Modified teaching-learning-based optimization algorithm for multi-objective optimization problems

Abstract

When solving multi-objective optimization problems, an important issue is how to promote convergence and distribution of solution set simultaneously. To address the above issue, a novel optimization algorithm, named as multi-objective modified teaching-learning-based optimization (MOMTLBO), is proposed. Firstly, a grouping teaching strategy based on pareto dominance relationship is proposed to strengthen the convergence efficiency. Afterward, a diversified learning strategy is presented to enhance the distribution. Meanwhile, differential operations are incorporated to the proposed algorithm. By the above process, the search ability of the algorithm can be encouraged. Additionally, a set of well-known benchmark test functions including ten complex problems proposed for CEC2009 is used to verify the performance of the proposed algorithm. The results show that MOMTLBO exhibits competitive performance against other comparison algorithms. Finally, the proposed algorithm is applied to the aerodynamic optimization of airfoils.

Keywords

Teaching-learning-based optimization (TLBO)multi-objective optimization convergence distribution airfoils

1 Introduction

Many of the issues in scientific research, engineering and management can be transformed into optimization problems, which may involve multiple conflicting optimization objectives and are called multi-objective optimization problems (MOPs). Nature-inspired algorithms are effective method to solve MOPs due to the fact that they have many advantages such as global optimization performance, strong versatility, easy parallel processing, can handle different types of decision variables, and they do not depend on whether the objective functions and constraints are convex or differentiable [1], which include genetic algorithm [2 –4], particle swarm optimization [5 –7], differential evolution [8 –10], etc. The existing nature-inspired multi-objective optimization algorithms can be mainly divided into three categories: algorithms based on pareto dominance relationship [11, 12], algorithms based on decomposition [13, 14] and algorithms based on performance metrics [15, 16]. The algorithms based on pareto dominance relationship is firstly to use pareto dominance relationship to prescreen the non-dominated solutions in the population. Then, other specific strategies are used to further select some excellent solutions from all non-dominated solutions. The algorithms based on decomposition transforms the original multi-objective optimization problem into several simple sub-optimization problems by using a set of uniform weight vectors in the objective space. These kinds of methods often have high efficiency. However, because the optimization process depends on the weight vectors and is affected by the distribution of vectors, the effect of these methods on the multi-objective optimization problems with irregular pareto front is not ideal. The algorithms based on performance metrics selects some excellent solutions and retains them by calculating the contribution of the solution to the specific performance metrics of the population. HV [17] is a commonly used metrics that can evaluate convergence and distribution simultaneously. Since HV can evaluate the convergence and distribution of the population at the same time, these methods can finally obtain a set of solutions with good convergence and distribution. Nevertheless, the calculation process of HV is relatively complex. Calculating HV frequently in the optimization process will significantly increase the time complexity of the algorithm.

Teaching-learning-based optimization (TLBO) is a well-known nature-inspired algorithm proposed by Rao et al. recently [18], which simulates the teaching and learning process of teachers and students. TLBO has the characteristics of fast convergence, high accuracy, few parameters, and is used in the fields of function optimization [19], engineering optimization [20 –22], clustering [23] and other optimization problems [24 –27]. In recent years, scholars have developed numerous improved TLBO algorithms to solve MOPs. Zou et al. [28] introduced the non-dominated sorting and crowding distance of NSGA-II [11] into TLBO and proposed multi-objective optimization using teaching-learning-based optimization (MOTLBO), which solved three-dimensional Two-Bar truss design problem and four-dimensional I-beam design problem respectively. Medina et al. [29] proposed a multi-objective teaching and learning optimization algorithm (MOTLA/D) based on decomposition strategy which solved the reactive power system optimization problem. Niknam et al. [30] proposed θ-multi-objective teaching-learning-based optimization which is applied to the dynamic economic emission dispatch problem.

However, few existing multi-objective TLBOs focus on solving high-dimensional complex problems. To address above issue, this article tries to develop a modified teaching-learning-based optimization (MOMTLBO) algorithm which is based on pareto dominance relationship to strengthen the convergence and distribution on high-dimensional complex problems. The main contributions of this paper are highlighted as follows.

A grouping teaching strategy based on pareto dominance relationship is proposed to strengthen the convergence efficiency.

A diversified learning strategy is presented to enhance the distribution.

Differential operations are incorporated to encourage the searchability of the algorithm.

