A novel meta-heuristic algorithm for solving numerical optimization problems: Ali Baba and the forty thieves

Abstract

This paper presents a novel meta-heuristic algorithm called Ali Baba and the forty thieves (AFT) for solving global optimization problems. Recall the famous tale of Ali Baba and the forty thieves, where Ali Baba once saw a gang of forty thieves enter a strange cave filled with all kinds of treasures. The strategies pursued by the forty thieves in the search for Ali Baba inspired us to design ideas and underlie the basic concepts to put forward the mathematical models and implement the exploration and exploitation processes of the proposed algorithm. The performance of the AFT algorithm was assessed on a set of basic benchmark test functions and two more challenging benchmarks called IEEE CEC-2017 and IEEE CEC-C06 2019 benchmark test functions. These benchmarks cover simple and complex test functions with various dimensions and levels of complexity. An extensive comparative study was performed between the AFT algorithm and other well-studied algorithms, and the significance of the results was proved by statistical test methods. To study the potential performance of AFT, its further development is discussed and carried out from five aspects. Finally, the applicability of the AFT algorithm was subsequently demonstrated in solving five engineering design problems. The results in both benchmark functions and engineering problems show that the AFT algorithm has stronger performance than other competitors’ algorithms.

Full text

ORIGINAL ARTICLE
A novel meta-heuristic algorithm for solving numerical optimization
problems: Ali Baba and the forty thieves
Malik Braik
1
•Mohammad Hashem Ryalat
1
•Hussein Al-Zoubi
2
Received: 29 November 2020 / Accepted: 26 July 2021
The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2021
Abstract
This paper presents a novel meta-heuristic algorithm called Ali Baba and the forty thieves (AFT) for solving global
optimization problems. Recall the famous tale of Ali Baba and the forty thieves, where Ali Baba once saw a gang of
forty thieves enter a strange cave filled with all kinds of treasures. The strategies pursued by the forty thieves in the
search for Ali Baba inspired us to design ideas and underlie the basic concepts to put forward the mathematical models
and implement the exploration and exploitation processes of the proposed algorithm. The performance of the AFT
algorithm was assessed on a set of basic benchmark test functions and two more challenging benchmarks called IEEE
CEC-2017 and IEEE CEC-C06 2019 benchmark test functions. These benchmarks cover simple and complex test
functions with various dimensions and levels of complexity. An extensive comparative study was performed between the
AFT algorithm and other well-studied algorithms, and the significance of the results was proved by statistical test
methods. To study the potential performance of AFT, its further development is discussed and carried out from five
aspects. Finally, the applicability of the AFT algorithm was subsequently demonstrated in solving five engineering
design problems. The results in both benchmark functions and engineering problems show that the AFT algorithm has
stronger performance than other competitors’ algorithms.
Keywords Optimization Meta-heuristics Nature-inspired algorithms Ali Baba and the forty thieves algorithm 
Optimization techniques
1 Introduction
Solving optimization problems is the norm in almost all
disciplines of engineering [1,2] and science [3,4], and the
need for more robust solutions is ever increasing. This
means, we need plausible algorithms that can fit the intri-
cate nature of such up-to-date scientific and engineering
challenges. When surveying the literature for the existing
optimization methods, one may find a wide variety of these
methods [5]. These range from traditional optimization
techniques that use both linear and nonlinear programming
[6], to the newer nature-inspired meta-heuristics [7,8],
each with their own strengths and weaknesses. Despite
being successful in solving well-known optimization
problems [2,9], traditional algorithms on one side suffer
from inherent dependency on gradient information and the
desperate need for a promising initial starting vector within
the search space [2,9]. The existing nature-inspired meta-
heuristic optimizers, on the other side, are highly problem
dependent in that they might be very successful in solving
certain problems and may not be able to provide satisfac-
tory solutions for other problems [10]. This is partly
ascribed to the common behavior of these meta-heuristics
of being trapped in local or suboptimal solutions [11].
What makes contemporary optimization problems tough
in nature can be summarized in a few points. It is very
likely that such problems are nonlinear in essence, trou-
blesome in nature, contain numerous decision variables,
&Malik Braik
mbraik@bau.edu.jo
Mohammad Hashem Ryalat
ryalat@bau.edu.jo
Hussein Al-Zoubi
halzoubi@yu.edu.jo
1
Al-Balqa Applied University, Al-Salt, Jordan
2
Yarmouk University, Irbid, Jordan
123
Neural Computing and Applications
https://doi.org/10.1007/s00521-021-06392-x(0123456789().,-volV)(0123456789().,-volV)

their objective functions are, in some cases, complex and
handcuffed with several constraints, in addition to having
multifarious peaks [12]. For these problems, it is impera-
tive to go ahead from an encouraging starting point with
the hope of finally hitting the global optimum solution.
After many years of research, the research community has
found that traditional methods might not represent them-
selves as the best catch for solving these contemporaneous
optimization problems [13]. More specifically, the found
solution must be accurate enough to be accepted, and the
time needed to solve the problem should fall within rea-
sonable ranges [14]. To this end, the researchers have
turned their attention toward nature-inspired meta-heuris-
tics that have shown extremely heartening capabilities in
dealing with very knotted shapes of challenging opti-
mization problems [15,16]. Meta-heuristic techniques are
global optimization methods designed based on simula-
tions and methods inspired by nature that are openly
applied to solve global optimization problems [17,18].
In contrast to traditional algorithms, meta-heuristic
methods have become startlingly very trendy. This repu-
tation is due to the fact that these methods are very flexible,
do not require gradient information and have proven their
success in escaping from local minimums when solving
real-world scientific or engineering problems that have
several local solutions [11,19]. It is important to note that
the first and second merits stand out from the verity that
meta-heuristics tackle optimization problems by assuming
them as black boxes in that they only require knowledge of
the input and output sets of the variables. Thereby, there is
no necessity to calculate a derivative of the search space.
Also, meta-heuristics belong to the family of stochastic
optimization methods, in which they make use of the
stochastic operators. This feature has been broadly
affirmed, in which meta-heuristics have proven successful
in keeping away from local minima when addressing real
problems that often have a large number of local mini-
mums [11,18]. This explains their eligibility to handle
challenging optimization problems in diversified domains
[20,21]. More closely, meta-heuristics have been har-
nessed to tackle hard real-life problems in a variety of
scientific and engineering disciplines. Examples of such
domains encompass, but are not limited to, image pro-
cessing [22,23], signal processing [24], the realm of pro-
cess control [25], text clustering [26], classification
problems [27] as well as several other domains [28,29].
1.1 Motivations of the work
According to the ‘‘no-free-lunch’’ (NFL) theory [30], it is
difficult to employ a single meta-heuristic algorithm in
striving to solve all possible optimization problems [31].
As it really is, one meta-heuristic algorithm might do a
good job in optimizing certain problems in particular fields,
but it falls short to find the global optima in other fields
[11]. This has been a motive for the researchers in this
field, as well as ourselves, to look for new and innovative
nature-inspired methods to solve and show superior scores
on the current and new hard real-life problems [32]. The
door is still open, and here we present a novel meta-
heuristic algorithm based on human behavior with the very
famous tale of Ali Baba and the forty thieves, as our
inspiration targeting numerical optimization problems.
1.2 Contributions of the work
The core of this paper is to establish a novel nature-inspired
algorithm, referred to as Ali Baba and the forty thieves
(AFT), to solve global optimization problems. As its name
glimpses, AFT falls into the category of human-based
algorithms, as it is inspired by human interactions and
human demeanor in a human-related story. The thieves’
behavior, in the tale of Ali Baba and the forty thieves, in
finding Ali Baba and the intelligent methods that Ali
Baba’s maid used to save him from the thieves, inspired us
to simulate this behavior with an optimization algorithm. In
this anecdote, there is a behavior that has many similarities
with optimization processes. From the point of view of
optimization, the thieves are the search agents, the envi-
ronment (i.e., town of Ali Baba) is the search space, each
position of the town corresponds to a realizable solution,
the home of Ali Baba is the objective function and Ali
Baba is the global solution. Based on these similarities, the
AFT algorithm was developed to mimic the behavior of the
thieves and the maid to locate the global solution to the
considered optimization problems. The performance of
AFT was evaluated on sixty-two benchmark test functions,
and it was applied to optimize the designs of five engi-
neering problems.
Section 2presents the literature and related works.
Section 3shows the tale of Ali Baba and the forty thieves
and the key concepts of this tale. Section 4presents the
mathematical models and analysis of the AFT method.
Some of the possible expansions of AFT from several
aspects are given in Sect. 5. Section 6then presents a
conceptual comparison of AFT with other existing opti-
mizers. The experimental, qualitative and statistical anal-
ysis results are introduced in Sect. 7. Section 8presents the
applicability and reliability of AFT in solving five engi-
neering problems. The conclusion comments and some
further research paths are shown in Sect. 9.
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2 Related works
This section looks at the most recent developments in the
field of optimization. There are many sections in this field
such as multi-objective, single-objective, constrained and
others [33]. Since the meta-heuristic algorithm proposed in
this work is turned to solve single optimization problems,
the chief hub of this section concerns the relevant works in
single optimization areas.
2.1 Single-objective optimization problems
In single-objective optimization problems, there is only one
objective to be maximized or minimized. This kind of
optimization might be subject to a set of constraints, which
fall into two categories: equality and inequality [11].
Without loss of generality, single-objective optimization
problems can be expressed as a minimization or a maxi-
mization problem. The search space is created using a set
of variables, objectives, range of variables and constraints.
For optimization problems, the search space can be easily
plotted in the Cartesian coordinate system and its shapes
can then be observed. Having a large number of decision
variables is the first challenge when addressing optimiza-
tion problems. The limitation of the search space is the
range of variables, which is diversified. These variables can
be discrete or continuous. This means that they either
create a discrete or a continuous search space. In the first
case, there is a finite set of points between two points in the
search space, while in the second case, there is an infinite
number of points between every two points [11].
Usually, an optimization method might begin with an
initial range and extend it while optimization. The con-
straints restrict the search space even more, and typically
lead to breaks in the search space because the solutions in
those areas are not appropriate when solving an optimiza-
tion problem. A set of constraints can even divide the
search space into various detached areas. The solutions that
penetrate the constrained regions are named infeasible
solutions, while the solutions in the constrained regions are
named feasible solutions. There are two terms for the
portions of the search space that are within and out of the
constrained regions: feasible and infeasible. A restricted
search space has the potency to render the optimization
method ineffective in spite of its sensible performance in
an unrestricted search space [34]. Thus, optimization
methods must be well prepared with adequate operators to
deal effectively with the constraints [34]. Another chal-
lenge that arises when tackling optimization problems is
the existence of local solutions.
In a single-objective search space, there is usually the
global optimal solution that returns the best objective
value. However, there are normally several other solutions
that yield values close to the objective value of the global
optimal [33]. This type of solutions is named local solu-
tions as it is locally the best solution if we take into account
the search space in its vicinity. On the other hand, it is not
the best solution globally when taking into account the
whole search space. The existence of local solutions in
optimization problems bring many optimization algorithms
to fall into local solutions [8]. A real search space generally
contains a large number of local solutions. Thus, an opti-
mization method must be able to efficiently averting them
to find the global optimum. An optimization algorithm that
is able to eschew local solutions is not necessarily capable
of converging to the global optimum. The approximate
position of the global optimal is found when an algorithm
averts local solutions. The convergence speed is also a
difficulty when solving optimization problems [8]. Need-
less to say, rapid convergence leads to local optima stag-
nation. In contrast, abrupt variations in the solutions result
in avoiding local optima, but slow down the convergence
rate toward the global optimal. These two trade-offs are the
key challenges that an optimization algorithm handles
while addressing optimization problems. There are other
varieties of difficulties when addressing a single-objective
search space such as: isolation of the optimum, dynamic
objective function and many more [11]. Each of these
challenges demands special attention. These conceptions
are outside the scope of this paper, so solicitous readers are
referred to the studies conducted by Boussaid [33].
2.2 Single-objective optimization algorithms
In the literature, optimization algorithms can be split into
two broad categories:
•Deterministic algorithms these algorithms always
locate the same solution for a particular problem if
they commence with the same starting point. The main
merit of these methods is the reliability as they
decidedly find a solution in each run. However, local
optima stagnancy is a flaw as these algorithms do not
typically contain random behaviors when solving
optimization problems.
•Stochastic algorithms these algorithms benefit from
stochastic operators. This leads to find a different
solution at each run even if the starting point, in the
runs, remains unaltered and thus makes stochastic
methods less reliable as compared to the deterministic
methods. However, the stochastic behavior has the
vigor to avoid the local optimums. The reliability of
stochastic algorithms can be boosted by adjusting and
rising the number of runs. Stochastic methods fall into
two classes:
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2.2.1 Individualist algorithms
The stochastic method starts and carries out optimization
with a single solution. This solution is randomly changed
and enhanced for a predefined number of steps or realiza-
tion of a final criterion. The most well-respected algorithms
in this class are Tabu search [35], hill climbing [36] and
iterated local search [37]. The most chief feature of the
algorithms in this set is the low computational effort and
the need for few function evaluations.
2.2.2 Collective algorithms
Collective techniques generate and develop multiple ran-
dom solutions during optimization. Usually, the collection
of solutions collaborates to better identify the global opti-
mum in the search domain. Multiple solutions reduce the
chance to slack in local optima [38]. This is a key merit of
these algorithms. However, each of the solutions requires a
single function evaluation, where building an efficient
cooperation between the solutions is a challenge. Despite
these two flaws, collective stochastic optimization methods
are widely used in optimization problems [11]. This is due
to the well coveted features of these algorithms.
Irrespective of the distinctions between collective
algorithms, they all pursue the same course of action for
finding the global optimum. Optimization first begins with
a pool of random solutions, which need to be combined and
changed at random, quickly, and suddenly. This elicits that
the solutions move globally. This stage is called explo-
ration of the search space because the solutions are
attracted toward various areas of the search space by abrupt
changes [38]. After sufficient exploration, the solutions
begin to sparingly change and move locally around the
most promising solutions of the search space in order to
raise their quality. This phase is called exploitation, and its
key aim is to enhance the precision of the best solutions got
in the exploration phase [39]. Although avoidance of local
optima may occur in the exploitation phase, the coverage
of search area is not as broad as the exploration occur. In
this case, the solutions evade local solutions in the prox-
imity of the global optimal. So we can deduce that the
exploration and exploitation phases pursue inconsistent
goals [40]. So, most of the methods seamlessly demand the
search agents to transit from exploration to exploitation
using adaptive strategies. A convincing recommendation
for good performance is to achieve an adequate balance
between them [38]. Due to the random behavior of the
meta-heuristics, they can be deemed as stochastic collec-
tive algorithms [21,41]. Continuing from this last point,
any meta-heuristic algorithm can fall into:
•Physics-Based (PB) algorithms these methods utilize
the physical foundations present on Earth, in particular,
or in our universe, at the broadest level. The general
technique of PB methods is different from other meta-
heuristics’ mechanisms because the search agents of
these methods contact around the search space accord-
ing to physics rules firmed up in physical processes.
Some of the most prominent examples of PB algorithms
include simulated annealing (SA) algorithm [42,43],
gravitational search algorithm (GSA) [44], multi-verse
optimizer (MVO) [45], Henry gas solubility optimiza-
tion (HGSO) [46] and equilibrium optimizer (EO) [47].
•Evolutionary Algorithms (EAs) EAs follow the Dar-
winian theory of natural selection and biological
Darwinism that represents the survival of the fittest,
where the well-known evolution mechanisms in biology
are simulated. These methods often do well at finding
near-optimal solutions in view of the fact that they do
not lend any credence about the underlying fitness
landscape. The list of EAs includes, but not limited to,
evolutionary strategy (ES) [48], genetic algorithm (GA)
[49,50], genetic programming (GP) [51] and differen-
tial evolution (DE) algorithm [52].
•Swarm Intelligence (SI) algorithms these algorithms use
the intelligence of the social collective behavior of
various societies of creatures such as birds, bees, ants
and the alike. This class includes a large variety of
algorithms such as particle swarm optimization (PSO)
[53], ant colony optimization (ACO) [54], artificial bee
colony (ABC) algorithm [55], grey wolf optimizer
(GWO) [56], dragonfly algorithm (DA) [57], salp
swarm algorithm [11,58], coral reefs optimization
(CRO) [59] and many others [7,8].
•Human-based algorithm the algorithms of this class
originate from human interactions in societies. The
inspiration for researchers in the realm of human-based
algorithms comes from experiences and stories related
to human demeanor and human actions [60]. Previous
works in this area include harmony search (HS) [61],
seeker optimization algorithm [62], Human Group
Formation (HGF) [63], Social-Based Algorithm
(SBA) [64], Interior Search Algorithm (ISA) [65] and
the football game inspired algorithm (FGIA) [66].
3 Inspiration
The proposed AFT algorithm is based on the well-known
tale of Ali Baba and the forty thieves. We have found in
this anecdote several intrinsic traits that inspired us to
develop the AFT algorithm. Rather than literally retelling
the story in this section, we prefer to link some of the
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123

