In this paper, based on the concept of the NFL theorem, that there is no unique algorithm that has the best performance for all optimization problems, a new human-based metaheuristic algorithm called Language Education Optimization (LEO) is introduced, which is used to solve optimization problems. LEO is inspired by the foreign language education process in which a language teacher trains the students of language schools in the desired language skills and rules. LEO is mathematically modeled in three phases: (i) students selecting their teacher, (ii) students learning from each other, and (iii) individual practice, considering exploration in local search and exploitation in local search. The performance of LEO in optimization tasks has been challenged against fifty-two benchmark functions of a variety of unimodal, multimodal types and the CEC 2017 test suite. The optimization results show that LEO, with its acceptable ability in exploration, exploitation, and maintaining a balance between them, has efficient performance in optimization applications and solution presentation. LEO efficiency in optimization tasks is compared with ten well-known metaheuristic algorithms. Analyses of the simulation results show that LEO has effective performance in dealing with optimization tasks and is significantly superior and more competitive in combating the compared algorithms. The implementation results of the proposed approach to four engineering design problems show the effectiveness of LEO in solving real-world optimization applications.
Numerous challenges in various sciences face several possible solutions. Such challenges are known as optimization issues. Hence, the operation of finding the best solution to such problems is called optimization [
Optimization techniques fall into two groups: deterministic and stochastic methods. Deterministic methods are efficient on optimization topics that have a linear, convex, continuous, differentiable objective function, and a continuous search space. However, as optimization problems become more complex, deterministic approaches lose their ability in real-world applications that have features, such as non-convex, discrete, nonlinear, non-differentiable objective functions, discrete search space, and high-dimensions [
The two most important factors influencing the performance of metaheuristic algorithms are exploration and exploitation. Exploration represents the power of the algorithm in the global search, and exploitation represents the power of the algorithm in the local search [
Natural phenomena, the behaviors of living things in nature, the laws of physics, the concepts of biology, and other evolutionary processes have been the sources of inspiration for the design of metaheuristic algorithms. The Genetic Algorithm (GA) [
The main research question is: Despite the numerous metaheuristic algorithms introduced till now, is there still any necessity for designing newer metaheuristic algorithms? The No-Free-Lunch Theorem (NFL) [
The aspects of novelty and innovation of this study are in the introduction of a new human-based metaheuristic algorithm called Language Education Optimization (LEO) that is efficient in optimization tasks. The key contributions of this paper are as follows:
LEO is introduced based on the simulation of the foreign language education process. The fundamental inspiration of LEO is to train students in language schools in language skills and rules. The LEO theory is described and then mathematically modeled in three phases. The performance of LEO in optimization tasks is assessed in dealing with fifty-two standard benchmark functions. The results of LEO are compared with the performance of ten well-known metaheuristic algorithms. The effectiveness of LEO in handling real-world applications is evaluated in the optimization of four engineering design problems.
The paper consists of the following sections: a literature review is provided in the section “Literature Review.’’ The proposed Language Education Optimization (LEO) approach is introduced and modeled in the section “Language Education Optimization.’’ LEO simulation and evaluation studies on the handling of optimization tasks are presented in the section “Simulation Studies and Results’’. A discussion of the results is provided in the section “Discussion.’’ The study evaluating the ability of the proposed LEO approach in CEC 2017 test suite optimization is presented in the section “Evaluation CEC 2017 Test Suite.” The analysis of LEO capabilities in real-world applications is presented in the section “LEO for Real-World Applications.” Conclusions and suggestions for further studies are expressed in the section “Conclusions and Future Researches.’’
Metaheuristic algorithms have been developed based on mathematical simulations of various phenomena, such as genetics and biology, swarm intelligences in the life of living organisms, physical phenomena, rules of games, human activities, etc. According to the main source of inspiration resulting in the design, metaheuristic algorithms fall into the following five groups: (i) swarm-based, (ii) evolutionary-based, (iii) physics-based, (iv) human-based, and (v) game-based methods.
Modeling the swarming behaviors and social and individual lives of living organisms (birds, aquatic animals, insects, animals, etc.) has led to the development of swarm-based metaheuristic algorithms. The major algorithms belonging to this group are PSO, ABC, and ACO. PSO is based on modeling the behavior of swarm movement of groups of fish and birds in which two factors, individuals' experience and group experience, affect the population displacement of the algorithm. ABC is based on simulating the social life of bees seeking food sources and extracting nectar from these food sources. ACO is based on the behavior of the ant colony searching the optimal path between the nest and the food sources. Artificial Hummingbird Algorithm (AHA) is a swarm-based method based on the simulation of intelligent foraging strategies and special flight skills of hummingbirds in nature [
Modeling of genetics and biology concepts has been the main source for evolutionary-based metaheuristic algorithms development. The reproduction process simulation, based on the concepts of Darwin’s theory of evolution and natural selection, has been the main source in the design of Differential Evolution (DE) [
Modeling of the physical laws and phenomena has been used in the physics-based metaheuristic algorithms development. Material engineers use the annealing method to achieve a state in which the solid is well organized, and its energy is minimized. This method involves placing the material in a high-temperature environment and following a gradual lowering of the temperature. The Simulated Annealing (SA) method simulates this solid-state annealing process to solve the optimization problem [
Modeling of human activities and interactions existing in society and individuals' life has led to the emergence of human-based metaheuristic algorithms. The educational environment of the classroom and the exchange of knowledge between the teacher and the students and also among students, have been a good inspiration source for Teaching-Learning Based Optimization (TLBO) [
Modeling the game rules and behavior of players, referees, and coaches brings tremendous inspiration to game-based metaheuristic algorithms development. Football League simulations and club performances resulted in the Football Game Based Optimization (FGBO) [
We have not found any metaheuristic algorithms simulating a foreign language education process in language schools. However, the process of teaching language skills decided by the teacher and applied to the learners is an intelligent structure with remarkable potential to be used in designing a new optimizer. In order to complete this research gap, a new human-based metaheuristic algorithm based on a simulation of the foreign language teaching process and the interactions of the people involved in it is designed and presented in this paper.
In this section, the metaheuristic algorithm LEO and its mathematical model based on the simulation of human activity in foreign language education is presented.
One of the most important ways human beings communicate with each other is by using their ability to speak. First, human beings acquire and empirically learn the official language of their society and country. With the advancement of societies and technology, communication between different nations has increased. This reality has led to the increasing importance of learning not only the native language if people are to be able to communicate with people living in other countries. As a consequence, foreign language schools have been established.
When a person decides to learn other languages, she/he has several options for choosing a school or language teacher. Choosing the appropriate school and teacher is one of the essential steps which has a great impact on the person’s success in the language learning process. After the learner chooses the language teacher, she/he also communicates with other students in the classroom environment. These learners make efforts to learn language skills from the teacher training them in the given classroom environment. Additionally, to improve their skills, the students talk and practice with each other. These interactions between students improve their level of language learning. In addition, each student improves foreign language skills by doing homework and individual practice.
There are three important phases in this intelligent process, which represent the basic specifics of human activity in foreign language teaching, which must be considered into account in the new design of the metaheuristic algorithm. These three phases are (see
LEO is a population-based approach that is able to provide the problem-solving process for an optimization task in an iteration-based procedure. Each member of LEO is a candidate solution of the optimization problem that proposes values for decision variables. From a mathematical point of view, each LEO member can be modeled using a vector, and the population of LEO members using a matrix according to the
The initial positions of all LEO members in the search space are randomly set-up by the
As the value of the objective function is the main criterion for measuring the goodness of a candidate solution, the minimal value in the set of values of objective function
By initializing the algorithm, candidate solutions are generated and evaluated. These candidate solutions in LEO are updated in three different phases to improve their quality.
Each person can choose one of the available teachers in order to learn a foreign language. In LEO, for each member of the population, members who have a better objective function value than that member are considered as suggested teachers. One of these suggested teachers is randomly selected for language teaching whose schematic is shown in
This strategy leads LEO members to move to different areas of the search space, which demonstrates the global search power of LEO in exploration. In order to mathematically model this phase, the set of suggested teachers for each member of LEO, thus, for the
Similar to the decision of the student in language school, who chooses a teacher from among the teachers who teach in the school, in the design of LEO, this concept has also been selected for choosing a teacher. Therefore, one teacher is randomly selected among the members who have been identified as possible teachers to teach the
In language school, the teacher tries to make positive changes in the student’s foreign language skill level by teaching the student. Inspired by this process, in the design of LEO, the number of changes in the position of the population members has been calculated based on the subtraction of the position of the teacher and the student to improve the position of the population members in the search space. According to this, new components of each LEO member are generated for
In the second phase of LEO, population members of LEO are updated based on modeling skills exchange between students. Students try to improve their skills based on their interactions with each other whose schematic is shown in
This affects the ability of LEO exploration to scan the search space. In language schools, students usually practice with each other and improve their skills. In this exercise, the student who has more skills tries to increase the scientific level of that student by teaching another student. Inspired by this interaction in language school, in LEO design, another member of the population is randomly selected for each member of the population. Then, based on the subtraction of the difference in the position of the two members, the changes in the displacement of the corresponding member are calculated. To mathematically model these interactions, for each LEO member another member of the population is randomly selected, and it is used for recomputing of its components. Thus, the new components of each member of LEO are calculated for
The third phase of LEO is motivated by learning approaches that are commonly called self-learning. This is how the students make efforts to identify their own learning needs. Set learning goals, find the additional study literature and self-study online platforms. In this phase of LEO, members of LEO are updated based on simulations of individual students’ practices to improve the skills they have acquired from the teacher in the first phase whose schematic is shown in
In fact, LEO scans the search space around members based on local search, seeking better solutions. A student who goes to a language school, after participating in the class and practicing with her/his classmates, tries to improve his skills as much as possible with individual practice, which leads to small but useful changes in the student’s language skills. Inspired by this student’s behavior in the language learning process, in the design of LEO, the students’ individual practice is modeled by making small changes in their position. To model the concepts of this LEO phase mathematically, a random position near each member is first generated using
Subsequently, a decision is made whether to update each LEO member
After updating all members of LEO based on all three phases, an iteration of the algorithm is completed. At the end of each iteration, the best candidate solution is updated. The iterative process of the algorithm based on
The computational complexity of LEO is analyzed in this subsection. LEO initialization has a computational complexity equal to
The TLBO algorithm updates the population members of the algorithm in two phases, teacher and student. On the other hand, the proposed LEO approach updates the population members in three stages: teacher selection and training, students learning from each other, and individual practice.
In the teacher’s phase of the TLBO algorithm, the best member is considered a teacher for the entire population, and the other members are considered students. But in LEO, for each member of the population, all the members with better fitness compared to that member are considered candidate teachers for the corresponding member. Among them, the teacher is randomly selected to train the corresponding member. Also, in the population update equation in TLBO, subtracting the teacher’s position from the average of the entire population is used. But in the population update equation in LEO, the difference between the selected teacher’s position and the corresponding member’s position. In the student phase of the TLBO algorithm, the update equation is modeled based on the subtraction of the position of two students. But in the design of LEO, the subtraction of the member with better fitness than the other member multiplied by the
Also, compared to TLBO, which only has two phases of population update, in LEO design, to increase the exploitation ability in local search, the third phase of an update called individual exercise is used.
