https://doi.org/10.31449/inf.v44i2.2385 Informatica 44 (2020) 183–198 183 
 
Hybrid Bees Approach Based on Improved Search Sites Selection by 
Firefly Algorithm for Solving Complex Continuous Functions 
Mohamed Amine Nemmich and Fatima Debbat 
Department of Computer Science, University Mustapha Stambouli of Mascara, Mascara, Algeria 
E-mail: amine.nemmich@gmail.com, debbat_fati@yahoo.fr 
 
Mohamed Slimane 
Université de Tours, Laboratoire d’Informatique Fondamentale et Appliquée de Tours (LIFAT), Tours, France 
E-mail: mohamed.slimane@univ-tours.fr 
Keywords: optimization, bees algorithm, firefly algorithm, memory scheme, adaptive neighborhood search, swarm 
intelligence, bio-Inspired computation 
Received: July 25, 2018 
The Bees Algorithm (BA) is a recent population-based optimization algorithm, which tries to imitate the 
natural behavior of honey bees in food foraging. This meta-heuristic is widely used in various 
engineering fields. However, it suffers from certain limitations. This paper focuses on improvements to 
the BA in order to improve its overall performance. The proposed enhancements were applied alone or 
in pair to develop enhanced versions of the BA. Three improved variants of BA were presented: BAMS-
AN, HBAFA and HFBA. The new BAMS-AN includes memory scheme in order to avoid revisiting 
previously visited sites and an adaptive neighborhood search procedure to escape from local optima 
during the local search process. HBAFA introduces the Firefly Algorithm (FA) in local search of BA to 
update the positions of recruited bees, thus increasing exploitation in each selected site. The third 
improved BA, i.e. HFBA, employs FA to initialize the population of bees in the BA for a best exploration 
and to start the search from more promising regions of the search space. The proposed enhancements to 
the BA have been tested using several continuous benchmark functions and the results have been 
compared to those achieved by the standard BA and other optimization techniques. The experimental 
results indicate that the improved variants of BA outperform the standard BA and other algorithms on 
most of the benchmark functions. The enhanced BAMS-AN performs particularly better than others 
improved BAs in terms of solution quality and convergence speed. 
Povzetek: Za reševanje kompleksnih zveznih funkcij so razvili nov pristop na osnovi hibridnega 
čebeljega algoritma (BA) in algoritma Firefly. 
1 Introduction 
Metaheuristic algorithms generally mimic the more 
successful behaviors in nature. These methods present 
best tools in reaching the optimal or near optimal 
solutions for complex engineering optimization problems 
[1]. 
As a branch of metaheuristic methods, swarm 
intelligence (SI) is the collective behavior of populated, 
self-organized and decentralized systems. It concentrates 
specifically on insects or animals behavior in order to 
develop different metaheuristics that can imitate the 
capabilities of these agents in solving their problems like 
nest building, mating and foraging. These interactions 
have been effectively appropriated to solve large and 
complex optimization problems [2]. For instance, the 
behaviors of social insects, such as ants and honey bees, 
can be patterned by the Ant Colony Optimization (ACO) 
[3] and Artificial Bee Colony (ABC) [4] algorithms. 
These methods are generally utilized to describe effective 
food search behavior through self-organization of the 
swarm. 
In SI, honey bees are one of the most well studied 
social insects. Furthermore, it is in a growing tendency in 
the literature for the past few years and it will continue. 
Many intelligent popular search algorithms are 
developed such as Honey Bee Optimization (HBO), 
Beehive (BH), Honeybees Mating Optimization 
(HBMO), Bee Colony Optimization (BCO), Artificial 
Bee Colony (ABC) and the Bees Algorithm (BA) [5].  
The Bees Algorithm (BA) is one of optimization 
technique that imitates the foraging behavior of honeybee 
in nature to solve optimization problems [6]. The main 
advantage of Bees Algorithm is has power and 
equilibrate in local search (exploitation) and 
global random search, (exploration), where both are 
completely decoupled, and can be clearly varied through 
the learning parameters [7]. It is very efficient in finding 
optimal solutions and overcoming the problem of local 
optima, easy to apply, and available for hybridization 
combination with other methods [5].  
The BA has been successfully applied in many  
different engineering problems, such as supply chain 

184 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
optimization [8], production scheduling [9], numerical 
functions optimization [5, 6, 7, 10], solving timetabling 
problems [11], control system tuning [12], protein 
conformation search [13], test form construction [14], 
Placements of FACTS devices [15], pattern recognition 
[16], robotic swarm coordination [17], data mining [18, 
19, 20], chemical process [21], mechanical design [22], 
wood defect classification [23], Printed Circuit Board 
(PCB) assembly optimization [24], image analysis [25], 
and many other applications [26]. 
The BA has attracted attention of many researchers 
in different fields since it has been proved to be efficient 
and robust optimization tool. It can be split up into four 
components: parameter tuning, initialization of 
population, exploitative neighborhood or local search 
(intensification) and exploratory global search 
(diversification) [26]. However, despite different 
successful applications of the BA, the algorithm has 
some limitations and weaknesses. The algorithm 
efficiency is much influenced by initialization of the 
different parameters that need to be tuned. Additionally, 
the BA suffers from slow convergence caused by many 
repeated iterations in local search [5].  
Many different investigations have been made to 
improve BA performance. Certain of these studies focus 
on the parameter tuning and setting component [27, 28]. 
Others developed different concepts and techniques for 
the local search neighborhood stage [5, 7], or global 
search stage [9] or for both the local and global stage 
[29]. Nonetheless, limited interest has been given to the 
improvement of the initialization stage.  
In order to deal with some weaknesses of BA, this 
paper considers different improvements based on other 
strategies and procedures. Hybridization of different 
techniques may improve the solutions quality and 
enhance the efficiency of the algorithms.  
The present work is an extension of the methods 
presented in [30]. The Firefly Algorithm (FA), which is 
swarm intelligence metaheuristic for optimization 
problems that based upon behavior and motion of 
fireflies [31], is applied to initialize the bee population of 
Basic BA for a best exploration of research space and 
start the search from more promising locations. FA is 
also introduced in the local search part of Basic BA in 
order to increase exploitation in each research zone. As a 
result, two BA variants called the Hybrid Firefly Bee 
Algorithm (HFBA) and Hybrid Bee Firefly Algorithm 
(HBAFA), respectively. 
We also investigate the behavior of local search and 
global search of the BA by introducing two strategies: 
memory scheme (local and global memories) to 
overcome repetition and unnecessarily spinning inside 
the same neighborhood and avoid revisiting previously 
visited sites, and adaptive neighborhood search 
procedure to jump from the sites of similar fitness values, 
thus improving the convergence speed of the BA to the 
optimal solution. This improved variant of BA is called, 
Bees Algorithm with Memory Scheme and Adaptive 
Neighborhood (BAMS-AN). 
The rest of this article is organized as follows. 
Section 2 provides a description of Basic BA, FA and 
three improved variants of BA with different strategies. 
Section 3 presents and discusses the computational 
simulations and results obtained on benchmark instances, 
while Section 4 presents our conclusions and highlights 
some suggestions and future research directions. 
2 Materials and methods 
2.1 The standard bees algorithm 
The Bees Algorithm (BA) is a bee swarm-based 
metaheuristic algorithm, which is derived from the food 
foraging behavior of honey bees in nature. It was 
originally proposed by Pham et al. [6]. 
The BA starts out by initializing the population of 
scouts randomly on the space of search. Then the fitness 
of the points (i.e. solutions) inspected by the scouts is 
evaluated. The scouts with the highest fitness are selected 
for neighborhood search (i.e. local search) as “selected 
bees” [6]. To avoid duplication, a neighborhood called a 
“flower patch” is created around every best solution; 
furthermore the forager bees are recruited and affected 
randomly within the neighborhoods. If one of the 
recruited bees lands on a patch of higher fitness than the 
scout, that recruited bee turns into the new scout and 
participates in the waggle dance in the next generation. 
This step is called local search or exploitation [7]. 
Finally, the remaining of the colony bees (i.e. population) 
is assigned around the space of search scouting in a 
random manner for new possible solutions. This activity 
is called global search (i.e. exploration). In order to 
establish the exploitation areas, these solutions with 
those of the new scouts are calculated with a number of 
better solutions being reserved for the succeeding 
learning cycle of the BA. This procedure is repeated in 
cycles until it is necessary to converge to the optimal 
global solution [5].  
The Bees Algorithm detects the most promising 
solutions, and explores in selective manner their 
neighborhoods to find the global minimum of the 
objective function. When the best solutions are selected, 
the BA in its basic version makes a good balance 
between a local (or exploitative) search and a global (or 
exploratory) search. The both develop random search [7]. 
The pseudo-code of the Standard Bees Algorithm is 
given below in Figure 1.  
A certain number of parameters are required for the 
BA, called: number of the scout bees or sites (n), (m) 
sites selected for local search among n sites, (e<m) elite 
sites chosen from the m selected site for a more intense 
neighborhood search, (nre) recruited bees for the elite 
sites, (nrb<nre) bees recruited for neighborhood search 
around other m-e best sites, initial size of all selected 
patch (ngh) around each scout for neighborhood search 
(i.e. flower patch), stopping criterion for the algorithm to 
terminate and number of iterations [5]. 
From the control settings of algorithm, the bees’ 
population size p can be determined as below: 
( ) nrb e m nre e ns p * * − + + =
     (1)      