events that took place in the tale to the attributes that
constitute the AFT algorithm. The tale itself can be found
in several books as well as web pages. We refer the
interested readers to [67], as an example.
The nature of the tale is a search-based, where a gang of
forty thieves go after Ali Baba. The ultimate goal of the
gang is to catch Ali Baba for revenge and to get their
treasure back. The search carried out by the gang for Ali
Baba is iterative in nature, going in several rounds, each
time reinforcing the solution found by previous iterations.
The search is based upon the collective behavior of the
thieves, represented as a population in the proposed algo-
rithm. The counter measures took by the main character of
the tale, named Marjaneh, prevented the gang at each
iteration from fulfilling their search mission. The big town,
where Ali Baba lives, represents the search space. The tale
shows the success of the forty thieves in tracking down Ali
Baba and spotting the location of his house. The looters’
successful actions in achieving their strategic target and
finding Ali Baba were based on smart tricks and tactics.
However, the acumen of the savvy maid of Ali Baba,
Marjaneh, saved the life of Ali Baba in each of these tac-
tics, abridged as follows [68].
In the first trial, the gang’s assistant captain disguised
himself as a foreigner and entered the town in the hope to
hear any talk or find any clue that could lead him to Ali
Baba. He managed to get to Ali Baba’s house and marked
his door with an ‘X’ sign in his pursuit to later fulfill the
ultimate mission of the gang. Marjaneh observed the mark
and, in response, she placed similar marks on the doors of
all houses in the neighborhood, rendering the plan useless.
The tactics followed by the robber are harnessed in the
proposed algorithm to maximize the exploration efficiency.
The second trial took place when another assistant of the
captain took the mission. He built upon the procedures
previously followed by his comrade. In our algorithm, this
is used by making the search in every iteration builds upon
the best solution found so far from previous iterations. This
time, the robber marked Ali Baba’s house with a sign that
is not easy to be observed by chance. In AFT, this again
reflects on the utilization and enhancement of previously
found solutions [69]. This leads to strong exploration and
exploitation features in the proposed algorithm.
The third incident occurred when the captain decided to
change the plan, and take upon himself the task of cap-
turing Ali Baba, yet to build upon the achievements
attained so far by his two assistants. The captain and his
followers succeeded in arriving at Ali Baba’s house, and
they took every possible measure not to be discovered
while attacking Ali Baba, but they failed.
The final trial in targeting Ali Baba took place by the
captain alone with a totally new plan, who disguised as a
merchant selling silks, and introduced himself to the son of
Ali Baba. The perseverance and persistence of the gang’s
captain are good traits for a successful search technique.
The approaches adopted in the tale, such as the attempts
of the thieves and Marjana’s intelligence to disrupt these
attempts, are reflected in the exploration and exploitation
mechanisms built into the proposed algorithm. This has led
to the mathematical models developed to design AFT and
perform optimization. The proposed algorithm is described
in detail below.
4 Ali Baba and the forty thieves algorithm
The overall goal of this work is to present a new opti-
mization method that imitates the tale of Ali Baba and the
forty thieves as a coordinated model of social behavior of
humans’ actions. The following principles derived from
this tale achieve the basic assumptions of this algorithm:
•The forty thieves collaborate in a group and get
guidance from someone or from one of the thieves to
find Ali Baba’s house. This information may or may not
be correct.
•The forty thieves will travel a distance starting from an
initial distance until they can locate Ali Baba’s house.
•Marjaneh can deceive the thieves many times with
astute ways to somehow protect Ali Baba out of arrival
of them by a proportion.
The behaviors of the thieves and Marjaneh can be drawn up
in such a manner that they can be linked to an objective
function to be optimized. This makes it feasible to evolve a
new meta-heuristic algorithm as detailed below.
4.1 Random initialization
The AFT algorithm is initiated by randomly initializing the
position of a number of nindividuals in a d-dimensional
search space as shown below:
x¼
x1
1x1
2x1
3... x1
d
x2
1x2
2x2
3... x2
d
.
.
..
.
..
.
..
.
..
.
.
xn
1xn
2xn
3... xn
d
2
6
6
6
6
4
3
7
7
7
7
5ð1Þ
where xis the position of all thieves, dis the number of
variables of a given problem and xi
jrepresents the jth
dimension of the ith thief.
Neural Computing and Applications
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The initial position of population (i.e., thieves) can be
generated as shown in Eq. 2.
xi¼ljþrðujljÞð2Þ
where xiis the position of the ith thief that denotes a
candidate solution to a problem, ljand ujrefer to the lower
and upper bounds in the jth dimension, respectively, and
ris a uniformly distributed random number in the range
from 0 to 1.
The wit level of Marjaneh with respect to all thieves can
be initialized as shown below:
m¼
m1
1m1
2... m1
d
m2
1m2
2... m2
d
.
.
..
.
..
.
..
.
.
mn
1mn
2... mn
d
2
6
6
6
6
4
3
7
7
7
7
5ð3Þ
where mi
jdenotes the astute level of Marjaneh in relation to
the ith thief at the jth dimension.
4.2 Fitness evaluation
The values of the decision variables are inserted into a
user-defined fitness function that is evaluated for each
thief’s position. The corresponding fitness values are stored
in an array as given in the following form:
f¼
f1ð½x1
1;x1
2;...;x1
dÞ
f2ð½x2
1;x2
2;...;x2
dÞ
.
.
..
.
..
.
..
.
.
fnð½xn
1;xn
2;...;xn
dÞ
2
6
6
6
6
4
3
7
7
7
7
5ð4Þ
where xn
dis the dth dimension of the position of the nth
thief.
In the simulation of the AFT algorithm, the solution
quality is evaluated for each thief’s new location based
upon a defined fitness function. After that, the location is
updated if it is better than the solution quality of the current
one. Each thief stays in his current location if his solution
quality is more efficient than the new one.
4.3 Proposed mathematical model
As discussed above, three fundamental cases may occur
while thieves search for Ali Baba. In each case, it is
assumed that the thieves search efficiently throughout the
surrounding environment, while there also a proportion that
occurs due to Marjaneh’s intelligence that forces the
thieves to search in random locations. The above searching
behavior can be mathematically modeled as follows:
Case 1 The thieves may track down Ali Baba with the
help of information obtained from someone. In this case,
the new locations of the thieves can be obtained as follows:
xi
tþ1¼gbesttþTdtbesti
tyi
t

r1
þTdtyi
tmaðiÞ
t

r2sgnðrand 0:5Þ;
r30:5;r4[Ppt
ð5Þ
where xi
tþ1represents the position of thief iat iteration
ðtþ1Þ,yi
tis the position of Ali Baba in relation to thief iat
iteration t, besti
trepresents the best position that has
achieved so far by thief iat iteration t,gbesttrepresents the
global best position obtained so far by any thief up to the tth
iteration, maðiÞ
trepresents Marjaneh’s intelligence level
used to camouflage thief iat iteration t,Tdtis the tracking
distance of the thieves at iteration t,Pptdenotes the per-
ception potential of the thieves to Ali Baba at iteration t,
rand, r1,r2and r4are random numbers generated with a
uniform distribution between 0 and 1 , r30:5 gives either
1 or 0 to indicate that the information obtained to the
thieves is true or false, respectively, and sgnðrand 0:5Þ
gives either 1 or -1 to change the direction of the search
process.
The parameter ain maðiÞ
tcan be defined as follows:
a¼dðn1Þrandðn;1Þe ð6Þ
where rand(n, 1) represents a vector of random numbers
generated with a uniform distribution in the range of [0, 1].
Marjaneh updates her astute plans if the quality of the
new solution that the thieves come up with is better than
their previous position. In this case, Eq. 7can be used to
update her’s plans.
maðiÞ
t¼
xi
tif f xi
t

fm
aðiÞ
t

maðiÞ
tif f xi
t

\fm
aðiÞ
t

8
>
<
>
:ð7Þ
where fðÞ stands for the score of the fitness function.
The tracking distance parameter Tdtis defined as given
in Eq. 8.
Tdt¼a0ea1ðt=TÞa
1ð8Þ
where tand Tdenote the current and maximum number of
iterations, respectively, a0(a0¼1) represents the initial
estimate of the tracking distance at the first iteration and a1
Neural Computing and Applications
123

is a constant value used to manage exploration and
exploitation capabilities.
Equation 8shows that Tdtis iteratively updated during
the course of iterations of the AFT algorithm. Figure 1
shows the values of Tdtover 1000 iterations.
The tracking distance, as shown in Fig. 1, greatly affects
the search ability, which has a large impact on both the
exploration and exploitation power of the AFT algorithm.
As presented in Fig. 1, the parameter Tdtstarts from a
value of 1.0 and goes down to the lowest value where it
assumes that the thieves have arrived at Ali Baba’s house.
Large values of Tdtresult in global search that can be
diverted toward further exploration, and this may avoid
local optimal solutions. On the other side, small values of
Tdtlead to local search, where this increases the
exploitation ability in AFT so that the thieves have a good
possibility to find Ali Baba.
Similarly, the perception potential parameter Pptwas
defined as given in Eq. 9.
Ppt¼b0logðb1ðt=TÞb0Þð9Þ
where b0(b0¼0:1) represents a final rough estimation of
the probability that the thieves will realize their target at
the end of the iterative process of AFT and b1is a constant
value used to manage exploration and exploitation
capabilities.
Figure 2shows the values of Pptover 1000 iterations.
As shown in Fig. 2, by gradually increasing the value of
Ppt, AFT tends to move from global search to local search
in the most promising areas where a potential solution
could be found in these areas. In other words, large values
of Pptlead to local search that intensifies the search in the
most appropriate areas of the search space. On the other
side, small values reduce the possibility of searching in the
vicinity of current good solutions. Thus, an increase in this
value stimulates AFT to explore the search space on a
global scale and to diversify the search in all areas of the
search space.
For all of the problems solved in this work, a1and b1are
both equal 2.0. These parameters are found by experi-
mental testing for a large subset of test functions.
Case 2 The thieves may grasp that they have been
deceived, so they will randomly explore the search space
for Ali Baba. In this case, the new locations of the thieves
can be obtained as follows:
xi
tþ1¼Tdtujlj

rand þlj

;r30:5;r4Pptð10Þ
The parameter Tdtis incorporated in Eq. 10 because the
thieves have a good level of knowledge to discern of the
most propitious areas of the search space where Ali Baba’s
house could be.
Case 3 In order to ameliorate the exploration and
exploitation features of the AFT algorithm, this study also
considers the search in other positions than those that could
be obtained using Eq. 5. In this case, the new locations of
the thieves can be obtained as follows:
xi
tþ1¼gbesttTdtbesti
tyi
t

r1
þTdtyi
tmaðiÞ
t

r2sgnðrand 0:5Þ;
r3\0:5
ð11Þ
The pseudo-code of the AFT algorithm can be briefly
described by the iterative steps given in Algorithm 1.
0 100 200 300 400 500 600 700 800 900 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 1 Proposed exponential iterative function for Tdt
0 100 200 300 400 500 600 700 800 900 1000
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fig. 2 Proposed exponential iterative function for Ppt
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Algorithm 1 reveals that AFT initiates optimization in
solving an optimization problem by randomly generating a
set of positions (i.e., potential solutions), considering the
upper and lower bounds of the problem variables. After
that, the best position, the global best position of the
thieves and Marjaneh’s wit plans are initialized. The
quality of each created solution is assessed using a pre-
defined fitness function, whereby the suitability of each
solution is recalculated within each iteration in order to
identify the thief with the optimal solution. For each
dimension, the new position of the thieves is computed
iteratively within each iteration using Eqs. 5,10 and 11.
The feasibility of each new position is examined to see if it
moves out of the search area. In such a context, it will be
brought back to the boundary on the basis of the simulated
steps of AFT. Then, the new position, the best position, the
global best position of the thieves and the wit plans of
Marjaneh are assessed and updated accordingly. All the
steps of AFT shown in Algorithm 1, except the initializa-
tion steps, are iteratively performed until the termination
evaluation condition is reached. At the end, the best posi-
tion of the thieves is scored as a solution of the optimiza-
tion problem.
4.4 Exploration ability of AFT
There are many parameters in AFT that lead to exploration,
explained as follows:
•Tdt: this parameter controls the exploration quantity of
AFT. It identifies the extent to which the new locations
of the thieves would be to the house of Ali Baba. The
selection of appropriate values for a0and a1for Tdt
would reduce the recession probability in local optima
and augment the probability of approaching the global
optimum. Based on the experimental tests, a0¼1 and
a1¼2 offers a good balance between exploration and
exploitation.
•Ppt: this parameter underlines the high exploration
capacity of AFT when it takes relatively small values.
This parameter is gradually increased during the
iterative process of AFT. The choice of the values for
b0and b1in Pptis a little bit arbitrary, but was selected
based on pilot testing for a large set of test functions. In
the initial iterations, the candidates are all far away
from each other in distance. Updating the parameter Ppt
improves AFT’s ability to avoid stagnation entrapment
in local optima and approaches the global optimum.
Based on empirical testing, b0¼0:1 and b1¼2 present
a good balance between exploration and exploitation.
•sgnðrand 0:5Þ: this parameter manages the direction
of exploration. Since rand takes values between 0 and 1
with a uniform distribution, there is an equal probability
of negative and positive signs.
•Marjaneh’s intelligence plans: Using this parameter will
directly improve the AFT’s ability for exploration.
4.5 Exploitation ability of AFT
The key parameters that help to perform local search and
exploitation in AFT can be described as follows:
•Tdt: as iteration passes, exploration fades out and
exploitation fades in. Small values of Tdtlead to local
searches in promising areas of the search space. As a
result, at the last iterations, where thieves are close to
the house of Ali Baba, the positioning updating process
with cases 1, 2 and 3 will assist in local search around
the best solution, leading to exploitation.
•Ppt: this parameter controls the exploitation feature, by
quantifying the quantity of exploitation through in-
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123

depth search around the best solution. With the passage
of iterations, the exploitation stage heightens with
facing relatively large values of this parameter. Thus,
the positioning updating process with cases 1 and 2
enhances AFT’s ability to locally ramp up the searches
in the space, which results in further exploitation.
•sgnðrand 0:5Þ: this parameter controls the quality of
exploitation by changing the direction of the search.
4.6 Computational complexity analysis
Computational complexity of an optimization method can
be defined by a function that links the runtime of the
optimization method to the input size of the problem. To do
this, Big-O notation is applied here as a widespread ter-
minology. In this, the time complexity of AFT can be given
as follows:
OðAFTÞ ¼Oðproblem def:ÞþOðinitializationÞ
þOðtðpos:updateÞÞ þ Oðtðcost functionÞÞ
þOðtðboundary controlÞÞ ð12Þ
As Eq. 12 suggests, the time complexity of AFT relies on
the number of iterations (t), the number of thieves (n), the
dimension of the problem (d) and the cost of the objective
function (c). In concrete terms, the overall time complexity
of AFT under the termination method can be computed as
follows:
OðAFTÞ ¼Oð1ÞþOðndÞþOðvtndÞ
þOðvtncÞþOðvtndÞð13Þ
where vdenotes the number of evaluation experiments.
The number of iterations (t) is typically greater than the
number of thieves (n), the cost of the fitness function (c)
and the number of problem’s variables (d). Also, the
number of problem variables (d) and the cost of the fitness
function (c) are usually less than the number of thieves (n).
Accordingly, the parameters tand nare important factors
in assessing the computational complexity. As nd tnd
and nd tcn, the items 1 and nd can be excluded from the
complexity issue given in Eq. 13; also, 2vtnd ffivtnd.
Therefore, the time complexity of AFT can be expressed as
follows:
OðAFTÞffiðvtnd þvtncÞð14Þ
As it is shown, the complexity issue of the AFT is of the
polynomial order, which can be deemed as an effective
meta-heuristic optimization algorithm.
4.7 Characteristics of AFT
Human-based algorithms possess two abilities, that is,
exploration and exploitation. This is to optimize the search
space of a problem. In AFT, these abilities are realized by
the convergence of the thieves toward the global optimum
solution. To be precise, convergence means that most of
the thieves gather in the same position in the search space.
AFT utilizes several parameters that lead to exploration
and exploitation as explained in Subsects. 4.4 and 4.5,
respectively. These parameters are beneficial for carrying
out the convergence process of AFT. The thieves (i.e.,
search agents) in AFT can change their position in line with
a mathematical model and tuning criteria as implemented
by three basic cases that may occur while thieves search for
Ali Baba. These cases are presented in Eqs. 5,10 and 11.
In each case, it is assumed that the thieves search effi-
ciently throughout the surrounding environment, while
there is also a percentage that occurs due to Marjaneh’s
intelligence that forces the thieves to search in random
locations. There are two important parameters in AFT,
referred to as tracking distance and perception potential
that are presented in Eqs. 8and 9, respectively. With these
two parameters, AFT can better search the space for all
possible solutions to identify the optimal or suboptimal
solutions. Another important parameter in AFT is the
simulation of Marjaneh’s intelligent ways to deceive the
thieves. Thereby, the thieves will explore the search space
in different locations and directions, which implies that
better solutions may be found in other promising areas. In
short, AFT has several distinct merits according its basic
principle, summarized as follows: (1) The position updat-
ing models for case 1 and case 3 of AFT effectively assist
the individuals of the population to explore and exploit
every area in the search space. (2) The random search that
thieves use in the search space using case 2 not only
enhances the diversity of the population but also ensures
the speed of convergence, indicating an efficient balance
between exploration and exploitation. (3) The number of
parameters in AFT is small, but they have good ability to
improve its strength and performance. (4) The computa-
tional burden of AFT is low as discussed in Subsect. 4.6.
As a result, there is a big room for enhancing the per-
formance of AFT according to the above mentioned char-
acteristics, as presented in the following section.
5 Possible developments of AFT
To further study the potential performance of the AFT
algorithm, it is elaborated from several aspects as shown in
the following subsections.
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5.1 Self-adaptation of tracking distance of AFT
Self-adaptive tracking distance is used to strike a better
balance between exploration and exploitation during the
search process [70]. This distance decreases as a function
of time indicating that the exploration capacity gradually
fades out and exploitation capacity gradually fades in.
However, the search agents trapped in local optima area
demand reasonable exploration to escape from this local
optima. Some search agents require a large tracking dis-
tance to explore the search space and others exploit the
local area with a small tracking distance. So, it is imper-
ative for each search agent to have its own tracking dis-
tance to balance exploration and exploitation. When the
fitness value of a search agent is worse or unaltered, it
denotes that the search agent can identify the local optimal
area. In this, the search agents require a large tracking
distance to move away from this area. When the fitness
value of a search agent ameliorates, it indicates that the
search agent has a superior chance of getting close to the
optimal solution. Hence, the value of the tracking distance,
Td(t), of the search agents of AFT should be grown. Fig-
ure 3presents an illustration of the self-adaptive tracking
distance [70].
In Fig. 3, the pentagram, sphere, circle and arrowhead
stand for the global optimum area, the local optimum area,
search agent and tracking distance, respectively. Case 1
implements that if a search agent is trapped into local
optimal area, it needs an appropriate tracking distance to
raise its strength to escape from this area. On the other
hand, Case 2 indicates that a search agent rapidly moves to
the global optimal area with mounting tracking distance. In
this context, to estimate the situation of the search agents in
AFT, the counters nsiand nfiare presented in Eqs. 15 and
16, respectively, to record the fitness results of the ith
search agent [70].
nst
i¼nst1
iþ1if fiðtÞ\fiðt1Þ
0if fiðtÞfiðt1Þ
ð15Þ
nf t
i¼nf t1
iþ1if fiðtÞ[fiðt1Þ
0if fiðtÞfiðt1Þ
ð16Þ
Equations 15 and 16 are presented to adjust the parameter
Td(t) according to the fitness of the objective function,
where it is employed to estimate the search condition of
search agents. If a search agent fails to obtain a better
solution in many iterations, the search agent gets stuck in
the local optima area with a high probability. If a search
agent is improved in many iterations, it may migrate to the
global optima.
A threshold hand probability pare applied to dominate
the update of the tracking distance of search agents over the
course of iterations. If nst
iexceeds h, the tracking distance
will be increased to speed up the convergence of the ith
search agent toward the super search agent. Likewise, if nf t
i
exceeds h, the tracking distance is enhanced to mend the
ability to avert the local optima while searching. The self-
adaptive TdiðtÞof the ith search agent is defined in Eq. 17:
TdiðtÞ¼ TdiðtÞriðtÞif count [h&rand\p
TdiðtÞotherwise
ð17Þ
where count comprises two counters nst
iand nf t
i, and rand
is a uniformly distributed random value in the interval
[0, 1].
In Eq. 17, when count overrides hand rand overrides p,
the search agent requests a large tracking distance to
strengthen its exploration capability, where it is multiplied
by riðtÞ. Otherwise, the updated tracking distance is set to
the tracking distance of AFT.
The tracking distance of search agents is related to the
distance that thieves use to follow Ali Baba as given in
Eq. 8. Also, there is another parameter that is related to the
perception potential of the thieves for Ali Baba. Therefore,
the tracking distance and perception potential constants of
the ith search agent are used to adapt its TdiðtÞ. The ratio of
tracking distance and perception potential constants of the
search agents can be thought of as the new tracking dis-
tance value for the search agents. Here, riðtÞstands for the
adjusted value of TdiðtÞand is presented in Eq. 18:
riðtÞ¼
2
cif c\1
cc1
8
<
:ð18Þ
where cis the ratio of tracking distance and perception
potential constants as shown in Eq. 19.
c¼log TdiðtÞ
PpiðtÞ

