Exploring the Effects of Attraction and Repulsion Parameters on the
Bacterial Foraging Algorithm through Benchmark Functions
RIOS-WILLARS ERNESTO*, REYES-ACOSTA ALFREDO VALENTIN
Faculty of Systems,
Autonomous University of Coahuila,
Blvd. Fundadores Km. 13, Ciudad Universitaria, Arteaga, Coahuila,
MÉXICO
*Corresponding Author
Abstract: - Metaheuristics are essential when working with complex problems from different fields. However, a suitable
tuning scheme for these parameters is necessary to facilitate the search for potential solutions. This tuning is a challenging
task. This work aims to develop a tuning method for the BFOA algorithm regarding attraction and repulsion values. In
some cases, the parameter values are taken from previous works, while in other cases, the parametrization scheme comes
from an automated or dynamic process. This work explores the Bacterial Foraging Algorithm (BFOA) within its
parameters related to attraction and repulsion among bacteria, using 18 well-known benchmark functions from the
literature. For this purpose, multiple BFOA executions were made, and averages were calculated for each test with
repetitions for 24k BFOA executions. The interest variables for contrasting performance were the number of evaluated
functions (NFE), the required time for the execution (time), and the associated cost to the achieved solution by BFOA
(cost). Results: BFOA produced a different performance corresponding to each benchmark function. From this, four tuning
schemes are proposed and validated by repetition, also contrasted by t-test. The conclusions show that the BFOA
algorithm is susceptible to tuning, and the attraction and repulsion parameters must be according to the optimization
problem. In terms of execution time, scheme III showed remarkable results. Regarding the obtained solution cost, scheme
II outperformed the other three schemes.
Key-Words: - BFOA, tuning, bacterial foraging algorithm, benchmark functions, bioinspired, metaheuristics
Received: May 16, 2022. Revised: August 19, 2023. Accepted: September 19, 2023. Available online: October 25, 2023.
1 Introduction
The parameter setting for nature-inspired algorithms
is an open problem, [1]. The control parameters
significantly influence the algorithm’s performance;
their correct setting is crucial when obtaining
optimal solutions. Setting them is difficult since
their values are problem-dependent, [2]. In this
work, we describe a method for exploring the
Bacterial Foraging Algorithm in terms of
performance based on benchmark functions in an
extensive field of possible parameter values and
then integrating a set of four schemas and a
statistical validation. In this context, the benefit of
this work is a novel tuning procedure based on
attraction and repel (d_attr, h_rep, w_attr, and
w_rep) BFOA parameters. The interest variables for
comparison are the number of evaluated functions
(NFE), the required time for execution (time), and
the cost of the obtained solutions (cost). This BFOA
paradigm has proven efficient as an optimization
strategy in different areas, although it has the
inconvenience of requiring several parameters
tuning, [3]. This study explores the parametrization
of the algorithm parting from the default considered
values reported by, [4]. Figure 1 is a conceptual
scheme of the process in this study.
Fig. 1: Conceptual scheme of the process in the
study.
Next is a list of the steps taken in this work:
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1. Establish a BFOA with default parameters
2. Run BFOA on 18 benchmark functions
3. From the results, define the range of
parameters and run BFOA on each range
value with repetitions.
4. From the results, generate four fixed
parameter schemes and run BFOA for
further analysis.
5. Validate results by t-test and generate the
final tuning scheme regarding NFE, time,
and cost variables.
1.1 BFOA Background
The Bacterial Foraging Optimization Algorithm
(BFOA) proposed by, [5], is based on the natural
movement of the Escherichia coli bacteria (E.coli),
which, aided by its scourges, can use a translatory
motion in pursuit of nutrients into the environment,
or if needed to get out of the way of adverse factors.
It is a unicellular organism that tends to form
colonies in which the individuals are attracted to and
repelled each other by chemical substances
segregated in a mutual interaction with swarming
characteristics. The scourges are an extension in the
form of tentacles that can rotate in both ways,
allowing the bacteria to move in the environment in
two ways: a) swim and b) tumble. A description of
the results of the translatory movement of the
bacteria in the function of its scourges’ action and
the environmental conditions can be found in the
original Passino document.
1.2 BFOA Usage Diversity
The BFOA is a widely used tool in engineering and
optimization areas. It helps to solve numerous
problems, making it an essential part of these fields,
for instance, In, [6], an application of the BFOA in
turbine design. Alternatively, the one presented in, [7],
where the author describes a BFOA application that
solves vehicle routing problems. In, [8], BFOA was
used as a strategy for designing active filters in
electrical engineering and in, [9], for the power
transference problem. In, [10], [11], it is applied to
creating a PID controller in a DC-DC electric
converter.