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 185 
 
where; ns is (n – m) remaining bees. 
For random scouting in the initialization phase as 
well as for the unselected bees, the following equation is 
adopted: 
( )
min max min
* x x rand x x
rand
− + = (1) 
where; rand is a random vector element between 0 to 1, 
x max, x min are the upper and lower bound to the solution 
vector, respectively. 
Two additional steps are called when neighborhood 
search does not improve any fitness enhancement in a 
neighborhood: the neighborhood shrinking and the site 
abandonment. The primary objective is to enhance the 
computation performance and the search accuracy. This 
implementation is called the Standard Bees Algorithm 
(Standard-BA) [7]. 
2.1.1 Neighborhood shrinking strategy 
A large value is defined for the initial size of the 
neighborhoods. For each ai of a = {a1, …,an} is set as 
follows: 
( ) ( ) ( ) i i i t ngh t a min max − =  
( ) 0 . 1 0 = ngh (2) 
where t represents the t-th iteration of the BA main loop. 
The initial size of flower patches is on a wide area to 
promote the exploration of the search space. Whilst the 
local search procedure gives better fitness, the size of 
patches is maintained unchanged, and is then gradually 
reduced to yield the search more exploited round the 
local optimal [7]. The ultimate objective of the 
neighborhood shrinking approach is to make the search 
progressively concentrated in the proximity of the local 
peak of fitness. The updating formula is given as follows: 
( ) ( ) t ngh t ngh  = + 8 . 0 1 (3) 
A new variant of BA can be obtained by applying 
the shrinking procedure over Basic BA. This 
implementation is named the Shrinking-based BA [26]. It 
can be deducted from the first paper of Pham et al. [6]. 
2.1.2 Site abandonment strategy 
To enhance the efficiency of the local search, the site 
abandonment procedure is introduced, in which there is 
no more enhancement of the fitness value of the fittest 
bee after a predetermined number (stlim) of successive 
stagnation cycles. If the abandoned site corresponds to 
the fitness best value, the location of the peak is 
registered. If no other flower patch will generate a better 
value of fitness in the rest of the search, the better 
registered fitness position is taken as the final solution 
[7]. 
To sum up, in the literature review of Bees 
Algorithm, three important implementations could be 
discovered, which are Basic-BA, Shrinking-based-BA, 
and Standard-BA. 
Bees Algorithm (BA) 
Initialize population with random solutions by sending (n) scout bees.  
Iterate until a stopping criterion is met 
1. Evaluate the fitness of the population. 
2. Sort the solutions according to their fitness. 
// Waggle dance 
3. Select the highest-ranking m solutions (i.e. sites or patches) for neighborhood search. 
4. Determine the patch of each selected site with an initial size of (ngh init).  
5. Recruit nr = nre foragers to each of the e≤m top-ranking elite solutions. 
6. Recruit nr = nrb≤nre foragers to each of the remaining m – e selected solutions. 
// Local search 
7. For each of the m selected solutions 
a. Create nr new solutions by perturbing the selected solution randomly or otherwise and evaluate their fitness. 
b. Retain the best solution amongst the selected and new solutions. 
// Neighborhood Shrinking Strategy 
c. Adjust the neighborhood of the retained solution (i.e. Shrink patches whose neighborhood has achieved no improvement 
by a shrinking factor (sf)).  
// Site Abandonment Strategy 
d. Abandon sites where no progress has been made for a number (stlim) of consecutive iterations and re-initialize them 
(Save the location of the abandoned site if it represents the current best solution). 
// Global search 
8. Assign the remaining bees (n - m) to search randomly and evaluate their fitness.  
9. Form new population (m solutions from local search, n – m from global search) 
Figure 1: Pseudo code for the Standard Bees Algorithm (BA). 

186 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
2.2 The standard firefly algorithm 
Firefly Algorithm (FA) is a recent biologically-inspired 
algorithm originally introduced by [31]. FA imitates the 
communication behavior of tropical fireflies and the 
idealized flashing features of fireflies [31]. 
The primary strengths of FA are namely exploitation 
and exploration. In addition, the proposal time of this 
algorithm is relatively short, the parameters needing to be 
adjusted are quite few due to its simple structure and less 
complexity without requiring complex evolutionary 
operations; and the optimizing search ability is also 
relatively good [32].  
FA has been successfully applied to solve different 
optimization problems such as software testing [33], 
dynamic multidimensional knapsack problems [34], 
combinatorial problem [35], mobile robotics [36], 
network route selection [37], linear antenna array failure 
correction [38], optimized gray-scale image 
watermarking [39], automatic control system 
optimization [40, 41], price budget [42], multi-objective 
economic emission dispatch solution [43], and so on. 
Generally, it can get the optimal solution with its 
exploitation [44]. 
According to [32], two important things need to be 
defined in FA: the change of the attractiveness (β) and 
the formulation of the light intensity or brightness (I). 
The strength of attraction is proportional to the 
intensity of the brighter firefly and the distance between 
the two fireflies, so the brightness reduces with the 
distance from its source and the media will absorb the 
light. The light intensity of a firefly is given according to 
the value of the objective function. For simplicity, 
different features of fireflies are idealized into diverse 
rules detailed in [32]. The descriptive pseudo code of the 
FA is given in Figure 2. 
The attractiveness β between two fireflies can be 
determined by the following equation: 
( )
2
0
* exp r    − = (4) 
where; β0 is the attractiveness at r = 0, r is the distance of 
two fireflies and γ is the light absorption coefficient. γ is 
the most crucial parameter in FA method, it performs a 
very important role in determining the convergence 
speed and how FA behaves. Hence, nearly all the 
applications it differs from 0.1 to 10 [45]. 
The distance among any two fireflies i and j at xi and 
xj, respectively, calculated by the Euclidean or Cartesian 
distance formulation as follows: 
( )

