Improved teaching–learning-based optimization algorithm with group learning

Abstract

The typical teaching–learning-based optimization (TLBO) algorithm tends to devolve into local optimization and suffers from the rapid loss of population diversity. In this study, an improved TLBO algorithm with group learning (GTLBO) is established to solve these problems. In the proposed algorithm, a class is divided into several groups. The individual with the highest level is selected as the teacher for each group. Then, the teacher implements the TLBO algorithm in each group. This strategy of group learning can maximize the time before the students reach the teacher’s level and effectively ensure population diversity. Given an effectively diverse population, the idea of reversing the beginning and ending is introduced to boost the convergence rate of the algorithm. Moreover, a matrix displacement method is provided to solve the premature termination phenomenon of the algorithm. Finally, the performance of the GTLBO is investigated across six complex high-dimensional benchmark functions. Results obtained through experiments show that the GTLBO conduces enhanced performance in solving problems of multimodal function optimization. The convergence speeds and solution accuracy of the proposed algorithm are significantly improved compared with those of the typical TLBO algorithm.

Keywords

Teaching–learning-based optimization group learning complex high-dimensional optimization problem

1 Introduction

In current industrial applications and research fields, multi-constraint and multi-objective optimization problems become increasingly difficult and complex [4 , 7]. Traditional techniques (e.g., linear programming, dynamic programming, and steepest descent) often fail to solve these problems, particularly those with nonlinear objective functions [21]. In recent years, several novel swarm intelligence optimization algorithms perform effectively in dealing with these problems. These algorithms have become the focus of increasing attention in the field of optimization algorithm research.

Many researchers have attempted to design new algorithms, modify existing algorithms, or hybridize different methods to enhance the performance of these algorithms [3 , 25]. These improved algorithms reduce calculation numbers and running time and obtain more precise results than that of previous algorithms. Setting or tuning different parameters properly for existing algorithms is a time-consuming work requiring experience, which is a major barrier encountered by researchers. To a certain degree, optimization can mitigate the difficulties, but the result is not completely satisfying. Recently, the difficulties have been resolved by an effective and efficient optimization algorithm called teaching–learning-based optimization (TLBO) developed by Rao et al. [19]. The TLBO has been widely used in industry production and daily life [13 , 23]. The TLBO uses simple mathematical operations to achieve the combined aim of reduced computational effort with easy implementation. Most importantly, the TLBO requires no extra parameters [10, 24]. However, the TLBO is discovered to have low efficiency in local extreme searching; this optimization algorithm may be premature. Given this problem, the originator improved the algorithm. In [18] an elitist TLBO (ETLBO) is proposed. The convergence ability of the ETLBO is increased by retaining the optimal N individuals in the next generation. The ETLBO algorithm improves the convergence speed of the original algorithm [16], but the solution accuracy is low when solving high-dimensional complex issues. The early loss of population leads to a rapid loss of population diversity.

In this study, the TLBO algorithm with group learning (GTLBO) is proposed to improve the original version of TLBO. In the GTLBO, students in a class are firstly divided into several groups by matrix, in which the group leaders are the highest-rated students in the class and they lead their group to explore for an optimal solution. We select the highest-ranked student in each group whose rank is preserved as an optimal value. As a teacher, the group leader teaches other students in the group. Each member in the group can learn from all other members. After finishing this step, the rank of the worst-performing student in the group is replaced by the preserved optimal value. When the level of the class reaches a balanced situation, matrix permutation is used for regrouping. The aforementioned process will be performed repeatedly until the termination condition is satisfied. The results show that the convergence speeds and solution accuracy of the GTLBO are significantly improved compared with those of the original TLBO.

This paper is structured as follows: Section 2 presents the related work. Section 3 gives the improved algorithm. Section 4 describes the results obtained and Section 5 concludes this paper.

2 State of the art

The swarm optimization algorithm is an optimal calculation technique based on the dynamics of the population. Social animals (e.g., birds, fish, and ants) can finish extremely complicated work through mutual cooperation [1]. Researchers are inspired to develop this optimization algorithm. Studies have demonstrated that these group systems have good robustness and high-efficiency performance in addressing the aforementioned problems. The swarm optimization algorithm is related only to fundamental math operations. Only the objective optimal solution is needed; the gradient information is not required in the calculation. Compared with the traditional optimization algorithm, the swarm optimization algorithm has three advantages, that is, flexibility, robustness, and self-organization. The swarm optimization algorithm is insensitive to environmental factors. In addition, if one of individuals fails in the calculation, the rest of the individuals can continue to complete the entire calculation process. Moreover, this algorithm is free of regulation.

