# A Modified Group Teaching Optimization Algorithm for Solving Constrained Engineering Optimization Problems

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## Abstract

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## 1. Introduction

- A modified GTOA is proposed based on three strategies: learning motivation (LM), random opposition-based learning (ROBL), and restart strategy (RS).
- The optimization of MGTOA in different dimensions (dim = 30/500) among 23 standard benchmark functions is evaluated, and the distribution of MGTOA in some benchmark functions is shown.
- Test the optimization performance of MGTOA in CEC2014.
- MGTOA is compared with seven different optimization algorithms.
- Six process problems verify the engineering practicability of MGTOA.

## 2. Group Teaching Optimization Algorithm (GTOA)

#### 2.1. Ability Grouping Phase

#### 2.2. Teacher Phase

#### 2.3. Student Phase

#### 2.4. Teacher Allocation Phase

#### 2.5. The Proposed Approach

_{max}, current evaluation times t (t = 0), population size N, upper and lower bounds ub and lb of decision variables, dimension dim, and fitness function f(·).

_{max}, the algorithm is terminated, and the optimal solution G is output.

## 3. Proposed Algorithm

#### 3.1. Learning Motivation

#### 3.2. Random Opposition-Based Learning

#### 3.3. Restart Strategy (RS)

_{1}and T

_{2}, respectively, by Equations (18) and (19) and performs boundary processing by Equation (20), and then selects a better solution to replace the original solution. After that, the corresponding trial is reassigned to 0. The formula of the restart strategy is as follows:

#### 3.4. MGTOA Complexity Analysis

- Initialization of problem definition demands O(1) time.
- Initialization of population creation demands O(N × dim) time.
- Updating the population position includes the teacher and student phases and the required time O(2 × T × N × dim).
- Time required for random opposition-based learning O(T × N × dim).
- Time required for restart strategy O(2 × T × N × dim/Limit).
- The cost time of the calculation function includes the calculation time cost of the algorithm itself, the calculation time cost of the random opposition-based learning strategy, and the calculation time cost of the restart strategy. The calculation time cost of the algorithm itself is O(T × N × C). The calculation time cost of the random opposition-based learning strategy is O(T × N × C). The calculation time cost of the restart strategy takes into account the change of the Limit value, so the time cost is O(T × N × C/Limit). The total time cost is O(2 × T × N × C + T × N × C/Limit).

#### 3.5. MGTOA Implementation

Algorithm 1 Pseudo-code of MGTOA |

1. Initialization parameters t, T_{max}, ub, lb, N, dim. |

2. Initialize population x according to Equations (9) and (10). |

3. The fitness values of all individuals are calculated, and the optimal solution G is selected. |

4. While t < T_{max} |

5. Define the teacher according to Equation (8). |

6. Students are divided into elite students (X_{good}) and ordinary students (X_{bad}). The number of elite students is N_{good}. |

7. for i = 1:N |

8. if i < N_{good} |

9. The teacher phase is achieved according to Equations (1)–(3), and (5). |

10. else |

11. The teacher phase is realized according to Equations (4) and (5). |

12. end |

13. Carry out boundary processing for the updated students. |

14. Calculate the average knowledge level of elite students (M) |

15. if i < N_{good} |

16. for j = 1:dim |

17. The elite students get the learning motivation D according to Equation (12) and carry out the student phase through Equations (13) and (7). |

18. end |

19. else |

20. Ordinary students carry out the student phase according to Equations (14) and (7). |

21. end |

22. end |

23. Carry out boundary processing for the updated students. |

24. An inverse solution is generated using a random opposition-based learning strategy by Equation (15), and the student position is updated according to Equation (16). |

25. Calculate the new fitness value of the students and judge whether it is better. If it is better, replace the fitness value and the corresponding trial = 0. Otherwise, trial will add 1 |

26. Define Limit according to Equation (17). |

27. for i = 1:N |

28. while trial(i) < Limit |

29. T_{1} and T_{2} are generated by Equations (18) and (19), and T_{2} is subjected to boundary processing using Equation (20). Assign a smaller position to x_{i}. |

30. trial(i) = 0 |

31. end |

32. end |

33. t = t + 1 |

34. end |

## 4. Experimental Results and Discussion

#### 4.1. Experiments on Standard Benchmark Functions

_{min}is the optimal value the corresponding function can achieve. Among the 23 standard benchmark functions, set the population number N = 30, the maximum number of iterations T = 500, and the dimension dim = 30/500. All the algorithms run independently 30 times to obtain the optimal fitness value, the average fitness value, and the standard deviation.

F | Metric | MGTOA | GTOA [28] | GA [11] | SCA [37] | BES [38] | ROA [10] | AOA [39] | WOA [9] | BTLBO [40] | TLBO [25] |
---|---|---|---|---|---|---|---|---|---|---|---|

F1 | min | 0 | 1.13 × 10^{−14} | 9.69 × 10^{−3} | 8.36 × 10^{−2} | 0 | 0 | 3.05 × 10^{−193} | 6.03 × 10^{−86} | 1.47 × 10^{−97} | 5.3 × 10^{−81} |

mean | 0 | 5.92 × 10^{−6} | 2.37 × 10^{−2} | 16.5 | 0 | 2.99 × 10^{−319} | 2.56 × 10^{−30} | 2.21 × 10^{−68} | 9.71 × 10^{−96} | 6.99 × 10^{−79} | |

std | 0 | 2.42 × 10^{−5} | 6.57 × 10^{−3} | 36.5 | 0 | 0 | 1.40 × 10^{−29} | 1.21 × 10^{−67} | 2.03 × 10^{−95} | 1.76 × 10^{−78} | |

F2 | min | 0 | 3.43 × 10^{−7} | 3.64 × 10^{−1} | 1.47 × 10^{−6} | 2.38 × 10^{−229} | 1.70 × 10^{−182} | 0 | 1.7 × 10^{−59} | 3.54 × 10^{−49} | 1.24 × 10^{−40} |

mean | 0 | 5.83 × 10^{−4} | 4.91 × 10^{−1} | 1.43 × 10^{−2} | 2.43 × 10^{−161} | 2.91 × 10^{−162} | 0 | 1.92 × 10^{−49} | 2.63 × 10^{−48} | 5.07 × 10^{−40} | |

std | 0 | 1.67 × 10^{−3} | 7.28 × 10^{−2} | 2.31 × 10^{−2} | 1.33 × 10^{−160} | 1.56 × 10^{−161} | 0 | 8.92 × 10^{−49} | 2.63 × 10^{−48} | 5.35 × 10^{−40} | |

F3 | min | 0 | 3.58 × 10^{−11} | 9.24 × 10^{3} | 1.82 × 10^{3} | 0 | 1.59 × 10^{−321} | 1.96 × 10^{−132} | 2.11 × 10^{4} | 2.29 × 10^{−37} | 1.15 × 10^{−19} |

mean | 0 | 5.10 × 10^{−4} | 2.16 × 10^{4} | 8.87 × 10^{3} | 1.76 × 10^{−27} | 1.02 × 10^{−284} | 2.9 × 10^{−3} | 4.08 × 10^{4} | 5.54 × 10^{−33} | 1.3 × 10^{−16} | |

std | 0 | 2.63 × 10^{−3} | 8.05 × 10^{3} | 5.64 × 10^{3} | 9.64 × 10^{−27} | 0 | 5.97 × 10^{−3} | 1.37 × 10^{4} | 1.51 × 10^{−32} | 5.06 × 10^{−16} | |

F4 | min | 0 | 3.04 × 10^{−7} | 2.15 × 10^{−1} | 9.56 | 9.19 × 10^{−237} | 7.35 × 10^{−180} | 7.99 × 10^{−51} | 8.24 | 3.52 × 10^{−40} | 2.09 × 10^{−33} |

mean | 0 | 4.8 × 10^{−4} | 2.82 × 10^{−1} | 36.4 | 9.34 × 10^{−171} | 4.87 × 10^{−152} | 2.25 × 10^{−2} | 54.0 | 2.27 × 10^{−39} | 1.43 × 10^{−32} | |

std | 0 | 8.52 × 10^{−4} | 3.86 × 10^{−2} | 13.1 | 0 | 2.63 × 10^{−151} | 2.12 × 10^{−2} | 24.3 | 3 × 10^{−39} | 1.30 × 10^{−32} | |

F5 | min | 5.39 × 10^{−6} | 28.9 | 25.5 | 44.6 | 9.27 × 10^{−4} | 26.6 | 27.8 | 2.73 × 10^{−1} | 2.29 × 10^{−1} | 23.4 |

mean | 8.90 × 10^{−1} | 28.9 | 70.7 | 8.17 × 10^{4} | 21.8 | 27.1 | 28.5 | 28.0 | 23.9 | 24.7 | |

std | 4.7 | 2.79 × 10^{−2} | 30.7 | 1.83 × 10^{5} | 12 | 3.63 × 10^{−1} | 3.08 × 10^{−1} | 4.64 × 10^{−1} | 6.67 × 10^{−1} | 6.32 × 10^{−1} | |

F6 | min | 1.4 × 10^{−5} | 4.42 | 7.83 | 5.52 | 3.21 × 10^{−4} | 1.69 × 10^{−2} | 2.69 | 1.14 × 10^{−1} | 4.84 × 10^{−9} | 2.66 × 10^{−8} |

mean | 1.11 × 10^{−3} | 5.67 | 8.07 | 25.8 | 1.35 | 1.04 × 10^{−1} | 3.24 | 4.47 × 10^{−1} | 1.06 × 10^{−6} | 1.17 × 10^{−6} | |

std | 1.18 × 10^{−3} | 7.71 × 10^{−1} | 1.38 × 10^{−1} | 49 | 2.8 | 1.21 × 10^{−1} | 2.74 × 10^{−1} | 3.27 × 10^{−1} | 2.64 × 10^{−6} | 3.77 × 10^{−6} | |

F7 | min | 4.97 × 10^{−8} | 1.06 × 10^{−4} | 8.88 × 10^{−2} | 1.7 × 10^{−2} | 2.03 × 10^{−3} | 4.43 × 10^{−6} | 2.47 × 10^{−7} | 9.51 × 10^{−5} | 3.60 × 10^{−4} | 5.57 × 10^{−4} |

mean | 3.79 × 10^{−5} | 4.49 × 10^{−4} | 1.98 × 10^{−1} | 1.1 × 10^{−1} | 6.19 × 10^{−3} | 1.85 × 10^{−4} | 8 × 10^{−5} | 3.32 × 10^{−3} | 8.10 × 10^{−4} | 1.13 × 10^{−3} | |

std | 3.29 × 10^{−5} | 2.6 × 10^{−4} | 7.02 × 10^{−2} | 1.08 × 10^{−1} | 3.54 × 10^{−3} | 1.71 × 10^{−4} | 8.71 × 10^{−5} | 3.65 × 10^{−3} | 4.16 × 10^{−4} | 5.20 × 10^{−4} | |

