Nowadays, due to the increase in information resources, the number of parameters and complexity of feature vectors increases. Optimization methods offer more practical solutions instead of exact solutions for the solution of this problem. The Emperor Penguin Optimizer (EPO) is one of the highest performing meta-heuristic algorithms of recent times that imposed the gathering behavior of emperor penguins. It shows the superiority of its performance over a wide range of optimization problems thanks to its equal chance to each penguin and its fast convergence features. Although traditional EPO overcomes the optimization problems in continuous search space, many problems today shift to the binary search space. Therefore, in this study, using the power of traditional EPO, binary EPO (BEPO) is presented for the effective solution of binary-nature problems. BEPO algorithm uses binary search space instead of searching solutions like conventional EPO algorithm in continuous search space. For this purpose, the sigmoidal functions are preferred in determining the emperor positions. In addition, the boundaries of the search space remain constant by choosing binary operators. BEPO's performance is evaluated over twenty-nine benchmarking functions. Statistical evaluations are made to reveal the superiority of the BEPO algorithm. In addition, the performance of the BEPO algorithm was evaluated for the binary feature selection problem. The experimental results reveal that the BEPO algorithm outperforms the existing binary meta-heuristic algorithms in both tasks.
Technological developments enable information to increase and spread easily. This situation makes it challenging to choose the relevant information among all the information. The increase in the number of information is often confusing as irrelevant knowledge is involved. It is misleading to think of this information as just literature. In real-world applications, the information obtained from the devices is increasing. The reason for this is the data provided by the technological devices added to the traditional production process. These data present all the details regarding the functioning of the system in almost every field from the production process to the health system. The reason for collecting such data is usually the control of the production process, classification, or making a decision [
The usage areas of optimization algorithms are not limited to a specific framework, it is obvious that optimization is required in almost every field today. As technological advances increase and automation becomes widespread, many complex problems arise that need to be considered. Optimization algorithms, which provide great support to the solutions of the mentioned problems, have been recommended since the past and still continue to be recommended. As the accumulation of knowledge increases and problems arise, drawbacks in current optimization techniques emerge. The historical development of optimization techniques is a very comprehensive topic and can be studied in detailed literature research [
Compared with exact search approaches, metaheuristic optimization approaches are more successful in terms of performance (also, they are less time-consuming). One of the most important reasons for this is that they do not have to analyze the entire search space during the research process. High computational complexity is an undesirable feature today, although exact search approaches guarantee to find the best solution when they have sufficient time and hardware power. Metaheuristic approaches are a kind of random search engine. But randomness here is usually in a directed form. Thanks to this manageability, they can determine the most suitable solution in a short time. The evolutionary-based algorithms are motivated by biological evolution. Some of these are Genetic Algorithm (GA) [
The above-mentioned optimization algorithms can be applied to the feature selection problem [
Binary versions of almost all meta-heuristic optimization algorithms are available in the literature. Our analysis on these algorithms clearly shows that it carries the problems that continuous optimization versions have. The proposed BEPO algorithm is proposed to eliminate current problems in the literature and for a better solution. The main difference between the original EPO and the proposed BEPO is that EPO works well in continuous search space. In contrast, BEPO works in discrete search space. For this, BEPO utilizes a sigmoidal transfer function that can change the positions of search agents between the values 0 and 1. Performance indicators of the proposed BEPO algorithms are calculated according to 29 benchmark test functions. Then they are compared with the existing binary metaheuristic algorithms in the literature. Furthermore, the feature selection problem is solved through BEPO. The main contributions of this paper can be concluded as follows.
The binary version of EPO and search space range values are proposed and analyzed deeply. The sigmoidal binarization function is proposed to convert search agent values to 0 and~1. The performance of the proposed BEPO algorithm is analyzed in detail using many benchmark test functions and benchmark datasets. Extensive feature selection experiments are concluded based on BEPO.
The remainder of this paper is organized as follows. First, the preliminary concepts of EPO are mentioned in Section 2. Then, the binary variant of the emperor penguin optimizer algorithm is presented in Section 3. Results and discussions are depicted in Section 4. Section 5 illustrates the application of the proposed algorithm on the feature selection problem. Finally, concluding remarks are explained in Section 6.
Aptenodytes forsteri is the scientific name of the Emperor penguin. As compared to all different types of penguins, they are the largest ones. With respect to the size, all the males and females are very much alike. They have a blackhead, white stomach, wings, and tail and live in Antarctica. In order to survive in the winters of Antarctica, they huddle together to keep themselves warm and protect themselves from cold wind. The huddling mechanism of penguins is emphasized into four main phases: creating a huddling boundary, evaluating the temperature profile, calculating the distance among emperor penguins, and relocating mover [
The above-mentioned four phases of emperor penguins are mathematically modeled. The foremost motivation behind this algorithm is to find the efficient mover and update the position of emperor penguins.
