Improved teaching-learning-based optimization algorithm based on fusion difference mutation 1

Abstract

Teaching-learning-based optimization algorithm (TLBO) is a swarm intelligence optimization algorithm that simulates classroom teaching phenomenon. In order to solve the problem that TLBO algorithm is easy to fall into local optimum and has poor stability, an improved teaching-learning-based optimization algorithm based on fusion difference mutation (IDMTLBO) is proposed. Firstly, adaptive teaching factors are introduced. Secondly, in the teaching stage, each student studies according to the gap between himself and the teacher, which improves the convergence speed and convergence accuracy of the algorithm. Finally, in the learning stage, students are divided into two levels according to their learning level, and two students are randomly selected to improve the iterative equation in the learning stage with the difference mutation strategy, It improves the disadvantage that the algorithm is easy to fall into local optimum. Numerical experiments show that the convergence speed and convergence accuracy of the algorithm are obviously better than TLBO algorithm, DMTLBO algorithm, DSTLBO algorithm.

Keywords

Teaching-learning-based optimization adaptive teaching factors the improved teaching stage learning stages differential mutation

1 Introduction

Teaching-learning-based optimization (TLBO) belongs to swarm intelligence algorithm [1, 2]. It was proposed by Indian scholar Rao in 2011. The principle of the algorithm is to simulate the teaching phenomenon in the class, which is divided into teaching stage and learning stage. Through the joint action of these two stages, the purpose of improving students is achieved. Compared with Genetic algorithm [3], Particle Swarm Optimization algorithm (PSO) [4], Ant Colony Optimization (ACO) [5], TLBO algorithm has a simple structure, is easy to implement, and has few parameters. It has been successfully applied to many practical optimization problems, such as job shop scheduling problem [6], distribution network reactive power optimization problem [7], and inbound aircraft scheduling and deployment problem [8].

TLBO algorithm has received extensive attention since it was proposed. In order to improve the performance of TLBO algorithm, researchers put forward improvements to the TLBO algorithm in [9 –20]. Matej crepinsek studied in [9] whether TLBO is a new star of meta heuristic algorithm, and whether the theory or practice supporting its dominant position is effective. It is proved that although TLBO performs well on unconstrained problems, it has a low success rate for some optimization problems, revealing some shortcomings of TLBO compared with other meta heuristic algorithms. Li Huirong proposed an adaptive teaching factors in [11], improved the iterative equation in the learning stage by using the mutation strategy in differential evolution algorithm, ensuring the diversity of the population. Bi Xiaojun fused mutation strategy of the differential evolution algorithm in learning stage [12]. A hybrid learning strategy is proposed, and added a perturbation strategy in the later stage of the algorithm to ensure the global optimality of the algorithm. Zhai Junchang proposed an improved TLBO algorithm to solve the global optimization problem in view of the disadvantage of premature convergence of TLBO [13]. In the learning phase, according to the adaptability of two learners, a poor student randomly selects an excellent student to learn, or an excellent student learns from the teacher, which improves the performance of the TLBO algorithm. Ma Yunpeng introduced a new population mechanism into the traditional teaching optimization algorithm in [14]. Chen D adopted the variable population size (VTTLBO) in the form of a triangle, which reduced the calculation cost of the original TLBO and extended it to optimize the parameters of the artificial neural network (ANN) [16]. Tuo, SH proposed a hybrid search algorithm, HSTLBO, which combines harmony search (HS) and teaching-learning-based optimization (TLBO), and uses adaptive selection strategy to solve complex optimization problems cooperatively in [17]. Huang Jida proposed an effective hybrid algorithm for continuous optimization problems, expanded the scope of application, modified it, and applied it to constrained optimization problems in [18]. Yu K combined feedback mechanism and differential evolution (DE) in [19], and introduced mutation crossover operation of DE to increase population diversity and prevent premature convergence. The hybrid perturbation mechanism is used to ensure that the algorithm can get rid of the local optimum. Liu Yin improved the performance of basic TLBO by using adaptive teaching factors and learning process integrating differential evolution in [20]. These improvements are roughly divided into two categories, namely algorithm performance and algorithm application improvements. However, these improved TLBO algorithms have slow convergence speed and are easy to fall into local optimum in the late iteration, which leads to premature convergence of the algorithm.

