On the Exploration and Exploitation Capabilities of the Artificial Bee Colony Algorithm

Abstract

In this paper, we investigate the exploration and exploitation capabilities of the Artificial Bee Colony (ABC) algorithm using novel attraction basin-based measures. Previous claims about the ABC’s weak exploitation and exploration capabilities have been scrutinized. These claims are not based on exploration and exploitation measurements and, as such, are questionable. Direct measurements are needed to get real insights into the exploration and exploitation capabilities of any search algorithm. The results show that indirect measurements based on diversity are not appropriate. Our newly developed attraction basin-based measurements allow us to differentiate between exploration types (successful, failed, deceptive, successful rejection) and exploitation types (successful, unsuccessful). Namely, it is not only important that an algorithm is in the exploration phase, but also that promising regions with better solutions are not abandoned and that regions with worse solutions are visited less frequently. Similarly, during the exploitation phase, it is important to discover better solutions in the neighborhood and not exploit in an unsuccessful direction. It has been shown that ABC’s exploration and exploitation capabilities are versatile, and can adapt to different fitness landscapes successfully.

1. Introduction

The Artificial Bee Colony (ABC) algorithm [1,2] is an interesting metaheuristic algorithm [3,4] inspired by honey bees’ foraging behavior, with three distinct phases: employed bee, onlooker bee, and scout bee. ABC has been applied to numerous real-world optimization problems, such as structural damage detection [5], searching for soil models’ parameters [6], determination of a hysteresis model’s parameters [7], mobile robot path planning [8], determination of DC motor and drive parameters [9], sales forecasting [10], and flow-shop scheduling [11]. ABC has been compared with other metaheuristic algorithms [12], and many variants of ABC exist [13]. Despite ABC’s simple structure and implementation, many researchers have misunderstood how many new solutions are generated in the scout bee phase and how to count the fitness evaluations consumed in that phase correctly. These issues have been reported and discussed in [14]. However, there is yet another widespread misunderstanding about how ABC explores and exploits the search space. The search process is in the exploration phase when new solutions are generated in the previously unvisited search space. Conversely, it is in the exploitation phase when new solutions are generated in the neighborhood of previously visited solutions [15]. It has been widely acknowledged that exploration and exploitation are fundamental and crucial processes of any search algorithm [16], and that the effectiveness of evolutionary algorithms (EAs) can be attributed to the good balance achieved between exploration and exploitation in EAs [17]. In many reports [18,19,20,21,22,23,24,25,26], the authors claim that ABC’s exploration capability is much better than its exploitation capability, despite the fact that the employed bee and onlooker bee phases use the same equation and should generate a new solution in the vicinity of the previous solution, whilst generating a random solution in the scout bee phase. As such, ABC seems to exploit the search space during the employed bee and onlooker bee phases, and to explore it during the scout bee phase [11]. However, without measuring exploration and exploitation directly, these are only assumptions. It is also interesting that the authors in [25] classified the search type for the employed bee and the onlooker bee phases differently, despite the same search equation (Equation (1)) appearing in both phases.

$$\begin{array}{cc}\hfill x_{ij}^{\prime }& =x_{ij}+{\varphi }_{ij}(x_{ij}-x_{kj})\hfill \end{array}$$

(1)

The authors in [25] wrote: “As for the employed bees, they are responsible for searching new food sources in the entire search space, and they can be considered to play the role of exploration. … To some extent, the onlooker bees play the role of exploitation.” The reasoning for the equation used in the employed bee and onlooker bee phases was as follows in [18]: “On the other hand, in Equation (1), the coefficient ${\varphi }_{ij}$ is a uniform random number in $[-1,1]$ and $x_{kj}$ is a random individual in the population, therefore, the solution search dominated by Equation (1) is random enough for exploration. To sum up, the solution search equation described by Equation (1) is good at exploration but poor at exploitation.” Since measuring exploration and exploitation capabilities directly has not been realized up to now, such claims are mere speculations.

The authors in [27] found out that the scout bee phase is activated rarely for high-dimensional problems when standard control parameter settings are used ($limit={\displaystyle \frac{SN}{2}}\ast D$; $SN$ is the population size and D is the dimension of a problem). Since dimension D is high, the control parameter limit is high as well. In the scout bee phase, a new solution is only randomly generated for a previous solution, which has a highest number of unsuccessful attempts on improvements that is equal to or greater than the control parameter limit. The authors in [27] found that in many cases, the best-so-far solutions have the highest number of unsuccessful attempts. Hence, the best-so-far solutions are abandoned in favor of random solutions. In our opinion, this is the main reason for the slow convergence of ABC, not low exploitation capability as claimed by many others [18,19,20,21,22,23,24,25,26]. The authors in [27] also questioned the claim of the good exploration capability of ABC [18,19,20,21,22,23,24,25,26] since the scout bee phase, which is responsible for exploration, is often not activated at all in high-dimensional problems due to the high value of the control parameter limit (when the standard setting is used for control parameter limit). According to [27], the exploration capability of ABC is also weak. Firstly, study [28] shows that the standard control parameter setting for the limit is not conclusively the best setting and should not be used in every scenario (e.g., for high-dimensional problems). Secondly, as the current study shows, exploration in ABC not only happens in the scout bee phase, but also in the employed bee and onlooker bee phases. This has been verified by measuring exploration and exploitation directly using attraction basin-based measures (see Section 3).

