Recent Advances in Butterfly Optimization Algorithm, Its Versions and Applications

Abstract

The butterfly optimization algorithm (BOA) is a recent successful metaheuristic swarm-based optimization algorithm. The BOA has attracted scholars’ attention due to its extraordinary features. Such as the few adaptive parameters to handle and the high balance between exploration and exploitation. Accordingly, the BOA has been extensively adapted for various optimization problems in different domains in a short period. Therefore, this paper reviews and summarizes the recently published studies that utilized the BOA for optimization problems. Initially, introductory information about the BOA is presented to illustrate the essential foundation and its relevant optimization concepts. In addition, the BOA inspiration and its mathematical model are provided with an illustrative example to prove its high capabilities. Subsequently, all reviewed studies are classified into three main classes based on the adaptation form, including original, modified, and hybridized. The main BOA applications are also thoroughly explained. Furthermore, the BOA advantages and drawbacks in dealing with optimization problems are analyzed. Finally, the paper is summarized in conclusion with the future directions that can be investigated further.

1 Introduction

Optimization is ordinarily concerned with finding the optimal configurations for a set of decision variables to minimize or maximize specific criteria formulated in an objective function [1]. Optimization problems can be utilized in different variants to deal with various types of search space, such as binary, continuous, discrete, and hierarchical [2]. The search space includes the set of all possible solutions to any optimization problem. An optimization problem can be unimodal or multimodal, where the front has only one niche while the rear has several search space niches with different local optima [3]. The optimization method can navigate the search space niches using two main processes: exploration and exploitation. Exploration is the ability to navigate not-yet-visited niches, while exploitation is the ability to drill down/up the already-visited niches [4]. Exploration and exploitation are contradictory, where the primary purpose is to find a suitable trade-off between them during the search for powerful utilization.

The optimization problems are initially tackled by calculus-based methods, such as linear programming [5], integer programming [6], branch and bound [7], and many others. These traditional methods are potent for problems with small dimension search spaces. However, their performance is reduced with larger dimensions of the search space [8]. Therefore, the heuristic and metaheuristic-based methods have gained substantial attention from the optimization research communities due to their efficiency and easy-to-use features. Heuristic-based methods are very fast in tackling optimization problems. However, they are problem-dependent and solve the problem as a constraint satisfaction method with less consideration of the solution quality [9]. The metaheuristic-based methods work with high dimensional search space trying to find a solution with good-enough quality [1, 10].

Metaheuristics are a group of algorithms that work as general optimization frameworks to explore and exploit problems in search spaces using intelligent operators making use of accumulative knowledge gained during the search controlled by tuned parameters and initiated by some algorithmic parameters [9]. They are conventionally categorized in different ways: natural-inspired vs non-natural-inspired, memory-use vs memory-less, static vs dynamic objective function, population-based vs trajectory-based [8]. Recently, the research community agreed to categorize the population-based meta-heuristic based on their inspiration into human-based, physical-based, swarm intelligence, and evolutionary-based [1, 11]. More information about these types of metaheuristics categories can be found in [2].

The swarm intelligence (SI) category has recently gained substantial attention from the optimization research community due to its simple features. SI algorithms are normally inspired by the natural living behavior of some animals that live in groups or swarms, such as birds, fishes, wolves, salps, ants, bees, bacteria, and insects [12]. The common behavior of the SI algorithms is the interaction between the swarm members, where normally they follow the leader(s) to reach the food source or to hunt the prey [13]. The base algorithm of SI is particle swarm optimization (PSO) which was proposed in 1996 by Kennedy and Eberhart [13]. It mimics the motion behavior of bird flocks and schooling fish. The ant colony optimization (ACO) is also a powerful SI algorithm proposed by Dorigo, Marco, Mauro Birattari, and Thomas Stutzle [14]. Other SI algorithms include krill herd optimization (KH) [15], cuckoo search (CS) [16], multi-verse optimizer (MVO) [17], group search optimizer [18], grey wolf optimizer (GWO) [19], Coronavirus herd immunity optimizer (CHIO) [20], and others reported in [13].

The butterfly optimization algorithm (BOA) is a recent SI algorithm that imitates the butterfly food foraging behaviour in nature [21]. The butterfly techniques in searching for food are formulated as an optimization method based on global and local movements. The BOA reveals powerful features as other SI algorithms: it is simple in adaptation for any optimization problem, easy to use, derivative-free, less tuned parameters, flexible, scalable, and sound-and-complete. Therefore, BOA has been adapted and utilized for a wide range of optimization problems, such as feature selection [22], photovoltaic models [23], early search blindness [24], energy consumption [25], image segmentation [25], scheduling [26], medical data classification [27], sentiment analysis [28], engineering problems [29] and many others as summarized in Table 2.

As stated by the No Free Lunch (NFL) theorem [30], no single algorithm can yield a superior outcome for all optimization problems or even for instances of the same problem. Therefore, the basic form of any SI algorithm, such as BOA, can be modified or hybridized to align with the problem-solving requirements. Thus, several problem-dependent versions of BOA has been recently proposed to enhance its searching behavior, such as binary BOA [31], discrete BOA [26], chaotic BOA [32], adaptive BOA [23], bidirectional BOA [33], dynamic BOA [34], and others [29, 35,36,37].

This review paper intensively analyzes all works related to the BOA in which six pillars are addressed as follows:

• 1. The growth of BOA in the literature in terms of the number of BOA-related articles, citations, authors, countries, institutions, and topics since 2016 is presented in Sect. 2.

• 2. The original foundation BOA is demonstrated in terms of inspirations, procedural steps, exploration, and exploitation. A simple example with step-by-step explanations is given in Sect. 3.

• 3. All versions of BOA are discussed with their related works to show the adaptability features of BOA in Sect. 4.

• 4. All applications tackled by BOA-versions are also illustrated in Sect. 5.

• 5. The behavior of BOA in tackling optimization problems is criticized and analyzed to show the pros and cons of BOA. In addition, the motivation behind the variations of BOA is shown in Sect. 6.

• 6. Finally, the fruitful review of the current works and the directions for future enhancements are presented in Sect. 7.

2 The Growth of Butterfly Optimization Algorithm in the Literature

To study the progression and growth of BOA in the literature, this section summarizes the usage of BOA in terms of scholarly publications. Scopus database is the most effective indexer nowadays to collect such materials. The search in Scopus is based on the following criteria:

• 1. The query in the search is “Butterfly optimization algorithm”

• 2. The dates are limited to the articles from 2016 until 2022.

• 3. The date of collecting the results is 15/02/2022.

