An Enhanced Artificial Bee Colony Algorithm Based on Elimination History and Elite Correction

An Enhanced Artificial Bee Colony Algorithm Based on Elimination History and Elite Correction

Yingduo Lei

School of Electrical and Control Engineering North University of China

Taiyuan, China 17312266275@163.com

Qinna Zhu

School of Computer Science and Technology North University of China

Taiyuan, China zqn15735747719@163.com

Haibo Yu^*

School of Computer Science and Technology North University of China

Taiyuan, China

* nucyuhaibo@nuc.edu.cn Jianchao Zeng^*

School of Computer Science and Technology North University of China

Taiyuan, China

*20160002@nuc.edu.cn Abstract—Artificial bee colony (ABC) algorithm is often

challenged by slow convergence, poor accuracy, and premature convergence in handling complex medium-scale optimization problems, due to its biased search equation and the high assimilation rate of bees within the colony. To trade off the ABC for global exploration and local exploitation of complex problem landscapes, this paper proposes an enhanced ABC based on elimination history and elite correction (HeCABC). Given the bias effects of the superior solutions and the historical inferior solutions eliminated on the search behavior of ABC, HeCABC separately formulates an exploration equation oriented by the historically eliminated inferior solutions and an exploitation equation upon multi-elite information fusion for employed bees and onlook bees, to regulate their exploration and exploitation intensity of the solution space. Meanwhile, HeCABC couples an elite correction strategy for fine-tuning the quality of the elites based on the update signal of these elites within the colony.

HeCABC is experimented on various complex CEC 2014 test functions of 30 dimensions. The experimental results showcase its superior performance over five state-of-the-art ABC variants and two advanced swarm optimizers.

Keywords-Artificial bee colony; Search equation; Diversity maintenance; Elimination history; Elite correction

I. INTRODUCTION

The artificial bee colony algorithm is a representative swarm intelligence optimizer first proposed by Karaboga et al.

[1] in 2005. ABC achieves the optimal solution to complex problems by simulating the division of labor among different bee species and the information exchange mechanism among bee species in the honey harvesting process of honey bees in the biological world [2]. ABC has attracted much attention since its inception and prevails for its excellent performance in solving complex optimization problems, such as vehicle path planning [3], node distribution in wireless sensor networks [4], continuous optimization [5, 6], and signal processing [7].

However, ABC features strong exploration and weak exploitation [8], making it often converge slowly and with poor convergence accuracy when tackling complex multimodal optimization problems. To effectively trade off the exploration and exploitation of ABC, a variety of ABC variants have been

proposed, including elite-driven ABC variants, ABC variants integrating multi-preference search equations, and ABC variants hybridized with other algorithms or techniques [9].

Most ABC variants emphasize using the historical elite solutions or the currently acquired global optimal solution to enhance the local exploitation of the potential regions the bee colony covers at the behavior learning stage. For instance, GABC [10] introduced globally optimal individuals into the search equation to guide the bee colony toward the globally optimal neighborhoods. Similarly, iqABC [11] also steered the direction of behavioral updating in bee colony with the help of globally optimal individuals. However, unlike GABC, iqABC dynamically regulates the diversity of the colony by restricting the usage frequency of the optimal individuals during the iteration. Given the greedy orientation of the elites on the exploitation of the bee colony search, for each bee individual, CABC [12] randomly assigned an individual within the colony to perturb it and correct its search orientation, improving the diversity of exploitation of the solution space. Moreover, NABC [13] utilized the best neighboring solutions to drive the state transfer of the colony individual at each iteration, significantly improving the convergence of ABC.

Despite the significant enhancement in the bee colony exploitation preference underlying the superior elite solutions, the single search equation plus the univariate perturbation learning severely limits the diversity of the bee colony. To address this issue, researchers suggested configuring a multi- level diversity search equation in different search stages of bee species (employed bee, onlook bee, and scout bee) to drive the state transition of the bee colony. Coupling different search equations can overcome the biased search of bee colonies and trade off the diversity and convergence of bee colonies in complex multi-modal landscape search [14, 15]. For instance, Yu et al. [16] adopted a set of probabilistically switched dual- preference search equations to update the position of the bee colony, which significantly improved the diversity of the bee colony at different search stages and the convergence performance. Gao et al. [17-18] formulated three search equations with different preferences following the search tendency of different bee species. At each iteration, the bee individual was adaptively paired with the most proper search 2024 6th International Conference on Data-driven Optimization of Complex Systems (DOCS)

equation to drive the bee colony toward the potential optimal region. Moreover, inspired by reinforcement learning, Zhao et al. [19] preferred four search equations from six mainstream mutation strategies at each iteration to update the position of the bee colony.

