[1, n\right], k_i\in \left[0, n\right]$$

(3)

$${{\displaystyle \sum }}_{i=1}^{n}i{k}_i=n$$

(4)

Cyclic form is the second-order partition of hierarchy in terms of the permutation of cyclic factors, and it consists of sequential combinations denoted by the cyclic form.

$$cyc=\left(s_1 s_2 \cdots \right) \cdots \left(\cdots s_{n-1} s_n\right)$$

(5)

Orbital plane is the third-order partition of hierarchy, and it is the collection of elements if the results of a sequence set $S$ are the same under two group actions $g_1$ and $g_2$.

$$obp=\left\{S_k\right\}, iff S_i\left(g_1\right)=S_i\left(g_2\right), \forall i\in \left[1,k\right]$$

(6)

Orbit is the fourth-order partition of hierarchy, and it is the collection of all elements by group action $g$ from a given group $G$ of a sequence set $S$.

$$obt=\left\{gs|g\in G\right\}, \forall s\in S$$

(7)

2.4. Moving Neighborhood

The moving neighborhood operations of a specific particle are manipulated under the guidance of different combinations of group actions to search the incumbent solutions. Four types of operations are designed based on four-layered hierarchical partitions. The illustration of neighborhood movement details with respect to particle update is illustrated in Figure 3.

The blue arrow is the operation of a conjugator that jumps over different conjugacy classes, and the red arrow is the operation of a cycler that moves across different cyclic forms within the same conjugacy class. These two operations belong to the stage of explorations during the optimization, and their goal is to enhance the ability of diversification.

The green arrow is the operation of a planer that changes from one orbital plane to another within the same cyclic form, and the yellow arrow is the operation of an orbiter that searches along sequential orbits within the orbital plane. These two operations belong to the stage of exploitations during the optimization, and their goal is to emphasize the capacity of intensification.

2.5. Swarm Topology

After the update of a single particle, the swarm topology for updating all particles is taken into account. Let $\nabla$ denote the operation generated by combinations of group actions through group multiplication $\otimes$. The evolutionary formulae of the whole swarm are deduced from the above.

$$\nabla =\left(r_1{\nabla }_{cnj}\right)\otimes \left(r_2{\nabla }_{cyc}\right)\otimes \left(r_3{\nabla }_{obp}\right)\otimes \left(r_4{\nabla }_{obt}\right)$$

(8)

$$v_{k+1}^i=\nabla \otimes v_k^i+r_5{c}_1\left(p_k^i-s_k^i\right)+r_6{c}_2\left(p_k^g-s_k^i\right)$$

(9)

$$s_{k+1}^i=s_k^i+v_{k+1}^i$$

(10)

where $r$ is the random number, $c$ is the acceleration coefficient, $k$ is the iteration number, $i$ is the particle index, and $p_k^i$ is the local best value of $i^{th}$ particle’s objective function after $k$ iterations, while $p_k^g$ is the global best value. Equation (8) shows that operations from different hierarchical layers would cooperate concurrently with each other and update the velocity of a particle simultaneously, which can also be observed in Figure 4. The new velocity in Equation (9) of a sequence is calculated via three terms. They are the inertia weight of group operations, the cognitive coefficient of local best attraction, and the social coefficient of global best guidance. Thereby, the final position of a sequence is added in Equation (10). The search stops when certain conditions are satisfied, for example, the maximal number of iterations is counted or the minimal error rate is reached. The flowchart of GT-PSO is shown in Figure 4.

2.6. Objective Function

Regarding the complexity of learning materials in the education system, it may not be adequate to model the ACS problem using only one single objective, and applying multi-objective optimization technology is important to defining ACS objectives. Therefore, the fitness function depicted here is the fusion of three objectives, with their sum as shown in Equation (11),

$$f\left(\mathit{s}\right)=\min {\displaystyle \sum }_{i=1}^3{f}_i\left(\mathit{s}\right)$$

(11)

where those objectives are concept coverage, preference adequacy, and estimated curriculum period.

$$f_1\left(\mathit{s}\right)=\left(1-\theta \right)\left(\left|R\left(\mathit{s}\right)\right|-\left|R\left(\mathit{s}\right){{\displaystyle \cap }}^{ }O\right|\right)-\theta \left(\left|O\right|-\left|R\left(\mathit{s}\right){{\displaystyle \cap }}^{ }O\right|\right)$$

(12)

We have Equation (12) for objective 1 where $\theta \in \left[0,1\right]$, $R\left(\mathit{s}\right)$ is the measure of concept percent in the sequence $\mathit{s}$, and $O$ is the set of learning contents. The objective function $f_1$ measures the concept coverage in one sequence by learning contents.