In summary, the whole query of searching executed in Scopus database is as follows: TITLE ( “Butterfly optimization algorithm” ) AND ( LIMIT-TO ( PUBYEAR, 2022 ) OR LIMIT-TO ( PUBYEAR, 2021 ) OR LIMIT-TO ( PUBYEAR, 2020 ) OR LIMIT-TO ( PUBYEAR, 2019 ) OR LIMIT-TO ( PUBYEAR, 2018 ) OR LIMIT-TO ( PUBYEAR, 2017 ) OR LIMIT-TO ( PUBYEAR, 2016 ) )

Initially, the growth of BOA in terms of publication per year is shown in Fig. 1. Researchers in different domains considered the BOA the main algorithm during the years mentioned. This consideration is sharply increased, where the number of BOA-based publications was two in 2016 and grew to 34 by the end of 2021. By the end of 2022, BOA will gain much attention from the research community, which might duplicate the number of published articles in 2021.

Fig. 1

The number of BOA-based publications per year

Regarding citation gain for BOA publications shown in Fig. 2, the number grew from 11 citations in 2017 to 411 in 2021. The total number of citations is expected to be duplicated by the end of 2022. This exponential growth in BOA citations results from the BOA’s efficiency and effectiveness when solving optimization problems.

Fig. 2

The number of citations per year for BOA-based publications

In the pie chart shown in Fig. 3, the growth of BOA is visualized concerning the research topics, and each slice in the pie chart reflects the number of articles per topic. As shown in the pie chart, the most significant percentage of articles is published on the computer science topic, which is equal to 30%. The second topic that gained much interest is engineering with 28%. The mathematics topic is ranked third with 12%. The remaining publication percentage is covered by other topics such as medicine, chemistry, business, energy, etc.

Fig. 3

The number of BOA-based publications published by each subject

The BOA-based articles are published in well-reputed journals, conferences, and book chapters on well-established publishers, such as Springer-Link, ScienceDirect, MDPI, IEEE, world scientific, Taylor & Francis, Hindawi, etc., as shown in Fig. 4. Publishing on these publishers with a high reputation is due to the solid theory of BOA and its impressive characteristics.

Fig. 4

The number of BOA-based publications published by each publisher

Using country as another dimension, the number of BOA-based articles published by Indian researchers is equal to 27, where BOA and its versions gained the highest interest, as shown in Fig. 5. China is the second country that considered BOA and its variants to solve optimization problems, published 25 articles since 2016. Other countries have published BOA in less than 15 articles each, as shown in Fig. 5. This shows the popularity of applying BOA by researchers from different countries.

Fig. 5

The number of BOA-based publications of each country

The top ten authors who published BOA articles are shown in Fig. 6. As can be noticed, the BOA founders have published five articles on various topics and different optimization problems.

Fig. 6

The number of BOA-based publications of top eight authors

Finally, the top ten institutions that considered the BOA-based articles in their research are bar-charted in Fig. 7. The National Natural Science Foundation of China produced the highest number of BOA-based articles, which are 13 articles. Guizhou Key Laboratory of Big Data Statistics Analysis is the second-highest institution that published BOA-based articles, which are two. Other institutions are shown in the bar-chart visualized in Fig. 7.

Fig. 7

The number of BOA-based publications per Institution

3 Basic Concepts of Butterfly Optimization Algorithm

The BOA is a robust swarm-based metaheuristic that mimics butterfly food foraging behaviour. This section illustrates the inspiration of BOA and its mathematical model. In addition, the BOA’s capabilities in exploring and exploiting search spaces are presented. Furthermore, an illustrative example that describes the algorithm searching steps is exhibited.

3.1 Butterfly Optimization Algorithm Inspiration

Butterflies have more the 18,000 species around the world [38]. These species can interact with each other using five senses that let them smell, sight, taste, touch, and hear. In addition, these senses can help them change home, reproduce, and find food. The smell sense is essential due to its effect on finding food and nectar from long-distance [39]. The butterflies have sense receptors scattered over the butterfly’s body used to smell. These receptors contain nerve cells, called chemoreceptors, which are also used to find the best mating partner [40].

Generally, butterflies can locate the fragrance source accurately and distinguish different fragrances. Butterflies move from their location to other locations with more nectar. The butterflies produce fragrance while moving to share their current location and personal information with other butterflies. Subsequently, butterflies will follow the produced fragrance to find a better location with more nectar [21]. The general behavior of butterflies is presented in Fig. 8.

Fig. 8

Butterflies movement behavior

3.2 Butterfly Optimization Algorithm Mathematical Model

The butterflies’ inspiration and movement behaviour can be formulated as an optimization method, where butterflies present the search agents and the produced fragrances denote the fitness values.

In BOA, the butterflies/search agents can generate fragrance/fitness value with some power to be distinguished from other fragrances. Such behaviour can help other search agents update their position in the search space. Once the butterfly which finds the best nectar food in the search space produces a fragrance, all neighbourhood butterflies will move towards the best butterfly location. Such an updating mechanism is called global search in BOA. On the other hand, the butterflies will move randomly in the search space if other butterflies’ fragrances are detected, known as local search in BOA.

The fragrance intensity is mathematically modeled as follows:

$$\begin{aligned} pf_i = cI^a, \end{aligned}$$

(1)

where $pf_i$ denotes the fragrance strength of ith butterfly, I presents the stimulus intensity, c denotes the sensory modality, and a is the power exponent depending on modality, which presents a varying absorption degree. Each butterfly’s location is presented as a vector of particular problem values. Such a location can be updated in attempting to find a better location using the following formula:

$$\begin{aligned} x_i^{t+1} = x_i^t + F_i^{t+1}, \end{aligned}$$

(2)

where $x_i^t$ denotes the current position of butterfly i in iteration t, $x_i^{t+1}$ is the next position of butterfly i and $F_i^{t+1}$ denotes fragrance that utilized by $x_i$ to update its position during iterations.

As mentioned previously, the updating mechanism can be in two phases, including local and global searches. In the global search, the butterfly i moves toward the fittest butterfly $g^*$, which can be represented as:

$$\begin{aligned} F_i^{t+1} = (r^2 \times g^* - x_i^t) \times pf_i, \end{aligned}$$

(3)

where r is a random number in [0, 1]. In local search, the updating movement can be formulated as follows:

$$\begin{aligned} F_i^{t+1} = (r^2 \times x_j^t - x_k^t) \times pf_i, \end{aligned}$$

(4)

where $x_j^t$ and $x_k^t$ denote the positions of jth and kth butterflies in the search space. A new parameter, called switch probability p, is utilized in BOA to switch the algorithm’s behaviour between local and global search to get the best balance between exploration and exploitation.

3.3 Butterfly Optimization Algorithm Optimization Steps

In this section, the BOA general optimization steps are presented and illustrated. The BOA consists of five main steps, which are graphically presented in Fig. 9 and thoroughly discussed below.

Fig. 9

Flowchart of the general BOA steps

Step 1: :

BOA and the problem parameters initialization.