On the other hand, the authors in [20] used a slightly different description of exploration and exploitation by writing: “The exploration refers to the ability to investigate the various unknown regions in the solution space to discover the global optimum. While the exploitation refers to the ability to apply the knowledge of previous solutions to find better solutions”, and claiming that ABC is poor in exploitation due to slow convergence (not finding better solutions). Similar definitions of exploration and exploitation also appear in [29]: “Exploration involves searching for new regions of the solution space to avoid local optima, while exploitation refines existing solutions to achieve better convergence.” A fast convergence rate was also used as a measure for exploitation in [21]. The same reasoning was also applied in [23], where the authors measured the number of better solutions obtained and used this as an indicator of ABC’s exploitation ability. However, we should distinguish between exploiting the search space and finding a better solution, and similarly, exploring the search space and finding a better region. The algorithm might be in the process of exploitation, but finding a better solution cannot be guaranteed. Similarly, we cannot claim that an algorithm is not performing exploration if the global optimum is not found. This actually happens in the Random Search Algorithm (RSA), where a global optimum is found rarely, despite RSA’s main search mode being exploration. We need to differentiate between how an algorithm performs the search (exploration and exploitation) and whether an improvement is achieved by finding a better solution or a better region. By our novel direct measure of exploration and exploitation based on attraction basins [30], this can indeed be achieved by differentiating further between exploration types (successful exploration, successful rejection, failed exploration, and deceptive exploration) and exploitation types (successful exploitation, unsuccessful exploitation). The exploitation measure used in [23] actually only measured successful exploitation.

The main contributions of this paper are:

• A direct measurement of exploration (successful, failed, deceptive, successful rejection) and exploitation (successful, unsuccessful) in the ABC algorithm.
• An analysis of the influence of ABC’s control parameters on exploration and exploitation.
• A demonstration and explanation of the limitations of indirect diversity measures for assessing exploration and exploitation using a specially designed EA.

Note that ABC was developed primarily for continuous optimization problems, and only later have ABC’s equations been adjusted for discrete optimization problems [31,32,33]. In this work, we focus solely on measuring ABC’s exploration and exploitation capabilities in continuous optimization problems, while discrete problems are out of scope.

The remainder of this paper is organized as follows: Related work is described in Section 2. The novel approach to measuring exploration and exploitation directly is outlined in Section 3. The experimental results and analysis are presented in Section 4. Finally, Section 5 provides the conclusions of the study and future work.

2. Related Work

Before controlling exploration and exploitation in EAs we need to know how to measure both processes. Unfortunately, this is an open problem in EAs, and researchers primarily use indirect measures, such as diversity- and entropy-based metrics. All these approaches measure exploration and exploitation indirectly under different assumptions. For example, if the diversity of the population is low (high), then exploitation (exploration) is declared [34], or similarly, when the entropy is low (high), then exploitation (exploration) is declared. The most popular diversity-based measure is the dimension-wise diversity measure [35,36,37,38,39], computing the amount of exploration (XPL%) and exploitation (XPT%) as follows [40] (n is the population size, and D is the dimension of a problem):

$$\begin{array}{cc}\hfill \mathrm{XPL}\%& =\left({\displaystyle \frac{\mathrm{Div}}{{\mathrm{Div}}_{\max }}}\right)\times 100\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \mathrm{XPT}\%& =\left({\displaystyle \frac{|\mathrm{Div}-{\mathrm{Div}}_{\max }|}{{\mathrm{Div}}_{\max }}}\right)\times 100\hfill \end{array}$$

(3)

where

$$\begin{array}{cc}\hfill {\mathrm{Div}}_j& ={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|\mathrm{mean}\left(x^j\right)-x_i^j\right|\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{Div}& ={\displaystyle \frac{1}{D}}\sum _{j=1}^D{\mathrm{Div}}_j\hfill \end{array}$$

(5)

However, these measures are very coarse and imprecise [30] and can be applied only at the population level. They cannot determine whether a particular new solution is generated in the exploration or exploitation zone.

On the other hand, useful direct measures are difficult to develop, since a neighborhood is difficult to define. In early attempts, a neighborhood is modeled as a hypercube with a user-defined diameter [24,41], or by a hypersphere with a user-defined radius [42]. Clearly, a user-defined diameter/radius is hard to define properly, because it depends on the fitness landscape. Even within the same search space, the ruggedness of the fitness landscape can change drastically [43], and the diameter/radius needs to be adjusted properly. Again, such direct measures, where a neighborhood is modeled by a hypercube or a hypersphere, are not precise enough, but can at least be applied at the individual level, not only at the population level. The authors in [44] suggested that attraction basins might be useful for measuring exploration and exploitation. Based on this recommendation, we proposed in [30] a direct measure for exploration and exploitation using attraction basins with the assumption that a neighborhood consists of all the solutions, which can be simply reached by a local search (e.g., gradient search). Therefore, the user-defined diameter/radius is no longer needed. If the parent and the new solution are in the same (different) attraction basin, then this is identified as exploitation (exploration). Although the idea is interesting, it is impractical, since attraction basins are very expensive to compute. As such, the approach in [30] can be used only for offline analysis. In our subsequent work, we improve the approach so that attraction basins no longer need to be computed, but it is still possible to identify if a parent and a new solution belong to the same (different) attraction basins. Every attraction basin [45] contains a point called the attractor, which corresponds to the basin’s local optimum and can be reached from any point in the attraction basin by a local search. To identify if a parent and a new solution belong to the same (different) attraction basin, we have to compute the parent’s attractor, as well as the new solution’s attractor. If they are the same (different), then we can declare the exploitation (exploration) phase. This approach is much less computationally expensive than computing attraction basins in advance, and is comparable to the computational costs of Memetic Algorithms [46,47]. When greedy survival selection is used, the four different types of exploration (the parent and a new solution are in different attraction basins) can be defined as [48]:

• Successful exploration (SE): The fitness of a new solution is better than the parent’s fitness, greedy survival selection will preserve a new solution, and the new solution is in a better attraction basin than the parent’s attraction basin. We find a better solution in a better attraction basin, and we can declare the exploration as successful.
• Successful rejection (SR): The fitness of a new solution is worse than the parent’s fitness, greedy survival selection will preserve the parent, and the new solution is in a worse attraction basin than the parent’s attraction basin. We find a worse solution in a worse attraction basin, and we reject it successfully.
• Failed exploration (FE): The fitness of a new solution is worse than the parent’s fitness, greedy survival selection will preserve the parent, and the new solution is in a better attraction basin than the parent’s attraction basin. We find a solution in a better attraction basin, but we fail to accept it.
• Deceptive exploration (DE): The fitness of a new solution is better than the parent’s fitness, greedy survival selection will preserve the new solution, and the new solution is in a worse attraction basin than the parent’s attraction basin. We find a better solution in a worse attraction basin and accept it, moving from an attraction basin with a better local optimum to one with a worse local optimum. This action is undesirable and should be avoided by the search algorithm.