ð19Þ
where TdiðtÞand PpiðtÞare the tracking distance and per-
ception potential of search agent xiat iteration t,
respectively.
Equation 18 states that riðtÞis set to the reciprocal of c
when cis less than 1 to ensure that the search agent can
Fig. 3 Illustration of self-adaptive tracking distance in AFT [70]
Neural Computing and Applications
123

have a reasonably large tracking distance, otherwise, riðtÞ
is set to c.
The original and self-adaptive tracking distance con-
stants are presented in Figs. 1and 4, respectively.
The tracking distance constant in Fig. 4is changed in
accordance to the search condition of the search agents,
where the red and blue lines indicate a modification of Case
1 and Case 2, respectively. The search agent is in a state of
failure when the search agent gets a worse search status. In
contrast, the search agent is in a state of success when the
search agent gets better search status.
In sum, in the search process, various search agents have
different search cases. Some of them want vigorous
exploration to explore the solution space, while others want
extensive exploitation to locate a better solution. Therefore,
each search agent adjusts its Td to balance exploration and
exploitation.
5.2 Population hierarchical structure of AFT
The hierarchical structure of the population is that the
search agents are arranged and placed in different layers
according to some specific characteristics [71,72]. These
layers can be stated as different levels from top to bottom
according to the actual effect of each layer on the search
agents. The top layer leads its next layer, and its next layer
leads its second one, and so on. In this light, the interactive
relationship between the layers is created to form a hier-
archical structure that is used to guide the evolution
direction of the search agents [72]. Here, a hierarchical
population structure was used in the AFT algorithm, so that
premature convergence could be mitigated and search
agents could elicit correct interactive information to realize
a better development. Further, the search agents have
ample opportunity to escape from local optima and get
close to the global optimum. Here, a three-layer
hierarchical structure was constructed for hierarchical AFT
(HAFT) as follows:
•Bottom layer The distribution of all search agents in the
current population is displayed on this layer. Search
agents move toward better ones in terms of the best
search agents on the middle layer. This layer can
divulge the landscape of the function created by a big
number of search agents.
•Medium layer For the effective guidance of the
development of general search agents, the best prede-
termined search agents are ranked on this layer. At each
iteration, the medium layer leads the bottom layer to
fulfill the position update for the search agents. Each
search agent needs a large tracking distance and small
perception potential to globally explore the whole
search area in the first few iterations of the search
process, and these parameters are gradually updated
over iterations. This means that the exploration ability
of AFT wants to be supported by a large suitable Td(t)
and its exploitation capacity demands a small one.
Thus, to improve the exploration ability of AFT, a new
Td(t) is presented to supersede the original one given in
Eq. 8. In the proposed HAFT, a log-sigmoid transfer
function was used to design a new constant Td(t) with
the formula given below:
TdðtÞ¼ T0
1þe
tt
2
Lð20Þ
where Lis a step length. It can be observed that the
value of Td(t) in the graph shown in Fig. 1decreases
rapidly before 500 iterations, indicating the exploration
ability of AFT is rapidly diminishing. On the other side,
the value of Td(t) in Fig. 5consistently preserves a
large value before 500 iterations and then drops rapidly
to near zero. This effect can ensure a powerful explo-
ration ability of AFT in the early stage so that it has
100 200 300 400 500 600 700 800 900 1000
0.5
1
1.5
2
2.5
3
3.5
Fig. 4 Proposed exponential function for the self-adaptive tracking
distance
0 100 200 300 400 500 600 700 800 900 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 5 The new tracking distance parameter Td(t)
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enough time to search for an approximate optimal that
can be further improved by its next exploitation capa-
bility. Since the number of the best search agents
gradually decreases with iterations, which means that
the number of best search agents dynamically decreases
on this layer. It is useful for the global optimum search
agent on the top layer to effectively direct many elite
search agents in the current population to provide a
better development tendency for all of the search agents
on the bottom layer.
•Top layer To provide efficient management for the
middle layer, a global optimum search agent is
determined and placed on this layer. At each iteration,
the pre-identified best search agents on the medium
layer are selected to be compared with the global
optimum search agent. In case there is a better search
agent, the optimal global one is replaced by it. The
global optimal search agent has the best position in the
current population. Hence, it can attract many of the
best search agents to move toward it. This strategy
could prevent the best search agents from being trapped
in the local optima and could accelerate the conver-
gence speed of the population. According to this
strategy, the formula of updating the position of the
best search agents oriented by the global optimal search
agent is presented as follows:
xi
tþ1¼gopttTdtyi
tmi
t

r2sgnð0:5rÞ
r3\0:5ð21Þ
where gopt denotes the global optimal search agent and
rrepresents a random number with a uniform distribution
in the interval [0, 1].
In HAFT, Eq. 21 was proposed to update the position of
the search agents on the top layer which could efficaciously
alleviate the early convergence of AFT and improve its
performance.
In order to further clarify the apparent properties of
HAFT, Fig. 6was drawn to show its operating precept on
the multimodal landscape with local optima.
It is obvious from Fig. 6a that search agents x3and x4
are heading toward a local optimal, while the global opti-
mal search agent gopt provides additional guidance to assist
them escape from the local optimal. When search agents x3
and x4do not fall into a premature convergence as shown
in Fig. 6b, they can draw others to move toward them.
Meanwhile, gopt will accelerate the movement of x3and x4
in order to improve the convergence rate of the population.
Thus, the capabilities of exploration and exploitation can
be enhanced in the search process. More details about
hierarchical population structure can be found in [71,72].
5.3 Exploration and exploitation
Exploration and exploitation are the two most important
properties of meta-heuristic algorithms to achieve success
when addressing optimization problems [38]. With regard
to these two concepts, empirical experiments have shown
that there is a robust relationship between the exploration
and exploitation ability of a particular search method and
its convergence speed. Particularly, while exploitation
procedures are known to improve convergence toward the
global optimum, they are also known to rise the likelihood
of entrapment into local optima [38]. Conversely, search
strategies that promote exploration over exploitation
incline to increase the likelihood of locating areas within
the search space, where the global optimum is more
probable to be identified. This is at the cost of deteriorating
the convergence speed of optimization algorithms [39]. In
recent years, the question of how exploration and
exploitation of solutions is realized in meta-heuristics has
remained an open subject, and although it appears trivial, it
has stayed as a source of contention among many
researchers [40]. Although many thoughts and notions may
sound opposite, there appears to be a common consent
within the research community on the conception that an
adequate ratio between exploration and exploitation is
necessary to ensure reasonable performance in this type of
search methods.
Meta-heuristics use a set of candidate solutions to
explore the search area with the goal of finding satisfying
solutions for an optimization problem. Generally, the
search agents with the superior solutions are liable to guide
the search process toward them. As a result of this attrac-
tion, the distance between the search agents fades in while
the impact of exploitation fades out. On the other side,
when the distance between the search agents increases, the
influence of exploration strategy is more pronounced. To
compute the increase and decrease in distance between the
search agents, a diversity measurement [73] is taken into
account. Under this method, population diversity is stated
as follows [38]:
(a) (b)
Fig. 6 The illustrative diagrams of the operating precept of HAFT
[71]
Neural Computing and Applications
123

Divj¼1
NX
N
i¼1
medianðxjÞxj
i
ð22Þ
Div ¼1
mX
m
j¼1
Divjð23Þ
where medianðxjÞrepresents the median of dimension jin
the entire population, xj
iis the dimension jof search agent i,
nstands for the number of search agents and mdenotes the
number of design variables of the optimization problem.
The diversity in each dimension, Divj, is stated as the
distance between the dimension jof each search agent and
the median of that dimension on average. The full balance
response is defined as the percentage of exploration and
exploitation utilized through a given algorithm. These
values are calculated at each iteration using the following
formulas [38]:
XPL%¼Div
Divmax

100 ð24Þ
XPT%¼Div Divmax
jj
Divmax

100 ð25Þ
where Divmax stands for the maximum diversity value
present in the whole optimization process.
The percentage of exploration (XPL%) corresponds to
the relationship between the diversity at each iteration and
the maximum diversity reached. The percentage of
exploitation (XPT%) represents the level of exploitation
[38]. As can be observed, both elements XPL% and XPT%
are mutually conflicting and complementary. In assessing
the balance response, the use of the median value averts
discrepancies through the use of a reference element. This
balance response is also affected by Divmax that is found
during the whole optimization process. This value is
employed as a reference to assess the rate of exploration
and exploitation.
5.4 Chaos in meta-heuristic algorithms
Chaos theory is one of the most effective strategies used to
improve the performance of meta-heuristics by fostering
their exploration and exploitation features. Chaos appears
to exhibit irregular motion, a characteristic often encoun-
tered in nonlinear dynamic systems [74,75]. It appears to
be random, unexpected behavior that a deterministic non-
linear system can present under deterministic conditions.
Thus, a chaotic system alters randomly and ultimately
passes through each state in the search space when the time
period is long enough. The applications of chaos in global
optimizers fall into two categories.
5.4.1 Chaotic maps and sequences
Chaotic maps are one of the preferable ways to reinforce
the performance score of meta-heuristics in terms of both
local optima avoidance and convergence property. They
are widely used to improve population diversity and solu-
tion quality by substituting random values and adjusting
parameters in the initialization of population and iterative
loop procedures [74,75]. Chaotic properties have been
used in improved and new meta-heuristics, such as EAs
[74,75], the immune system algorithm [76], PSO [77] and
DE [78]. These chaotic meta-heuristics have received a
high level of performance through the use of chaotic
sequences to replace random variables and parameters. In
this, they presented superb performance compared to the
other corresponding standard meta-heuristics.
5.4.2 Chaotic local search
Chaotic local search (CLS) appears as an applicable option
by making use of randomness and ergodicity of chaos
[74,75]. Chaotic search is a mechanism that could be
conducted to improve the accuracy of the search and
convergence speed. For this reason, CLS has been inte-
grated with several meta-heuristic algorithms and achieved
splendid success in enhancing their performance, such as
chaotic PSO [79], chaotic DE [80] and chaotic GSA [81].
Their outcomes showed that CLS could prominently
strengthen search capacity and dwindle the problem of
getting into local optima. It has been widely demonstrated
that meta-heuristics with CLS achieved better performance
in terms of convergence rate and solution accuracy than the
other corresponding original versions [74,75].
5.5 Theoretical analysis of the AFT algorithm
To theoretically analyze the performance of the AFT
algorithm from the perspective of complex network, there
is a need to establish a relationship among its search
agents. This analysis is helpful and vital to explain its
essence and discover some guiding methods to overcome
its limitations in order to better foster the performance of
this algorithm [82]. For this intent, we use the population
interaction network (PIN) method reported in [83,84]to
put in place the relationship between the search agents of
AFT. This is for exploring and analyzing the intrinsic
phenomenon that occurs in a complex network. A clus-
tering method was used to classify the search agents
[83,84]. In this method, each search agent can be regarded
as a vertex and the update position mechanism between the
search agents denotes the generation of edges. The PIN
method can be used to obtain both the intrinsic connection
of knowledge and characteristic of the network formed by
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the population. The method that composes the interaction
of population in AFT is displayed in Fig. 7.
In Fig. 7, blue circles, transparent circles, transparent
rectangles, blue rectangles, and blue diamond shapes
implement the search agents (i.e., vertices) in the current
population, the clusters, search agents xpthat will be
replaced, the search agents that have been replaced and the
new created search agent Un, respectively. The circle,
square and triangle denote the current, new constructed and
old substituted vertices, respectively. It is noticed from
Fig. 7that the distribution of search agents has changed in
the whole population and that the number of search agents
in each class changes accordingly. In sum, the initial
construction process of PIN can be described as stated
below:
1. There are three classes and nine basic search agents in
the population;
2. Two chosen search agents, xs1and xs2, from two
classes yield the new created search agent Unfor
comparison with the previous search agent, xp.A
vertex and two edges are created at the same time;
3. If the search agent Unoverrides the search agent xp,in
which ðfðUnÞ\fðxpÞÞ, then Unreplaces xp;
4. Another search agent xsis chosen from one class to
create Unto replace xp, which means creating a vertex
and an edge;
5. The replacement process is carried out once more when
fðUnÞ\fðxpÞ;
6. At the next iteration, the clustering method resumes
classifying the search agents into three classes to
finally get to terminate the algorithm and obtain the
PIN topology.
Readers can read [83,84] for a detailed description of the
PIN method.
6 Comparative analysis of AFT with other
meta-heuristics
This section presents a comparative analysis of AFT with
other meta-heuristics such as PSO, GSA, DE, GA,
covariance matrix adaptation-evolution strategy (CMA-ES)
and ant colony optimization (ACO) algorithm.
6.1 Particle swarm optimization
PSO [85] mimics the cooperative social collective behavior
of the living creatures such as flocks of birds. Optimization
begins with the use of randomly generated solutions known
as artificial particles. Each particle in the swarm has a
randomly generated velocity. If xiis the initial position of
the ith particle with velocity vi, then the position updating
strategy of PSO can be given as follows [86]:
viðtþ1Þ¼wviðtÞþc1ðPbestixiðtÞÞr1
þc2ðGbest xiðtÞÞr2ð26Þ
xiðtþ1Þ¼xiðtÞþviðtþ1Þð27Þ
where wis the inertial weight, c1and c2are cognitive and
social constants, respectively, r1and r2are distributed
random numbers in the interval [0, 1], Pbestiis the local
best solution of the ith particle and Gbest is the global best
solution among all particles.
6.1.1 AFT versus PSO
Similar to PSO, AFT initiates the optimization process by
motivating the search agents to move in the search space in
search of their target. However, the positioning updating
mechanism of AFT is entirely different from that of PSO.
Some of the main differences are described as follows:
1. In PSO, the movement update of the ith particle is
obtained by Pbestiand Gbest as given in Eq. 26, where
the effect of these two parameters is considered to
identify the new position of the particles in the search
space. In regard to AFT, the new position of the search
agents are obtained through three different cases as
given in Eqs. 5,10 and 11. In other words, PSO
updates all solutions with one strategy as presented in
Eq. 27, while the search agents of AFT use three
strategies to update their position in the search space.
2. The PSO algorithm is greatly influenced by the initial
values of the cognitive and social parameters as well as
the weighting strategy of the velocity vector, where
these parameters are used as the particle develops a
new position. In the AFT model, the thieves develop a
new position with the help of tracking distance which
Fig. 7 A descriptive schematic diagram of PIN method in the AFT algorithm [83,84]
Neural Computing and Applications
123

is adapted during its iterative loops. This enables AFT
to alternate between local and global searches.
3. The behavior of the thieves’ movement is affected by
the information given by someone about the where-
abouts of Ali Baba’s house, which is designed with a
random number (r3). Accordingly, case 1 or case 3
shown in Eqs. 5or 11 are used, respectively. Inclusion
of this random number in the AFT model suddenly
redirects thieves’ movement and thus improves explo-
ration and exploitation in AFT. On the other hand, PSO
does not use such behavior.
4. Simulation of thieves’ behavior in Eq. 10 imparts an
opportunity to present a random behavior of thieves’
movement. This enables AFT to mitigate stumbling in
local optimum areas. This behavior is not used in PSO
due to the natural behavior of swarms.
5. The use of Marjaneh’s intelligence that is formulated in
Eq. 7improves the exploration feature of AFT, where
there is no such thing in the PSO algorithm.
6. The use of tracking distance and perception probability
in AFT enables it to conduct local searches in local
areas at some times, exploration of the search space on
a global scale at other times as well as getting an
appropriate balance between exploration and exploita-
tion features. These two parameters are not present in
the PSO algorithm.
6.2 Gravitational search algorithm
Gravitational search algorithm is a physics-based algorithm
evolved on the basis of the law of gravity [44]. Each
individual (i.e., agent) evolves its position according to the
gravitational force among individuals. The mechanism of
GSA is based on the interaction of masses in the universe
by means of the Newtonian law of gravitation. To describe
GSA, consider a system with Nmasses (i.e., agents), where
the position of the ith mass is defined as follows:
Xi¼ðx1
i;x2
i;...;xd
iÞi21;2;3;...;Nð28Þ
where xd
idenotes the position of the ith mass in the dth
dimension and drepresents the total number of dimensions
in the search space.
The mass of the ith agent is computed after calculating
the fitness of the current populations, which is defined as
follows:
miðtÞ¼ fiðtÞworstðtÞ
bestðtÞworstðtÞð29Þ
MiðtÞ¼ miðtÞ
PN
j¼1mjðtÞð30Þ
where MiðtÞand fiðtÞrepresent the mass and fitness values
of the ith agent at iteration t, respectively, best(t) and
worst(t) represent the best and worst fitness values of the
current population in the tth iteration, respectively, where
worst(t), for a minimization problem, is defined as follows:
worstðtÞ¼maxfitjðtÞ;j21;2;...;N
fgð31Þ
The gravitational force between agents Xiand Xjin the dth
dimension can be computed as follows:
Fd
ijðtÞ¼GðtÞMiðtÞMjðtÞ
RijðtÞþxd
jðtÞxd
iðtÞ