$$f_2\left(\mathit{s}\right)=\left(1-\frac{{{\displaystyle \sum }}_{j=i}^n\left|R\left(s_i\right){{\displaystyle \cap }}^{ }O\right|}{{{\displaystyle \sum }}_{j=1}^n\left|R\left(s_i\right){{\displaystyle \cap }}^{ }O\right|}\right){\displaystyle \sum }_{i=1}^n{s}_i$$

(13)

We have Equation (13) for objective 2 where $s_i$ is each element in sequence $\mathit{s}$. The objective function $f_2$ checks the average adequacy of users’ preferences included in the concept coverage.

$$f_3\left(\mathit{s}\right)=\max \left(\overline{T}-{\displaystyle \sum }_{i=1}^n{T}_i{s}_i,0\right)+\max \left({\displaystyle \sum }_{i=1}^n{T}_i{s}_i-\underset{\_}{T},0\right)$$

(14)

We have Equation (14) for objective 3 where T_i is the expected time for learning the element $s_i$ in the curriculum sequence, and $\overline{T}$ and $\overline{T}$ are the upper and lower limits of learning time, respectively. The objective function $f_3$ evaluates the period between the upper and lower limits of total time.

We select the Open University Learning Analytics Dataset (OULAD) [21], a real-life dataset, as the experimental material to test the effectiveness of our proposed GT-PSO method. OULAD is a part of the Open University Online Learning Platform (OUOLP) and contains diverse learning materials such as assessment marks, forum discussions, course contents, demographics, and interactions of college students under the Virtual Learning Environment (VLE). Computational experiments are conducted on OULAD with a variety of learning materials. Comparative experiments are performed among our proposed method and other conventional metaheuristic algorithms including GA [11], ACO [9], PSO [13], DE [16], and the state-of-the-art metaheuristic algorithms including the Adaptive Tunicate Swarm Algorithm (ATSA) [22] and the Improved Chimp Optimization Algorithm (IChOA) [23]. In order to determine the statistical significance of the meaningful comparisons between the proposed and alternative methods, the pairwise non-parametric Wilcoxon’s rank sum test is implemented between them. In this case, the results of 30 independent runs of each comparative method are considered for quality evaluations for solutions of those algorithms.

The parameter configuration for the compared methods is presented in

Table 1. The parameter configuration for the compared methods.

Method	Parameter	Value
GA	Population size	30
	Crossover rate	0.5
	Mutation rate	0.2
ACO	Population size	30
	Pheromone factor	1.0
	Heuristic factor	1.5
	Evaporation rate	0.3
PSO	Population size	30
	Inertial weight	0.8
	Acceleration coefficient	2.0
DE	Population size	30
	Crossover rate	0.5
	Low bound	0.3
	High bound	0.6
ATSA	Population size	30
	Initial speed	1
	Subordinate speed	4
IChOA	Population size	30
	Chaotic rate	0.5
	Turbulence factor 1	1.5
	Turbulence factor 2	0.3
GT-PSO	Population size	30
	Random 1 and 2	0.4
	Random 3 and 4	0.5
	Random 5 and 6	0.3
	Acceleration coefficient	2.0

. The selection of these appropriate parameters is an empirical study from a wide range of research works. For GA and DE, they are all based on an evolutionary process, so a proper mutation rate is needed to ensure diversification. Meanwhile, a good crossover rate is required to generate offspring to keep the intensification. PSO and GT-PSO are both based on swarm intelligence. Thus, the coefficients of acceleration are set rather high to trace the local best and global optimum, and inertial weight is set normal to keep the search behavior of the particle itself. The parameter configuration of ACO is conventional as usual and the situation is also the same for ATSA when compared to its original version of TSA. The difference between two turbulence factors of IChOA should be big enough to maintain the balance of both diversification and intensification.

3. Results

In the case study of the real-life application of OULAD, the fitness scores of objective function and convergence of the compared algorithms are the main factors to measure the performance. The number of learning materials varies from 50 to 1000, which means that the difficulty of finding the optimal curriculum path covering users’ preferences, learning contents, and knowledge concepts increases severely. The quantitative results of fitness scores returned by the fused objective functions of multiple metaheuristics are tabulated in

Table 2. The fitness scores of the compared methods with different dataset sizes and evaluation metrics of mean, standard deviation, best value, and p-value.