In this section, the capability of the proposed LEO algorithm in optimization applications is studied. A set of twenty-three objective functions including seven unimodal functions, six multimodal functions, and ten fixed-dimensional multimodal functions have been utilized to analyze LEO performance. Details and full description of these benchmark functions are provided in [
Objective function | Range | Dimensions ( |
||
---|---|---|---|---|
1. | 30 | 0 | ||
2. | 30 | 0 | ||
3. | 30 | 0 | ||
4. | 30 | 0 | ||
5. | 30 | 0 | ||
6. | 30 | 0 | ||
7. | 30 | 0 |
Objective function | Range | Dimensions ( |
||
---|---|---|---|---|
8. | 30 | |||
9. | 30 | 0 | ||
10. | 30 | 0 | ||
11. | 30 | 0 | ||
12. | 30 | 0 | ||
13. | 30 | 0 |
Objective function | Range | Dimensions ( |
||
---|---|---|---|---|
14. | ||||
15. | ||||
16. | ||||
17. | [−5, 10] |
|||
18. | ||||
19. | ||||
20. | ||||
21. | ||||
22. | ||||
23. |
LEO’s ability in optimization applications is compared with the performance of ten well-known metaheuristic algorithms. The reasons for choosing these competitor algorithms are explained below. The first group includes the widely used and well-known GA and PSO algorithms. The second group includes highly cited algorithms TLBO, MVO, GSA, GWO, and WOA, which have been employed by researchers in many optimization applications. The third group includes MPA, RSA, and TSA algorithms that have recently been published and have received a lot of attention. The control parameters of the competitor algorithms are set according to
Algorithm | Parameter | Value |
---|---|---|
GA | ||
Type | Real coded | |
Selection | Roulette wheel (Proportionate) | |
Crossover | Whole arithmetic (Probability = 0.8, |
|
Mutation | Gaussian (Probability |
|
Population size | 50 | |
PSO | ||
Topology | Fully connected | |
Cognitive and social constant | ( |
|
Inertia weight | Linear reduction from 0.9 to 0.1 | |
Velocity limit | 10% of dimension range | |
Population size | 50 | |
GSA | ||
Alpha, |
20, 100, 2, 1 | |
Population size | 50 | |
TLBO | ||
random number | where |
|
Population size | 50 | |
GWO | ||
Convergence parameter ( |
||
Population size | 50 | |
MVO | ||
Wormhole existence probability (WEP) | ||
Exploitation accuracy over the iterations ( |
||
Population size | 50 | |
WOA | ||
Convergence parameter ( |
||
Population size | 50 | |
TSA | ||
Pmin and Pmax | 1, 4 | |
random numbers from the interval |
||
Population size | 50 | |
MPA | ||
Constant number | ||
Random vector | ||
Fish Aggregating Devices ( |
||
Binary vector | ||
Population size | 50 | |
RSA | ||
Sensitive parameter | ||
Sensitive parameter | ||
Evolutionary Sense (ES) | ES: randomly decreasing values between 2 and −2 | |
Population size | 50 | |
LEO | ||
Population size | 30 |
The LEO method and ten competing algorithms are each employed in twenty independent executions, while each execution contains 1000 iterations to optimize the objective functions. The results of these simulations are reported using indicators: best, mean, median, standard deviation (std), execution time (ET), and rank.
The results of recruiting LEO and competitor algorithms to handle the benchmark functions F1 to F7 are reported in
LEO | RSA | MPA | TSA | WOA | MVO | GWO | TLBO | GSA | PSO | GA | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | Mean | 0 | 0 | 2.91E-49 | 1.19E-46 | 3.10E-149 | 0.148926 | 6.92E-59 | 7.88E-75 | 1.42E-16 | 0.076198 | 32.8966 |
best | 0 | 0 | 7.27E-52 | 2.48E-51 | 7.40E-175 | 0.085859 | 5.81E-62 | 1.06E-77 | 5.40E-17 | 1.80E-05 | 21.06668 | |
std | 0 | 0 | 6.05E-49 | 4.64E-46 | 1.40E-148 | 0.040488 | 1.22E-58 | 1.35E-74 | 9.22E-17 | 0.162114 | 11.34771 | |
median | 0 | 0 | 4.82E-50 | 1.22E-48 | 7.30E-158 | 0.1438 | 1.34E-59 | 1.32E-75 | 1.06E-16 | 0.00377 | 28.87703 | |
ET | 2.8937191 | 15.362978 | 4.0495352 | 1.1430597 | 0.5090984 | 3.0899855 | 1.3499048 | 1.7468798 | 4.0656678 | 0.5208846 | 0.7732775 | |
rank | 1 | 1 | 5 | 6 | 2 | 9 | 4 | 3 | 7 | 8 | 10 | |
F2 | mean | 0 | 0 | 9.36E-28 | 2.73E-28 | 1.80E-105 | 0.237577 | 1.29E-34 | 7.37E-39 | 5.70E-08 | 0.748007 | 2.952664 |
best | 0 | 0 | 7.18E-32 | 2.63E-30 | 4.40E-111 | 0.132067 | 4.73E-36 | 2.95E-41 | 3.45E-08 | 0.034689 | 1.88159 | |
std | 0 | 0 | 1.40E-27 | 7.71E-28 | 4.50E-105 | 0.065336 | 1.81E-34 | 7.01E-39 | 2.75E-08 | 0.491811 | 0.710477 | |
median | 0 | 0 | 1.88E-28 | 2.57E-29 | 2.30E-107 | 0.244521 | 8.18E-35 | 4.99E-39 | 4.94E-08 | 0.738133 | 2.883687 | |
ET | 2.9539787 | 15.467449 | 2.4832809 | 1.1412211 | 0.5172495 | 2.61118 | 1.3202898 | 1.880538 | 3.8135901 | 0.5116 | 0.8022538 | |
rank | 1 | 1 | 6 | 5 | 2 | 8 | 4 | 3 | 7 | 9 | 10 | |
F3 | mean | 0 | 0 | 3.59E-12 | 2.33E-10 | 16661.72 | 13.9069 | 4.85E-16 | 2.62E-25 | 424.2214 | 1330.795 | 2369.284 |
best | 0 | 0 | 7.75E-18 | 2.28E-20 | 1815.858 | 3.460768 | 9.76E-20 | 4.02E-28 | 204.953 | 39.58518 | 1050.732 | |
std | 0 | 0 | 1.18E-11 | 8.32E-10 | 9734.605 | 5.176504 | 7.98E-16 | 6.12E-25 | 179.0495 | 2194.333 | 786.8941 | |
median | 0 | 0 | 9.15E-14 | 7.97E-14 | 16053.15 | 13.61052 | 1.62E-16 | 1.28E-26 | 379.9541 | 252.6239 | 2356.537 | |
ET | 8.016984 | 17.348643 | 5.80953 | 2.8731348 | 2.2469567 | 6.2164555 | 3.0022204 | 7.1115726 | 5.5055033 | 2.2323942 | 2.7865939 | |
rank | 1 | 1 | 4 | 5 | 10 | 6 | 3 | 2 | 7 | 8 | 9 | |
F4 | mean | 0 | 0 | 2.02E-19 | 0.008548 | 40.69107 | 0.548397 | 7.80E-14 | 2.23E-30 | 1.848961 | 5.795349 | 3.327428 |
best | 0 | 0 | 2.18E-20 | 7.87E-05 | 0.015711 | 0.308784 | 1.01E-15 | 1.06E-31 | 0.115681 | 3.024605 | 2.1599 | |
std | 0 | 0 | 1.39E-19 | 0.012793 | 30.82539 | 0.128869 | 3.06E-13 | 4.42E-30 | 1.561592 | 1.597663 | 0.551305 | |
median | 0 | 0 | 1.68E-19 | 0.005147 | 38.12119 | 0.518 | 4.49E-15 | 9.56E-31 | 1.708118 | 5.705651 | 3.446797 | |
ET | 2.8576027 | 15.674336 | 2.278244 | 1.113035 | 0.4876719 | 2.6937605 | 1.2761624 | 1.9604716 | 3.7837563 | 0.5207023 | 0.7268312 | |
rank | 1 | 1 | 3 | 5 | 10 | 6 | 4 | 2 | 7 | 9 | 8 | |
F5 | mean | 14.96979 | 15.775355 | 23.61681 | 28.30852 | 27.35899 | 455.0556 | 26.7365 | 26.74216 | 29.41177 | 244.4892 | 413.4251 |
best | 0 | 7.33E-29 | 22.93308 | 26.24623 | 26.5709 | 26.22482 | 26.09765 | 25.67999 | 24.60701 | 11.84862 | 196.9711 | |
std | 12.17512 | 11.85089 | 0.446521 | 0.765354 | 0.682608 | 797.5517 | 0.720383 | 0.919737 | 13.49202 | 660.584 | 131.1147 | |
median | 24.92267 | 1.03E-28 | 23.55474 | 28.64422 | 27.11864 | 34.62966 | 26.3345 | 26.37507 | 26.37678 | 81.73046 | 386.2732 | |
ET | 3.5164898 | 15.768983 | 2.792387 | 1.3642896 | 0.8270417 | 3.1662504 | 1.528469 | 2.5528939 | 3.9652233 | 0.7641116 | 1.0784773 | |
rank | 1 | 2 | 3 | 7 | 6 | 11 | 4 | 5 | 8 | 9 | 10 | |
F6 | mean | 0 | 6.730564 | 1.86E-09 | 3.633548 | 0.054408 | 0.152641 | 0.734942 | 1.118305 | 1.29E-16 | 0.968726 | 32.57338 |
best | 0 | 3.580407 | 7.14E-10 | 2.818033 | 0.010794 | 0.096246 | 0.251521 | 0.3022 | 6.39E-17 | 3.58E-05 | 15.90207 | |
std | 0 | 0.941189 | 1.01E-09 | 0.448715 | 0.067093 | 0.03576 | 0.237898 | 0.428271 | 6.03E-17 | 3.76892 | 12.48423 | |
median | 0 | 7.077314 | 1.58E-09 | 3.558282 | 0.020215 | 0.161175 | 0.751677 | 1.16301 | 1.20E-16 | 0.006687 | 29.24295 | |
ET | 2.7485223 | 15.385006 | 2.2584292 | 1.1126183 | 0.4742653 | 2.7653587 | 1.2809443 | 1.9332615 | 3.7752441 | 0.5209432 | 0.7771004 | |
rank | 1 | 10 | 3 | 9 | 4 | 5 | 6 | 8 | 2 | 7 | 11 | |
F7 | mean | 1.82E-05 | 4.91E-05 | 0.000687 | 0.005114 | 0.001269 | 0.010608 | 0.000741 | 0.001977 | 0.070721 | 0.171905 | 0.009377 |
best | 4.48E-07 | 5.84E-07 | 0.000194 | 0.000959 | 3.88E-05 | 0.005372 | 0.000129 | 0.000503 | 0.032415 | 0.082012 | 0.003451 | |
std | 1.53E-05 | 5.97E-05 | 0.000331 | 0.003311 | 0.001309 | 0.003574 | 0.000458 | 0.001321 | 0.048188 | 0.07817 | 0.003122 | |
median | 1.30E-05 | 3.46E-05 | 0.000668 | 0.004969 | 0.00118 | 0.011065 | 0.000684 | 0.001384 | 0.055562 | 0.168856 | 0.009215 | |
ET | 4.8368757 | 16.256141 | 3.7829544 | 1.8270088 | 1.1957864 | 4.4459639 | 2.0153875 | 3.9675918 | 4.4816348 | 1.1786659 | 2.0212563 | |
rank | 1 | 2 | 3 | 7 | 5 | 9 | 4 | 6 | 10 | 11 | 8 | |
Sum rank | 7 | 16 | 27 | 44 | 39 | 54 | 29 | 29 | 48 | 61 | 66 | |
Mean rank | 1 | 2.2857 | 3.857143 | 6.285714 | 5.571429 | 7.714286 | 4.142857 | 4.142857 | 6.857143 | 8.714286 | 9.428571 | |
Total rank | 1 | 2 | 3 | 6 | 5 | 8 | 4 | 4 | 7 | 9 | 10 |
The optimization results for the multimodal functions F8 to F13 using LEO and competitor algorithms are released in
LEO | RSA | MPA | TSA | WOA | MVO | GWO | TLBO | GSA | PSO | GA | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
F8 | mean | −12036.5 | −5492.29 | −9682.86 | −6071.81 | −11575 | −8125.01 | −5789.44 | −5205.39 | −2699.69 | −6733.77 | −8780.06 |
best | −12569.5 | −5680.5 | −10592.5 | −7029.82 | −12569.4 | −9232.93 | −7112.15 | −5843.69 | −3817.06 | −9667.69 | −9669.95 | |
std | 1301.688 | 174.523 | 398.4401 | 535.9141 | 1490.39 | 743.772 | 1053.789 | 345.395 | 481.9165 | 1106.838 | 534.6354 | |
median | −12569.5 | −5532.59 | −9609.38 | −6133.59 | −12345.4 | −8199.62 | −6053.67 | −5126.91 | −2711.14 | −6652.61 | −8830.14 | |