To solve the above problems, this paper proposes an improved teaching-learning-based optimization algorithm based on fusion difference mutation (IDMTLBO). First, the teaching factor T of the basic algorithm is improved, and a linear decreasing adaptive teaching factor is proposed to enhance the local search ability of the algorithm. Secondly, in the teaching stage, it is proposed to learn through the difference between students’ own level and teachers, which improves the accuracy and convergence speed of the algorithm. Finally, differential mutation strategy is applied in the learning stage to make the results approach the optimal solution. Students are divided into excellent students and inferior students according to their learning level. At the same time, two learning objects with different levels are selected to enhance the global search ability. Compared with other improvements, the improvement of TLBO algorithm in this paper not only speeds up the convergence speed and improves the convergence accuracy of the algorithm, but also compared with the improvement made by other scholars, the method in this paper is simpler, more practical and more efficient. Numerical experiments show that IDMTLBO algorithm is superior to TLBO algorithm, DMTLBO algorithm and DSTLBO algorithm in convergence speed and convergence accuracy.

2 TLBO Algorithm

In the TLBO algorithm, the subjects that students learn represent the dimensions of the population, and the number of middle school students in the class represents the size of the population. Teachers are defined as the students with the best performance in the class, and students’ academic performance is equivalent to the value of fitness. The specific process of the teaching optimization algorithm is described below.

2.1 Class initialization

Assuming that the number of students in the class is n and the subject is d, the initial population is generated by formula (1). $X_{k} = {Min}_{X} + rand ({Max}_{X} - {Min}_{X}) .$ (1)

Among them, k = 1, 2, ⋯ , n, X_k denotes the kth student in the class, rand functions denote the random number between [0, 1], Min_X and Max_X represent the minimum and maximum values in the solution space X.

2.2 Teacher phase

In the actual teaching, students learn according to the gap between the average level of the class and the teacher. In the ith iteration, the gap between the kth student in the jth discipline and the teacher is given by Equation (2). ${Difference}_{i, j, k} = rand (X_{i, j, kBest} - T \times {Mean}_{i, j}) .$ (2) where X_i,j,kBest denotes the best learner in the class, T = round (1 + rand) represents the teaching factor, ${Mean}_{i, j} = \sum_{k = 1}^{n} X_{i, j, k} / n$ represents the average level of class students in the jth course during the ith iteration, round function representation rounded, update the kth student in the class according to formula (3). $X_{i, j, k}^{new 1} = X_{i, j, k}^{old} + {Difference}_{i, j, k} .$ (3)

In formula (3), $X_{i, j, k}^{old}$ and $X_{i, j, k}^{new 1}$ represent the knowledge level of the kth student before and after the j course in the first iteration. If $f (X_{i, j, k}^{new 1}) > f (X_{i, j, k}^{old})$ , update $f (X_{i, j, k}^{old})$ , otherwise, no updates.

2.3 Learner phase

Teachers ‘teaching is not necessarily able to effectively improve students ’ ability, students can also learn from each other to enhance. In this phase, random selection of two different students k₁, k₂, Produce new students during the ith iteration as formula (4).

$X_{i, k_{1}}^{new 2} = {\begin{matrix} X_{i, k_{1}}^{old} + rand \times (X_{i, k_{1}}^{old} - X_{i, k_{2}}), f (X_{i, k_{1}}^{old}) > f (X_{i, k_{2}}) \\ X_{i, k_{1}}^{old} + rand \times (X_{i, k_{2}} - X_{i, k_{1}}^{old}), f (X_{i, k_{1}}^{old}) ⩽ f (X_{i, k_{2}}) \end{matrix}} .$ (4)

Where f (X_i,k) represents the fitness value for each student. Update if $f (X_{i, k_{1}}^{new 2})$ is better than $f (X_{i, k_{1}}^{old})$ , otherwise, no updates.