=
− = − =
d
k
k j k i j i ij
x x x x r
1
2
, ,
 (5) 
Firefly i is attracted to another more attractive 
(brighter) firefly j and its movement is calculated by 
equation (7) [31]. In this equation the second part is the 
attraction, whilst the third is randomization with the 
control parameter α ∈ [0, 1], and εi is a vector of random 
numbers being generated from different distributions 
such as uniform distribution, Gaussian distribution and 
Lévy flight. The most basic form is εi could be changed 
by (rand − ½) where rand is a random real number 
between interval [0 1] [32]. It is interesting that (7) is a 
random-walk partial to the brighter fireflies. If β0 = 0, it 
turns into a simple random walk. Furthermore, the 
randomization term can easily be prolonged to different 
distributions such as Lévy flights [31]. 
( )
t
i
t
j
t
i
r
t
i
t
i
x x e x x
ij
 

+ − + =
−
+
2
0
1
 (6) 
2.3 The improved bees algorithms 
In this subsection, three improved variants of BA are 
presented: Bees Algorithm with Memory Scheme and 
Adaptive Neighborhood (BAMS-AN), Hybrid Firefly-
Bee Algorithm (HFBA) and Hybrid Bee Firefly 
Algorithm (HBAFA). 
2.3.1 Improved bees algorithm with memory 
scheme and adaptive neighborhood  
In this improved variant, two modifications of BA are 
introduced. The first one is to improve the BA in more 
efficient and natural manner with memory integration 
(local and global memories) to avoid revisiting 
previously visited sites and overcome repetition and 
unnecessarily spinning inside the same neighborhood. 
Thus, the search in the BA can prevent searching within 
infertile areas and can jump from potential local 
optimums. The second modification is to deploy an 
adaptive neighborhood search procedure to escape from 
local optima during the local search process. 
The memory is part of the honey bees’ nature, but 
was not included in basic BA. The strategy is to force the 
bees to stay away from the neighborhoods studied and 
experienced with no beneficial results. This will enforce 
the bees to visit different solutions for investigation. This 
needs the integration of a memory mechanism into the 
bees to allow them to respect the experiences and not 
waste their energy and time. This type of memory will be 
Firefly Algorithm (FA) 
Define    the    objective    function    of    f(x),    where 
x=(x1,........,xd)
T 
Generate initial population of fireflies xi (i = 1, 2,..., n) 
randomly. 
Determine the light intensity of Ii at xi via f(xi) 
Define α, β and γ 
While (t <MaxIterFA)  
For i = 1 :n (all n fireflies) 
For j = 1 : n (all n fireflies) 
If (Ii <Ij ), Move firefly i towards 
j by using equation (7); end if 
Vary attractiveness with distance 
r via exp[− γr] 
Evaluate new solutions and 
update light intensity 
Endfor j 
Endfor i 
Rank the fireflies according to light intensity and find 
the current global best 
End while 
Post-process results and visualization 
Figure 2: Pseudo code for the Firefly Algorithm (FA). 

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 187 
 
integrated to the scout and the follower bees where these 
bees could employ information and details about 
previously visited sites (i.e. positions and finesses). 
Different structures of arrays are used to save and handle 
the memory of each bee (scout and follower). The 
memory contains the information obtained during direct 
communication with the environment that used to decide 
the way of patch visiting based on a waggle dance or 
depend on it. This information is very important and 
affects the way forager bees (both scout and follower 
bees) choose the patch to visit and whether it will be 
based on a waggle dance or not (i.e., choosing a familiar 
food source). Foragers depend on their own memories to 
discover particular locations during the visit of food 
patches repeatedly [16, 19]. 
The memorized experiences are continuously 
updated and utilized to lead the further foraging of the 
bees, contributing to a more effective search procedure 
than basic BA. 
The pseudo-code below shows the procedures added. 
It is included in the local neighborhood search for 
follower bees and in the global search for scout bees. 
This procedure decides if the new position (in different 
dimensions) of any bee is close to its previous one 
recorded in its memory. 
where r is the radius, x
d
max and x
d
min are the upper and the 
lower bound to the solution vector respectively, itmax is 
the maximum number of iterations and ngh is the 
neighborhood size. 
The radius is the Euclidean distance from the field 
center to the border. It defines the size of field. It is 
applied so that different patches do not congregate in the 
same area. As can be seen from Figure 3, the value of 
radius r is based on the size of the solution space for 
scouts and the size of ngh for followers, which involves 
that r, will be greater than ngh for the scouts and less 
than ngh for followers. 
In addition, this enhanced variant of BA introduces 
an improvement to the Basic-BA by using two 
approaches simultaneously; adaptive change to the size 
of neighborhood of a selected patch and abandonment of 
site procedure, in the local search part. The neighborhood 
shrinking method bears similarities with Simulated 
Annealing procedure. The flower patches are initially set 
to a large region to foster the exploration and favor large 
“jumps” across the fitness landscape. Then, the 
neighborhood shrinking method is applied to a patch 
after predefined number of consecutive stagnation cycles. 
If shrinking the patch size does not result in any 
improvement in the local search after a predefined 
number of iterations, an enhancement procedure is 
applied. The enhancement procedure is applied for a 
number of consecutive iterations of no progress. Finally, 
after this number of iterations, the patch is assumed to be 
trapped within a local minimum (or peak) and it is 
abandoned, and the scout bee is randomly re-initialized. 
This process is known as site abandonment. If the 
abandoned location corresponds to the best-so-far 
solution, it is stored temporarily. If no better solution is 
subsequently found, the saved best solution is taken as 
the final one. The objective is to improve the local 
intensification, evade from local optima and speed the 
convergence to the optimal solution. 
2.3.2 Hybrid firefly-bee algorithm 
In initialization step of BA, the scout bees fly at random 
to initial resources. This stage is performed by 
distributing the scouts uniformly in random manner on 
search space. The initial position of scout bees relative to 
the optimal source may influence the level of optimality 
of other algorithm steps. Therefore, the population 
initialization of BA is critical and important factor 
because it can significantly influence the convergence 
speed to the optimal target, the quality of the resource 
found and the stability [26]. 
In the second variant of BA, an enhancement is 
introduced based on the Firefly Algorithm (FA) to 
improve the initialization stage of the BA. The proposed 
approach is called Hybrid Firefly Bee Algorithm 
(HFBA).  
Although the BA has a combination of both local 
and global search abilities, it nonetheless has some 
drawbacks such as a high level of randomness and 
processing time. The results generated by the standard 
BA are subject to the random initialization step. This 
may not contain a sufficient amount of information and 
details about the space surrounding. Nevertheless, due to 
its randomness, it is infeasible for random initialization 
to obtain a high-quality of initial solution regularly. It 
affects convergence speed, accuracy of final solution and 
stability. Practically, we should be searching in all 
directions simultaneously; put differently, initial 
solutions must be distributed uniformly through the 
overall search space. Hence, to obtain a better view of 
the surroundings of all random positions visited by each 
scout bee and to begin a local search from more 
promising sites, a FA search step is introduced during the 
first scouting procedure. The aim of the FA is to enhance 
the performance of the initialization stage of the BA, 
which will offer a better exploration of the search-space 
and empower the global performance of the method. 
High exploration would provide a high chance to obtain a 
solution which is close to global optima. FA based search 
process encourages the fireflies to look for new areas 
including some lesser explored areas and improves the 
fireflies’ ability to explore the large search space. In 
other words, the FA not only integrates the self-
If the bee is a scout then 
r = x1 * x
d
max - x
d
min; 
Else if the bee is a follower bee then  
r = x2 * ngh; 
End If 
While (number_iterations<itmax) do 
If the new position is close to its previous position 
recorded in the private memory (by using the Euclidian 
distance) then 
Find a new position randomly; 
End If 
End While 
Figure 3: The pseudo code of the procedures added. 