A set of well-known benchmark test functions including ten complex high-dimensional problems proposed for CEC2009 is used to verify the performance of the proposed algorithm.

The aerodynamic optimization of airfoils is calculated to verify the effectiveness of MOMTLBO on real-world complex optimization problems.

The remainder of this paper is organized as follows. Section 2 briefly introduces the basic TLBO. The proposed algorithm, MOMTLBO, is elaborated in Section 3. Numerical experiments are presented in Section 4. Section 5 applies the proposed algorithm to the aerodynamic optimization of airfoils. Finally, Section 6 concludes this paper.

2 Teaching-learning-based optimization (TLBO)

TLBO is a nature-inspired optimization algorithm that simulates the teaching and learning process of teachers and students. Solutions in the population are given two concepts of “teacher” and “student”. “Teacher” is the best solution selected from all the solutions, represented by ${\vec{X}}_{teacher}$ and “student” are the other solutions. TLBO includes teacher phase and learner phase. In teacher phase, solutions update according to the teacher and the average position of the population. All solutions move according to Equation (1): ${\vec{X}}_{new, i} = {\vec{X}}_{old, i} + \vec{r_{i}} \cdot ({\vec{X}}_{teacher} - T_{F} \cdot {\vec{X}}_{mean})$ (1) where ${\vec{X}}_{new, i}$ is the new position of the ith solution, ${\vec{X}}_{old, i}$ is the original position of the ith solution, $\vec{r_{i}}$ is a vector and its elements are uniformly distributed random numbers in the range of [0,1], T_F is the teaching factor, which is either 1 or 2 randomly, and ${\vec{X}}_{mean}$ is the average position of all the solutions. In learner phase, two different solutions $\vec{X_{i}}$ and $\vec{X_{j}} (i \neq j)$ are selected randomly and move according to Equation (2):

$\begin{matrix} {\vec{X}}_{new, i} \\ = {\begin{matrix} {\vec{X}}_{old, i} + \vec{r_{i}} \cdot ({\vec{X}}_{old, i} - {\vec{X}}_{old, j}) if \vec{X_{i}} is better than \vec{X_{j}} \\ {\vec{X}}_{old, i} + \vec{r_{i}} \cdot ({\vec{X}}_{old, j} - {\vec{X}}_{old, i}) otherwise \end{matrix} \end{matrix}$ (2)

The selection operator implements the greedy law after teacher phase and learner phase. When reaching the termination condition, the best solution is outputted as the optimization result.

3 Multi-objective modified teaching-learning-based optimization (MOMTLBO)

A MOMTLBO algorithm is described in this section.

3.1 Algorithm framework

Algorithm 1 shows the procedure of the proposed algorithm.

Algorithm 1

: Procedure of MOMTLBO

Input: population size NP and maximum number of function evaluations MaxFEs

Output: population P

1 The number of function evaluations FEs = 0;

2 Initialize population P and evaluate P , FEs = FEs + NP ;

3 While FEs ⩽ MaxFEs do

4 Apply the teacher phase in Algorithm 2 on P to generate an

offspring population offP and evaluate offP ,

FEs = FEs + NP ;

5 mixP ← P ∪ offP ;

6 Update P : P ← mixP ;

7 Apply the learner phase in Algorithm 3 on P to generate an

offspring population offP and evaluate offP ,

FEs = FEs + NP ;

8 mixP ← P ∪ offP ;

9 Update P : P ← mixP ;

10 End while

: Procedure of MOMTLBO
Input: population size NP and maximum number of function evaluations MaxFEs
Output: population P
1 The number of function evaluations FEs = 0;
2 Initialize population P and evaluate P , FEs = FEs + NP ;
3 While FEs ⩽ MaxFEs do
4 Apply the teacher phase in Algorithm 2 on P to generate an
offspring population offP and evaluate offP ,
FEs = FEs + NP ;
5 mixP ← P ∪ offP ;
6 Update P : P ← mixP ;
7 Apply the learner phase in Algorithm 3 on P to generate an
offspring population offP and evaluate offP ,
FEs = FEs + NP ;
8 mixP ← P ∪ offP ;
9 Update P : P ← mixP ;
10 End while

Initially, uniform sampling is applied to initialize the population P with NP solutions. Then, the teacher phase in Algorithm 2 is executed on P and offP is generated. Afterward, the best NP solutions in P and offP are retained, which constitute a new P . Next, the learner phase in Algorithm 3 is applied on P to generate offP . Finally, the best NP solutions in P and offP are retained, which constitute a new P . It is noteworthy that the teacher phase and the learner phase are repeated until the termination condition is reached.