3 IDMTLBO algorithm

3.1 Adaptive teaching factors

In teaching-learning-based optimization algorithm, teaching factor T is derived from formula T = round (1 + rand), and teaching factor T determines the change of mean value. According to the formula of teaching factor, the value of teaching factor can only be 1 or 2, This means that students either have not learned knowledge from teachers, or have learned all knowledge. But this is often not the case in actual teaching. The knowledge learned by students is generally between 1 and 2. Formula (5) is now used to solve this problem. $T = T_{min} + (I_{max} - i) / I_{max} .$ (5) where T_min represents the minimum value of the teaching factor, I_max represents the maximum number of iterations, and i represents the current number of iterations. It can be seen from Formula (5) that the value of teaching factor at the beginning is 2, and at this time, students have stronger ability to accept knowledge with less difficulty in learning. As the number of iterations increases, the value of the teaching factor approaches from 2 to 1, which indicates that students ’ ability to accept knowledge is decreasing due to the increase of learning difficulty. Therefore, it is conducive to enhancing the local search ability.

3.2 Improved teacher phase

In the teacher stage of TLBO algorithm, each student learns according to the difference between teacher and class average. Among them, the gap between the kth student in the jth discipline and the teacher in the ith iteration is given by the formula Difference_i,j,k = rand (X_i,j,kBest - T × Mean_i,j). However, in the actual teaching, students often learn according to their own learning level and the difference between teachers. Therefore, the teacher stage of TLBO algorithm is improved as follows. ${Difference}_{i, j, k}_new = rand (X_{i, j, kBest} - T \times X_{i, j, k}^{old}) .$ (6)

Updated according to formula (7). $X_{i, j, k}^{new 3} = X_{i, j, k}^{old} + {Difference}_{i, j, k}_new .$ (7)

In formula (7), each student in the class learns according to their own learning level and the difference between teachers, which improves the limitation of learning by middle school students in Equation (2) according to the average level of the class and the difference between teachers, and improves the accuracy and convergence speed of the algorithm.

3.3 Differential mutation strategy

Differential evolution algorithm (DE) is in a random population, which makes the results approximate to the optimal solution through mutual cooperation and competition among individuals. The algorithm has simple structure and is often used to deal with practical optimization problems.

Students can learn knowledge from teachers and improve themselves through mutual learning. In the basic algorithm, students in the learning stage are only affected by two students, and this stage ignores the influence of teachers on students ’ learning. According to the mutation strategy in the differential evolution algorithm, we improve the equation in the learning stage as follows in [11]. $X_{i, k}^{new 4} = X_{i, kBest} + rand (X_{i, k_{1}} - X_{i, k_{2}}) .$ (8)

X_i,k₁, X_i,k₂ in the above equation represents two different students in the ith iteration process. If $X_{i, k}^{new 4}$ is better than X_i,k, update X_i,k. If $X_{i, k}^{new 4}$ is better than X_i,kBest, update X_i,kBest.

3.4 Improved differential mutation strategy

If only one student is randomly selected, the student may be poor or even the worst in learning. This choice may make the learning effect poor. Therefore, formula (8) should be improved. If two students are randomly selected in the class, the influence of only one poor student on learning effect can be avoided. The improvement method is as follows.

${\begin{matrix} if : X_{i, k_{1}} > {Mean}_{i, j}, \\ X_{i, k}^{new 5} = X_{i, k, Best} + rand (X_{i, j, kBest} - T \times X_{i, j, k}^{old}) \\ if : X_{i, k_{1}} < {Mean}_{i, j}, \\ X_{i, k}^{new 5} = X_{i, k, Best} + rand (X_{i, k_{1}} - X_{i, k_{2}}) + rand (X_{i, k_{1}} - X_{i, k_{3}}) \end{matrix}$ (9)

In this formula, X_i,kBest is a teacher, used to guide students to learn, so that students take teachers as the goal of self - promotion. In the original algorithm, the limitation of $X_{i, k_{1}}^{old}$ on learning guidance is solved, and the problem of poor local search ability of TLBO algorithm is also solved, and the convergence speed of the algorithm is improved.