In recent years, many excellent optimization algorithms have been proposed. The well-known algorithms are genetic algorithm (GA) [1, 22], which is inspired by the behavior of bees finding honey and is one of the most commonly used evolutionary optimization techniques [2, 14]; particle swarm optimization [11], which is inspired by artificial life and evolutionary computation; artificial bee colony [1], which is inspired by the behavior of bees finding honey; ant colony optimization [12], which is enlightened by the action in which ants search for food; grenade explosion method [18]; and harmony search [9]. All of them are based on the principles of different natural phenomena. They have been applied to many engineering optimization problems and are proved to be effective in solving specific types of problems [5]. The aforementioned optimization algorithms are all population based and derivative free. However, none of these algorithms can effectively handle problems with multimodality, large scale, or discontinuity of objective function.

The TLBO is proposed to obtain global solutions for continuous nonlinear functions with reduced computational effort and high consistency. The TLBO algorithm is based on the effect of the influence of a teacher on the output of learners in a class. The teacher is the top learner in her class who communicates her knowledge to other students. The competence of the teacher’s teaching influences the students’ outcome. Evidently, learners with a good teacher can have enhanced results in terms of their marks or grades. As a swarm intelligence-based global optimization algorithm, TLBO is suitable for solving combinatorial optimization problems. The TLBO has the advantages of simple code, fast calculating speed, good population diversity, and simple and easy implementation. The TLBO has already been tested in many applications and shown better performance [8, 17].

In a prior study of TLBO, the author only considered two common controlling parameters: number of generations and population size. The conception of “elitism” had yet to be considered. Later the author brought the conception of “elitism” into TLBO, resulting in the ETLBO [18]. Before each execution, the best optimal N individuals in the current population are selected as the elite solution. After optimization, the worst optimal N individuals are replaced by the elite solution. The ETLBO can improve the convergence ability of the algorithm to retain the best adaptation value of the individuals to the next generation. However, the ETLBO has its shortcomings and drawbacks especially when solving a multi-constrained problem.

3 Methodology

3.1 Basic concepts of TLBO

The TLBO algorithm simulates the teaching–learning phenomenon of a classroom to solve multidimensional, linear, and nonlinear problems with good efficiency [20]. The various subjects studied by a student are analogous to the different design variables of the optimization problem, namely, a student learns from his teacher and classmates. The TLBO process involves two phases, namely, the teacher phase and the student phase. A teacher shares his/her knowledge to improve the students’ grades, and the mutual interaction between students is called collaborative learning. In TLBO, a class of students is called a population and different subjects offered to students are considered discrete design variables of the optimization problem. Meanwhile, a student’s output is considered a measure of a solution’s fitness to a problem, and the best solution in the population is regarded as the teacher.

This study assumes that the number of students and subjects are denoted as M and N, respectively, and lets x_i = (x_i,1 x_i,2 … x_i,n) ^T, i = 1, 2, …, m be the ith student, where N is the number of subjects (or dimensions of the problem) that the student has taken. M_j denotes the mean output of the students in the subject. In addition, x_best = (x_best,1, x_best,2, …, x_best,n) ^T denotes the set of best solutions of the students. A teacher is defined as a highly learned person who trains students such that they can have better outcomes. In this study, the best student is selected as the teacher. The whole teaching and learning processes are described as follows:

3.1.1 Teacher phase

A teacher strives to enhance the students’ achievement. However, the effort cannot reach the optimal result where the students reach the level of their teacher, due to the limitations of the individuals and the class as a whole. Mathematically, the mean level of the subjects’ learning is expressed as: $M (t) = (M_{1} (t) M_{2} (t) \dots M_{n} {(t)}^{T}),$ (1) $M_{j} (t) = \frac{1}{m} \sum_{i = 1}^{m} x_{i, 1} (t), j = 1, 2, \dots, n,$ (2) where t denotes the generation counter. The difference between the teacher’s level and the mean level is defined as follows: $d x_{j} (t) = r_{j} \cdot (x_{best} (t) - TF \cdot M_{j} (t)) .$ (3)

