A Novel Group-Based Firefly Algorithm with Adaptive
Intensity Behaviour
Adam Robson
1a
, Kamlesh Mistry
2b
and Wai Lok Woo
2c
1
School of Computer Science, Faculty of Technology, University of Sunderland, Sunderland, U.K.
2
Department of Computer and Information Sciences, Northumbria University, Newcastle upon Tyne, U.K.
Keywords: Firefly Algorithm (FA), Intensity Behaviour, Attraction, Optimisation, Swarm Intelligence.
Abstract: This paper presents novel modifications to the Firefly Algorithm (FA) that manipulate the functionality of the
intensity and attractiveness of fireflies through the incorporation of grouping behaviours into the movement
of the fireflies. FA is one of the most well-known and actively researched swarm-based algorithms, gaining
notoriety for the powerful search capability offered and overall computational simplicity. While the FA is an
effective optimisation algorithm, it is unfortunately susceptible to the issue of premature convergence and
oscillations within the swarm, which can lead to suboptimal performance. In the original FA formulation, at
each iteration fireflies will instinctively move towards the most intensely bright firefly which is in closest
proximity to them. The algorithm proposed in this paper manipulates the movement of the fireflies through
modification of this intensity and attraction relationship, allowing the swarm to move in different ways,
ultimately increasing the search diversity within the swarm. While group-based FAs have been proposed
previously, the group-based FAs presented in this paper utilise a different approach to creating groups,
implementing groupings based upon firefly performance at each iteration, resulting in continually varying
groupings of fireflies, to further increase search diversity and maintain computational simplicity.
1 INTRODUCTION
Swarm intelligence has been an increasingly
important and popular field throughout the last
decade and is inspired by the collective behaviours of
social swarms of animals and insects that occur
within nature (Qi et al., 2017). These swarms are
typically made up of a collection of unsophisticated
agents that demonstrate a coordinated behaviour to
achieve the desired goal of the swarm. Agents within
the swarm interact with each other, creating a
decentralised and self-organising swarm. Examples
of notable and frequently used Swarm Intelligence
optimisation methods are Ant Colony Optimisation
(ACO), Artificial Bee Colony (ABC), Firefly
Algorithm (FA) and Particle Swarm Optimisation
(PSO) (Wang and Liu, 2019).
Swarm intelligence methods have been applied in
a variety of optimisation problem areas, such as
forecasting (Altherwi, 2020), scheduling (Bacanin et
a
https://orcid.org/0000-0003-2752-3381
b
https://orcid.org/0000-0001-9371-7833
c
https://orcid.org/0000-0002-8698-7605
al., 2022), medical diagnosis (Nayak et al., 2020) and
structural design (Chou and Ngo, 2017),
demonstrating good performance and successful
outcomes (Wang and Liu, 2019). FA has shown itself
to be an effective and powerful optimisation
technique in a variety of optimisation problems, but it
is susceptible to issues such as oscillations in the
swarm during the search process (Wang et al., 2017),
and premature convergence (Qi et al., 2017). This
paper proposes a novel Group-based Firefly
Algorithm (GBFA) to overcome these issues. In the
original FA, at each iteration fireflies will
instinctively move toward the most intensely bright
firefly which is in closest proximity to them. The FA
variant proposed in this paper manipulates the
behaviour of fireflies within the swarm through
modification of the intensity and attraction
relationship. This modification allows the swarm to
move in different ways, ultimately increasing search
diversity within the swarm, which prevents the
Robson, A., Mistry, K. and Woo, W.
A Novel Group-Based Firefly Algorithm with Adaptive Intensity Behaviour.
DOI: 10.5220/0011672200003393
In Proceedings of the 15th International Conference on Agents and Artificial Intelligence (ICAART 2023) - Volume 1, pages 233-239
ISBN: 978-989-758-623-1; ISSN: 2184-433X
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
233