3.2 Teacher phase

Algorithm 2 shows the procedure of the teacher phase. In this phase, a grouping teaching strategy is executed to strengthen the convergence efficiency.

Algorithm 2

: Teacher phase

Input: population P

Output: offspring population offP

1 Use the pareto dominance on objective functions of solutions in

P and select all non-dominated solutions, the number of non-

dominated solutions is k ;

2 If k ⩾ 10

3 The population is divided into k groups, each non-dominated

solution is the teacher of each group, and the other solutions

defined as students are grouped according to the Euclidean

distance from the teachers;

4 Update students by using Equation (3);

5 Update teachers by simulated binary crossover and polynomi-

al mutation;

6 End if

7 If k < 10

8 Update all solutions by simulated binary crossover and poly-

nomial mutation;

9 End if

10 All updated solutions form an offspring population offP .

: Teacher phase
Input: population P
Output: offspring population offP
1 Use the pareto dominance on objective functions of solutions in
P and select all non-dominated solutions, the number of non-
dominated solutions is k ;
2 If k ⩾ 10
3 The population is divided into k groups, each non-dominated
solution is the teacher of each group, and the other solutions
defined as students are grouped according to the Euclidean
distance from the teachers;
4 Update students by using Equation (3);
5 Update teachers by simulated binary crossover and polynomi-
al mutation;
6 End if
7 If k < 10
8 Update all solutions by simulated binary crossover and poly-
nomial mutation;
9 End if
10 All updated solutions form an offspring population offP .

Firstly, the pareto dominance is used on objective functions of all solutions to find all non-dominated solutions and the number of the non-dominated solutions is k . It is confirmed that the grouping teaching is an effective strategy to improve the performance of TLBO [31], but the type of grouping is a crucial issue. In our approach, the population is divided into k groups, each non-dominated solution is the teacher of each group, and the other solutions defined as students are grouped according to the Euclidean distance from the teachers. Afterward, the students in each group move according to Equation (3):

$\begin{matrix} {\vec{X}}_{new, i} = & {\vec{X}}_{old, i} + \vec{r_{1, i}} \cdot ({\vec{X}}_{teacher} - T_{F} \cdot {\vec{X}}_{groupmean}) \\ + \vec{r_{2, i}} \cdot ({\vec{X}}_{r 1} - {\vec{X}}_{r 2}) \end{matrix}$ (3) where ${\vec{X}}_{r 1}$ and ${\vec{X}}_{r 2}$ are different solutions randomly selected from other different groups, ${\vec{X}}_{groupmean}$ is the average position of all the solutions in the group, and ${\vec{X}}_{teacher}$ is selected from all teachers by roulette wheel selection according to crowding-distance [11]. As shown in Equation (3), a differential vector between two solutions which are selected from different groups are introduced to increase information exchange between groups, so as to improve the search ability of algorithm. It is noteworthy that the teacher is not fixed and randomly selected from all teachers. Afterward, the teachers are executed simulated binary crossover and polynomial mutation [11] to update themselves, for the purpose of exchanging information between excellent solutions, which is more promising to produce better solutions. It is necessary to emphasize that when the number of groups is small, the effect of group teaching strategy will be limited simultaneously. Note that a small number of groups means that there are few non-dominated solutions in the population. Under this condition, few teachers in population causes that algorithm may not be able to guide the population to the optimal region effectively. To address this issue, we set that if k < 10, all the solutions in population are executed simulated binary crossover and polynomial mutation, which can probably produce more non-dominated solutions so as to be defined as teachers in the next teacher phase. In contrast, if k ⩾ 10, group teaching strategy is applied as above mentioned. By the above process, the convergence efficiency of the algorithm can be encouraged.

3.3 Learner phase

The procedure of the learner phase is shown in Algorithm 3. In this phase, a diversified learning strategy is adopted to strength the distribution.

Algorithm 3

: Learner phase

Input: population P

Output: offspring population offP

1 Use the pareto dominance on objective functions of solutions

in P ;

2 For each solution in P do

3 if solution is non-dominated

4 Update solution by using Equation (4);

5 end if

6 if solution is dominated

7 Update solution by using Equation (5)–Equation (7);

8 end if

9 End for

10 All updated solutions form an offspring population offP .

: Learner phase
Input: population P
Output: offspring population offP
1 Use the pareto dominance on objective functions of solutions
in P ;
2 For each solution in P do
3 if solution is non-dominated
4 Update solution by using Equation (4);
5 end if
6 if solution is dominated
7 Update solution by using Equation (5)–Equation (7);
8 end if
9 End for
10 All updated solutions form an offspring population offP .