Instead of laying emphasis solely on the superior elite individuals when refreshing the bee colony, some researchers referred to the inferior individuals in the bee colony to enhance the search diversity of ABC. Depending on the division of labor and the switching of roles between these two types of individuals in updating the bee colony behavior, a commendable equilibrium of exploration and exploitation can be achieved. For instance, Cui et al. [20] proposed a two colony-driven ABC variant. The proposal constructed a convergent sub-colony by dynamically selecting part of the superior individuals in the bee colony, which was then used to deeply exploit the local optimal regions of the solution space.

Concurrently, a diversified sub-colony composed of the remaining inferior individuals in the bee colony was then used for breadth exploration of potential optimal regions in the solution space. The collaborative search between these two sub-colonies results in a favorable trade-off between search diversity and convergence of ABC. To mitigate the rapid loss of population diversity, Cui et al. [21] proposed a mechanism to regulate the numbers of employed and onlooker bees, and an opposition-based learning method was employed on inferior employed bees forgotten to generate their new positions.

Nevertheless, as aforementioned, most existing ABC variants focus on selecting different elite individuals to assist in constructing biased search equations to regulate the exploitation intensity in the iterative bee colony while ignoring the guidance role of the historically eliminated inferior individuals in the maintenance of colony diversity. The eliminated inferior solutions alongside the optimization preserve abundant a priori knowledge of the solution space. If these solutions are used to update the bee colony, it will help to aggregate more prior knowledge of global solution space during the optimization process, thus mitigating the loss of colony diversity. In such a way, the bee colony is expected to achieve a broad exploration of the solution space. In addition, the elite individual is limited to the uniqueness of its feature information, which hinders the search diversity of the bee colony once a specific elite is chosen to induce the state shift of the bee individual. This is especially the case when the elite solutions trap in the local optima, making the bee colony intractable to reach the global optimal region. Thus, fusing and correcting the feature information of different elite individuals and strengthening the intercommunication between different elites are of great necessity in enhancing the local escape ability and the search robustness of the bee colony.

Given the above consideration, in this work, we propose an enhanced ABC based on elimination history and elite correction, named HeCABC, to further optimize the balance between the diversity and convergence of ABC search. In HeCABC, the eliminated inferior individuals along with bee colony evolution are collected to comprise a history pool of the eliminated inferior solutions. During the employed bee search phase, the eliminated inferior solutions randomly chosen from the history pool of the eliminated inferior solutions and the

iterative bee individuals are jointly used to direct the state transfer of the employed bees, to enhance the search diversity of employed bees for the global solution space. Meanwhile, to strengthen the exploitation of potential optimal local regions and improve the convergence of the iterative bee colony, a search equation integrating multiple elite feature information is constructed for the onlooker bee search. In addition, given the different distribution properties of the elite individuals in the solution space, HeCABC couples an elite correction strategy for fine-tuning the quality of the elites based on the update signal of these elites within the colony. By comparing the proposed HeCABC against five state-of-the-art ABC variants as well as two advanced swarm optimizers on the CEC'2014 [22] test set, the experimental results disclosed the remarkable performance of the proposed HeCABC in striking the commendable equilibrium of diversity and convergence.

II. ARTIFICIALBEECOLONYALGORITHM

Artificial bee colony algorithm hunts for the optimal solution to a specific problem by simulating the intelligent nectar-gathering process of the bee colony. Its search process mainly consists of an initialization phase and a sequential search phase for three types of bee species: employed bee, onlooker bee, and scout bee. The employed bees are responsible for exploring and recording the locations of nectar sources and sharing excellent nectar source information with onlooker bees. The onlooker bees then prefer to select high- quality nectar sources for deep exploitation based on the nectar source information provided by the employed bees. When stagnation occurs in the update of the nectar sources, the employed bees allied to the nectar sources will be transformed into scout bees to re-explore the new nectar source. An overview of the four main search phases in ABC is as follows:

1) Initialization phase

Assuming the existence of N nectar sources

ζ |

X_{i i<1,2,...,N}, let

,1 ,2 ,

( , ,..., )

i i i i D

X < x x x be the i^th nectar source, _D denotes the problem dimension. Each bee within the colony occupies the position of a specific nectar source. A counter trial is configured for each bee, which is used to track the updates to the position of the nectar source. If the nectar source gets updated, then trial<0 ; otherwise trial trial< ∗1. The bee colony

ζ |

^X_{i i<1,2,...,N} is initialized by (1):

min max min

, ( )

i j j j j

x <x ∗ ≥r x ,x , (1)

min

xj and x^max_j indicate the upper and lower bounds of the j^th dimension, respectively. r U⊆ (0,1).