F8 | min | −1.26 × 10^{4} | −6.16 × 10^{3} | −5.94 × 10^{3} | −4.73 × 10^{3} | −9.97 × 10^{3} | −1.26 × 10^{4} | −6.19 × 10^{3} | −1.26 × 10^{4} | −9.87 × 10^{3} | −9.45 × 10^{3} |

mean | −1.26 × 10^{4} | −5.09 × 10^{3} | −4.73 × 10^{3} | −3.78 × 10^{3} | −5.80 × 10^{3} | −1.24 × 10^{4} | −5.41 × 10^{3} | −1.03 × 10^{4} | −7.60 × 10^{3} | −7.67 × 10^{3} | |

std | 6.00 × 10^{−2} | 6.87 × 10^{2} | 6.85 × 10^{2} | 3.61 × 10^{2} | 3.23 × 10^{3} | 3.26 × 10^{2} | 4.37 × 10^{2} | 1.68 × 10^{3} | 1.01 × 10^{3} | 1.21 × 10^{3} | |

F9 | min | 0 | 5.7 × 10^{−11} | 1.37 | 1.25 × 10^{−1} | 0 | 0 | 0 | 0 | 8.04 | 9.95 |

mean | 0 | 2.44 × 10^{−5} | 2.74 | 34.7 | 0 | 0 | 0 | 7.58 × 10^{−15} | 19.0 | 14.4 | |

std | 0 | 8.93 × 10^{−5} | 8.29 × 10^{−1} | 30.5 | 0 | 0 | 0 | 2.47 × 10^{−14} | 8.82 | 6.97 | |

F10 | min | 8.88 × 10^{−16} | 2.55 × 10^{−8} | 9.25 × 10^{−2} | 4.78 × 10^{−2} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 4.44 × 10^{−15} | 4.44 × 10^{−15} |

mean | 8.88 × 10^{−16} | 3.36 × 10^{−4} | 1.33 × 10^{−1} | 15.4 | 1.13 × 10^{−15} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 4.8 × 10^{−15} | 5.03 × 10^{−15} | 6.34 × 10^{−15} | |

std | 0 | 7.52 × 10^{−4} | 3.09 × 10^{−2} | 7.96 | 9.01 × 10^{−16} | 0 | 0 | 2.35 × 10^{−15} | 1.35 × 10^{−15} | 1.8 × 10^{−15} | |

F11 | min | 0 | 1.06 × 10^{−12} | 6.76 × 10^{−4} | 1.83 × 10^{−2} | 0 | 0 | 2.66 × 10^{−2} | 0 | 0 | 0 |

mean | 0 | 1.11 × 10^{−5} | 8.6 × 10^{−2} | 9.91 × 10^{−1} | 0 | 0 | 1.88 × 10^{−1} | 1.18 × 10^{−2} | 0 | 2.76 × 10^{−6} | |

std | 0 | 4.1 × 10^{−5} | 2.7 × 10^{−1} | 3.07 × 10^{−1} | 0 | 0 | 1.59 × 10^{−1} | 6.45 × 10^{−2} | 0 | 1.51 × 10^{−5} | |

F12 | min | 3.52 × 10^{−7} | 3.44 × 10^{−1} | 1.55 | 8.5 × 10^{−1} | 1.48 × 10^{−6} | 2.78 × 10^{−3} | 3.98 × 10^{−1} | 7.12 × 10^{−3} | 2.49 × 10^{−10} | 5.78 × 10^{−11} |

mean | 2.15 × 10^{−5} | 6.21 × 10^{−1} | 1.72 | 5.51 × 10^{4} | 1.33 × 10^{−1} | 1.07 × 10^{−2} | 5.24 × 10^{−1} | 4.39 × 10^{−2} | 1.02 × 10^{−8} | 1.26 × 10^{−8} | |

std | 2.01 × 10^{−5} | 1.88 × 10^{−1} | 5.17 × 10^{−2} | 1.74 × 10^{5} | 3.28 × 10^{−1} | 6.41 × 10^{−3} | 4.53 × 10^{−2} | 1 × 10^{−1} | 2.68 × 10^{−8} | 3.63 × 10^{−8} | |

F13 | min | 1.17 × 10^{−6} | 2.45 | 1.37 × 10^{−3} | 4.82 | 4.66 × 10^{−5} | 7.92 × 10^{−2} | 2.65 | 1.72 × 10^{−1} | 2.13 × 10^{−7} | 1.59 × 10^{−7} |

mean | 2.65 × 10^{−4} | 2.87 | 4.46 × 10^{−3} | 1.89 × 10^{5} | 1.22 | 2.4 × 10^{−1} | 2.82 | 5.69 × 10^{−1} | 9.96 × 10^{−2} | 8.4 × 10^{−2} | |

std | 3.56 × 10^{−4} | 2.26 × 10^{−1} | 3.29 × 10^{−3} | 5.58 × 10^{5} | 1.47 | 1.38 × 10^{−1} | 9.65 × 10^{2} | 2.44 × 10^{−1} | 8.27 × 10^{−2} | 1.23 × 10^{−1} |

F | Metric | MGTOA | GTOA [28] | GA [11] | SCA [37] | BES [38] | ROA [10] | AOA [39] | WOA [9] | BTLBO [40] | TLBO [25] |
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F1 | min | 0 | 3.10 × 10^{−13} | 67.2 | 9.04 × 10^{4} | 0 | 0 | 5.60 × 10^{−1} | 2.86 × 10^{−81} | 3.95 × 10^{−85} | 1.12 × 10^{−68} |

mean | 0 | 1.37 × 10^{−4} | 70.6 | 2.03 × 10^{5} | 0 | 2.17 × 10^{−318} | 6.43 × 10^{−1} | 1.49 × 10^{−67} | 9.97 × 10^{−84} | 1.34 × 10^{−67} | |

std | 0 | 6.23 × 10^{−4} | 2.63 | 8.08 × 10^{4} | 0 | 0 | 4.45 × 10^{−2} | 8.06 × 10^{−67} | 1.03 × 10^{−83} | 1.78 × 10^{−67} | |

F2 | min | 0 | 2.76 × 10^{−7} | 1.35 × 10^{2} | 31.8 | 6.28 × 10^{−225} | 9.78 × 10^{−177} | 3.81 × 10^{−12} | 9.26 × 10^{−55} | 4.58 × 10^{−43} | 7.22 × 10^{−35} |

mean | 0 | 5.98 × 10^{−3} | 1.40 × 10^{2} | 1.07 × 10^{2} | 2.93 × 10^{−153} | 3.10 × 10^{−151} | 1.82 × 10^{−3} | 1.3 × 10^{−47} | 2.06 × 10^{−42} | 2.16 × 10^{−34} | |

std | 0 | 1.28 × 10^{−2} | 3.03 | 57.5 | 1.6 × 10^{−152} | 1.70 × 10^{−150} | 1.7 × 10^{−3} | 6.57 × 10^{−47} | 1.53 × 10^{−42} | 1.35 × 10^{−34} | |

F3 | min | 0 | 9.38 × 10^{−9} | 4.86 × 10^{5} | 5.09 × 10^{6} | 0 | 7.16 × 10^{−299} | 13.7 | 1.88 × 10^{7} | 2.77 × 10^{−13} | 4.58 × 10^{−3} |

mean | 0 | 1.23 × 10^{−1} | 7.17 × 10^{5} | 6.75 × 10^{6} | 8.68 × 10^{5} | 5.08 × 10^{−261} | 3.42 × 10^{3} | 3.32 × 10^{7} | 5.59 × 10^{−6} | 3.8 × 10^{−1} | |

std | 0 | 5.99 × 10^{−1} | 1.39 × 10^{5} | 1.42 × 10^{6} | 4.43 × 10^{6} | 0 | 1.85 × 10^{4} | 1.23 × 10^{7} | 2.41 × 10^{−5} | 1.27 | |

F4 | min | 0 | 5.2 × 10^{−8} | 9.52 × 10^{−1} | 98.6 | 8.66 × 10^{−217} | 2.23 × 10^{−173} | 1.63 × 10^{−1} | 55.1 | 2.13 × 10^{−35} | 5.69 × 10^{−28} |

mean | 0 | 2.8 × 10^{−4} | 9.71 × 10^{−1} | 99.0 | 1.3 × 10^{−128} | 7.74 × 10^{−152} | 1.81 × 10^{−1} | 81.3 | 1.22 × 10^{−34} | 1.42 × 10^{−27} | |

std | 0 | 4.51 × 10^{−4} | 1.04 × 10^{−2} | 3.42 × 10^{−1} | 7.13 × 10^{−128} | 4.13 × 10^{−151} | 1.65 × 10^{−2} | 22.1 | 8.42 × 10^{−35} | 9.58 × 10^{−28} | |

F5 | min | 4.12 × 10^{−8} | 4.99 × 10^{2} | 4.90 × 10^{3} | 1.12 × 10^{9} | 1.13 × 10^{2} | 4.94 × 10^{2} | 4.99 × 10^{2} | 4.96 × 10^{2} | 4.96 × 10^{2} | 4.96 × 10^{2} |

mean | 1.16 × 10^{2} | 4.99 × 10^{2} | 5.14 × 10^{3} | 1.95 × 10^{9} | 4.32 × 10^{2} | 4.95 × 10^{2} | 4.99 × 10^{2} | 4.96 × 10^{2} | 4.97 × 10^{2} | 4.97 × 10^{2} | |

std | 2.14 × 10^{2} | 3.20 × 10^{−2} | 1.76 × 10^{2} | 5.05 × 10^{8} | 1.67 × 10^{2} | 2.94 × 10^{−1} | 1.03 × 10^{−1} | 4.20 × 10^{−1} | 6.30 × 10^{−1} | 4 × 10^{−1} | |

F6 | min | 9.08 × 10^{−5} | 1.22 × 10^{2} | 3.35 × 10^{2} | 7.45 × 10^{4} | 1.14 × 10^{−2} | 7.29 | 1.14 × 10^{2} | 20.3 | 70 | 71.6 |

mean | 16.8 | 1.23 × 10^{2} | 3.45 × 10^{2} | 2.42 × 10^{5} | 30.6 | 15.3 | 1.16 × 10^{2} | 32.6 | 75.4 | 75.5 | |

std | 35.3 | 7.36 × 10^{−1} | 5.84 | 9.01 × 10^{4} | 53 | 6.51 | 1.38 | 9.53 | 2.39 | 2.11 | |

F7 | min | 2.6 × 10^{−8} | 7.82 × 10^{−5} | 4.30 × 10^{3} | 9.38 × 10^{3} | 8.2 × 10^{−4} | 5.87 × 10^{−6} | 1.21 × 10^{−5} | 8.52 × 10^{−5} | 7.16 × 10^{−4} | 8.04 × 10^{−4} |

mean | 3.43 × 10^{−5} | 6.1 × 10^{−4} | 4.56 × 10^{3} | 1.44 × 10^{4} | 5.75 × 10^{−3} | 2.08 × 10^{−4} | 1.06 × 10^{−4} | 4.37 × 10^{−3} | 1.3 × 10^{−3} | 1.66 × 10^{−3} | |

std | 3.22 × 10^{−5} | 6.31 × 10^{−4} | 2.74 × 10^{2} | 3.4 × 10^{3} | 4.01 × 10^{−3} | 1.66 × 10^{−4} | 9.94 × 10^{−5} | 5.6 × 10^{−3} | 4.27 × 10^{−4} | 5.37 × 10^{−4} | |