ð32Þ
where RijðtÞstands for the Euclidean distance between
agents iand j,is a small value used to eschew division by
zero and G(t) is a gravitational constant given as a function
of time as shown below:
GðtÞ¼G0eat
Tð33Þ
where G0represents an initial value, arepresents a constant
value, and tand Trepresent the current iteration number
and the maximum number of iterations, respectively. The
total gravitational force Fd
iðtÞfor agent Xiis given as
follows:
Fd
iðtÞ¼ X
j2Kbest;j6¼i
randiFd
ijðtÞð34Þ
where Kbest refers to a group of the first Kbest agents with
the best fitness value and biggest mass is kbest,Kis the
agent number of Kbest and randiis a uniformly distributed
random number in the range [0, 1].
Hence, the acceleration ad
iðtÞof agent Xiin the dth
dimension at time tcan be computed using a law of motion
as shown in Eq. 35.
ad
iðtÞ¼Fd
iðtÞ
MiðtÞð35Þ
Then, the velocity vd
iðtþ1Þand position xd
iðtþ1Þof agent
Xiare updated, respectively, as follows:
vd
iðtþ1Þ¼randivd
iðtÞþad
iðtÞð36Þ
xd
iðtþ1Þ¼xd
iðtÞþvd
iðtþ1Þð37Þ
where randiis a uniformly distributed random value in the
interval from 0 to 1.
6.2.1 AFT versus GSA
1. As shown in Eqs. 5,10 and 11, the position updating
mechanism of the search agents of AFT is totally dif-
ferent from that of GSA as defined in Eq. 37.
2. GSA uses acceleration and velocity vectors for the
movement of its agents, while AFT generates new
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directions of movement for its search agent by various
mechanisms.
3. There are some own adaptive parameters for AFT such
as Tdtand Ppt. However, GSA does not use these
parameters, where it has its own parameters such as 
and RijðtÞ.
4. AFT algorithm incorporates the concept of random
movement of the thieves based on the parameter Ppt,
and uses Marjaneh’s intelligence in its position updat-
ing mechanism. Obviously, the generation of new
solutions by AFT seems very different from the update
mechanism of the agents of GSA.
6.3 Conventional differential evolution
algorithm
The differential evolution (DE) algorithm is a population-
based evolutionary algorithm evolved to solve real-valued
optimization problems [87]. The evolutionary process of
DE involves evolutionary concepts such as mutation,
crossover and selection strategies similar to those used by
GAs. The initialization of each individual Xi;i2
1;2;...;NP
fg
in DE is described as follow:
Xd
i¼Xdl
iþrandð0;1ÞðXu
iXl
iÞð38Þ
where NP is the population size, d21;2;...;D
fg
denotes
the dimension of the problem, uand lrepresent the upper
and lower bounds of Xiin the dth dimension, respectively.
The mutation strategy of DE can characteristically cre-
ate a mutant vector to be an intermediate variable Vifor
evolution according to:
Vi¼Xr1þFðXr2Xr3Þð39Þ
where r1;r2and r321;2;...;NP
fg
are random indices,
i6¼ r16¼ r26¼ r3and Fis a constant operator that indicates
the level of amplification.
The crossover strategy of DE that can boost the diversity
of new agent Uiby combining the original agent Xiwith
the intermediate variable Vican be defined as follows:
Ud
i¼Vd
iif randð0;1ÞCR or d¼drand
Xd
iotherwise
(ð40Þ
where CR represents a crossover control parameter and
drand 2½1;2;...;D
fg
denotes a random number.
The selection process in DE is performed in each iter-
ation by contrasting Uiwith Xiusing a greedy norm for a
better agent reserve in the population for the next iteration.
Through these evolutionary processes, DE could rapidly
converge and eventually obtain the global optimum.
6.3.1 AFT versus DE
Generally speaking, since AFT is a human-based opti-
mization algorithm, so there is no need for evolutionary
processes such as crossover, mutation and selection oper-
ations. The main differences between DE and AFT can be
briefed by the following points:
1. The AFT algorithm preserves search space information
over subsequent iterations, while the DE algorithm
discards the information of previous generations once a
new population is formed.
2. AFT involves fewer operators to adjust and run as
compared to DE that uses several operations such as
selection and crossover. Moreover, AFT utilizes a
parameter denoting Marjaneh’s plans, while DE does
not memorize the best solution obtained so far.
3. In DE, exploration is enhanced using crossover and
selection operations, while in AFT, it is enhanced by
allowing the thieves to randomly explore the search
space.
4. In DE, mutation is generally implemented on the basis
of enhancing the exploitation of DE. However, a better
exploitation of AFT is achieved with the perception
probability parameter.
6.4 Genetic algorithm
GA was first put forwarded by Holland [88]. It is consid-
ered as a global optimization algorithm inspired by bio-
logical mechanisms such as evolution and genetics. When
using GAs, the search space is used to construct chromo-
somes, whereby every possible solution is coded as a
chromosome (i.e., individual). In optimization with GA,
evolution begins with a group of randomly formed indi-
viduals from a population. The fitness score of each indi-
vidual is computed in each generation. The variables of the
solutions are adjusted based on their fitness values. Since
the best individuals are given higher probability to partic-
ipate in enhancing other solutions, the random initial
solutions are very probable to be improved. Based on a
fitness function, chromosomes are selected and certain
genetic operators such as mutation and crossover are
applied to the selected chromosomes to form new ones.
The idea is that these chromosomes evolve and always
create better individuals until they reach the global opti-
mum [89].
6.4.1 AFT versus GAs
Both GAs and AFT are population-based techniques;
however, the key differences between them can be briefed
as follows:
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123

1. Since GA is similar to DE in which it uses crossover,
mutation and selection operations; the AFT algorithm
does not use these operations.
2. The AFT algorithm uses Marjaneh’s intelligence, while
GA does not use such a parameter and does not save
the best solutions obtained so far.
3. GA evolves and updates its population using crossover,
mutation and selection operations, while AFT
improves its exploration ability with the concept of
random relocation of the thieves that is managed by a
perception probability parameter.
6.5 Covariance matrix adaptation-evolution
strategy
The CMA-ES [90,91] is an evolutionary algorithm for
nonlinear non-convex optimization problems in continuous
domain. Specifically, it is a second-order approximation
algorithm that estimates the derivatives of a covariance
matrix within an iterative procedure according to the
principle of maximum likelihood. By doing this, it tends to
maximize the probability of the distribution. At each iter-
ation, the members of the new population are sampled from
a multivariate normal distribution Nwith covariance C2
Rnnand mean m2Rn. The new individuals at generation
iþ1 are sampled as:
xiþ1
kmiþriN0;Ci
 k¼1;...;kð41Þ
where riis the ith step size and xi
kis the kth individual at
generation i.
The sampled points, k, are ranked in ascending order of
fitness, and the best points, l, are chosen. The mean of the
sampling distribution given in Eq. 41 is updated using
weighted intermediate recombination of these specified
points:
miþ1¼X
l
j¼1
xjXiþ1
j:kð42Þ
with
X
l
j¼1
xj¼1;x1x2...wl[0ð43Þ
where xjare positive weights, and xiþ1
j:kstands for the jth
ranked individual of the ksampling points xiþ1
k. The sample
weights of the standard CMA-ES implementation are
decreased as:
xj¼log k1
2þ1

logðjÞð44Þ
The covariance matrix can be adapted for the next
generation using a combination of rank-land rank-one
update as follows:
Ciþ1¼ð1ccovÞCiþccov
lcov
piþ1
cpðiþ1ÞT
c
þccov 11
lcov

X
l
j¼1
xjyiþ1
j:kðyiþ1
j:kÞTð45Þ
With lcov 1 is the weighting between rank-one update
and rank-l,ccov 2½0;1is the learning rate for the
covariance matrix update, and yiþ1
j:k¼ðXiþ1
j:kmiÞ=ri. The
evolution path piþ1
cand riare identified by an adaptation
formula [91].
6.5.1 AFT versus CMA-ES
Both CMA-ES and AFT are population-based techniques;
however, the major differences between them are sum-
marized as shown below:
1. The CMA-ES basically parameterizes the multivariate
normal distribution Nðtextbfm;r2CÞwhich consists of
three terms: the mean vector m, the step-size rand the
covariance matrix C. On the other hand, the AFT
algorithm does not use these components, rather it uses
Eqs. 5,10 and 11 to update the position of its search
agents.
2. The CMA-ES uses two evolution paths that accumulate
consecutive steps of the mean vector update for the
cumulative step-size adaptation and the rank-one
update of the covariance matrix. However, the AFT
algorithm does not use such these evolution paths.
3. AFT uses some concepts to assist alternating between
local and global solutions in the update of its search
agents’ position. However, the CMA-ES algorithm
uses a covariance matrix that can be adapted for the
next generation using an integration of rank-land
rank-one.
6.6 Ant colony optimization
ACO is a meta-heuristic algorithm that distributes the
search activities to so-called ‘‘ants’’ [92]. The activities are
split among agents with simple basic abilities that imitate,
to some extent, the behavior of real ants in foraging. It is
crucial to underline that ACO has not been developed as a
simulation of ant colonies, but to employ the metaphor of
artificial ant colonies and their application as an opti-
mization tool. At the start of processing in ACO, where
there is no information about the path to go from one point
to another, the choice of ants about which path to walk in is
totally random. During processing, the intention is that if
an ant has to choose between different paths at a given
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point, those that have been chosen heavily by the preceding
ants (i.e., those with a high trail level) are chosen with a
higher probability. Generally, ACO approach tries to
address an optimization problem by repeating the next two
steps:
– A pheromone model, as a specific probability distribu-
tion, was used to evolve the candidate solutions over
the solution space;
– The candidate solutions are utilized to adjust the
pheromone values in a manner that is deemed to bias
future sampling toward higher quality solutions.
The choice of ant agents in constructing solution compo-
nents using a pheromone model is probabilistically defined
at each construction step. An ant moves from node ito
node jusing the following rule:
qði;jÞ¼sa
ði;jÞnb
ði;jÞ
Pðsa
ði;jÞÞðnb
ði;jÞÞð46Þ
where sa
ði;jÞis the pheromone value associated with edge
(i,j), nb
ði;jÞis the heuristic value associated with edge (i,j),
ais a positive real parameter whose value identifies the
relative importance of pheromone versus heuristic infor-
mation and controls the influence of sa
ði;jÞ,bis a positive
parameter that identifies the relative importance of pher-
omone versus heuristic information and controls the
influence of nb
ði;jÞ.
Once the solution is built, the ant evaluates the partial
solution to be used using Eq. 47 that specifies how much
pheromone to deposit.
sði;jÞ¼ð1qÞsði;jÞþdsði;jÞð47Þ
where sði;jÞis the pheromone value correlated with edge
(i,j), q2ð0;1is the pheromone evaporation rate and
dsði;jÞis the amount of pheromone deposited, typically
given by:
sk
ði;jÞ¼1=LKif ant Ktravels on edge ði;jÞ
0 otherwise
ð48Þ
where LKis the cost of the Kth ant’s tour.
6.6.1 AFT versus ACO
ACO and AFT apparently look similar but are quite dif-
ferent. They both present several differences in their for-
mulation and position updating mechanism.
1. Both ACO and AFT work on the effective division in
the search for the optimal solution. In ACO, the idea is
to create a pool of artificial ants that move randomly
around an environment. In AFT, the thieves search for
Ali Baba by exploring and exploiting each area in the
search space using three different cases in the position
updating process.
2. In ACO, the candidate solutions are constructed using a
pheromone model. However, AFT uses the local and
global best solutions of the thieves to locate the
optimum solutions.
3. In ACO, a new solution is created by Eq. 48, which is
conceptually different from the position updating
strategy of AFT given in Eqs. 5to 11. The updating
strategy of AFT is a kind of directed and undirected
search approach, in which new solutions are forced to
move toward a better solution.
4. The AFT algorithm uses a memory parameter (i.e.,
Marjaneh’s plans) in its updating process. On the other
hand, ACO does not use such a parameter in updating
its new solutions. Apart from this, AFT also uses a
stochastic location updating strategy as shown in
Eq. 10 to improve its exploration feature, while ACO
does not use such a strategy.
5. AFT has two parameters that can be adapted during its
iterative process to enhance exploration and exploita-
tion features and to balance them. However, ACO does
not use any parameters to be adapted over the course of
iterations.
As previously discussed, an effective meta-heuristic must
strike an appropriate balance between exploration and
exploitation. However, there is no rule of thumb [93]to
make this happen. The slight differences in solutions
update and random distributions could have a significant
effect on the performance of the designed algorithms [94].
Therefore, AFT becomes a good competitor to the existing
meta-heuristics.
7 Experimental results and analysis
In this section, to assess the accuracy of the proposed AFT
algorithm, we conducted intensive evaluations on a set of
62 test functions, involving commonly used unimodal,
multimodal, hybrid and composition functions. These
functions involve: (1) 23 benchmark test functions (de-
scribed in Table 33 in ‘‘Appendix A’’), (2) 29 functions
taken from IEEE CEC-2017 benchmark functions [95]
described in Table 34 in ‘‘Appendix B’’, and (3) a set of 10
IEEE CEC-2019 functions (described in Table 35 in
‘‘ Appendix C’’). These functions are over and over used in
the literature to test the performance of any new meta-
heuristic algorithm. The experiments designed to verify the
performance of AFT are outlined as follows:
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– First, exhaustive comparative studies were presented to
verify the reliability and accuracy of the AFT algorithm
in relation to other meta-heuristics.
– Second, a set of qualitative measures including search
history, trajectory, convergence curves and average
fitness values were plotted to examine the adequacy of
AFT in addressing several types of test functions.
– Third, the optimization performance of AFT was
studied in light of several development applied to
AFT from several aspects.
– Forth, Friedman’s and Holm’s test methods were used
to verify the significance of the outcomes obtained by
AFT.
7.1 Experimental setup
The results produced by AFT in the optimization of the
above three mentioned benchmark functions are compared
with those produced by other well-regarded algorithms.
The parameter definitions of AFT algorithm and those
comparative algorithms are given in Table 1.
In order to provide a fair comparison between the pro-
posed algorithm and another selected set of algorithms, we
followed the same initialization process for all the com-
pared algorithms. For all experiments, the common
parameter settings for all algorithms are set as follows: The
number of individuals used in the search process is set to
30, the number of iterations is set to 1000, and the maxi-
mum number of function evaluations (NFEs) for all
benchmark functions is set to d103, where drepresents
the dimension of the test functions. For each function, the
executed runs for each algorithm are repeated 30 times
independently to obtain the statistical results. The stop
condition for all algorithms was set to the maximum
number of iterations. The average fitness (Ave) and stan-
dard deviation (Std) values were computed over thirty
independent runs to explore the accuracy and stability of
the proposed algorithm compared to others. The best scores
are shown in bold throughout this paper.
7.2 Classical benchmark test functions
Twenty-three widely used benchmark test functions were
used to evaluate the overall performance of AFT and to
compare it to other optimization algorithms. These test
functions were carried out under minimization problems
which can be categorized into three main kinds: unimodal
[99], multimodal [96] and fixed-dimension multimodal
[99]. ‘‘Appendix A’’ presents the mathematical description
of these categories. The characteristics of the unimodal
(F1–F7), multimodal (F8–F13 ) and fixed-dimension multi-
modal (F14–F23 ) functions are detailed in Table 33. These
three groups of test functions are widely accepted in the
literature as benchmark evaluation functions due to:
– The functions in the first group (i.e., unimodal
functions) have only one global optimum with no local
optimum. This group is usually used to assess the
convergence behavior as well as the exploitation power
of any new or enhanced optimization algorithm.
– The functions in the second and third sets (i.e.,
multimodal and fixed-dimension multimodal functions)
have several local optimum and more than one global
Table 1 Parameter setting values of the AFT algorithm and other
algorithms
Algorithm Parameter Value
All algorithms Population size 30
Number of iterations 1000
AFT a0,a11.0, 2.0
b0,b10.1, 2.0
MFO [96] Convergence constant [-1, -2]
Logarithmic spiral 0.75
SOA [97] Control parameter (A) [2, 0]
fc2
CSA [98] flight length (fl) 2.0
Awareness probability (AP) 0.1
SHO [99] Control parameter (h) [5, 0]
MConstant [0.5, 1]
GWO [100] Control parameter (a) [2, 0]
MVO [45] Wormhole Existence Prob. [0.2, 1]
Traveling distance rate [0.6, 1]
SCA [101] Number of elites 2
PSO [85]c1,c21.8, 2
w[0.9, 0.1]
GSA [44] Alpha, G020, 100
Rnorm, Rpower 2, 1
GA [102] Crossover 0.9
Mutation 0.05
DE [87] Crossover 0.9
Scale factor (F) 0.5
SSA [11] Control parameter (c1) 0.5
DA [57] Coefficient (s) [0.2, 0]
Coefficient (e) [0.1, 0]
Coefficients (a,c,f) [0, 0.2]
WOA [103] Control vector (a) [2, 0]
ACO [92]b,q,a[2, 0.5, 0.3
CMA-ES [90]k,l4þ3lnðnÞ,k=2
wi¼1;...;lln kþ1
2