In [48], these four different types of exploration were defined using only the Rastrigin function, since the attraction basins were obtained analytically. Our work demonstrates how the proposed direct measures for exploration and exploitation can be used to analyze the behavior of the ABC algorithm across different optimization problems, and investigates the effects of the control parameter settings. Such an analysis, based on direct exploration and exploitation measures, has not been conducted before. It is fundamentally different from the analysis in [40], which was based on imprecise diversity-based measures of exploration and exploitation, and found that most EAs produced the best results when the balance between exploitation and exploration was approximately 90% to 10%, respectively.

In the literature, we observe many works proposing new approaches intended to improve the balance between exploration and exploitation. However, since exploration and exploitation have not been measured directly, the effect of new proposals is unknown. For example, the authors in [26] proposed a new variant of ABC, called BDLDABC, where new search equations were designed in a manner to promote exploration; coexisting exploration–exploitation; and exploitation for employed bees, onlooker bees, and scout bees, respectively. Although the amount of exploration and exploitation was not measured, it is interesting that the authors acknowledged that, in the proposed onlooker bee phase, both exploration and exploitation could happen.

3. Material and Methods

To analyze exploration in a landscape-aware manner, we propose an attraction basin-based classification of offspring solutions. The approach builds on the concept of attraction basins induced by local search, and aims to quantify both the extent and the outcome of the exploration performed by an algorithm. An attraction basin $A\left(x^{*}\right)$ is defined as a subset of the search space S associated with a local optimum $x^{*}$, referred to as the attractor, such that any solution $y\in A\left(x^{*}\right)$ converges to $x^{*}$ when a local search procedure is applied [45,49,50,51,52]. Formally, for a deterministic local search operator $\mathcal{L}(·)$, all solutions in $A\left(x^{*}\right)$ satisfy $\mathcal{L}\left(y\right)=x^{*}$.

Let $f\left(x\right)$ denote the objective function to be minimized, and consider a population of n parent–offspring pairs $(p_i,x_i)$ in a given generation. In the strict sense, exploration is assumed to occur only when an offspring reaches a different attraction basin than its parent. This is captured by the basin-transition indicator:

$${\delta }_i=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathcal{L}\left(x_i\right)\ne \mathcal{L}\left(p_i\right),\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(6)

where ${\delta }_i=1$ indicates that $x_i\in A\left(x_k^{*}\right)$ and $p_i\in A\left(x_l^{*}\right)$ with $x_k^{*}\ne x_l^{*}$. Only such cases are considered genuine exploration events. For each basin transition, the exploration outcome is determined based on the relative quality of the offspring and the parent before and after the local search. A transition is classified as successful exploration if the offspring improves upon its parent both prior to and after local optimization, corresponding to a move to a strictly better attraction basin:

$$I_i^{SE}=\left\{\begin{array}{cc}1,\hfill & {\delta }_i=1\wedge f\left(x_i\right)<f\left(p_i\right)\wedge f\left(\mathcal{L}\left(x_i\right)\right)<f\left(\mathcal{L}\left(p_i\right)\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(7)

Deceptive exploration occurs when an offspring is superior before the local search, but converges to a worse attractor, reflecting misleading information in the global search space:

$$I_i^{DE}=\left\{\begin{array}{cc}1,\hfill & {\delta }_i=1\wedge f\left(x_i\right)<f\left(p_i\right)\wedge f\left(\mathcal{L}\left(x_i\right)\right)>f\left(\mathcal{L}\left(p_i\right)\right),\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(8)

In contrast, failed exploration refers to cases where an initially inferior offspring converges to a superior attractor after local optimization:

$$I_i^{FE}=\left\{\begin{array}{cc}1,\hfill & {\delta }_i=1\wedge f\left(x_i\right)>f\left(p_i\right)\wedge f\left(\mathcal{L}\left(x_i\right)\right)<f\left(\mathcal{L}\left(p_i\right)\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(9)

Finally, if the offspring is inferior both before and after the local search, the transition is categorized as a successful rejection:

$$I_i^{SR}=\left\{\begin{array}{cc}1,\hfill & {\delta }_i=1\wedge f\left(x_i\right)>f\left(p_i\right)\wedge f\left(\mathcal{L}\left(x_i\right)\right)>f\left(\mathcal{L}\left(p_i\right)\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(10)

The relative frequencies of the different exploration outcomes are defined as

$$\begin{array}{cccc}\hfill SE& ={\displaystyle \frac{1}{E}}\sum _{i=1}^n{I}_i^{SE},\hfill & \hfill DE& ={\displaystyle \frac{1}{E}}\sum _{i=1}^n{I}_i^{DE},\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill FE& ={\displaystyle \frac{1}{E}}\sum _{i=1}^n{I}_i^{FE},\hfill & \hfill SR& ={\displaystyle \frac{1}{E}}\sum _{i=1}^n{I}_i^{SR},\hfill \end{array}$$

(12)

where

$$E=\sum _{i=1}^n{\delta }_i$$

(13)

denotes the total number of basin transitions observed in the population. By construction, these quantities satisfy $SE+DE+FE+SR=1$.

Unlike traditional diversity measures based solely on spatial dispersion, this metric captures movement directly across the attraction basins, and therefore provides a landscape-aware characterization of the exploration. It is evident that a multimodal search space can be decomposed into multiple attraction basins, each associated with a local optimum, whereas a unimodal search space consists of a single attraction basin containing the global optimum. Illustrations of attraction basins can be found in [30,53]. In the context of evolutionary search, transitions between different attraction basins correspond to exploration, while movements within the same basin represent exploitation. As discussed in the previous section, successful exploration and successful rejection are considered desired exploration ($D{E}_{xpl}$) outcomes, whereas deceptive and failed exploration are considered undesired exploration ($U{E}_{xpl}$). A well-performing evolutionary algorithm is therefore expected to exhibit a higher proportion of desired exploration. While [48] focused exclusively on different exploration types, in this work we additionally distinguish between two types of exploitation, defined for parent–offspring pairs that remain within the same attraction basin.