When the students ’ learning level is higher than the average level of the class, the students only learn according to the gap between themselves and teachers; when the students ’ learning level is lower than the average level of the class, the student not only learns according to the gap between themselves and teachers, but also selects two learning objects with different levels. According to the gap between the two different learning objects, the influence of poor students on learning is reduced, and the randomness of rand function is used to maintain the characteristics of TLBO algorithm. In addition, the algorithm of teaching factor is also improved to enhance the global search ability.

3.5 Implementation process of IDMTLBO algorithm

In summary, we will introduce the steps of IDMTLBO as follows.

Step 1: Sets the total number of the whole class as n, the population dimension as x, and the maximum number of iterations as I_max, so that the current number of iterations is i = 1.

Step 2: Initializes the class in the entire search space.

Step 3: Teacher stage, according to the formula (5–7) to produce new students, keep good grades.

Step 4: Learning stage, all students in the class and the average level of students in the class to do a comparison, according to (9) learning, get $X_{i, k_{1}}^{new 5}$ , choose to update.

Step 5: Repeat the teaching process for all students throughout the class.

4 Simulation experiments and results analysis

4.1 Test function

In order to verify the effectiveness of the improved TLBO algorithm, eight different standard test functions are used for simulation experiments. These standard test functions are universal in life, and the global optimal values are 0. The test functions are as follows.

Sphere function: There is a unique global minimum value, which is obtained by summing the squares of the minimum values of d independent variables with the same domain.

Schwefel’s Problem 22 Function: When the independent variable approaches infinity, the function will form a large number of local extreme value regions. The global optimal value is located at the boundary of the definition domain.

Generalized Rastigin function: The position of the minimum value is regular, which is used to detect a situation where the solution is regular, and the algorithm is practical.

Schwefel’s Problem 12 function: The independent variable of this function is epistatic, so its gradient direction will not change along the axis direction, which is difficult to find the best.

Generalized Rosenbrock function: Used to test the performance of optimization algorithm. Its global minimum is located in a narrow parabolic valley. However, although the valley is easy to find, it is difficult to converge to the minimum.

Step function: When the function approaches infinity in the definition domain, different step phenomena will occur at the given interval, and a large number of local extremum will be generated between each step, which makes it difficult to find the optimal solution.

Generalized Griewank function: The function solution changes with the quantity change, and there are a lot of local extremum in the real data distribution of the function, so the detection algorithm has the ability to jump out of the local.

Ackely function: When using gradient method to optimize a multi-dimensional point, there are often multiple directions. This function detects the global convergence rate of an algorithm. When the dimension increases, its direction gradient and direction of advance are various.

4.2 Analysis of effect

In order to verify the effectiveness of IDMTLBO algorithm, the original algorithm is expressed as (TLBO) the algorithm with adaptive teaching factor is denoted as (ATLBO), the improved algorithm in the teaching stage is denoted as (MTLBO), and the improved algorithm in the learning stage is denoted as (LTLBO). Set the class number to 50, the population dimension d = 50, the maximum number of iterations I_max = 500. Each test function runs 20 times independently, and the minimum, average and standard deviation of the results are shown in Table 2.