Moreover, in, [12], the proposal regards an
application for solving the problem of robotic
movement planning. On the other hand, in ,[13], an
application is used as a placement strategy in a laser
radar. In, [14], the BFOA is used in the Hydro Power
Dispatch problem. In, [15], The BFOA application
optimizes picture recording on multi-core hardware
processors. In, [16], there is an application for the
pattern recognition problem. The Passino's algorithm
has generated numerous proposals and adaptations in
various contexts. Hereafter, they are classified and
briefly described. Adaptations and improvements: For
example, in, [17], the authors describe a modified
version of the Bacterial Foraging Optimization
Algorithm (BFOA) developed to expedite the
convergence to the optimal solution. However, it has
been observed that this algorithm sometimes keeps
oscillating when it is close to the objective. To tackle
this issue, the researchers have proposed an operator
that dynamically adjusts the chemotaxis steps based on
the fitness values obtained. In, [18], the reproduction
phase of the BFOA is analyzed as decisive in the
convergence through two differential equations and in
two bacteria in a one-level environment. In, [19], a
version of the BFOA is proposed, where the best
bacteria remain intact (elitism) while the others, which
are a few, reboot themselves. In, [20], a parallelization
of the BFOA is applied to the task planning problem.
Mathematical Models: In [3], [21], [22], a detailed
description of the algorithm is made from the
mathematical point of view. Performance
Benchmarking: In, [12], [23], [24], [25], [26], a
performance benchmarking of the BFOA is made,
comparing it with other optimization algorithms and
within different problem contexts. Hybridizations: In,
[27], there is a hybridization description between PSO,
BFOA, and Differential Evolution (DE). In, [13] the
authors present a hybrid between BFOA and Firefly
Optimization proposed for the vehicle routing
problem. In, [28], BFOA and ant colony hybridization
are described and applied to the labor programming
problem. The work in, [29], is a hybrid between GA
and BFOA applied to the manufacturing cells
distribution problem. In, [30], another hybridization
between BFOA and PSO was used for the global
numeric optimization problem.
1.3 Algorithm Tuning
The BFOA algorithm has the characteristic of
requiring precise tuning, and this may be a severe
disadvantage. However, every metaheuristic
algorithm requires a set of parameters tuning
according to the problem in the case. In this area, the
central part of the metaheuristic applications uses the
default values proposed by the metaheuristic
developer. In other cases, the parameters are adjusted
based on different strategies, looking for better
solutions in an intuitive or automated iteration
process.
For the BFOA algorithm, a case of auto-tuning is
made on, [31], where the search process is made by
enabling the bacterial foraging algorithm to adjust the
run-length unit parameter dynamically during
algorithm execution to balance the
exploration/exploitation tradeoff. However, the
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authors point out that developing and experimenting
with various methods and tuning them for each
particular problem is necessary. Several procedures
have been made to parametrize other algorithms,
such as GRASP, genetic algorithms, and scatter
search, [32], with significant findings in this context.
However, tuning or performing a parametrization of a
metaheuristic is still under experimentation,
especially for relatively new metaheuristics such as
BFOA. In this context, a proposed method is based
on the “functions reuse” considering parameters
standard to several metaheuristics, for example, the
number of elements in the initial set or the number of
cycles without improving the best solution in the end
condition, [33]. In other cases, metaheuristics are
built from blocks, taking from other algorithms those
parts that might be useful in a particular problem,
[34]. Optimizing a BFOA remains complex due to
the varying types of problems it addresses. A
contribution is generated in this field through a base
model with an Adaptive Neuro-Fuzzy Inference
System, [35]. Also, in the search for efficient
parameter tuning, a computational tool has been
developed. In this case, it implements the iterated
racing procedure for automatic algorithm
configuration, [36]. However, in some cases, it is
feasible to adopt parameter values from prior
research where they have been established based on
the tuning conducted by others, which allows for the
rapid development of new features and strategies,
[37]. In this work, we explore an extensive area of
possible parameter values, which allows us to
contrast and choose the most adequate.
Table 1 lists the most common practices for
tuning a metaheuristic like BFOA.
Table 1. Most common tuning strategies.
Dynamic auto-
tuning and self-
adaptive process
The BFOA can change its parameters
along the execution based on rules
relative to the optimization problem.
Fuzzy Logic
A series of fuzzy rules are associated
with the BFOA to explore parameter
values.
Independent
Computational tool
A tool consists of three phases: sampling
new parameter configurations with a
specific distribution, selecting the most
competitive configurations through
racing, and then updating the sampling
distribution to favor better
configurations.
Manual
experimentation
It consists of a manual trial and error
process where the parameter values are
explored first.
Taken from others
The parameter values are taken from
other implementations where a tuning
process is already made.
2 Bacterial Foraging Algorithm
The optimization in the BFOA algorithm is based on
the chemotactic behavior of the Escherichia Coli
Bacteria (E. Coli). Although using chemotaxis as a
model for optimization was proposed first in, [38],
Passino's work includes some modifications, such as
agents' reproduction and dispersion. E. Coli is the
best-understood microorganism, [5], since its
behavior and genetic structure are well-studied. This
unicellular organism consists of a capsule with its
organs and scourges used for locomotion; it can
reproduce by dividing itself and exchanging genetic
information with its peers. In addition, it can detect
food (nutrients) and avoid toxic substances, making
a random search based on two locomotion states: the
translation (swim) and the rotation (tumble). The
decision to remain in one of these states depends on
the nutrients or toxic substances concentration in the
environment. This behavior is called chemotaxis.