An offspring is considered to be generated through exploitation if it converges to the same attractor as its parent under a local search. Formally, exploitation events are identified using the basin-preservation indicator:

$${\gamma }_i=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathcal{L}\left(x_i\right)=\mathcal{L}\left(p_i\right),\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(14)

where ${\gamma }_i=1$ indicates that both solutions belong to the same attraction basin.

For each exploitation event, two outcomes are distinguished based on the fitness relation between the offspring and the parent. Successful exploitation ($S{E}_{xpt}$) occurs when the offspring improves upon its parent while remaining within the same basin, representing a desirable local refinement:

$$I_i^{S{E}_{xpt}}=\left\{\begin{array}{cc}1,\hfill & {\gamma }_i=1\wedge f\left(x_i\right)<f\left(p_i\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(15)

In contrast, unsuccessful exploitation ($U{E}_{xpt}$) corresponds to cases where the offspring does not improve upon the parent, resulting in no progress despite consuming fitness evaluations:

$$I_i^{U{E}_{xpt}}=\left\{\begin{array}{cc}1,\hfill & {\gamma }_i=1\wedge f\left(x_i\right)\ge f\left(p_i\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(16)

Let

$$X=\sum _{i=1}^n{\gamma }_i$$

(17)

denote the total number of exploitation events in the population. The relative frequencies of successful and unsuccessful exploitation are then given by

$$\begin{array}{cc}\hfill S{E}_{xpt}& ={\displaystyle \frac{1}{X}}\sum _{i=1}^n{I}_i^{S{E}_{xpt}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill U{E}_{xpt}& ={\displaystyle \frac{1}{X}}\sum _{i=1}^n{I}_i^{U{E}_{xpt}},\hfill \end{array}$$

(19)

with $S{E}_{xpt}+U{E}_{xpt}=1$. These measures quantify the effectiveness of local refinement within attraction basins, complementing the basin-based exploration analysis presented earlier.

In contrast to the previously introduced diversity-based measures XPL% and XPT%, which quantify exploration and exploitation in terms of population diversity, the measures XPL and XPT proposed here capture basin-based exploration and exploitation derived from local search behavior. Since each parent–offspring pair either results in a transition to a different attraction basin or remains within the same basin, exploration and exploitation form a complete partition of the population. Consequently,

$$\mathrm{XPL}={\displaystyle \frac{E}{n}},\mathrm{XPT}={\displaystyle \frac{X}{n}},$$

(20)

and therefore

$$\mathrm{XPL}+\mathrm{XPT}=1.$$

(21)

In this work, we apply a direct attraction basin-based measure for exploration and exploitation using the ABC algorithm [1,2]. From the pseudocode of the ABC (Algorithm 1), three distinct phases can be observed: employed bee, onlooker bee, and scout bee. The ABC has a few control parameters: maximum fitness evaluations (MFEs) (in some implementations, a maximum number of cycles—MNC), population size (SN), and maximum number of trials for abandoning a solution (limit). The complexity of ABC and its variants has been discussed in [54]. The main goal of this paper is to measure ABC’s exploration and exploitation capabilities and determine the influence of control parameters on these behaviors.

Algorithm 1 Pseudocode for Artificial Bee Colony (ABC)

Input: $SN$: number of food sources (population size) Input: $limit$: maximum number of trials before abandonment Input: $MFE$: maximum number of fitness evaluations Output:  $bestSolution$ Initialize $num\_eval\leftarrow 0$ for $s=1$ to $SN$ do       $X\left(s\right)\leftarrow$ generate random solution (Equation (1)) [2]       $f_s\leftarrow f\left(X\left(s\right)\right)$      $trial\left(s\right)\leftarrow 0$      $num\_eval\leftarrow num\_eval+1$ end for while  $num\_eval<MFE$  do      for $s\leftarrow 1$ to $SN$ do                                                                ▹ Employed Bees Phase            $x^{\prime }\leftarrow$ generate new solution by Equation (2) [2]            $f_{x^{\prime }}\leftarrow f\left(x^{\prime }\right)$           $num\_eval\leftarrow num\_eval+1$           if $f_{x^{\prime }}<f_s$ then                  $X\left(s\right)\leftarrow x^{\prime }$                  $f_s\leftarrow f_{x^{\prime }}$                  $trial\left(s\right)\leftarrow 0$           else                  $trial\left(s\right)\leftarrow trial\left(s\right)+1$           end if           if $num\_eval=MFE$ then                  break           end if     end for     Update $bestSolution$     Compute selection probabilities $p_i$ using fitness values (Equations (3) and (4)) [2]     $s\leftarrow 1$, $t\leftarrow 1$     while $t<SN$ and $num\_eval<MFE$ do                                           ▹ Onlooker Bees Phase           $r\leftarrow rand(0,1)$           if $r<p\left(s\right)$ then              $t\leftarrow t+1$              $x^{\prime }\leftarrow$ generate new solution by Equation (2) [2]              $f_{x^{\prime }}\leftarrow f\left(x^{\prime }\right)$              $num\_eval\leftarrow num\_eval+1$              if $f_{x^{\prime }}<f_s$ then                   $X\left(s\right)\leftarrow x^{\prime }$                   $f_s\leftarrow f_{x^{\prime }}$                   $trial\left(s\right)\leftarrow 0$              else                   $trial\left(s\right)\leftarrow trial\left(s\right)+1$              end if           end if           $s\leftarrow (smodSN)+1$     end while     Update $bestSolution$     $mi\leftarrow \{s:trial(s)=max(trial\left)\right\}$                                                                ▹ Scout Bee Phase     if $trial\left(mi\right)\ge limit$ then          $X\left(mi\right)\leftarrow$ generate random solution (Equation (1)) [2]          $f_{mi}\leftarrow f\left(X\left(mi\right)\right)$          $num\_eval\leftarrow num\_eval+1$          $trial\left(mi\right)\leftarrow 0$     end if     Update $bestSolution$ end while

To support our claim of the imprecision of diversity-based measures for exploration and exploitation experimentally, we developed a variant of an EA that generates a new solution in the parent’s vicinity using a small mutation step. We take an extra step to ensure that the newly generated solution is always in the parent’s neighborhood (in the same attraction basin as the parent). By definition, we are always exploiting the search space. However, when exploration and exploitation are quantified using the XPL% and XPT% diversity-based equations, the algorithm is always classified as being in the exploration phase due to the high population diversity (Figure 1). This is a contradiction, since it is clear that we are only exploiting the search space (Figure 2). This example shows, in an extreme case, how imprecise XPL% and XPT% measures can be, contrary to the findings in [53]. Therefore, we advise researchers to avoid using indirect measures for exploration and exploitation.