F8 | min | −2.09 × 10^{5} | −2.85 × 10^{4} | −3.63 × 10^{4} | −1.73 × 10^{4} | −2.08 × 10^{5} | −2.09 × 10^{5} | −2.58 × 10^{4} | −2.09 × 10^{5} | −7.28 × 10^{4} | −6.17 × 10^{4} |

mean | −2.09 × 10^{5} | −2.15 × 10^{4} | −3.29 × 10^{4} | −1.53 × 10^{4} | −1.61 × 10^{5} | −2.05 × 10^{5} | −2.3 × 10^{4} | −1.7 × 10^{5} | −3.39 × 10^{4} | −4.39 × 10^{4} | |

std | 1.18 | 3.12 × 10^{3} | 1.91 × 10^{3} | 1.23 × 10^{3} | 2.49 × 10^{4} | 1.02 × 10^{4} | 1.58 × 10^{3} | 3.1 × 10^{4} | 1.10 × 10^{4} | 1.21 × 10^{4} | |

F9 | min | 0 | 0 | 2.26 × 10^{3} | 4.19 × 10^{2} | 0 | 0 | 0 | 0 | 0 | 0 |

mean | 0 | 9.27 × 10^{−5} | 2.41 × 10^{3} | 1.14 × 10^{3} | 0 | 0 | 8.08 × 10^{−6} | 9.09 × 10^{−14} | 0 | 0 | |

std | 0 | 3.52 × 10^{−4} | 72.5 | 5.78 × 10^{2} | 0 | 0 | 7.61 × 10^{−6} | 3.66 × 10^{−13} | 0 | 0 | |

F10 | min | 8.88 × 10^{−16} | 2.26 × 10^{−8} | 2.85 | 10.7 | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 7.09 × 10^{−3} | 8.88 × 10^{−16} | 7.99 × 10^{−15} | 7.99 × 10^{−15} |

mean | 8.88 × 10^{−16} | 2.84 × 10^{−4} | 2.91 | 18.4 | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.02 × 10^{−3} | 4.91 × 10^{−15} | 7.99 × 10^{−15} | 2.22 | |

std | 0 | 5.35 × 10^{−4} | 2.92 × 10^{−2} | 4.11 | 0 | 0 | 4.3 × 10^{−4} | 2.23 × 10^{−15} | 0 | 4.17 | |

F11 | min | 0 | 2.10 × 10^{−12} | 2.28 × 10^{−1} | 1.13 × 10^{3} | 0 | 0 | 6.52 × 10^{3} | 0 | 0 | 0 |

mean | 0 | 4.72 × 10^{−5} | 3.05 × 10^{−1} | 2.08 × 10^{3} | 0 | 0 | 9.99 × 10^{3} | 3.7 × 10^{−18} | 0 | 3.7 × 10^{−18} | |

std | 0 | 2.56 × 10^{−4} | 2.66 × 10^{−1} | 7.11 × 10^{2} | 0 | 0 | 3.09 × 10^{3} | 2.03 × 10^{−17} | 0 | 2.03 × 10^{−17} | |

F12 | min | 1.69 × 10^{−8} | 1.09 | 2.73 | 4.03 × 10^{9} | 1.24 × 10^{−5} | 9.45 × 10^{−3} | 1.07 | 3.97 × 10^{−2} | 3.84 × 10^{−1} | 3.67 × 10^{−1} |

mean | 1.08 × 10^{−5} | 1.14 | 2.80 | 6.28 × 10^{9} | 2.42 × 10^{−1} | 4.54 × 10^{−2} | 1.08 | 1.01 × 10^{−1} | 4.3 × 10^{−1} | 4.27 × 10^{−1} | |

std | 2.10 × 10^{−5} | 3.37 × 10^{−2} | 4.75 × 10^{−2} | 1.36 × 10^{9} | 4.89 × 10^{−1} | 2.79 × 10^{−2} | 1.2 × 10^{−2} | 5.11 × 10^{−2} | 2.75 × 10^{−2} | 2.96 × 10^{−2} | |

F13 | min | 5.42 × 10^{−11} | 50 | 10.2 | 6.62 × 10^{9} | 3.46 × 10^{−3} | 3.08 | 50.1 | 10.7 | 49.8 | 49.8 |

mean | 1.87 × 10^{−3} | 50 | 10.8 | 1.06 × 10^{10} | 12.2 | 8.69 | 50.2 | 19.3 | 49.8 | 49.8 | |

std |
5.68 × 10^{−3} | 4.52 × 10^{−3} | 4.73 × 10^{−1} | 2.11 × 10^{9} | 21.1 | 3.88 | 4.54 × 10^{−2} | 6.02 | 1.01 × 10^{−2} | 8.82 × 10^{−3} |

F | Metric | MGTOA | GTOA [28] | GA [11] | SCA [37] | BES [38] | ROA [10] | AOA [39] | WOA [9] | BTLBO [40] | TLBO [25] |
---|---|---|---|---|---|---|---|---|---|---|---|

F14 | min | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 2.98 | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 9.98 × 10^{−1} |

mean | 9.98 × 10^{−1} | 1.16 | 9.43 | 1.83 | 3.06 | 5.75 | 10.2 | 3.68 | 9.98 × 10^{−1} | 9.98 × 10^{−1} | |

std | 3.38 × 10^{−11} | 5.27 × 10^{−1} | 3.58 | 1.89 | 1.39 | 5.05 | 4.05 | 3.47 | 0 | 0 | |

F15 | min | 3.07 × 10^{−4} | 3.07 × 10^{−4} | 4.35 × 10^{−4} | 3.94 × 10^{−4} | 3.14 × 10^{−4} | 3.08 × 10^{−4} | 3.54 × 10^{−4} | 3.19 × 10^{−4} | 3.07 × 10^{−4} | 3.07 × 10^{−4} |

mean | 3.08 × 10^{−4} | 1.89 × 10^{−3} | 1.29 × 10^{−2} | 9.83 × 10^{−4} | 7.02 × 10^{−3} | 4.32 × 10^{−4} | 1.22 × 10^{−2} | 7.49 × 10^{−4} | 3.25 × 10^{−4} | 3.50 × 10^{−4} | |

std | 2.54 × 10^{−7} | 5.34 × 10^{−3} | 2.75 × 10^{−2} | 3.98 × 10^{−4} | 8.72 × 10^{−3} | 2.27 × 10^{−4} | 2.66 × 10^{−2} | 5.11 × 10^{−4} | 7.35 × 10^{−5} | 1.19 × 10^{−4} | |

F16 | min | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 |

mean | −1.03 | −1.03 | −1.00 | −1.03 | −9.54 × 10^{−1} | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | |

std | 5.25 × 10^{−16} | 6 × 10^{−16} | 2.07 × 10^{−2} | 8.02 × 10^{−5} | 2.25 × 10^{−1} | 1.25 × 10^{−7} | 1.21 × 10^{−7} | 2.04 × 10^{−9} | 6.78 × 10^{−16} | 6.65 × 10^{−16} | |

F17 | min | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 4.50 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} |

mean | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 1.50 | 3.99 × 10^{−1} | 5.81 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | 3.98 × 10^{−1} | |

std | 6.21 × 10^{−7} | 0 | 1.48 | 1.6 × 10^{−3} | 6.48 × 10^{−1} | 6.34 × 10^{−6} | 7.93 × 10^{−8} | 1.48 × 10^{−5} | 0 | 0 | |

F18 | min | 3 | 3 | 3.2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

mean | 3 | 3 | 29.5 | 3 | 4.39 | 3 | 9.3 | 3 | 3 | 3 | |

std | 2.06 × 10^{−5} | 8.4 × 10^{−15} | 22.2 | 1.33 × 10^{−4} | 1.51 | 1.61 × 10^{−4} | 11.6 | 1.71 × 10^{−4} | 1.31 × 10^{−15} | 9.79 × 10^{−16} | |

F19 | min | −3.86 | −3.86 | −3.86 | −3.86 | −3.85 | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 |

mean | −3.86 | −3.86 | −3.71 | −3.85 | −3.65 | −3.86 | −3.85 | −3.86 | −3.86 | −3.86 | |

std | 9.49 × 10^{−6} | 2.63 × 10^{−15} | 5.48 × 10^{−1} | 1.09 × 10^{−2} | 1.89 × 10^{−1} | 2.8 × 10^{−3} | 5.9 × 10^{−3} | 7.31 × 10^{−3} | 2.71 × 10^{−15} | 2.71 × 10^{−15} | |

F20 | min | −3.32 | −3.32 | −3.32 | −3.11 | −3.19 | −3.32 | −3.17 | −3.32 | −3.32 | −3.32 |

mean | −3.29 | −3.26 | −3.28 | −2.70 | −2.91 | −3.22 | −3.04 | −3.22 | −3.3 | −3.31 | |

std | 5.27 × 10^{−2} | 7.66 × 10^{−2} | 5.55 × 10^{−2} | 5.27 × 10^{−1} | 2.4 × 10^{−1} | 1.24 × 10^{−1} | 8.89 × 10^{−2} | 1.71 × 10^{−1} | 3.67 × 10^{−2} | 3.71 × 10^{−2} | |

F21 | min | −10.2 | −10.2 | −5.05 | −7.36 | −10.2 | −10.2 | −6.85 | −10.2 | −10.2 | −10.2 |

mean | −10.2 | −8 | −1.47 | −2.27 | −6.55 | −10.1 | −3.93 | −7.94 | −10.1 | −9.56 | |

std | 2.90 × 10^{−4} | 2.75 | 1.49 | 2.06 | 2.76 | 1.6 × 10^{−2} | 1.72 | 2.79 | 7.74 × 10^{−2} | 1.81 | |

F22 | min | −10.4 | −10.4 | −5.07 | −5.62 | −10.4 | −10.4 | −6.05 | −10.4 | −10.4 | −10.4 |

mean | −10.4 | −8.04 | −1.67 | −3.29 | −5.43 | −10.4 | −3.38 | −7.76 | −10.4 | −10 | |

std | 1.29 × 10^{−4} | 2.97 | 1.25 | 1.85 | 2.61 | 2.41 × 10^{−2} | 1.49 | 2.91 | 7.38 × 10^{−16} | 1.35 | |

F23 | min | −10.5 | −10.5 | −5.13 | −5.93 | −10.5 | −10.5 | −6.69 | −10.5 | −10.5 | −10.5 |

mean | −10.5 | −7.98 | −1.76 | −3.73 | −5.55 | −10.5 | −3.52 | −6.27 | −10.3 | −10.1 | |

std | 1.28 × 10^{−4} | 3.24 | 1.40 | 1.54 | 2.73 | 2.1 × 10^{−2} | 1.26 | 3.26 | 1.21 | 1.74 |

F | dim | MGTOA vs. GTOA [28] | MGTOA vs. GA [11] | MGTOA vs. SCA [37] | MGTOA vs. BES [38] | MGTOA vs. ROA [10] | MGTOA vs. AOA [39] | MGTOA vs. WOA [9] | MGTOA vs. BTLBO [40] | MGTOA vs. TLBO [25] |
---|---|---|---|---|---|---|---|---|---|---|