In this step, all BOA and the problem parameters are initialized. BOA has five parameters, including population size ( N ), number of iterations ( Itr ), c, a, and p.

Step 2: :

Population initialization.

All solutions are generated randomly by the BOA in this step. The solutions are presented as vectors of length equal to the problem dimension d.

All solutions are located in a matrix to create the population, as shown in Eq. 5.

$$\begin{aligned} {{{{Population}}}}=&\left[ \begin{matrix} x_{1}^{1} &{} x_{2}^{1} &{} \cdots &{} x_{d}^{1}\\ x_{1}^{2} &{} x_{2}^{2} &{} \cdots &{} x_{d}^{2}\\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ x_{1}^{N} &{} x_{2}^{N} &{} \cdots &{} x_{d}^{N}\\ \end{matrix} \right] , \end{aligned}$$

(5)

Step 3::

Fitness value calculation.

In this step, all solutions are evaluated based on the objective function of the optimization problem. Subsequently, the best solution is assigned to $g^*$.

Step 4::

Update the population.

In this step, all solutions are updated using the BOA to find better solutions based on the fitness values obtained in Step 3. In the BOA, a random number r is generated and compared with p to lead the searching behaviour locally or globally. In case r is less than p, the butterfly moves globally using Eq. 3; otherwise, it moves locally using Eq. 4. Subsequently, if the new solution is better than the old one, it replaces the old solution. Finally, $g^*$ will be updated.

Step 5::

Check the stop condition

Steps 3 and 4 are reproduced until the maximum number of iterations is reached.

Algorithm 1 shows the pseudo-code of the general BOA steps. The BOA is coded in MATLAB, which is released as open-source [41]. The code addressed standard benchmark problems, which increased the BOA popularity.

3.4 Exploration vs. Exploitation in Butterfly Optimization Algorithm

This section analyzes the BOA exploration and exploitation behaviour due to its significant effect on optimizing problems. The success of metaheuristics stands for their capabilities in achieving the best balance between exploration and exploitation. Such two terms are conflicting in their searching behaviour. In the explorations phase, the algorithm has a high ability to explore and navigate not-visited search space regions, whereas, in the exploitation phase, the algorithm’s processes concentrate on searching deeply in the navigated search space regions. Typically, the algorithm searches randomly in the exploration phase and accumulatively in the exploitation phase.

In the context of BOA, r and p operators are in charge of determining the searching phase, whether exploration or exploitation, as discussed in Sect. 3.2. In the initial stage of the search, the distances between the decision variables’ values are usually high due to the randomness of the initial population; thus, the exploration capability is high. In the last stage of the search, the distances are lower due to the similar population situation; thus, the exploitation capability is high. Figure 10 shows an example of the distance reduction between the decision variables in five solutions from the initial iteration to the last one. Notably, during the first few iterations, the average distance between the decision variables is high; thus, the exploration is high. Subsequently, the average distance is reduced iteratively until the last iteration, where the distance is low.

Furthermore, the convergence behaviour is studied and evaluated for the same problem and solutions to show the BOA efficiency in optimizing optimization problems, as shown in Fig. 11. Note that the BOA reduced most of the objective function values significantly while maintaining the balance between exploration and exploitation.

Fig. 10

The exploration and exploitation of the BOA

Fig. 11

The convergence of the BOA

3.5 Illustrative Example of Butterfly Optimization Algorithm

The Ackley 30-dimensional function is utilized to provide a complete picture of the convergence behaviour of BOA. This function can be represented as:

$$\begin{aligned} min f (x)= & {} -20\exp (-0.2\sqrt{\frac{1}{n}\sum _{i=1}^{n}x_{i}^{2}})\nonumber \\&-\exp \left( \frac{1}{n}\sum _{i=1}^{n}cos\left( 2\pi x_{i}\right) \right) +20+e \end{aligned}$$

(6)

where $x_{i} \in [1, 30]$ and the best objective optima value is 20.6678. The population size is 5 as shown in Table 1. For each solution, the 1st, 2nd, 3th, and 30th dimensions were extracted in the 1st, 2nd, 10th, 20th, 30th, and 40th iterations with their objective function values. Table 1 elaborates that the solutions in the initial population are very scattered and far from each other. This shows that the BOA tends to explore many regions in the search space in the initial search. Later after iteration #30, the BOA exploits the search space by reducing the distance between the solutions (i.e., following the best solution) until the global optima is reached.

Table 1 BOA results

4 Recent Variants of Butterfly Optimization Algorithm

The BOA is a manageable optimization algorithm where it has a simple searching process with high-efficiency [21]. The BOA is primarily proposed and applied for continuous search spaces. Several studies have adjusted its processes to suit different search spaces. In addition, the BOA behaviour is modified to improve its searching rules and obtain better solutions. The modifications are proposed in two variants, including improved and hybridized versions. In the following sections, several BOA variants, including original, improved, and hybridized, are discussed for different problems. The studies that are proposed in each variant are presented in Fig. 12.

Fig. 12

The BOA variants

4.1 Original Butterfly Optimization Algorithm

This section presents the adaptation of the original BOA to show its performance in addressing optimization problems and achieving the best of their results.

Fadaee et al. [42] investigated the capability of the original BOA by comparing its performance with the genetic algorithm (GA) to increase the accuracy of machine learning models, including adaptive neuro-fuzzy inference system, multiple linear regression, and artificial neural network. Several independent parameters were added to the machine learning models to increase prediction accuracy. The experimental results showed that the original machine learning models obtained similar results, whereas integrating the models with the BOA and GA could improve the results. The integration results proved the BOA essential performance compared with that of the GA.

The original BOA was adapted to find the maximum power point tracking of the photovoltaic (PV) systems under partial shading conditions [43]. The BOA was implemented for three different scenarios to investigate its performance using various situations. Also, the performance of BOA was compared with that of PSO, GWO, and the gravitational search algorithm to verify its efficiency. The results showed that the BOA performed better, where it outperformed all compared algorithms in terms of accuracy and tracking speed.

Abdul-Rashid and Alawode [44] adapted the BOA to optimize the designed Lead-Lad Controller parameters. The main aim of such an optimization is to enhance the stability of the control system. In the evaluation results, a numerical analysis was provided to evaluate the BOA efficiency in optimizing the objective function. In addition, the BOA’s results were compared with that of the GA and differential evolution algorithm to investigate their performance. The BOA outperformed all compared algorithms in optimizing the control system’s damping coefficient.

The BOA was used by Arora and Singh [45] to estimate the position of sensor nodes in wireless sensor networks (WSNs). The researchers described the BOA-based localization approach and summarized the findings by comparing the algorithm to others, such as the firefly algorithm and PSO, based on the computing time, localized nodes, and localization error. The computation results demonstrated that the proposed method is an efficient refinement method for node localization. In terms of accuracy and computing time, BOA beat the other algorithms tested.