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 compare our measures of exploration and exploitation with those based on diversity, further demonstrating the differences between the two approaches on the ABC. The graphs in Figure 3, Figure 5, and Figure 7 look very similar, since they are based on diversity, which declines gradually.

4. Results

To provide a meaningful assessment of the exploration and exploitation behavior of the ABC algorithm, the test suite was deliberately composed of both classical benchmark functions and more challenging benchmark problems with diverse landscape characteristics. The Rastrigin, Ackley, and Griewank functions were selected as they are well-established baseline problems that are used widely in the evaluation of metaheuristic optimizers. These functions exhibit different forms of multimodality, non-separability, and interaction between variables, allowing for a transparent and interpretable analysis of algorithmic behavior on canonical optimization landscapes. In addition, three functions from the CEC 2017 benchmark [55] were included to extend the evaluation to more complex and realistic scenarios. Specifically, F03 represents a highly multimodal landscape, F10 is a hybrid function combining multiple sub-functions with distinct properties, and F20 is a composition function that integrates several basic functions into a single, highly irregular search space. Together, these problems capture a broad spectrum of difficulty levels and landscape features, enabling a more comprehensive investigation of how the control parameter influences the balance between exploration and exploitation under increasing problem complexity. Regarding ABC’s control parameters, a full factorial design has been applied with the following settings that explore the ABC’s behavior under different resource constraints:

• $lim=\{0,K,100,250,750,\infty \}$, where $K={\displaystyle \frac{SN}{2}}\ast D$;
• $SN=\{24,50,100\}$;
• $MFE=\{50,000;100,000;250,000\}$;
• $D=\{5,10,30\}$.

As such, the number of different scenarios was 162 ($6\times 3\times 3\times 3$), with 10 independent runs, due to the CRS4EAs (Chess Rating System for Evolutionary algorithms) method, which is appropriate when only a small number of independent runs are available [56]. The local step size $ϵ$ in the local search was as follows: Rastrigin ($ϵ=0.01$), Ackley ($ϵ=0.006$), and Griewank ($ϵ=0.12$), and for F03, F10, and F20 ($ϵ=0.02$).

Due to an immense amount of data, only representative graphs of exploration (successful, failed, deceptive, successful rejection) and exploitation (successful, unsuccessful) are shown in the continuation. A summary of the amounts of different types of exploration and exploitation over all problems and for all different control parameter settings (except $limit=\infty$) is shown in

Table 1. Exploration and exploitation summary metrics of ABC for selected problems.

Problem	D	SE	FE	DE	SR	${\mathit{SE}}_{\mathit{xpt}}$	${\mathit{UE}}_{\mathit{xpt}}$	XPL	XPT
Rastrigin	5	0.0702 ± 0.0014	0.1875 ± 0.0084	0.0316 ± 0.0010	0.7107 ± 0.0101	0.1469 ± 0.0354	0.8531 ± 0.0354	0.2896 ± 0.0573	0.7104 ± 0.0573
Rastrigin	10	0.0567 ± 0.0029	0.1700 ± 0.0108	0.0147 ± 0.0013	0.7586 ± 0.0146	0.1685 ± 0.0373	0.8315 ± 0.0373	0.2678 ± 0.0438	0.7322 ± 0.0438
Rastrigin	30	0.0986 ± 0.0147	0.0746 ± 0.0662	0.0033 ± 0.0028	0.8235 ± 0.0636	0.2211 ± 0.0794	0.7789 ± 0.0794	0.4409 ± 0.1855	0.5591 ± 0.1855
Ackley	5	0.1525 ± 0.0039	0.1410 ± 0.0045	0.0481 ± 0.0014	0.6584 ± 0.0094	0.1995 ± 0.0371	0.8005 ± 0.0371	0.1713 ± 0.0344	0.8287 ± 0.0344
Ackley	10	0.1546 ± 0.0079	0.1139 ± 0.0066	0.0312 ± 0.0022	0.7003 ± 0.0162	0.2214 ± 0.0355	0.7786 ± 0.0355	0.1702 ± 0.0363	0.8298 ± 0.0363
Ackley	30	0.1932 ± 0.0234	0.0717 ± 0.0633	0.0197 ± 0.0172	0.7154 ± 0.0583	0.2713 ± 0.0583	0.7287 ± 0.0583	0.3225 ± 0.2439	0.6775 ± 0.2439
Griewank	5	0.0412 ± 0.0095	0.1210 ± 0.0251	0.0140 ± 0.0033	0.8238 ± 0.0376	0.1235 ± 0.0289	0.8765 ± 0.0289	0.4880 ± 0.0424	0.5120 ± 0.0424
Griewank	10	0.0562 ± 0.0160	0.1282 ± 0.0286	0.0201 ± 0.0057	0.7954 ± 0.0498	0.1382 ± 0.0419	0.8618 ± 0.0419	0.3851 ± 0.0381	0.6149 ± 0.0381
Griewank	30	0.2087 ± 0.0367	0.0454 ± 0.0406	0.0056 ± 0.0049	0.7403 ± 0.0511	0.2384 ± 0.0736	0.7616 ± 0.0736	0.3684 ± 0.1665	0.6316 ± 0.1665
F03	5	0.0502 ± 0.0120	0.4273 ± 0.0197	0.0442 ± 0.0096	0.4783 ± 0.0115	0.0811 ± 0.0135	0.9189 ± 0.0135	0.8217 ± 0.0232	0.1783 ± 0.0232
F03	10	0.0557 ± 0.0150	0.1134 ± 0.0135	0.0167 ± 0.0045	0.8142 ± 0.0281	0.2190 ± 0.0512	0.7810 ± 0.0512	0.9319 ± 0.0268	0.0681 ± 0.0268
F03	30	0.0920 ± 0.0437	0.0323 ± 0.0289	0.0090 ± 0.0080	0.8667 ± 0.0379	0.0549 ± 0.0096	0.9451 ± 0.0096	0.9268 ± 0.0049	0.0732 ± 0.0049
F10	5	0.0213 ± 0.0092	0.4336 ± 0.0171	0.0185 ± 0.0077	0.5266 ± 0.0035	0.0662 ± 0.0211	0.9338 ± 0.0211	0.9243 ± 0.0039	0.0757 ± 0.0039
F10	10	0.0384 ± 0.0144	0.1390 ± 0.0227	0.0060 ± 0.0017	0.8166 ± 0.0109	0.0745 ± 0.0145	0.9255 ± 0.0145	0.9296 ± 0.0254	0.0704 ± 0.0254
F10	30	0.0680 ± 0.0263	0.0148 ± 0.0161	0.0024 ± 0.0024	0.9148 ± 0.0243	0.0687 ± 0.0233	0.9313 ± 0.0233	0.9431 ± 0.0035	0.0569 ± 0.0035
F20	5	0.0135 ± 0.0072	0.2882 ± 0.0084	0.0086 ± 0.0041	0.6897 ± 0.0130	0.0789 ± 0.0312	0.9211 ± 0.0312	0.7613 ± 0.0135	0.2387 ± 0.0135
F20	10	0.0150 ± 0.0066	0.2966 ± 0.0072	0.0065 ± 0.0028	0.6819 ± 0.0118	0.0746 ± 0.0253	0.9254 ± 0.0253	0.6959 ± 0.0089	0.3041 ± 0.0089
F20	30	0.0469 ± 0.0218	0.0491 ± 0.0587	0.0025 ± 0.0030	0.9016 ± 0.0519	0.0634 ± 0.0160	0.9366 ± 0.0160	0.9440 ± 0.0235	0.0560 ± 0.0235