ET | 3.3654446 | 15.793495 | 2.7434296 | 1.3305275 | 0.7168849 | 2.5087217 | 1.5015152 | 2.918874 | 4.015091 | 0.8559844 | 1.4428962 | |
rank | 1 | 9 | 3 | 7 | 2 | 5 | 8 | 10 | 11 | 6 | 4 | |
F9 | mean | 0 | 0 | 0 | 161.7428 | 0 | 114.4433 | 8.53E-15 | 0 | 28.40606 | 67.1125 | 61.93813 |
best | 0 | 0 | 0 | 91.14753 | 0 | 79.67151 | 0 | 0 | 17.90926 | 34.825 | 28.30534 | |
std | 0 | 0 | 0 | 46.30094 | 0 | 21.49045 | 2.08E-14 | 0 | 7.07584 | 22.56296 | 18.83339 | |
median | 0 | 0 | 0 | 151.9278 | 0 | 110.5188 | 0 | 0 | 27.85884 | 62.19637 | 58.77781 | |
ET | 2.8669459 | 17.212279 | 2.376211 | 1.2647991 | 0.5323911 | 3.0123371 | 1.3347335 | 2.1450715 | 3.8513696 | 0.6159458 | 1.0649503 | |
rank | 1 | 1 | 1 | 7 | 1 | 6 | 2 | 1 | 3 | 5 | 4 | |
F10 | mean | 8.88E-16 | 8.88E-16 | 4.09E-15 | 1.195842 | 4.09E-15 | 0.670089 | 1.58E-14 | 4.26E-15 | 8.17E-09 | 3.030108 | 3.637002 |
best | 8.88E-16 | 8.88E-16 | 8.88E-16 | 7.99E-15 | 8.88E-16 | 0.082962 | 7.99E-15 | 8.88E-16 | 5.73E-09 | 1.963289 | 2.996306 | |
std | 0 | 0 | 1.09E-15 | 1.525827 | 2.28E-15 | 0.570127 | 3.75E-15 | 7.94E-16 | 1.85E-09 | 0.869371 | 0.374535 | |
median | 8.88E-16 | 8.88E-16 | 4.44E-15 | 2.22E-14 | 4.44E-15 | 0.627017 | 1.51E-14 | 4.44E-15 | 7.89E-09 | 2.836942 | 3.589696 | |
ET | 3.0094849 | 17.125813 | 2.4113186 | 1.2760752 | 0.5678977 | 3.1039018 | 1.3526236 | 2.1633398 | 3.8715898 | 0.6312308 | 1.0342777 | |
rank | 1 | 1 | 2 | 7 | 2 | 6 | 4 | 3 | 5 | 8 | 9 | |
F11 | mean | 0 | 0 | 0 | 0.005483 | 0 | 0.367801 | 0.000419 | 0 | 6.910967 | 0.085506 | 1.51316 |
best | 0 | 0 | 0 | 0 | 0 | 0.198274 | 0 | 0 | 2.816413 | 0.000312 | 1.254768 | |
std | 0 | 0 | 0 | 0.006307 | 0 | 0.097641 | 0.001874 | 0 | 2.991608 | 0.112577 | 0.136013 | |
median | 0 | 0 | 0 | 0 | 0 | 0.351523 | 0 | 0 | 6.936742 | 0.038802 | 1.545753 | |
ET | 3.5200459 | 16.768273 | 2.9110024 | 1.4053481 | 0.7795856 | 3.6463012 | 1.5800646 | 3.6000582 | 4.2177367 | 0.8763135 | 1.1001583 | |
rank | 1 | 1 | 1 | 3 | 1 | 5 | 2 | 1 | 7 | 4 | 6 | |
F12 | mean | 1.57E-32 | 1.285311 | 2.01E-10 | 6.893611 | 0.006199 | 0.796597 | 0.034746 | 0.071769 | 0.453499 | 1.072484 | 0.171397 |
best | 1.57E-32 | 0.894772 | 5.04E-11 | 2.099116 | 0.000818 | 0.00085 | 0.013112 | 0.028785 | 4.45E-19 | 0.1061 | 0.06194 | |
std | 2.81E-48 | 0.270639 | 1.64E-10 | 3.433371 | 0.004561 | 0.816088 | 0.015606 | 0.018791 | 0.763806 | 0.957253 | 0.149226 | |
median | 1.57E-32 | 1.111042 | 1.63E-10 | 7.188082 | 0.005082 | 0.535963 | 0.033764 | 0.071666 | 0.103669 | 0.920981 | 0.117873 | |
ET | 9.6997483 | 20.339866 | 7.1015617 | 3.5621885 | 2.8451768 | 8.1251023 | 3.6690062 | 9.844877 | 6.0840876 | 2.7898242 | 2.9549824 | |
rank | 1 | 10 | 2 | 11 | 3 | 8 | 4 | 5 | 7 | 9 | 6 | |
F13 | mean | 1.35E-32 | 5.86E-28 | 0.004674 | 3.170441 | 0.180748 | 0.03488 | 0.476456 | 1.0485 | 0.013185 | 5.6271 | 2.497715 |
best | 1.35E-32 | 6.28E-32 | 1.06E-09 | 2.163469 | 0.010754 | 0.012523 | 0.198365 | 0.60571 | 6.08E-18 | 0.169616 | 1.233345 | |
std | 2.81E-48 | 2.62E-27 | 0.011071 | 0.54377 | 0.136257 | 0.019653 | 0.18628 | 0.230574 | 0.025606 | 5.095354 | 0.874627 | |
median | 1.35E-32 | 5.15E-31 | 2.93E-09 | 3.19416 | 0.141566 | 0.030787 | 0.436048 | 1.081058 | 1.46E-17 | 4.537058 | 2.204447 | |
ET | 9.5685345 | 20.25093 | 7.1125807 | 12.459267 | 2.8430157 | 7.3899077 | 3.6610298 | 8.8516304 | 6.1925442 | 2.7882383 | 3.1005814 | |
rank | 1 | 2 | 3 | 10 | 6 | 5 | 7 | 8 | 4 | 11 | 9 | |
Sum rank | 6 | 24 | 12 | 45 | 15 | 35 | 27 | 28 | 37 | 43 | 38 | |
Mean rank | 1 | 4 | 2 | 7.5 | 2.5 | 5.833333 | 4.5 | 4.666667 | 6.166667 | 7.166667 | 6.333333 | |
Total rank | 1 | 4 | 2 | 11 | 3 | 7 | 5 | 6 | 8 | 10 | 9 |
The results of recruiting LEO and ten competitor algorithms to tackle fixed-dimensional multimodal functions F14 to F23 are reported in
LEO | RSA | MPA | TSA | WOA | MVO | GWO | TLBO | GSA | PSO | GA | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
F14 | Mean | 0.998004 | 4.717972 | 0.998004 | 9.568298 | 1.34532 | 0.998004 | 4.91193 | 0.998004 | 2.75256 | 2.622134 | 0.998599 |
Best | 0.998004 | 1.002247 | 0.998004 | 0.998004 | 0.998004 | 0.998004 | 0.998004 | 0.998004 | 1.135726 | 0.998004 | 0.998004 | |
Std | 0 | 3.552871 | 0 | 4.734825 | 0.739327 | 2.16E-12 | 4.46652 | 5.07E-07 | 1.072373 | 2.683317 | 0.002326 | |
Median | 0.998004 | 2.982105 | 0.998004 | 10.76318 | 0.998004 | 0.998004 | 2.982105 | 0.998004 | 2.916217 | 0.998004 | 0.998004 | |
ET | 17.7462491 | 8.0923956 | 11.635424 | 5.4224719 | 5.4019958 | 11.709339 | 5.3290457 | 17.22723 | 6.2926096 | 5.2216184 | 5.5117843 | |
Rank | 1 | 8 | 1 | 10 | 5 | 2 | 9 | 3 | 7 | 6 | 4 | |
F15 | Mean | 0.000307 | 0.001291 | 0.000307 | 0.008292 | 0.000622 | 0.004623 | 0.00637 | 0.003568 | 0.002911 | 0.003688 | 0.005315 |
Best | 0.000307 | 0.000682 | 0.000307 | 0.000308 | 0.000316 | 0.000308 | 0.000307 | 0.000308 | 0.001046 | 0.000307 | 0.000716 | |
std | 1.07E-30 | 8.10E-15 | 2.15E-30 | 1.44E-13 | 2.93E-15 | 8.08E-14 | 9.40E-14 | 7.25E-14 | 1.72E-14 | 7.20E-14 | 6.55E-14 | |
median | 0.000307 | 0.000977 | 0.000307 | 0.000481 | 0.000579 | 0.000721 | 0.000308 | 0.000372 | 0.002269 | 0.000672 | 0.002452 | |
ET | 2.77643102 | 3.1187178 | 1.6154843 | 0.4930209 | 0.4477875 | 1.2104194 | 0.5224414 | 1.8548539 | 1.619234 | 0.3907623 | 0.6492235 | |
rank | 1 | 4 | 2 | 11 | 3 | 8 | 10 | 6 | 5 | 7 | 9 | |
F16 | mean | −1.03163 | −1.02994 | −1.03163 | −1.02847 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 |
best | −1.03163 | −1.03161 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | |
std | 2.28E-27 | 2.65E-14 | 2.22E-27 | 9.74E-14 | 8.93E-22 | 2.62E-19 | 3.80E-20 | 1.74E-17 | 1.44E-27 | 1.44E-27 | 4.35E-17 | |
median | −1.03163 | −1.03092 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | |
ET | 2.48395666 | 1.8550119 | 1.5035697 | 0.4205441 | 0.4086445 | 1.0698828 | 0.4390887 | 1.6525417 | 1.4601808 | 0.2923966 | 0.5743327 | |
rank | 1 | 7 | 1 | 8 | 2 | 4 | 3 | 5 | 1 | 1 | 6 | |
F17 | mean | 0.397887 | 0.424048 | 0.397887 | 0.397917 | 0.397888 | 0.397887 | 0.397888 | 0.397975 | 0.397887 | 0.727026 | 0.43026 |
best | 0.397887 | 0.398061 | 0.397887 | 0.397888 | 0.397887 | 0.397887 | 0.397887 | 0.397888 | 0.397887 | 0.397887 | 0.397887 | |
std | 0 | 6.52E-13 | 0 | 3.52E-16 | 5.45E-18 | 1.20E-18 | 7.38E-18 | 1.17E-15 | 0 | 6.97E-12 | 1.43E-12 | |
median | 0.397887 | 0.405146 | 0.397887 | 0.397902 | 0.397887 | 0.397887 | 0.397888 | 0.397934 | 0.397887 | 0.397887 | 0.397904 | |
ET | 2.27247994 | 1.9477333 | 1.5244566 | 0.3927561 | 0.3839855 | 1.0321453 | 0.4026287 | 1.5180712 | 1.4767552 | 0.2382841 | 0.5140409 | |
rank | 1 | 7 | 1 | 5 | 3 | 2 | 4 | 6 | 1 | 9 | 8 | |
F18 | mean | 3 | 3.000035 | 3 | 4.35002 | 3.000008 | 3.000001 | 3.000007 | 3.000001 | 3 | 3 | 4.384222 |
best | 3 | 3 | 3 | 3.000001 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
std | 3.06E-27 | 6.45E-16 | 9.50E-27 | 6.04E-11 | 1.41E-16 | 5.94E-18 | 7.51E-17 | 1.30E-17 | 3.94E-26 | 2.83E-26 | 6.13E-11 | |
median | 3 | 3.000017 | 3 | 3.000005 | 3.000002 | 3 | 3.000005 | 3.000001 | 3 | 3 | 3.000653 | |
ET | 2.32502184 | 1.6320922 | 1.3984629 | 0.3793201 | 0.3472154 | 0.9753871 | 0.369263 | 1.4492838 | 1.3355731 | 0.2360337 | 0.4987929 | |
rank | 1 | 8 | 1 | 9 | 7 | 4 | 6 | 5 | 3 | 2 | 10 | |
F19 | mean | −3.86278 | −3.83729 | −3.86278 | −3.86274 | −3.85965 | −3.86278 | −3.8625 | −3.86168 | −3.86278 | −3.86278 | −3.86233 |
best | −3.86278 | −3.86107 | −3.86278 | −3.86278 | −3.86278 | −3.86278 | −3.86278 | −3.86269 | −3.86278 | −3.86278 | −3.86278 | |
std | 2.28E-26 | 2.45E-13 | 2.28E-26 | 2.90E-16 | 3.95E-14 | 6.19E-19 | 8.28E-15 | 2.35E-14 | 1.97E-26 | 1.99E-26 | 1.64E-14 | |
median | −3.86278 | −3.84502 | −3.86278 | −3.86275 | −3.86192 | −3.86278 | −3.86276 | −3.86243 | −3.86278 | −3.86278 | −3.86277 | |
ET | 6.82789298 | 2.4637427 | 1.6723552 | 0.5636936 | 0.5091525 | 1.3600367 | 0.5640221 | 2.1025086 | 1.6371261 | 0.4054243 | 0.7175955 | |
rank | 1 | 8 | 1 | 3 | 7 | 2 | 4 | 6 | 1 | 1 | 5 | |
F20 | mean | −3.322 | −2.68989 | −3.322 | −3.23413 | −3.2744 | −3.25053 | −3.21956 | −3.26134 | −3.322 | −3.25512 | −3.20621 |
best | −3.322 | −3.08344 | −3.322 | −3.32134 | −3.32193 | −3.322 | −3.32199 | −3.31429 | −3.322 | −3.322 | −3.31857 | |
std | 4.56E-27 | 3.95E-12 | 4.20E-27 | 2.00E-12 | 8.27E-13 | 5.99E-13 | 7.69E-13 | 5.76E-13 | 4.08E-27 | 7.90E-13 | 1.00E-12 | |
median | −3.322 | −2.84683 | −3.322 | −3.32021 | −3.32126 | −3.20308 | −3.20286 | −3.30078 | −3.322 | −3.322 | −3.20919 | |
ET | 3.2087629 | 4.3023587 | 1.7909341 | 0.6611978 | 0.5274411 | 1.3858897 | 0.644364 | 2.2403645 | 1.8011172 | 0.4276574 | 0.702935 | |
rank | 1 | 9 | 1 | 6 | 2 | 5 | 7 | 3 | 1 | 4 | 8 | |
F21 | mean | −10.1532 | −5.0552 | −10.1532 | −4.87294 | −8.6648 | −8.26233 | −9.90012 | −7.0017 | −5.43537 | −5.02652 | −4.87562 |
best | −10.1532 | −5.0552 | −10.1532 | −10.0977 | −10.153 | −10.1532 | −10.1531 | −9.34076 | −10.1532 | −10.1532 | −8.95019 | |
std | 3.58E-26 | 3.06E-18 | 2.31E-26 | 2.83E-11 | 2.77E-11 | 3.04E-11 | 1.13E-11 | 1.65E-11 | 3.41E-11 | 3.49E-11 | 2.35E-11 | |