2) Employed bee search phase

The employed bee performs state transfer according to (2) and greedily updates the nectar source location.

, , , ( , , )

i j i j i j i j k j

v <x ∗ε ≥ x ,x , (2)

where V_i <(v_i,1,v_i,2, ,√√√v_{i D,} ) denotes the new nectar source.

Xi indicates the i^th employed bee (nectar source location), while X_k(i k÷ ) indicates a randomly selected employed bee from the bee colony.ε ⊆_{i j,} U( 1,1), .

3) Onlooker bee search phase

After the employed bee search, the roulette wheel selection is used to figure out promising nectar sources for onlooker bees according to the nectar source quality (fitness), where the nectar source quality is calculated according to (3).

1 ( ) 0

1 ( )

1 ( ( ))

i i i

i

if f X fit f X

abs f X otherwise

 ″

<  ∗

 ∗

, (3)

where f( )√ represents the objective function value, fit( )√ indicates the nectar source quality (fitness). The higher the fitness, the higher the quality of the nectar source. The selection probability of each nectar source is calculated according to (4).

1

( ) ( )

i

i N

j j

fit X p

f it X

<

<



^{, (4)}

where p_i denotes the probability that X_i is selected. As might be expected, high-quality nectar sources have a high selection probability. The onlooker bees then exploit the new nectar sources around their neighborhoods according to (2).

4) Scout bee search phase

In this phase, once the counter trial associated with the nectar source exceeds a predefined threshold, the employed bee allied to this nectar source is transformed into a scout bee, which thereafter re-generates a new nectar source according to (1).

III. THEPROPOSEDMETHOD

Algorithm 1 showcases the pseudocode of HeCABC. As shown in Algorithm 1, HeCABC follows the basic principle asserted in the standard ABC. Different from the standard ABC, HeCABC deposits all the eliminated inferior individuals in the optimization process into a historically eliminated inferior solution pool Pool and dynamically updates it with iterations.

The size of thePool is managed the same as the population size. During the employed and onlooker bees search, HeCABC utilizes the randomly chosen solutions from the Pool and part of the elite individuals to regulate the search bias of employed and onlooker bees, respectively.

Concurrently, at each iteration, HeCABC additionally adds an elite correction auxiliary strategy to correct the elite individuals within the bee colony to avoid premature convergence.

Algorithm 1:Pseudocode of HeCABC Input:N,D, MaxFEs,MaxTrial^.

Output: The optimal solution.

1: Generate{ }X_{i i<1,2, ,≈N} by (1), set{trial_i <0 |i<1, 2, , }ϑ N

2: Compute f X( _i),i<1, 2, ,≈ N, set FEs N<

3: Set Pool<{ }X_{i i<1,2, ,≈N},{stop_i<0 |i<1, 2, , }ϑ N

4:while FEs MaxFEs′ do 5: //* Employed bee phase * //

6: fori< ↑1 N do

7: Generate V_i by (5), compute f V( )_i

8: FEs FEs< ∗1,s ceil rand< ( (0,1)≥N)

9: if f V( )_i ; f X( _i) then

10: pool_s ↔X_i, X_i ↔V_i,trial_i <0,stop_i<0

11: else

12: pool_s↔V_i,triali<triali∗1^,stop_i<stop_i∗1

13: end if 14: end for

15: Compute the selection probability p_i by (4) 16:// * Onlook bee phase * //

17: fori< ↑1 N do 18: ifrand(0,1);p_i then

19: GenerateV_i by (6)-(8), compute f V( )_i

20: FEs<FEs∗1^, s ceil rand< ( (0,1)≥N)

21: if f V( )_i ; f X( _i) then

22: pool_s↔X_i,X_i ↔V_i,trial_i <0,stop_i<0

23: else

24: pool_s ↔V_i,trial_i<trial_i∗1^,stop_i<stop_i∗1

25: end if 26: end if 27: end for

28:// * Scout bee phase * //

29: fori< ↑1 N do

30: iftrial_i =MaxTrial ^then

31: Refresh X_i by (1), compute f X( _i)

32: trial_i <0, FEs FEs< ∗1

33: end if 34: end for

35:// * Elite refinement phase * //

36: Identify the top-tier elite solutions{ }e_{i i<1,2, , /10≈N} 37: fori< ↑1 N/10 do

38: if stop_i =Maxstop then

39: Compute the Euclidean distances from e_i to othere_j 40: GenerateV_i by (9), compute f V( )_i

41: FEs FEs< ∗1,s ceil rand< ( (0,1)≥N)

42: pool_s ↔X_i,X_i ↔V_i,trial_i <0,stop_i<0

43: end if 44: end for 45:end while

46: Output the best solution.