Table 2
TLBO, ATLBO, MTLBO, LTLBO and IDMTLBO test results

Function Algorithm Minimum Value Optimal Value Standard Deviation

f ₁ TLBO 5.2968e-76 3.7860e-75 4.0189e-75

ATLBO 1.0353e-83 2.8261e-82 4.8194e-82

MTLBO 2.9553e-274 3.2722e-271 0

LTLBO 1.4927e-130 1.4060e-128 1.6775e-128

IDMTLBO 0 0 0

f ₂ TLBO 4.8066e+10 1.2516e+15 3.2587e+15

ATLBO 2.6995e-37 1.3997e-36 8.2305e-37

MTLBO 8.2888e-139 4.5066e-138 4.3544e-138

LTLBO 1.4109e-65 5.5174e-64 9.7199e-64

IDMTLBO 0 0 0

f ₃ TLBO 610.6532 645.1730 21.8390

ATLBO 0 0.1003 0.4485

MTLBO 0 0 0

LTLBO 0 0 0

IDMTLBO 0 0 0

f ₄ TLBO 1.4535e+03 1.6165e+03 103.9176

ATLBO 1.6046e-37 4.7637e-37 2.8662e-37

MTLBO 3.0989e-139 3.3610e-138 2.6389e-138

LTLBO 1.9764e-65 2.9716e-64 2.6617e-64

IDMTLBO 0 0 0

f ₅ TLBO 1.7209e+08 2.3837e+08 3.6114e+07

ATLBO 45.4257 46.6164 0.3868

MTLBO 46.7614 47.7873 0.7046

LTLBO 44.1152 45.6274 0.9715

IDMTLBO 0 0 0

f ₆ TLBO 5.4818e+04 7.4533e+04 9.1746e+03

ATLBO 0.0839 0.3186 0.1438

MTLBO 1.2247 2.4163 0.5608

LTLBO 3.0111 3.6131 0.4597

IDMTLBO 0 0 0

f ₇ TLBO 572.0629 696.0259 76.5645

ATLBO 0 0 0

MTLBO 0 0 0

LTLBO 0 0 0

IDMTLBO 0 0 0

f ₈ TLBO 20.0312 20.1652 0.1184

ATLBO 10.6668 15.8256 3.3059

MTLBO 2.6645e-15 2.6645e-15 0

LTLBO 4.4409e-15 4.4409e-15 0

IDMTLBO 8.8818e-16 8.8818e-16 0

Function	Algorithm	Minimum Value	Optimal Value	Standard Deviation
f ₁	TLBO	5.2968e-76	3.7860e-75	4.0189e-75
	ATLBO	1.0353e-83	2.8261e-82	4.8194e-82
	MTLBO	2.9553e-274	3.2722e-271	0
	LTLBO	1.4927e-130	1.4060e-128	1.6775e-128
	IDMTLBO	0	0	0
f ₂	TLBO	4.8066e+10	1.2516e+15	3.2587e+15
	ATLBO	2.6995e-37	1.3997e-36	8.2305e-37
	MTLBO	8.2888e-139	4.5066e-138	4.3544e-138
	LTLBO	1.4109e-65	5.5174e-64	9.7199e-64
	IDMTLBO	0	0	0
f ₃	TLBO	610.6532	645.1730	21.8390
	ATLBO	0	0.1003	0.4485
	MTLBO	0	0	0
	LTLBO	0	0	0
	IDMTLBO	0	0	0
f ₄	TLBO	1.4535e+03	1.6165e+03	103.9176
	ATLBO	1.6046e-37	4.7637e-37	2.8662e-37
	MTLBO	3.0989e-139	3.3610e-138	2.6389e-138
	LTLBO	1.9764e-65	2.9716e-64	2.6617e-64
	IDMTLBO	0	0	0
f ₅	TLBO	1.7209e+08	2.3837e+08	3.6114e+07
	ATLBO	45.4257	46.6164	0.3868
	MTLBO	46.7614	47.7873	0.7046
	LTLBO	44.1152	45.6274	0.9715
	IDMTLBO	0	0	0
f ₆	TLBO	5.4818e+04	7.4533e+04	9.1746e+03
	ATLBO	0.0839	0.3186	0.1438
	MTLBO	1.2247	2.4163	0.5608
	LTLBO	3.0111	3.6131	0.4597
	IDMTLBO	0	0	0
f ₇	TLBO	572.0629	696.0259	76.5645
	ATLBO	0	0	0
	MTLBO	0	0	0
	LTLBO	0	0	0
	IDMTLBO	0	0	0
f ₈	TLBO	20.0312	20.1652	0.1184
	ATLBO	10.6668	15.8256	3.3059
	MTLBO	2.6645e-15	2.6645e-15	0
	LTLBO	4.4409e-15	4.4409e-15	0
	IDMTLBO	8.8818e-16	8.8818e-16	0