188 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
improving approach with the current search space, but it 
also integrates the enhancement among its own space 
from the prior phases [30, 44]. This contributes to the 
discovery of better fitness valued patches from where the 
neighborhood search will be performed since if the 
method begins its search from a promising location, it is 
evident that it will have better chance to converge to the 
global optima. In addition, FA has fast convergence 
capability.  
In the proposed algorithm, FA explores the search 
place to either isolate the more advantageous region of 
the search space, and then to escape from trapping into 
local optima and enhance global search, it is introduced 
BA to explore search space (starting with the best 
solutions found by FA) and discover new better 
solutions. The pseudo-code and flow chart of the 
proposed hybrid HFBA are given in Figures 4 and 5.  
The main steps of the novel hybrid algorithm 
(HFBA) can be summarized as follows: First, The initial 
population of n fireflies should be generated, and then, 
each firefly ought to be assigned the random position and 
the fitness (light intensity) for that position should also 
be calculated. Then, a new function is introduced to 
increase the convergence by calculating the step α as 
mentioned in Equation (8). This additional function is 
designated to modify the basic value of α parameter and 
design a dynamic adjusting scheme. The idea to 
minimize randomness is to maximize the algorithm’s 
search capability [30, 46]. The step α can be determined 
as following: 
( ) ( ) ( ) ( ) 3 015 . 0 exp 1 4 . 0 MaxIterFA t t −  + = 
(8)   
where; t and MaxIterFA represent current and maximum 
number of iterations, respectively. The next step is the 
process of moving each firefly in research space towards 
other brighter and attractive firefly. The position of each 
firefly is updated by Equation (7). The research process 
of firefly is controlled by the maximum number of 
iterations MaxIterFA. When the main loop of FA ends, 
the n best results generated from FA are communicated 
to the BA as an initial population of n scout bees. Thus, 
the BA will start with a population, and the rest of the 
method behaves like the standard (normal) BA algorithm 
in Figure 1. 
To sum up, the main difference between the novel 
proposed HFBA and the Standard BA lies in the first step 
of initialization, with the introduction of a new 
mechanism (FA) for generating the initial population of 
scout bees. Initialization phase of the improved BA is 
controlled by firefly fitness and optimized by α and 
MaxIterFA, which are important HFBA control 
parameters. 
2.3.3 Hybrid bee firefly algorithm 
In the third improved variant of BA, the FA algorithm is 
called to improve the local search of Bees Algorithm. We 
name this proposed algorithm by Hybrid Bee Firefly 
Algorithm (HBAFA). 
Disadvantage of BA is that it can be relatively weak 
in neighborhood search activities, and it suffers from 
slow convergence phenomena caused by the repetition of 
unneeded similar process in the local search. The 
repetitive iteration of the algorithm triggers a 
supplementary computational time in producing the 
solution. This makes the whole process slow. To 
overcome these problems, the intelligent algorithm FA is 
introduced, so as to increase the speed of convergence 
and avoid running the method with additional iteration 
without any improvements. 
The basic idea behind the proposed HBAFA is to 
introduce a FA search strategy into local search of BA. 
The main difference between the Standard BA and the 
hybrid HBA-FA is the manner of how the different 
selected patches types to be exploited. The Standard BA 
employs random mutations as the only means of 
neighborhood search, whereas in HBAFA, FA has been 
applied to improve the local search ability of BA and 
reconfigures the neighborhood dance search as a FA 
search. FA enhances the capability to exploit the whole 
local search space in order to either isolate the most 
promising solution of each selected site. Figure 6 shows 
the flow chart of the Hybrid HBAFA. The pseudo code 
of the improved local search part of BA is presented in 
Figure 7.  
The general steps of HBAFA algorithm are as 
follows. The proposed algorithm uses the Standard BA as 
a core, where FA works as a local search to refine the 
best sites found by the scout bees. Firstly, the best sites 
of BA are given and the control parameters of FA are set. 
Then, the FA algorithm starts its search process in each 
best site and it is run (as a normal FA algorithm in 
Figure. 2) until its stopping criterion is met. The initial 
population of nr fireflies (i.e. nre fireflies per site for elite 
sites and nrb fireflies per site for best sites) should be 
generated and then, each firefly ought to be assigned the 
random position in the neighborhood of each selected 
site, and the fitness (light intensity) for that position 
should also be calculated. The next step is the process of 
moving each firefly in research space of each selected 
site towards other brighter and attractive fireflies. The 
position of each firefly is updated by equation (7). The 
value of the parameter alpha (α) impacts the algorithm 
convergence. Therefore, the initial value of step size α is 
modified and reduced according to equation (8) to 
maximize the algorithm’s search [30]. 
The FA search process is essential because there 
might be better solutions than the original solution in the 
neighborhood region. The research procedure of FA is 
controlled by the maximum number of iterations 
MaxIterFA. When the main loop of FA ends, it returns a 
single global best position. If it lands in a position of 
higher fitness than the scout bee of the selected site, that 
forager is maintained as the new scout bee. When local 
search does not improve any fitness enhancement in a 
neighborhood, the neighborhood shrinking and the site 
abandonment procedures are called. 
 

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 189 
 
  
 
Figure 4: Flow chart of Hybrid Firefly-Bee Algorithm (HFBA). 
Assign the (n–m) Remaining Bees to 
Random Search 
 
Fitness Evaluation 
Random initialisation of n Scout Bees = 
Final Best population of Fireflies 
Fitness Evaluation 
Local Search 
Global Search 
New Population of Scout Bees 
Stop? 
Solution 
Sort the initial population 
 
Generate n random fireflies 
Calculate light intensity 
Calculate the relative distance and 
attractiveness between each couple of 
fireflies in the population 
Update the positions of fireflies 
Evaluate new solutions and update light 
intensity 
Reduce alpha 
Rank the fireflies and find the current 
best 
Stop? 
Final Best population of Fireflies 
Selection 
 
Elite Site (e)  
- nre bees per patch 
 
Best Site (m - e)  
- nrb bees per patch 
 
Fitness Evaluation 
 
Fitness Evaluation 
 
Select Patch Fittest 
 
Select Patch Fittest 
 
Initialisation Phase by Firefly Algorithm 
 
Note: Dotted box is Firefly Algorithm. 
 