. A small standard deviation (e.g., Ackley $D=5$) indicates that the amounts of different types of exploration and exploitation are stable across all the different parameter settings. For example, the portion of successful exploration is $15\%$, failed exploration is $14\%$, deceptive exploration is $5\%$, and successful rejection is $66\%$, regardless of the different control parameter settings. Similarly, this is the case for different types of exploitation (successful exploitation is $20\%$, unsuccessful exploitation is $80\%$). A large standard deviation indicates that the balance between exploration and exploitation depends heavily on the control parameter setting (e.g., Ackley $D=30$ where exploration is $32\%\pm 24\%$ and exploitation is $68\%\pm 24\%$). From Table 1 it can also be observed that the exploration for F03, F10, and F20 is very high, between $70\%$ and $95\%$, while the ratio of exploration to exploitation for the Rastrigin, Ackley, and Griewank problems is distributed more evenly. For Griewank $D=5$, it is $49\%:51\%$.

By examining the exploration and exploitation capabilities of ABC, the following conclusions can be derived from Table 1:

• Successful rejection is high, above $50\%$.
• Deceptive exploration is not problematic, and is usually below $4\%$.
• Successful exploration is low, between $2\%$ and $20\%$.
• The amount of failed exploration is usually higher than successful exploration, between $5\%$ and $40\%$.
• Successful exploitation is low, between $5\%$ and $25\%$.
• Unsuccessful exploitation is high, between $75\%$ and $95\%$.

Although these conclusions are descriptive, in our opinion, the ratios between the different types of exploration and exploitation are even more important (

Table 2. Ratios derived from the exploration and exploitation summary metrics of ABC for selected problems.

Problem	D	SE/DE	SE/FE	${\mathit{DE}}_{\mathit{xpl}}$/${\mathit{UE}}_{\mathit{xpl}}$	SE/${\mathit{SE}}_{\mathit{xpt}}$	${\mathit{SE}}_{\mathit{xpt}}$/${\mathit{UE}}_{\mathit{xpt}}$
Rastrigin	5	2.2215	0.3744	3.5641	0.4779	0.1722
Rastrigin	10	3.8571	0.3335	4.4142	0.3365	0.2026
Rastrigin	30	29.8788	1.3217	11.8370	0.4460	0.2839
Ackley	5	3.1705	1.0816	4.2882	0.7644	0.2492
Ackley	10	4.9551	1.3573	5.8918	0.6983	0.2844
Ackley	30	9.8071	2.6946	9.9409	0.7121	0.3723
Griewank	5	2.9429	0.3405	6.4074	0.3336	0.1409
Griewank	10	2.7960	0.4384	5.7424	0.4067	0.1604
Griewank	30	37.2679	4.5969	18.6078	0.8754	0.3130
F03	5	1.1357	0.1175	1.1209	0.6190	0.0883
F03	10	3.3353	0.4912	6.6864	0.2543	0.2804
F03	30	10.2222	2.8483	23.2131	1.6758	0.0581
F10	5	1.1514	0.0491	1.2119	0.3218	0.0709
F10	10	6.4000	0.2763	5.8966	0.5154	0.0805
F10	30	28.3333	4.5946	57.1395	0.9898	0.0738
F20	5	1.5698	0.0468	2.3693	0.1711	0.0857
F20	10	2.3077	0.0506	2.2992	0.2011	0.0806
F20	30	18.7600	0.9552	18.3818	0.7397	0.0677