The BOA was adapted to optimize the network lifetime and overall energy consumption by selecting an optimal/near-optimal cluster head from a nodes’ group in WSNs [25]. The optimal cluster head was selected considering several criteria, including the nodes’ residual energy, distance to the base station, distance to the neighbours, node centrality, and node degree. In the experimental results, the BOA performance was evaluated and analyzed based on alive nodes, dead nodes, data packets, and energy consumption. The BOA showed a significant performance in optimizing the objectives.

Yıldız et al. [46] implemented the BOA to obtain a lighter vehicle design. Firstly, BOA optimized the coupling with a bolted rim problem. Secondly, the vehicle suspension arm shape is optimized using BOA. The Kriging metamodeling method was used to obtain constraint function and objective equations in shape optimization. The BOA reduced the suspension arm’s weight significantly. The findings demonstrated the BOA’s potential to create better optimal automotive components.

Fathy [47] adapted the BOA to extract the global maximum power based on optimizing the reconfigure shaded PV array under partial shadow conditions. The outcomes of BOA were compared to shade dispersion using GWO and novel structure-based arrangements, NS puzzle pattern, and series-parallel total-cross-tied. The proposed BOA achieved a maximum improvement compared to the series-parallel total-cross-tied system. Furthermore, the Wilcoxon test was performed for the proposed GWO and BOA results. Additionally, both approaches’ statistical parameters were computed. The obtained findings revealed that the proposed BOA is suitable for optimally reconfiguring the PV array under PSCs.

Wen and Cao [48] introduced a novel intelligent model that combines prediction of driving forces and investigation. The BOA was used to optimize one classifier parameter (i.e., least-square support vector machine) and predict residential CO2 emissions. By comparing prediction accuracy with existing models, the final experiments demonstrated the new model’s superior performance.

A new classification approach was developed by Jalali et al. [49] to combine the BOA and artificial neural networks (ANN). As a new training approach, the BOA was used to optimize the biases and weights of ANN. Based on several evaluation measures and two well-known datasets, the proposed classification approach was compared with other state-of-the-art methods. The experimental findings demonstrated that the proposed method is superior to the other approaches.

The BOA was adapted to optimize the control horizon and prediction horizon of a new approach-based model predictive control and motion cueing algorithm by Qazani et al. [50]. The primary purpose of adapting the BOA is to reduce the tuning process’s time of the proposed approach. In the experimental results, the BOA showed a significant performance in optimizing the proposed approach objectives and achieving the best results.

Parambil et al. [51] adapted the BOA to solve the economic load dispatch problem. The authors used the advantages of BOA in solving this kind of problem. To evaluate performance, the BOA was compared with GA and PSO. In terms of lowering fuel costs and increasing efficiency, the BOA strategy outperformed GA. BOA has a faster run time than PSO, which is an advantage in practical applications.

The BOA was utilized by Tamilarasan et al. [52] to optimize the parameters of the roller burnishing process. Three process parameters were considered: burnishing depth to achieve a higher surface hardness value, burnishing feed, and burnishing speed. Single-objective optimization was used to evaluate the BOA solutions to determine the optimal burnishing parameters. The percentage error for surface hardness is $\pm 2\%$, according to a comparison of experimental and optimization results.

The BOA was used by Bagchi et al. [53] to propose an Amended Adaptive Equalizer. The proposed method was investigated on different Adaptive Equalizers based on evaluating the Bit Error Rate and the convergence rate. To evaluate the proposed approach, the constant modulus algorithm and least mean square algorithm was adapted and compared with the proposed approach. The proposed approach using the BOA showed better performance than all compared approaches.

The multi-layer radar absorbing material problem was addressed utilizing the BOA by Toktas and Ustun [54]. The BOA was used due to its high performance in optimizing this optimization problem. The problem was designed as a multi-objective optimization problem to optimize all objectives simultaneously, including total reflection and total thickness. The Pareto optimality approach was used to handle such a multi-objective problem. The experimental results proved the significant performance of the proposed method in optimizing the problem compared with other state-of-the-art methods.

ALTİRAİKİ and TEZEL [55] used the BOA to solve one of the critical challenges in the 5G network called environmental pollution. The authors utilized the BOA to maximize the average signal-to-noise index of the network and user-related vectors for service in a single cell to improve the communication power and output in the 5G networks. The researchers found the other methods, such as SPRS+WGC-PD, WGC-PD, SPRS, and Random, less efficient than the proposed BOA. In addition, each user’s bit rate was higher than with previous methods.

The BOA was utilized by Mahboob and Moghaddam [56] to find the best features of the intrusion detection system. The NSL-KDD dataset was used in the evaluation step to investigate the system’s efficiency with the BOA in detecting four types of attacks, including DoS, Probe, R2L, and U2R. The results proved the robust performance of the BOA in addressing the problem with high accuracy. In addition, the BOA performance was compared with that of state-of-the-art methods. The BOA showed significant improvement compared with other methods.

4.2 Modified Versions of Butterfly Optimization Algorithm

The BOA, like other algorithms, has drawbacks in the process of obtaining an optimal solution which may lead to not obtaining the global optima. Therefore, several studies have been proposed to modify the BOA searching processes to overcome low convergence speed, local optima stagnation, and imbalance between exploitation and exploration. In addition, the BOA is modified to deal with different kinds of search spaces, where it was initially proposed for only continuous search spaces.

In this section, the modified BOA versions are reviewed based on the modification types, where each type of modified BOA is presented in a separate section.

4.2.1 Binary Butterfly Optimization Algorithm

As discussed previously, the BOA was initially proposed to address continuous optimization problems. A new version of BOA was proposed by modifying its searching features to be suitable for binary optimization problems [31].

Zhang et al. [31] proposed an improved version of binary BOA to address one of the most popular binary optimization problems, known as the feature selection (FS) problem. The primary objectives of the proposed method are to improve dimension reduction, classification accuracy, and reliability in the FS problem. A new differential evolution initialization strategy was applied to minimize the binary BOA randomness. In addition, a new parameter is added to enhance the balance between exploration and exploitation of the binary BOA. Subsequently, the behaviour of BOA is modified to improve the quality of solutions by employing an evolution population dynamics mechanism. In the evaluation results, the proposed method was tested for twenty datasets and compared with seven algorithms to investigate its performance. The results proved the robust performance of the proposed method in classification accuracy maximization and the number of features selected minimization.

Two proposed binary variations of BOA were used by Arora and Anand [57] to find the best feature combination that reduces the number of features involved while maximizing classification accuracy. In these modifications, the basic BOA was used. Four high-performance optimization algorithms and five state-of-art methods were compared with the proposed binary algorithms. Various evaluation indicators were used to correctly evaluate these algorithms’ performance and compare them over 21 datasets. The experiments confirmed that BOA is efficient in searching for the search space in the most informative classification attributes and that the proposed techniques are efficient compared to previous wrapper-based algorithms.