At the beginning of learner phase, the pareto dominance is used on objective functions of all solutions. Afterward, two kinds of solutions move according to different strategies. To be specific, non-dominated solutions update by using Equation (4):

{\vec{X}}_{n e w, i} = {\begin{matrix} {\vec{X}}_{o l d, i} + {\vec{s}}_{1, i} \cdot ({\vec{X}}_{o l d, i} - {\vec{X}}_{o l d, j}) i f {\vec{X}}_{i} i s b e t t e r t h a n {\vec{X}}_{j} \\ {\vec{X}}_{o l d, i} + {\vec{r}}_{i} \cdot ({\vec{X}}_{o l d, j} - {\vec{X}}_{o l d, i}) + {\vec{s}}_{2, i} \cdot ({\vec{X}}_{r 1} - {\vec{X}}_{r 2}) o t h e r w i s e \end{matrix}

(4)

where $\vec{s_{1, i}}$ is a vector and its elements are uniformly distributed random numbers in the range of [0,0.2], $\vec{s_{2, i}}$ is a vector and its elements are uniformly distributed random numbers in the range of [0.5,1], $\vec{X_{j}}$ is the solution selected from all non-dominated solutions by roulette wheel selection according to crowding-distance, which is different from X_i^→. ${\vec{X}}_{r 1}$ and ${\vec{X}}_{r 2}$ are different solutions randomly selected from population which are different from $\vec{X_{i}}$ and $\vec{X_{j}}$ . Note that when comparing two solutions, $\vec{X_{i}}$ is better than $\vec{X_{j}}$ if and only if the following conditions are satisfied: $\vec{X_{i}}$ pareto dominates $\vec{X_{j}}$ or $\vec{X_{i}}$ and $\vec{X_{j}}$ are not pareto dominated each other but the crowding-distance of $\vec{X_{i}}$ is larger than that of $\vec{X_{j}}$ . Actually, as a non-dominated solution in population, $\vec{X_{i}}$ is relatively excellent, so in order to find better solution, the algorithm prefers to search in its adjacent region. Hence, if $\vec{X_{i}}$ is better than $\vec{X_{j}}$ , the range of $\vec{s_{1, i}}$ is set to [0,0.2], which is different from Equation (2). On the contrary, if $\vec{X_{j}}$ is better than $\vec{X_{i}}$ , as shown in Equation (4), $\vec{X_{i}}$ will move to $\vec{X_{j}}$ , which means $\vec{X_{i}}$ will move to more sparse region. As a result, the distribution can be improved. However, if $\vec{X_{i}}$ only moves to the direction of $\vec{X_{j}}$ , when $\vec{X_{j}}$ falls into local optimal, the population may be guided to a bad region. To remedy this issue, a random differential vector is added and the range of $\vec{s_{2, i}}$ is set to [0.5,1], for the propose of utilizing the overall population information to increase search ability. By this process, population is more like to reach global optimal region. On the contrary, dominated solutions update by using following strategy: for each solution $\vec{X_{i}}$ , another solution $\vec{X_{j}}$ is randomly selected from the whole population at first. Then, if $\vec{X_{j}}$ is better than $\vec{X_{i}}$ , $\vec{X_{i}}$ moves to $\vec{X_{j}}$ by using Equation (5):

\begin{matrix} {\vec{X}}_{new, i} = & {\vec{X}}_{old, i} + \vec{r_{1, i}} \cdot ({\vec{X}}_{old, j} - {\vec{X}}_{old, i}) \\ + \vec{r_{2, i}} \cdot ({\vec{X}}_{r 1} - {\vec{X}}_{r 2}) \end{matrix}

(5)

Similarly, a differential vector is introduced. In addition, if $\vec{X_{i}}$ is better than $\vec{X_{j}}$ , $\vec{X_{i}}$ will not require searching in its adjacent region like Equation (4), because the $\vec{X_{i}}$ in Equation (4) is the non-dominated solution which is high quality in population, searching in its adjacent region are more promising to obtain better solution. While in Equation (5), the $\vec{X_{i}}$ is dominated solution, it needs to move to a better region. Therefore, to promptly achieve the above purpose, the mutation and crossover operations in differential evolution are used to update $\vec{X_{i}}$ :