The following describes the optimization steps
with the algorithm, [10]. Step 1: The inner cycle of
the chemotaxis is illustrated in Figure 2. In this
process, the E. coli movement is simulated. The
move is made in two ways: lurching (tumbling) or
swimming. One operation at a time. The value of
the objective function is calculated. The bacteria
change its position if the value of the modified
objective function is worse than before. Once the
chemotaxis is completed, the bacteria will circulate
a new interest point in the search space.
Fig. 2: Chemotaxis process in the BFOA algorithm,
[5].
Step 2: In the reproduction process, the value of
the objective function is calculated for each of the
bacteria in the organized population. The worst half of
the population is disregarded, and the best half is
duplicated. The chemotaxis process starts and
continues for this new bacteria generation according
to the number of reproductive cycles. Step 3: In the
elimination-dispersion cycle, some bacteria are
eliminated with low probability and dispersed
randomly. This process maintains the bacteria
numbers constant. The entry parameter is the number
of bacteria (Sb) Chemotaxis steps limit (Nc), Swim-steps
limit (Ns), Reproductive cycles limit (Nre), bacteria
number aimed to produce elimination-dispersion
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cycle limit (Ned), step size (Ci), and elimination and
dispersion probability (Ped). The cost of each bacteria
is optimized by its interaction with other bacteria. The
interaction function is calculated according to the
expression g().
Where the cell is the bacteria of the issue, and the
other is a neighboring cell. dattr y wattr are attraction
coefficients. hrepel and wrepel are repulsion coefficients.
S is the number of bacteria in the population. P is the
number of dimensions in the optimization problem,
[4].
2.1 Bacterial Interaction Parameters
These are significant since this interaction sustains
the optimization process in the paradigm that states
that bacterial colonies aim to move towards new
positions where nutrients are in better availability.
These parameters represent the attraction extent and
depth between bacteria (d_attr y w_attr) and the
repulsion extent and depth between bacteria (h_rep y
w_rep). These four parameters are part of the tuning
assembly of the BFOA. In this sense, [3], expresses
that the BFOA has the disadvantage of requiring
more parameters than other optimization algorithms,
which makes it more complicated from this point of
view. However, [11], points out that researchers use
the BFOA because it does not require precise
mathematical models for tuning. Notably, [5], points
out that if the extent of the attraction signal is long
and very deep, the cells will have a solid tendency to
swarm. Moreover, in the opposite sense, each cell
will look for the optimization independently.
3 Methodology
For the development of this project, two phases are
established.
1) First Phase:
a) BFOA algorithm implantation in a set of
18 benchmark functions and result
evaluation.
b) Algorithm sensibility exploration within
the bacterial interaction parameters on a
benchmark function.
2) Second Phase:
a) Algorithm Parameterization proposal
from the sensibility exploration.
b) Perform Results validations by repetition
and t-test.
3.1 Parameters and Resources
The BFOA algorithm was developed in Ruby
language with computer equipment, including an
Intel Core i7 Processor, 10Gb in RAM Memory, and
a Windows 7 operative system. The graphics were
obtained from the Excel Microsoft Software and the
Minitab Statistic Software. The BFOA parameters
considered by default and reported by, [4], are
described in Table 2.
Table 2. Default parameters in the BFOA algorithm.
Population size
50
Step Size (Ss)
0.1
Elimination – dispersion cycles (Ned)
1
Reproductive cycles (Nre)
4
Quimiotaxis cycles (Nc)
70
Swim length (Ns)
4
Elimination probability (Ped)
0.25
Attraction depth (d_attr)
0.1
Attraction wide (w_attr)
0.2
Repeland depth (h_rep)
d_attr
Repeland wide (w_ewp)
10
3.2 First Phase: Benchmark Functions
To ensure evidence diversity, in this study, we use a
group of 18 functions relative to known problems in
the optimization area, [39], [40]. These are
classified according to similarity and search space
form, [41]. Moreover, some functions have multiple
local minimums, others have a plane form, valley
form, or bowl-like depth, and others are staggered
and of various shapes. For this benchmark, all
functions are two-dimensional (d=2) and carried out
on 30 repetitions.
Cross In Tray: Has multiple global minima. The
function is commonly evaluated on the square xi
[-10, 10] for all i = 1, 2, [42].
(2)
Search Space: xi [-10, 10]
Global Min: -2.06261
Drop Wave: This is a multimodal test function.
The given form of function has only two variables
and the following definition. The test area is usually
restricted to the square −5.12≤x1≤5.12,−5.12≤x2≤
5.12, [40].
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(3)
Search Space: xi [-5.12, 5.12]
Global Min: -1
Holder Table: Is a Continuous, Differentiable,
Separable, Non-Scalable, Multimodal function. The
four global minima are at x
=f(±9.646168,
±9.646168), f(x
)= −26.920336, [41].