occurrence of issues such as premature convergence
or oscillations within the swarm. While group-based
Firefly Algorithms have been proposed previously in
work such as (Tong et al, 2017), (Suganya and
Murugavalli, 2019) and (Cao et al, 2022), the group-
based FA presented in this paper utilises a different
approach to creating groups. Groupings are
implemented at each iteration, based upon individual
firefly performance, resulting in continually varying
groupings of fireflies, allowing increased search
diversity within the swarm.
This paper is organised as follows: Section 2
presents the standard FA, and a brief review of
previous applications and research. Section 3 contains
an overview of the GBFA proposed in this paper and
briefly discusses related work. Section 4 shows the
experiment design and describes the optimisation
benchmark functions used to test the algorithm.
Section 5 presents and discusses the results obtained
from the proposed algorithm, and a comparison with
the standard FA and recent research within the field.
Section 6 provides a summary of the paper and the
findings.
2 FIREFLY ALGORITHM
The Firefly Algorithm (FA) was originally developed
in 2008 by Yang, with advances and applications
noted in (Yang and He, 2013). It is a relatively new
swarm intelligence algorithm and has been
successfully applied to a variety of optimisation
problems such as vehicle route planning, data fitting,
scheduling, resource allocation (Ariyaratne et al.,
2019). FA has also generated promising results for
optimisation in engineering systems with applications
in optimising power flow, micro-hydro applications,
and the optimisation of electromagnetic devices
(Parwanti et al., 2021).
FA is a population-based stochastic search
algorithm, with similarities in functionality to Particle
Swarm Optimisation (PSO). FA is a nature inspired
algorithm, and behind the algorithmic design
concepts of FA are the luminescence attribute,
behaviour, and movement of tropical fireflies (Jain et
al., 2021). Subsequently, FA has become a well-
known optimisation technique for complex
optimisation problems (Napalit and Ballera, 2021).
Fireflies are individual agents within the swarm, and
each uses its luminescence property to indicate their
position within a search domain. Attractiveness of
fireflies and movements within the swarm are based
around the attractiveness of a firefly. FA is based on
the following idealised rules (Yang and He, 2013):
(i) All fireflies within the swarm are unisex and
therefore all fireflies will be attracted to one
and other regardless of gender.
(ii) The attractiveness of a firefly is proportional
to the brightness, with brightness decreasing
as distance increases.
(iii) For any two flashing fireflies, the less bright
of the two will move toward the brighter one.
(iv) If there is no brighter firefly, it will move
randomly.
(v) The brightness of an individual firefly is
determined by the objective function.
The attractiveness of a firefly is directly
proportional to the intensity of brightness visible to
adjacent fireflies, and attractiveness of a firefly is
defined in (1),
𝛽=𝛽



(1
)
with 𝛽 as the attractiveness, 𝑟 as the distance, and
where 𝛽

is the attractiveness at 𝑟=0. Firefly
movement is based on the level of attraction to
another firefly, as shown in (2),
𝑥


=𝑥


+𝛽

𝑒



𝑥


−𝑥


+𝛼

𝜖


(2
)
where the 𝑖

firefly is attracted to move towards the
𝑗 firefly if it has a higher intensity of brightness. With
the first term representing the current location of
firefly 𝑖 and the second term representing movement
from one position to another, due to attraction to
firefly 𝑗. With the parameter 𝑟

being the Euclidian
distance between the two fireflies. The light
absorption coefficient is represented by 𝛾, where 𝛾=
1. 𝛽

is the original light attractiveness of each firefly
at 𝑟=0, and in the event that 𝛽

=0, the firefly will
take a simple random walk. The third term of (2) is a
randomisation with 𝛼

acting as the randomisation
parameter, and 𝜖


is a vector containing random
numbers drawn from a Gaussian distribution. The
framework of FA can be broken down into the three
stages. The first stage is initialisation. In this stage,
the objective function, 𝑓(𝑥), is defined and a
population of 𝑛 fireflies are generated through the
expression shown in (3), where 𝑖 is the population
(𝑖=1,2,…𝑁), 𝑑 is the dimension, with 𝑙𝑜𝑤 and 𝑢𝑝
representing the upper and lower bounds of the
dimension.
𝑥
,
=𝑙𝑜𝑤 + 𝑟𝑎𝑛𝑑(0,1)(𝑢𝑝 − 𝑙𝑜𝑤)
(3
)
The second stage focuses on firefly movement based
upon attraction. Each solution generated ( 𝑥