{\vec{V}}_{i} = {\vec{X}}_{r 1} + \vec{r_{i}} \cdot ({\vec{X}}_{best} - {\vec{X}}_{r 1}) + F \cdot ({\vec{X}}_{r 2} - {\vec{X}}_{r 3})

(6)

\begin{matrix} X_{{new}_{i, j}} \\ = {\begin{matrix} V_{i, j}, if {rand}_{j} < CR or j = j_{rand} \\ X_{{old}_{i, j}} otherwise \end{matrix} j = 1, . . ., D \end{matrix}

(7)

where ${\vec{X}}_{best}$ is selected from all non-dominated solutions by roulette wheel selection according to crowding-distance, ${\vec{X}}_{r 1}$ , ${\vec{X}}_{r 2}$ and ${\vec{X}}_{r 3}$ are mutually different solutions randomly selected from population, V_i,j , X _{old_i,j}, X _{new_i,j} are the j th dimension of ${\vec{V}}_{i}$ , ${\vec{X}}_{old, i}$ and ${\vec{X}}_{new, i}$ , respectively. F is the scaling factor, CR is the crossover control parameter, rand_j is a uniformly distributed random number in the range of [0,1], and j _rand is a random integer selected from {1, . . . , D }. In our approach, according to the suggestions in literature [32], F is randomly selected from the pool {0.6, 0.8, 1.0}, and CR is randomly selected from the pool {0.1, 0.2, 1.0}. After the above processes, the distribution of the solutions can be strengthened.

4 Numerical experiments

In order to show the superiority of proposed approach, the MOMTLBO algorithm is compared with NSGA-II [11], MOEA/D [13], GDE3 [33], SMS-EMOA [15], and MOTLBO [28]. NSGA-II is one of the most popular algorithms based on pareto dominance relationship, which is widely used in engineering optimization because of its good performance. MOEA/D is a classical algorithm based on decomposition strategy. Due to the differential evolution operation is integrated into our approach, GDE3, which is a differential evolution algorithm based on pareto dominance relationship with good performance, is selected as one comparison algorithm in the experiments. SMS-EMOA is an algorithm based on HV metric and MOTLBO is a modified TLBO based on pareto dominance relationship. A set of well-known benchmark test functions UF from CEC2009 [34] is selected for performance testing, which contains 10 extremely complex high-dimensional unconstrained problems due to the fact that there is a strong correlation between decision variables.

In the experiment, the population size is set to 100 and the maximum number of function evaluations is set to 300000 [34]. The additional parameters of algorithms are set as follows. MOMTLBO and NSGA-II use simulated binary crossover which the crossover index is 20 and crossover probability is 1 and use polynomial mutation which the mutation index is 20 and mutation probability is 1/d (d is the dimension of the decision space). To avoid the influence of randomness, each algorithm runs 30 times independently for each problem. Experiments are executed on the optimization platform PlatEMO [35]. Two performance metrics are used in experiments: inverted generational distance (IGD) [36] and hypervolume (HV) [17], which can evaluate the quality of obtained solution set in terms of both convergence and distribution. Note that a smaller IGD value and a larger HV value means a better quality of solution set obtained by an algorithm. The comparison results of experiments on IGD value are shown in Table 1.

Table 1
Results of IGD

Problem NSGA-II MOEA/D GDE3 SMS-EMOA MOTLBO MOMTLBO

UF1 9.15E-02(3.80E-02)- 2.58E-01(9.05E-02)- 1.81E-02(4.49E-03)≈ 1.04E-01(1.64E-02)- 7.59E-02(2.48E-02)- 1.98E-02(2.58E-03)

UF2 3.59E-02(9.27E-03)- 1.16E-01(4.53E-02)- 1.66E-02(2.22E-03)+ 3.88E-02(1.82E-02)- 2.96E-02(2.46E-03)- 1.84E-02(9.51E-04)

UF3 2.24E-01(5.86E-02)- 3.21E-01(1.78E-02)- 3.77E-02(2.25E-02)+ 2.62E-01(4.75E-02)- 1.66E-01(2.04E-02)- 5.29E-02(1.49E-02)

UF4 4.46E-02(7.79E-04)- 5.77E-02(2.09E-03)- 6.17E-02(5.13E-03)- 4.01E-02(8.03E-04)+ 6.12E-02(5.15E-03)- 4.25E-02(6.18E-04)