A. Global exploration equation based on historically eliminated inferior solution pool

In HeCABC, the employed bees explore new nectar sources according to (5).

, , , ( , , )

i j a j i j a j r j

v <x ∗ε √ x ,pool , (5) where X_a is a randomly selected individual from the current iterative bee colony. pool_r denotes a randomly selected individual from the Pool. The search equation (5) takes into account both the individual prior of the current iterative bee colony and the historical prior of the bee colony. Thus, the bee colony can not only explore the neighborhood covered by the current iterative bee colony but also refer to the historical prior of the colony, to drive it to explore the global solution space landscape in breadth. As the optimization proceeds, the Pool members are constantly updated by the newly eliminated inferior solutions, so that the comprehensive quality level of the eliminated inferior solutions in Pool is gradually improved. In such a way, the employed bees can adaptively regulate their breadth exploration and depth exploitation intensity to the solution space. In addition, to avoid the degradation of the convergence speed of the search caused by unidimensional perturbations [23], HeCABC adopts a fixed probability q<0.3 to control the number of perturbation dimensions of the bee colony individuals to accelerate the convergence.

B. Local exploitation equation upon multi-elite information fusion

To strengthen the information exchange of the elites and weaken the greedy steer of specific elite individuals on the bee colony search, in the onlooker search phase, HeCABC first corrects the feature information of each elite individual e_i based on the subtraction average correction strategy [24] and then generates its trial elite individualTe_i following (6).

1

1 ⁿ ( ), 1, 2, ,

i i i j

j

Te e r e ve i n

n <

< ∗ ≥



, < ^{ϑ , (6)}

where e_i and e_j separately represent the i^th elite and j^th elite among the bee colony. n< √θ N indicates the size of elite group. e_i,ve_j <sign f e( ( )_i ,f e( ))(_j e_i,v e* )_j , where ( )f ei

and f e( )_j denote the objective function values of e_i and e_j, respectively. v⊆{1, 2}^D and the operator “*” represents the Hadamard product [24] of the two vectors. r U⊆ (0,1) and θ <0.1 [25]. As shown in (6), the trial elite individual Te_i integrates the excellent information of all elite individuals and thereby weakens the greedy guidance of single elite individuals.

After that, the elite individual e_i gets updated by competing with its trial solution Te_i, as shown in (7).

( ) ( )

i i i

i i

Te if f Te f e e e Otherwise

 ;

< 

 , (7)

Finally, the onlooker bee performs a state transfer according to (8) to obtain a new nectar source.

, , , ,

, ,

( ) (0,1)

r j i j r j i j i

i j i j

e e x rand CR

v x Otherwise

ε

∗ √ , ;

< 

 , (8)

where CR indicates the selection probability of dimensional inheritance. X_i is an individual selected by roulette, and e_r is an elite randomly selected from the elite group.

C. Elite correction strategy

While ABC uses the scout bees re-initializing the stagnant individuals to avoid premature convergence of the bee colony, re-initialization on the one hand increases the risk of redundant search, on the other hand, it discards the existing excellent history prior and weakens the convergence efficiency of the bee colony. To fully retain and utilize the historical elite information and enhance the local escape capability of the bee colony, HeCABC uses an elite tuning auxiliary strategy to correct the stagnant elite individuals after the scout bee search, as shown in (9).

1 2

(0,1) ( )

(0,1) ( ( ) ( ))

i j i

i

i mean i best i

e rand e e if case

V e rand r X e r X e Otherwise

∗ ≥ ,

<  ∗ ≥ ≥ , , ≥ , , (9)

where e ii( <1,2,...,N/ 10) denotes the i^th elite individual, case represents the condition that the elite individual e_i lies in a different local attraction domain than the other elite individuals.

( 1,2,... /10, )

e jj < N j i÷ denotes a randomly selected elite individual from the elite group. X_mean and X_best are the mean position of the current bee colony and the optimal individual, respectively. r r₁, ₂⊆[0,1], meeting r₁∗ <r₂ 1.

HeCABC is paired with a stagnation counter stop for each elite individual e_i. Ife_i is not updated during the iteration, then

1

stop stop< ∗ ; otherwise, stop<0. If the stagnation counter stop exceeds the predefined threshold Maxstop, it indicates that the elites within the bee colony occurs stagnation, at which point the elite correction strategy is triggered. Note that HeCABC only takes into account the best N/10 individuals in the bee colony in the elite correction phase.