median | −10.1532 | −5.0552 | −10.1532 | −4.39643 | −10.1511 | −10.1531 | −10.1528 | −7.33007 | −3.37362 | −2.68286 | −4.62892 | |
ET | 3.18573257 | 3.0584338 | 2.0165565 | 0.7852542 | 0.6399693 | 1.6992735 | 0.7098701 | 2.4809084 | 1.7481269 | 0.5359069 | 0.8200472 | |
rank | 1 | 7 | 1 | 10 | 3 | 4 | 2 | 5 | 6 | 8 | 9 | |
F22 | mean | −10.4029 | −5.08767 | −10.4029 | −7.5522 | −7.72926 | −8.58232 | −10.4025 | −7.294 | −10.2257 | −6.86804 | −7.26614 |
best | −10.4029 | −5.08767 | −10.4029 | −10.3694 | −10.4027 | −10.4029 | −10.4028 | −10.239 | −10.4029 | −10.4029 | −10.1895 | |
std | 2.97E-26 | 7.73E-18 | 3.65E-26 | 3.41E-11 | 3.11E-11 | 2.92E-11 | 2.96E-15 | 1.70E-11 | 7.93E-12 | 3.70E-11 | 2.20E-11 | |
median | −10.4029 | −5.08767 | −10.4029 | −10.0003 | −10.3963 | −10.4029 | −10.4025 | −7.67937 | −10.4029 | −7.76588 | −7.7724 | |
ET | 3.49341681 | 3.4076357 | 2.1893865 | 0.8162337 | 0.7114147 | 1.6880477 | 0.8373461 | 2.7606977 | 1.8845902 | 0.6828267 | 0.9331872 | |
rank | 1 | 10 | 1 | 6 | 5 | 4 | 2 | 7 | 3 | 9 | 8 | |
F23 | mean | −10.5364 | −5.12847 | −10.5364 | −5.74677 | −8.85422 | −8.95456 | −10.5359 | −7.90185 | −10.5364 | −5.99368 | −7.44674 |
best | −10.5364 | −5.12848 | −10.5364 | −10.4859 | −10.5363 | −10.5364 | −10.5363 | −9.55081 | −10.5364 | −10.5364 | −10.219 | |
std | 1.82E-26 | 1.71E-17 | 2.61E-26 | 3.61E-11 | 3.01E-11 | 2.87E-11 | 1.86E-15 | 1.60E-11 | 1.63E-26 | 3.87E-11 | 1.94E-11 | |
median | −10.5364 | −5.12847 | −10.5364 | −4.34441 | −10.5325 | −10.5363 | −10.5359 | −8.31455 | −10.5364 | −3.83543 | −8.09458 | |
ET | 3.85466265 | 3.4715726 | 2.4489502 | 0.9250013 | 0.833995 | 1.9249851 | 0.9267729 | 3.3723531 | 1.9306753 | 0.7938576 | 1.0732158 | |
rank | 1 | 10 | 2 | 9 | 5 | 4 | 3 | 6 | 2 | 8 | 7 | |
Sum rank | 10 | 78 | 12 | 77 | 42 | 39 | 50 | 52 | 30 | 55 | 74 | |
Mean rank | 1 | 7.8 | 1.2 | 7.7 | 4.2 | 3.9 | 5 | 5.2 | 3 | 5.5 | 7.4 | |
Total rank | 1 | 11 | 2 | 10 | 5 | 4 | 6 | 7 | 3 | 8 | 9 |
Boxplot diagrams of the proposed LEO and competitor algorithms to handle the functions F1 to F23 are shown in
This subsection is devoted to statistical analysis to determine whether LEO has a statistically significant superiority over competitor algorithms. To this end, the non-parametric Wilcoxon rank sum test is utilized to determine this issue [
The results obtained from the statistical analysis using the Wilcoxon rank sum test of the performance of the LEO and competitor algorithms are presented in
Compared algorithm | Objective function type | ||
---|---|---|---|
Unimodal | multimodal | Fixed-dimensional multimodal | |
LEO |
0.054684 | 1.63E-11 | 1.44E-34 |
LEO |
1.56E-13 | 1.15E-11 | 0.014404 |
LEO |
1.01E-24 | 1.28E-19 | 1.44E-34 |
LEO |
1.01E-24 | 1.26E-11 | 1.44E-34 |
LEO |
1.01E-24 | 1.97E-21 | 1.44E-34 |
LEO |
1.01E-24 | 1.66E-15 | 1.44E-34 |
LEO |
8.73E-24 | 1.04E-14 | 1.44E-34 |
LEO |
6.63E-24 | 1.97E-21 | 6.79E-14 |
LEO |
2.6E-23 | 1.97E-21 | 4.13E-17 |
LEO |
1.01E-24 | 2.48E-20 | 1.44E-34 |
The LEO method performs the optimization process using a random search of its population members in the problem-solving space in an iteration-based procedure. As a result, the LEO population size (
In the first analysis, the sensitivity of the LEO to the parameter N is evaluated. For this purpose, LEO is used for different values of the parameter
Objective functions | Number of population members | |||
---|---|---|---|---|
20 | 30 | 50 | 100 | |
F1 | 0 | 0 | 0 | 0 |
F2 | 0 | 0 | 0 | 0 |
F3 | 0 | 0 | 0 | 0 |
F4 | 0 | 0 | 0 | 0 |
F5 | 18.81451 | 14.96979 | 14.11042 | 5.062263 |
F6 | 0 | 0 | 0 | 0 |
F7 | 1.83E-05 | 1.82E-05 | 1.02E-05 | 6.54E-06 |
F8 | −10200.7 | −12036.5 | −12391.8 | −12569.5 |
F9 | 0 | 0 | 0 | 0 |
F10 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
F11 | 0 | 0 | 0 | 0 |
F12 | 1.57E-32 | 1.57E-32 | 1.57E-32 | 1.57E-32 |
F13 | 0.002243 | 1.35E-32 | 1.35E-32 | 1.35E-32 |
F14 | 1.14691 | 0.998 | 0.998 | 0.998004 |
F15 | 0.000307 | 0.000307 | 0.000307 | 0.000307 |
F16 | −1.03163 | −1.03163 | −1.03163 | −1.03163 |
F17 | 0.3978 | 0.3978 | 0.3978 | 0.397887 |
F18 | 4.35 | 3 | 3 | 3 |
F19 | −3.86278 | −3.86278 | −3.86278 | −3.86278 |
F20 | −3.32141 | −3.322 | −3.322 | −3.322 |
F21 | −10.1532 | −10.1532 | −10.1532 | −10.1532 |
F22 | −10.1372 | −10.4029 | −10.4029 | −10.4029 |
F23 | −10.5364 | −10.5364 | −10.5364 | −10.5364 |
In the analysis, the sensitivity of LEO to the
Objective functions | Maximum number of iterations | |||
---|---|---|---|---|
200 | 500 | 800 | 1000 | |
F1 | 2.1E-171 | 0 | 0 | 0 |
F2 | 5.54E-85 | 5.2E-219 | 0 | 0 |
F3 | 2.6E-108 | 5.8E-307 | 0 | 0 |
F4 | 2.89E-85 | 4.4E-215 | 0 | 0 |
F5 | 20.87501 | 18.88817 | 17.25557 | 14.96979 |
F6 | 0 | 0 | 0 | 0 |
F7 | 9.39E-05 | 3.92E-05 | 2.67E-05 | 1.82E-05 |
F8 | −10970.6 | −11378.9 | −11858.9 | −12036.5 |
F9 | 0 | 0 | 0 | 0 |
F10 | 8.88E-16 | 8.88E-16 | 8.88E-16 | 8.88E-16 |
F11 | 0 | 0 | 0 | 0 |
F12 | 1.57E-32 | 1.57E-32 | 1.57E-32 | 1.57E-32 |
F13 | 1.35E-32 | 1.35E-32 | 1.35E-32 | 1.35E-32 |
F14 | 0.998 | 0.998 | 0.998 | 0.998 |
F15 | 0.000308 | 0.000307 | 0.000307 | 0.000307 |
F16 | −1.03163 | −1.03163 | −1.03163 | −1.03163 |
F17 | 0.3978 | 0.3978 | 0.3978 | 0.3978 |
F18 | 3 | 3 | 3 | 3 |
F19 | −3.86278 | −3.86278 | −3.86278 | −3.86278 |
F20 | −3.32199 | −3.322 | −3.322 | −3.322 |
F21 | −9.8983 | −10.1532 | −10.1532 | −10.1532 |
F22 | −10.1372 | −10.4029 | −10.4029 | −10.4029 |
F23 | −10.3518 | −10.5364 | −10.5364 | −10.5364 |
Metaheuristic algorithms provide an optimization process based on a random search in the problem-solving space. The optimization operation will be successful when, first, the problem-solving space is scanned well at the global level, and second, it is scanned around the solutions discovered at the local level.
Metaheuristic algorithms based on local search, which indicates the exploitation ability of an algorithm, scan around existing solutions to achieve a better possible solution. Exploitation gives this capability to the metaheuristic algorithm to be able to converge towards the global optimal. The exploitation power of a metaheuristic algorithm in local search is well measured in unimodal problems. These types of issues have only one optimal solution, and the goal of optimizing them is to get as close as possible to the global optimal based on the power of exploitation. The results obtained from LEO on the unimodal functions of F1 to F7 indicate the exploitation capability of the proposed method in converging towards global optimal. This ability is especially evident in handling the functions F1, F2, F3, F4, F5, and F6, as LEO is converged precisely to the global optimum. Therefore, the simulation finding of unimodal functions is the high exploitation capability of LEO in local search.
Metaheuristic algorithms based on global search, which indicates the exploration ability of the algorithm, scan different parts of the problem-solving space with the aim of identifying the main optimal area without getting caught up in local solutions. In fact, exploration gives this metaheuristic algorithm the ability to break out of local optimal solutions. The exploitation power of a metaheuristic algorithm in global search is well measured in multimodal problems. In addition to the main solution, these types of problem have several local solutions, and the purpose of optimizing them is to identify the area related to the main optimal solution based on the power of exploration. The results obtained from the use of LEO in the multimodal functions of F8 to F13 indicate the exploration ability of the proposed method in identifying the main optimal region and not getting caught in local solutions. This ability is especially evident in the handling of the functions F9 and F11, as LEO has been able to both discover the local optimal region well and converge precisely to the global optimal of these functions. Thus, the finding that simulates multimodal functions is the high exploration capability of LEO in the global search.
Although exploration and exploitation capabilities are crucial to the performance of metaheuristic algorithms, a more successful algorithm can balance these two capabilities during the optimization process. Creating this balance will lead to: first, the algorithm being able to discover the main optimal region based on exploration, and second, to converge towards the global optimal based on exploitation. The ability of a metaheuristic algorithm to strike a balance between exploration and exploitation is well measured in fixed-dimensional multimodal problems. The results obtained from the implementation of LEO on fixed-dimensional multimodal functions from F14 to F23 indicate the ability of the proposed method to strike a balance between exploration and exploitation. In addition, LEO showed the capability to explore the main optimal region and converge towards the global optimal. Therefore, the simulation finding of fixed-dimensional multimodal functions is a high capability of LEO in balancing exploration and exploitation.