UF5 3.09E-01(1.28E-01)- 4.49E-01(1.31E-01)- 4.46E-01(1.00E-01)- 2.90E-01(8.19E-02)- 1.47E+00(4.16E-01)- 2.02E-01(1.24E-01)

UF6 1.98E-01(1.27E-01)- 4.59E-01(1.66E-01)- 1.39E-01(1.23E-01)≈ 2.29E-01(1.17E-01)- 4.77E-01(1.39E-01)- 1.41E-01(1.51E-01)

UF7 1.36E-01(1.48E-01)- 3.52E-01(1.97E-01)- 1.07E-02(1.09E-03)+ 1.69E-01(1.42E-01)- 3.87E-02(1.08E-02)- 1.48E-02(9.31E-04)

UF8 2.67E-01(8.84E-02)+ 3.57E-01(2.68E-01)+ 6.50E-01(1.70E-01)- 4.65E-01(3.87E-02)- 3.05E-01(3.29E-02)+ 3.57E-01(1.00E-01)

UF9 3.60E-01(1.19E-01)+ 2.52E-01(2.56E-02)+ 6.80E-01(1.79E-01)≈ 4.23E-01(1.98E-01)+ 3.88E-01(8.82E-02)+ 6.92E-01(2.89E-01)

UF10 6.04E-01(2.14E-01)+ 6.93E-01(1.72E-01)+ 1.68E+00(4.02E-01)- 4.54E-01(1.07E-01)+ 8.24E-01(2.21E-01)+ 9.63E-01(2.56E-01)

- 7 7 4 7 7 /

+ 3 3 3 3 3 /

≈ 0 0 3 0 0 /

Problem	NSGA-II	MOEA/D	GDE3	SMS-EMOA	MOTLBO	MOMTLBO
UF1	9.15E-02(3.80E-02)-	2.58E-01(9.05E-02)-	1.81E-02(4.49E-03)≈	1.04E-01(1.64E-02)-	7.59E-02(2.48E-02)-	1.98E-02(2.58E-03)
UF2	3.59E-02(9.27E-03)-	1.16E-01(4.53E-02)-	1.66E-02(2.22E-03)+	3.88E-02(1.82E-02)-	2.96E-02(2.46E-03)-	1.84E-02(9.51E-04)
UF3	2.24E-01(5.86E-02)-	3.21E-01(1.78E-02)-	3.77E-02(2.25E-02)+	2.62E-01(4.75E-02)-	1.66E-01(2.04E-02)-	5.29E-02(1.49E-02)
UF4	4.46E-02(7.79E-04)-	5.77E-02(2.09E-03)-	6.17E-02(5.13E-03)-	4.01E-02(8.03E-04)+	6.12E-02(5.15E-03)-	4.25E-02(6.18E-04)
UF5	3.09E-01(1.28E-01)-	4.49E-01(1.31E-01)-	4.46E-01(1.00E-01)-	2.90E-01(8.19E-02)-	1.47E+00(4.16E-01)-	2.02E-01(1.24E-01)
UF6	1.98E-01(1.27E-01)-	4.59E-01(1.66E-01)-	1.39E-01(1.23E-01)≈	2.29E-01(1.17E-01)-	4.77E-01(1.39E-01)-	1.41E-01(1.51E-01)
UF7	1.36E-01(1.48E-01)-	3.52E-01(1.97E-01)-	1.07E-02(1.09E-03)+	1.69E-01(1.42E-01)-	3.87E-02(1.08E-02)-	1.48E-02(9.31E-04)
UF8	2.67E-01(8.84E-02)+	3.57E-01(2.68E-01)+	6.50E-01(1.70E-01)-	4.65E-01(3.87E-02)-	3.05E-01(3.29E-02)+	3.57E-01(1.00E-01)
UF9	3.60E-01(1.19E-01)+	2.52E-01(2.56E-02)+	6.80E-01(1.79E-01)≈	4.23E-01(1.98E-01)+	3.88E-01(8.82E-02)+	6.92E-01(2.89E-01)
UF10	6.04E-01(2.14E-01)+	6.93E-01(1.72E-01)+	1.68E+00(4.02E-01)-	4.54E-01(1.07E-01)+	8.24E-01(2.21E-01)+	9.63E-01(2.56E-01)
-	7	7	4	7	7	/
+	3	3	3	3	3	/
≈	0	0	3	0	0	/

+, -, and ≈ represent the performance of the competitor is significantly better than, worse than, and similar to the MOMTLBO based on the Wilcoxon rank sum test at 0.05 significance level. The data in Table 1 represent mean (standard deviation) of IGD value.