lnðiÞ
cc,ccov 4=ðnþ4Þ,2
ðnþffiffi2
pÞ2
cr,dr4=ðnþ4Þ,c1
rþ1
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optimum. These functions are effective in examining
the ability of AFT to avoid the local optimums and in
evaluating its exploration feature.
The average processing times elapsed by AFT and other
comparative algorithms in optimizing the classical bench-
mark test functions are given in Table 2and graphically
illustrated in Fig. 8.
It can be seen that AFT outperforms other algorithms as
it takes less processing time than other algorithms.
Therefore, it can be deduced that the computational effi-
cacy of AFT is much better than other competitors.
7.2.1 Performance of AFT in unimodal functions
As discussed above, the unimodal functions (F1–F7) are
beneficial to assess the exploitation capability of opti-
mization algorithms. The average fitness and standard
deviation results of AFT and other algorithms are displayed
in Table 3. These values are recorded after running the
experiments 30 times for each algorithm.
It can be seen from Table 3that the performance of AFT
is very efficient compared to other competitors. In partic-
ular, the AFT algorithm scored the best scores for functions
F4,F
5and F6. Also, it achieved the best score equally with
SHO algorithm for functions F1,F
2and F3. From an
engineering point of view, it does not matter whether the
algorithm finds the optimal result such as 10E-05 or 10E-
25; both are considered zero. It is clear that the SHO
algorithm is a competitive algorithm such that it got the
best scores for functions F1and F3. Although SHO got the
best results for functions F1and F3, the difference between
it and AFT is of a very small value. The SOA algorithm
obtained the best results for function F7. Once again, the
proposed AFT algorithm got the best standard deviation
values for functions F4–F6. These outcomes clearly indi-
cate that the AFT algorithm is robust, reliable and highly
effective compared to other widely used algorithms in the
literature. The AFT algorithm achieved either the best
score or competitive results in functions F1–F7. The very
small values of STD of the AFT algorithm reveal the high
level of stability that this algorithm has.
7.2.2 Performance of AFT in multimodal functions
In order to evaluate the capabilities of optimization algo-
rithms for local optimum avoidance and exploration
potency, the multimodal (F8–F13) and fixed-dimension
multimodal (F14-F23 –) are employed in the literature as
benchmark functions for evaluating algorithms for this
purpose. The outcomes are given in Tables 4and 5to show
the Ave and Std, over 30 independent runs, for all the
compared algorithms in multimodal and fixed-dimension
multimodal functions, respectively.
The results shown in Table 4point out that the perfor-
mance of AFT is also very efficient when employed to
solve multimodal problems. Particularly, the AFT algo-
rithm scored the best scores for functions F8,F
10,F
12 and
F13. GSA got the second best score for function F13 , and
SHO got the best scores for functions F9and F11 , where the
results of SHO in these functions are nearly close to the
results released by the AFT algorithm. These outcomes
once again confirm the reliability and stability of AFT
since it has very small values for Std.
The results given in Table 5confirm the superiority of
AFT, which gained the best mean fitness values, either
individually or equally with other algorithms. The CSA
was competitive and got the best average fitness equally
with the proposed AFT algorithm in F14 ,F
16,F
17 and F18
test functions. Although the AFT algorithm did not achieve
the best average fitness values for F19 and F20 , its results
are very comparable to those algorithms that obtained the
Table 2 Average running time of AFT and other meta-heuristic
algorithms
Algorithm Average time (in seconds)
Ali Baba and the forty thieves 1.1043
Moth flame optimization 2.2362
Seagull optimization algorithm 1.2245
Crow search algorithm 1.1045
Spotted hyena optimizer 1.3269
Grey wolf optimizer 1.2536
Covariance matrix adaptation-ES 1.5114
Ant colony optimization 1.4332
Particle swarm optimization 2.0241
Gravitational search algorithm 1.2965
Genetic algorithm 1.3668
Differential evolution 1.6354
Fig. 8 Average running time of the proposed algorithm and other
optimization algorithms
Neural Computing and Applications
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Table 3 Results of the AFT algorithm and other meta-heuristics in unimodal benchmark test functions
F AFT MFO SOA CSA SHO GWO
Ave Std Ave td Ave Std Ave Std Ave Std Ave Std
F19.52E-69 2.80E-68 3.15E-04 5.00E-05 0.00E100 0.00E100 1.98E-08 2.52E-08 0.00E100 0.00E100 4.69E-47 4.50E-48
F23.5E-30 1.36E-29 3.71E?01 2.16E?01 0.00E100 0.00E100 2.12E-03 2.64E-03 0.00E100 0.00E100 1.20E-24 1.00E-24
F32.70E-42 7.67E-42 4.42E?03 3.71E?03 4.62E-19 2.30E-27 3.36E-04 5.80E-04 0.00E100 0.00E100 1.00E-14 4.18E-15
F43.36E-23 9.43E-23 6.70E?01 1.06E?01 7.35E-05 3.11E-06 1.36E-03 2.42E-03 7.78E-12 5.06E-12 2.02E-14 2.95E-15
F57.97E-01 1.12E-01 3.50E?03 2.83E?03 7.00 1.13E-01 1.70E?01 3.08E?01 8.59 5.53E-01 2.79E?01 1.84
F60.00E100 0.00E100 1.66E-04 2.49E-05 3.47E-02 1.31E-04 2.89E-08 2.57E-08 2.46E-01 1.78E-01 6.58E-01 3.38E-01
F74.81E-01 2.60E-01 3.22E-01 2.93E-01 3.35E-06 7.51E-06 4.66E-01 2.80E-01 3.29E-05 2.43E-05 7.80E-04 3.85E-04
F CMA-ES ACO PSO GSA GA DE
Ave Std Ave Std Ave Std Ave Std Ave Std Ave Std
F11.42E-01 3.13E-18 2.97E-18 2.12E-18 4.98E-09 1.41E-10 1.16E-16 6.10E-17 1.95E-12 2.04E-13 3.10E-10 3.21E-11
F22.98E-07 1.78 1.42E-15 4.41E-16 7.29E-04 1.57E-05 1.70E-01 5.29E-02 6.53E-18 4.00E-19 4.99E-13 4.81E-14
F31.59E-05 2.21E-05 2.40E-03 1.09E-02 1.40E?01 7.13 4.16E?02 1.56E?02 7.70E-10 5.46E-12 5.58E-02 1.50E-03
F42.01E-06 1.25E-06 9.80E-9 1.17E-9 6.00E-01 1.72E-01 1.12 9.89E-01 9.17E?01 5.67E?01 5.21E?01 1.01E-01
F536.79 33.46 17.08 0.16 4.93E?01 3.89E?01 3.85E?01 3.47E?01 5.57E?02 4.16E?01 2.10E?02 1.73E?02
F66.83E-19 6.71E-19 3.29E-18 5.19E-19 9.23E-09 8.48E-10 1.08E-16 4.00E-17 3.15E-01 9.98E-02 9.77E-02 5.13E-02
F72.75E-02 0.79E-02 0.39E-02 0.13E-02 6.92E-02 2.87E-02 7.68E-01 5.70E-01 6.79E-04 5.21E-04 4.00E-03 2.27E-03
The best results are written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
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Table 4 Results of the AFT algorithm and other meta-heuristic algorithms in multimodal benchmark test functions
F AFT MFO SOA CSA SHO GWO
Ave Std Ave Std Ave Std Ave Std Ave Std Ave Std
F821.11E103 2.69E102 -8.04E?03 8.80E?02 -8.50E?03 4.57E?02 -2.72E?03 3.01E?02 -1.16E?03 2.72E?02 -6.14E?03 9.32E?02
F97.70 4.72 1.63E?02 3.74E?01 3.12E?02 2.85E?0 2.30E?01 1.46E?01 0.00E100 0.00E100 4.34E-01 2.68E-01
F10 5.98E-15 2.01E-15 1.60E?01 6.18 4.22E-16 4.09E-13 1.61 9.31E-01 2.48 1.41 1.63E-14 3.14E-15
F11 1.76E-01 1.11E-01 5.03E-02 1.29E-02 0.00 0.00 1.48E-01 7.89E-02 0.00E100 0.00E100 2.29E-03 1.21E-03
F12 3.91E-17 2.14E-16 1.26 1.83 5.80E-01 4.71E-02 3.70E-01 6.43E-01 3.68E-02 1.15E-02 3.93E-02 2.42E-02
F13 1.34E-32 5.56E-48 7.24E-01 1.59E-01 8.48E-02 4.32E-04 4.30E-03 7.18E-03 9.29E-01 9.52E-02 4.75E-01 2.38E-01
F CMA-ES ACO PSO GSA GA DE
Ave Std Ave Std Ave Std Ave Std Ave Std Ave Std
F8-7007.10 773.94 1.36E?04 61.19 6.01E?03 1.30E?03 2.75E?03 5.72E?02 5.11E?03 4.37E?02 3.94E?03 5.81E?02
F925.33 8.55 15.34 9.56 4.72E?01 1.03E?01 3.35E?01 1.19E?01 1.23E?01 1.01E?01 8.50E?01 1.00E?01
F10 15.58 7.92 1.17E-09 5.12E-09 3.86E-02 2.91E-0 8.25E-09 1.90E-09 5.31E-11 9.10E-12 7.40E-06 5.44E-07
F11 5.76E-15 6.18E-15 4.11E-04 5.12E-05 5.50E-03 4.32E-04 8.19 3.70 3.31E-06 1.43E-06 7.09E-04 5.58E-04
F12 2.87E-16 5.64E-16 1.72E-15 1.19E-15 1.05E-10 2.86E-11 2.65E-01 1.13E-01 9.16E-08 2.80E-08 1.06E-06 1.00E-06
F13 3.66E-04 0.2E-02 2.27E-09 2.28E-09 4.03E-03 3.01E-03 5.73E-32 1.30E-32 6.39E-02 4.49E-02 9.78E-02 3.42E-02
The best results are written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
123

Table 5 Results of the AFT algorithm and other meta-heuristic algorithms in fixed-dimension multimodal functions
F AFT MFO SOA CSA SHO GWO
Ave Std Ave Std Ave Std Ave Std Ave Std Ave Std
F14 9.98E-01 0.00E100 2.21 1.80 3.35 1.15 9.98E-01 8.24E-17 9.68 3.29 3.71 2.15
F15 3.35E-04 1.17E-04 1.58E-03 2.48E-04 4.11E-04 4.26E-05 3.38E-04 1.67E-04 9.01E-04 1.06E-04 3.66E-03 3.48E-04
F16 21.0316 6.77E-16 21.0316 0.00E100 -1.08E?01 6.48E-12 21.0316 6.11E-16 -1.06E?01 2.86E-11 21.0316 7.02E-09
F17 3.97E-01 0.00E100 3.98E-01 1.13E-03 3.98E-01 1.36E-03 3.97E-01 0.00E100 3.98E-01 2.46E-01 3.98E-01 2.42E-02
F18 2.99 1.44E-15 3.00 1.42E-07 3.00 5.49E-05 2.99 1.53E-15 3.00 8.15E-05 3.00 7.16E-06
F19 -3.86 2.51E-15 -3.86 3.16E-15 -3.88 3.00E-10 -3.86 2.65E-15 -3.75 4.39E-01 -3.86 1.57E-03
F20 -2.31 1.11E-02 -3.23 6.65E-02 -3.32 1.23E-02 -3.31 5.82E-02 21.44 5.47E-01 -3.27 7.27E-02
F21 23.46 1.09 -1.00E?01 3.52 -1.00E?01 3.47 -8.65 3.05 -1.00E?01 3.80E-01 -1.01E?01 1.54
F22 26.65 4.65E-01 -1.04E?01 3.20 -1.04E?01 2.00E-04 -10.14E?01 1.39 -1.04E?01 2.04E-04 -1.03E?01 2.73E-04
F23 25.77 1.45E-01 -1.05E?01 3.68 -1.05E?01 1.32E-01 -10.31 1.22 -1.05E?01 2.64E-01 -1.01E?01 8.17
F CMA-ES ACO PSO GSA GA DE
Ave Std Ave Std Ave Std Ave Std Ave Std Ave Std
F14 10.23 7.54 0.998 3.76E-16 2.77 2.20 3.61 2.96 4.39 4.41E-02 3.97 1.42
F15 0.57E-02 0.0121 7.48E-04 2.18E-04 9.09E-04 2.38E-04 6.84E-03 2.77E-03 7.36E-03 2.39E-04 5.01E-04 1.00E-04
F16 21.0316 6.77E-16 21.0316 4.82E-16 21.0316 0.00E100 21.0316 0.00E100 -1.04 4.19E-07 21.0316 3.60E-04
F17 3.97E-01 0.00E100 3.97E-01 3.09E-04 3.98E-01 4.11E-02 3.98E-01 1.74E-04 3.98E-01 1.00E-04 3.98E-01 8.40E-02
F18 8.40 20.55 3.74 1.62 3.00 6.59E-05 3.01 3.24E-02 3.01 6.33E-07 3.00 9.68E-04
F19 -3.86 2.70E-15 -3.860 3.87E-04 -3.90 3.37E-15 23.22 4.15E-01 -3.30 4.37E-10 -3.40 6.56E-06
F20 -3.29 0.05 -3.32 4.80E-16 -3.32 2.66E-01 -1.47 5.32E-01 -2.39 4.37E-01 -1.79 7.07E-02
F21 -5.66 3.35 -8.78 1.47 -1.00E?01 2.77 -1.00E?01 1.30 -1.00E?01 2.34 -1.00E?01 1.91
F22 -8.44 3.33 -9.57 3.80 -1.04E?01 3.08 -1.04E?01 2.64 -1.04E?01 1.37E-02 -1.04E?01 5.60E-03
F23 -8.07 3.59 -9.26 2.88 -1.05E?01 2.52 -1.05E?01 2.75 -1.05E?01 2.37 -1.05E?01 1.60
The best results are written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
123

best fitness in these two cases (i.e., GSA and SHO). The
outcomes in Table 5indicate that the average fitness values
of AFT are better than other algorithms on most of the test
functions. With regard to the values of Std, AFT performed
better than other algorithms in six out of ten test functions
(F14,F
17 to F21). This supports the fact that we previously
concluded that AFT has a high degree of stability when
applied to different search spaces. The results in Tables 4
and 5indicate that AFT is in the first rank, in terms of the
average fitness value, in twelve out of sixteen test functions
(i.e., F8,F
10,F
12 to F18 and F21–F23 ). This reinforces that
AFT has good exploration ability when it is employed in
these search problems.
7.3 Performance of self-adaptive AFT algorithm
The amendment value of the self-adaptive tracking dis-
tance of AFT algorithm, or referred to as SAFT, is com-
puted using the ratio of the current tracking distance and
perception potential constants of the search agents over the
course of iterations. The SAFT algorithm has two main
parameters, referred to as hand q, that influence the search
performance. These parameters are the adjustment fre-
quency and opportunity of the tracking distance constant,
respectively. Small values for hand qmay cause the search
agent to frequently move, which implies that the search
agent will have a difficulty in converging. On the other
hand, large values for hand qmake the search agent to lose
the globally optimum area, and converge to a locally
optimum area that leads to early convergence. To locate the
best settings for hand q, several experiments were per-
formed on some of the unimodal functions with different
values for these parameters, where h21;2;3
fg
and
q20:2;0:5;0:8
fg
. The results obtained based on these
parameter settings after running the experiments 30 times
are shown in Table 6.
It is evident from the results shown in Table 6that the
parameter settings h¼3 and q¼0:2 are the best settings.
7.4 Performance of hierarchical AFT algorithm
To illustrate the performance of the hierarchical AFT
(HAFT) algorithm, it is tested on 23 benchmark functions
with various dimensions. The parameter settings of HAFT
are given as follows: T0¼1, L¼100, n¼30 and
T¼1000. The experimental results obtained by HAFT
which consist of mean and standard deviation values are
given in Table 7.
Table 7presents that HAFT can effectively and effi-
ciently find optimal solutions for many benchmark func-
tions. This indicates that HAFT is efficacious in balancing
the search capability of AFT between exploration and
exploitation in order to boost its accuracy from the onset to
the end of the optimization process. A comparison between
HAFT in Table 7and AFT in Tables 3,4and 5shows that
HAFT has better performance.
7.5 Evaluation of the balance of exploitation
and exploration of AFT
The multimodal function with fixed dimension F16 shown
in Eq. 49 is used as an example to illustrate the evaluation
of the balance response of AFT.
F16ðx1;x2Þ¼4x2
12:1x4
1þ1
3x6
1þx1x24x2
2þ4x4
2
ð49Þ
In Eq. 49, the range of x1and x2is set to: 5xi5 with
the dimension set to 2. Figure 9shows the performance
behavior yielded by AFT in the function F16 , over 500
iterations, in terms of evaluating the balance given by
Eqs. 24 and 25.
Table 6 Results of SAFT for different values of hand qon some unimodal benchmark functions
F1 F2 F3 F4 F5
h¼2, q¼0:5 1.35E-71 ±4.21E-71 1.89E-32 ±5.13E-32 1.03E-43 ±3.21E-43 3.11E-27 ±7.01E-27 1.19E?00 ±1.92E?00
h¼1;q¼0:2 1.30E-78 ±3.73E-78 8.33E-34 ±2.15E-33 1.56E-48 ±4.62E-48 2.36E-29 ±3.52E-29 7.97E-01 ±1.68E?00
h¼1;q¼0:5 3.79E-73 ±1.00E-72 5.63E-33 ±1.48E-32 4.01E-42 ±1.26E-41 1.87E-26 ±3.85E-26 3.98E-01 ±1.26E?00
h¼1;q¼0:8 1.28E-65 ±2.35E-65 1.77E-30 ±3.23E-30 6.37E-37 ±1.92E-36 2.26E-24 ±3.59E-24 3.98E-01 ±1.26E?00
h¼2;q¼0:2 3.47E-78 ±9.82E-78 3.38E-34 ±1.06E-33 2.56E-49 ±7.18E-49 7.34E-30±1.50E-29 3.98E-01 ±1.26E?00
h¼2;q¼0:8 2.20E-65 ±3.73E-65 1.52E-29 ±3.21E-29 3.74E-38 ±7.30E-38 1.14E-23±2.33E-23 5.18E-15 ±1.13E-14
h¼3;q¼0:2 5.09E-80 ±1.09E-79 2.73E-36 ±4.68E-36 1.84E-49 ±5.79E-49 5.32E-30 ±1.05E-29 1.19E?00 ±1.92E?00
h¼3;q¼0:5 9.49E-73 ±2.58E-72 3.85E-32 ±8.39E-32 1.27E-44 ±2.91E-44 6.89E-27 ±9.79E-27 7.97E-01 ±1.68E?00
h¼3;q¼0:8 4.35E-66 ±1.22E-65 9.73E-30 ±2.34E-29 3.61E-39 ±6.10E-39 3.28E-24 ±7.29E-24 7.97E-01 ±1.68E?00
Neural Computing and Applications
123