), is
compared with all other solutions within the
population of fireflies (𝑥

). Firefly 𝑥

will change
position based upon (2), if the objective function
ICAART 2023 - 15th International Conference on Agents and Artificial Intelligence
234

result of 𝑥

is better than 𝑥

. Each firefly is then
evaluated based upon updated positions and sorted.
At the third stage, the stopping criteria is checked, and
the algorithm will end if the stopping criteria is
satisfied. If the stopping criteria is not satisfied, the
second stage will repeat. Pseudo code describing the
functionality of the standard FA algorithm is as
follows:
/* Define objective function
𝑓
(
𝑥

)
,𝑥

=(𝑥
,
,…,𝑥
,
)
/* Initialise population of fireflies
𝑥

(𝑖=1,2,…,𝑛)
/* Begin
while (t < MaxIteration)
for 𝑖=1 to 𝑛 do
for 𝑗=1 to 𝑛 do
/* Calculate brightness
𝑏

=𝑓(𝑥

), 𝑏

=𝑓(𝑥

)
/* Determine movement
if (𝑏

>𝑏

)
Move 𝑥

towards 𝑥

end if
end for 𝑗
end for 𝑖
Rank fireflies and set current best
end while
/* End
While the FA is a powerful optimisation
algorithm, it is still susceptible to issues such as
swarm oscillations and premature convergence,
usually occurring because of fireflies moving towards
non-optimal solutions within their local vicinity
(Suganya and Murugavalli, 2019). The next section
of this paper discusses the proposed algorithm to
address these issues.
3 PROPOSED FIREFLY
ALGORITHM
This paper proposes a novel Group-based Firefly
Algorithm (GBFA) to alleviate the issues standard FA
is susceptible to, such as oscillations within the
swarm during the search process, resulting in
decreased search diversity (Wang et al., 2017), and
premature convergence (Qi et al., 2017). FA
implementations which utilise grouping behaviours
have been previously proposed in research such as
(Tong et al, 2017) and (Cao et al, 2022) and have
demonstrated positive results and successes in
preventing premature convergence or swarm
oscillations. Oscillations within the swarm is usually
caused by too many or too few attractions within the
swarm during the search process (Wang et al., 2017),
and therefore manipulation of attraction of fireflies
within the swarm is an important area for research.
In their work, (Tong et al, 2017) attempted to
solve the problem of premature convergence by
creating a modified evolutionary mechanism, with the
swarm divided into fixed groups, each with different
model parameters. While they achieved positive
results, showing a good balance between exploration
and exploitation, the implementation was not without
flaws, the main being that the overall computational
simplicity of the FA was reduced through these
augmentations. In other work, (Cao et al, 2022)
proposed groupings based upon visual fields and an
observer strategy and encouraged collaboration
between groups by having individual fireflies existing
in multiple groups. Again, while the work of Cao et
al. shows promising results, the implied overhead of
the additional behaviours added to the algorithmic
design of FA sacrifices computational simplicity for
more powerful search results.
While it is important to increase search diversity
within the swarm, reductions in computational
simplicity can have negative impacts on real-time
systems implementing swarm intelligence algorithms
that require updating to changes in the problem
domain, such as vehicle route planning (Chandrawati
and Sari, 2018), or controlling drone swarms
(Siemiatkowska and Stecz, 2021). It is therefore
important to try and obtain a balance between
ensuring computational simplicity and algorithmic
performance. The GBFA proposed in this paper seeks
to alleviate issues with the standard FA, by increasing
search diversity with the addition of dynamic groups,
whilst maintaining computational simplicity.
3.1 Group-Based Firefly Algorithm
Functionality
In the standard FA, fireflies will instinctively move
toward the most intensely bright firefly that is in
closest proximity to them. In the GBFA algorithm, the
intensity and movement relationship is manipulated
through the addition of dynamic groupings to each
iteration. At each iteration, fireflies are ranked based
on their brightness. Groupings are then dynamically
allocated based upon the position of fireflies within
this ranking and will fluctuate at each iteration based
upon the new rankings. For example, if the group size
is set to five, the highest ranked firefly (the current
best) will be the leader of the first group, which
contains the fireflies ranked second to fifth. The
second group leader will be the firefly ranked at the
sixth position, with the fireflies ranked seventh to
A Novel Group-Based Firefly Algorithm with Adaptive Intensity Behaviour
235