Sadeghian et al. [22] proposed a new version of binary BOA with Information Gain to maximize the mean of mutual information between class labels and sample features as well as the accuracy of classification. The proposed method was utilized in a three-phase method, called ensemble information theory-based bBOA, which aimed to reduce the number of selected features. Using the minimal redundancy-maximum new classification information FS, 80% of redundant and irrelevant features were eliminated in the first phase. The best feature subset was selected using the proposed method in the second phase. Finally, the final features subset was selected using a similarity-based ranking method. Six standard datasets were used to conduct the experiments. The results showed that the proposed strategy effectively selects the best optimal features subset with the lowest number of features and improves classification accuracy in most cases.

Shahbandegan and Naderi [58] modified the BOA searching behaviour to deal with a binary multidimensional knapsack optimization problem. The binary BOA was proposed in six versions using three S-shaped and three V-shaped transfer functions to determine the most effective version. In addition, an initial population generator and a repair operator were proposed to enhance the method’s performance. In the evaluation step, 11 medium-scale and large-scale benchmark problems were used to investigate the proposed method’s performance. Also, a comparison study was conducted with other competitive methods.

4.2.2 Discrete Butterfly Optimization Algorithm

The BOA was also modified to suit the discrete search spaces of different optimization problems.

Hosseinzadeh et al. [26] proposed a discrete version of BOA to deal with the discrete values in the search space of the workflow scheduling problem. The primary purpose of addressing such a problem is to find the optimal task execution order in the scientific workflows. The Levy flight method was combined with the proposed discrete BOA for further enhancement in addressing the problem by improving the BOA local search ability and convergence speed. The evaluation results proved the significant performance of the proposed method in outperforming the compared methods and achieving the results.

Another discrete version of BOA was proposed by Lansky et al. [59] to address a workflow scheduling problem and achieve the best schedule optimally. In addition, the Levy flight was utilized with the BOA to improve the BOA convergence speed and prevent the local optima problem. In the experimental evaluation step, the proposed method was examined using well-known scientific workflow scheduling problems with different dimensions. The results proved the proposed method’s robust performance, where it outperformed all compared methods in optimizing the objectives.

4.2.3 Chaotic Butterfly Optimization Algorithm

New BOA versions were proposed by integrating the BOA searching behaviour with the chaotic maps method to improve its performance and searching speed. The chaotic maps method is a mathematical method that deals with nonlinear dynamical systems and plays an essential role in improving optimization methods’ performance [32].

A new modified version of BOA based on the chaotic maps method was proposed by Awad et al. [60] to increase its diversity and avoid stagnation in local optima. The new method was mainly proposed to address the optimization problems with high complexity and dimensionality, such as the FS optimization problem. The proposed chaotic BOA’s performance was investigated using 16 benchmark datasets and evaluated by comparing it with six optimization algorithms. The chaotic BOA outperformed all compared algorithms in optimizing the objectives.

The BOA was modified by Arora and Singh [61] by introducing the chaotic maps method into its searching behaviour to improve its performance in terms of convergence speed and local optima avoidance. To improve the BOA’s performance, ten chaotic maps are used. On engineering design problems and multimodal and unimodal benchmark test functions. According to the findings, chaotic maps can considerably improve BOA’s performance.

A modified version of the BOA based on the chaotic map method was proposed by Zhi et al. [62] for improving the heat recovery system’s performance. The primary objective of modifying the BOA is to increase the BOA convergence speed. Based on a multi-criteria assessment, the authors proposed an optimal heat recovery system for power generation, heating, and combined cooling, a gas compressor, a humidifier, a small absorption chiller, and a 5 kW proton exchange membrane stack. In the experiment results, the modified BOA proved its efficiency in improving the system and achieved the best performance. In addition, the BOA reduced the annual operation cost.

4.2.4 Adaptive Butterfly Optimization Algorithm

The adaptive parameter tunning concept in the optimization context refers to automatically tuning parameters during the search process by the algorithm. This approach could help the optimization method to find the values for the parameters and find the best solutions.

Long et al. [23] proposed a new improved version of the BOA, called Enhanced adaptive BOA, to improve the searching behaviour of the original BOA and achieve a better balance between exploration and exploitation by adding a new position search equation. The proposed method was evaluated to achieve accurate, reliable, and quick parameter identification for PV models. In the experimental results, the performance and efficiency of the proposed method were tested using 12 classical benchmark test problems and three benchmark test PV models, including single diode, double diode, and PV module models. The proposed method proved its high performance in obtaining the results, where it outperformed most of the compared methods in terms of reliability and accuracy.

An adaptive BOA was proposed by Hu et al. [24] to address the early search blindness and the relatively poor adaptability of the sensory modality. The performance of the adaptive BOA was evaluated using 14-benchmark test functions. The results showed that the proposed method obtained accuracy and stability competitively. In addition, the adaptive BOA was tested using the support-vector-machine prediction model based on an engineering problem. Also, the proposed method’s performance was compared with other optimization algorithms. The adaptive BOA performed better than all compared algorithms.

Ustun [63] proposed an adaptive BOA to enhance the BOA searching capability by emphasizing local and global search and thus optimize the balance between exploration and exploitation. The proposed adaptive BOA was tested using CEC-2017 benchmark functions. In addition, the adaptive BOA was tested using three constrained engineering problems, including welded beam design, spring design, and gear train design. In the evaluation results, the proposed method was evaluated by comparing its performance with that of the other four methods. The adaptive BOA outperformed all compared methods in achieving their optimal solutions.

An adaptive BOA was proposed by Singh and Anand [64], which utilized a phenomenon of adjusting the sensory modality of BOA throughout the optimization process to produce better results than standard BOA. Seventeen common benchmark functions were investigated with the proposed method. The new method’s performance was compared with existing standard optimization algorithms, such as BOA, FA, and artificial bee colony (ABC) algorithm. The findings show that the proposed method with enhanced factor calculation procedure outperformed the competition to achieve the global optimal solution and convergence rate efficiently.

4.2.5 Bidirectional Butterfly Optimization Algorithm

The bidirectional modification could improve the algorithms’ searching performance by allowing the parameters to perform the local search in forward and backward directions [65]. Therefore, this type of improvement attracts the scholar’s attention to modify optimization algorithms, such as BOA, in different domains.

Sharma et al. [33] integrated the bidirectional approach into the original BOA to improve the convergence rate and avoid stagnation in local optima. The proposed method was tested using three real-life problems and seven unimodal benchmark datasets. The simulation results demonstrated the improved version of BOA competency in optimizing the objectives. In addition, the proposed method outperformed most of the compared state-of-the-art methods.