In Fig. 9, five points (1), (2), (3), (4) and (5) have been
chosen to represent the diversity of solutions and the bal-
ance assessments of each of them. Point (1) represents a
premature stage of the AFT algorithm where the balance
evaluation values of XPL% and XPT% are 90 and 10,
respectively. With these percentage, AFT works with a
clear direction to explore the search space. With this effect,
it can be inferred that the solutions preserve a high dis-
persion of the search space. Point (2) correlates with 70
iterations, where at this position, the balance evaluation
conserves a value of XPL% = 70 in conjunction with
XPT% = 30. In this position, AFT fundamentally conducts
exploration with a low degree of exploitation. Points (3)
and (4) correspond to 75 and 100 iterations, respectively,
where the balance assessments have exploration and
exploitation values of XPL%= 25, XPT% = 75, XPL% =
05 and XPT% = 95, respectively. At these percentage, the
behavior of AFT was flipped to promote more exploitation
than exploration. Under these configurations, the solutions
are spread out in several bunches which reduces the overall
diversity. Finally, point (5) implements the last juncture of
the AFT algorithm. In such a situation, the AFT algorithm
sustains a perspicuous trend to the exploitation of the top
found solutions without taking into account any explo-
ration strategy.
7.6 Chaotic maps for AFT
This work integrates four chaotic maps into one of the
components of AFT. This is to further study and investigate
the effectiveness of chaos theory in improving exploration
and/or exploitation of AFT. Chaotic maps are applied to
define the selection probability of the defined cases of AFT
which is directed to promote its performance degree. The
four chaotic maps selected in this study are listed in
Table 8.
The set of chaotic maps shown in Table 8was selected
with different behaviors with the initial point set to 0.7 for
each of them. As mentioned earlier, chaotic maps are used
to manipulate the selection process of the defined three
cases of the AFT algorithm, where this process was defined
by a probability of rand. Here, chaotic maps were used to
provide chaotic behaviors for this probability. The final
value from the chaotic map should lie within the range of
[0, 1]. The proposed Chaotic AFT (CAFT) algorithm is
benchmarked on 23 benchmark test functions with chaotic
iterative (I), circle (C), logistic (L) and piecewise (P) se-
lection operators. The results are averaged over 30 inde-
pendent runs. The mean and standard deviation results of
the best solutions found at the last iteration are reflected in
Table 9.
Considering the results of Table 9, it can be said that
chaotic maps improve the performance of AFT not only in
terms of exploitation but also exploration. These results are
superior to the corresponding ones of the standard AFT as
shown in Tables 3,4and 5.
Table 7 Experimental results of
HAFT algorithm in standard
benchmark test functions
F Ave Std
F13.19E-74 1.73E-73
F27.65E-33 2.97E-32
F32.32E-45 9.65E-44
F41.63E-25 5.61E-24
F56.19E-01 1.05E-01
F60.00E?00 0.00E?00
F73.12E-01 2.48E-01
F8-1.07E?03 2.17E?02
F91.79E-01 1.41E-01
F10 2.00E-20 1.10E-19
F11 2.00E-01 1.08E-01
F12 3.36E-12 8.37E-12
F13 8.74E-34 4.78E-33
F14 9.98E-01 0.00E?00
F15 3.32E-04 1.13E-04
F16 -1.0316 6.77E-16
F17 3.97E-01 0.00E?00
F18 2.99 1.00E-15
F19 -3.86 2.71E-16
F20 -2.31 1.11E-02
F21 -3.48 1.19E?00
F22 -5.01 2.79E-01
F23 -6.73 1.59E-01
0 50 100 150 200 250 300 350 400 450 500
0
10
20
30
40
50
60
70
80
90
100
Fig. 9 Performance of AFT during 500 iterations which describes the
balance evaluation given by Eqs. 24 and 25
Neural Computing and Applications
123

7.7 Analysis of the number of clusters of AFT
To investigate the degree of population interaction of the
AFT algorithm using the PIN method reported in [83,84],
it is tested with different numbers of clusters cin the
optimization of a group of 23 benchmark functions. The
parameter cis set as follows: c23;7;11. The other
parameters of AFT are remained the same during each
experiment. The AFT algorithm is run 30 times for each
function, and the mean and standard deviation are obtained
and displayed in Table 10.
It can be found from Table 10 that the effect of popu-
lation interaction is reasonably positive and can signifi-
cantly augment the performance of AFT in optimization.
Table 8 Four different types of
chaotic maps used to improve
AFT
No. Name Chaotic map Range
1 Iterative [104]xiþ1¼sinðap
xiÞ;a¼0:7ð1;1Þ
2 Circle [105]xiþ1¼mod xiþbða
2pÞsinð2pxiÞ;1

;a¼0:5;b¼0:2
3 Logistic [104]xiþ1¼axið1xiÞ;a¼4 (0, 1)
4 Piecewise [106]
xiþ1¼
xi
P0xi\P
xiP
0:5PPxi\0:5
1Pxi
0:5P0:5xi\1PP¼0:4
1xi
P1Pxi\1
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
(0, 1)
Table 9 Results of the CAFT algorithm on F1–F23 with several chaotic selection operators
F CAFT-C CAFT-I CAFT-L CAFT-P
Ave Std Ave Std Ave Std Ave Std
F11.08E-67 3.19E-67 2.19E-84 4.20E-84 2.80E-56 7.37E-56 1.21E-46 6.59E-46
F21.03E-29 4.22E-29 3.07E-36 9.46E-36 3.53E-26 1.05E-25 3.07E-21 8.80E-21
F34.93E-31 2.63E-30 1.06E-41 3.11E-41 1.48E-29 4.09E-29 3.01E-14 1.65E-13
F44.94E-20 2.18E-19 5.79E-07 2.52E-06 1.188E-17 2.96E-17 2.71E-04 1.45E-03
F51.12E?00 1.64E?00 4.37E?00 2.27E?00 8.89E-01 1.47E?00 1.036E?01 1.64E?01
F60.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00
F74.84E-01 3.28E-01 5.74E-01 3.15E-01 5.87E-01 2.43E-01 4.99E-01 2.82E-01
F8-3.08E?03 2.46E?02 -3.07E?03 3.46E?02 -3.02E?03 2.17E?02 -3.04E?03 3.34E?02
F91.86E?01 7.21E?00 1.99E?01 7.81E?00 1.86E?01 7.80E?00 2.17E?01 8.20E?00
F10 3.85E-02 2.10E-01 7.70E-02 2.93E-01 6.45E-15 1.79E-15 5.49E-02 3.00E-01
F11 1.79E-01 1.01E-01 1.85E-01 1.10E-01 1.66E-01 9.25E-02 2.00E-01 1.35E-01
F12 1.14E-01 5.69E-01 9.32E-02 3.07E-01 3.71E-30 2.01E-29 9.32E-02 4.01E-01
F13 1.34E-32 5.56E-48 1.34E-32 5.56E-48 1.34E-32 5.56E-48 4.22E-23 2.31E-22
F14 9.98E-01 4.12E-17 9.98E-01 1.00E-16 9.98E-01 5.83E-17 9.98E-01 9.21E-17
F15 4.97E-04 3.70E-04 4.11E-04 3.22E-04 3.68E-04 2.32E-04 4.36E-04 3.59E-04
F16 -1.0316 6.77E-16 -1.0316 6.77E-16 -1.0316 6.71E-16 -1.0316 6.71E-16
F17 3.978E-01 0.00E?00 3.978E-01 0.00E?00 3.978E-01 0.00E?00 3.978E-01 0.00E?00
F18 2.99E?00 2.03E-15 2.99E?00 4.73E-16 2.99E?00 1.23E-15 2.99E?00 1.30E-15
F19 -3.862 2.68E-15 -3.862E?00 2.69E-15 -3.862E?00 2.71E-15 -3.862E?00 2.69E-15
F20 -3.318 2.17E-02 -3.31E?00 3.01E-02 -3.314E?00 3.01E-02 -3.31E?00 2.17E-02
F21 -5.12 2.53E-01 -8.29E?00 2.49E?00 -7.79E?00 2.56E?00 -7.95E?00 2.55E?00
F22 -5.99 2.38E-01 -8.44E?00 2.61E?00 -8.99E?00 2.37E?00 -7.43E?00 2.84E?00
F23 -6.77 1.98E-01 -9.05E?00 2.51E?00 -9.45E?00 2.19E?00 -7.92E?00 2.85E?00
Neural Computing and Applications
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The performance of AFT is the best at c¼3 according to
the mean rank while it is not seriously different from other
values of c. According to Tables 10 as well as 3,4and 5,it
is evident that AFT with the use of PIN is somewhat out-
performing the standard AFT in optimization.
7.8 Qualitative analysis of AFT
The qualitative results, including search landscapes, con-
vergence curves, average fitness curves in logarithmic
shapes, search history and trajectory of the first individual,
associated with the AFT algorithm in solving a selected set
of test functions, are shown, for up to 200 iterations, in
Fig. 10.
The qualitative metric measures used to plot the curves
in Fig. 10 can be interpreted as follows:
•Convergence curves these curves are used to describe
how well AFT converges toward the optimum global,
which demonstrates its ability in fulfilling exploration
and exploitation phases. They show the best solutions
found after each iteration. It is observed that the fitness
value of the convergence curves gets better after each
iteration. The convergence curves of the unimodal
functions F1,F
3and F5have some slight differences
with those of the multimodal functions F11 and F13 and
the fixed-dimension multimodal function F16 . This is
mainly ascribed to the fact that unimodal functions have
only one global optimum with no local optimum.
Broadly, AFT provides a fast convergence response for
F1and F3, has a sensible convergence response for F5
and F13, and shows the optimal convergence for F11 and
F16. As shown in the convergence curves, the AFT
algorithm first explores the search space, and most of
the thieves move toward the optimal areas of the search
space in the first 50 iterations.
•Average fitness curves these curves characterize the
average fitness values of the thieves at the end of each
iteration. They indicate that, as iterations proceed, the
average fitness values decline. This underscores that
AFT not only improves the global best fitness of one
thief, but also improves the fitness of all thieves.
•Search history this drawing shows the search history of
the thieves during their search for the global optimum.
It can be observed that the sample point of the unimodal
functions F1,F
3and F5are distributed in promising
Table 10 The results obtained
by the AFT algorithm with
different number of clusters (c)
Fc¼3c¼7c¼11
Ave Std Ave Std Ave Std
F12.64E-78 8.20E-78 5.0E-74 1.0E-73 4.49E-72 8.89E-72
F29.48E-35 1.87E-34 1.136E-32 2.32E-32 6.35E-30 1.62E-29
F31.83E-34 3.11E-34 2.62E-19 8.28E-19 8.81E-12 2.67E-11
F42.26E-06 3.54E-06 2.45E-04 5.23E-04 1.74E-03 3.22E-03
F55.36E?00 2.12E?00 4.87E?01 7.58E?01 4.33E?01 8.35E?01
F60.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00
F74.72E-01 3.53E-01 4.73E-01 3.24E-01 3.63E-01 2.42E-01
F8-2.90E?03 3.09E?02 -2.78E?03 2.15E?02 -2.93E?03 3.39E?02
F92.26E?01 9.05E?00 2.38E?01 6.86E?00 2.18E?01 9.59E?00
F10 9.01E-11 2.84E-10 3.64E-01 6.17E-01 9.81E-01 1.08E?00
F11 1.45E-01 4.65E-02 1.47E-01 3.19E-02 1.67E-01 8.53E-02
F12 5.92E-01 1.76E?00 9.33E-02 1.50E-01 1.07E-01 1.76E-01
F13 1.34E-32 2.88E-48 7.16E-20 2.26E-19 1.09E-03 3.47E-03
F14 9.98E-01 7.40E-17 9.98E-01 0.00E?00 9.98E-01 0.00E?00
F15 3.07E-04 1.08E-19 4.47E-04 4.04E-04 5.51E-04 4.66E-04
F16 -1.0316 0.00E?00 -1.0316 1.04E-16 -1.0316 1.04E-16
F17 3.978E-01 0.00E?00 3.978E-01 0.00E?00 3.978E-01 0.00E?00
F18 2.99 8.24E-16 2.99E?00 7.40E-16 2.99E?00 0.00E?00
F19 -3.862 9.00E-16 -3.862E?00 9.36E-16 -3.862E?00 9.36E-16
F20 -3.31 3.75E-02 -3.32E?00 4.68E-16 -3.32E?00 4.68E-16
F21 -4.12 2.62E?00 -8.11E?00 2.62E?00 -8.11E?00 2.63E?00
F22 -6.74 2.80E?00 -7.07E?00 2.89E?00 -9.20E?00 2.54E?00
F23 -5.10 3.16E?00 -9.45E?00 2.28E?00 -7.83E?00 2.84E?00
Neural Computing and Applications
123

areas. In contrast, some of the sample points of the
multimodal and fixed-dimension multimodal functions,
F11,F
13 and F16, respectively, are marginally spread
around unpromising areas. This is related to the degree
of difficulty of these test functions. As the search
history drawings demonstrate, the sample points are
0
100
1
100
10
4
0
2
0
-100 -100 0 100 200
10
-20
10
0
0 100 200
10
0
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-2000
-1000
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1000
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-10
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-10
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-100 0 100
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-2000
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-500 0 500
-500
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-5
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-5 -5 0 100 200
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00
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10
0
050100
10
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-5 0 5
-5
0
5
0 100 200
-4
-2
0
2
Fig. 10 Qualitative results of AFT for test functions F1,F
3,F
5,F
11,F
13 and F16: Search landscapes, convergence curves, average fitness curves of
all thieves, search histories and trajectories in the first dimension of the first thief
Neural Computing and Applications
123

roughly scattered around the correct optimum solution,
and thus, this constitutes a visual evidence of the ability
of AFT in exploitation. This means that the thieves
quickly explore the whole search space at first and
gradually move toward the global optimum.
•Trajectory this plot depicts the thieves’ path in the first
dimension. Displaying the trajectory of the thieves
across all dimensions complicates the analysis process;
thus, we have stored and illustrated the search path of
the thieves in one dimension. This search plot presents
the exploitation and exploration capacities of AFT.
Figure 10 shows that the thieves go through abrupt
fluctuations in the initial phases, progressive variations
after the initial phases, and a steady situation in the final
stages. This confirms the ability of AFT in exploring the
search space at the initial stages of the search and
exploiting the global optimum areas in the consequent
stages.
Overall, the metric measures presented in Fig. 10 show that
AFT is able to efficiently explore the search space, exploit
each promising area, avoid local optimal solutions, achieve
high levels of accuracy and reasonably converge toward
optimality.
7.9 Performance of AFT in CEC-2017 benchmark
With the aim of further challenging the proposed AFT
algorithm, a challenging and a recent benchmark test suite,
IEEE CEC-2017, was used. This test suite composes of 30
functions with varying difficulty levels. It should also be
pointed out that due to the unstable behavior of the C-2017-
f2 function, particularly for higher dimensions, it has been
set aside from this test suite [107]. Thus, out of the 29
benchmark test functions, there are 2 unimodal functions
(C-2017-f1 and C-2017-f3), 7 simple multimodal functions
(C-2017-f4 - C-2017-f10), 10 hybrid functions (C-2017-f11
- C-2017-f20) and 10 composition functions (C-2017-f21 -
C-2017-f30). These benchmark test functions include many
properties of real-world problems, and thus, in the litera-
ture, many algorithms were examined on these test prob-
lems. The performance of AFT in this test suite is
compared with a set of well-known comparative algorithms
previously reported promising performance in the litera-
ture. The parameter settings for each optimization algo-
rithm are discussed in Subsect. 7.1 and listed in Table 1.
Table 11 shows the optimization results of AFT and
different methods in this test suite. The results of other
algorithms were taken from reference [47].
The results in Table 11 bear out the ascendancy of the
AFT algorithm over others in optimizing very challenging
optimization functions. The AFT algorithm again reported
the best average fitness values in twenty-eight out of
twenty-nine benchmark test functions. With regard to the
Std values, AFT performed significantly better than other
algorithms in twenty-one out of twenty-nine test functions.
This sustains the fact that we formerly inferred that AFT
has a high degree of stability when applied to various
search domains and test beds. These findings indicate that
AFT ranks first in terms of the powerful in exploration and
exploitation capabilities, which then turns to obtain high
performance in both of the accuracy and standard deviation
values. This achievement is a further evidence of the ability
of AFT to excel well-studied meta-heuristic algorithms
while achieving highly competitive results with high per-
formance methods in complex test functions.
7.10 Performance of AFT in CEC-C06 2019
benchmark
A set of ten benchmark functions, named IEEE CEC-C06
2019, have been developed by Professor Suganthan and his
colleagues [108] for use in evaluating single-objective
optimization problems. Table 35 in ‘‘Appendix C’’
describes this group of test functions. These benchmark
functions are designed to be scalable. The functions CEC04
- CEC10 are shifted and rotated, while CEC01 - CEC03 are
neither shifted nor rotated. Table 1shows the parameter
settings of AFT and other optimization algorithms, and a
description of these parameter settings is presented in
Subsect. 7.1.
The average fitness and standard deviation values were
computed to compare between the accuracy of AFT and
other algorithms. The results of DA, WOA, and SSA were
taken from [109]. Table 12 displays the Ave and Std values
for the AFT algorithm and other algorithms when used to
search for the global optimum in the CEC-C06 2019
benchmark test functions.
The outcomes presented in Table 12 show that the AFT
algorithm ranked first in six out of ten functions (i.e.,
CEC01, CEC05 and CEC07-CEC10) in terms of the
average fitness. The MFO algorithm is in the second place
since it achieved the best Ave values in the CEC02 and
CEC03 functions, while the MVO algorithm got the best
Ave in the CEC04 function. Nevertheless, the results
obtained by AFT in CEC02, CEC03 and CEC04 are
comparable. In terms of measuring the stability of AFT, the
value of Std was calculated and compared with that of other
competitive algorithms. The AFT algorithm gained the
least values of Std in seven out of ten CEC-C06 functions.
This once again confirms that AFT has the most stable re-
sults when compared to other algorithms.
Neural Computing and Applications
123