). The ratio between successful exploration and deceptive exploration showed that the amount of successful exploration was always higher than that of deceptive exploration, indicating that it moved more often to better than to worse attraction basins, and deceptive exploration was not problematic. This ratio was exceptionally high for $D=30$. On the other hand, the ratio between successful exploration and failed exploration was mixed, and more often, the amount of failed exploration was higher than the amount of successful exploration. This is especially problematic, since better attraction basins have been found but rejected due to greedy survival selection. Again, this ratio was always higher for $D=30$ than for $D=5$ and $D=10$. The ratio between desired exploration (successful exploration and successful rejection) and undesired exploration (failed exploration and deceptive exploration) was always very high due to a high amount of successful rejection. This ratio was extremely low for $F03$ ($D=5$) and $F10$ ($D=5$). The ratio between successful exploration and successful exploitation showed that the amount of successful exploitation was mostly higher than that of successful exploration, which was the desired behavior, as the global optimum could be reached mostly by the exploitation of a neighborhood. Only in a few cases was the amount of successful exploration higher (or almost equal) than the amount of successful exploitation ($F03$ ($D=30$) and $F10$ ($D=30$)). The ratio between successful and unsuccessful exploitation was, surprisingly, very low, indicating that searching the neighborhood with existing equations is often not efficient.

The results in Table 1 show a bird’s-eye view of the exploration and exploitation capabilities of ABC on specific problems. It would be interesting to delve deeper and examine the exploration and exploitation graphs for specific problems and control parameter settings through different stages of the evolutionary run. In Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40, we examine the differences between $limit=100$ and $limit=750$, as well as between $SN=24$ and $SN=100$ when $MFE=50,000$. For the selected problems, we were interested in the differences in exploration/exploitation behavior when the scout bee phase was triggered more ($limit=100$) and less frequently ($limit=750$) for a small population ($SN=24$) and a larger population ($SN=100$).

The results for the Rastrigin optimization problem are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. Comparing Figure 9 and Figure 10, the effect of the scout bee phase is clearly visible. The increase in exploration (red line) occurs in waves, with unsuccessful trials exceeding $limit$. When $limit$ is small, $limit=100$, the number of waves is bigger than when $limit=750$. With an increase in population size, from $SN=24$ to $SN=100$, we can observe that the amount of exploitation increased, since, at most, one new solution was generated in the scout bee phase (exploration zone), while the new solutions generated in the employed and onlooker bee phases were generated mainly in the exploitation zone (Figure 11 and Figure 12). The previously clearly visible waves are now only observable in Figure 11. In all four examples the amount of exploitation (blue line) is larger than the amount of exploration (red line). When the dimension is increased from $D=5$ to $D=10$, from Figure 13, Figure 14, Figure 15 and Figure 16, we can observe that the waves are still visible for a small population ($SN=24$), while Figure 15 and Figure 16 are similar to Figure 12.

The results for the Ackley optimization problem are shown in Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24, and the conclusions are very similar to those of the Rastrigin optimization problem.

The results for the Griewank optimization problem are shown in Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32. It is interesting how ABC’s behavior adapted to the new problem, leading to increased exploration. The ratio between exploration and exploitation for $D=5$ is $49\%$:$51\%$. Waves are still observable for the small population size $SN=24$.

The optimization problems F03, F10 and F20 are more difficult than the previously tested problems. Their exploration/exploitation behavior is very similar. Therefore, graphs are presented only for the F03 problem. From Figure 33, Figure 34, Figure 35, Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40, we can observe that exploration prevails over exploitation. For a small population ($SN=24$ and $D=5$), the wave from the scout bee phase is still visible (Figure 33 and Figure 34). In all other cases, the amount of exploration is overwhelming, and the ratio between exploration and exploitation is around $90\%$:$10\%$, indicating that solutions generated in the employed and onlooker bee phases are mainly in the exploration zone.

From Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40, we might expect there to be a direct relationship between the performance and exploration/exploitation curves. However, when exploration is performed primarily, there is no guarantee that a region with a better solution will be discovered and accepted. Similarly, when exploitation is performed primarily, there is no guarantee of fitness improvements. Note that failed exploration, successful rejection and unsuccessful exploitation do not improve solutions. The amount of exploitation and exploration might be high, but there will be no improvements if the aforementioned exploration/exploitation types occur. It is essential to know the proportions of successful, failed, and deceptive exploration and successful rejection, as well as successful and unsuccessful exploitation.

Figure 41, Figure 42, Figure 43 and Figure 44 show the occurrence of different types of exploration and exploitation and their amounts for the Rastrigin problem $D=5$. It can be observed that successful rejection (orange line) prevails over other types of exploration. ABC often finds another region with a worse attractor (local optimum) than the current region. Since the solution is also worse than the current parent solution, it is rejected due to greedy survival selection. The amount of deceptive exploration (green line), where a worse region is accepted, is minimal (around $3\%$) and not problematic. More problematic is the amount of failed exploration (blue line), where a better region is discovered but not accepted due to greedy survival selection. On average, almost every fifth solution in the exploration zone is rejected due to this problem (Table 1). It is also interesting to observe that the amount of failed exploration is almost always larger than successful exploration (red line). On the other hand, the amount of unsuccessful exploitation (gray line) is far larger than successful exploitation (black line). A rough ratio between successful and unsuccessful exploitation is $20\%$:$80\%$ (Table 1). A meticulous reader may find a gap in Figure 42 where only exploitation is performed due to a bigger $limit=750$. However, successful exploitation does not happen during this exploration gap. Only after successful exploration, when better solutions are found, do further improvements occur during the exploitation phase, implying that the previous solution is a local optimum.

Different types of exploration and exploitation on the Ackley problem, $D=10$, show somewhat different behavior than for the Rastrigin problem (Figure 45, Figure 46, Figure 47 and Figure 48). For a small population ($SN=24$), exploration comes in waves. The amount of successful exploration is slightly larger than that of failed exploration. An interesting observation from Figure 45 and Figure 46 follows: there is an increase in successful exploration culminating in an increase in successful exploitation. This situation depicts the critical role of exploration and its main purpose: to escape a local optimum, a better region must be found.

Different types of exploration and exploitation on the Griewank problem, $D=10$, are shown in Figure 49, Figure 50, Figure 51 and Figure 52. The main type of exploration is successful rejection, which prevails over other types overwhelmingly. It is interesting that the amount of successful explorations is initially higher than that of failed explorations, implying that a good attraction basin is found quickly. For a small population of $SN=24$ and a small $limit=100$, the effect of the scout bee phase can be noticed, with an increased amount of successful exploration (Figure 49), while the amount of successful exploitation is larger than the amount of failed exploration.