F1 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1.25 × 10^{−1} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 3.13 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F2 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F3 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.5 × 10^{−1} | 3.79 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 3.1 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F4 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F5 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.38 × 10^{−3} | 6.16 × 10^{−4} | 1.73 × 10^{−6} | 6.16 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F6 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.6 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.89 × 10^{−1} | 4.72 × 10^{−2} | 1.73 × 10^{−6} | 8.97 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F7 | 30 | 3.18 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 6.64 × 10^{−4} | 2.7 × 10^{−2} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.36 × 10^{−5} | 2.22 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F8 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 3.59 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 3.18 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F9 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1 | 5 × 10^{−1} | 1.73 × 10^{−6} | 3.79 × 10^{−6} |

500 | 2.56 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1.32 × 10^{−4} | 1 | 1 | 1 | |

F10 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1 | 2.29 × 10^{−5} | 2.57 × 10^{−7} | 7.86 × 10^{−7} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1.73 × 10^{−6} | 2.04 × 10^{−5} | 4.32 × 10^{−8} | 1.11 × 10^{−6} | |

F11 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1.73 × 10^{−6} | 6.25 × 10^{−2} | 1 | 1 |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 | 1.73 × 10^{−6} | 1 | 1 | 1 | |

F12 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} |

500 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.48 × 10^{−3} | 2.77 × 10^{−3} | 1.73 × 10^{−6} | 2.77 × 10^{−3} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F13 | 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.6 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.11 × 10^{−3} | 2.41 × 10^{−4} |

500 | 1.73 × 10^{−6} | 2.8 × 10^{−3} | 1.73 × 10^{−6} | 2.26 × 10^{−3} | 2.77 × 10^{−3} | 1.73 × 10^{−6} | 1.71 × 10^{−3} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | |

F14 | 2 | 4.07 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 4.07 × 10^{−5} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

F15 | 4 | 2.41 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 5.75 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 7.71 × 10^{−4} | 1.41 × 10^{−1} |

F16 | 2 | 1 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1 | 1 |

F17 | 2 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 5.79 × 10^{−5} | 4.53 × 10^{−4} | 6.64 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

F18 | 2 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 5.79 × 10^{−5} | 1.73 × 10^{−6} | 1.04 × 10^{−3} | 9.75 × 10^{−1} | 3.61 × 10^{−3} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

F19 | 3 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.84 × 10^{−5} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

F20 | 6 | 3.38 × 10^{−3} | 3.68 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.48 × 10^{−2} | 2.13 × 10^{−6} | 3.85 × 10^{−3} | 3.59 × 10^{−4} | 1.48 × 10^{−2} |

F21 | 4 | 1.48 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.77 × 10^{−3} |

F22 | 4 | 6.16 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.48 × 10^{−2} |

F23 | 4 | 9.27 × 10^{−3} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} | 1.73 × 10^{−6} | 3.59 × 10^{−4} |

#### 4.2. Experiments on CEC2014 Test Suite

## 5. Constrained Engineering Design Problems

#### 5.1. Welded Beam Design Problem

#### 5.2. Pressure Vessel Design Problem

_{s}, head thickness T

_{h}, inner radius R, and container length L regardless of the head. Where T

_{s}and T

_{h}are integers of 0.625 and R and L are continuous variables. The specific constraints are referred to in [43]. The schematic diagram of optimal structure design is shown in Figure 9.

_{s}= 0.754364, T

_{h}= 0.366375, R = 40.42809, L = 198.5652 of MGTOA. While for GTOA, T

_{s}= 0.778169, T

_{h}= 0.38465, R = 40.3196, L = 200, the variables obtained by MGTOA are obviously better. The final cost of MGTOA is 5752.402458. Compared with other algorithms, the cost of MGTOA is greatly reduced, and good results are achieved, which shows that MGTOA has excellent effects on this problem.

#### 5.3. Tension/Compression Spring Design Problem

_{1}), shear stress (g

_{2}), impact frequency (g

_{3}), and outer diameter limit (g

_{4}). The specific constraints are referred to in [48]. Corresponding decision variables include wire diameter d, average coil diameter D, and adequate coil number N. f(x) is the minimum spring mass.

#### 5.4. Three-Bar Truss Design Problem

_{1}(=x

_{1}) and A

_{2}(=x

_{2}). Among them, the constraint conditions refer to literature [4].

#### 5.5. Car Crashworthiness Design Problem

_{1}, x

_{3}, x

_{4}, and x

_{7}all reached 0.5, and the final weight obtained the best solution compared with other algorithms.

#### 5.6. Gear Train Design Problem

_{A}, n

_{B}, n

_{C}, n

_{D}). The constraints refer to literature [60]. Specific schematic diagram is shown in Figure 13.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Convergence curves for the optimization algorithms for standard benchmark functions (F1–F13) with dim = 30.

**Figure 5.**Convergence curves for the optimization algorithms for standard benchmark functions (F1–F13) with dim = 500.

**Figure 6.**Convergence curves for the optimization algorithms for standard benchmark functions (F14–F23).

Algorithm | Parameters | Value |
---|---|---|

BTLBO [40] | T_{F} | 1 or 2 |

θ | 0 or 1 | |

TLBO [25] | T_{F} | 1 or 2 |

WOA [9] | Coefficient vectors $\overrightarrow{A}$ | 1 |

Coefficient vectors $\overrightarrow{C}$ | [−1, 1] | |

Helical parameter b | 0.75 | |

Helical parameter l | [−1, 1] | |

AOA [39] | MOP_Max | 1 |

MOP_Min | 0.2 | |

A | 5 | |

Mu | 0.499 | |

ROA [10] | C | 0.1 |

BES [38] | α | [1.5, 2.0] |

r | [0, 1] | |

SCA [37] | α | 2 |

GA [11] | Type | Real coded |

Selection | Roulette wheel (Proportionate) | |

Crossover | Whole arithmetic | |

(Probability = 0.7) | ||

Mutation | Gaussian | |

(Probability = 0.01) | ||

GTOA [28] | - | - |

MGTOA | Limit | lg(t) |

Type | F | dim | Range | F_{min} |
---|---|---|---|---|

Unimodal benchmark functions | ${F}_{1}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}$ | 30/100/500 | [−100, 100] | 0 |

${F}_{2}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}|{x}_{i}|}\text{}+\text{}{\prod}_{i\text{}=\text{}1}^{n}|{x}_{i}|$ | 30/100/500 | [−10, 10] | 0 | |

${F}_{3}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}{({\displaystyle {\sum}_{j-1}^{i}{x}_{j}})}^{2}}$ | 30/100/500 | [−100, 100] | 0 | |

${F}_{4}(x)\text{}=\text{}\mathrm{max}\{|{x}_{i}|,1\le i\le n\}$ | 30/100/500 | [−100, 100] | 0 | |

${F}_{5}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n\text{}-\text{}1}[100{({x}_{i\text{}+\text{}1}\text{}-\text{}{x}_{i}^{2})}^{2}\text{}+\text{}{({x}_{i}\text{}-\text{}1)}^{2}]}$ | 30/100/500 | [−30, 30] | 0 | |

${F}_{6}(x)\text{}=\text{}{{\displaystyle {\sum}_{i=1}^{n}({x}_{i}\text{}+\text{}5)}}^{2}$ | 30/100/500 | [−100, 100] | 0 | |

${F}_{7}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}i\text{}\times \text{}{x}_{i}^{4}\text{}+\text{}random[0,1)}$ | 30/100/500 | [−1.28, 1.28] | 0 | |

Multimodal benchmark functions | ${F}_{8}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}\text{}-{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})}$ | 30/100/500 | [−500, 500] | −418.9829 × dim |

${F}_{9}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{n}[{x}_{i}^{2}\text{}-\text{}10\mathrm{cos}(2\pi {x}_{i})\text{}+\text{}10]}$ | 30/100/500 | [−5.12, 5.12] | 0 | |

${F}_{10}(x)\text{}=\text{}-20\mathrm{exp}(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}}\text{}-\text{}\mathrm{exp}(\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\mathrm{cos}(2\pi {x}_{i})})\text{}+\text{}20\text{}+\text{}e)$ | 30/100/500 | [−32, 32] | 0 | |

${F}_{11}(x)\text{}=\text{}\frac{1}{400}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}\text{}-\text{}{\mathrm{\Pi}}_{i\text{}=\text{}1}^{n}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}})\text{}+\text{}1}$ | 30/100/500 | [−600, 600] | 0 | |

$\begin{array}{l}{F}_{12}(x)\text{}=\text{}\frac{\pi}{n}\{10\mathrm{sin}(\pi {y}_{1})\text{}+\text{}{\displaystyle {\sum}_{i=1}^{n\text{}-\text{}1}{({y}_{i}\text{}-\text{}1)}^{2}[1\text{}+\text{}10{\mathrm{sin}}^{2}(\pi {y}_{i\text{}+\text{}1})]\text{}+\text{}{({y}_{n}\text{}-\text{}1)}^{2}}\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\text{}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},10,100,4),\mathrm{where}\text{}{y}_{i}\text{}=\text{}1\text{}+\text{}\frac{{x}_{i}\text{}+\text{}1}{4}},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}u({x}_{i},a,k,m)\text{}=\text{}\left\{\begin{array}{l}k{({x}_{i}\text{}-\text{}a)}^{m}\hspace{1em}{x}_{i}a\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-a{x}_{i}a\\ k{(-{x}_{i}\text{}-\text{}a)}^{m}\text{}{x}_{i}-a\end{array}\right.\end{array}$ | 30/100/500 | [−50, 50] | 0 | |

$\begin{array}{l}{F}_{13}(x)\text{}=\text{}0.1({\mathrm{sin}}^{2}(3\pi {x}_{1})\text{}+\text{}{\displaystyle {\sum}_{i=1}^{n}{({x}_{i}\text{}-\text{}1)}^{2}}[1\text{}+\text{}{\mathrm{sin}}^{2}(3\pi {x}_{i}\text{}+\text{}1)]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\text{}+{({x}_{n}\text{}-\text{}1)}^{2}[1\text{}+\text{}{\mathrm{sin}}^{2}(2\pi {x}_{n})])\text{}+\text{}{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},5,100,4)}\end{array}$ | 30/100/500 | [−50, 50] | 0 | |

Fixed-dimension multimodal benchmark functions | ${F}_{14}(x)\text{}=\text{}{(\frac{1}{500}\text{}+\text{}{\displaystyle {\sum}_{j=1}^{25}\frac{1}{j\text{}+\text{}{\displaystyle {\sum}_{i=1}^{2}{({x}_{i}\text{}-\text{}{a}_{ij})}^{6}}}})}^{-1}$ | 2 | [−65, 65] | 1 |

${F}_{15}(x)\text{}=\text{}{\displaystyle {\sum}_{i=1}^{11}{\left[{a}_{i}\text{}-\text{}\frac{{x}_{1}({b}_{i}^{2}\text{}+\text{}{b}_{i}{x}_{2})}{{b}_{i}^{2}\text{}+\text{}{b}_{i}{x}_{3}\text{}+\text{}{x}_{4}}\right]}^{2}}$ | 4 | [−5, 5] | 0.00030 | |