Penguins create a huddle boundary of polygon shape. The wind flow is also considered during the generation of huddle boundaries. The complex variable is used to illustrate the behavior of huddle boundary creation. Let us assume that the velocity of wind and its gradient are denoted by
The penguins create a huddle to maintain the high temperature inside the huddle for survival in winter. It is assumed that when the radius of the polygon (
The best-fit emperor penguin plays the most critical role in determining the location of other emperor penguins. The locations of all other emperor penguins are updated accordingly. During the huddle boundary creation, the distance among the emperor penguins is calculated. The mathematical formulation of distance computation is given below.
The mathematical formulation of
According to the mover (i.e., best-fit emperor penguin), the positions of emperor penguins are updated. The mover is used to vacate the current position of penguins and change the positions of other penguins. The position update function is given below.
Many of the optimization problems we encounter in real life require the discrete/binary search field. For this, problems having continuous variables can be transformed into binary variables. The binary metaheuristic algorithms help in solving the discrete problems that are feature selection [
EPO is easy to understand and apply in a real-life scenario. Due to these advantages, a novel binary variant of EPO (BEPO) is developed. In the proposed algorithm, the position updating mechanism has been modified. As per the knowledge of the authors, there is no such algorithm present so far.
The proposed algorithm uses discrete search space instead of uninterrupted search space. The discrete search space can be deliberated as hyperspace. The penguins of BEPO are either moved to closer or extreme corners of the hypercube by switching the binary bits. In BEPO, the position updating mechanism has been modified.
The Original EPO algorithm is designed to allow penguins to move in continuous search space. However, BEPO uses discrete search space to solve binary problems. For this purpose, updating the position updating mechanism ensures that only 0 and 1 are produced. The sigmoidal transfer function is used to perform this process. The mathematical formulation of the sigmoidal transfer function is given below:
It forces penguins to transfer in binary space. It should be lying in the range of
In order to show the superior performance of the proposed BEPO algorithm fairly, comparisons are made with recently developed binary metaheuristic algorithms. In this process, benchmarking standard test functions are used. In addition, 29 independent simulations are performed for each metaheuristic algorithm. Among these independent experiments, the best mean and standard deviation parameters for each method are selected. Finally, comparison tables are created according to this approach.
BEPO is validated on twenty-nine benchmark functions such as unimodal (F1-F7) [
The BEPO algorithm is evaluated on the above-mentioned test functions in Section 4.1 and compared with four binary metaheuristic algorithms, namely Binary Grey Wolf Optimizer (BGWO) [
Algorithms | Parameters | Values |
---|---|---|
BEPO | Search Agents | 30 |
Temperature Profile ( |
[1,1000] | |
Constant |
[−1.5, 1.5] | |
Function |
[0,1.5] | |
Parameter |
2 | |
Parameter |
[2,3] | |
Parameter |
[1.5,2] | |
Runs | 30 | |
Iterations | 1000 | |
BGWO [ |
Runs | 30 |
Search Agents | 30 | |
Control Parameter ( |
[0,2] | |
Iterations | 1000 | |
BGSA [ |
Runs | 30 |
Search Agents | 30 | |
Gravitational Constant |
100 | |
Iterations | 1000 | |
BPSO [ |
Runs | 30 |
Search Agents | 30 | |
Maximum velocity | 6 | |
Constants |
2,2 | |
Inertia weight | An increase from 0.4 to 0.9 | |
Iterations | 1000 | |
BBA [ |
Runs | 30 |
Search Agents | 30 | |
Frequency Range | [0,2] | |
Loudness | 0.25 | |
Pulse rate | 0.5 | |
0.9 | ||
0.9 | ||
Iterations | 500 |
BEPO is evaluated on test functions that are mentioned in Section 4.1. In addition, the convergence analysis of BEPO and other metaheuristics are also compared.