190 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
  
Hybrid Firefly-Bees Algorithm (HFBA) 
Define the objective function of f(x), where x=(x1,........,xd)
T 
Define the FA parameters α, β, γ andMaxItrFA 
Define the BA parameters n, m, e, nre, nrb, nghinit, sf and stlim 
Begin FA 
Generate initial population of fireflies xi (i = 1, 2,..., n) randomly. 
Determine the light intensity of Ii at xi via f(xi) 
While (t <MaxIterFA)  
For i = 1 : n (all n fireflies) 
For j = 1 : n (all n fireflies) 
If (Ii <Ij ), Move firefly i towards j by using equation (7); end if 
Vary attractiveness with distance r via exp[− γr] 
Evaluate new solutions and update light intensity 
Endfor j 
Endfor i 
Reduce a lp h a (α ) according to Eq. (8)   
Rank the fireflies according to light intensity and find the current global best 
End while 
Final Best population of n Fireflies 
End begin FA 
Begin BA 
// Initialize n scout bees for BA based on the best solution found by FA 
Initialize population of BA = Final Best population of n Fireflies found by FA 
While (stopping criterion not met) 
Evaluate the fitness of the population. 
Sort the solutions according to their fitness. 
// Waggle dance 
Select the highest-ranking m solutions (i.e. sites or patches) for neighborhood search. 
Determine the patch of each selected site with an initial size of (nghinit).  
Recruit nr = nreforagers to each of the e≤m top-ranking elite solutions. 
Recruit nr = nrb≤nre foragers to each of the remaining m – e selected solutions. 
// Local search 
For each of the m selected solutions 
Create nr new solutions by perturbing the selected solution randomly or otherwise and evaluate their fitness. 
Retain the best solution amongst the selected and new solutions. 
Adjust the neighborhood of the retained solution (i.e. Shrink patches whose neighborhood has achieved no improvement 
by a shrinking factor (sf)). 
Abandon sites where no progress has been made for a number (stlim) of consecutive iterations and re-initialize them 
(Save the location of the abandoned site if it represents the current best solution). 
// Global search 
Assign the remaining bees (n - m) to search randomly and evaluate their fitness.  
Form new population (m solutions from local search, n – m from global search) 
End while 
End begin BA 
Figure 5: Pseudo code of Hybrid Firefly-Bee Algorithm (HFBA). 
 
Figure 6: Flow chart of Hybrid Bee Firefly Algorithm (HBAFA). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Note: Dotted box is improved local search of BA by FA. 
Random initialisation of n 
Scout Bees 
Fitness Evaluation 
Local Search 
Global Search 
New Population of Scout Bees 
Stop? 
 
Select m sites for neighbourhood search 
Generate initial population for each Site 
- nre fireflies per site for Elite sites (e) 
- nrb fireflies per site for Best sites (m-e) 
Calculate the relative distance and 
attractiveness between each couple of 
fireflies in the population 
Update the positions of fireflies 
Evaluate new solutions and update light 
intensity 
Reduce alpha 
Rank the fireflies and find the current best 
Stop? 
 
Select Site Fittest 
Neighbourhood Search Phase by Firefly Algorithm 
Solution 
Sort the initial population 
Assign the (n–m) Remaining Bees to 
Random Search 
 
Fitness Evaluation 

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 191 
 
3 Experimental setup 
3.1 Benchmark functions 
To measure the performance of the improved BAs, some 
well known continuous type benchmark problems were 
selected. Each of these functions has its own properties 
whether it unimodal, multimodal, separable, non-
separable, and multidimensional, so obtained results 
illustrate strengths and weaknesses of the algorithm in 
different situations. 
Some functions are unimodal which mean there is 
only one global optimum. It can easily locate the point; 
thus they are useful for evaluating the exploitation ability 
of an optimization algorithm. If a function has two or 
more local optima, this function is called multimodal. 
Multimodal functions are suitable for benchmarking the 
global exploration capability of an optimization 
algorithm. If the exploration operation of a method is 
mediocre that it cannot perform an efficient search in 
whole space, this method gets trapped in local minima. 
Multimodal benchmarks are difficult for algorithms as 
well as benchmarks having flat surfaces because the 
flatness of the benchmark does not offer the algorithm 
any kind of information to lead the search process 
towards the minima. A function is regarded as separable 
if it can be expressed as a sum of function just from one 
variable. Non-separable benchmarks cannot be expressed 
in this form. Consequently, non-separable benchmarks 
are considerably more difficult compared to the separable 
benchmarks. Meanwhile, dimensionality of the search 
area reflects the number of parameters to be optimized. It 
is an essential characteristic that determines the difficulty 
of the problem [4]. 
The Martin and Gaddy and Hypersphere are 
unimodal benchmarks which are fairly simple for 
optimization tasks. The Easom function is a unimodal 
test function. It is characterized by a single narrow mode 
and the global minima have a small region relative to the 
search space. The Goldstein & Price is a multimodal 
function (i.e., including multiple local optima, but just 
one global optimum) and it is easy optimization task with 
only two variables. Schaffer benchmark is symmetric 
complex multimodal function that has a global optimum 
very close to a local optimum. The Rosenbrock function 
is complex and unimodal and the minimum is at the 
lowest part of a long, narrow and parabolic shaped 
valley. To find the valley is trivial and it can be easily 
discovered with a few repetitions of an optimization 
methods, however, to locate the exact global minima is 
quite difficult. Rastrigin, Griewank and Ackley have an 
overall unimodal behavior and local multi-modal pockets 
created by a cosinusoidal “noise” component. The 
Schwefel function is well-known to be a complex and 
tough problem. It is a multimodal function and the global 
minimum is geometrically distant from the second best 
local minima. Consequently, the optimization algorithms 
are potentially susceptible to convergence in the incorrect 
direction. Thus, the algorithm never gets the same value 
on the same position. Algorithms that fail to tackle this 
function will do poorly in real world problems with noisy 
data. The combination of these characteristics determines 
the complexity of the continuous functions.  
// Improved Local Search phase of BA 
7. For each of the m selected solutions 
a. // Perform neighborhood search using Firefly Algorithm 
Generate initial population of fireflies xi(i = 1, 2,…, nr) randomly (i.e. nre fireflies per site for elite sites and nrb fireflies per site 
for best sites).  
Determine the light intensity of Ii at xi via the objective function f(xi) 
Define α, β and γ 
While (t <MaxGeneration)  
For i = 1 :nr (all nr fireflies) 
For j = 1 : nr (all nr fireflies) 
If (Ii <Ij), Move firefly i towards j by using equation (7); end if 
Vary attractiveness with distance r via exp[− γr] 
Evaluate new solutions and update light intensity 
Endfor j 
Endfor i 
Reduce alpha (α) according to Eq. (8) 
Rank the fireflies according to light intensity and find the current global best 
End while 
Output optimum solution value and its locate 
b. Retain the best solution amongst the selected and new optimum solution. 
c. Adjust the neighborhood of the retained solution (i.e. Shrink patches whose neighborhood has achieved no improvement 
by a shrinking factor (sf)). 
d. Abandon sites where no progress has been made for a number (stlim) of consecutive iterations and re-initialize them 
(Save the location of the abandoned site if it represents the current best solution). 
Figure 7: Pseudo code of the Improved Local Search part of BA. 