(4)
Search Space: xi [-10, 10]
Global Min: -19.2085
EggHolder: This has a deceptive landscape and
is a challenging function to optimize because it is
characterized by an uneven plane having several
dozen local minimums that easily mislead the search
agents. The function is usually evaluated on the
square xi [-512, 512], for all i = 1, 2, [42].
(5)
Search Space: xi [-512, 512]
Global Min: -959.6407
Rastrigin: This is based on the function of De
Jong with the addition of cosine modulation to
produce frequent local minima. The test area is
usually restricted to hypercube
−5.12≤xi≤5.12,i=1,...,n. Its global minimum equal
f(x)=0 is obtainable for xi=0,i=1,...,n., [40].
(6)
Search Space: xi [-5.12, 5.12]
Global Min: 0
Schwefel: It is deceptive in that the global
minimum is geometrically distant, over the
parameter space, from the next best local minima.
Therefore, the search algorithms are potentially
prone to convergence in the wrong direction. The
function has the following definition. The test area
is usually restricted to a hypercube
−500 ≤ xi ≤ 500, i=1,...,n.
Its global minimum f(x)=−418.9829n is
obtainable for xi=420.9687,i= 1,...,n., [43].
(7)
Search Space: xi [-500, 500]
Global Min: 0
Schaffer2: The function is usually evaluated on
the square xi [-100, 100], for all i = 1, 2, [41].
(8)
Search Space: xi [-100, 100]
Global Min: 0
Ackley has a flat outer region and a large hole at
the center. The function poses a risk for
optimization algorithms, particularly hill climbing
algorithms, to be trapped in one of its many local
minima, [44].
(9)
Var: α=20, b=0.2, c=2π
Search Space: xi [-32.768, 32.768]
Global Min: 0
Booth: Is usually evaluated on the square xi [-
10, 10], for all i = 1, 2, [41].
(10)
Search Space: xi [-10, 10]
Global Min: 0
Matyas: Has no local minima except the global
one. The function is usually evaluated on the square
xi
[-10, 10] for all i = 1, 2, [41].
(11)
Search Space: xi [-10, 10]
Global Min: 0
Zakharov: Is continuous and unimodal. The
suggested search area is the hypercube [−10, 10]D.
The global minimum is f28(x*) = 0 at x* = {0, 0,
…, 0}. The general formulation of the function is,
[45].
(12)
Search Space: xi [-5, 10]
Global Min: 0
Sphere: Is continuous, convex, unimodal,
differentiable, separable, highly symmetric, and
rotationally invariant. The suggested search area is
the hypercube [−100, 100]D. The global minimum
is f01(x*) = 0 at x* = {0, 0, …, 0}, [45].
(13)
Search Space: xi [-5.12, 5.12]
Global Min: 0
Rosenbrok: This is often used as a test problem
for optimization algorithms (where a variation with
100 replaced by 105 is sometimes used. It has a
global minimum of 0 at the point (1, 1), [46].
(14)
Search Space: xi [-5, 10]
Global Min: 0
Michaelwicz: Has d! local minima, and it is
multimodal. The parameter m defines the steepness
of the valleys and ridges; a larger m leads to a more
complex search. The recommended value of m is m
= 10, [41].
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(15)
Var: m=10
Search Space: xi [0, π]
Global Min: -1.8013
Easom: This is an unimodal test function where
the global minimum has a
small area relative to the search space. The
function was inverted for minimization, [40].
It has only two variables
(16)
Search Space: xi [-100, 100]
Global Min: 0
Beale: is multimodal, with sharp peaks at the
corners of the input domain. The function is usually
evaluated on the square xi
[-4.5, 4.5] for all i = 1,
2, [41].
(17)
Search Space: xi [-4.5, 4.5]
Global Min: 0
Styblinski-Tang: Has three local minimums in
addition to its global minimum. The function is
continuous, not convex, defined on n-dimensional
space, and multimodal, [41].
(18)
Search Space: xi [-5, 5]
Global Min: -39.16599d
3.2.1 Sensibility Analysis
As pointed out before, the w_attr, h_rep, d_attr, and
w_rep parameters represent the attraction extent and
depth between bacteria (d_attr, w_attr) and the
repulsion extent and depth between bacteria (h_rep,
w_rep). In this section, we describe how parameter
values are explored to observe the results for the
benchmark function when running on each
parameter setting.
Consider the set
{(w_attr, h_rep, d_attr, w_rep) | w_attr = i,
h_rep = i, d_attr = i, w_rep = k}
Where w_attr, h_rep, and d_attr take values from
integer i {i | -999.9 ≤ i ≤ 1000.1} and w_rep take
values from integer k in the established range {k | -
990 ≤ k ≤ 1010}. From the definition of (i,k) values,
we run a "scenario", a BFOA algorithm running on
the Michalewicz benchmark function for a
workspace with multiple local minima and
multimodal characteristics.