Given the diverse distribution of elite individuals trapped in the local attraction domains, we first calculate the Euclidean distance Dis_i↑j between each elite e_i and the remaining elite

( 1,2,... /10, )

e jj < N j i÷ . If _Disi↑j=_(X^max_j ,_X^min_{j ) /10} , it indicates that e_i and e_j are located in different local attraction domains. e_i is then corrected according to the first equation of (9). Otherwise, it means that e_i and e_j are likely to be located in the same local attraction domain. e_i is then corrected according to the second equation of (9).

IV. EXPERIMENTALVERIFICATIONS

To assess the effectiveness and efficacy of the proposed HeCABC, it is compared with five state-of-the-art ABC variants including BPLABC [16], CABC [12], NABC [13], iqABC [11], and KFABC [26] as well as two advanced swarm intelligence algorithms including SABO [24] and DBO [27] on the CEC'2014 benchmark problems with 30 dimensions. The parameter configuration of each comparison algorithm is shown in Table I. All compared algorithms are coded in a Matlab R2022b numerical simulation platform and are executed on a PC equipped with an AMD Ryzen9 7900X 12- Core Processor CPU @ 4.70GHz and 32.0GB RAM. To be fair, all algorithms are run independently 30 times and the termination condition is set toMaxFEs<10000≥D.

A. Exploration-Exploitation Performance Analysis

To verify the effectiveness of the proposed search equations (5) and (8) in balancing exploration and exploitation of the bee colony, here we take three functions F10, F23, and F30, of which the landscape complexity increases, as examples. Figure 1(a-c) showcases the exploration and exploitation ratio against the number of iterations of HeCABC and ABC in a run. The vertical axis represents the percentage of exploration and exploitation of the bee colony, and the horizontal axis manifests the number of iterations. The exploration ratioEG(%) and the exploitation ratio EL(%) are separately calculated according to (10) and (11) [28], where Dt gen( )computed by (12) and Dt_max measure the dimension-wise diversity at each generation and the maximum dimension-wise diversity throughout the optimization, respectively. As shown in Figure 1, compared to ABC, HeCABC can better trade off the exploration and exploitation intensities with good generalization and robustness on the selected test instances, indicating the high efficiency of search equations (5) and (8) in managing the search preference of the bee colony.

max

( )

(%) Dt gen 100

EG < Dt ≥ , (10)

max max

| ( ) |

(%) Dt gen Dt 100

EL Dt

< , ≥ , (11)

1 1

1 1

( ) ^{D N} | ( (_j )) _ij( ) |

j i

Dt gen m dian xe gen n

D x e

N g

< <

<

 

, ^{. (12)}

B. Comparison Results of HeCABC Against Other Swarm Optimizers

Table II summarizes the Mean(std.) results of HeCABC against seven advanced swarm optimizers on the CEC'2014 benchmark problems with 30 dimensions, as well as their Friedman rankings and the Wilcoxon's rank sum test results at

=0.05

 significance level. The ‘+’, ‘-’, and ‘=’ indicate that the compared algorithm wins, loses, or performs competitively over HeCABC, respectively. The best solutions obtained by the comparison algorithms on each instance are highlighted in bold.

TABLE Ⅰ Parameter Setting of HeCABC and The Compared Algorithms.

Algorithm Parameter settings

BPLABC ^{N<100,MaxTrial<200,q<0.8,p<0.5} CABC N<100,MaxTrial<200 NABC N<100,MaxTrial<200,Neighbor<5 iqABC N<100,MaxTrial<200,bestLimit<1.5≥ ≥N D KFABC ^{N<100,MaxTrial<200,θ<0.1}

SABO N<100

DBO N<100,k< <κ 0.1,b<0.3,S<0.5

HeCABC ^{N<100,MaxTrial<200,q<0.3,CR<0.5,n<10,Maxstop<20}

As shown in Table II, HeCABC performs significantly better than the other seven compared algorithms on 19 problems. For the unimodal problems F1-F3, HeCABC performs the best, demonstrating the strong exploitation capability of HeCABC. For the multimodal problems F4-F16, compared to the other seven compared algorithms, HeCABC obtained the best results on problems F4, F6, F9, F13, and F15- F16. The absolute winning rate of HeCABC on this set of test instances accounts for 6/13. For problems F5, F7-F8, F10-F12, and F14, the performance of HeCABC is slightly worse than that of BPLABC, iqABC, and KFABC. Here, it is worth noting that the diversity of the bee colony in HeCABC is significantly constrained by the diversity of the individuals in the historically eliminated inferior solution pool Pool. If the diversity of the individuals in Pool losses too rapidly, the global exploration of the bee colony on the complex solution space then degrades greatly, resulting in a high risk of falling into the pseudo-global optimum.