192 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
Interval, full equation, dimensions, theory global 
optimal solutions and properties of these functions are 
shown in Table 1 as reported in [4]. Readers are referred 
to [47] for more details on the benchmark functions used 
in present investigation and its characteristics. 
3.2 Performance measures 
Performance assessment of the different algorithms is 
based on two metrics: namely, the accuracy and the 
average evaluation numbers and results were compared 
to Standard Bees Algorithm (BA), Firefly Algorithm 
(FA), and other well known optimization techniques such 
as Particle Swarm Optimization (PSO), Evolutionary 
Algorithm (EA). These are given in Tables 3, 4 and 5. 
The accuracy of algorithms (E) was defined as the 
difference of the fittest solution obtained by algorithms 
and the value of the global optimum. It was chosen as a 
performance indicator to ascertain the quality of the 
solutions obtained. According to this approach, the more 
accurate results are closer to zero. However, the number 
of function evaluations (NFE), which is the number of 
times any benchmark problem had to be assessed to 
grasp the optimal solution, was used to evaluate the 
convergence speed. If the algorithm could not find E less 
than 0.001, the final fitness value is recorded with 
maximum function evaluations. Lower value of NFE for 
a method on a problem means the method is faster in 
solving that problem. 
The process will run until stopping criteria are met. 
Each time, the optimization algorithm is run until the 
accuracy (error) is less than 0.001, or the maximum 
number of iterations (5000) is elapsed. For each 
configuration, 50 independent minimization trials are 
performed on each benchmark function. 
3.3 Parameters settings 
In order to obtain a reliable and fair comparison amongst 
standard and novel improved BA, the same parameters 
and values were utilized for all benchmarks to achieve 
acceptable results within the required tolerance without 
careful tuning, except for the parameter α, β and γ which 
were only set for the two variants of BA based on FA 
and FA. The Firefly Algorithm (FA) was implemented in 
this study according to the method described in [31]. The 
parameter configuration recommended by [32] is used. 
The parameters of the implemented BA used in this 
paper have been empirically tuned and the optimal 
parameter settings are used to find the optimal solution. 
The Standard BA implemented in this work is called 
BA1. 
The simulation results of our experiment are 
compared with ones reported in [7]. This comparison was 
carried out between the BA1, FA and the proposed 
No Benchmark Function Min Interval P 
1 Easom 2D 
( ) ( )
( ) ( )  
2
2
2
1
2 1
cos cos min
  − − −
− =
x x
e x x F 
0 [-100, 100] UN 
2 
Goldstein & 
Price 2D 
( ) ( )  
2
2
2 2 1 2
2
1 1
2
2 1
3 6 14 3 14 19 1 1 min x x x x x x x x F + + − + − + + + = 
( ) ( )  
2
2 2 1 2
2
1 1
2
2 1
27 36 48 12 32 18 3 2 30 x x x x x x x x + − + + − − +  
3 [-2, 2] MN 
3 
Martin & Gaddy 
2D 
( ) ( ) ( )
2
2 1
2
2 1
3 / 10 min − + + − = x x x x F 0 [-20, 20]  
4 Schaffer 2D 
( )
( )  
2
2
2
2
1
2
2
2
2
1
001 . 0 0 . 1
5 . 0 sin
5 . 0 min
x x
x x
F
+ +
− +
+ = 
0 [-100, 100] MN 
5 Schwefel 2D ( ) ( )

=
− =
2
1
1
sin min
i
i
x x F 0 [-500, 500] MS 
6 Ackley 10D 
( )
e e e F
i
i
i
i
X X
+ +

−

− =
= =
−
20 20 min
10
2 cos
10
2 . 0
10
1
10
1
2

 
0 [-32, 32] MN 
7 Griewank 10D ( )
 
= =






+
−
− − =
10
0
10
0
2
1
100
cos 100
4000
1
min
i i
i
i
i
x
x F 0 [-600, 600] MN 
8 Hypersphere 10D 

=
=
10
1
2
min
i
i
x F 0 [-100, 100] US 
9 Rastrigin 10D ( ) ( )

=
− + =
10
1
2
2 cos 10 100 min
i
i i
x x F  0 [-5.12, 5.12] MS 
10 Rosenbrock 10D ( ) ( )  

=
+
− + − =
10
1
2
2
1
2
1 100 min
i
i i i
x x x F 0 [-50, 50] UN 
Table 1: Summary characteristics of test functions used. D: Dimension, Min: Global Minimum, P: Properties, U: 
Unimodal, M: Multimodal, S: Separable, N: Non-Separable. 

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 193 
 
variants of BA with the standard BA introduced in [7] 
(noted by BA2) and other well-known optimization 
techniques such as EA and PSO. Pham and Castellani 
(2009) were analyzed in [7] the learning results of EA, 
PSO and BA2 algorithms, and the parameter setting that 
gives the most consistent performance across all the 
benchmarks was found for each algorithm. The 
parameter settings of the algorithms are given in Table 2. 
Given an optimization algorithm, the No Free Lunch 
Theorem of optimization (NFLT) entails that there is no 
universal tuning that guarantees top performance on all 
possible optimization problems [48]. 
The algorithms developed in this study were 
implemented using Octave programming language and 
all experiments were performed on Intel Core i3-370M 
2.4 GHz and 4 GB RAM running a 64-bit operating 
system. 
4 Results and discussion 
In this work, initialization stage, local search and global 
search parts of BA were investigated. The attention was 
on improving the performance of the BA by increasing 
the accuracy and the speed of the search. 
The comparisons were carried out between improved 
variant of BAs (BAMS-AN, HFBA and HBAFA), the 
standard BA (BA1) and FA. Then, the performance of 
these algorithms is compared against other well-known 
optimization techniques presented in [7] such as standard 
BA (BA2), EA and PSO. The same stopping criterion is 
used for all the algorithms (see Section. 3.2). 
The Average accuracy (μE) and their standard 
deviation (σE), and the mean numbers of function 
evaluations (Mean) and standard deviation (Std) for 50 
runs are compared in Tables 3 and 4, respectively. Bold 
values represent the best results. The second best results 
are in italics. Additionally, Table 5 displays the 
percentage of improvements in term of reducing the 
number of evaluations (NFE) when comparing the 
variants BAs with FA and the Standard BA. 
First, we compare the performance of our 
implemented BA (BA1) to the Standard BA exposed by 
Pham and Castellani (BA2) in [7]. The two algorithms 
use different combinations of parameters (Table 2). After 
finishing the simulation, it is found from Table 3 that the 
two parameter combinations are capable to find the 
global optimum on 7 cases out of 10 benchmark 
functions. For the rest, the average accuracy (μE) found 
for the Rastrigin function with 10 dimensions was better 
for the BA1 compared with the BA2 with 7.8601 against 
8.8201 respectively. However, BA2 performed slightly 
better than BA1 on Rosenbrock 10D with 0.0293 against 
0.1093, respectively. For Griewank 10D, the result is 
almost slightly the same for both. 
In order to examine and compare the convergence 
speed of these algorithms, the average number of 
function evaluations (Mean) is considered. On the other 
hand, evaluate which parameter combination is better 
than another. Parameter combination with the minimum 
number of function evaluations achieving global 
optimum over all benchmarks and all dimensions is 
distinguished as the best. It is immediately obvious from 
Table 4, that BA1 outperformed BA2, except for 
Schaffer 2D and Griewank 10D functions. Hence, the 
combination of parameters used for our implementation 
of BA (BA1) algorithm gave the best results in terms of 
number of function evaluations. For this reason, it has 
been selected for the proposed variants of BA. This 
suggests that if a user faces a problem, this parameter 
combination ought to be used as default setting. 
The second comparison of performances is achieved 
between the improved algorithms (BAMS-AN, HBAFA, 
HFBA), BA1 and FA and others well-known 
optimization algorithms EA, PSO and BA2. 
Algorithm Parameters Value 
BA1, 
BAMS-AN, 
HBAFA, 
HFBA -   
Common 
Parameters 
Scout bees (n) 
Elite sites (e) 
Best sites (m) 
Recruited elite (nre) 
Recruited best (nrb) 
Stagnation limit (stlim) 
Neighborhood size (nghinit) 
 