${F}_{16}(x)\text{}=\text{}4{x}_{1}^{2}\text{}-\text{}2.1{x}_{1}^{4}\text{}+\text{}\frac{1}{3}{x}_{1}^{6}\text{}+\text{}{x}_{1}{x}_{2}\text{}-\text{}4{x}_{2}^{2}\text{}+\text{}{x}_{2}^{4}$ | 2 | [−5, 5] | −1.0316 | |

${F}_{17}(x)\text{}=\text{}{({x}_{2}\text{}-\text{}\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}\text{}+\text{}\frac{5}{\pi}{x}_{1}\text{}-\text{}6)}^{2}\text{}+\text{}10(1\text{}-\text{}\frac{1}{8\pi})\mathrm{cos}{x}_{1}\text{}+\text{}10$ | 2 | [−5, 5] | 0.398 | |

$\begin{array}{l}{F}_{18}(x)\text{}=\text{}[1\text{}+\text{}{({x}_{1}\text{}+\text{}{x}_{2}\text{}+\text{}1)}^{2}(19\text{}-\text{}14{x}_{1}\text{}+\text{}3{x}_{1}^{2}\text{}-\text{}14{x}_{2}\text{}+\text{}6{x}_{1}{x}_{2}\text{}+\text{}{3}_{2}^{2})]\\ \text{}\times \text{}[30\text{}+\text{}{(2{x}_{1}\text{}-\text{}3{x}_{2})}^{2}\text{}\times \text{}(18\text{}-\text{}32{x}_{2}\text{}+\text{}12{x}_{1}^{2}\text{}+\text{}48{x}_{2}\text{}-\text{}36{x}_{1}{x}_{2}\text{}+\text{}27{x}_{2}^{2})]\end{array}$ | 5 | [−2, 2] | 3 | |

${F}_{19}(x)\text{}=\text{}-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle {\sum}_{j=1}^{3}{a}_{ij}{({x}_{j}\text{}-\text{}{p}_{ij})}^{2}})}$ | 3 | [−1, 2] | −3.86 | |

${F}_{20}(x)\text{}=\text{}-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle {\sum}_{j=1}^{6}{a}_{ij}{({x}_{j}\text{}-\text{}{p}_{ij})}^{2}})}$ | 6 | [0, 1] | −3.32 | |

${F}_{21}(x)\text{}=\text{}-{\displaystyle {\sum}_{i=1}^{5}{[(X\text{}-\text{}{a}_{i}){(X\text{}-\text{}{a}_{i})}^{T}\text{}+\text{}{c}_{i}]}^{\text{}-\text{}1}}$ | 4 | [0, 10] | −10.1532 | |

${F}_{22}(x)\text{}=\text{}-{\displaystyle {\sum}_{i=1}^{7}{[(X\text{}-\text{}{a}_{i}){(X\text{}-\text{}{a}_{i})}^{T}\text{}+\text{}{c}_{i}]}^{\text{}-\text{}1}}$ | 4 | [0, 10] | −10.4028 | |

${F}_{23}(x)\text{}=\text{}-{\displaystyle {\sum}_{i=1}^{10}{[(X\text{}-\text{}{a}_{i}){(X\text{}-\text{}{a}_{i})}^{T}\text{}+\text{}{c}_{i}]}^{\text{}-\text{}1}}$ | 4 | [0, 10] | −10.5363 |

Name | NO. | Functions | F_{min} |
---|---|---|---|

Unimodal Functions | CEC 1 | Rotated High Conditioned Elliptic Function | 100 |

CEC 2 | Rotated Bent Cigar Function | 200 | |

CEC 3 | Rotated Discus Function | 300 | |

Simple Multimodal Functions | CEC 4 | Shifted and Rotated Rosenbrock’s Function | 400 |

CEC 5 | Shifted and Rotated Ackley’s Function | 500 | |

CEC 6 | Shifted and Rotated Weierstrass Function | 600 | |

CEC 7 | Shifted and Rotated Griewank’s Function | 700 | |

CEC 8 | Shifted Rastrigin’s Function | 800 | |

CEC 9 | Shifted and Rotated Rastrigin’s Function | 900 | |

CEC 10 | Shifted Schwefel’s Function | 1000 | |

CEC 11 | Shifted and Rotated Schwefel’s Schwefel’s Function | 1100 | |

CEC 12 | Shifted and Rotated Katsuura Function | 1200 | |

CEC 13 | Shifted and Rotated HappyCat Function | 1300 | |

CEC 14 | Shifted and Rotated HGBat Function | 1400 | |

CEC 15 | Shifted and Rotated Expanded Griewank’splus Rosenbrock’s Function | 1500 | |

CEC 16 | Shifted and Rotated Expanded Scaffer’s F6 Function | 1600 | |

Hybrid Function 1 | CEC 17 | Hybrid Function 1 (N = 3) | 1700 |

CEC 18 | Hybrid Function 2 (N = 3) | 1800 | |

CEC 19 | Hybrid Function 3 (N = 4) | 1900 | |

CEC 20 | Hybrid Function 4 (N = 4) | 2000 | |

CEC 21 | Hybrid Function 5 (N = 5) | 2100 | |

CEC 22 | Hybrid Function 6 (N = 5) | 2200 | |

Composition Functions | CEC 23 | Composition Function 1 (N = 5) | 2300 |

CEC 24 | Composition Function 2 (N = 3) | 2400 | |

CEC 25 | Composition Function 3 (N = 3) | 2500 | |

CEC 26 | Composition Function 4 (N = 5) | 2600 | |

CEC 27 | Composition Function 5 (N = 5) | 2700 | |

CEC 28 | Composition Function 6 (N = 5) | 2800 | |

CEC 29 | Composition Function 7 (N = 3) | 2900 | |

CEC 30 | Composition Function 8 (N = 3) | 3000 | |

Search Range: [−100, 100] ^{dim} |

CEC | Metric | MGTOA | GTOA [28] | GA [11] | SCA [37] | BES [38] | ROA [10] | AOA [39] | WOA [9] | BTLBO [40] | TLBO [25] |
---|---|---|---|---|---|---|---|---|---|---|---|

CEC 1 | min | 1.13 × 10^{7} | 3.30 × 10^{8} | 4.18 × 10^{8} | 3.58 × 10^{8} | 5.35 × 10^{8} | 1.81 × 10^{8} | 8.42 × 10^{8} | 1.27 × 10^{8} | 1.76 × 10^{6} | 1.35 × 10^{6} |

mean | 7.02 × 10^{7} | 7.36 × 10^{8} | 9.95 × 10^{8} | 5.44 × 10^{8} | 9.45 × 10^{8} | 4.23 × 10^{8} | 1.36 × 10^{9} | 2.56 × 10^{8} | 5.36 × 10^{6} | 6.54 × 10^{6} | |

std | 4.06 × 10^{7} | 3.16 × 10^{8} | 3.86 × 10^{8} | 1.84 × 10^{8} | 3.72 × 10^{8} | 2.1 × 10^{8} | 4.95 × 10^{8} | 1.08 × 10^{8} | 5.46 × 10^{6} | 4.13 × 10^{6} | |

CEC 2 | min | 1.69 × 10^{8} | 3.19 × 10^{10} | 3.12 × 10^{10} | 2.43 × 10^{10} | 4.8 × 10^{10} | 2.11 × 10^{10} | 5.93 × 10^{10} | 5.05 × 10^{9} | 3.79 × 10^{3} | 6.05 × 10^{3} |

mean | 4.21 × 10^{9} | 5.18 × 10^{10} | 4.36 × 10^{10} | 3.1 × 10^{10} | 6.67 × 10^{10} | 3.2 × 10^{10} | 7.33 × 10^{10} | 7.99 × 10^{9} | 9.18 × 10^{5} | 9.76 × 10^{5} | |

std | 3.56 × 10^{9} | 1.3 × 10^{10} | 9.53 × 10^{9} | 5.62 × 10^{9} | 1.54 × 10^{10} | 1.12 × 10^{10} | 1.12 × 10^{10} | 3.85 × 10^{9} | 4.92 × 10^{6} | 5.05 × 10^{6} | |

CEC 3 | min | 1.80 × 10^{4} | 6.36 × 10^{4} | 6.11 × 10^{4} | 6.04 × 10^{4} | 8.35 × 10^{4} | 5.72 × 10^{4} | 7.61 × 10^{4} | 6.93 × 10^{4} | 1.37 × 10^{3} | 2.14 × 10^{4} |

mean | 4.61 × 10^{4} | 7.68 × 10^{4} | 1.74 × 10^{6} | 7.68 × 10^{4} | 1.82 × 10^{5} | 6.89 × 10^{4} | 8.55 × 10^{4} | 1.46 × 10^{5} | 6.31 × 10^{3} | 3.44 × 10^{4} | |

std | 6.52 × 10^{3} | 1.86 × 10^{4} | 6.76 × 10^{6} | 1.81 × 10^{4} | 1.46 × 10^{5} | 8.82 × 10^{3} | 1.59 × 10^{4} | 7.68 × 10^{4} | 4.53 × 10^{3} | 1.13 × 10^{4} | |

CEC 4 | min | 4.06 × 10^{2} | 4.33 × 10^{3} | 3.31 × 10^{3} | 2.04 × 10^{3} | 7.52 × 10^{3} | 1.38 × 10^{3} | 8.54 × 10^{3} | 9.5 × 10^{2} | 4.82 × 10^{2} | 4.78 × 10^{2} |

mean | 5.23 × 10^{2} | 8.75 × 10^{3} | 6.11 × 10^{3} | 3.01 × 10^{3} | 1.34 × 10^{4} | 3.57 × 10^{3} | 1.54 × 10^{4} | 1.47 × 10^{3} | 5.38 × 10^{2} | 5.34 × 10^{2} | |

std | 33.4 | 3.32 × 10^{3} | 3.13 × 10^{3} | 1.07 × 10^{3} | 3.91 × 10^{3} | 1.93 × 10^{3} | 4.11 × 10^{3} | 5.22 × 10^{2} | 48.1 | 40 | |

CEC 5 | min | 5.2 × 10^{2} | 5.21 × 10^{2} | 5.2 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} |

mean | 5.2 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | 5.21 × 10^{2} | |

std | 1.26 × 10^{−1} | 8.36 × 10^{−2} | 1.35 × 10^{−1} | 8.9 × 10^{−2} | 7.53 × 10^{−2} | 9.95 × 10^{−2} | 6.16 × 10^{−2} | 1.1 × 10^{−1} | 6.04 × 10^{−2} | 6.73 × 10^{−2} | |