BEPO Std | Avg | BGWO Std | Avg | BGSA Std | Avg | BPSO Std | Avg | BBA Std | Avg | |
---|---|---|---|---|---|---|---|---|---|---|
F1 | 0.14E+01 | 0.57E+01 | 9.54E+03 | 1.35E+05 | 0.07E+01 | 0.18E+01 | 0.12E+01 | 0.13E+01 | ||
F2 | 0.14E+01 | 0.57E+01 | 1.50E+22 | 7.05+E21 | 0.06E+01 | 0.17E+01 | 0.11E+01 | 0.10E+01 | ||
F3 | 1.56E+01 | 2.88E+02 | 2.67E+06 | 8.90E+06 | 1.11E+01 | 2.47E+01 | 2.23E+01 | 1.55E+01 | ||
F4 | 1.82E−01 | 8.33E−01 | 2.53E−01 | 8.65E−02 | 9.98E+01 | 0.10E+01 | 0.03E+01 | 0.08E+01 | ||
F5 | 2.90E+01 | 0.01E+00 | 6.11E+07 | 7.27E+08 | 9.93E+01 | 3.04E+02 | 1.63E+02 | 3.39E+01 | ||
F6 | 0.25E+01 | 1.92E+01 | 7.93E+03 | 1.34E+05 | 0.11E+01 | 1.11E+01 | 0.19E+01 | 0.93E+01 | ||
F7 | 1.57E+01 | 1.74E+01 | 4.19E+01 | 4.11E+02 | 0.52E+01 | 1.43E+01 | 2.42E+01 | 3.75E+01 |
The results obtained from BEPO and other binary metaheuristic techniques on multimodal test functions are depicted in
BEPO Std | Avg | BGWO Std | Avg | BGSA Std | Avg | BPSO Std | Avg | BBA Std | Avg | |
---|---|---|---|---|---|---|---|---|---|---|
F8 | 5.61E+02 | −2.52E+01 | 5.22E+02 | −2.44E+03 | 4.64E+02 | −1.90E+03 | 0.11E+01 | −1.71E+01 | ||
F9 | 0.14E+01 | 0.45E+01 | 2.85E+01 | 6.79E+02 | 0.07E+01 | 0.16E+01 | 0.10E+01 | 0.08E+01 | ||
F10 | 0.02E+01 | 0.16E+01 | 7.43E−02 | 2.15E+01 | 0.01E+01 | 0.09E+01 | 0.04E+01 | 0.05E+01 | ||
F11 | 7.92E−02 | 0.02E+01 | 7.29E+01 | 1.18E+03 | 0.18E−01 | 0.48E−01 | 0.31E−01 | 0.45E+01 | ||
F12 | 0.03E+01 | 0.26E+01 | 2.08E+08 | 2.02E+09 | 0.96E−01 | 0.19E+01 | 0.01E+01 | 0.18E+01 | ||
F13 | 8.58E−02 | 8.76E−01 | 3.01E+08 | 3.42E+09 | 0.53E−01 | 0.01E+01 | 1.15E−01 | 0.99E+00 |
BEPO Std | Avg | BGWO Std | Avg | BGSA Std | Avg | BPSO Std | Avg | BBA Std | Avg | |
---|---|---|---|---|---|---|---|---|---|---|
F14 | 3.61E−15 | 1.26E+01 | 2.01E−04 | 4.99E+02 | 1.26E+01 | 3.61E−15 | 1.26E+01 | |||
F15 | 0.11E+01 | 1.48E−01 | 3.32E+10 | 6.09E+09 | 1.02E+01 | 1.48E−01 | 2.33E+01 | 1.48E−01 | ||
F16 | 0.10E+01 | 9.08E+01 | 8.14E+02 | 3.77E+03 | 1.01E+02 | 2.20E+02 | 4.28E+02 | 1.01E+02 | ||
F17 | 2.04E+02 | 3.49E+01 | 1.24E+00 | 1.09E+03 | 5.10E−01 | 4.30E+00 | 5.41E−01 | 9.10E−01 | ||
F18 | 0.23E+03 | 1.60E+03 | 1.94E+07 | 5.61E+06 | 1.05E+04 | 0.60E+04 | 2.03E+03 | 0.60E+03 | ||
F19 | 1.69E−16 | −3.34E−01 | 2.06E−20 | −3.77E−05 | 1.69E−16 | 3.34E−01 | 1.69E−16 | 3.34E−01 | ||
F20 | 0.43E−01 | −1.45E−01 | 6.7E−08 | −4.53E−08 | 2.82E−17 | −1.65E−01 | 0.17E−01 | −1.57E−01 | ||
F21 | 4.52E−01 | −5.57E−01 | 0.22E−02 | −4.54E−01 | 0.12E+01 | −0.46E+01 | ||||
F22 | 2.08E+01 | 0.50E+01 | 0.41E−02 | −0.61E−01 | 2.44E+01 | 0.50E+01 | 7.62E−01 | 0.49E+01 | ||
F23 | 3.81E−01 | −8.26E−01 | 0.53E−02 | −0.95E−01 | 0.12E+01 | −0.49E+01 |
BEPO Std | Avg | BGWO Std | Avg | BGSA Std | Avg | BPSO Std | Avg | BBA Std | Avg | |
---|---|---|---|---|---|---|---|---|---|---|
F24 | 1.13E+02 | 5.20E+02 | 1.50E+01 | 6.46E+02 | 6.19E+01 | 6.38E+02 | 3.31E+01 | 7.21E+02 | ||
F25 | 1.11E+01 | 2.11E+01 | 5.51E+01 | 3.32E+02 | 4.02E+02 | 3.78E+02 | 1.99E+03 | 9.21E+03 | ||
F26 | 9.89E+02 | 2.18E+03 | 9.33E+03 | 1.65E+03 | 4.13E+04 | 1.98E+03 | 1.75E+02 | 2.22E+03 | ||
F27 | 5.11E+01 | 2.41E+03 | 4.56E+04 | 7.12E+03 | 4.32E+02 | 1.98E+04 | 3.32E+03 | 3.27E+03 | ||
F28 | 3.37E+02 | 1.07E+03 | 1.67E+02 | 2.55E+03 | 7.46E+01 | 1.03E+03 | 3.87E+01 | 1.19E+03 | ||
F29 | 0.37E+01 | 0.88E+01 | 0.37E+02 | 4.62E+03 | 9.29E+01 | 0.20E+02 | 3.37E+00 | 5.71E+00 |