tenth forming the rest of that group. Example
groupings are visualised in Figures 1 and 2. Each of
the groupings is assigned a group leader, which all
other fireflies within the group will move toward.
Movement is handled in the same way as the standard
FA, as shown in equation (2), except with the caveat
that group members move only toward the leader of
their group. If a group member has the same value
returned from the objective as the group leader, the
group member will complete the same random walk
as in the standard FA. This modification increases
search diversity and allows the swarm to move in
different ways, overcoming issues such as premature
convergence and oscillations. Groups are dynamically
allocated at each iteration, based on sizing parameter
𝑔, which must be defined before the algorithm begins.
The pseudo code for the GBFA can be seen below,
where the group size is set to 5 (𝑔=5).
/* Define objective function
𝑓
(
𝑥

)
,𝑥

=(𝑥
,
,…,𝑥
,
)
/* Initialise population of fireflies
𝑥

(𝑖=1,2,…,𝑛)
/* Define group size
𝑔=5
/* Begin
while (t < MaxIteration)
for 𝑖=1 to 𝑛 do
Create groupings based on size 𝑔
Assign group lead as 𝑔

/* Calculate brightness of 𝑔

𝑏


=𝑓(𝑔

)
for 𝑗=1 to 𝑔 do
/* Calculate brightness
𝑏

=𝑓(𝑔

)
/* Determine movement
if (𝑏


>𝑏

)
Move 𝑔

towards 𝑏


end if
end for 𝑗
end for 𝑖
Rank fireflies and set current best
end while
/* End
Group sizes can be set to allow groups to operate
independently of each other, Figures 1 and 2 show a
visualisation of how groups are organised. With
Figure 1 showing a group size (𝑔) set to five (𝑔=5),
with a population (𝑛) of 50 (𝑛=50), where each
colour represents a different group of fireflies within
the swarm, and the visible group leaders are 𝑥

, 𝑥

,
𝑥

and 𝑥

. Figure 2 shows a group size of ten (𝑔=
10), with a population of 80 (𝑛=80), again where
each group within the swarm is represented by a
different colour and the visible group leaders are 𝑥

,
𝑥

, 𝑥

and 𝑥

.
Figure 1: GBFA group example where 𝑔=5 and 𝑛=50.
Figure 2: GBFA group example where 𝑔=10 and 𝑛=
80.
4 EXPERIMENT DESIGN
To review the performance of the proposed GBFA, a
standard FA and the GBFA were implemented using
Python 3.9, along with eight bound constrained global
optimisation problems that are commonly used to test
the performance of optimisation algorithms (Ackley,
Easom, Griewank, Michalewicz, Rastrigin,
Rosenbrock, Schwefel and Sphere). These were
chosen based on the optimisation problems noted by
(Fister et al., 2013), which have seen continued usage
in modern optimisation algorithm testing in work
such as (Wang et al., 2017) and (Zivkovic et al.,
2022). Functions were implemented using the
NumPy Python library and each of the benchmark
functions are outlined within this section.
All experiments were conducted using 30 runs,
each with 200 iterations and a population of 𝑛=40
fireflies. The group size for the GBFA is 𝑔=5 and
the number of dimensions is 𝑑=2.
ICAART 2023 - 15th International Conference on Agents and Artificial Intelligence
236

4.1 Ackley
The Ackley function is shown in (4), it is highly
multimodal and has a global minimum of 𝑓=0 at
𝑠=(0,0,…,0), where 𝑠

∈
[
−32.768,32.768
]
,𝑖=
1,2,…,𝐷.
𝑓

(
𝑠
)
(20 + 𝑒− 20𝑒
.