CEC 6 | min | 6.2 × 10^{2} | 6.34 × 10^{2} | 6.32 × 10^{2} | 6.35 × 10^{2} | 6.38 × 10^{2} | 6.33 × 10^{2} | 6.37 × 10^{2} | 6.35 × 10^{2} | 6.18 × 10^{2} | 6.16 × 10^{2} |

mean | 6.31 × 10^{2} | 6.38 × 10^{2} | 6.35 × 10^{2} | 6.38 × 10^{2} | 6.41 × 10^{2} | 6.35 × 10^{2} | 6.39 × 10^{2} | 6.4 × 10^{2} | 6.21 × 10^{2} | 6.2 × 10^{2} | |

std | 2.6 | 3.07 | 3 | 2.67 | 3.17 | 3.91 | 2.79 | 3.85 | 3.27 | 3.17 | |

CEC 7 | min | 7.02 × 10^{2} | 1.02 × 10^{3} | 9.78 × 10^{2} | 8.99 × 10^{2} | 1.1 × 10^{3} | 7.95 × 10^{2} | 1.25 × 10^{3} | 7.3 × 10^{2} | 7 × 10^{2} | 7 × 10^{2} |

mean | 7.25 × 10^{2} | 1.21 × 10^{3} | 1.07 × 10^{3} | 9.69 × 10^{2} | 1.28 × 10^{3} | 9.09 × 10^{2} | 1.39 × 10^{3} | 7.51 × 10^{2} | 7 × 10^{2} | 7.01 × 10^{2} | |

std | 22.1 | 1.35 × 10^{2} | 87.4 | 53.2 | 1.33 × 10^{2} | 92 | 1.29 × 10^{2} | 25.7 | 1.32 | 3.21 | |

CEC 8 | min | 8.98 × 10^{2} | 1.03 × 10^{3} | 1.07 × 10^{3} | 1.06 × 10^{3} | 1.1 × 10^{3} | 1 × 10^{3} | 1.11 × 10^{3} | 9.97 × 10^{2} | 8.62 × 10^{2} | 8.6 × 10^{2} |

mean | 9.69 × 10^{2} | 1.07 × 10^{3} | 1.11 × 10^{3} | 1.09 × 10^{3} | 1.13 × 10^{3} | 1.05 × 10^{3} | 1.16 × 10^{3} | 1.06 × 10^{3} | 8.91 × 10^{2} | 8.91 × 10^{2} | |

std | 23.8 | 31.9 | 30.6 | 28.1 | 26.8 | 35.2 | 37.9 | 51.2 | 21.6 | 19 | |

CEC 9 | min | 9.83 × 10^{2} | 1.16 × 10^{3} | 1.16 × 10^{3} | 1.19 × 10^{3} | 1.23 × 10^{3} | 1.14 × 10^{3} | 1.2 × 10^{3} | 1.15 × 10^{3} | 9.86 × 10^{2} | 9.75 × 10^{2} |

mean | 1.05 × 10^{3} | 1.2 × 10^{3} | 1.21 × 10^{3} | 1.22 × 10^{3} | 1.25 × 10^{3} | 1.17 × 10^{3} | 1.23 × 10^{3} | 1.23 × 10^{3} | 1.02 × 10^{3} | 1.01 × 10^{3} | |

std | 28.7 | 33.2 | 34.5 | 26.8 | 38 | 32.8 | 27.8 | 68.1 | 27.3 | 35.4 | |

CEC 10 | min | 2.81 × 10^{3} | 6.52 × 10^{3} | 6 × 10^{3} | 7.37 × 10^{3} | 7.6 × 10^{3} | 5.49 × 10^{3} | 6.8 × 10^{3} | 5.8 × 10^{3} | 3.79 × 10^{3} | 3.06 × 10^{3} |

mean | 4.33 × 10^{3} | 7.3 × 10^{3} | 6.54 × 10^{3} | 8.18 × 10^{3} | 8.36 × 10^{3} | 6.44 × 10^{3} | 7.56 × 10^{3} | 6.6 × 10^{3} | 4.96 × 10^{3} | 4.93 × 10^{3} | |

std | 7.19 × 10^{2} | 7.33 × 10^{2} | 6.39 × 10^{2} | 4.82 × 10^{2} | 5.9 × 10^{2} | 7.79 × 10^{2} | 6.27 × 10^{2} | 8.51 × 10^{2} | 9.58 × 10^{2} | 1.71 × 10^{3} | |

CEC 11 | min | 1.26 × 10^{3} | 1.83 × 10^{3} | 2.64 × 10^{3} | 2.32 × 10^{3} | 2.32 × 10^{3} | 1.73 × 10^{3} | 1.8 × 10^{3} | 1.82 × 10^{3} | 1.33 × 10^{3} | 1.42 × 10^{3} |

mean | 1.96 × 10^{3} | 2.19 × 10^{3} | 3.08 × 10^{3} | 2.59 × 10^{3} | 2.68 × 10^{3} | 2.18 × 10^{3} | 2.17 × 10^{3} | 2.39 × 10^{3} | 1.56 × 10^{3} | 2.04 × 10^{3} | |

std | 2.79 × 10^{2} | 3.24 × 10^{2} | 3.38 × 10^{2} | 2.65 × 10^{2} | 3.04 × 10^{2} | 4.18 × 10^{2} | 3.04 × 10^{2} | 3.53 × 10^{2} | 1.41 × 10^{2} | 4.34 × 10^{2} | |

CEC 12 | min | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} |

mean | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | 1.2 × 10^{3} | |

std | 9.13 × 10^{−2} | 3.46 × 10^{−1} | 8.91 × 10^{−1} | 3.53 × 10^{−1} | 3.9 × 10^{−1} | 4.11 × 10^{−1} | 3.44 × 10^{−1} | 4.41 × 10^{−1} | 1.32 × 10^{−1} | 3.2 × 10^{−1} | |

CEC 13 | min | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} |

mean | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | 1.3 × 10^{3} | |

std | 7.62 × 10^{−2} | 9.25 × 10^{−1} | 1.06 | 1.8 × 10^{−1} | 1.27 | 8.07 × 10^{−1} | 1.27 | 2.31 × 10^{−1} | 6.65 × 10^{−2} | 9.29 × 10^{−2} | |

CEC 14 | min | 1.4 × 10^{3} | 1.4 × 10^{3} | 1.41 × 10^{3} | 1.4 × 10^{3} | 1.41 × 10^{3} | 1.4 × 10^{3} | 1.42 × 10^{3} | 1.4 × 10^{3} | 1.4 × 10^{3} | 1.4 × 10^{3} |

mean | 1.4 × 10^{3} | 1.41 × 10^{3} | 1.42 × 10^{3} | 1.4 × 10^{3} | 1.42 × 10^{3} | 1.4 × 10^{3} | 1.43 × 10^{3} | 1.4 × 10^{3} | 1.4 × 10^{3} | 1.4 × 10^{3} | |

std | 8.69 × 10^{−2} | 6.63 | 9.5 | 1.21 | 9.72 | 5.12 | 12.2 | 3.22 × 10^{−1} | 1.63 × 10^{−1} | 1.42 × 10^{−1} | |

CEC 15 | min | 1.5 × 10^{3} | 1.5 × 10^{3} | 1.6 × 10^{3} | 1.51 × 10^{3} | 1.53 × 10^{3} | 1.5 × 10^{3} | 1.64 × 10^{3} | 1.5 × 10^{3} | 1.5 × 10^{3} | 1.5 × 10^{3} |

mean | 1.5 × 10^{3} | 1.64 × 10^{3} | 2.38 × 10^{4} | 1.57 × 10^{3} | 4.03 × 10^{3} | 1.63 × 10^{3} | 4.77 × 10^{3} | 1.51 × 10^{3} | 1.5 × 10^{3} | 1.5 × 10^{3} | |

std | 1 | 5.46 × 10^{2} | 1.1 × 10^{5} | 2.2 × 10^{2} | 5.63 × 10^{3} | 4.47 × 10^{2} | 6.16 × 10^{3} | 7.08 | 7.64 × 10^{−1} | 8.25 × 10^{−1} | |

CEC 16 | min | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} |

mean | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | 1.6 × 10^{3} | |

std | 1.8 × 10^{−1} | 4.4 × 10^{−1} | 3.08 × 10^{−1} | 2.87 × 10^{−1} | 2.99 × 10^{−1} | 3.03 × 10^{−1} | 3.38 × 10^{−1} | 4.32 × 10^{−1} | 4.31 × 10^{−1} | 3.3 × 10^{−1} | |

CEC 17 | min | 1.89 × 10^{3} | 2.47 × 10^{3} | 2.12 × 10^{6} | 1.6 × 10^{4} | 5 × 10^{4} | 4.42 × 10^{3} | 8.19 × 10^{4} | 1.83 × 10^{4} | 1.98 × 10^{3} | 2.65 × 10^{3} |

mean | 5.49 × 10^{3} | 7.59 × 10^{3} | 1.13 × 10^{7} | 1.03 × 10^{5} | 1.05 × 10^{6} | 1.55 × 10^{5} | 5.67 × 10^{5} | 4.66 × 10^{5} | 2.5 × 10^{3} | 4.47 × 10^{3} | |

std | 2.83 × 10^{3} | 1.55 × 10^{4} | 1.48 × 10^{7} | 1.57 × 10^{5} | 2.75 × 10^{6} | 2.25 × 10^{5} | 5.48 × 10^{5} | 7.91 × 10^{5} | 1.01 × 10^{3} | 2.12 × 10^{3} | |

CEC 18 | min | 1.86 × 10^{3} | 1.89 × 10^{3} | 1.02 × 10^{6} | 1.19 × 10^{4} | 1.49 × 10^{4} | 2.94 × 10^{3} | 2.35 × 10^{3} | 2.81 × 10^{3} | 1.83 × 10^{3} | 1.95 × 10^{3} |

mean | 5.73 × 10^{3} | 1.97 × 10^{3} | 5.55 × 10^{7} | 7.88 × 10^{4} | 1.15 × 10^{6} | 1.41 × 10^{4} | 1.76 × 10^{4} | 1.93 × 10^{4} | 1.89 × 10^{3} | 5.9 × 10^{3} | |

std | 3.25 × 10^{3} | 7.04 × 10^{1} | 5.88 × 10^{7} | 1.05 × 10^{5} | 4.68 × 10^{6} | 9.68 × 10^{3} | 1.26 × 10^{4} | 3.48 × 10^{4} | 65.4 | 4.92 × 10^{3} | |

CEC 19 | min | 1.9 × 10^{3} | 1.9 × 10^{3} | 1.91 × 10^{3} | 1.91 × 10^{3} | 1.91 × 10^{3} | 1.9 × 10^{3} | 1.91 × 10^{3} | 1.9 × 10^{3} | 1.9 × 10^{3} | 1.9 × 10^{3} |

mean | 1.9 × 10^{3} | 1.91 × 10^{3} | 1.95 × 10^{3} | 1.91 × 10^{3} | 1.92 × 10^{3} | 1.91 × 10^{3} | 1.95 × 10^{3} | 1.91 × 10^{3} | 1.9 × 10^{3} | 1.9 × 10^{3} | |

std | 7.98 × 10^{−1} | 2.7 | 3.22 × 10^{1} | 1.04 | 14.2 | 9.77 | 30.4 | 2.1 | 9.63 × 10^{−1} | 8.29 × 10^{−1} | |