The sensibility analysis consists of executing a
set of BFOA algorithms n=8000 such that the
values of the bacterial interaction parameters are
different in each one. Establishing as a start the
default values in, [4]. Then, gradual and unitary
changes are made to each parameter (one at a time)
incrementally in one case and decremental in the
other, considering three repetitions for each
experiment. See Figure 3 for a graphic description.
From the repetitions, the averages are calculated.
Fig. 3: Graphic description of the variation
parameters in the BFOA sensibility analysis.
This series of combinations produces 2000
adjustments for each parameter, 1000 for the
incremental variation case, and 1000 for the
decremental variation. Each experiment had three
repetitions. In sum, 24,000 BFOA algorithm
executions. For this test, the interest variables to be
measured in each run were the execution time (t),
the number of evaluated functions (NFE), and the
cost of the found solution by the algorithm (cost) to
measure the algorithm’s efficiency and robustness,
[47].
Figure 4 describes the workflow used for this
exploration.
3.3 Second Phase
From the results graphic in the sensibility
exploration, we selected the regions where the
algorithm has a noticeable performance regarding
the interest variables. (NFE, t, cost). For this, the
surface graphics are also generated to determine by
observation a beneficial point to tune the algorithm
and validate. The validation consists of executing
new runs for the algorithm in the 18 benchmark
functions and a group (n=4) of parametrization
schemes proposed from the analysis.
As a part of the validation process, the t-tests are
executed for each parametrization scheme and in
each optimization benchmark function for the
results of the 30 runs in terms of the cost variable. In
the same way, box graphics are presented as a
contrast of the results before and after
parametrization with the proposed scheme.
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Fig. 4: The workflow used for BFOA algorithm
parameters exploration. Each of the w_attr, h_rep,
d_attr, and w_rep parameters are updated in turn,
and three algorithm executions are performed for
average value calculation.
4 Results
The results from the algorithm executions from each
benchmark function are presented.
4.1 First Phase: Benchmark Results
Table 3 shows the BFOA algorithm's results in 18
benchmark functions. Each box corresponds to the
average result of the 30 algorithm’s repetitions in
the respective function and with the default
parameters.
Table 3. BFOA Results in the 18 benchmark
functions.
Opt-Cost
Óptimo
Function
NFE
t
Cost
0
Ackley
28616.8
333
3764.51
527
9.344239
34
-1.80
Michaelwi
kcz
29372.3
667
3873.65
487
-
1.801165
383
0
Rastrigin
28588.8
3762.24
857
0.710712
312
0
Rosenbro
k
32261.7
667
4275.71
127
0.000652
085
0
Schwefel
40398.7
333
2474.17
49
201.1314
457
0
Sphere
38576.7
667
4499.39
07
2.74135E-
05
-78.33
Styblinski
32229.9
1959.00
273
-
78.33226
261
0
Zakharov
38901.1
333
5398.30
883
5.33258E-
05
0
Beale
31692.7
5485.38
043
4.7815E-
05
0
Booth
37141.1
333
4843.21
027
6.12336E-
05
-2.06
Cross In
Tray
43420.5
667
5238.76
627
-
2.062583
605
-1
Drop
Wave
33297.1
667
4514.52
487
-
0.870848
792
-1
Easom
56335.3
333
6101.51
563
-
0.033333
363
-959.64
Egg
Holder
40293.5
667
5974.64
177
-
719.0540
648
-19.20
Holder
Table
35989.3
333
4941.58
257
-
19.20810
283
0
Matyas
40177
5371.40
723
1.69966E-
05
0
Schaffer 2
26183.6
3376.09
31
0.002326
416
-186.73
Shubert
28487.7
667
3935.19
177
-
178.8116
76
4.2 Sensibility Analysis Results
Regarding the four variables of bacterial interactions
and the three interest variables. (time, NFE, cost). A
noticeable difference can be observed between the
sides of the graphic parting from the center, where
the default value corresponds. To the left are the
incremental values from the default, and to the right
are the decremental ones. That is to say that the
value 1 is the most positive and the value 2000 the
most negative. These tests were made,
simultaneously varying one of the four parameters
and leaving the other three in default value.
In the execution time comparison between the
four parameters in Figure 5, an increase for the
range 987 to 1161 stands out in the values from
w_attr, while d_attr and h_rep remain in stable
fields. In the same way, w_rep presents an increase
in the range 1103 to 1451 to stabilize subsequently.
Fig. 5: Time results comparison for the parameters
d_attr, h_rep, w_attr, w_rep.
Figure 6 compares the number of the evaluated
function (NFE) for the four parameters d_attr,
h_rep, w_attr, and w_rep. The range 987 to 1161
stands out because the evaluated functions increase
between the parameters w_rep and w_attr. In
contrast, the parameter h_rep demonstrates a
decrease in the functions being assessed.
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Fig. 6: Comparison of the evaluated functions
number (NFE) for the parameters d_attr, h_rep,
w_attr, w_rep.