Shrinking factor (sf) 
Evolution cycles (ec) 
12 
2 
6 
29 
9 
10 
(Search 
range)/2 
0.8 
5000 
FA, HFBA, 
HBAFA - 
Common 
Parameters 
Population size 
Initial attractiveness (β0) 
Minimum value of beta (βmin) 
Light absorption coefficient (γ) 
Control parameter (α) 
FA cycles (MaxIterFA)   
40 
1 
0.2 
1 
0.2 
12500 
HFBA & 
HBAFA 
FA cycles in inner loop of 
HFBA 
FA cycles in inner loop of 
HBA-FA 
[3 10] 
 
5 
BA2 [7] Scout bees (n) 
Elite sites (e) 
Best sites (m) 
Recruited elite (nre) 
Recruited best (nrb) 
Stagnation limit (stlim) 
Neighborhood size (nghinit) 
 
Shrinking factor (sf) 
Evolution cycles (ec) 
11 
2 
6 
30 
10 
10 
Not 
presented 
0.8 
5000 
EA [7] Population size 
Evolution cycles (max number) 
Children per generation 
Mutation rate (variables) 
Mutation rate (mutation width) 
Initial mutation interval width a 
(variables) 
Initial mutation interval width p 
(mutation width) 
100 
5000 
99 
0.8 
0.8 
0.1 
 
0.1 
PSO [7] Population size 
Connectivity (no. of neighbors) 
Maximum particle velocity (u) 
c1 
c2 
wmax 
wmin 
PSO cycles 
100 
2 
0.05 
2.0 
2.0 
0.9 
0.4 
5000 
Table 2: Parameters setting of the algorithms. 

194 Informatica 44 (2020) 183–198  M.A. Nemmich et al. 
 
The algorithms are tested first for accuracy, and 
compared on the benchmark functions. According   to 
Table 3, BA and its variants found the minimum results 
for the most of the functions with good standard 
deviations. BAMS-AN is the best performing approach 
which have been successful in achieving the minimum 
error in 8 functions out of 10 benchmarks followed by 
standard HBAFA, HFBA and BA in 7 out of 10 
functions. Each of PSO, EA and FA has been capable of 
obtaining the theoretical value of 0 in, respectively, 6, 5 
and 4 functions out of 10 benchmarks. 
BAMS-AN did not reach the minimum average 
accuracy only in Schwefel 2D, Rastrigin 10D functions 
while the other variants of BA (Standard BA, HBAFA 
and HFBA) are not capable of finding function minimum 
on Griewank, Rastrigin and Rosenbrock benchmarks 
with 10 dimensions. PSO cannot find the global 
minimum for Schwefel 2D, Rastrigin 10D, Griewank 
10D and Rosenbrock 10D functions. In addition to these 
benchmarks, EA is not capable of finding function 
minimum on Schaffer 2D benchmark and FA on Easom 
2D, Schwefel 2D and Ackley 10D benchmark functions. 
All BA algorithms managed to achieve the 
theoretical optimal value with the Schwefel 2D function 
except for BAMS-AN where the mean value obtained is 
0.0003 which is very close to the theoretical optimum 
value. However, BAMS-AN can find the optimal values 
for Griewank 10D and Rosenbrock10D functions or near 
optimal value for Rastrigin 10D function with 0.0002, 
while the other variants of BA and other algorithms fail. 
HBAFA comes in second place (after BAMS-AN) 
and outperformed other compared methods (EA, PSO, 
FA, BA1, BA2 and HFBA) when solving the 
Rosenbrock 10D function with 0.0234 which is closest to 
the theoretical optimum value of 0, followed by HFBA 
with 0.1062. BA2 and BA1 become the third and the 
fourth best algorithm with 0.0241 and 0.1093, 
respectively. However, the results found using the FA 
and EA were not good and far from the theoretical 
optimal value (0.00) with 10.1469 and 61.5213, 
respectively. The Rosenbrock 10D benchmark was hard 
for the EA and the FA algorithms. 
The Rastrigin 10D function proved to be difficult 
tasks, particularly for the EA and none of the algorithms 
achieved the best minimization performance. HBAFA 
has become the second best algorithm after BAMS-AN 
algorithm with 2.9848 and PSO has become the third 
best with 4.8162. The best fourth one is FA where it 
managed to achieve 6.0693. 
The second best results for the Griewank benchmark 
function in 10 dimensions are when applying BA2 with 
0.0089. HFBA, HBAFA and BA1 perform almost the 
same on Griewank 10D function with 0.0120 but slightly 
better than FA with 0.0178.   
The results in Goldstein & Price 2D, Martin & 
Gaddy 2D and Hypersphere 10D functions indicated that 
all algorithms have achieved the optimal values. 
Therefore, it is noticeable that BA, HBAFA and HFBA 
share the only performing approaches in Schwefel 2D 
function where all of them managed to find the 
theoretical optimal solution.  
For Easom 2D function, all algorithms managed to 
achieve the theoretical optimal value, except for FA 
where the mean value obtained is -0.7996. 
Besides the average error, the assessment of the 
performance involves also the average number of 
evaluations. To compare the convergence speed of the 
algorithms (FA, BA1, BAMS-AN, HBAFA and HFBA), 
we calculated and compared the means and standard 
deviation of number of evaluations (Mean and Std.) 
generated by each algorithm after 50 runs.  
It can be clearly observed in Table 4 that the 
implementation of BA that utilizes the memory scheme 
and adaptive neighborhood search (BAMS-AN) achieved 
the smallest expected numbers of function evaluations 
(the smallest values of Mean) on 6 out of the 10 tested 
problems, followed by the modified BA by improving 
the local search part with FA (HBAFA) on 4 cases out of 
10. The proposed BAMS-AN performed significantly 
better than the other methods on most of the test 
functions such as Easom 2D, Goldstein & Price 2D, 
Griewank 10D, Hypersphere 10D, Rastrigin 10D and 
Rosenbrock 10D (see Table 4). However, the results 
found from Martin & Gaddy 2D, Schaffer 2D, Schwefel 
2D and Ackley 10D using the HBAFA algorithm were 
better than the proposed BAMS-AN, basic BA and other 
approaches. Thus, it can be concluded that BAMS-AN 
and HBAFA algorithms were able to converge to the 
optimal solution much faster than HFBA, BA, FA and 
other approaches.  
The third best result is for HFBA algorithm which 
was capable to reduce NFE and performed better in 8 and 
7 out of 10 benchmark functions compared to BA1 and 
BA2, respectively. BA2 produces better results than 
HFBA on Schwefel 2D, Griewank 10D and Rosenbrock 
10D benchmarks.  In the meantime, BA1 is selected as 
the best performing method in only Schwefel 2D and 
Griewank 10D functions compared to HFBA algorithm.  
Comparing HFBA, BA1 and with FA, PSO and EA 
algorithms, it is apparent that on the most benchmarks, 
the majority of the results for HFBA and BA were found 
to be better than the results of the other algorithms, 
except for Rastrigin 10D function where PSO followed 
by FA excelled better. 
On the whole, BAMS-AN and HBAFA can be 
classified as the first and the second best performers in 
terms of reduced number of evaluations, respectively, 
followed by HFBA and BA. From the observed Table, it 
is also obvious that PSO, EA and FA have a fairly low 
convergence rate during the entire process compared 
with BA variants but PSO performs slightly better than 
EA and FA. 
Table 5 presents the percentage of improvements in 
term of reducing NFE when comparing the enhanced 
variants of BA with the Standard BA.  
Based on these results, BAMS-AN is the most 
efficient approach. If NFE of all the functions were 
totaled, BAMS-AN is capable to minimize the total NFE 
to 86.39% and 52.62% compared to FA and Standard 
BA, respectively. The improvement percentage shows 
the superiority of the BAMS-AN algorithm and how the 
integration of memory scheme in local and global 