Index j denotes the subject being learned and TF is a number, either 1 or 2, calculated as follows: $TF = round (1 + rnd (1)),$ (4) where rnd is a random number in (0, 1). TF is randomly selected by the algorithm according to Equation (4). TF reflects the random learning process. Then, the new knowledge value of a student is updated as follows: $x_{i, j}^{'} (t) = x_{i, j} (t) + {dx}_{j} (t) .$ (5)

Without loss of generality, generation counter t will be skipped for clarity in the following discussion.

3.1.2 Student phase

In addition to learning from the teacher, students increase their knowledge through interaction and cooperation with their classmates in the form of group discussion, presentations, and formal communication. In these activities, a student can learn new information from other students who have more knowledge. Based on this process, if student p performs better than student i, then the difference in learning is conceptually expressed in Equation (6), which is expressed as follows: $x_{i, j}^{″} (t) = x_{i, j}^{'} (t) + r \cdot (x_{p, j}^{'} (t) - x_{i, j}^{'} (t)) .$ (6)

Otherwise, $x_{i, j}^{″} (t) = x_{i, j}^{'} (t) + r \cdot (x_{i, j}^{'} (t) - x_{p, j}^{'} (t)),$ (7) where p is a randomly selected student, $x_{p}^{'}$ denotes the student’s new value after the teacher phase, and r_j is a random number in (1, 0). Finally, if the function value or fitness of $x_{i}^{″}$ is better than that of $x_{i}^{'}$ , then replace $x_{i}^{'}$ with $x_{i}^{″}$ .

3.2 Definition

By analyzing TLBO, we find that students’ achievement levels soon become close to their teacher’s achievement level, such that population diversity is likely to be limited. The previously presented simulation indicates that our primary basis, that is, TLBO, tends to fall into the local search when dealing with multidimensional problems, resulting in a local optimal solution rather than a global optimal solution. In this study, we propose a new teaching method using group teaching by dividing a class into several groups and appointing the group’s best student as the teacher. Then, we run the teaching and learning program in each group. This type of grouping does not allow students to select the incorrect student as the best of the class, thereby ensuring population diversity. In addition, we have considered the problem of premature termination of the algorithm. We suggest the use of matrix permutation to reduce the chance of premature termination. If the same result appears thrice in a row, then the grouping should be adjusted by matrixpermutation.

To solve the local search and premature termination problems, we have improved the algorithm of TLBO in the following subsections.

3.2.1 Group teaching

In the TLBO algorithm, students will approach their teacher’s knowledge rapidly, leading to the loss of population diversity. For the grouping, we divide the class into the appropriate number of groups and represent them with a matrix of dimension equal to the number of groups. Each row of the matrix is a group. Then, the teaching and learning operations of TLBO are implemented in each group. In the learning process, group members’ achievement levels are close to the best of the group rather than the best of the class. Hence, population diversity does not decrease rapidly.

3.2.2 Regrouping of matrix permutation

Our method regards the entire class as a one-dimensional matrix. Then, we transform the one-dimensional matrix into a matrix of dimension equal to the number of groups. If the same result appears thrice in a row, then the algorithm tends to terminate prematurely. When this situation occurs, the groups should be adjusted by matrix permutation.

Suppose the class has nine students denoted A, B, C, D, E, F, G, H, and I. They are regarded as a collection in a one-dimensional matrix. We divide them into three groups, as follows: $Students = [A B C D E F G H I],$ (8) $Stu = reshape (Students, groups, scale),$ (9) $Scale = select \cdot Classsize / groups,$ (10) $Stu = [\begin{matrix} A & B & C \\ D & E & F \\ G & H & I \end{matrix}] .$ (11)

From the matrix, we can determine the following groupings:

A, B and C are the first group.

D, E and F are the second group.

G, H and I are the third group.

When the grouping must be adjusted, we use the matrix permutation, as follows: $Stu = {Stu}^{'} .$ (12)

Then, the matrix becomes: $Stu = [\begin{matrix} A & D & G \\ B & E & H \\ C & F & I \end{matrix}] .$ (13)

In the new matrix, we can observe the following changes in the groupings:

A, D and G become the first group.