In this section, the performance of the proposed LEO approach in optimization tasks is evaluated on the CEC 2017 test suite. This set has thirty standard benchmark functions, including three unimodal functions C17-F1 to C17-F3, seven multimodal functions C17-F4 to C17-F10, ten hybrid functions C17-F11 to C17-F20, and ten composition functions C17-F21 to C17-F30. Full details of the CEC 2017 test suite are explained in [
The proposed LEO approach and competitor algorithms are employed in handling the CEC 2017 test suite. The simulation results are reported in
LEO | RSA | MPA | TSA | WOA | MVO | GWO | TLBO | GSA | PSO | GA | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
C17-F1 | mean | 100.00002 | 1.32E + 10 | 113895.2 | 2.872E + 09 | 8690927.8 | 9430.0349 | 230476.22 | 76686529 | 245.4564 | 3328.3076 | 16493242 |
best | 100.00001 | 9.16E + 09 | 175.65384 | 393204569 | 2039380.1 | 4426.4772 | 19273.455 | 53825370 | 100.19451 | 703.20384 | 10463278 | |
std | 9.97E−06 | 3.87E + 09 | 2.25E + 05 | 2.17E + 09 | 1.04E + 07 | 4.38E + 03 | 3.12E + 05 | 2.98E + 07 | 1.99E + 02 | 2.67E + 03 | 6.26E + 06 | |
median | 100.00001 | 1.269E + 10 | 1824.5957 | 2.867E + 09 | 4222785 | 9232.082 | 110843.31 | 66917665 | 176.41254 | 3489.0256 | 15334932 | |
ET | 0.3172958 | 1.8727244 | 0.8779954 | 0.3733181 | 0.2978872 | 0.4491082 | 0.3701036 | 1.1978118 | 0.8354247 | 0.3481572 | 0.3980948 | |
rank | 1 | 11 | 5 | 10 | 7 | 4 | 6 | 9 | 2 | 3 | 8 | |
C17-F3 | mean | 300 | 10123.783 | 338.60644 | 12744.19 | 3540.6629 | 300.0554 | 4659.2923 | 763.2455 | 8808.339 | 300 | 24812.862 |
best | 300 | 6884.7583 | 300.16743 | 9428.5035 | 1237.6601 | 300.01055 | 2582.6579 | 586.05927 | 4051.73 | 300 | 18042.268 | |
std | 1.54E−10 | 4.88E + 03 | 4.40E + 01 | 2.98E + 03 | 2.60E + 03 | 5.76E−02 | 2.71E + 03 | 1.76E + 02 | 4.56E + 03 | 2.44E−12 | 1.02E + 04 | |
median | 300 | 8109.7558 | 329.75584 | 13075.684 | 2913.1907 | 300.03781 | 3796.1123 | 769.47312 | 8108.2321 | 300 | 20617.655 | |
ET | 0.2993556 | 1.8736643 | 0.8212701 | 0.3644942 | 0.2867802 | 0.4102439 | 0.3556395 | 1.2062873 | 0.8082796 | 0.3183468 | 0.4028248 | |
rank | 2 | 9 | 4 | 10 | 6 | 3 | 7 | 5 | 8 | 1 | 11 | |
C17-F4 | mean | 400 | 1089.4244 | 403.84745 | 637.73782 | 423.53626 | 404.77441 | 417.73193 | 412.94333 | 406.64192 | 407.30478 | 416.29336 |
best | 400 | 681.4032 | 400.05471 | 408.20739 | 408.00923 | 403.92719 | 407.25852 | 409.57264 | 406.52228 | 401.28055 | 412.93487 | |
std | 6.944E−09 | 5.66E + 02 | 2.722759 | 3.19E + 02 | 1.80E + 01 | 9.76E−01 | 1.49E + 01 | 5.02E + 00 | 0.1756922 | 4.26E + 00 | 5.30E + 00 | |
median | 400 | 876.77377 | 404.72508 | 526.19315 | 422.4949 | 404.64257 | 412.30938 | 410.93454 | 406.57131 | 408.30336 | 414.07128 | |
ET | 0.3043906 | 1.8715125 | 0.828421 | 0.3301639 | 0.2831765 | 0.4272641 | 0.3519701 | 1.1745109 | 0.7738175 | 0.312928 | 0.3816822 | |
rank | 1 | 11 | 2 | 10 | 9 | 3 | 8 | 6 | 4 | 5 | 7 | |
C17-F5 | mean | 509.4521 | 570.97852 | 520.39673 | 555.36156 | 557.14932 | 516.91702 | 514.92238 | 539.19304 | 548.00649 | 539.22352 | 532.43573 |
best | 507.95967 | 560.71035 | 511.9395 | 525.77926 | 530.35745 | 510.94663 | 508.56508 | 531.29651 | 536.81332 | 523.57033 | 527.03431 | |
std | 1.28E + 00 | 1.31E + 01 | 5.66E + 00 | 3.13E + 01 | 2.88E + 01 | 5.14E + 00 | 4.69E + 00 | 7.62E + 00 | 1.08E + 01 | 2.19E + 01 | 4.61E + 00 | |
median | 509.45211 | 566.65115 | 522.88405 | 552.43704 | 551.16319 | 516.91766 | 515.72158 | 537.9668 | 546.76286 | 530.84366 | 532.23219 | |
ET | 0.3023736 | 1.8887621 | 0.8763099 | 0.3562381 | 0.291399 | 0.4501801 | 0.3750951 | 1.2508191 | 0.8026902 | 0.3304977 | 0.4101859 | |
rank | 1 | 11 | 4 | 9 | 10 | 3 | 2 | 6 | 8 | 7 | 5 | |
C17-F6 | mean | 600.00055 | 649.12886 | 600.43167 | 628.19803 | 632.02369 | 601.00316 | 601.3355 | 605.4803 | 624.93122 | 603.25733 | 607.67247 |
best | 600.00022 | 642.83648 | 600.03648 | 613.46525 | 616.78611 | 600.48046 | 600.11559 | 604.02203 | 613.54637 | 601.12871 | 604.66041 | |
std | 3.03E−04 | 5.31E + 00 | 6.69E−01 | 1.58E + 01 | 1.49E + 01 | 7.54E−01 | 2.28E + 00 | 1.30E + 00 | 1.05E + 01 | 2.28E + 00 | 3.34E + 00 | |
median | 600.00051 | 649.00457 | 600.12907 | 625.7324 | 631.1373 | 600.70446 | 600.23533 | 605.37232 | 623.64711 | 602.96419 | 607.60822 | |
ET | 0.3619768 | 1.9709199 | 0.9639959 | 0.409375 | 0.3515911 | 0.5038948 | 0.4302542 | 1.4059159 | 0.8681172 | 0.4037424 | 0.4727251 | |
rank | 1 | 11 | 2 | 9 | 10 | 3 | 4 | 6 | 8 | 5 | 7 | |
C17-F7 | mean | 721.53846 | 806.79054 | 725.52924 | 792.43089 | 792.82487 | 727.82661 | 740.81091 | 758.78859 | 717.50492 | 746.01076 | 736.89383 |
best | 718.98051 | 796.93456 | 714.18747 | 768.65568 | 766.28166 | 723.10768 | 731.66864 | 755.46415 | 714.398 | 730.73613 | 728.78023 | |
std | 1.97E + 00 | 7.11E + 00 | 1.19E + 01 | 2.37E + 01 | 2.14E + 01 | 6.05E + 00 | 7.39E + 00 | 4.64E + 00 | 3.12E + 00 | 2.10E + 01 | 5.42E + 00 | |
median | 721.69834 | 808.92042 | 722.87693 | 789.0756 | 796.38176 | 725.76904 | 741.21091 | 757.01329 | 716.90536 | 738.54308 | 739.42632 | |
ET | 0.3230003 | 1.9364146 | 0.8928027 | 0.3756039 | 0.3177452 | 0.4635729 | 0.3910292 | 1.2818439 | 0.7851255 | 0.3475454 | 0.4267207 | |
rank | 2 | 11 | 3 | 9 | 10 | 4 | 6 | 8 | 1 | 7 | 5 | |
C17-F8 | mean | 808.20841 | 858.60323 | 809.70134 | 849.07577 | 832.91949 | 831.03479 | 816.08247 | 833.47787 | 819.15294 | 824.87392 | 823.20364 |
best | 804.9748 | 855.82927 | 806.9649 | 843.52729 | 814.19138 | 819.90228 | 812.69591 | 827.68976 | 815.91933 | 812.93446 | 816.71073 | |
std | 2.21E + 00 | 3.08E + 00 | 2.62E + 00 | 5.12E + 00 | 1.58E + 01 | 2.07E + 01 | 3.69E + 00 | 3.95E + 00 | 2.97E + 00 | 1.11E + 01 | 9.28E + 00 | |
median | 808.95463 | 858.21387 | 809.95008 | 848.9468 | 832.36631 | 821.12256 | 815.29334 | 834.83538 | 818.90419 | 823.87898 | 819.66 | |
ET | 0.3250002 | 1.8945481 | 0.8614258 | 0.354857 | 0.3171315 | 0.4475804 | 0.3828746 | 1.2504869 | 0.7580395 | 0.3373446 | 0.4180784 | |
rank | 1 | 11 | 2 | 10 | 8 | 7 | 3 | 9 | 4 | 6 | 5 | |
C17-F9 | mean | 900 | 1439.2536 | 926.16775 | 1598.9883 | 1562.1942 | 900.34594 | 900.78655 | 948.97968 | 900 | 958.54298 | 905.0812 |
best | 900 | 1139.663 | 901.0761 | 1009.0175 | 1043.5495 | 900.00299 | 900.01409 | 928.3364 | 900 | 901.81727 | 901.98992 | |
std | 2.60E−08 | 2.98E + 02 | 4.44E + 01 | 7.15E + 02 | 6.50E + 02 | 4.36E−01 | 1.32E + 00 | 2.70E + 01 | 0.00E + 00 | 5.12E + 01 | 2.16E + 00 | |
median | 900 | 1383.0983 | 905.49128 | 1431.7658 | 1345.5542 | 900.23299 | 900.18809 | 939.43633 | 900 | 953.61704 | 905.78519 | |
ET | 0.3562098 | 1.9355973 | 0.8955252 | 0.3693074 | 0.3140052 | 0.4698967 | 0.3918213 | 1.2679357 | 0.7862795 | 0.3677462 | 0.4245071 | |
rank | 2 | 9 | 6 | 11 | 10 | 3 | 4 | 7 | 1 | 8 | 5 | |
C17-F10 | mean | 1448.2414 | 2472.9544 | 1840.8555 | 2330.0521 | 2254.8178 | 1612.9939 | 1706.457 | 2155.1391 | 2665.9463 | 1977.6016 | 1778.7483 |
best | 1339.1179 | 2202.7083 | 1122.4927 | 1523.226 | 1886.5293 | 1495.6607 | 1610.8705 | 2065.3083 | 2227.3131 | 1837.465 | 1530.4446 | |
std | 1.19E + 02 | 2.73E + 02 | 5.13E + 02 | 5.53E + 02 | 4.09E + 02 | 1.31E + 02 | 7.72E + 01 | 8.45E + 01 | 3.51E + 02 | 2.07E + 02 | 2.21E + 02 | |
median | 1421.8198 | 2455.6405 | 2010.586 | 2527.3279 | 2189.5993 | 1587.4155 | 1712.9007 | 2160.193 | 2699.8038 | 1897.4982 | 1783.6486 | |
ET | 0.2921257 | 1.9344708 | 0.8899964 | 0.3861561 | 0.3123344 | 0.4670793 | 0.3881009 | 1.3143316 | 0.8104788 | 0.3601149 | 0.4339666 | |
rank | 1 | 10 | 5 | 9 | 8 | 2 | 3 | 7 | 11 | 6 | 4 | |
C17-F11 | mean | 1102.4081 | 3988.9348 | 1115.9881 | 1288.4482 | 1220.2127 | 1126.1077 | 1143.8912 | 1151.1927 | 1167.7559 | 1138.16 | 5155.6612 |
best | 1101.035 | 2128.0718 | 1112.5021 | 1151.0867 | 1124.5424 | 1113.657 | 1134.6187 | 1132.2017 | 1132.3212 | 1114.3392 | 1359.2219 | |