However, for the highly complex hybrid problems F17-F22 and the composition problems F23-F30, HeCABC performs significantly better than the competitors on 10 test problems, with exceptions where it performs slightly worse than SABO, DBO and BPLABC on problems F23-F25 and F29. This indicates that HeCABC is capable of efficiently balancing the global exploration and local exploitation of bee colony for highly complex multimodal solution space.

Figure 2(a-c) presents the convergence profiles of HeCABC and the seven comparison algorithms on the complex multimodal problems F9, F17, and F27, respectively. As shown in Figure 2(a-b), HeCABC maintains favorable convergence performance and rapid convergence speed on these three multimodal problems and performs the best among the competitors. In particular, as shown in Figure 2(a), the seven comparison algorithms all fall into local optimality and converge prematurely on the multimodal problem F9, while HeCABC effectively escapes from the pseudo-optimal neighborhood in the later search stage and quickly explores the new global optimal region. Overall, the above results demonstrate that the proposed HeCABC has stronger convergence and robustness in solving complex optimization problems compared to other ABC variants and swarm intelligence algorithms.

TABLE II. THESTATISTICALRESULTS OFCOMPAREDALGORITHMS ON THE SELECTEDBENCHMARKPROBLEMS

NO. BPLABC CABC NABC iqABC KFABC SABO DBO HeCABC

F01 1.82E+07

(5.44E+06) - 2.05E+07

(6.91E+06) - 3.16E+07

(1.57E+07) - 3.59E+06

(2.00E+06) - 2.09E+07

(6.98E+06) - 1.22E+08

(6.01E+07) - 2.83E+07

(2.40E+07) - 1.77E+06 (1.32E+06)

F02 3.72E+02

(1.78E+02) - 7.39E+02

(7.10E+02) - 3.45E+07

(8.67E+07) - 4.31E+02

(2.17E+02) - 5.52E+02

(4.86E+02) - 3.63E+09

(2.39E+09) - 1.56E+05

(6.97E+05) - 2.87E+02 (4.35E+03)

F03 9.78E+02

(4.62E+02) - 1.91E+03

(1.18E+03) - 2.18E+03

(1.39E+03) - 9.46E+02

(4.47E+02) - 1.18E+03

(1.07E+03) - 4.45E+04

(1.01E+04) - 7.95E+03

(1.09E+04) - 3.06E+02 (9.42E+01)

F04 4.49E+02

(2.06E+01) = 4.71E+02

(1.43E+01) - 5.24E+02

(2.68E+01) - 4.69E+02

(3.76E+01) - 4.74E+02

(2.45E+01) - 9.02E+02

(1.56E+02) - 5.47E+02

(6.64E+01) - 4.44E+02 (3.14E+01) F05 5.20E+02 (4.51E-

02) + 5.20E+02 (2.79E-

02) + 5.20E+02 (3.34E-

02) + 5.20E+02 (2.58E-

02) + 5.20E+02 (4.73E-

02) + 5.21E+02 (5.23E-

02) - 5.21E+02 (2.44E-

01) - 5.21E+02 (5.55E- 02)

F06 6.14E+02

(1.32E+00) - 6.15E+02

(1.07E+00) - 6.17E+02

(1.26E+00) - 6.15E+02

(1.17E+00) - 6.14E+02

(1.24E+00) - 6.28E+02

(2.86E+00) - 6.26E+02

(3.09E+00) - 6.01E+02 (1.05E+00) F07 7.00E+02 (3.33E-

04) + 7.00E+02 (9.44E-

04) - 7.01E+02

(1.86E+00) - 7.00E+02 (4.00E-

03) - 7.00E+02 (5.14E-

05) + 7.33E+02

(2.51E+01) - 7.00E+02 (3.35E-

01) - 7.00E+02 (8.43E- 03) F08 8.00E+02 (1.27E-

15) + 8.00E+02 (2.99E-

14) = 8.04E+02

(2.90E+00) - 8.00E+02 (4.07E-

04) = 8.00E+02 (1.10E-

13) = 1.02E+03

(2.83E+01) - 9.32E+02

(3.62E+01) - 8.00E+02 (1.01E- 13)

F09 9.70E+02

(8.17E+00) - 9.49E+02

(6.94E+00) - 9.74E+02

(1.26E+01) - 9.83E+02

(8.42E+00) - 9.43E+02

(8.14E+00) - 1.15E+03

(2.24E+01) - 1.09E+03

(4.99E+01) - 9.14E+02 (3.05E+00) F10 1.00E+03 (2.80E-

02) + 1.00E+03 (2.79E-

01) + 1.08E+03

(6.54E+01) - 1.00E+03 (5.34E-

01) + 1.00E+03

(1.17E+00) + 7.78E+03

(6.00E+02) - 4.21E+03

(9.31E+02) - 1.00E+03 (1.39E+00)