CEC 20 | min | 2.04 × 10^{3} | 2.1 × 10^{3} | 3.63 × 10^{4} | 3.21 × 10^{3} | 4.53 × 10^{3} | 3.79 × 10^{3} | 5.79 × 10^{3} | 3.18 × 10^{3} | 2.02 × 10^{3} | 2.1 × 10^{3} |

mean | 5.12 × 10^{3} | 6.43 × 10^{3} | 3.6 × 10^{7} | 1.16 × 10^{4} | 3.03 × 10^{5} | 1.03 × 10^{4} | 1.37 × 10^{4} | 1.78 × 10^{4} | 2.08 × 10^{3} | 2.55 × 10^{3} | |

std | 2.65 × 10^{3} | 1.15 × 10^{4} | 6.32 × 10^{7} | 9.7 × 10^{3} | 9.12 × 10^{5} | 4.92 × 10^{3} | 9.2 × 10^{3} | 2.44 × 10^{4} | 61.3 | 6.62 × 10^{2} | |

CEC 21 | min | 2.28 × 10^{3} | 2.55 × 10^{3} | 2.8 × 10^{5} | 7.79 × 10^{3} | 7.91 × 10^{3} | 3.58 × 10^{3} | 6.76 × 10^{3} | 1.54 × 10^{4} | 2.12 × 10^{3} | 2.27 × 10^{3} |

mean | 6.23 × 10^{3} | 4.47 × 10^{3} | 5.37 × 10^{6} | 2.07 × 10^{4} | 4.94 × 10^{5} | 1.84 × 10^{4} | 1.74 × 10^{6} | 9.71 × 10^{5} | 2.29 × 10^{3} | 2.52 × 10^{3} | |

std | 4.38 × 10^{3} | 4.9 × 10^{3} | 7.59 × 10^{6} | 1.03 × 10^{4} | 9.91 × 10^{5} | 4.06 × 10^{4} | 2.64 × 10^{6} | 2.23 × 10^{6} | 1.51 × 10^{2} | 2.11 × 10^{2} | |

CEC 22 | min | 2.22 × 10^{3} | 2.24 × 10^{3} | 2.42 × 10^{3} | 2.25 × 10^{3} | 2.3 × 10^{3} | 2.23 × 10^{3} | 2.27 × 10^{3} | 2.24 × 10^{3} | 2.21 × 10^{3} | 2.22 × 10^{3} |

mean | 2.32 × 10^{3} | 2.33 × 10^{3} | 2.63 × 10^{3} | 2.29 × 10^{3} | 2.44 × 10^{3} | 2.29 × 10^{3} | 2.43 × 10^{3} | 2.33 × 10^{3} | 2.23 × 10^{3} | 2.24 × 10^{3} | |

std | 5.34 × 101 | 8.78 × 10^{1} | 1.79 × 10^{2} | 4.29 × 10^{1} | 1.21 × 10^{2} | 90.3 | 1.24 × 10^{2} | 96.9 | 6.67 | 30.3 | |

CEC 23 | min | 2.5 × 10^{3} | 2.5 × 10^{3} | 2.66 × 10^{3} | 2.64 × 10^{3} | 2.5 × 10^{3} | 2.5 × 10^{3} | 2.5 × 10^{3} | 2.63 × 10^{3} | 2.63 × 10^{3} | 2.63 × 10^{3} |

mean | 2.5 × 10^{3} | 2.5 × 10^{3} | 2.74 × 10^{3} | 2.65 × 10^{3} | 2.6 × 10^{3} | 2.5 × 10^{3} | 2.5 × 10^{3} | 2.64 × 10^{3} | 2.63 × 10^{3} | 2.63 × 10^{3} | |

std | 0 | 5.32 × 10^{−1} | 1.1 × 10^{2} | 9.53 | 95.5 | 0 | 2.66 × 10^{−1} | 26.9 | 2.42 × 10^{−12} | 2.02 × 10^{−12} | |

CEC 24 | min | 2.51 × 10^{3} | 2.54 × 10^{3} | 2.57 × 10^{3} | 2.55 × 10^{3} | 2.57 × 10^{3} | 2.6 × 10^{3} | 2.57 × 10^{3} | 2.56 × 10^{3} | 2.51 × 10^{3} | 2.51 × 10^{3} |

mean | 2.59 × 10^{3} | 2.58 × 10^{3} | 2.60 × 10^{3} | 2.56 × 10^{3} | 2.6 × 10^{3} | 2.6 × 10^{3} | 2.59 × 10^{3} | 2.58 × 10^{3} | 2.52 × 10^{3} | 2.53 × 10^{3} | |

std | 2.31 × 10^{1} | 2.94 × 10^{1} | 2.06 × 10^{1} | 1.18 × 10^{1} | 9.04 | 14.2 | 18.5 | 27.5 | 16.4 | 35.5 | |

CEC 25 | min | 2.63 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.69 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.69 × 10^{3} | 2.63 × 10^{3} | 2.63 × 10^{3} |

mean | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.71 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.66 × 10^{3} | 2.66 × 10^{3} | |

std | 5.77 | 1.09 × 10^{1} | 6.1 | 5.58 | 6.77 | 0 | 1.38 | 13.4 | 30.6 | 30.9 | |

CEC 26 | min | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} |

mean | 2.7 × 10^{3} | 2.71 × 10^{3} | 2.71 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.72 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} | |

std | 6.25 × 10^{−2} | 3.03 × 10^{1} | 3.11 × 10^{1} | 2.91 × 10^{−1} | 1.3 | 18.1 | 33.4 | 18.2 | 5.84 × 10^{−2} | 7.31 × 10^{−2} | |

CEC 27 | min | 2.7 × 10^{3} | 2.9 × 10^{3} | 3.13 × 10^{3} | 2.73 × 10^{3} | 2.88 × 10^{3} | 2.9 × 10^{3} | 2.9 × 10^{3} | 3.11 × 10^{3} | 2.7 × 10^{3} | 2.7 × 10^{3} |

mean | 2.89 × 10^{3} | 3 × 10^{3} | 3.25 × 10^{3} | 3.04 × 10^{3} | 3.17 × 10^{3} | 2.88 × 10^{3} | 2.91 × 10^{3} | 3.16 × 10^{3} | 2.75 × 10^{3} | 2.96 × 10^{3} | |

std | 4.86 × 10^{1} | 1.45 × 10^{2} | 1.04 × 10^{2} | 1.47 × 10^{2} | 1.86 × 10^{2} | 64.3 | 80.2 | 1.72 × 10^{2} | 1.1 × 10^{2} | 1.75 × 10^{2} | |

CEC 28 | min | 3 × 10^{3} | 3 × 10^{3} | 3.65 × 10^{3} | 3.24 × 10^{3} | 3 × 10^{3} | 3 × 10^{3} | 3 × 10^{3} | 3.23 × 10^{3} | 3.18 × 10^{3} | 3.18 × 10^{3} |

mean | 3 × 10^{3} | 3.16 × 10^{3} | 3.99 × 10^{3} | 3.32 × 10^{3} | 3.47 × 10^{3} | 3 × 10^{3} | 3.09 × 10^{3} | 3.45 × 10^{3} | 3.22 × 10^{3} | 3.24 × 10^{3} | |

std | 0 | 2.07 × 10^{2} | 3.04 × 10^{2} | 7.3 × 10^{1} | 1.92 × 10^{2} | 0 | 2.71 × 10^{2} | 2.02 × 10^{2} | 47.3 | 71.2 | |

CEC 29 | min | 3.1 × 10^{3} | 3.33 × 10^{3} | 5.69 × 10^{5} | 5.56 × 10^{3} | 8.13 × 10^{3} | 3.41 × 10^{3} | 3.1 × 10^{3} | 3.54 × 10^{3} | 3.22 × 10^{3} | 3.37 × 10^{3} |

mean | 3.1 × 10^{3} | 1.87 × 10^{6} | 1.06 × 10^{7} | 3.17 × 10^{4} | 1.63 × 10^{6} | 3.17 × 10^{5} | 2.53 × 10^{6} | 4.71 × 10^{5} | 3.38 × 10^{3} | 2.55 × 10^{5} | |

std | 5.47 | 3.59 × 10^{6} | 1.49 × 10^{7} | 4.6 × 10^{4} | 3.1 × 10^{6} | 8.49 × 10^{5} | 1.11 × 10^{7} | 1.26 × 10^{6} | 2.03 × 10^{2} | 6.54 × 10^{5} | |

CEC 30 | min | 3.2 × 10^{3} | 4.13 × 10^{3} | 1.26 × 10^{4} | 4.56 × 10^{3} | 5.93 × 10^{3} | 3.99 × 10^{3} | 5.49 × 10^{3} | 4.42 × 10^{3} | 3.53 × 10^{3} | 3.5 × 10^{3} |

mean | 3.2 × 10^{3} | 1.76 × 10^{4} | 1.2 × 10^{5} | 5.73 × 10^{3} | 2.7 × 10^{4} | 5.49 × 10^{3} | 1.58 × 10^{5} | 7.62 × 10^{3} | 3.86 × 10^{3} | 3.8 × 10^{3} | |

std | 5.44 | 4.25 × 10^{4} | 1.64 × 10^{5} | 1.5 × 10^{3} | 5.38 × 10^{4} | 2.21 × 10^{3} | 7 × 10^{5} | 9.8 × 10^{3} | 3.12 × 10^{2} | 4.13 × 10^{2} |

CEC | MGTOA vs. GTOA [28] | MGTOA vs. GA [11] | MGTOA vs. SCA [37] | MGTOA vs. BES [38] | MGTOA vs. ROA [10] | MGTOA vs. AOA [39] | MGTOA vs. WOA [9] | MGTOA vs. BTLBO [40] | MGTOA vs. TLBO [25] |
---|---|---|---|---|---|---|---|---|---|

CEC 1 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.88 × 10^{−6} | 1.73 × 10^{−6} | 3.72 × 10^{−5} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 2 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 6.16 ×10 ^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 3 | 3.88 × 10^{−6} | 2.13 × 10^{−6} | 6.98 × 10^{−6} | 1.73 × 10^{−6} | 4.29 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} |

CEC 4 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 1.36 × 10^{−4} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 5 | 1.73 × 10^{−6} | 3.18 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} | 2.35 × 10^{−6} | 9.63 × 10^{−4} |

CEC 6 | 7.51 × 10^{−5} | 5.71 × 10^{−4} | 2.6 × 10^{−5} | 2.35 × 10^{−6} | 3.06 × 10^{−4} | 8.47 × 10^{−6} | 2.88 × 10^{−6} | 1.92 × 10^{−6} | 1.73 × 10^{−6} |

CEC 7 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} | 1.73 × 10^{−6} | 2.30 × 10^{−2} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 8 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 2.35 × 10^{−6} | 1.73 × 10^{−6} | 9.32 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 9 | 2.88 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 3.18 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 10 | 1.73 × 10^{−6} | 4.29 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.6 × 10^{−4} | 1.73 × 10^{−6} | 3.88 × 10^{−6} | 4.29 × 10^{−6} | 3.72 × 10^{−5} |

CEC 11 | 1.31 × 10^{−1} | 2.13 × 10^{−6} | 6.98 × 10^{−6} | 6.32 × 10^{−5} | 6.29 × 10^{−1} | 4.72 × 10^{−2} | 4.72 × 10^{−2} | 3.52 × 10^{−6} | 3.29 × 10^{−1} |