Number of Materials	Metric	GA	ACO	PSO	DE	ATSA	IChOA	GT-PSO
50	Mean	9.642	9.607	9.424	9.285	9.280	9.323	9.266
	Std	0.317	0.325	0.217	0.111	0.253	0.186	0.128
	Best	9.307	9.254	9.241	9.042	8.904	9.006	8.953
	p-value	0.000	0.000	0.000	0.006	0.024	0.017	/
200	Mean	16.750	16.608	16.686	16.418	16.378	16.403	16.214
	Std	0.371	0.377	0.248	0.166	0.305	0.234	0.174
	Best	16.480	16.276	16.308	16.055	15.873	15.947	15.847
	p-value	0.000	0.000	0.001	0.016	0.037	0.021	/
500	Mean	27.354	27.151	27.141	27.032	26.875	27.068	26.907
	Std	0.435	0.418	0.277	0.206	0.326	0.364	0.304
	Best	27.133	26.937	26.829	26.550	26.106	26.248	26.218
	p-value	0.001	0.000	0.001	0.009	1.000	0.018	/
1000	Mean	65.138	64.176	63.778	63.411	63.399	63.514	62.805
	Std	0.462	0.459	0.396	0.327	0.338	0.406	0.473
	Best	64.392	63.863	63.414	62.758	62.478	62.303	61.882
	p-value	0.001	0.001	0.001	0.019	0.033	0.028	/
	+/=/-	4/0/0	4/0/0	4/0/0	4/0/0	3/0/1	4/0/0	4/0/0

with respect to corresponding numbers of learning materials. Best results are in bold for emphasis. In most situations, all methods can obtain good solutions and it can be inferred that the proposed GT-PSO achieves the best averaged fitness score among them. For instance, GT-PSO generates the superior averaged fitness score of 9.266 on OULAD with 50 learning materials, which is almost 0.4 higher than that of GA with its fitness score of 9.642. ATSA is the second best of all, followed by IChOA and DE. The fitness score of PSO occupies the position almost equivalent to ACO because there is a slight variation between them, and the worst results are produced by GA. The fitness scores of metaheuristics become higher with the incremental extension of material size, and the trend underlying this seems to be proportional.

In addition to mean value of the fitness score, the standard deviation and best value of the fitness score and p-value of the hypothesis test are shown in Table 2 to evaluate solution qualities as well. The significance level $\alpha =0.05$, and if the p-value of the given method is less than that, then there is indeed a significant difference between those two comparative algorithms, and vice versa. It can be revealed from Table 2 that the proposed GT-PSO provides the best performance among all except for one case, i.e., the pairwise comparison between GT-PSO and ATSA under the circumstance of 500 learning materials in a path of the curriculum sequence. Its p-value is much greater than 0.05 and there exists no significant difference between them, and GT-PSO has inferior performance to ATSA, which is further evidenced by the statistical results of mean and best values of the fitness score. Please also note that the standard deviation of GT-PSO fitness score diverges as the number of learning materials increases. In spite of this, DE shows a relatively converged standard deviation regardless of material numbers, while GA acts the opposite of DE.

Figure 5 displays the visualizations of fitness scores divided by different numbers of learning materials. Result curves of the compared methods are represented in different colors. The total number of iterations is set to be 1000 since nearly all methods would converge after 800 iterations in the later stage of the search procedure. From all subfigures in Figure 5, we can observe that searching difficulty also rises while the number of learning materials increases, which is reflected in the convergence speed of the compared algorithms as well. In Figure 5a of 50 materials, almost all methods converge at the iterations less than 300 except for DE and the convergence situation happens around 400 iterations in Figure 5b of 200 materials except for DE and IChOA. In Figure 5c of 500 materials, the number of iterations is near 400 to 500 and finally in Figure 5d of 1000 materials it is around 800. From Figure 5a,b it can be observed that the convergence speed of ACO is relatively slow, on the contrary, the trend of GT-PSO decreases rapidly, and the other five algorithms have intermediate results. However, Figure 5c,d indicate that GT-PSO has relatively lower convergence, and DE and ATSA show the relatively fair performance.

4. Discussion

The experimental results show that, in general, all selected algorithms tested in the experiment could handle the ACS problem with the feature of NP hardness using their own search strategies. Basically, the population-based metaheuristic can be regarded as the trial-and-error method of multi-agent collaborative search. On the other hand, it contains the approximation approach of random technology. It is worth mentioning that GT-PSO is built based on the mathematical foundation of symmetric group theory and its search paradigm is classified into the categories of diversification and intensification, and the adaptive control of the balanced relationship between these two is maintained by parameter configuration and hierarchical operation. When the dataset size is relatively small, GT-PSO focuses on the stage of intensification (operations of planer and orbiter) and the corresponding convergence rate is fast, but stagnation at the local optimum is prevented by the balance of diversification (operations of conjugator and cycler), which is proven by the results in Figure 5a,b. However, from Figure 5c to Figure 5d, as the dataset size grows, diversification (operations of conjugator and cycler) dominates the early search to provide more possible candidates and the intensification (operations of planer and orbiter) guarantees that GT-PSO would finally converge in the late search. Thus, the implication of adaptive control of GT-PSO convergence is interpreted and summarized.