Figure 7 compares the costs of the functions
according to the four parameters d_attr, h_rep,
w_attr, and w_rep. An increase is to be noted from
the value 1009 for the parameters h_rep y d_attr. In
the same way, the parameter w_attr presents an
increase parting from said value to stabilize parting
from the value 1121 subsequently.
Fig. 7: Cost comparison according to parameters
d_attr, h_rep, w_attr, w_rep.
Surface Graphics. Figure 8 shows the surface
generated by parametrizations in terms of interest
variables. The favorable point is located in the
region where the cost, execution time, and evaluated
function number are minimum.
Fig. 8: Surface graphics in the sensibility analysis.
4.3 Second Phase: Proposed Parameters
The sensibility analysis and surface graphics
observation establish the following parametrization
schemes for validation by repetition and t-test. See
Table 4.
Table 4. Proposed schemes.
Scheme
d_attr
h_rep
w_attr
w_rep
I
995.1
0.1
0.2
10
II
0.1
-0.9
0.2
10
III
0.1
0.1
655.2
10
IV
0.1
0.1
0.2
967
4.4 Validation by Repetition and T-Test
Once the four schemes for tuning BFOA are
established, the next step is to run the algorithm for
the 18 benchmark functions with 30 repetitions to
have a general panorama of the algorithm's
performance in terms of the cost value.
Also, as a part of the validation process, it is
known that a t-test can be used to determine whether
two groups differ from each other in terms of
independent samples. At this point, we use the test
to determine whether the average result (n=30)
differs from the execution on the default parameter
values vs. the execution with each proposed scheme.
Table 5 shows results from the validation by
repetition and t-test of the parametrization proposed
scheme number I. The value in the t column
corresponds to the p significance of the statistical
test from the cost with default values vs. the value
achieved with the scheme mentioned.
Table 5. Validation by repetition in the scheme I.
Cost w/
Scheme I
Optimal
Function
defaults
Cost
t
0.0000
Ackley
9.3442
7.7122
0.0600
-1.8013
Michaelwikcz
-1.8012
-1.7987
0.0010
0.0000
Rastrigin
0.7107
0.0424
0.0000
0.0000
Rosenbrok
0.0007
0.0744
0.0140
0.0000
Schwefel
201.1314
213.7595
0.6610
0.0000
Sphere
0.0000
0.0814
0.3220
-78.3320
Styblinski
-78.3323
-75.9263
0.0040
0.0000
Zakharov
0.0001
1.0591
0.0050
0.0000
Beale
0.0000
0.1003
0.0060
0.0000
Booth
0.0001
3.1139
0.0000
-2.0636
Cross In Tray
-2.0626
-2.0491
0.0260
-1.0000
Drop Wave
-0.8708
-0.9873
0.0000
-1.0000
Easom
-0.0333
-0.0407
0.8730
-959.6407
Egg Holder
-719.0541
-677.5126
0.2330
-19.2085
Holder Table
-19.2081
-18.1454
0.1000
0.0000
Matyas
0.0000
0.1210
0.0000
0.0000
Schaffer 2
0.0023
0.0026
0.7420
-186.7309
Shubert
-178.8117
-175.8331
0.6160
Table 6 shows results from the validation by
repetition and t-test of the parametrization proposed
scheme number II. The value in the t column
corresponds to the p significance of the statistical
test from the cost with default values vs. the value
achieved with the scheme mentioned.
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Table 6. Validation by repetition in the scheme II.
Cost w/
Scheme II
Optimal
Function
defaults
Cost
t
0.0000
Ackley
9.3442
8.7900
0.5640
-1.8013
Michaelwikcz
-1.8012
-1.8005
0.0000
0.0000
Rastrigin
0.7107
0.9730
0.2170
0.0000
Rosenbrok
0.0007
0.0009
0.3540
0.0000
Schwefel
201.1314
171.9608
0.2430
0.0000
Sphere
0.0000
0.0001
0.0170
-78.3320
Styblinski
-78.3323
-78.3307
0.0000
0.0000
Zakharov
0.0001
0.0001
0.0110
0.0000
Beale
0.0000
0.0001
0.0080
0.0000
Booth
0.0001
0.0004
0.0050
-2.0636
Cross In Tray
-2.0626
-2.0586
0.0810
-1.0000
Drop Wave
-0.8708
-0.8939
0.3530
-1.0000
Easom
-0.0333
-0.1333
0.1680
-959.6407
Egg Holder
-719.0541
-777.8452
0.0950
-19.2085
Holder Table
-19.2081
-19.2077
0.1380
0.0000
Matyas
0.0000
0.0005
0.2750
0.0000
Schaffer 2
0.0023
0.0013
0.1400
-186.7309
Shubert
-178.8117
-167.4650
0.1680
Table 7 shows results from the validation by
repetition and t-test of the parametrization proposed
scheme number III. The value in the t column
corresponds to the p significance of the statistical
test from the cost with default values vs. the value
achieved with the scheme mentioned.
Table 7. Validation by repetition in the scheme III.