Different types of exploration and exploitation on the F03 problem, $D=5$, are shown in Figure 53, Figure 54, Figure 55 and Figure 56. The amount of failed exploration (the blue line) is on par with the successful rejection (the orange line) and is considered very problematic. A better region is found, but is rejected due to greedy survival selection. Another problematic issue is deceptive exploration (the green line), which is on par with successful exploration (the red line). Furthermore, in the beginning, the exploitation is not successful and only increases in later stages. All these observations imply a more difficult landscape.

The influence of the control parameter $limit$ on exploration is shown in Figure 57, Figure 58, Figure 59, Figure 60, Figure 61 and Figure 62. For the simple optimization problems (Rastrigin, Ackley, Griewank), there is a clear trend: by increasing the $limit$, the amount of exploration drops significantly (roughly by 10%). On these problems, the scout bee phase is the main source for the exploration of the search space. By increasing the limit, the scout bee phase no longer activates often. As a consequence, ABC spends less time on exploration. However, it is clear that the employed and onlooker bee phases contribute to the exploration as well.

For the more complex problems (F03, F10, F20), increasing $limit$ had little impact on the amount of exploration. By increasing $limit$, the scout bee phase was indeed not activated often. However, the employed and onlooker bee phases generated many solutions in the exploration zone, without affecting the different settings of the control parameter $limit$. Hence, the ruggedness of the fitness landscape culminated in complex attraction basins.

5. Conclusions

Exploration and exploitation are the cornerstones of every search algorithm. Until now, exploration and exploitation have been measured mostly indirectly for EAs using diversity- or entropy-based measures. In this paper, we have shown empirically that these indirect measures are imprecise by developing an EA that exploits the search space by definition. A new solution is always generated in the neighborhood of the parent solution. Although the algorithm was always in the exploitation zone, the diversity-based measure reported 100% exploration because the population was highly diverse. Researchers are advised to stop using imprecise indirect measures for exploration and exploitation. The other major shortcoming of diversity- and entropy-based measures for exploration and exploitation is their inability to measure exploration and exploitation at the individual level. They can only measure exploration and exploitation at the population level indirectly. As such, these measures cannot differentiate between the different types of exploration and exploitation. It is not only important that the EA is in the exploration (exploitation) zone; an algorithm’s ability to move to a region (attraction basin) with a better optimum, or to improve a neighborhood solution, is also very important. In this paper, we discussed four different types of exploration: successful, failed, deceptive, and successful rejection. We also introduced two types of exploitation: successful and unsuccessful.

Based on our newly developed attraction basin-based exploration and exploitation measures, the ABC algorithm has been investigated to assess its exploration and exploitation capabilities. Previously, many researchers [18,19,20,21,22,23,24,25,26] have reported the weak exploitation capability of ABC, while the work in [27] also pointed to its weak exploration capability. However, these claims were not based on direct measurements of exploration and exploitation, but on assumptions about which zone a newly generated solution fell into, and on the observation that the scout bee phase was triggered rarely when the control parameter $limit$ was set to high values. While the scout bee phase indeed contributes to exploration, the line is blurred between exploration and exploitation between the employed and onlooker bee phases. Especially for hard problems (F03, F10, F20), many new solutions generated in the employed and onlooker bee phases contributed towards exploration.

To summarize, the most important observations about ABC’s exploration and exploitation capabilities are:

• The amount of deceptive exploration is small and not problematic.
• Often, the amount of failed exploration is higher than the amount of successful exploration. This is especially problematic, since better attraction basins have been found but rejected due to greedy survival selection.
• The ratio between successful and unsuccessful exploitation is surprisingly very low, indicating that searching the neighborhood with existing equations is often not efficient.

While conducting experiments using ABC, several important insights emerged regarding exploration and exploitation:

• Measuring exploration and exploitation alone is not sufficient and cannot be related directly to metaheuristics performance. Two different metaheuristics might have the same ratio between exploration and exploitation, yet very different performance. Hence, identification is needed of the different types of exploration and exploitation. Important is the ratio between desired and undesired explorations, as well as between successful and unsuccessful exploitation, although a general pattern of exploration and exploitation is descriptive and important (see the patterns in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40).
• Despite exploring the search space, it is not guaranteed that a better region has been discovered. This situation is depicted by successful rejection, deceptive exploration and failed exploration. In the latter case, a better region is discovered, but it is abandoned due to greedy survival selection.
• Despite exploiting the search space, it is not guaranteed that a better solution has been discovered, due to unsuccessful sampling of the neighborhood. This situation is depicted by unsuccessful exploitation.
• The amount of successful exploitation should be higher than that of successful exploration, as the global optimum can be reached mostly by exploitation of a neighborhood.
• A combination of successful rejection and unsuccessful exploitation depicts algorithm stagnation.
• When an increase in successful exploration precedes an increase in successful exploitation, investment in exploration has paid off.
• Unsuccessful exploitation is undesired, since the neighborhood has been searched, but no better solution has been found.

In our future work, we would like to investigate the differences and similarities in exploration and exploitation across different EAs (e.g., GA, GWO, and PSO), including adaptive strategies (e.g., adaptive Differential Evolution and adaptive ABC), and to measure the effects of different selection methods on exploration and exploitation. By using our direct attraction basin-based measure for exploration and exploitation, we can also verify several claims about the algorithm’s performance. For example, the work in [20] claimed that the newly developed EABC achieved an ideal balance between exploration and exploitation. Without measuring exploration and exploitation directly, such a claim is hard to verify. Once explicit control of exploration and exploitation is understood well in single-objective optimization, further research is needed to apply explicit exploration measures to constrained, noisy, multi-objective, and dynamic optimization problems. Yet another direction of research is to investigate which approaches lead to fewer undesired explorations (failed and deceptive), as well as to determine how to control exploration and exploitation in EAs.