It can be seen from Table 2 that the minimum, average and standard deviation of ATLBO, MTLBO, LTLBO and IDMTLBO running 20 are better than the basic TLBO algorithm. It can be seen that the introduction of adaptive teaching factors and the improvement of teaching and learning stages have improved the performance of the algorithm. By comparing TLBO and MTLBO, it can be concluded that learning according to the gap between students and teachers in the teaching stage has significantly improved students’ learning level, improved the stability of the algorithm, and prevented the algorithm from falling into local optimization prematurely. Compared with TLBO and LTLBO, we can get the improved differential mutation strategy in the learning stage, which further expands the search scope of the space, maintains the diversity of the population, and improves the solution accuracy.

In order to verify the advancement of IDMTLBO algorithm, IDMTLBO algorithm, DMTLBO algorithm and DSTLBO algorithm are run 20 times independently under the conditions of 50 class members, d = 30 of population dimension and I_max = 200 of maximum iteration number. The results are shown in Table 3.

Table 3

DMTLBO, DSTLBO and IDMTLBO test results

Function	Algorithm	Minimum Value	Optimal Value	Standard Deviation
f ₁	DMTLBO	1.2266e-58	3.0779e-57	3.0430e-57
	DSTLBO	4.0677e-98	7.6261e-51	3.3285e-50
	IDMTLBO	0	0	0
f ₂	DMTLBO	2.0356e-74	2.7400e-73	2.3312e-73
	DSTLBO	4.1814e-117	1.1667e-83	3.9454e-83
	IDMTLBO	5.4554e-173	8.1352e-167	0
f ₃	DMTLBO	0	0	0
	DSTLBO	0	1.0957e-08	3.9610e-08
	IDMTLBO	0	0	0
f ₄	DMTLBO	1.3833e-74	1.7017e-73	1.5197e-73
	DSTLBO	6.1729e-120	3.3678e-80	1.5060e-79
	IDMTLBO	6.6433e-172	3.2854e-165	0
f ₅	DMTLBO	22.3565	23.7484	0.6643
	DATLBO	0	0	0
	IDMTLBO	0	0	0
f ₆	DMTLBO	0.7209	0.8521	0.0983
	DSTLBO	0	0	0
	IDMTLBO	0	0	0
f ₇	DMTLBO	0	0	0
	DSTLBO	0	1.1123e-08	2.6307e-08
	IDMTLBO	0	0	0
f ₈	DMTLBO	4.4409e-15	4.4409e-15	0
	DSTLBO	8.8818e-16	8.8818e-16	0
	IDMTLBO	8.8818e-16	8.8818e-16	0

The results of Table 3 are analyzed, and the three algorithms are analyzed from the minimum, average and standard deviation, so as to verify the convergence accuracy, advancement and stability of IDMTLBO algorithm. The global optimal values of the eight standard test functions proposed in Table 1 are 0. Therefore, the closer the experimental results are to 0, the more effective the algorithm is. The smaller the standard deviation is, the smaller the floating range of the operation result is, and the more stable the algorithm is.