For the rest of alternative methods in Figure 5, the performance curve of PSO (in black) is similar to GT-PSO (in red) because GT-PSO is an extended and refined version of PSO, so naturally our proposed method outperforms the original PSO. GA (brown curve) suffers from the drawback of early maturing performance, which is stagnation at the local optimum and inadequate solution. The reason behind this phenomenon is probably the complex procedure of selection, crossover, mutation, and termination, with enormous parameters in each step. Sometimes, ACO (green curve) has a slower convergence rate than others and this may be caused by the positive feedback of accumulated pheromones on the particular paths, and there are few escape strategies for ants to create a potential new path to the solution, which further causes bad solutions from the local optimum. The performance of DE (blue curve) is very stable and its results are fairly decent due to the foundation of differential evolution approach and the adaptive dynamic turbulence strategy for stagnation at the local optimum. ATSA (purple curve) performs with a rapid convergence rate because of two main adaptive phases at each iteration: the randomly selected tunicate for searching all the space and the best tunicate for improving the search navigation. They cooperate with each other adaptively to boost the search quality. IChOA (yellow curve) has the slowest convergence rate due to non-deterministic turbulence factors caused by the initialization of a Halton list and enhanced non-linear convergence operators. Its performance becomes worse when the complexity is higher.

The time complexity of GT-PSO for the ACS problem is discussed as follows. Assume that $n$ is the number of learning materials in a curriculum sequence, $cof$ is the evaluating cost of the objective function, $p$ is the size of the swarm population, and $k$ is the number of iterations. According to Equations (12) to (14), $co{f}_i=O\left(n\right)$, since each one calculates the sum of every element in the sequence, so its time complexity is linear to $O\left(n\right)$, and thus $cof=co{f}_1+co{f}_2+co{f}_3=O\left(n\right)$. In the initialization of GT-PSO in Figure 4, the time complexity is $T_{ini}+T_{eva}+T_{set}=O\left[p\ast \left(n+cof\right)\right]$. In each iteration, the time complexity of each operation is $T_{opt}=T_{cnj}=T_{cyc}=T_{obp}=T_{obt}=O\left(p\ast n\right)$ and the time complexity of one single iteration is $T_{opt}+T_{eva}+T_{upd}=O\left[p\ast \left(n+cof\right)\right]$. Finally, the time complexity of GT-PSO is $O\left[k\ast p\ast \left(n+cof\right)\right]=O\left(k\ast p\ast n\right)\le O\left(n^3\right)$, and it belongs to polynomial complexity and is acceptable as the complexity of the solutions to NP-hard problems.

The space complexity of GT-PSO for the ACS problem is discussed as follows. Similarly, the space complexity of the objective function is $O\left(n\right)$. In the initialization of GT-PSO, the space complexity is $S_{ini}+S_{eva}+S_{set}=O\left(p\ast n+p\ast n+1\right)=O\left(p\ast n\right)$, and in each iteration the space complexity of each operation is $S_{opt}=S_{cnj}=S_{cyc}=S_{obp}=S_{obt}=O\left(p\ast n\right)$. No extra space is needed for the rest of the search, so the time complexity of GT-PSO is $O\left(p\ast n\right)\le O\left(n^2\right)$. It is less complex than the time cost and in practice the storage space is always sufficient. Therefore, the time complexity is the main concern of the GT-PSO algorithm.

Despite the advantages achieved by GT-PSO, there are still some limitations of the existing algorithm. Firstly, the global optimum is not guaranteed to be reached because of the inherent randomness of all metaheuristics. Secondly, it can only deal with the combinatorial problem of discrete optimization. Thirdly, it may cost huge computational time when the objective function is extremely complex.

5. Conclusions

The Adaptive Curriculum Sequencing (ACS) problem remains a continuous challenge and a longstanding issue to be addressed in the field of education management system research. We proposed the novel metaheuristic algorithm named Group-Theoretic Particle Swarm Optimization (GT-PSO) based on the solid mathematical foundation of the theory of symmetric groups to provide a new search paradigm. It consists of four systematic parts: particle representation, landscape decomposition, moving neighborhood, and swarm topology. The ACS problem was formulated and the fused objective function was designed correspondingly. Our proposed method was tested by using a real-life dataset and was compared with other classic, effective optimization algorithms in experiments, and the results show that GT-PSO outperforms all other selected conventional metaheuristic methods, which proves that GT-PSO has good potential for further research and development as well as value of related theoretical studies and real-life applications.

The future works may include but are not limited to the following tasks. A further investigation could be conducted of mapping functions to transfer discrete and continuous representations so that the continuous optimization problems can be solved by GT-PSO. The balanced relationship between diversification and intensification could be enhanced by the incremental strategy of parameter configuration. As a step forward, the theories of Lie group and fiber bundles can be applied to enrich the mathematical foundations of metaheuristic frameworks. The improved GT-PSO would be utilized to address multi-objective optimization problems.