B, E and H become the second group.

C, F and I become the third group.

3.2.3 Replacing the last with the first

In the ETLBO algorithm, the top EN students are selected before starting. After the implementation of the teaching and learning programs, the lowest EN students will be replaced by the top EN students. When investigating the application of ETLBO, we first directly replaced the lowest EN students by the top EN students. Although the convergence speed of the algorithm was increased, the algorithm had the local optimum problem. If we do not save the top students initially, then, after the teaching and learning programs, the new top EN students differ from the original top EN students. (The new top EN students are better than the original top EN students). Then, the degree of similarity increases and the population diversity decreases. The algorithm can easily fall into the local optimum.

We have also added a simplification step to ETLBO. This step modifies a result where two students have the same score. However, after the change, their grades remain similar, which leads to a decline in population diversity. The key point is to limit the number of elites to maintain the population diversity while increasing the convergence speed of the algorithm.

3.3 GTLBO procedure

Step 1: Define the optimized parameter.

Step 2: Initialize the population.

Step 3: Group the population (set the appropriate number of groups).

Step 4: Regroup the matrix by permutation, if the same result appears thrice in a row.

Step 5: Appoint the best student as the teacher. Then, run the teaching and learning programs in each group. Subsequently, update the student’s results.

Step 6: Run the group learning program (the learning process from each other). Student i can choose j by random selection for the differentiated instruction. Then, update the student’s results.

Step 7: Replace the worst students with the best students.

Step 8: If the result satisfies the termination condition (i.e., maximum iterations), then end the procedure; otherwise, return to Step 3.

4 Result analysis and discussion

To evaluate the performance of the GTLBO, we select the following six high-dimensional complex benchmark functions:

F1: Shifted Sphere Function;

F2: Shifted Schwefel Function;

F3: Shifted Rosenbrock Function;

F4: Shifted Rastrigin Function;

F5: Shifted Griewank Function;

F6: Shifted Ackley Function.

We set the population size to 16, the number of variables to 100, the maximum number of cycles to 5,000, and the number of elites to 4 for maintaining consistency in the comparison. We conducted 30 independent tests on the TLBO algorithm and the proposed algorithm in this study. Table 1 shows the simulation results, including the optimal solution (best), the worst solution (worst), and the average (mean).

For functions F1, F2, F3, F4, F5, and F6 with population size of 16 and iterations of 5,000, the GTLBO produces the best result among the strategies. This finding demonstrates that the GTLBO shows excellent performance in complex function optimization. In the following, Figs. 1 6 illustrate the convergence curves for certain complex functions. Specially, we show the convergence speed of some functions when the number of iterations is from 1 to 3,000.

From the experiment results of all the functions, we can see that the convergence speed of the GTLBO tends to be faster than that of the original TLBO and ETLBO. Moreover, GTLBO also exhibits higher accuracy than the TLBO and ETLBO algorithms. In TLBO, only a single teacher exists. Thus, the students’ levels increase to the teacher’s level rapidly. This situation leads to the loss of population diversity and the algorithm quickly devolves into the local search. In ETLBO, the replacement by elites also causes a loss of population diversity. By contrast, the GTLBO significantly preserves population diversity.

5 Conclusion

This paper proposes the GTLBO for improving the performance of the original TLBO algorithm to provide an enhanced solution to multidimensional complex problems of high dimension. The class is divided into several groups and is represented via a matrix of dimension equal to the number of groups. Subsequently, the TLBO algorithm is executed in each group. Matrix permutation is performed when a certain group falls into the local search. The result of the tests shows that the GTLBO has higher accuracy and quicker solution speed than the TLBO algorithm. Particularly, the searching speed of the GTLBO is fast and does not easily fall into the local optimum. Moreover, the GTLBO can obtain improved performance, scalability, and convergence, leading to a global optimal policy. The proposed algorithm can effectively shorten calculation time and improve efficiency and accuracy. Further research is warranted to achieve high precision and processing efficiency.

Footnotes

Acknowledgments

This work was supported in part by the Department of Science and Technology Project of Jiangsu Province under Grant No. BY2014028-09 and the National Natural Science Foundation of China under Grant No. 51404258.

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