A new modified variant of BOA was introduced by Sharma [65] based on a bidirectional searching behaviour. The primary purpose of the introduced method is to address the BOA drawbacks in emphasizing exploitation, where the BOA is usually stagnant in local optima. In addition, the greedy selection method was applied in selecting the search direction. The proposed method was tested using CEC2006 and CEC2014 benchmark problems. The results showed the significant performance of the algorithm’s improvements, where it almost outperformed all compared methods.

The bidirectional BOA was proposed by Elangovan and Subedha [28] for sentiment analysis based on a kernel extreme learning machine. The kernel extreme learning machine method was utilized as a classifier to classify the samples of positive or negative user reviews. In addition, the proposed bidirectional BOA was used to set up the parameters to improve the classifier performance. To evaluate the outcome of the proposed method, substantial testing was carried out on a benchmark dataset, with the outcomes examined from many perspectives. The proposed method showed encouraging results in addressing the problem and optimizing the objective.

4.2.6 Butterfly Optimization Algorithm with Dynamic Parameters

As most of the optimization algorithm’s parts can be modified and improved, the behaviour of its parameters can be changed in attempting to enhance results. The parameters of the BOA can be modified to operate dynamically to move flexibly in search spaces and find better solutions.

Dynamic BOA was proposed by Tubishat et al. [34] to address the FS problem. In this paper, the invention of a local search based on mutation was used to improve the diversity of BOA solutions and avoid local optima problems. Twenty benchmark datasets were used to illustrate the efficiency and quality of the proposed method. The method and its competitor algorithms are notified of convergence curves, statistical results, the number of selected functions, fitness values, and classification accuracy. The results showed that DBOA outperformed most performance metrics significantly with comparing algorithms.

Tan et al. [66] proposed an improved BOA for data training of one of the unsupervised machine learning approaches, called wavelet neural networks. The proposed improved BOA was constructed based on combining a local search equation and a dynamic switch probability with BOA parameters to enhance its performance in achieving optimal/near-optimal results. The proposed method’s performance was evaluated based on experimental comparisons with three robust algorithms, including the original BOA, PSO, and momentum backpropagation algorithm. Also, the compared algorithms’ efficiency was evaluated using statistical analyses. The obtained results proved the high performance of the proposed method in terms of accuracy, robustness, and convergence.

4.2.7 Improved Versions of Butterfly Optimization Algorithm

Several other improved versions of BOA were proposed to enhance its search performance.

A modified version of BOA was proposed by Arora et al. [35] to address mechanical design optimization problems. The improvement emphasized a more intensive exploitation phase. The proposed method provided the solutions with a better ability to improve. The proposed algorithm’s performance was validated using three engineering design problems and fifteen benchmark test functions with different decision variables, constraints, and objectives. The obtained results were compared to those published in the literature. The results showed that the proposed method outperformed other known optimization methods in terms of performance.

In order to overcome low accuracy and to stack in local optima problems of the basic BOA, an improved BOA was proposed by Arora and Singh [36] that utilized a variable sensory modality parameter strategy. The proposed algorithm’s performance was evaluated using a variety of benchmark functions. The proposed method was evaluated against three other metaheuristic algorithms: PSO, FA, and ABC, as well as the standard BOA. The results obtained demonstrated that the proposed method is superior to, or at least equivalent to, the existing algorithm in terms of convergence rate and final solution quality.

Arora and Anand [67] utilized learning automata in the BOA to increase the potential of the BOA, which focuses on exploration in the early stages and exploitation in the later stages of optimization. To ensure the right balance between the local and global search, a learning automatic takes the function of configuring the butterfly’s behaviour. The proposed algorithm was evaluated on three classic engineering design problems with various real-world applications and seventeen benchmark test functions to validate its effectiveness. The findings show that the learning automata enhanced the global convergence to the true global optimum while maintaining the basic BOA’s fundamental feature. The findings also demonstrated that including learning automata into BOA considerably improved BOA’s performance in avoiding local optimum trapping and achieving true global optimums.

Li et al. [29] used the cross-entropy approach to enhance the original BOA. Based on a co-evolution strategy, the modification strategy fund an appropriate balance between exploitation and exploration to avoid stacking into an optimum local while improving global search efficiency. The proposed method was tested on three classic engineering design problems and 19 well-known benchmark test functions. The findings of the engineering problems demonstrated that the BOA fits to solve complex problems with unknown search space and constraints. Besides, test function results showed that the proposed algorithm could produce extremely competitive results concerning enhancing the convergence rate, optimal local avoidance, and global search.

Fan et al. [37] modified the BOA searching behaviour to address its drawbacks, including parameter complexities and poor exploration, by improving the iteration capability and the searching accuracy. The method was proposed with a simple structure, fewer parameters, and strong precision. The proposed method was tested using various optimization problems, including low-high dimensional optimization functions and constrained problems. To show the methods’ efficiency, its achieved results were compared with well-known optimization algorithms. The results proved the high performance of the proposed method in achieving the objectives compared with the other algorithms.

Shams et al. [68] proposed a new modified BOA for partial shading patterns, solar intensity, uniform shading, and load variation conditions with fast convergence speed. The BOA was modified using one dynamic parameter to reduce the algorithm’s complexity. In addition, a new skipping method was used with the BOA to reduce the high-dimension of the problem search space. In the evaluation results, the proposed method showed high achievements in tracking time, steady-state efficiency, and response to load variation.

Chen [69] used the recurrent neural network model to address one of the motivating research problems, known as the emotion recognition problem. The data in the used model was trained using an improved BOA version due to the BOA’s high performance in handling such training data. The proposed improved BOA was constructed to handle the BOA drawbacks and enhance its searching capability. The experimental evaluation showed encouraging results, where the proposed method addresses the problem with a maximum accuracy rate.

A modified version of the BOA based on gbest-guided search strategy and pinhole-imaging-based learning was proposed by Long et al. [70]. The gbest-guided search strategy was embedded in the update position equation of BOA, and then the pinhole-imaging-based learning was utilized to avoid premature convergence and to search for unknown regions effectively. The efficiency of the proposed method was investigated further using 60 complex optimization tasks from CEC 2014 and CEC 2017 and 23 classic problems. On benchmark test functions, the comparative findings showed that the proposed method outperformed the majority of the compared algorithms. Finally, in a real-world wind turbine, the proposed method was used to tackle fault diagnosis and FS problems. In terms of classification accuracy, the results demonstrated that the proposed method outperformed other competitors.

The percentage of modified BOA versions is shown in Fig. 13, where the improved BOA has the highest percentage.

Fig. 13

The BOA modified versions

4.3 Hybridized Versions of Butterfly Optimization Algorithm

The BOA searching performance can be improved using the hybridization approach. The BOA parameters and searching behaviour can be combined with components of another method(s) to be modified and enhanced for solutions’ quality improvements. Several studies are proposed to hybridize the BOA’s components with other methods to enhance solutions’ quality.