Hybrid Bees Approach Based on Improved Search Sites ... Informatica 44 (2020) 183–198 195 
 
searches with adaptive neighborhood search procedure in 
basic BA affects the results. The second best algorithm is 
HBAFA with the average percentage improvement of 
67.84% and 23.94% compared to FA and BA, 
respectively. This variant of BA uses FA to improve the 
local search stage of Basic BA. On average, the 
capability to reduce the total NFE when applying Firefly 
Algorithm in initialization step of BA is 64.42% and 
3.95% compared to FA and Standard BA respectively, 
which ranks HFBA in the third place. 
Therefore, the enhanced variants of BA delivered a 
highly significant improvement in terms of overall 
performance. An interesting finding is that the introduced 
strategies and procedures helped BA to converge to good 
solutions quickly and robustly. 
No Benchmark EA PSO FA BA2 BA1 BAMS-AN HFBA HBAFA 
  μE σE μE σE μE σE μE σE μE σE μE σE μE σE μE σE 
1 Easom 2D 0.0000 0.0000 0.0000 0.0000 -0.7996 0.4039 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 
2 Goldstein & 
Price 2D 
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 
3 Martin &Gaddy 
2D 
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 
4 Schaffer 2D 0.0009 0.0025 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 
5 Schwefel 2D 9.4751 32.4579 4.7376 23.4448 59.2194 64.4283 0.0000 0.0000 0.0000 0.0000 0.0003 0.0007 0.0000 0.0000 0.0000 0.0000 
6 Ackley 10D 0.0000 0.0000 0.0000 0.0000 0.0010 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 0.0000 0.0000 
7 Griewank 10D 0.0210 0.0130 0.0199 0.0097 0.0178 0.0223 0.0089 0.0059 0.0120 0.0081 0.0000 0.0001 0.0120 0.0075 0.0127 0.0080 
8 Hypersphere 10D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 
9 Rastrigin 10D 17.4913 7.3365 4.8162 1.4686 6.0692 2.7579 8.8201 2.2118 7.8601 2.1010 0.0002 0.0004 7.2632 2.1010 2.9848 0.8287 
10 Rosenbrock 10D 61.5213 132.6307 1.7879 1.5473 10.1469 39.2052 0.0293 0.0068 0.1093 0.5627 0.0000 0.0003 0.1062 0.5637 0.0234 0.0022 
Table 3: Accuracy of improved BA algorithms compared with FA and BA and other well-known optimization techniques. 
No Benchmark EA PSO FA BA2 BA1 BAMS-AN HFBA HBAFA 
  Mean Std. Mean Std. Mean Std. Mean Std. Mean Std. Mean Std. Mean Std. Mean Std. 
1 Easom 2D 36440 28121 97136 45642 189158.40 159102.21 3866 819 3326.9 415.2 2126 2260 3316.90 469.69 2658 858 
2 Goldstein & 
Price 2D 
5816 2259 4836 2361 57360 26870.34 2714 454 2658.4 383.4 1 522 660 2568.4 372.1 2332 479 
3 Martin 
&Gaddy 2D 
3248 1602 2512 781 12729.60 9644.23 2248 329 2180.3 363.7 2130 3034 1992.8 585.4 880 532 
4 Schaffer 2D 219376 183373 35474 27151 121040.80 100516.67 27890 27335 41472.7 38237.9 12618 4330 37077.2 34805.7 10298 11777 
5 Schwefel 
2D 
51468 133632 84572 90373 301726.40 193824.47 5006 2110 3963.2 590.1 124108 0 4027.1 397.9 2390 1080 
6 Ackley 10D 50344 3949 261608 9165 497459.20 4413.18 12186 3553 11836.2 3532.3 18398 7302 10402.2 516.4 9938 422 
7 Griewank 
10D 
490792 65110 497714 16164 474366.40 43760.27 447064 126512 460894.8 102457.2 245 426,34 198326,6 470787.2 118.6821 455307 127061 
8 Hypersphere 
10D 
36376 2736 223082 10872 356958.40 6627.86 8288 403 8212 425 5944 3926 7670.02 384.4 6962 332 
9 Rastrigin 
10D 
500000 0 500000 0 500000 0.00 500000 0.00 500180 9.9 15 118 14186 500300 7.92 500000 0.00 
10 Rosenbrock 
10D 
500000 0 500000 0 500000.00 0.00 500000 0 500000 0 19 642 6666 491924.1 57954.4 500000 0.00 
Table 4: Mean number of function evaluations of improved BA algorithms compared with FA and BA  
and other well-known optimization techniques. 

196 Informatica 44 (2020) 183–198 M.A. Nemmich et al. 
 
5 Conclusion 
In this paper, enhancements to the Bees Algorithm (BA) 
have been presented for solving continuous optimization 
problems. The basic BA was modified first to find the 
most promising patches, by using memory scheme in 
order to avoid revisiting previously visited sites, thus 
increasing the accuracy and the speed of the search, 
followed by adaptive neighborhood search procedure to 
escape from local optima during the local search process. 
This implementation is called BAMS-AN. The second 
improved version of BA, called HBAFA, uses Firefly 
Algorithm (FA) to update the positions of recruited bees 
in the local search part of BA, thus improving the 
convergence speed of the BA to good solutions. The 
third variant of BA, i.e. HFBA, developed a new strategy 
based on Firefly Algorithm (FA) to initialize the 
population of bees in the Basic BA, enhance the 
population diversity and start the search from more 
promising locations.  
We evaluated the improved algorithms on several 
widely used benchmark functions and compared the 
results with those from the basic BA, FA and other state- 
of-the-art algorithms found in the literature. These 
benchmarks cover a range of characteristics including 
unimodal, multimodal, separable, and inseparable. The 
results have shown that BAMS-AN followed by HBAFA 
could track the optimal solution and give reasonable 
solutions most of the time. By including the 
improvements, both the search speed was improved and 
more accurate results were obtained. The comparisons 
among BA-based algorithms showed that BAMS-AN 
outperformed HBAFA, HFBA and the conventional BA. 
The experiments have also indicated that the improved 
variants of BA performed much better than the standard 
BA, PSO, FA and EA algorithms.  
Testing the improved algorithms further on real 
world optimization problems and looking for algorithmic 
enhancements remains as future work. 
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