(.(





)


−𝑒
.(
(


)

(


)
)
(4)
4.2 Easom
The Easom function is shown in (5) has several local
minimum and global minimum, 𝑓=−1 at 𝑠=
(
𝜋,𝜋,…,𝜋
)
, where 𝑠

=[−2𝜋,2𝜋].
𝑓
(
𝑠
)
=
(
−1
)

cos

(
𝑠

)



exp−(𝑠

−𝜋)




(5)
4.3 Griewank
The Griewank function, shown in (6), has a global
minimum of 𝑓=0 at 𝑥=(0,0,…,0), where 𝑠

∈
[
−600,600
]
,𝑖=1,2,…,𝐷. It is also important to
note that when the number of variables is higher than
30, this function is highly multimodal.
𝑓
(
𝑠
)
=−cos
𝑠

√
𝑖




+
𝑠


4000
+1



(6)
4.4 Michalewicz
The Michalewicz function, in two-dimensional
parameter space, has the global minimum of 𝑓=
−1.8013 at 𝑠=(2.20319,1.57049) and is shown in
(7).
𝑓
(
𝑠
)
=−sin
(
𝑠

)



sin
𝑖𝑠


𝜋

⋅
(7)
4.5 Rastrigin
The Rastrigin function is shown in (8), where 𝑠

∈
[
−15,15
]
,𝑖=1,2,…,𝐷. It has a global minimum of
𝑓=0 at 𝑥=(0,0,…,0) and is highly multimodal.
𝑓
(
𝑠
)
=𝐷∗10+(𝑠


− 10cos(2𝜋𝑠

))



(8)
4.6 Rosenbrock
The Rosenbrock function is also commonly known as
the ‘banana function’. It has several local optima and
is shown in (9), where 𝑠

∈
[
−15,15
]
,𝑖=1,2,…,𝐷.
The function has a global minimum of 𝑓=0 at 𝑠=
(1,1,…,1).
𝑓
(
𝑠
)
=100(𝑠

−𝑠


)




+(𝑠

−1)

(9)
4.7 Schwefel
The Schwefel function is shown in (10), where 𝑠

∈
[
−500,500
]
,𝑖=1,2,…,𝐷 . This is a highly
multimodal function and has a global minimum of
𝑓=0 at 𝑠=(1,1,…,1).
𝑓
(
𝑠
)
=418.9829 ∗ 𝐷− s

sin

|
𝑠

|



(10)
4.8 Sphere
De Jong’s Sphere function is shown in (11), where
𝑠

∈
[
−600,600
]
,𝑖=1,2,…,𝐷. This is a unimodal
and convex function and has a global minimum of
𝑓=0 at 𝑠=(0,0,…,0).
𝑓
(
𝑠
)
=𝑠





(11)
5 RESULTS
The primary goal of the experiments conducted was
to show that the GBFA could outperform a standard
FA implementation through the incorporation of
grouping behaviours into the movement of the
fireflies. The data shown in Table 1 is the average best
results for the FA and GBFA when benchmarked
using each of the eight optimisation functions
described in Section 4. The experiments were
conducted over 30 runs at 200 iterations each, in 2
dimensions, with a population of 40. Based upon the
population size used, GBFA was configured to run
with a group size defined as 5, meaning there would
be a total of eight groups dynamically allocated at
each iteration of execution.
A Novel Group-Based Firefly Algorithm with Adaptive Intensity Behaviour
237