Cost w/
Scheme
III
Optimal
Function
defaults
Cost
t
0.0000
Ackley
9.3442
8.3033
0.237
0
-1.8013
Michaelwikc
z
-1.8012
-1.7891
0.001
0
0.0000
Rastrigin
0.7107
0.7487
0.843
0
0.0000
Rosenbrok
0.0007
0.0363
0.000
0
0.0000
Schwefel
201.1314
191.4861
0.693
0
0.0000
Sphere
0.0000
0.0250
0.000
0
-78.3320
Styblinski
-78.3323
-78.3307
0.020
0
0.0000
Zakharov
0.0001
0.0235
0.000
0
0.0000
Beale
0.0000
0.0071
0.000
0
0.0000
Booth
0.0001
0.0161
0.006
0
-2.0636
Cross In Tray
-2.0626
-1.9844
0.000
0
-1.0000
Drop Wave
-0.8708
-0.8301
0.168
0
-1.0000
Easom
-0.0333
0.0000
0.326
0
-
959.6407
Egg Holder
-
719.0541
-738.2348
0.584
0
-19.2085
Holder Table
-19.2081
-19.1903
0.000
0
0.0000
Matyas
0.0000
0.1508
0.000
0
0.0000
Schaffer 2
0.0023
0.0073
0.000
0
-
186.7309
Shubert
-
178.8117
-159.4473
0.050
0
Table 8 shows results from the validation by
repetition and t-test of the parametrization proposed
scheme number IV. The value in the t column
corresponds to the p significance of the statistical
test from the cost with default values vs. the value
achieved with the scheme mentioned.
Table 8. Validation by repetition in the scheme IV.
Cost w/
Scheme
IV
Optimal
Function
defaults
Cost
t
0.0000
Ackley
9.3442
9.5079
0.829
0
-1.8013
Michaelwikc
z
-1.8012
-1.–
0.000
0
0.0000
Rastrigin
0.7107
0.6810
0.865
0
0.0000
Rosenbrok
0.0007
0.0011
0.153
0
0.0000
Schwefel
201.1314
184.9300
0.546
0
0.0000
Sphere
0.0000
0.0000
0.000
0
-78.3320
Styblinski
-78.3323
-78.3323
0.637
0
0.0000
Zakharov
0.0001
0.0000
0.000
0
0.0000
Beale
0.0000
0.0000
0.020
0
0.0000
Booth
0.0001
0.0000
0.003
0
-2.0636
Cross In Tray
-2.0626
-2.0626
0.682
0
-1.0000
Drop Wave
-0.8708
-0.8681
0.920
0
-1.0000
Easom
-0.0333
-0.0395
0.918
0
-
959.6407
Egg Holder
-
719.0541
-767.3098
0.195
0
-19.2085
Holder Table
-19.2081
-19.1103
0.327
0
0.0000
Matyas
0.0000
0.0001
0.122
0
0.0000
Schaffer 2
0.0023
0.0023
0.993
0
-
186.7309
Shubert
-
178.8117
-159.7021
0.033
0
The average results for the NFE and time values
are calculated for the 30 runs on each of the four
schemes.
Table 9 shows the values for schemes I and II on
each benchmark function.
Table 9. Average results of the different schemes
regarding the NFE variables and execution time.
Scheme I
Scheme II
Function
NFE
time
NFE
time
Ackley
55527.7
4860.7
31556.0
3270.5
Michaelwikcz
40781.2
4406.9
38876.2
3967.6
Rastrigin
41811.0
4540.4
28611.6
3273.9
Rosenbrok
42265.7
4664.5
32620.5
3618.4
Schwefel
56504.8
4155.0
41796.4
3414.1
Sphere
41535.9
3330.1
42232.6
3428.6
Styblinski
41545.8
3307.1
39536.2
3443.3
Zakharov
43772.7
3561.1
41164.5
3497.8
Beale
41089.8
3839.0
35473.4
3553.2
Booth
47924.9
4277.0
40768.2
3878.8
Cross In Tray
48067.7
4306.3
53430.4
4688.1
Drop Wave
41860.6
3905.2
45815.9
4311.8
Easom
56376.8
4742.3
57087.8
4945.9
Egg Holder
56469.9
5403.0
41484.3
4075.8
Holder Table
47909.8
4967.1
45461.4
4359.0
Matyas
48012.7
4944.9
47669.3
4530.5
Schaffer 2
47086.3
4844.4
28157.3
3075.7
Shubert
41740.7
3412.9
28503.6
2617.2
Table 10 shows schemes III and IV values on
each benchmark function.
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Table 10. Average results of the different schemes
regarding the NFE variables and execution time.