A new hybrid optimization algorithm was proposed by Arora and Singh [71], which combined the basic BOA with the basic ABC algorithm. To establish a balance between exploration and exploitation, the proposed method used the strengths of both algorithms. Ten benchmark functions with a wide variety of dimensions and difficulty were used to evaluate the proposed algorithm. The simulation results showed that the proposed method outperformed the basic BOA and ABC in terms of accuracy and convergence speed in finding optimal solutions.

Sharma et al. [72] enhanced the BOA performance by hybridizing its searching behaviour with another optimizing algorithm, called symbiotic organisms search algorithm. The primary aim of proposing such hybridization was to improve the BOA searching behaviour by emphasizing both exploration and exploitation and optimizing the balance between them. The proposed method was evaluated using twenty-five benchmark functions. The proposed method’s performance was tested by comparing its results with well-known state-of-the-art methods. In addition, statistical tests based on the Friedman rank test and t-test were provided. The proposed method showed a significant performance in optimizing the problem and achieving the objectives compared with the other methods.

The FS optimization problem was addressed by Dubey [73] by adapting and hybridizing the BOA components with a recent swarm-based optimization algorithm, called the lion algorithm. In addition, a deep learning method was used to solve the multi-disease prediction problem. In the evaluation results, different standard datasets were used to evaluate the proposed method and prove its performance compared with traditional and original methods.

Zhang et al. [74] proposed a new hybrid method based on BOA and PSO to address the BOA drawbacks, including low accuracy and slow convergence. In addition, a new approach was introduced to enhance the BOA’s performance, including a cubic one-dimensional map and a nonlinear parameter control strategy. The proposed methods were evaluated using 26 benchmark functions. The comparison evaluation proved the high efficiency of the proposed hybrid method in addressing optimization problems, where it outperformed all compared methods.

The BOA was hybridized with one of the recent popular swarm-based optimization algorithms, known as symbiosis organisms search, to emphasize the exploration and exploitation capabilities of the BOA and achieve the best balance between them [75]. The proposed method was tested using 26 classical benchmark functions for a global optimization approach. The proposed method was evaluated by comparing its performance with well-known optimization algorithms. The proposed method obtained the best results, where it outperformed all compared methods.

Sharma and Saha [76] hybridized the BOA with the sine-cosine algorithm to improve the optimization strategy of the BOA by developing a balance between local exploitation and global exploration abilities. The researchers used 25 benchmark functions to investigate the robustness and efficiency of the proposed method. In addition, comprehensive analysis with state-of-art algorithms was conducted. Besides, the researchers applied the proposed algorithm to two real-world problems to investigate the performance of the proposed method in an unknown search space. The results of these real-world problems were compared with various other algorithms, and in real-world optimization problems, the proposed method proved superior in most cases.

Zhou et al. [77] proposed a new hybrid method based on the components of the BOA and differential evolution algorithms. The primary purposes of proposing the method were to emphasize the exploitation and exploration capabilities and avoid stagnation in local optima. The proposed method was constructed in two main steps: the first step is an improvement based on the differential evolution parameters, and the second step introduces dynamic adaptive operators into the BOA. In the evaluation results, the proposed method was tested using different datasets. The proposed method showed high and robust performance in optimizing the objectives compared with the original versions.

Utama et al. [78] proposed a new hybrid version of BOA by combining the searching behaviour of the BOA with one of the most popular local search algorithms, known as the tabu search algorithm. Such hybridization aimed to enhance the BOA exploitation capability and the solutions’ quality. The proposed HBOA addressed the green vehicle routing problem and achieved its objectives optimally, including minimizing the distribution costs considering carbon costs, fuel costs, and vehicle use costs. In the evaluation results, the proposed method was compared with other methods to investigate its performance. The obtained results proved the proposed method’s significant performance, where it outperformed all compared methods in optimizing the objectives.

The BOA searching parameters were hybridized with that of the multi-layer perception model to address one of the healthcare optimization problems [27]. In addition, the proposed method was combined with the oppositional-based learning model for performance enhancement. The primary aim of this study was to find the best classification for medical data, which was passed into three steps, including preprocessing, classification, and parameter tuning. In the evaluation results, the proposed method proved its performance in optimizing objectives.

5 Applications of Butterfly Optimization Algorithms

The BOA effectiveness and efficiency allow it to flourish when utilized for tackling different real-world applications. The algorithm tackled a wide range of problems: FS [22, 31, 34, 57, 60, 60], numerical optimization [33, 37, 64, 70, 71, 75], PV models [23, 47], rarly search blindness [24], energy consumption [25], image segmentation [72], scheduling [26, 59], multi disease prediction [73], medical data classification [27], optimal capacity of gas production [76], sentiment analysis [28], roller burnishing process parameters [52], pilot contamination in massive systems for 5G communication networks [55] , optimum shape design [46], combined cooling heating and power generation system [62], household CO2 emissions mitigation strategies [48], solving elliptic partial differential equations [66], reliability optimization problems [65], suspended sediment prediction [42], maximum power point tracking [43, 68], engineering problems [29, 61, 63, 67], green vehicle routing problem [78], high-dimensional optimization problems [74], model predictive control [50], crowd behaviour recognition [69], structural damage detection [77], anomaly-based intrusion detection [56], multidimensional knapsack problem [58], amended adaptive design [53], multilayer radar absorbing composite material [54], economic load dispatch problem [51], optimization of artificial neural networks [49], robustness evaluation of a control system [44], mechanical design optimization problems [35], node localization in wireless sensor networks [45], and global optimization [36, 55].

The list of the BOA versions used to tackle optimization problems is presented in Table 2. The table shows the problems and their types, the utilized methods and variants, the compared methods, and the published year.

The major applications addressed by the BOA are thoroughly briefly illustrated below.

FS means selecting the most favorable features that can be used to present the entire data. Solving the FS problem can help tackle a wide range of real-world applications, such as classifications and pattern recognition. BOA and modified BOA are used to tackle this problem [22, 31, 34, 57, 60, 60]. Numerical optimization is considered one of the mathematical techniques that are used to solve many engineering problems. The BOA and its variations are used to tackle these kinds of problems [33, 37, 64, 70, 71, 75]. The optimization of PV cells is vital to provide clean energy. Several researchers utilize BOA for developing PV models [23, 47].

The field of communication has several problems that are addressed by BOA, such as the efficient consumption of energy in WSNs [25], pilot contamination in massive MIMO systems for 5G communication networks [55], green vehicle routing problem [78], and node localization in wireless sensor networks [45].

BOA and its modifications undertake power and electrical system optimization problems. This encompasses economic emission dispatch [51], maximum power point tracking [43, 68], and combined cooling heating and power generation system [62].