As shown in Table 1, according to the result of Wilcoxon rank sum test, the proposed algorithm is significantly better than other comparison algorithms on seven, seven, four, seven, and seven test functions respectively, which indicates that MOMTLBO achieves superior performance against other competitors. Next, to further analyze experimental results, the results of Friedman test on IGD value are shown in Table 2.

Table 2

Results of Friedman test on IGD

Algorithm	Ranking
MOMTLBO	2.75
NSGA-II	3
GDE3	3.2
SMS-EMOA	3.7
MOTLBO	3.8
MOEA/D	4.55

As shown in Table 2, it is obvious that MOMTLBO gains the first ranking among six algorithms, followed by NSGA-II, which means MOMTLBO obtains the best results among all comparison algorithms. In addition, the comparison results of experiments on HV value are shown in Table 3.

Table 3

Results of HV

Problem	NSGA-II	MOEA/D	GDE3	SMS-EMOA	MOTLBO	MOMTLBO
UF1	6.14E-01(3.38E-02)-	4.75E-01(5.80E-02)-	6.99E-01(6.46E-03)≈	5.98E-01(1.98E-02)-	6.16E-01(2.98E-02)-	6.97E-01(3.40E-03)
UF2	6.85E-01(6.11E-03)-	6.47E-01(2.19E-02)-	7.03E-01(2.66E-03)+	6.85E-01(1.11E-02)-	6.86E-01(2.92E-03)-	7.00E-01(1.22E-03)
UF3	4.61E-01(5.39E-02)-	3.72E-01(3.05E-02)-	6.72E-01(2.88E-02)+	4.26E-01(4.22E-02)-	4.96E-01(2.36E-02)-	6.55E-01(1.89E-02)
UF4	3.85E-01(1.32E-03)-	3.58E-01(2.99E-03)-	3.60E-01(7.14E-03)-	3.93E-01(1.10E-03)+	3.59E-01(6.85E-03)-	3.89E-01(9.44E-04)
UF5	2.33E-01(7.77E-02)-	1.96E-01(7.68E-02)-	5.66E-02(4.85E-02)-	2.60E-01(4.79E-02)-	1.91E-03(1.05E-02)-	3.25E-01(7.01E-02)
UF6	3.23E-01(5.61E-02)≈	2.17E-01(8.47E-02)-	3.40E-01(6.84E-02)≈	3.06E-01(6.39E-02)-	1.10E-01(9.74E-02)-	3.43E-01(6.04E-02)
UF7	4.64E-01(1.06E-01)-	3.09E-01(1.27E-01)-	5.70E-01(1.67E-03)+	4.36E-01(1.01E-01)-	5.30E-01(1.31E-02)-	5.64E-01(1.33E-03)
UF8	2.81E-01(7.32E-02)+	3.06E-01(1.04E-01)+	2.46E-02(3.60E-02)-	3.37E-01(1.51E-03)+	2.62E-01(3.87E-02)+	1.69E-01(8.52E-02)
UF9	3.76E-01(1.21E-01)+	5.45E-01(2.22E-02)+	1.16E-01(8.42E-02)-	4.33E-01(1.52E-01)+	4.10E-01(9.28E-02)+	1.29E-01(1.97E-01)
UF10	5.50E-02(5.36E-02)+	1.30E-01(7.20E-02)+	0.00E+00(0.00E+00)-	1.68E-01(8.35E-02)+	1.11E-01(4.59E-02)+	2.20E-03(6.89E-03)
-	6	7	5	6	7	/
+	3	3	3	4	3	/
≈	1	0	2	0	0	/

As far as Wilcoxon rank sum test in Table 3 is concerned, MOMTLBO exhibits significantly better than other competitors on six, seven, five, six, and seven test functions respectively. Then, the results of Friedman test on HV value are shown in Table 4.

Table 4

Results of Friedman test on HV

Algorithm	Ranking
MOMTLBO	2.7
SMS-EMOA	3.05
GDE3	3.3
NSGA-II	3.65
MOTLBO	3.9
MOEA/D	4.4

In terms of Friedman test in Table 4, MOMTLBO achieves the first ranking among all algorithms similarly. The above analysis demonstrates that MOMTLBO exhibits better performance against other five competitors.