The proposed BEPO algorithm has better diversification and intensification than the existing algorithms. This is due to the adaptive nature of the proposed BEPO algorithm.
In this section the performance of the BEPO algorithm is analyzed in detail on the feature selection problem. Feature selection is a well-known optimization problem that uses discrete search space. The main purpose of feature selection techniques in the literature is to increase prediction performance. In order to demonstrate the performance of BEPO in the feature selection problems, it is compared with the well-known existing binary metaheuristics.
The eight UCI datasets [
Datasets | No. of instances | No. of attributes |
---|---|---|
Wine | 178 | 13 |
Zoo | 101 | 17 |
Glass | 214 | 09 |
Haberman | 306 | 03 |
Bupa | 345 | 06 |
WDBC | 576 | 30 |
PenglungEW | 325 | 73 |
CMC | 1473 | 09 |
The performance of the proposed BEPO algorithm is compared with the current binary metaheuristic algorithms in
Datasets | BEPO | BGWO | BGSA | BPSO | BBA |
---|---|---|---|---|---|
Wine | 9.05E−01 | 8.82E−01 | 8.76E−01 | 9.00E−01 | |
Zoo | 0.98E−01 | 0.79E−01 | 0.78E−01 | 1.37E−01 | |
Glass | 3.08E−02 | 3.01E−02 | 2.95E−02 | 3.04E−02 | |
Haberman | 2.60E−01 | 2.61E−01 | 2.63E−01 | 2.62E−01 | |
Bupa | 3.42E−01 | 3.34E−01 | 3.28E−01 | 3.50E−01 | |
WDBC | 1.41E−01 | 2.31E−01 | 8.23E−02 | 8.65E−02 | |
PenglungEW | 7.41E−02 | 2.38E−02 | 3.96E−02 | 7.15E−02 | |
CMC | 4.78E−01 | 4.88E−01 | 4.80E−01 | 4.96E−01 |
Nowadays, optimization solutions that enable us to reach the most suitable solutions in almost every field in the shortest time are mostly offered in the continuous search space. However, studies in the literature show that many real-life problems are more appropriate in binary search space and in a discrete way. For this purpose, binary versions of existing optimization algorithms are being studied extensively. In this study, the binary version of the well-known EPO algorithm, which is dubbed BEPO, is proposed to solve binary problems in the binary search space. The position updating mechanism of the original EPO is modified to switch to the binary search space. For this purpose, binarization is performed using the sigmoidal transfer function. The performance of the proposed BEPO algorithm is tested for general optimization and feature selection problems in two separate sections. First, the performance of the proposed BEPO algorithm is evaluated using 29 benchmark test functions. Experimental results reveal that BEPO outperforms the existing binary metaheuristics. The superior results of the proposed method are presented according to different evaluation criteria and compared with SOTA methods. The statistical significance of the proposed algorithm is analyzed through ANOVA. In the second stage of the experiments, the effects of BEPO algorithm on the solution of the feature selection problem are analyzed. The results obtained show that BEPO can be used effectively in feature selection tasks with high performance. The detailed analysis performed in this study reveals that the proposed BEPO algorithm is the most suitable method for both real-life general optimization and real-life feature selection problems. In future studies, the use of BEPO algorithm with hybrid structures will be studied. Effects of the BEPO algorithm hybridization on performance will be investigated in detail. Also, BEPO variants will be studied to solve different real-world problems.
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work.