CEC 12 | 2.88 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 4.29 × 10^{−6} | 8.94 × 10^{−4} | 1.92 × 10^{−6} | 4.72 × 10^{−2} | 1.92 × 10^{−6} |

CEC 13 | 2.61 × 10^{−4} | 1.73 × 10^{−6} | 3.88 × 10^{−6} | 1.73 × 10^{−6} | 1.89 × 10^{−4} | 1.73 × 10^{−6} | 3.32 × 10^{−4} | 5.75 × 10^{−6} | 4.11 × 10^{−3} |

CEC 14 | 1.83 × 10^{−3} | 1.73 × 10^{−6} | 1.49 × 10^{−5} | 1.73 × 10^{−6} | 1.29 × 10^{−3} | 1.73 × 10^{−6} | 2.13 × 10^{−1} | 7.66 × 10^{−1} | 5.44 × 10^{−1} |

CEC 15 | 3.52 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.02 × 10^{−5} | 1.73 × 10^{−6} | 7.69 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 16 | 4.49 × 10^{−2} | 1.73 × 10^{−6} | 6.89 × 10^{−5} | 5.79 × 10^{−5} | 1.65 × 10^{−1} | 4.29 × 10^{−6} | 2.22 × 10^{−4} | 1.73 × 10^{−6} | 3.18 × 10^{−6} |

CEC 17 | 5.67 × 10^{−3} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 5.32 × 10^{−3} | 1.73 × 10^{−6} | 2.60 × 10^{−6} | 6.34 × 10^{−6} | 7.16 × 10^{−4} |

CEC 18 | 1.92 × 10^{−6} | 1.73 × 10^{−6} | 2.13 × 10^{−6} | 1.73 × 10^{−6} | 3 × 10^{−2} | 3.71 × 10^{−1} | 3.68 × 10^{−2} | 1.73 × 10^{−6} | 4.53 × 10^{−4} |

CEC 19 | 3.11 × 10^{−5} | 1.73 × 10^{−6} | 3.88 × 10^{−6} | 1.92 × 10^{−6} | 2.26 × 10^{−3} | 1.73 × 10^{−6} | 1.80 × 10^{−5} | 1.92 × 10^{−6} | 2.35 × 10^{−6} |

CEC 20 | 2.43 × 10^{−2} | 1.73 × 10^{−6} | 3.50 × 10^{−2} | 1.64 × 10^{−5} | 9.84 × 10^{−3} | 4.72 × 10^{−2} | 7.73 × 10^{−3} | 1.73 × 10^{−6} | 2.60 × 10^{−6} |

CEC 21 | 2.16 × 10^{−5} | 1.73 × 10^{−6} | 3.16 × 10^{−3} | 5.75 × 10^{−6} | 2.96 × 10^{−3} | 1.83 × 10^{−3} | 1.92 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 22 | 1.04 × 10^{−2} | 1.92 × 10^{−6} | 4.49 × 10^{−2} | 4.11 × 10^{−3} | 8.94 × 10^{−1} | 3.72 × 10^{−5} | 4.41 × 10^{−1} | 1.73 × 10^{−6} | 8.47 × 10^{−6} |

CEC 23 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.22 × 10^{−4} | 1 | 1 | 5.61 × 10^{−6} | 4.32 × 10^{−8} | 4.32 × 10^{−8} |

CEC 24 | 8.59 × 10^{−2} | 2.29 × 10^{−1} | 3.72 × 10^{−5} | 6.89 × 10^{−1} | 6.45 × 10^{−2} | 4.54 × 10^{−1} | 3.49 × 10^{−1} | 1.73 × 10^{−6} | 3.52 × 10^{−6} |

CEC 25 | 7.73 × 10^{−3} | 5.75 × 10^{−6} | 2.41 × 10^{−3} | 3.91 × 10^{−2} | 4.38 × 10^{−1} | 1.88 × 10^{−1} | 9.91 × 10^{−1} | 1.73 × 10^{−6} | 1.80 × 10^{−5} |

CEC 26 | 1.48 × 10^{−4} | 3.11 × 10^{−5} | 3.11 × 10^{−5} | 3.11 × 10^{−5} | 2.22 × 10^{−4} | 2.84 × 10^{−5} | 1.83 × 10^{−3} | 1.73 × 10^{−6} | 2.88 × 10^{−6} |

CEC 27 | 1.32 × 10^{−2} | 7.69 × 10^{−6} | 4.73 × 10^{−6} | 7.03 × 10^{−6} | 3.75 × 10^{−1} | 1.88 × 10^{−1} | 2.35 × 10^{−6} | 6.34 × 10^{−6} | 4.99 × 10^{−3} |

CEC 28 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 3.79 × 10^{−6} | 1 | 2.50 × 10^{−1} | 3.79 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} |

CEC 29 | 1.64 × 10^{−5} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 4.73 × 10^{−6} | 1.86 × 10^{−2} | 1.73 × 10^{−6} | 1.11 × 10^{−2} | 8.19 × 10^{−5} |

CEC 30 | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.73 × 10^{−6} | 1.92 × 10^{−6} | 2.35 × 10^{−6} | 1.73 × 10^{−6} | 7.52 × 10^{−2} | 3.39 × 10^{−1} |

Algorithm | Optimal Values for Variables | Best Weight | |||
---|---|---|---|---|---|

h | l | t | b | ||

MGTOA | 0.205351 | 3.268419 | 9.069875 | 0.205621 | 1.701633939 |

GTOA [28] | 0.20573 | 3.470489 | 9.036624 | 0.20573 | 1.724852 |

TSA [41] | 0.244157 | 6.223066 | 8.29555 | 0.244405 | 2.38241101 |

MFO [42] | 0.2057 | 3.4703 | 9.0364 | 0.2057 | 1.72452 |

MVO [21] | 0.205463 | 3.473193 | 9.044502 | 0.205695 | 1.72645 |

RO [22] | 0.203687 | 3.528467 | 9.004233 | 0.207241 | 1.735344 |

Algorithm | Optimal Values for Variables | Best Cost | |||
---|---|---|---|---|---|

T_{s} | T_{h} | R | L | ||

MGTOA | 0.754364 | 0.366375 | 40.42809 | 198.5652 | 5752.402458 |

GTOA [28] | 0.778169 | 0.38465 | 40.3196 | 200 | 5885.333 |

CPSO [44] | 0.8125 | 0.4375 | 42.0913 | 176.7465 | 6061.0777 |

HPSO [45] | 0.8125 | 0.4375 | 42.0984 | 176.6366 | 6059.7143 |

CS [46] | 0.8125 | 0.4375 | 42.09845 | 176.6366 | 6059.714335 |

AO [47] | 1.054 | 0.182806 | 59.6219 | 39.805 | 5949.2258 |

Algorithm | Optimal Values for Variables | Best Weight | ||
---|---|---|---|---|

d | D | V | ||

MGTOA | 0.05 | 0.374396 | 8.549078 | 0.009875 |

IROA [49] | 0.053799 | 0.46951 | 5.811 | 0.010614 |

HHO [50] | 0.051796 | 0.359305 | 11.13886 | 0.012665 |

GWO [7] | 0.05169 | 0.356737 | 11.28885 | 0.012666 |

MFO [42] | 0.051994 | 0.364109 | 10.86842 | 0.012667 |

DE [16] | 0.051609 | 0.354714 | 11.41083 | 0.01267 |

Algorithm | Optimal Values for Variables | Best Weight | |
---|---|---|---|

x_{1} | x_{2} | ||

MGTOA | 0.788413 | 0.408121 | 263.8523 |

PSO-DE [51] | 0.788675 | 0.408248 | 263.8958 |

Tsa [52] | 0.788 | 0.408 | 263.68 |

DEDS [53] | 0.788675 | 0.408248 | 263.8958 |

GOA [54] | 0.788898 | 0.40762 | 263.8959 |

RSA [55] | 0.78873 | 0.40805 | 263.8928 |

Algorithm | MGTOA | GTOA | MPA [56] | HHOCM [57] | ROLGWO [58] | MALO [59] |
---|---|---|---|---|---|---|

x_{1} | 0.5 | 0.662833 | 0.5 | 0.500164 | 0.501255 | 0.5 |

x_{2} | 1.227894 | 1.217247 | 1.22823 | 1.248612 | 1.245551 | 1.2281 |

x_{3} | 0.5 | 0.734238 | 0.5 | 0.659558 | 0.500046 | 0.5 |

x_{4} | 1.203472 | 1.11266 | 1.2049 | 1.098515 | 1.180254 | 1.2126 |

x_{5} | 0.5 | 0.613197 | 0.5 | 0.757989 | 0.500035 | 0.5 |

x_{6} | 1.065913 | 0.670197 | 1.2393 | 0.767268 | 1.16588 | 1.308 |

x_{7} | 0.5 | 0.615694 | 0.5 | 0.500055 | 0.500088 | 0.5 |

x_{8} | 0.345 | 0.271734 | 0.34498 | 0.343105 | 0.344895 | 0.3449 |

x_{9} | 0.192 | 0.23194 | 0.192 | 0.192032 | 0.299583 | 0.2804 |

x_{10} | 0.367345 | 0.174933 | 0.44035 | 2.898805 | 3.59508 | 0.4242 |

x_{11} | 0.969872 | 0.462294 | 1.78504 | - | 2.29018 | 4.6565 |

Best Weight | 23.19125 | 25.70607 | 23.19982 | 24.48358 | 23.22243 | 23.2294 |

Algorithm | Optimal Values for Variables | Best Gear Ratio | |||
---|---|---|---|---|---|

n_{A} | n_{B} | n_{C} | n_{D} | ||

MGTOA | 43.90536 | 16.01273 | 19.59159 | 49.11997 | 2.70086 × 10^{−12} |

GTOA | 54.68955 | 37.07689 | 12 | 57.13786 | 8.88761 × 10^{−10} |

CS [46] | 43 | 16 | 19 | 49 | 2.7009 × 10^{−12} |

GA [11] | 49 | 16 | 19 | 43 | 2.7019 × 10^{−12} |

ABC [6] | 49 | 16 | 19 | 43 | 2.7009 × 10^{−12} |

MBA [61] | 43 | 16 | 19 | 49 | 2.7009 × 10^{−12} |

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## Share and Cite

**MDPI and ACS Style**

Rao, H.; Jia, H.; Wu, D.; Wen, C.; Li, S.; Liu, Q.; Abualigah, L.
A Modified Group Teaching Optimization Algorithm for Solving Constrained Engineering Optimization Problems. *Mathematics* **2022**, *10*, 3765.
https://doi.org/10.3390/math10203765

**AMA Style**

Rao H, Jia H, Wu D, Wen C, Li S, Liu Q, Abualigah L.
A Modified Group Teaching Optimization Algorithm for Solving Constrained Engineering Optimization Problems. *Mathematics*. 2022; 10(20):3765.
https://doi.org/10.3390/math10203765

**Chicago/Turabian Style**

Rao, Honghua, Heming Jia, Di Wu, Changsheng Wen, Shanglong Li, Qingxin Liu, and Laith Abualigah.
2022. "A Modified Group Teaching Optimization Algorithm for Solving Constrained Engineering Optimization Problems" *Mathematics* 10, no. 20: 3765.
https://doi.org/10.3390/math10203765