5 Real-world application

In this section, MOMTLBO is applied to deal with the aerodynamic optimization of RAE2822 airfoils, with the aim of investigating its effectiveness in solving the real-world MOPs.

5.1 Problem formulation

Because its objectives are highly nonlinear and multimodal, aerodynamic optimization of airfoils is one of the most challenging problems in the aerospace engineering. In this problem, lift-drag ratios in two states are selected as optimization objectives. The first state is Ma = 0.6, Re = 6.5 × 10⁶ and the second state is Ma = 0.75, Re = 6.5 × 10⁶, where Ma represents Mach number and Re represents Reynolds number. The mathematical model of the problem can be formulated as:

$\begin{matrix} maximize : f_{1} (\vec{x}) = (C_{L} / C_{D})_{1} \\ f_{2} (\vec{x}) = (C_{L} / C_{D})_{2} \end{matrix}$ where $\vec{x}$ is airfoil shape parameters, C_L is lift coefficient, C_D is drag coefficient, ( C_L/C_D ) ₁ is lift-drag ratio in the first state, and ( C_L/C_D ) ₂ is lift-drag ratio in the second state. In addition, CST (Class function/Shape function Transformation) parameterization method is used to process the airfoil with a total of 18 design variables. In the optimization process, the RBF-based grid deformation technology is adopted and the flow field grid with 256 × 120 grid scale is used for calculation, which is shown in Fig. 1.

Fig. 1

The flow field grid of RAE2822 airfoil.

5.2 Experimental results

As same as Section 4, five other algorithms are adopted as compared algorithms in this experiment. For all algorithms, the population size is set to 100 and the maximum number of function evaluations is set to 10000. The optimization results are shown in Fig. 2.

Fig. 2

The optimization results obtained by six algorithms.

It can be seen in Fig. 2 that this problem has an irregular Pareto front. Obviously, the convergence of MOMTLBO is the best among six algorithms. Although the distribution of GDE3 seems more satisfactory to some extent, but the convergence of that is the worst among all algorithms. However, the results of MOMTLBO can obtain better distribution while acquire superior convergence against other compared algorithms at the same time, especially in ( C_L/C_D ) ₁ ⩾ 68. Afterward, one solution in the results of MOMTLBO is selected from Fig. 2 which is marked with arrows to analyze specifically. The comparison data are shown in Table 5.

Table 5

Comparison of baseline and optimum

Airfoil	State	C_L/C_D
Baseline	1	62.8
	2	28.9
Optimum	1	68.8(+9.5%)
	2	42.2(+46.0%)

It can be seen from Table 5 that after optimization, the lift-drag ratio in two states increases 9.5% and 46.0%, respectively. Finally, to further compare performance of six algorithms, the optimization processes at the number of function evaluations (FEs) is 2000, 4000, 6000 and 8000 are shown in Fig. 3.

Fig. 3

Optimization processes comparisons among six algorithms.

As shown in Fig. 3, the convergence speed of MOMTLBO is more prompt than other algorithms, which can also indicate that for complex engineering optimization problems, MOMTLBO can obtain the pareto front by consuming fewer FEs.

6 Conclusions

In this work, a multi-objective modified teaching-learning-based optimization algorithm (MOMTLBO) is proposed. Firstly, a grouping teaching strategy is proposed to strengthen the convergence efficiency. Afterward, a diversified learning strategy is presented to enhance the distribution. Additionally, differential operations are incorporated to increase the search ability of the algorithm.

From the comparative experiments based on CEC2009 benchmark test functions, the performance of MOMTLBO is better than that of five well-known comparison algorithms in terms of IGD and HV. Moreover, MOMTLBO is applied to the aerodynamic optimization of RAE2822 airfoils to verify its capability. The results show that MOMTLBO can obtain pareto front with better convergence and distributions simultaneously against other five competitors. Furthermore, after optimization by MOMTLBO, the lift-drag ratios of RAE2822 airfoil in two states are improved 9.5% and 46.0% respectively.

In engineering optimization problems, there usually exist problems with more than three conflicting objectives that need to be optimized at the same time. Those problems are commonly known as many-objective optimization problems. For these problems, the simple selection method based on pareto dominance relationship will no longer be applicable, and the large search space requires strong search performance of the algorithm. Therefore, designing new selection methods and better search strategies for MOMTLBO to solve many-objective optimization problems will be an important direction of our future work.

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