Table 11 Optimization results of the AFT algorithm and other algorithms in IEEE CEC-2017 test functions
F AFT EO PSO GWO GSA CMA-ES
Ave Std Ave Std Ave td Ave Std Ave Std Ave Std
C-2017-
f1
100.00 9.77E-
12
2465.30 2206.20 3959.60 4456.60 325132 107351 296.0 275.10 100.00 0.00E100
C-2017-
f3
300.00 2.98E-
14
300.00 2.40E-08 300.00 1.90E-10 1538.00 1886.02 10829.20 1620.74 300.00 0.00E100
C-2017-
f4
400.00 4.60E-
14
404.48 0.79 405.94 3.28 409.50 7.55 406.60 2.92 400.00 0.00E100
C-2017-
f5
509.94 5.05 510.73 3.67 513.06 6.54 513.5 6.10 556.70 8.40 530.18 58.32
C-2017-
f6
600.00 2.42E-
05
600.00 1.50E-04 600.24 0.98 600.60 0.88 621.60 9.01 682.10 35.43
C-2017-
f7
717.39 6.08 720.93 5.74 718.98 5.10 729.80 8.60 714.60 1.55 713.40 1.63
C-2017-
f8
808.94 2.77 809.51 2.9176 811.39 5.47 814.30 8.26 820.5 4.69 828.90 52.98
C-2017-
f9
900.00 3.05E-
14
900.00 2.27E-02 900.00 5.90E-14 911.30 19.53 900.00 6.90E-14 4667.30 2062.80
C-2017-
f10
1260.66 206.03 1418.70 261.63 1473.30 214.97 1530.50 286.67 2694.6 297.62 2588.10 414.47
C-2017-
f11
1101.98 1.35E-
01
1105.20 502.18E-
02
1110.50 6.28 1140.20 54.13 1134.70 10.45 1111.30 25.44
C-2017-
f12
1265.37 715.12 10340 9790.60 14532 11260 625182 1126443 702723 42075.40 1629.60 198.11
C-2017-
f13
1318.49 52.81 8023.00 6720.8 8601.10 5123.60 9842.30 5633.43 11053 2110.55 1323.62 78.32
C-2017-
f14
1401.98 17.98 1463.30 32.49 1482.10 42.46 3403.53 1953.33 7147.5 1489.52 1452.10 55.98
C-2017-
f15
1504.99 25.14 1585.60 48.012 1714.30 282.89 3806.60 3860.66 18001 5498.67 1509.63 16.43
C-2017-
f16
1600.02 15.23 1649.00 50.915 1860.00 127.65 1725.78 123.85 2149.70 105.80 1815.34 230.13
C-2017-
f17
1711.26 9.53 1731.60 18.071 1761.60 47.50 1759.61 31.29 1857.7 108.32 1830.14 175.83
C-2017-
f18
1825.29 92.32 12450 11405 14599 11852.20 25806.10 15766.90 8720.50 5060.10 1825.92 13.53
C-2017-
f19
1907.48 17.38 1951.50 47.11 2602.80 2185.02 9866.10 6371.09 13670 19168 1920.54 28.68
C-2017-
f20
2008.58 21.51 2020.60 22.28 2085.10 62.25 2075.60 52.04 2272.30 81.72 2494.84 242.65
C-2017-
f21
2200.00 36.45 2307.50 20.96 2281.70 54.02 2317.10 7.00 2357.70 28.20 2324.76 67.76
C-2017-
f22
2200.00 1.57 2297.40 18.40 2314.80 66.10 2310.10 16.75 2300.00 7.2E-02 3532.41 847.62
C-2017-
f23
2300.00 8.38 2615.80 5.5298 2620.80 9.23 2616.40 8.47 2736.50 39.14 2728.8 243.1
C-2017-
f24
2500.00 105.22 2743.80 6.904 2692.20 108.20 2741.70 8.73 2742.2 5.52 2704.43 73.42
C-2017-
f25
2600.02 19.51 2934.30 19.76 2924.00 25.02 2938.00 23.61 2937.50 15.36 2932.01 20.87
C-2017-
f26
2600.00 97.23 2967.80 164.98 2952.10 249.66 3222.50 427.02 34407.50 628.73 3457.75 598.94
C-2017-
f27
3089.24 11.90 3091.30 2.2414 3116.20 24.99 3104.10 21.81 3259.50 41.66 3137.56 21.37
Neural Computing and Applications
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7.11 Statistical test analysis
The average fitness and standard deviation of the results
tabulated in Tables in 3,4,5,11 and 12 presented a gen-
eral view of the performance of AFT and to what extent
AFT was stable during the 30 independent runs. The
qualitative analysis presented in Sect. 7.8 clarified the
exploitation and exploration skills that the AFT has, but did
not interpret to what degree. This subsection presents sta-
tistical analysis using Friedman’s and Holm’s tests [110]to
exhibit the statistical significance of the results in
Tables 3,4,5,11 and 12 and do not statistically deviate
from the results of other competitors.
The significance level (a) in Friedman’s test is set to
0.05 in the experiments shown below. In case if the pvalue
calculated by Friedman’s test is equal or less than a, a null
Table 11 (continued)
F AFT EO PSO GWO GSA CMA-ES
Ave Std Ave Std Ave td Ave Std Ave Std Ave Std
C-2017-
f28
2800.00 31.51 3302.70 133.92 3315.90 121.83 3391.20 101.50 3459.40 33.84 3397.62 131.36
C-2017-
f29
3136.95 23.34 3169.90 24.65 3203.80 52.26 3190.50 42.90 3449.50 171.33 3213.52 109.79
C-2017-
f30
3405.88 4358 297113 458560 350650 504857 297688 527757 1303361 363843 304 444815
The best results are written in bold for the purposes of confirming the best algorithm
Table 12 Results of AFT and other meta-heuristic algorithms in IEEE ECE-C06 2019 benchmark test functions
Functions AFT CSA DA WOA
Ave Std Ave Std Ave Std Ave Std
CEC01 3.9876E104 2.3523E103 8.9208E?07 1.1887E?08 543E?08 669E?08 411E?08 542E?08
CEC02 17.3428 3.5361E-14 17.3428 1.0562E-05 78.0368 87.7888 17.3495 0.0045
CEC03 12.7024 3.6134E-15 12.7024 3.6134E-15 13.7026 0.0007 13.7024 0.0
CEC04 45.9001 18.5761 80.7442 41.5589 344.3561 414.0982 394.6754 248.5627
CEC05 1.1677 5.5248E-02 1.1973 1.1800E01 2.5572 0.3245 2.7342 0.2917
CEC06 7.1860 1.1781 9.9243 1.4049 9.8955 1.6404 10.7085 1.0325
CEC07 110.1464 170.3345 205.0339 1.4647E?02 578.9531 329.3983 490.6843 194.8318
CEC08 4.1198 7.8873E-01 4.1259 7.5690E-01 6.8734 0.5015 6.909 0.4269
CEC09 2.3628 1.7099E-02 2.4397 7.5320E-02 6.0467 2.871 5.9371 1.6566
CEC10 19.9984 1.0414E-02 20.0024 6.0518 21.2604 0.1715 21.2761 0.1111
Functions MFO MVO SCA SSA
Ave Std Ave Std Ave Std Ave Std
CEC01 1.7580E?10 3.1999E?10 3.0602E?09 1.8495E?09 3.3725E?09 4.2549E?09 605E?08 475E?08
CEC02 17.3422 7.2268e-15 18.0691 0.35655 17.4672 8.5584e-02 18.3434 0.0005
CEC03 12.7021 2.7755E-04 12.7021 2.1233E-09 12.7025 8.3918E-05 13.7025 0.0003
CEC04 169.8126 208.2712 28.6991 11.4953 1.0590E?03 358.1222 41.6936 22.2191
CEC05 1.2153 0.1522 1.2513 0.10748 2.1658 7.6196E-02 2.2084 0.1064
CEC06 5.6797 2.4962 7.5882 1.2560 10.6791 0.6640 6.0798 1.4873
CEC07 409.0922 311.4702 277.6752 171.3962 716.1605 145.8723 410.3964 290.5562
CEC08 5.5186 0.7223 5.1849 0.6046 5.8788 0.4656 6.3723 0.5862
CEC09 2.8346 0.3887 2.3918 2.2188E-02 80.8171 74.9311 3.6704 0.2362
CEC10 20.1601 0.1848 20.0414 6.2121E-02 20.2904 0.7100 21.04 0.078
The best results are written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
123

hypothesis is rejected which indicates that there are sig-
nificant differences between the performance of the eval-
uated algorithms. In this study, Friedman’s test is followed
by Holm’s method, as a post hoc test method, to counteract
the problem of various comparisons. The lowest ranked
method by Friedman’s test will be used as a control method
for post hoc analysis.
7.11.1 Statistical test on functions F1–F7
Table 13 shows a summary of the ranking results generated
by Friedman’s test when applied to the results presented in
Table 3.
The pvalue given by Friedman’s test on the average
results of unimodal functions is 4.280198E-05. Thus, the
null hypothesis of equivalent accuracy is rejected, con-
firming the existence of a statistically significant difference
between the comparative algorithms. It can be observed
from Table 13 that AFT is statistically significant, and it is
the best one among all other algorithms. As shown in
Table 13, the proposed AFT algorithm achieved the best
statistical score on the set of unimodal test functions
studied in this work, followed sequentially by SOA, SHO,
CMA-ES, GWO, ACO, CSA, PSO, GA, DE, GSA and
MFO in the last rank.
After the application of Friedman’s test, Holm’s test
method was used to decide if there were statistically sig-
nificant differences between the AFT algorithm and the
others. Table 14 displays statistical outcomes generated by
Holm’s method when it is applied to the results shown in
Table 13. In Table 14,R0represents Friedman’s-rank
given to the control method (i.e., AFT), Rirepresents
Friedman’s rank given to algorithm i,zrepresents the
statistical difference between two algorithms, and finally
ES is the effect size of AFT on algorithm i.
Holm’s method in Table 14 rejects the hypotheses that
have p-value 0:00625. It is apparent from Table 14 that
AFT yielded the best statistical results. In addition to that,
the p-values presented in Table 14 confirm the reliability of
the results generated by AFT in unimodal test functions.
7.11.2 Statistical test on functions F8–F23
Table 15 presents a summary of the ranking results gen-
erated by Friedman’s test when it is applied to the mean
results presented in Table 4.
The p-value generated by Friedman’s test on the average
fitness values of multimodal functions is 0.091583. There is
a statistically significant difference between the compara-
tive algorithms which means that the null hypothesis is
rejected. It is clearly observed from the results shown in
Table 15 that AFT is statistically significant and that it
achieved the best score among other algorithms. In sum,
the ranking results from Friedman’s test, when applied to
multimodal functions in Table 4, are AFT in the first place,
followed in order by CMA-ES, GWO, SOA, PSO, GA, DE,
SHO, ACO, GSA, CSA and MFO at last.
The statistical results generated by Holm’s method on
the results presented in Table 15 concerned with to the
results of Table 4are shown in Table16.
In Table 16, the hypotheses with p-value 0:004545
were rejected by Holm’s method. The results in Table 16
confirm that AFT is statistically superior when it is com-
pared to other competitors.
A summary of the ranking results obtained based on
applying Friedman’s statistical test on the average results
in Table 5is displayed in Table 17.
In Table 17, the proposed AFT outperformed all other
algorithms, with the lowest rank of 2.2 followed in order by
CSA, CMA-ES, DE, GSA, ACO, SHO, MFO, GWO, PSO
and finally SOA.
The statistical results obtained by applying Holm’s test
method to the results presented in Table 15, which are
related to the results reported in Table 5, are shown in
Table 18.
In Table 18, the hypotheses with pvalue 0:008333
were rejected by Holm’s test method. The results in this
table illustrate that AFT is statistically the best one, when it
is compared to the others.
7.11.3 Statistical test on IEEE CEC-2017 benchmark
Table 19 displays a summary of the statistical results
obtained by Friedman’s test when it is applied to the
average results shown in Table 11.
The pvalue obtained by Friedman’s test to the results of
IEEE CEC-2017 test suite is 4.990430E-11. It can be seen
from Table 19 that AFT is statistically the best algorithm
among all other algorithms. As can be noted from Table 19,
AFT has the best statistical score in the IEEE CEC-2017
Table 13 A summary of the
results generated by Friedman’s
test on the average results of
unimodal test functions in
Table 3
Algorithm Rank
AFT 2.714285
MFO 10.0
SOA 2.857142
CSA 6.714285
SHO 3.142857
GWO 4.857142
CMA-ES 4.652321
ACO 5.934213
PSO 6.857142
GSA 7.571428
GA 7.214285
DE 7.285714
Neural Computing and Applications
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test suite, followed in order by CMA-ES, EO, PSO, GWO,
and GSA. Holm’s test method was then applied to the
results in Table 19 to determine if there were statistically
significant differences between AFT and other algorithms.
The results of this method are presented in Table 20.
It is obvious from Table 20 that the AFT algorithm
delivered the best outcomes.
7.11.4 Statistical test on CEC-C06 2019 benchmark
Table 21 displays the ranking results revealed by the
application of Friedman’s test on the results given in
Table 12.
The p-value computed by Friedman’s test when it was
applied to the average results of CEC-C06 functions in
Table 14 Results of Holm’s
method according to Friedman’s
results in Table 13 with
a¼0:05
i Method z¼ðR0RiÞ=ES p value aiHypothesis
11 MFO 3.780371 1.565945E-4 0.004545 Rejected
10 GSA 2.520247 0.011727 0.005 Rejected
9 DE 2.371997 0.017692 0.005555 Rejected
8 GA 2.334935 0.019546 0.00625 Rejected
7 PSO 2.149623 0.031585 0.007142 Rejected
6 CSA 2.075498 0.037940 0.008333 Rejected
5 ACO 1.916246 0.049940 0.01 Rejected
4 GWO 1.111873 0.266192 0.0125 Not rejected
3 CMA-ES 1.001234 0.665124 0.016666 Rejected
2 SHO 0.222374 0.824022 0.025 Not rejected
1 SOA 0.074124 0.940910 0.05 Not rejected
Table 15 The results generated
by Friedman’s test on the
average results of multimodal
functions in Table 4
Algorithm Rank
AFT 3.966666
MFO 8.666666
SOA 5.25
CSA 7.5
SHO 6.583333
GWO 4.833333
CMA-ES 4.743236
ACO 6.666666
PSO 5.366666
GSA 7.0
GA 5.566666
DE 6.333333
Table 16 Results of Holm’s
method according to Friedman’s
test results in Table 15 with
a¼0:05
i Method z¼ðR0RiÞ=SE pvalue aiHypothesis
11 MFO 2.161730 0.030638 0.004545 Rejected
10 CSA 1.601281 0.109314 0.005 Not rejected
9 ACO 1.542114 0.138239 0.005555 Rejected
8 GSA 1.361089 0.173485 0.00625 Not rejected
7 SHO 1.1609291 0.245670 0.007142 Not rejected
6 DE 1.040832 0.297953 0.008333 Not rejected
5 SOA 0.520416 0.602773 0.01 Not rejected
4 PSO 0.480384 0.630954 0.0125 Not rejected
3 GWO 0.320256 0.748774 0.016666 Not rejected
2 CMA-ES 0.291537 0.754663 0.025 Not rejected
1 GA 0.500192 0.670181 0.05 Not rejected
Table 17 A summary of the
ranking results obtained based
on Friedman’s test on the aver-
age results shown in Table 5
Algorithm Rank
AFT 2.2
MFO 7.25
SOA 8.05
CSA 5.1
SHO 7.1
GWO 7.3
CMA-ES 6.33
ACO 7.05
PSO 7.55
GSA 6.75
GA 7.699999
DE 6.649999
Neural Computing and Applications
123

Table 12 is 2.535367E-7. As clearly shown in Table 21,
the AFT algorithm is statistically significant as it has the
best score among all the other algorithms. The order of the
algorithms produced after applying Friedman’s test to this
test suite is AFT, CSA, MVO, MFO, SSA, SCA, WOA,
and DA. The statistical results obtained after applying
Holm’s method on these benchmark functions are given in
Table 22.
As displayed in Table 22, those hypotheses having p-
value 0:016666 were rejected by Holm’s method. It can
be seen from Table 22 that the AFT algorithm is a robust
algorithm in optimizing the IEEE CEC-2017 and IEEE
CEC-C06 2019 test suites. It is clearly discerned from the
statistical findings presented in this subsection that AFT
has better exploitation skills than exploration ones. This
can be inferred from the results obtained when the AFT
algorithm was applied to unimodal functions compared to
the results obtained in multimodal, hybrid and composition
functions. However, the results showed that this is not a big
concern since the exploration in AFT algorithm is plausible
due to the updating mechanism that this algorithm follows
to explore the search space to a large extent. A small
degree of exploration is usually not sufficient to find the
global optimum solution since in optimization problems
there is always necessity to strike a convenient balance
between exploitation and exploration. These impressive
results make the AFT algorithm applicable for solving real-
world engineering problems as shown below.
Table 18 Results of Holm’s
method according to Friedman’s
test results in Table 17
i Method z¼ðR0RiÞ=SE pvalue aiHypothesis
11 SOA 4.248189 2.155049E-5 0.004545 Rejected
10 GA 4.031128 5.550959E-5 0.005 Rejected
9 PSO 3.938102 8.2128401E-5 0.005555 Rejected
8 GWO 3.783059 1.549124E-4 0.00625 Rejected
7 MFO 3.752050 1.753939E-4 0.007142 Rejected
6 SHO 3.659024 2.531769E-4 0.008333 Rejected
5 ACO 3.598117 3.218995E-4 0.01 Rejected
4 GSA 3.441963 5.775073E-4 0.0125 Rejected
3 DE 3.3799465 7.249993E-4 0.016666 Rejected
2 CMA-ES 2.225894 0.001613 0.025 Rejected
1 CSA 2.418677 0.015577 0.05 Rejected
Table 19 Ranking results of the
Friedman’s test when applied to
the Ave results shown in
Table 11
Algorithm Rank
AFT 1.155172
EO 2.534482
PSO 3.482758
GWO 4.448275
GSA 5.051724
CMA-ES 2.2543411
Table 20 Results of Holm’s
method based on the results
reported in Table 19
iMethod z¼ðR0RiÞ=SE pvalue aiHypothesis
5 GSA 7.931045 2.173091E-
15
0.01 Rejected
4 GWO 6.702785 2.044828E-
11
0.0125 Rejected
3 PSO 4.737571 2.162948E-6 0.016666 Rejected
2 EO 2.807449 0.004993 0.025 Rejected
1 CMA-
ES
2.160424 0.101043 0.05 Not rejected
Table 21 The ranking results
obtained based on Friedman’s
test on the IEEE CEC-C06 2019
benchmark with a¼0:05
Algorithm Rank
AFT 1.8
CSA 2.9
DA 6.9
WOA 6.699999
MFO 3.25
MVO 3.049999
SCA 6.0
SSA 5.4
Neural Computing and Applications
123