F11 3.27E+03

(2.62E+02) +

3.50E+03 (2.86E+02) +

3.52E+03 (2.67E+02) +

3.13E+03 (2.44E+02) +

3.11E+03 (3.14E+02) +

8.22E+03 (3.29E+02) -

5.22E+03 (7.48E+02) -

4.42E+03 (3.03E+02) F12 1.20E+03 (6.37E-

02) + 1.20E+03 (4.98E-

02) + 1.20E+03 (4.69E-

02) + 1.20E+03 (4.86E-

02) + 1.20E+03 (4.60E-

02) + 1.20E+03 (2.30E-

01) - 1.20E+03 (7.10E-

01) = 1.20E+03 (7.46E- 02) F13 1.30E+03 (2.12E-

02) - 1.30E+03 (2.70E-

02) - 1.30E+03 (2.91E-

02) - 1.30E+03 (3.54E-

02) - 1.30E+03 (4.08E-

02) - 1.30E+03 (5.86E-

01) - 1.30E+03 (1.44E-

01) - 1.30E+03 (2.98E- 02) F14 1.40E+03 (1.90E-

02) + 1.40E+03 (2.18E-

02) = 1.40E+03 (3.27E-

02) = 1.40E+03 (1.72E-

02) = 1.40E+03 (3.08E-

02) = 1.41E+03

(8.12E+00) - 1.40E+03 (3.26E-

01) - 1.40E+03 (2.15E- 02) F15 1.51E+03 (9.78E-

01) - 1.51E+03 (6.84E-

01) - 1.51E+03

(5.23E+00) - 1.51E+03

(1.46E+00) - 1.51E+03 (8.55E-

01) - 2.22E+03

(1.33E+03) - 1.52E+03

(7.12E+00) - 1.50E+03 (9.83E- 01) F16 1.61E+03 (2.22E-

01) - 1.61E+03 (2.97E-

01) - 1.61E+03 (2.93E-

01) - 1.61E+03 (6.39E-

01) - 1.61E+03 (4.01E-

01) = 1.61E+03 (2.71E-

01) - 1.61E+03 (4.76E-

01) - 1.61E+03 (5.30E- 01)

F17 3.13E+06

(1.40E+06) - 4.23E+06

(1.78E+06) - 3.75E+06

(1.74E+06) - 1.57E+06

(1.37E+06) - 2.97E+06

(1.35E+06) - 5.05E+06

(3.16E+06) - 1.24E+06

(1.04E+06) - 4.22E+05 (2.45E+05)

F18 1.14E+04

(6.91E+03) - 6.57E+04

(3.97E+04) - 8.47E+04

(1.94E+05) - 5.42E+03

(2.40E+03) - 5.09E+03

(3.35E+03) - 2.37E+06

(3.93E+06) - 7.34E+04

(2.24E+05) - 2.79E+03 (6.22E+02) F19 1.91E+03 (5.06E-

01) -

1.91E+03 (4.57E- 01) -

1.93E+03 (1.79E+01) -

1.91E+03 (1.24E+01) -

1.91E+03 (8.85E- 01) -

2.02E+03 (2.10E+01) -

1.93E+03 (2.74E+01) -

1.91E+03 (1.20E+00)

F20 6.58E+03

(2.24E+03) - 7.87E+03

(2.37E+03) - 7.73E+03

(3.45E+03) - 9.72E+03

(8.50E+03) - 8.17E+03

(2.92E+03) - 2.93E+04

(1.37E+04) - 7.23E+03

(3.60E+03) - 3.49E+03 (1.18E+03)

F21 6.21E+05

(2.92E+05) - 6.59E+05

(2.98E+05) - 7.43E+05

(3.72E+05) - 1.02E+06

(7.24E+05) - 6.97E+05

(4.76E+05) - 3.44E+06

(2.57E+06) - 4.28E+05

(4.12E+05) - 1.29E+05 (3.75E+04)

F22 2.52E+03

(8.34E+01) - 2.61E+03

(9.46E+01) - 2.57E+03

(1.06E+02) - 2.47E+03

(1.08E+02) - 2.49E+03

(1.02E+02) - 3.34E+03

(1.87E+02) - 2.84E+03

(2.16E+02) - 2.33E+03 (2.76E+01) F23 2.62E+03 (1.81E-

01) - 2.62E+03 (5.37E-

01) - 2.63E+03

(5.94E+00) - 2.62E+03 (2.99E-

02) - 2.62E+03 (3.21E-

01) - 2.51E+03

(2.55E+01) + 2.62E+03

(8.77E+00) - 2.62E+03 (9.72E- 03)