Scheme III
Scheme IV
Function
NFE
time
NFE
time
Ackley
28397.0
2642.3
34970.3
3378.5
Michaelwikcz
27624.6
3258.6
37688.3
3982.1
Rastrigin
28518.5
3399.1
29088.9
3284.6
Rosenbrok
31482.9
3555.4
33113.6
3678.9
Schwefel
38489.8
2882.3
42143.2
3058.0
Sphere
30898.6
2607.4
37131.7
2695.0
Styblinski
29133.1
2450.5
38598.5
2861.5
Zakharov
34235.4
2752.8
43078.6
3141.3
Beale
30177.7
2788.7
35717.8
3263.7
Booth
35541.1
3108.8
38808.1
3424.3
Cross In Tray
28032.6
2611.1
44224.6
3737.6
Drop Wave
28062.0
2667.1
47192.7
4012.9
Easom
28000.0
2563.6
50800.7
4254.3
Egg Holder
38283.2
3848.3
41806.9
4239.0
Holder Table
28787.8
3346.8
45688.5
4369.4
Matyas
31181.9
3522.4
48440.2
4600.1
Schaffer 2
28123.9
3317.0
27974.6
3082.4
Shubert
28444.3
2424.9
28712.0
2186.1
4.5 Box Diagrams
Here are the box diagrams for the cases where the
tuning changed the results (Figure 9).
Fig. 9: Cases where the parametrization proposal
improved the cost variable results in contrast with
the default parametrization results in the
independent runs group of 30 repetitions for the
BFOA algorithm.
5 Discussion
Phase 1, developed in this study, allowed us to
observe an overview of the behavior and
performance of BFOA in the 18 benchmark
functions described. From this phase, it was possible
to make a performance comparison considering the
variables NFE, time, and cost. As a next step of
phase 1, additional tests were made with the
algorithm considering a set of parameterization
values in a benchmark function. This part of the
process allowed us to observe a new panorama of
performance concerning the multiplicity of values
between the four parameters of attraction and
repulsion in BFOA and proceed to phase 2, in which
from the results obtained between the 18 benchmark
functions and the observation of the results of
parameter values on landscape graphs, four work
schemes were integrated.
With these schemes, more specific tests were
carried out with the fact that 30 runs were again
made for each benchmark function in each proposed
scheme.
This finding is consistent with the literature that
the performance of an optimization algorithm
depends on the problem to be attacked, as well as
the parameter values and limits that direct the search
for solutions. Likewise, as part of phase 2, t-tests
were carried out to determine significant differences
between sets of 30 runs of each benchmark function,
being a set corresponding to the runs of the
algorithm with the default values, in contrast to
another independent set of runs made with the
different values of the proposed schemes. This
contrast was made by calculating the significance
value of the t-test, with which it is possible to reject
the null hypothesis that there is no significant
difference between the mean of the independent
samples. In this sense, Table 11 shows those cases
in which a significant difference was found between
said pair of sets of runs. Likewise, the values of
significance p marked in bold for the corresponding
cases are observed.
Table 11. The p-value for each benchmark function
where any scheme made a significant difference.
Function
Scheme
I
Scheme
II
Scheme
III
Scheme
IV
Michaelwikc
z
0.001
0
0.001
0
Rastrigin
0
0.217
0.843
0.865
Rosenbrok
0.014
0.354
0
0.153
Sphere
0.322
0.017
0
0
Styblinski
0.004
0
0.02
0.637
Zakharov
0.005
0.011
0
0
Beale
0.006
0.008
0
0.02
Booth
0
0.005
0.006
0.003
Cross In Tray
0.026
0.081
0
0.682
Drop Wave
0
0.353
0.168
0.92
Holder Table
0.1
0.138
0
0.327
Matyas
0
0.275
0
0.122
Schaffer 2
0.742
0.14
0
0.993
Shubert
0.616
0.168
0.05
0.033
It is noteworthy in the case of benchmark
functions in which no difference was found. As are
the functions Ackley, Schwefel, Easom, and Egg
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Holder. This finding might be due to the search
space and the nature of the algorithm based on
foraging bacteria. Finally, the box diagrams visually
describe cases from the EggHolder, Zarackarov, and
Rastrigin functions in which an improvement was
found regardless of the statistic t-test. It is
recommended to perform a more extensive test on
the EggHolder function.
6 Conclusion
The results presented in this study conclude that the
BFOA algorithm is susceptible to particular tuning
according to the problem aimed to solve. The
benchmark functions used for this experiment can
be taken as a reference for future tunings. As a
prospective study, the proposed schemes can be
used as a reference. It is essential to notice that in
terms of time and NFE, scheme III has remarkable
results regarding the other three. In regards to the
cost, by adding the absolute differences of the
averages throughout the 18 benchmark functions
from the optimal vs. the result of each scheme,
results outstanding scheme II; however, to exploit
the usefulness of this study, the results of the
interest function must be considered (any of the 18
presented) and in the light of the explored
parametrization schemes.
The limitations of this study are related to the
parameter values explored. Only one of the four
parameters was adjusted, while the other three
remained on the default value. It is recommended to
perform a more extensive test for the generated
schemes in this work. In future directions, BFOA
might be tested on schemes III and II, leaving
behind benchmark functions to work on problems
from a specific field.
Acknowledgement:
The authors would like to acknowledge the Systems
Faculty from the Autonomous University of
Coahuila for research support.
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