Scheduling and planning tasks are considered NP combinatorial problems that try assigning a set of resources while adhering to a set of constraints to accomplish a predefined objective. Many research works targeted toward resolving such problems, such as data placement and scheduling in edge computing environments [26] and scientific workflow scheduling in mobile edge computing [59].

Design optimization is a famous engineering design problem that formulates such a problem mathematically, intending to achieve an optimal design from different possible alternatives. The BOA and its variations is used to tackle these problems such as optimum shape design [46], amended adaptive design [53], and mechanical design optimization problems [35].

Table 2 The BOA applications on optimization problems

6 Critical Analysis of BOA Theory

This section provides the analysis of BOA. The BOA’s versions and applications are summarized in the previous sections. BOA has attracted much attention since it can solve a wide range of optimization problems efficiently. Such high performance is because of the BOA’s features, including few control parameters, simplicity in construction, efficacy in tackling real-time problems, and ease of implementation. However, the BOA has a few limitations and disadvantages despite all of these advantages.

The primary disadvantage of the BOA is associated with the NFL theorem of optimization [30, 79,80,81] which is also a problem facing other optimization algorithms. According to the NFL, no superior optimization algorithm can outperform all other optimization algorithms for all varieties of optimization problems or perhaps all cases of the same optimization problem. Thus, the convergence of the BOA is significantly related to the nature of the problem search space. Therefore, it is necessary to hybridize or modify the BOA, particularly in an unknown search space, to address the optimization problem.

The second disadvantage is related to the problem domain. As noted previously, the original BOA was designed for optimization problems with continuous search spaces, unconstrained objective functions, and single objectives. However, for various fields and nature, such as multi and many objectives, dynamic, combinatorial, binary, and discrete, BOA must expand its applications [26, 31].

The third disadvantage of BOA is related to the behaviour of its population. BOA can expand the search to the best global solution during a search process and closely move into an extensive range and variety of search regions via its operators, which decreases the distance with the best solutions. The BOA searching behaviour is based on inherited values, where its operators do not pay much attention to each region in the search space. Such an issue reduces the exploitation capabilities when dealing with multi-modal landscape problems. Thus, the BOA should be combined with other local search-based algorithms to improve its exploitation capabilities. BOA’s combination of local search algorithms allows it to stay focused on promising search space regions [78].

Like the other optimization algorithms, the performance of the BOA is closely associated with the level of dimensionality (i.e., the size of the search space). As the number of problem explanatory variables, the algorithm’s performance degrades. Therefore, dimensionality reduction is appropriate for specific problems to produce more efficient outcomes. Accordingly, other techniques for scalability and transformation should be considered [27].

Another recurrent drawback is BOA’s reliance on stochastic operators that are essentially pseudo-random. The pseudo-randomness has an artificial cyclic aspect, which implies that some patterns will repeat themselves in the long term, and the search pattern will adapt a cyclic behaviour as well, potentially leading to an unending loop of search. However, BOA’s behaviour is comparable to that of Simplex algorithms and Nelder-Mead. On the other hand, the BOA has stochastic components that limit its ability to cover multidimensional search spaces without cyclic behaviour [54].

Finally, one of the major BOA drawbacks is related to diversity control. Due to the loss of variety in the first course of execution, the BOA convergence behaviour tends to stagnate at local minima early. This is because the BOA’s operators only select one solution (the best) per generation and ignore the other characteristics of the population’s other solutions. Thus, the best solution will be affected by other solutions early. It will be attracted to the optimal solution while avoiding the drawbacks of the worst solution. In optimization, it is common knowledge that the search direction to the global optima is not always the strike to the optimal solution. As a result of ignoring most other solutions in the population, immaturity occurs, and loss of diversity. Several studies propose splitting the population into sub-populations and treating each sub-population with different BOA to limit diversity. It is also worth noting that parallel BOA can substantially speed up convergence without compromising solution quality [60].

7 Conclusion and Future Trends

The BOA has powerful and dynamic parameters that make it flexible and efficient to address various optimization problems in different domains. Accordingly, several studies have proposed using BOA to address optimization problems, particularly engineering optimization problems. However, BOA was initially proposed to address problems with continuous search spaces. In addition, although the BOA has robust parameters, it is still facing several problems that may lead not to obtaining optimal/near-optimal solutions in some cases. These drawbacks are low convergence speed, local optima stagnation, and an imbalance between exploitation and exploration. Therefore, several versions have been proposed to address these drawbacks and make the BOA handle different search spaces. These versions are classified into modified and hybridized classes. The modified class improved the BOA searching performance by modifying its components, whereas the hybrid class improved the algorithm performance by combining it with a component from another algorithm.

In this paper, the BOA is studied and evaluated by reviewing recent studies from 2016 to 2022. Initially, general information regarding the optimization algorithms and the reasons behind the growth of BOA usage by the research community. Then, most of the studies that utilized the BOA in all its forms, including original, modified, and hybridized, are reviewed and summarized. For the modified class, the reviewed papers are categorized based on their modification approach, including binary BOA, discrete BOA, chaotic BOA, adaptive BOA, bidirectional BOA, dynamic BOA, and other modified versions. Each modified version was proposed to handle a drawback, such as the binary BOA and discrete BOA was proposed to change the BOA searching behaviour to deal with binary search spaces. In other versions, some approaches were added to the algorithm to improve its performance, such as chaotic.

This paper also provides the main applications addressed by the algorithm. In addition, a critical analysis is presented to show the BOA’s advantages and disadvantages and its strong points.

Possible future directions can be considered to enhance the BOA’s performance quality and achieve better optimal/near-optimal solutions in different domains as follows:

• Selection mechanisms The BOA focuses mainly on the best solution in the population in terms of global search. In this matter, several good solutions and directions that might improve the current solution are neglected. Accordingly, other solutions can be selected instead of the best solution using a selection method, such as tournament, proportional, and roulette wheel selection, for further improvement [82, 83].

• Adaptive Parameters In the BOA searching processes, three main parameters must be initialized before starting the searching processes, including c, a, and p. Some parameters are fixed, such as a and p, which may make the searching mechanism rigid. Therefore, modifying these parameters’ nature can enhance the algorithm’s performance in covering all search space regions [84].

• BOA Modification The BOA searching behavior can be improved to enhance the obtained solutions’ qualities. Such improvement can be achieved by modifying or hybridizing the BOA’s searching processes. These enhancement approaches were proposed in several studies in attempting to achieve the optimal solution to optimization problems. However, most of these studies obtained good and encouraging solutions but not the optimal ones. Therefore, an enhancement approach can be proposed for further solutions improvements, particularly in the exploitation phase, where the BOA has low exploitation capability in the initial searching stage, as shown in Sect. 3.4. Various local search optimization methods can be combined with the BOA to improve its local searchability, such as the Min-conflict algorithm [85].