std | 1.0913706 | 1728.279 | 4.0348047 | 141.03395 | 125.71804 | 1.46E + 01 | 9.882597 | 1.74E + 01 | 31.540223 | 16.776248 | 3985.9942 | |
median | 1102.4457 | 4053.552 | 1114.9148 | 1258.449 | 1175.5186 | 1123.2158 | 1141.9369 | 1149.9879 | 1166.8796 | 1142.2867 | 4377.88 | |
ET | 0.3082884 | 1.9053695 | 0.8705133 | 0.3753039 | 0.291397 | 0.4437908 | 0.3689143 | 1.2571838 | 0.8021757 | 0.3487779 | 0.426534 | |
rank | 1 | 10 | 2 | 9 | 8 | 3 | 5 | 6 | 7 | 4 | 11 | |
C17-F12 | mean | 1209.4469 | 67700546 | 3037.4471 | 89572425 | 9353082.3 | 517891.28 | 151943.54 | 2295970.5 | 895980.25 | 14707.321 | 1692251.5 |
best | 1200.0549 | 30532622 | 1669.0281 | 330069.7 | 57983.924 | 8014.0696 | 41866.656 | 493223.58 | 9831.5883 | 1562.5217 | 171741.04 | |
std | 1.80E + 01 | 3.33E + 07 | 1.67E + 03 | 1.76E + 08 | 9.76E + 06 | 6.17E + 05 | 2.10E + 05 | 1.42E + 06 | 1.19E + 06 | 1.00E + 04 | 2.74E + 06 | |
median | 1200.6076 | 67910730 | 2527.3454 | 1923878.3 | 8475087.7 | 384471.94 | 49316.617 | 2526439.2 | 492469.14 | 16361.009 | 400976.52 | |
ET | 0.3032189 | 1.9319159 | 0.8627085 | 0.3800094 | 0.2951645 | 0.4692286 | 0.372782 | 1.2642304 | 0.8014802 | 0.3495984 | 0.4363488 | |
rank | 1 | 10 | 2 | 11 | 9 | 5 | 4 | 8 | 6 | 3 | 7 | |
C17-F13 | mean | 1305.6389 | 40894980 | 1343.4927 | 16224.82 | 20438.622 | 13523.206 | 11757.028 | 7095.1454 | 12468.177 | 5580.4752 | 70502.24 |
best | 1301.7989 | 115399.57 | 1311.2063 | 7208.9788 | 8268.3414 | 2358.1337 | 7429.7441 | 3996.5404 | 7359.7815 | 2151.9689 | 12200.378 | |
std | 3.86E + 00 | 5.66E + 07 | 2.21E + 01 | 6.98E + 03 | 1.10E + 04 | 1.05E + 04 | 5.04E + 03 | 3.39E + 03 | 3.79E + 03 | 3.06E + 03 | 7.31E + 04 | |
median | 1305.1608 | 20451941 | 1352.2554 | 17905.32 | 19461.528 | 12026.242 | 10471.052 | 6346.504 | 13268.961 | 5291.5555 | 50644.828 | |
ET | 0.3159577 | 1.8878708 | 0.9128838 | 0.3918017 | 0.3109009 | 0.4621428 | 0.4100802 | 1.293707 | 0.7846826 | 0.3845192 | 0.4292407 | |
rank | 1 | 11 | 2 | 8 | 9 | 7 | 5 | 4 | 6 | 3 | 10 | |
C17-F14 | mean | 1402.2394 | 5470.9716 | 1427.7437 | 4765.1136 | 1584.9334 | 1435.7655 | 2833.0291 | 1521.9175 | 5567.3905 | 7002.6586 | 6763.8244 |
best | 1400.9954 | 2190.0306 | 1401.9899 | 2573.429 | 1496.5768 | 1429.0875 | 1476.9492 | 1475.783 | 2070.8126 | 3681.5587 | 1870.5221 | |
std | 1.8833096 | 2.52E + 03 | 22.172178 | 1.46E + 03 | 8.71E + 01 | 5.06E + 00 | 1.58E + 03 | 3.70E + 01 | 3060.1102 | 3.26E + 03 | 5.16E + 03 | |
median | 1401.4934 | 5804.6809 | 1427.4043 | 5441.0149 | 1577.9782 | 1436.7638 | 2528.55 | 1525.2724 | 5361.7578 | 6433.3565 | 6801.1672 | |
ET | 0.3127391 | 1.9024018 | 0.888269 | 0.3900152 | 0.309189 | 0.4749437 | 0.4114049 | 1.3180878 | 0.7786291 | 0.3770875 | 0.4499625 | |
rank | 1 | 8 | 2 | 7 | 5 | 3 | 6 | 4 | 9 | 11 | 10 | |
C17-F15 | mean | 1500.432 | 9246.3665 | 1508.1166 | 14722.578 | 5770.572 | 1556.754 | 5410.7223 | 1793.1079 | 15392.044 | 4459.8968 | 4356.8363 |
best | 1500.3617 | 4958.6671 | 1502.3574 | 4159.502 | 2029.7502 | 1535.9023 | 1810.0088 | 1692.966 | 6548.9487 | 2271.1247 | 1868.5744 | |
std | 6.52E−02 | 5.19E + 03 | 6.89E + 00 | 1.09E + 04 | 6.24E + 03 | 1.84E + 01 | 2.49E + 03 | 1.48E + 02 | 6.13E + 03 | 1.91E + 03 | 2.81E + 03 | |
median | 1500.4366 | 7766.8841 | 1506.3314 | 15203.76 | 2981.3282 | 1557.1446 | 6299.4192 | 1733.9539 | 17366.648 | 4350.8524 | 4234.0085 | |
ET | 0.294 | 1.9022906 | 0.8479029 | 0.361073 | 0.2894658 | 0.4514907 | 0.3667248 | 1.2349542 | 0.8035952 | 0.341312 | 0.4035415 | |
rank | 1 | 9 | 2 | 10 | 8 | 3 | 7 | 4 | 11 | 6 | 5 | |
C17-F16 | mean | 1601.417 | 2049.3493 | 1693.6086 | 2139.1819 | 1859.9833 | 1875.4528 | 1751.4474 | 1702.8066 | 2212.7605 | 1872.4306 | 1822.0213 |
best | 1601.0584 | 2022.0876 | 1602.8932 | 1993.8675 | 1659.0682 | 1722.138 | 1608.1802 | 1640.8401 | 2163.3077 | 1720.5276 | 1749.2084 | |
std | 2.69E−01 | 2.01E + 01 | 1.15E + 02 | 1.70E + 02 | 2.08E + 02 | 1.26E + 02 | 1.74E + 02 | 1.03E + 02 | 5.96E + 01 | 1.18E + 02 | 4.86E + 01 | |
median | 1601.4494 | 2053.9084 | 1664.7854 | 2097.6784 | 1847.8764 | 1876.5461 | 1697.9799 | 1656.6107 | 2194.891 | 1897.6183 | 1845.052 | |
ET | 0.3153007 | 1.8969569 | 0.8690144 | 0.3843161 | 0.3049045 | 0.4727657 | 0.3788271 | 1.282693 | 0.8012009 | 0.3413454 | 0.4216952 | |
rank | 1 | 9 | 2 | 10 | 6 | 8 | 4 | 3 | 11 | 7 | 5 | |
C17-F17 | mean | 1718.215 | 1860.355 | 1740.9724 | 1863.2917 | 1865.2555 | 1798.9236 | 1770.9041 | 1758.3665 | 1966.631 | 1861.4076 | 1752.9279 |
best | 1706.6847 | 1817.3276 | 1732.6829 | 1802.7291 | 1824.6212 | 1734.4969 | 1743.3297 | 1757.5236 | 1759.2691 | 1772.4206 | 1749.6637 | |
std | 7.71E + 00 | 5.07E + 01 | 8.21E + 00 | 7.19E + 01 | 4.07E + 01 | 6.19E + 01 | 2.84E + 01 | 1.01E + 00 | 1.60E + 02 | 9.52E + 01 | 2.46E + 00 | |
median | 1721.7195 | 1846.9176 | 1741.4033 | 1841.5942 | 1858.1224 | 1802.7053 | 1771.7776 | 1758.0739 | 1978.2256 | 1845.205 | 1753.2717 | |
ET | 0.3658334 | 1.9520999 | 0.9804134 | 0.4225763 | 0.3617183 | 0.5147969 | 0.4358819 | 1.4915146 | 0.8211044 | 0.401581 | 0.4841767 | |
rank | 1 | 7 | 2 | 9 | 10 | 6 | 5 | 4 | 11 | 8 | 3 | |
C17-F18 | mean | 1800.7237 | 94816134 | 1833.5988 | 28939.221 | 24569.47 | 17408.62 | 23074.566 | 40341.67 | 11965.988 | 11312.675 | 9290.5265 |
best | 1800.0758 | 1566346.8 | 1810.6191 | 11253.046 | 3349.5232 | 4178.411 | 7853.7434 | 15409.717 | 8162.1711 | 3072.7269 | 4531.9869 | |
std | 5.71E−01 | 1.83E + 08 | 1.73E + 01 | 1.22E + 04 | 1.55E + 04 | 1.63E + 04 | 1.16E + 04 | 1.77E + 04 | 4.81E + 03 | 6.63E + 03 | 5.92E + 03 | |
median | 1800.7107 | 4220229.1 | 1837.6971 | 33389.233 | 28156.391 | 13637.271 | 24346.02 | 44840.006 | 10615.389 | 12384.862 | 7518.9237 | |
ET | 0.3256107 | 1.8980687 | 0.8925264 | 0.3567493 | 0.3006547 | 0.4739268 | 0.3785245 | 1.280987 | 0.7779809 | 0.3552352 | 0.4123118 | |
rank | 1 | 11 | 2 | 9 | 8 | 6 | 7 | 10 | 5 | 4 | 3 | |
C17-F19 | mean | 1901.0291 | 679394.81 | 1909.263 | 73494.592 | 292988.36 | 2023.3414 | 6135.8892 | 2157.1287 | 28942.073 | 11413.638 | 20057.439 |
best | 1900.9488 | 161328.47 | 1902.5189 | 2005.9509 | 7901.4191 | 1925.8671 | 1933.5347 | 2043.9118 | 8744.8689 | 5501.8386 | 8219.752 | |
std | 6.50E−02 | 8.04E + 05 | 5.99E + 00 | 1.35E + 05 | 5.51E + 05 | 1.68E + 02 | 5.04E + 03 | 1.43E + 02 | 1.99E + 04 | 6.29E + 03 | 1.13E + 04 | |
median | 1901.0328 | 341107.87 | 1908.7373 | 8081.1832 | 22321.465 | 1946.7148 | 5287.4278 | 2113.0137 | 26612.935 | 10074.195 | 21059.965 | |
ET | 0.6015283 | 2.1930547 | 1.4988764 | 0.6585444 | 0.6057556 | 0.7472846 | 0.682842 | 2.1939116 | 1.0763017 | 0.6634356 | 0.7332974 | |
rank | 1 | 11 | 2 | 9 | 10 | 3 | 5 | 4 | 8 | 6 | 7 | |
C17-F20 | mean | 2010.0534 | 2301.7385 | 2024.9164 | 2213.9835 | 2256.847 | 2155.3699 | 2054.9244 | 2084.6507 | 2329.6939 | 2235.7275 | 2052.2351 |
best | 2001.9904 | 2248.0003 | 2020.3081 | 2089.0765 | 2066.7782 | 2026.6169 | 2030.6709 | 2064.3507 | 2201.9677 | 2196.8025 | 2036.0513 | |
std | 1.02E + 01 | 5.15E + 01 | 8.87E + 00 | 1.49E + 02 | 1.29E + 02 | 9.73E + 01 | 2.42E + 01 | 2.56E + 01 | 9.38E + 01 | 3.50E + 01 | 1.84E + 01 | |
median | 2006.6239 | 2298.0865 | 2020.5675 | 2170.523 | 2303.7862 | 2165.9516 | 2055.9022 | 2076.3213 | 2347.968 | 2240.2324 | 2047.3626 | |
ET | 0.3549563 | 1.9792104 | 1.0031495 | 0.4189973 | 0.3629973 | 0.5175917 | 0.4502586 | 1.4757591 | 0.8379548 | 0.4152999 | 0.4851776 | |
rank | 1 | 10 | 2 | 7 | 9 | 6 | 4 | 5 | 11 | 8 | 3 | |
C17-F21 | mean | 2200 | 2308.2706 | 2290.0245 | 2348.6744 | 2337.4394 | 2297.0061 | 2292.0723 | 2304.9949 | 2361.3391 | 2322.6768 | 2308.7322 |
best | 2200 | 2245.3264 | 2209.0492 | 2336.2904 | 2318.0377 | 2201.9281 | 2201.4314 | 2205.0065 | 2357.5108 | 2309.5763 | 2223.4109 | |
std | 1.221E−05 | 59.029993 | 54.252421 | 18.961418 | 14.666468 | 6.36E + 01 | 60.473011 | 6.68E + 01 | 5.8686381 | 11.367088 | 57.267239 | |