Table 1

Standard test function expression and search space

Function name	Function expressions	Character of function	Search space
Sphere Model	$f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	Nonlinear, Single Peak	[-100, 100]
Schwefel’s Problem 22	$f_{2} (x) = \sum_{i = 1}^{n} \| x_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	Non Separable, Single Peak	[-10, 10]
Generalized Rastrigin	$f_{3} (x) = \sum_{i = 1}^{n} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$	Multiple Peak	[-5.12, 5.12]
Schwefel’s Problem 12	f₄ (x) = max {\|x_i\|, 1 ⩽ i ⩽ n}	Non Separable, Single Peak	[-100, 100]
Generalized Rosenbrock	$f_{5} (x) = \sum_{i = i}^{n - 1} [100 (x_{i + 1} - x_{i}^{2}]^{2} + (x_{i} - 1)^{2}$	Nonconvex, Single Peak	[-30, 30]
Step function	$f_{6} (x) = \sum_{i = 1}^{n} (\| x_{i} + 0.5 \|)^{2}$	Progressive Decrease	[-100, 100]
Generalized Griewank	$f_{7} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	Non Separable, Multiple Peak	[-600, 600]
Ackley function	$\begin{matrix} f_{8} (x) = - 20 exp [- 0.2 \sqrt{\sum_{i = 1}^{n} x_{i}^{2} / n}] \\ - exp (\sum_{i = 1}^{n} cos (2 π x_{i}) / n) + 20 + e \end{matrix}$	Continuous, Rotating, Non Separable, Multiple peak	[-32, 32]

It can be seen from Table 3 that the minimum and average values of IDMTLBO algorithm are closer to the global optimal value 0, which shows that the performance of the algorithm is greatly improved by adding adaptive teaching factors and the improved differential mutation strategy. From the perspective of standard deviation, the test results of IDMTLBO algorithm are smaller than those of other algorithms, which indicates that the improved differential mutation strategy further expands the search scope of space, effectively maintains the diversity of the population, and improves the accuracy of the solution. Therefore, it can be concluded that the convergence accuracy, progressiveness and stability of this algorithm are obviously superior to the other two algorithms.

Figure 1 shows the evolution curve of DMTLBO, DSTLBO and IDMTLBO algorithms. It can be seen from the figure that IDMTLBO algorithm converges to the global optimal solution quickly, DMTLBO algorithm and DSTLBO algorithm converge slowly, and TLBO algorithm falls into the local optimal solution. Therefore, IDMTLBO algorithm is obviously superior to DMTLBO algorithm and DSTLBO algorithm in terms of both convergence speed and convergence accuracy, and converges to the best in a small number of iterations. This is because students improve the stability of the algorithm according to their learning gap with teachers, and the improved differential mutation strategy effectively maintains the diversity of the population, with strong optimization ability. To sum up, IDMTLBO algorithm has higher optimization precision and stronger optimization ability, and the algorithm has simple structure and short running time. Therefore, the proposed algorithm is progressiveness.

Fig. 1

Evolution curve of DMTLBO, DSTLBO and IDMTLBO algorithms.

5 Conclusion

In view of the shortcomings of TLBO algorithm, IDMTLBO first introduces an adaptive teaching factor, which makes up for the shortcomings of TLBO algorithm teaching factor can only take 1 or 2. Secondly, the learning process in the learning stage is improved to enable students to learn according to the gap between themselves and teachers, and enhance the stability of the algorithm. Finally, in the learning stage, the improved differential variation strategy is integrated, and students are divided into excellent students and inferior students. The excellent students learn according to the gap between themselves and teachers, and the inferior students learn according to the gap between themselves and teachers and two students randomly selected. At the same time, two students are selected to avoid the limitation of selecting only one student as the inferior student, effectively maintaining the diversity of the population, and improving the solution accuracy. Through the simulation experiments on the eight standard test functions in Table 1, it can be seen from the results that the IDMTLBO algorithm proposed in this paper has good convergence performance, and the main advantages of this algorithm are simple structure, easy to understand, few super parameters, and high convergence performance. Therefore, this algorithm has good prospects for development and can be further applied to other practical optimization problems.

Footnotes

Acknowledgments

The authors would like to thank the anonymous reviewers, the Editor-in-chief and the Associate Editor, for providing valuable comments and helpfull suggestions.

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