Table 1: Comparison of the average best results of FA and
GBFA, with the best result in bold.
Function FA GBFA
Ackle
y
1.59E-01 1.32E-01
Easo
m
-1.00E+00 -1.00E+00
Griewan
k
2.31E-02 2.12E-02
Michalewicz
-1.80E+00 -1.80E+00
Rastri
g
in
5.78E-02 4.86E-02
Rosenbroc
k
2.93E-03 1.36E-03
Schwefel
8.60E-01 6.05E-01
Sphere
5.83E-01 4.36E-01
The results shown in Table 1 are promising, with
the GBFA outperforming the standard FA in six of
the eight benchmark functions: Ackley, Griewank,
Rastrigin, Rosenbrock, Schewefel and Sphere, and
performing equally as well as the standard FA in the
remaining benchmark functions: Easom and
Michalewicz. This shows us that the increased search
diversity offered by the GBFA is capable of
addressing issues with FA that lead to suboptimal
performance, such as premature convergence or
oscillations within the swarm. Table 2 shows a
comparison of results with a recent study (Wahid et
al. 2018), which presents a hybrid FA. The standard
FA is combined with a Genetic Algorithm (GA) and
an embedded search pattern and is referred to as GA-
FA-PS. The GA is used to modify the positions of the
fireflies within the swarm after the fireflies have been
randomly placed within the search domain, before the
first execution of the FA. At the end of each iteration,
the embedded search pattern is used to increase
search diversity, through the introduction of further
exploitation and exploration.
Table 2: Comparison of the average best of the GBFA and
a modified FA presented by (Wahid et al. 2018), with the
best results in bold.
Function GBFA GA-FA-PS
Ackle
y
1.32E-01 2.52E-01
Rosenbroc
k
1.36E-03 -1.92E+00
Sphere
4.36E-01 -3.01E+00
The work presented by Wahid et al. attempts to
address the premature convergence issues within the
swarm by modifying the functionality of the FA,
much like the algorithm presented in this paper.
Wahid et al. conducted their research study using only
three optimisation benchmark functions: Ackley,
Rosenbrock and Sphere. The data is again quite
positive, as while the hybrid FA proposed by Wahid
et al. showed the capability to outperform the
standard FA, the GBFA presented in this paper was
able to significantly outperform the hybrid FA,
particularly in the Rosenbrock and Sphere
optimisation benchmark functions.
Table 3 shows a comparison of results with
another recent study (Gamao et al., 2019), which
presents a modified mutated FA, to attempt to address
the premature convergence issue of the standard FA
and improve results. In this study three FA variants
have been proposed: Mutated Firefly Algorithm
(MFA), Modified Mutated Firefly Algorithm-Las
Vegas (MMFA-LV) and Modified Mutated Firefly
Algorithm-Monte Carlo (MMFA-MC). These
algorithms have been implemented in an attempt to
increase search diversity of the swarm through
mutation of the lower performing fireflies and higher
performing fireflies.
Table 3: Comparison of the average best results of GBFA
and (Gamao et al., 2019), with the best result in bold.
Function GBFA MFA
MMFA-
LV
MMFA-
MC
Ackle
y
1.32E-01 5.97E+00 5.59E+00 3.87E+00
Rosenbroc
k
1.36E-03 7.54E+01 7.21E+01 4.69E+01
S
p
here
4.36E-01 3.98E+00 1.80E+00 1.77E+00
The proposed mutation process enhances features
and attractiveness of the bottom forty percent of the
swarm, by mutating them with the top forty percent
of the swarm. The algorithms presented by Gamao et
al. also attempt to improve the search capabilities of
FA by combining the mutation principles with a Las
Vegas (LV) search algorithm and a Monte Carlo
(MC) search algorithm. Gamao et al. used only three
optimisation benchmark functions: Ackley,
Rosenbrock and Sphere to test their algorithms.
Again, while the algorithm variants proposed by
Gamao et al. were able to show improvements on the
standard FA, the GBFA presented in this paper shows
significantly improved results.
6 CONCLUSIONS
The attraction behaviour of FA has an extremely
important role within the search process of the FA and
controls how the swarm moves and finds candidate
solutions. As previously noted, modification of the
attraction behaviour within the FA is an important area
to research when trying to alleviate issues such as
premature convergence or oscillations within the
swarm, as it can result in having the firefly and swarm
move in different ways to the standard FA, ultimately
increasing search diversity also. The GBFA presented
in this study has shown that positive results can be
achieved through modification of this attractiveness
relationship and can allow for fireflies to move toward
ICAART 2023 - 15th International Conference on Agents and Artificial Intelligence
238

positions that they would normally not, allowing for
greater search capabilities and enhanced performance.
While the results observed in this study are
extremely positive, further experimentation with the
GBFA is required. The next stage of research for this
algorithm is to tune the group size parameter and the
number of groups within a swarm, to evaluate the
performance when using larger or smaller group
sizes. Additionally, concepts such as the cross-group
communication behaviour seen in other research
within the area, such as (Cao et al., 2022), that allows
individual fireflies to exist across multiple groups can
be incorporated into the GBFA to investigate the
impact that this has on the performance of the
algorithm.
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