median | 2200 | 2301.2791 | 2313.7008 | 2340.7329 | 2341.0693 | 2325.2005 | 2321.3294 | 2335.6805 | 2358.9014 | 2321.9666 | 2334.4727 | |
ET | 0.3670121 | 1.9871627 | 0.9961948 | 0.431126 | 0.3589478 | 0.5171054 | 0.4407936 | 1.4386133 | 0.8538204 | 0.4021599 | 0.4880179 | |
rank | 1 | 6 | 2 | 10 | 9 | 4 | 3 | 5 | 11 | 8 | 7 | |
C17-F22 | mean | 2300.4086 | 2960.4831 | 2306.5754 | 2388.4242 | 2318.9205 | 2303.6665 | 2305.9506 | 2323.9978 | 2300.2574 | 2313.3484 | 2321.8836 |
best | 2300 | 2756.6595 | 2302.8892 | 2310.0835 | 2313.2141 | 2302.3375 | 2300.6213 | 2315.17 | 2300 | 2300.6448 | 2315.3845 | |
std | 4.77E−01 | 2.11E + 02 | 3.48E + 00 | 8.99E + 01 | 7.76E + 00 | 9.58E−01 | 6.58E + 00 | 7.23E + 00 | 1.77E−01 | 2.34E + 01 | 5.73E + 00 | |
median | 2300.3665 | 2917.2594 | 2306.5182 | 2386.4565 | 2316.1003 | 2303.905 | 2303.8566 | 2325.2645 | 2300.3161 | 2302.1961 | 2322.693 | |
ET | 0.3999132 | 2.0271331 | 1.0408899 | 0.4446991 | 0.3889436 | 0.5590779 | 0.4744016 | 1.5425062 | 0.8605875 | 0.4368888 | 0.5218687 | |
rank | 2 | 11 | 5 | 10 | 7 | 3 | 4 | 9 | 1 | 6 | 8 | |
C17-F23 | mean | 2608.667 | 2697.6138 | 2648.389 | 2690.7613 | 2658.5666 | 2613.7637 | 2623.5065 | 2637.0749 | 2733.7617 | 2642.8041 | 2663.4253 |
best | 2606.5232 | 2673.1877 | 2624.4481 | 2672.5407 | 2613.963 | 2607.0408 | 2608.1065 | 2623.829 | 2724.4199 | 2612.6221 | 2652.9578 | |
std | 2.05E + 00 | 2.01E + 01 | 2.07E + 01 | 1.78E + 01 | 3.38E + 01 | 6.12E + 00 | 1.60E + 01 | 1.09E + 01 | 9.84E + 00 | 2.02E + 01 | 1.44E + 01 | |
median | 2608.5333 | 2697.5253 | 2648.1834 | 2687.7457 | 2663.6208 | 2614.1411 | 2624.2502 | 2636.9896 | 2732.5358 | 2651.4089 | 2658.0759 | |
ET | 0.4175118 | 1.9807793 | 1.0808874 | 0.4609724 | 0.431678 | 0.5594673 | 0.4829631 | 1.5702774 | 0.8909863 | 0.4357085 | 0.5259606 | |
rank | 1 | 10 | 6 | 9 | 7 | 2 | 3 | 4 | 11 | 5 | 8 | |
C17-F24 | mean | 2500.0002 | 2889.4365 | 2641.6992 | 2821.9182 | 2795.2084 | 2751.4754 | 2750.8673 | 2765.2163 | 2740.8134 | 2778.2175 | 2773.0636 |
best | 2500.0001 | 2839.3557 | 2500.5995 | 2796.5576 | 2751.5338 | 2746.0029 | 2737.1449 | 2758.0342 | 2500 | 2773.298 | 2765.1812 | |
std | 4.785E−05 | 3.70E + 01 | 162.78876 | 2.24E + 01 | 2.95E + 01 | 3.95E + 00 | 2.02E + 01 | 5.82E + 00 | 162.61032 | 3.85E + 00 | 1.03E + 01 | |
median | 2500.0002 | 2895.5602 | 2640.796 | 2822.5121 | 2806.3359 | 2752.3181 | 2743.0222 | 2765.5625 | 2802.961 | 2778.592 | 2769.4377 | |
ET | 0.4417598 | 1.989241 | 1.098033 | 0.4703164 | 0.4488538 | 0.5772698 | 0.5009254 | 1.5867588 | 0.8846244 | 0.445734 | 0.5386319 | |
rank | 1 | 11 | 2 | 10 | 9 | 5 | 4 | 6 | 3 | 8 | 7 | |
C17-F25 | mean | 2897.7429 | 3383.1819 | 2934.6471 | 3128.9324 | 2954.6209 | 2898.191 | 2941.7004 | 2930.7864 | 2932.5136 | 2934.9204 | 2954.9786 |
best | 2897.7429 | 3352.0262 | 2899.7425 | 2943.9861 | 2948.9808 | 2897.8394 | 2918.7097 | 2909.7743 | 2899.585 | 2899.61 | 2952.1894 | |
std | 3.04E−07 | 4.90E + 01 | 2.33E + 01 | 2.54E + 02 | 5.12E + 00 | 2.46E−01 | 1.54E + 01 | 1.60E + 01 | 2.20E + 01 | 2.36E + 01 | 2.69E + 00 | |
median | 2897.7429 | 3362.6645 | 2945.8363 | 3033.9803 | 2954.1309 | 2898.2616 | 2948.6451 | 2932.2998 | 2943.4085 | 2945.6644 | 2954.6376 | |
ET | 0.418846 | 1.9986182 | 1.0641379 | 0.4549042 | 0.4141905 | 0.5427762 | 0.465997 | 1.5381446 | 0.8526837 | 0.4317387 | 0.5142647 | |
rank | 1 | 11 | 5 | 10 | 8 | 2 | 7 | 3 | 4 | 6 | 9 | |
C17-F26 | mean | 2875.0004 | 4236.0105 | 3263.5095 | 3883.8521 | 3260.1768 | 2900.1421 | 2956.2263 | 3287.936 | 4125.3708 | 2851.8668 | 3024.4918 |
best | 2800.0013 | 3815.9884 | 2900.0298 | 2911.5703 | 2833.7931 | 2900.1055 | 2900.235 | 2989.7582 | 3570.8634 | 2600 | 2907.1933 | |
std | 5.00E + 01 | 4.53E + 02 | 4.72E + 02 | 7.64E + 02 | 4.96E + 02 | 3.94E−02 | 3.77E + 01 | 5.87E + 02 | 3.81E + 02 | 1.94E + 02 | 9.89E + 01 | |
median | 2900 | 4168.6271 | 3098.7525 | 3925.0346 | 3116.6752 | 2900.133 | 2971.5624 | 2997.1261 | 4250.1054 | 2894.9058 | 3028.4579 | |
ET | 0.4662288 | 2.0127518 | 1.1348664 | 0.5017804 | 0.4605082 | 0.5896628 | 0.5151757 | 1.6685732 | 0.9167256 | 0.4923115 | 0.5441807 | |
rank | 2 | 11 | 7 | 9 | 6 | 3 | 4 | 8 | 10 | 1 | 5 | |
C17-F27 | mean | 3089.278 | 3174.6119 | 3111.5464 | 3166.2836 | 3173.718 | 3093.1151 | 3092.4668 | 3109.3677 | 3295.5918 | 3117.8607 | 3150.7429 |
best | 3088.978 | 3139.1529 | 3098.0647 | 3137.1017 | 3127.7165 | 3089.6471 | 3089.0864 | 3093.6744 | 3218.4998 | 3098.4758 | 3133.3309 | |
std | 2.23E−01 | 4.12E + 01 | 1.59E + 01 | 3.22E + 01 | 3.36E + 01 | 3.08E + 00 | 2.63E + 00 | 2.96E + 01 | 7.80E + 01 | 1.89E + 01 | 1.76E + 01 | |
median | 3089.3081 | 3165.578 | 3106.8009 | 3166.5004 | 3180.6311 | 3093.1165 | 3092.9317 | 3095.0128 | 3291.7639 | 3114.8985 | 3147.2637 | |
ET | 0.432171 | 2.0431644 | 1.1940608 | 0.5065569 | 0.4503224 | 0.6019193 | 0.5331149 | 1.7209385 | 0.918766 | 0.4850874 | 0.5712062 | |
rank | 1 | 10 | 5 | 8 | 9 | 3 | 2 | 4 | 11 | 6 | 7 | |
C17-F28 | mean | 3025.0006 | 3912.0323 | 3307.722 | 3471.8196 | 3307.8554 | 3332.3379 | 3244.7641 | 3439.0869 | 3471.5812 | 3319.6545 | 3195.8925 |
best | 2800.0022 | 3867.8031 | 3100.0812 | 3402.2513 | 3187.9788 | 3150.0575 | 3177.6079 | 3227.0805 | 3421.3532 | 3182.82 | 3168.5063 | |
std | 1.50E + 02 | 3.69E + 01 | 1.51E + 02 | 9.02E + 01 | 1.21E + 02 | 1.22E + 02 | 1.04E + 02 | 2.12E + 02 | 4.06E + 01 | 1.03E + 02 | 2.18E + 01 | |
median | 3100.0001 | 3914.5034 | 3343.1136 | 3440.3562 | 3315.6409 | 3383.7349 | 3201.7703 | 3398.7271 | 3472.0867 | 3341.8726 | 3200.6847 | |
ET | 0.4020287 | 2.0094507 | 1.1278612 | 0.4821352 | 0.4198695 | 0.5727171 | 0.502574 | 1.6352087 | 0.8968212 | 0.4629656 | 0.5554748 | |
rank | 1 | 11 | 4 | 10 | 5 | 7 | 3 | 8 | 9 | 6 | 2 | |
C17-F29 | mean | 3144.724 | 3339.5801 | 3165.6945 | 3314.6405 | 3324.3104 | 3231.0752 | 3187.9911 | 3232.8143 | 3520.4547 | 3287.4708 | 3278.031 |
best | 3134.1607 | 3207.6108 | 3148.2017 | 3288.9693 | 3264.6542 | 3177.2234 | 3173.9741 | 3177.7522 | 3333.3629 | 3208.0319 | 3224.1938 | |
std | 8.54E + 00 | 1.15E + 02 | 1.65E + 01 | 2.43E + 01 | 5.93E + 01 | 5.19E + 01 | 1.16E + 01 | 6.03E + 01 | 1.61E + 02 | 5.95E + 01 | 5.08E + 01 | |
median | 3145.445 | 3346.043 | 3163.2563 | 3313.3322 | 3317.8657 | 3223.867 | 3188.4674 | 3217.3081 | 3523.3351 | 3302.4084 | 3272.3932 | |
ET | 0.4219881 | 2.0567265 | 1.1564944 | 0.4921447 | 0.4352592 | 0.589355 | 0.5099219 | 1.6727754 | 0.8960592 | 0.4754221 | 0.5467503 | |
rank | 1 | 10 | 2 | 8 | 9 | 4 | 3 | 5 | 11 | 7 | 6 | |
C17-F30 | mean | 3407.9782 | 5675648.8 | 5604.9984 | 4719850.6 | 3203269.4 | 378962.03 | 836407.72 | 31117.588 | 1814995.5 | 631248.38 | 2253452.4 |
best | 3395.1259 | 984371.01 | 3643.8625 | 2489911.1 | 28614.714 | 14708.39 | 8096.1658 | 21195.57 | 306189.65 | 3865.2596 | 228474.19 | |
std | 1.80E + 01 | 8.54E + 06 | 3.63E + 03 | 2.74E + 06 | 2.89E + 06 | 7.25E + 05 | 9.54E + 05 | 1.07E + 04 | 2.52E + 06 | 8.70E + 05 | 2.00E + 06 | |
median | 3401.7333 | 1625841.3 | 3868.5572 | 3954248.4 | 3366679.2 | 17686.312 | 819283.54 | 29368.199 | 692721.36 | 333588.26 | 2288979.6 | |
ET | 0.6317577 | 2.271566 | 1.6151179 | 0.7461312 | 0.6762314 | 0.81334 | 0.7527132 | 2.3858616 | 1.1454059 | 0.7137589 | 0.7972583 | |
rank | 1 | 11 | 2 | 10 | 9 | 4 | 6 | 3 | 7 | 5 | 8 | |
Sum rank | 34 | 292 | 93 | 270 | 238 | 119 | 134 | 170 | 210 | 166 | 188 | |
Mean rank | 1.1724138 | 10.068966 | 3.2068966 | 9.3103448 | 8.2068966 | 4.1034483 | 4.6206897 | 5.862069 | 7.2413793 | 5.7241379 | 6.4827586 | |
Total rank | 1 | 11 | 2 | 10 | 9 | 3 | 4 | 6 | 8 | 5 | 7 | |
1.972E−21 | 1.289E−19 | 1.972E−21 | 1.972E−21 | 3.406E−20 | 3.881E−21 | 1.972E−21 | 1.803E−20 | 7.408E−20 | 1.972E−21 |
In this section, LEO’s ability to solve real-world optimization applications is evaluated.