F24 2.62E+03

(9.53E+00) = 2.63E+03 (4.91E-

01) - 2.63E+03

(5.86E+00) - 2.63E+03

(4.21E+00) - 2.63E+03 (7.58E-

01) - 2.60E+03 (1.41E-

04) + 2.60E+03 (1.71E-

04) + 2.63E+03 (8.07E- 01)

F25 2.71E+03

(1.13E+00) -

2.71E+03 (1.15E+00) -

2.71E+03 (1.85E+00) -

2.71E+03 (2.67E+00) -

2.71E+03 (8.85E- 01) -

2.70E+03 (2.07E- 13) +

2.70E+03 (3.95E+00) =

2.71E+03 (5.45E- 01) F26 2.70E+03 (3.92E-

02) - 2.70E+03 (6.52E-

02) - 2.70E+03 (4.15E-

01) - 2.73E+03

(4.50E+01) - 2.70E+03 (5.42E-

02) - 2.74E+03

(3.91E+01) - 2.70E+03

(1.81E+01) - 2.70E+03 (1.94E- 01)

F27 3.11E+03

(3.78E+00) - 3.12E+03

(7.75E+00) - 3.15E+03

(2.85E+01) - 3.11E+03

(5.13E+00) - 3.11E+03

(6.77E+00) - 3.85E+03

(7.27E+01) - 3.12E+03

(1.24E+01) - 3.05E+03 (9.94E+00)

F28 3.68E+03

(3.15E+01) - 3.66E+03

(2.52E+01) - 3.87E+03

(1.38E+02) - 3.76E+03

(5.11E+01) - 3.63E+03

(2.28E+01) = 7.04E+03

(1.16E+03) - 4.40E+03

(3.28E+02) - 3.62E+03 (3.36E+01)

F29 4.28E+03

(2.21E+02) + 7.51E+03

(3.61E+03) - 9.52E+04

(1.48E+05) - 4.46E+03

(4.73E+02) = 1.41E+04

(9.36E+03) - 5.66E+07

(2.79E+07) - 1.21E+07

(9.83E+06) - 4.65E+03 (1.31E+00)

F30 8.44E+03

(1.23E+03) - 1.05E+04

(1.87E+03) - 2.55E+04

(8.80E+03) - 6.46E+03

(1.13E+03) - 8.98E+03

(1.41E+03) - 2.83E+05

(1.73E+05) - 5.65E+04

(5.88E+04) - 5.58E+03 (1.27E+02)

+/=/- 8/2/20 4/2/24 3/1/26 4/3/23 5/4/21 3/0/27 1/2/27 —

Friedman

rank 3.27 4.43 5.83 3.70 3.23 7.33 5.87 2.33

0 100 200 300 400

Iteration 0

20 40 60 80 100

Exploration-ABC Exploitation-ABC Exploration-HeCABC Exploitation-HeCABC

(a) 30D-F10 (b) 30D- F23 (c) 30D- F30 Figure 1. The exploration and exploitation performance of HeCABC and ABC on the selected benchmark instances.

(a) 30D-F9 (b) 30D-F17 (c) 30D-F27 Figure 2. The convergence curves of compared algorithms on problems F9, F17 and F27 with 30 dimensions.

V. CONCLUSIONS

To address the issue of slow convergence and low accuracy of ABC due to the negative effects of the single search equation and the high assimilation rate in the bee colony, this paper proposed an enhanced ABC based on elimination history and elite correction, named HeCABC. HeCABC drives the state transition of bee colony by leveraging and iteratively updating the historically eliminated inferior solutions alongside the optimization while merging the feature information of the elite solutions to correct the quality of promising elites, which effectively regulates the search bias of ABC and improves the convergence speed and convergence accuracy. Meanwhile, to avoid the search from falling into a pseudo-global optimum, at the end of each iteration, HeCABC couples an elite correction auxiliary strategy to activate the stagnant elite solutions by leveraging the distribution features of the elites in the solution space, effectively enhancing their escape ability from the local traps for breadth exploitation. Experimental results show that our proposed HeCABC can effectively balance global exploration and local exploitation on various problems and exhibits excellent convergence and robustness over five other state-of-the-art ABC variants and two widely used swarm intelligence optimizers. In the future, we will try to resize the historically eliminated inferior solution pool adaptively to improve the performance of HeCABC further.

ACKNOWLEDGMENT

This work was supported in part by the National Natural Science Foundation of China under Grant 62106237, and the Joint Funds of the National Natural Science Foundation of China under Grant U21A20524.

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