The tension/compression spring problem is a design challenge in real-world applications, the goal of which is to minimize the weight of the tension/compression spring. The schematic of this design is shown in
The results of the implementation of LEO and competitor algorithms in the optimization of tension/compression spring design are reported in
Algorithm | Optimum variables | Optimum cost | ||
---|---|---|---|---|
LEO | 0.051689 | 0.356718 | 11.28897 | 0.012665 |
RSA | 0.05 | 0.310539 | 15 | 0.013198 |
MPA | 0.051684 | 0.356585 | 11.29675 | 0.012665 |
TSA | 0.051905 | 0.361628 | 11.0584 | 0.012723 |
WOA | 0.051486 | 0.351849 | 11.58021 | 0.012666 |
MVO | 0.05 | 0.313567 | 14.55586 | 0.012978 |
GWO | 0.05113 | 0.343162 | 12.14438 | 0.012689 |
TLBO | 0.069268 | 0.939835 | 2 | 0.018038 |
GSA | 0.0576 | 0.505834 | 6.273568 | 0.013885 |
PSO | 0.068994 | 0.933432 | 2 | 0.017773 |
GA | 0.069326 | 0.939615 | 2 | 0.018064 |
Algorithm | mean | best | std | median | ET | rank |
---|---|---|---|---|---|---|
LEO | 0.012665 | 0.012665 | 1.38E−18 | 0.012665 | 14.54877 | 1 |
RSA | 0.018891 | 0.013198 | 0.009397 | 0.01332 | 1.054796 | 8 |
MPA | 0.012666 | 0.012665 | 6.11E−07 | 0.012665 | 3.067909 | 2 |
TSA | 0.013023 | 0.012723 | 0.000339 | 0.012881 | 2.232119 | 4 |
WOA | 0.014042 | 0.012666 | 0.001447 | 0.013513 | 3.335292 | 5 |
MVO | 0.016781 | 0.012978 | 0.001966 | 0.017902 | 1.136586 | 6 |
GWO | 0.01274 | 0.012689 | 6.52E−05 | 0.012723 | 3.502764 | 3 |
TLBO | 0.018521 | 0.018038 | 0.000405 | 0.018374 | 18.83583 | 7 |
GSA | 0.019435 | 0.013885 | 0.003219 | 0.019385 | 4.320089 | 9 |
PSO | 4.41E + 13 | 0.017773 | 1.22E + 14 | 0.017773 | 3.289217 | 11 |
GA | 5.43E + 12 | 0.018064 | 1.42E + 13 | 0.02413 | 5.853759 | 10 |
The welded beam design problem is an engineering issue in real-world applications to minimize the fabrication cost of the welded beam. The schematic of this design is shown in
The results of welding beam design optimization using LEO and competitor algorithms are reported in
Algorithm | Optimum variables | Optimum cost | |||
---|---|---|---|---|---|
LEO | 0.20573 | 3.470489 | 9.036624 | 0.20573 | 1.724852 |
RSA | 0.236392 | 3.235121 | 8.568725 | 0.237878 | 1.889845 |
MPA | 0.20573 | 3.470489 | 9.036624 | 0.20573 | 1.724852 |
TSA | 0.205581 | 3.499014 | 9.022691 | 0.20658 | 1.73255 |
WOA | 0.161832 | 4.691246 | 9.365763 | 0.204143 | 1.855028 |
MVO | 0.204638 | 3.487916 | 9.059375 | 0.205619 | 1.728592 |
GWO | 0.205665 | 3.472964 | 9.03602 | 0.205822 | 1.725685 |
TLBO | 0.248204 | 3.662307 | 8.684848 | 0.365402 | 2.945841 |
GSA | 0.250196 | 3.715003 | 7.942199 | 0.277498 | 2.135258 |
PSO | 0.330894 | 4.892002 | 7.839737 | 0.468597 | 3.930698 |
GA | 0.329815 | 3.245087 | 6.377571 | 0.417296 | 2.597963 |
Algorithm | mean | best | std | median | ET | rank |
---|---|---|---|---|---|---|
LEO | 1.724852 | 1.724852 | 6.83E−16 | 1.724852 | 3.363431 | 1 |
RSA | 2.249565 | 1.889845 | 0.242249 | 2.209624 | 1.427522 | 6 |
MPA | 1.724853 | 1.724852 | 7.34E−07 | 1.724852 | 3.909917 | 2 |
TSA | 1.745368 | 1.73255 | 0.006329 | 1.747237 | 3.143402 | 5 |
WOA | 2.582368 | 1.855028 | 0.761705 | 2.281472 | 4.759158 | 8 |
MVO | 1.7386 | 1.728592 | 0.008201 | 1.737193 | 435.4079 | 4 |
GWO | 1.727195 | 1.725685 | 0.001082 | 1.726991 | 4.935636 | 3 |
TLBO | 9.02E + 12 | 2.945841 | 2.67E + 13 | 5.109645 | 25.56709 | 9 |
GSA | 2.455682 | 2.135258 | 0.249943 | 2.422385 | 5.715263 | 7 |
PSO | 1.58E + 14 | 3.930698 | 2.79E + 14 | 5.54E + 13 | 4.841792 | 11 |
GA | 5.16E + 13 | 2.597963 | 1.82E + 14 | 4.954044 | 8.077649 | 10 |
The speed reducer design problem is a real-world application in engineering studies with the aim of minimizing the weight of the speed reducer. The schematic of this design is shown in
The results of using LEO and competitor algorithms in optimizing of speed reducer design are released in
Algorithm | Optimum variables | Optimum cost | ||||||
---|---|---|---|---|---|---|---|---|
LEO | 3.5 | 0.7 | 17 | 7.3 | 7.8 | 3.350215 | 5.286683 | 2996.348 |
RSA | 3.54876 | 0.700327 | 17 | 7.3 | 7.8 | 3.354293 | 5.287113 | 3018.311 |
MPA | 3.5 | 0.7 | 17 | 7.3 | 7.8 | 3.350215 | 5.286683 | 2996.348 |
TSA | 3.517395 | 0.7 | 17 | 7.3 | 7.8 | 3.367129 | 5.287918 | 3008.294 |
WOA | 3.533787 | 0.7 | 17 | 7.486762 | 7.88665 | 3.350644 | 5.286713 | 3013.296 |
MVO | 3.502135 | 0.7 | 17 | 7.541675 | 8.049195 | 3.358798 | 5.28743 | 3007.468 |
GWO | 3.501526 | 0.7 | 17.0011 | 7.387738 | 7.804499 | 3.350782 | 5.28704 | 2998.382 |
TLBO | 3.592592 | 0.713244 | 19.94265 | 7.668475 | 8.297272 | 3.841732 | 5.297945 | 3844.25 |
GSA | 3.569804 | 0.700714 | 17.58068 | 7.319542 | 8.192896 | 3.479603 | 5.33575 | 3205.71 |
PSO | 3.555742 | 0.703886 | 21.66698 | 7.837641 | 8.111648 | 3.40429 | 5.431289 | 4085.292 |
GA | 3.564938 | 0.710523 | 22.56662 | 7.917575 | 7.973648 | 3.426693 | 5.308706 | 4269.365 |
Algorithm | mean | best | std | median | ET | rank |
---|---|---|---|---|---|---|
LEO | 2996.348 | 2996.348 | 8.97E−13 | 2996.348 | 3.29166 | 1 |
RSA | 3259.403 | 3018.311 | 80.46219 | 3258.554 | 1.513922 | 7 |
MPA | 2997.327 | 2996.348 | 2.86026 | 2996.364 | 3.601877 | 2 |
TSA | 3037.677 | 3008.294 | 14.77695 | 3039.229 | 2.595855 | 5 |
WOA | 3110.617 | 3013.296 | 86.29035 | 3093.958 | 3.980838 | 6 |
MVO | 3031.118 | 3007.468 | 11.34626 | 3032.71 | 4.210757 | 4 |
GWO | 3004.897 | 2998.382 | 4.296095 | 3005.444 | 4.027501 | 3 |
TLBO | 6.47E + 13 | 3844.25 | 6.06E + 13 | 4.48E + 13 | 20.44886 | 10 |
GSA | 3512.39 | 3205.71 | 248.8387 | 3443.385 | 5.315226 | 8 |
PSO | 2.03E + 14 | 4085.292 | 3.26E + 14 | 6.86E + 13 | 3.850949 | 11 |
GA | 5.58E + 13 | 4269.365 | 6.6E + 13 | 2.54E + 13 | 6.526017 | 9 |
The pressure vessel design problem is an optimization challenge in real-world applications to minimize the design cost. The schematic of this design is shown in
The results obtained from the implementation of LEO and competitor algorithms on the pressure vessel design problem are reported in
Algorithm | Optimum variables | Optimum cost | |||
---|---|---|---|---|---|
LEO | 0.778027 | 0.384579 | 40.31228 | 200 | 5882.901 |
RSA | 0.86013 | 0.593171 | 43.04395 | 168.5946 | 6865.859 |
MPA | 0.778027 | 0.384579 | 40.31228 | 200 | 5882.901 |
TSA | 0.781524 | 0.393985 | 40.37834 | 200 | 5946.404 |
WOA | 0.888888 | 0.468339 | 45.65375 | 136.9744 | 6253.655 |
MVO | 0.779395 | 0.387108 | 40.3343 | 199.8642 | 5900.834 |
GWO | 0.778602 | 0.385041 | 40.31623 | 199.9834 | 5888.694 |
TLBO | 1.243025 | 2.27283 | 43.75321 | 193.3142 | 16567.11 |
GSA | 1.135128 | 0.561094 | 58.81491 | 112.4656 | 10086.81 |
PSO | 1.404574 | 1.147427 | 71.4385 | 43.86409 | 16221.83 |
GA | 1.441419 | 0.823568 | 59.47861 | 129.9565 | 15421.79 |
Algorithm | mean | best | std | median | ET | rank |
---|---|---|---|---|---|---|
LEO | 5882.901 | 5882.901 | 1.87E-12 | 5882.901 | 3.66951 | 1 |
RSA | 11252.55 | 6865.859 | 3227.668 | 10925.66 | 1.168503 | 7 |
MPA | 5883.013 | 5882.901 | 0.426155 | 5882.901 | 2.989616 | 2 |
TSA | 6313.121 | 5946.404 | 523.8611 | 6022.13 | 2.169944 | 4 |
WOA | 8554.521 | 6253.655 | 2130.801 | 8233.353 | 3.369625 | 6 |
MVO | 6740.322 | 5900.834 | 408.5954 | 6851.96 | 3.450834 | 5 |
GWO | 6227.057 | 5888.694 | 516.3003 | 5930.408 | 3.286289 | 3 |
TLBO | 34608.64 | 16567.11 | 14549.03 | 31967.28 | 17.56472 | 10 |
GSA | 22843.39 | 10086.81 | 8114.234 | 22923.1 | 4.310473 | 8 |
PSO | 42178.43 | 16221.83 | 12122.96 | 42055.87 | 3.356884 | 11 |
GA | 34057.02 | 15421.79 | 12025.37 | 32102.06 | 5.450805 | 9 |
Real-Time Applications (RTAs) are applications that operate in specific time frames that users sense as current or immediate. Typically, RTAs are employed to process streaming data. Without ingesting and storing the data in a back-end database, real-time software should be able to sense, analyze and act on streaming data as it enters the system. RTAs usually use event-driven architecture to handle streaming data [
Metaheuristic algorithms, including the proposed LEO approach, are effective tools for managing real-time applications. In many RTAs, a combination of metaheuristic algorithms and neural networks have been employed to optimize the performance of real-time systems. The proposed LEO approach has applications in various fields of RTAs, including sensor networks, medical applications, IoT systems, military applications, electric vehicle control, fuel injection system control, robotics applications, clustering, etc.
This paper introduced a new human-based metaheuristic algorithm called Language Education Optimization (LEO), which has applications in optimization tasks. The fundamental inspiration behind LEO design is the process of teaching a foreign language in language schools where the teacher teaches skills to students. According to exploration and exploitation abilities, LEO was mathematically modeled in three phases (i) teacher selection, (ii) students learning from each other, and (iii) individual practice. The performance of LEO in optimization applications was tested on fifty-two benchmark functions of unimodal, multimodal, fixed-dimensional multimodal types and the CEC 2017 test suite. The optimization results showed that LEO, with its high power of exploration and exploitation, and its ability to balance exploration and exploitation, has a compelling performance in solving optimization problems. Ten well-known metaheuristic algorithms were employed to compare the results of the LEO implementation. The analysis of the simulation results showed that LEO has an effective performance in handling optimization tasks and providing solutions, and is far superior and more competitive than the competitor algorithms. The implementation results of the proposed LEO approach on four engineering design problems showed the high ability of LEO in optimizing real-world applications.
The most special advantage of the proposed LEO approach is that it does not have any control parameters and therefore does not need a parameter adjustment process. The high ability in exploration and exploitation and balancing them during the search process is another advantage of the proposed LEO. However, LEO also has limitations and disadvantages. First, as with all metaheuristic algorithms, there is no claim that LEO is the best optimizer for all optimization problems. The second disadvantage of LEO is that there is always a possibility that newer algorithms will be designed that perform better in solving optimization problems compared to the proposed approach. The third disadvantage of LEO is that, similar to other stochastic approaches. It does not provide any guarantee to provide the global optima for all optimization problems.
Following the design of LEO, several research tasks are activated for future work, the most important of which is the design of binary and multimodal versions of LEO. Employing LEO in optimization tasks in various sciences, real-time applications, and implementing LEO in real-world applications are other research suggestions of this study.
The authors thank Dušan Bednařík from the University of Hradec Kralove for our fruitful and informative discussions.
The research was supported by the
The authors declare that they have no conflicts of interest to report regarding the present study.