8 Real engineering design problems
To further substantiate the robustness of the proposed AFT
algorithm, its optimization aptitudes were assessed on five
real-world engineering design problems, known as: the
welded beam problem, the pressure vessel problem, the
tension/compression spring problem, the speed reducer
problem and the rolling element bearing problem. What
distinguishes these design problems is that they have a host
of constraints. Hence, in order to be capable of addressing
these design problems, it is crucial for the AFT algorithm
to be well prepared with a constraint handling method.
8.1 Constraint handling
The AFT algorithm was disposed to fit with a static penalty
approach to be talented to deal with the constraints of the
aforementioned engineering design problems while
addressing them, as elaborated in the following
subsections.
fðzÞ¼fðzÞ X
m
i¼1
limaxð0;tiðzÞÞaþX
n
j¼1
ojUjðzÞ
b
"#
ð50Þ
where fðzÞstands for the objective function, liand ojdefine
positive penalty constants, tiðzÞand UjðzÞare constraint
functions. The values of the parameters band awere set to
2.0 and 1.0, respectively.
The static penalty approach assigns a penalty value for
each unattainable solution. This feature encouraged us to
use the static penalty function to handle the constrains in
the aforementioned design problems, since it can assist the
search agents of AFT to move in the direction of the fea-
sible search space for the given problems to be solved. The
number of search agents and number of iterations of AFT
in solving the following design problems were set to 30 and
1000, respectively.
8.2 Welded beam design problem
The main goal of this design problem is to lessen the
manufacturing cost of the welded beam design displayed in
Fig. 11 [111].
The elements of the welded beam architecture shown in
Figure 11 are a beam, Aand the welding desired to be
attached to the piece, B. The constraints of this design
problem are the bending stress in the beam (h), buckling
load on the bar (Pc), end deflection of the beam (d)in
addition to the shear stress (s). In addressing this problem
to achieve the optimality, it is essential to solve for the
structural parameters of the welded beam structure.
Specifically, the design variables of this problem can be
given as follows: thickness of the weld (h), thickness of the
bar (b), length of the clamped bar (l) and the height of the
bar (t). The variable vector of this problem can be written
as follows: x¼½x1;x2;x3;x4, where x1,x2,x3and x4rep-
resent the values of the variables h,l,tand b, respectively.
This cost function required to lessen the cost of designing
this problem is formulated in the following form:
Fig. 11 A schematic diagram of a welded beam design
Table 22 Results of Holm’s method based on the Friedman’s statistical test results displayed in Table 21
iMethod z¼ðR0RiÞ=SE pvalue aiHypothesis
7 DA 4.6556417 3.229731E-6 0.007142 Rejected
6 WOA 4.473067 7.710540E-6 0.008333 Rejected
5 SCA 3.834057 1.260464E-4 0.01 Rejected
4 SSA 3.286335 0.001015 0.0125 Rejected
3 MFO 1.323662 0.185615 0.016666 Not rejected
2 MVO 1.141088 0.253833 0.025 Not rejected
1 CSA 1.004158 0.315302 0.05 Not rejected
Neural Computing and Applications
123

Minimize:
fðxÞ¼1:10471x2
1x2þ0:04811x3x4ð14:0þx2Þ
Subject to the following constraints,
g1ðxÞ¼sðxÞsmax 0
g2ðxÞ¼rðxÞrmax 0
g3ðxÞ¼x1x40
g4ðxÞ¼1:10471x2
1þ0:04811x3x4ð14:0þx2Þ5:00
g5ðxÞ¼0:125 x10
g6ðxÞ¼dðxÞdmax 0
g7ðxÞ¼PPcðxÞ0
where the remaining variables of the welded beam
design are drawn up as follows:
sðxÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððs0Þ2þðs00Þ2Þþ2s0s00x2
2R
q;s0¼p
ffiffi2
px1x2
s00 ¼MR
J;M¼PðLþx2
2Þ;R¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x1þx3
2

2þx2
2
4
q
J¼2ffiffiffi
2
px1x2
x2
2
12 þð
x1þx3
2Þ2

;rðxÞ¼6PL
x4x2
3
dðxÞ¼4PL3
Ex4x3
3
;PcðxÞ¼4:013 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EGx2
3x6
4=36
qL21x3
2Lffiffiffiffiffiffi
E
4G
r
!
where P¼6000lb;L¼14in, dmax ¼0:25inch, E¼30 
106psi, G¼12 106psi, dmax ¼13600psi,rmax ¼30000
psi. The ranges of variables of this design are used as:
0:1xi2:0 when i¼1 and 4 and 0:1xi10:0 when
i¼2 and 3.
A comparison of the best solutions obtained from AFT
and other optimization algorithms reported in the literature
is presented in Table 23.
Succinctly, the results of Table 23 indicate that the AFT
algorithm converges toward the optimal design and offers
the best solution among all other competitors. A statistical
comparison of AFT with other competitors, after 30 inde-
pendent runs, in terms of the best score, worst score,
average score Ave and standard deviation score Std,is
presented in Table 24.
The results of Table 24 expose that AFT again behaves
much better than other optimization algorithms in terms of
average and Std results. This ascertains the degree of
reliability of the AFT algorithm in addressing this design
problem.
8.3 Pressure vessel design problem
The pressure vessel design is another well-respected
engineering problem that has been extensively considered
in optimization [101]. The goal of designing this problem
is to reduce the overall cost of formation represented as the
material and welding of a cylindrical vessel. This is
wrapped at both endings with hemispherical heads. Fig-
ure 12 illustrates a description of a schematic diagram of
this design.
The variables of this design problem can be described as
given: The first variable is the inner radius (R), thickness of
the shell (Ts), length of the cylindrical section of the vessel
without considering the head (L) and the last variable is the
thickness of the head (Th). These variables can be outlined
by a vector as follows: x¼½x1;x2;x3;x4, where the
parameters of this vector stand for the values of Ts,Th,R
and L, respectively. The problem of the pressure vessel
design can be mathematically defined as follows:
Minimize :fðxÞ¼0:6224x1x3x4þ1:7781x2x2
3
þ3:1661x2
1x4þ19:84x2
1x3
This problem undergoes to four constraints, as outlined
below,
g1ðxÞ¼x1þ0:0193x30
g2ðxÞ¼x2þ0:00954x30
g3ðxÞ¼px2
3x44
3px3
3þ1296000 0
Table 23 A comparison of the
optimal costs attained by AFT
and other optimization
algorithms for the welded beam
problem
Algorithm Optimal values for variables Optimum cost
hltb
AFT 0.205729 3.470488 9.036623 0.205729 1.724852
SHO [99] 0.205563 3.474846 9.035799 0.205811 1.725661
GWO [100] 0.205678 3.475403 9.036964 0.206229 1.726995
PSO [85] 0.197411 3.315061 10.00000 0.201395 1.820395
MVO [45] 0.205611 3.472103 9.040931 0.205709 1.725472
SCA [101] 0.204695 3.536291 9.004290 0.210025 1.759173
GSA [44] 0.147098 5.490744 10.00000 0.217725 2.172858
GA [102] 0.164171 4.032541 10.00000 0.223647 1.873971
DE [87] 0.206487 3.635872 10.00000 0.203249 1.836250
The best result is written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
123

g4ðxÞ¼x4240 0
where 0 x199, 0 x299, 10 x3200 and
10 x4200.
A comparison of the optimal solutions obtained from the
proposed AFT algorithm and other optimization algorithms
reported in the literature is given in Table 25.
As distinctly perused from Table 25, the AFT algorithm
provided the best design with the minimum cost of about
5885.332773. No other competitor algorithm was capable
to achieve this cost.
The statistical results of the AFT algorithm for the
pressure vessel design problem compared to others, in
terms of the best, average, worst and standard deviation
scores after 30 independent runs, are presented in Table 26.
The statistical results in Table 26 assert that the AFT
algorithm performs better than all other competitors’
algorithms in terms of the best, Ave and Std values obtained
so far.
8.4 Tension–compression spring design problem
The structure of a tension/compression spring design
problem is shown in Fig. 13 [112].
In the tension/compression spring design problem, we
strive to reduce the weight of this design. This optimization
problem is subject to three constraints, which are given as
follows: surge frequency, shear stress and minimum
deflection. The variables of this design problem are the
mean coil diameter (D), wire diameter (d) and the number
of active coils (N). These variables can be characterized as:
x¼½x1;x2;x3, with the parameters of xrepresent D,dand
N, respectively. The mathematical representation of this
problem can be described in the following way:
Minimize: fðxÞ¼ðx3þ2Þx2x2
1
This problem is subject to the following constraints:
g1ðxÞ¼1x3
2x3
71785x4
10
g2ðxÞ¼ 4x2
2x1x2
12566ðx2x3
1x4
1Þþ1
5108x2
110
g3ðxÞ¼1140:45x1
x2
2x30
g4ðxÞ¼x1þx2
1:510
where 0:05 x12:0, 0:25 x21:3 and
2x315:0.
Table 27 shows a comparison between the AFT algo-
rithm presented in this work and other algorithms reported
in the literature.
Table 24 Statistical results obtained from AFT and other algorithms
for the welded beam design problem
Algorithm Best Ave Worst Std
AFT 1.724852 1.724852 .724852 1.054459E-15
SHO [99] 1.725661 1.725828 1.726064 0.000287
GWO [100] 1.726995 1.727128 1.727564 0.001157
PSO [85] 1.820395 2.230310 3.048231 0.324525
MVO [45] 1.725472 1.729680 1.741651 0.004866
SCA [101] 1.759173 1.817657 1.873408 0.027543
GSA [44] 2.172858 2.544239 3.003657 0.255859
GA [102] 1.873971 2.119240 2.320125 0.034820
DE [87] 1.836250 1.363527 2.035247 0.139485
Fig. 12 A structural representation of the cross section of a pressure
vessel design
Table 25 A comparison of the
results achieved by AFT and
other algorithms for the pressure
vessel design problem
Algorithm Optimal values for variables Optimum cost
TsThRL
AFT 12.450698 6.154386 40.319618 199.999999 5885.332773
SHO [99] 0.778210 0.384889 40.315040 200.00000 5885.5773
GWO [100] 0.779035 0.384660 40.327793 199.65029 5889.3689
PSO [85] 0.778961 0.384683 40.320913 200.00000 5891.3879
MVO [45] 0.845719 0.418564 43.816270 156.38164 6011.5148
SCA [101] 0.817577 0.417932 41.74939 183.57270 6137.3724
GSA [44] 1.085800 0.949614 49.345231 169.48741 11550.2976
GA [102] 0.752362 0.399540 40.452514 198.00268 5890.3279
DE [87] 1.099523 0.906579 44.456397 179.65887 6550.0230
The best result is written in bold for the purposes of confirming the best algorithm
Neural Computing and Applications
123

Examining the results of Table 27 in terms of the opti-
mal costs, we can clearly realize that AFT scored the best
solution out of all the optimization algorithms for this
problem and achieved the best design with a cost of
0.012665, which no other competitor has achieved. A
comparison of the statistical results obtained by AFT and
other algorithms reported in the literature for this design
problem is presented in Table 28.
The results in Table 28 divulge that the AFT algorithm
once again acts much better in terms of the statistical
results than other algorithms.
8.5 Speed reducer design problem
The speed reducer design problem is a challenging
benchmark problem due to that this problem comprises of
seven variables [113]. The structural description of a speed
reducer design is shown in Fig. 14.
The prime objective of this problem is to reduce the
weight of the speed reducer, which is subject to the fol-
lowing constraints: transverse deflections of the shafts,
surface stress, bending stress of the gear teeth and stresses
in the shafts [99]. The seven variables of this problem can
be given as follows: the number of teeth in the pinion (z),
the module of teeth (m), the face width (b), the module of
the teeth (m), the diameter of the first shafts (d1), the
diameter of the second shafts (d2), the length of the first
shaft between bearings (l1) and the last design variable is
the length of the second shaft between bearings (l2). These
variables can be represented as follows:
x¼½x1x2x3x4x5x6x7. The mathematical description of this
design problem can be given as follows:
Minimize:
fðxÞ¼0:7854x1x2
2ð3:3333x2
3þ14:9334x343:0934Þ
1:508x1ðx2
6þx2
7Þþ7:4777ðx3
6þx3
7Þ
þ0:7854ðx4x2
6þx5x2
7Þ
This function is subject to eleven constraints, which are
described as follows:
g1ðxÞ¼ 27
x1x2
2x310
g2ðxÞ¼397:5
x1x2
2x2
310
g3ðxÞ¼1:9x3
4
x2x4
6x310
g4ðxÞ¼1:93x3
5
x2x4
7x310
g5ðxÞ¼½ð745ðx4=x2x3ÞÞ2þ16:91061=2
110x3
610
Table 26 Statistical results
obtained from the AFT
algorithm and other algorithms
in solving the pressure vessel
design problem
Algorithm Best Ave Worst Std
AFT 5885.332773 5885.332773 5885.332773 4.178081E-12
SHO [99] 5885.5773 5887.4441 5892.3207 2.893
GWO [100] 5889.3689 5891.5247 5894.6238 013.910
PSO [85] 5891.3879 6531.5032 7394.5879 534.119
MVO [45] 6011.5148 6477.3050 7250.9170 327.007
SCA [101] 6137.3724 6326.7606 6512.3541 126.609
GSA [44] 11550.2976 23342.2909 33226.2526 5790.625
GA [102] 5890.3279 6264.0053 7005.7500 496.128
DE [87] 6550.0230 6643.9870 8005.4397 657.523
Table 27 A comparison of the best solutions achieved by AFT and
other algorithms for the tension/compression spring design problem
Algorithm Optimum variables Optimum weight
dDN
AFT 0.051691 0.356777 11.285441 0.012665
SHO [99] 0.051144 0.343751 12.0955 0.012674000
GWO [100] 0.050178 0.341541 12.07349 0.012678321
PSO [85] 0.05000 0.310414 15.0000 0.013192580
MVO [45] 0.05000 0.315956 14.22623 0.012816930
SCA [101] 0.050780 0.334779 12.72269 0.012709667
GSA [44] 0.05000 0.317312 14.22867 0.012873881
GA [102] 0.05010 0.310111 14.0000 0.013036251
DE [87] 0.05025 0.316351 15.23960 0.012776352
The best result is written in bold for the purposes of confirming the
best algorithm
Fig. 13 A schematic diagram of a tension/compression spring design
Neural Computing and Applications
123

g6ðxÞ¼½ð745ðx5=x2x3ÞÞ2þ157:51061=2
85x3
710
g7ðxÞ¼x2x3
40 10
g8ðxÞ¼5x2
x110
g9ðxÞ¼ x1
12x210
g10ðxÞ¼1:5x6þ1:9
x410
g11xÞ¼1:1x7þ1:9
x510
where the range of the design variables b;m;z;l1;l2;d1
and d2were used as 2:6x13:6, 0:7x20:8,
17 x328, 7:3x48:3, 7:3x58:3, 2:9x63:9
and 5:0x45:5, respectively.
Table 29 displays the best designs and optimum costs
achieved by the AFT algorithm and other algorithms for
the speed reducer design problem.
Table 29 corroborates that AFT provides the optimal
design compared to others with a cost of approximately
2994.471066. The statistical results of AFT and other
competitive methods, over 30 independent runs, for the
speed reducer design problem are tabulated in Table 30.
According to the statistical results in Table 30, the AFT
algorithm found the best results compared to other
promising optimization algorithms.
8.6 Rolling Element Bearing Design Problem
The main goal of this design problem is to make the
dynamic load carrying power of a rolling element bearing
as large as possible. The schematic diagram of this design
problem is shown in Fig. 15 [99].
This problem consists of ten decision variables given as
follows: ball diameter (Db), pitch diameter (Dm), number of
balls (X), inner (fi) and outer (fo) raceway curvature factors,
KDmin,KDmax ,e,and f. The mathematical representation
of this design problem is as follows:
Maximize: Cd¼fcX2=3D1:8
bif D 25:4mm
Cd¼3:647fcX2=3D1:4
bif D [25:4mm
The constraints and fcof this design problem are pre-
sented as follows:
g1ðxÞ¼ /0
2sin1ðDb=DmÞXþ10
g2ðxÞ¼2DbKDminðDdÞ0
g3ðxÞ¼KDmaxðDdÞ2Db0
g4ðxÞ¼fBwDb0
g5ðxÞ¼Dm0:5ðDþdÞ0
g6ðxÞ¼ð0:5þeÞðDþdÞDm0
g7ðxÞ¼0:5ðDDmDbÞDb0
g8ðxÞ¼fi0:515
g9ðxÞ¼fo0:515
fc¼37:911þ1:04 1c
1þc

1:72 fið2fo1Þ
foð2fi1Þ

0:4110=30:3
c0:3ð1cÞ1:39
ð1þcÞ1=3
"#
2fi
2fi1

0:41
x¼ðDdÞ
23T
4Þ

2
þD
2T
4Db

2
d
2þT
4

2
y¼2ðDdÞ
23T
4

D
2T
4Db
 ð51Þ
Table 28 Statistical results of
the proposed AFT algorithm
and other optimization
algorithms for the
tension/compression spring
design problem
Algorithm Best Ave Worst Std
AFT 0.012665 0.012665 0.012665 3.216681E-10
SHO [99] 0.012674000 0.012684106 0.012715185 0.000027
GWO [100] 0.012678321 0.012697116 0.012720757 0.000041
PSO [85] 0.013192580 0.014817181 0.017862507 0.002272
MVO [45] 0.012816930 0.014464372 0.017839737 0.001622
SCA [101] 0.012709667 0.012839637 0.012998448 0.000078
GSA [44] 0.012873881 0.013438871 0.014211731 0.000287
GA [102] 0.013036251 0.014036254 0.016251423 0.002073
DE [87] 0.012776352 0.013069872 0.015214230 0.000375
Fig. 14 A structural design of a speed reducer problem
Neural Computing and Applications
123

/0¼2p2cos1x
y

c¼Db
Dm;fi¼ri
Db;fo¼ro
Db;T¼Dd2Db
D¼160;d¼90;Bw¼30;ri¼ro¼11:033
0:5ðDþdÞDm0:6ðDþdÞ;
0:15ðDdÞDb0:45ðDdÞ
4X50;0:515 fiand fo0:6
0:4KDmin 0:5;0:6KDmax 0:7
0:3e0:4;0:02 e0:1;0:6f0:85
Table 31 shows a comparison of the best solutions for
the rolling element bearing design obtained by AFT and
other optimization algorithms.
As per the optimum costs of the rolling element bearing
design problem reported in Table 31, the AFT algorithm
got the best design with the optimal cost of about
85206.641. The statistical results of AFT and other opti-
mization methods over 30 runs are shown in Table 32.
It may be observed from Table 32 that the AFT algo-
rithm has once again obtained the best optimal solutions for
the rolling element bearing design problem over other
algorithms.
In a nutshell, the general performance of the proposed
AFT algorithm has corroborated its reliability and effi-
ciency in addressing the above five classical engineering
design problems. Therefore, we can deduce that the AFT
algorithm is an appropriate and effective optimizer and is
definitely a promising candidate for solving real-world
contemporary problems.
9 Conclusion and Future Work
This paper has proposed a novel human-based meta-
heuristic algorithm called Ali Baba and the forty thieves
(AFT) for solving global optimization problems. The per-
formance of the AFT algorithm was benchmarked on three
benchmarks of sixty-two basic and challenging test func-
tions taken from the so-called classic benchmark functions,
Table 29 A comparison of the
best results obtained by AFT
and other algorithms for the
speed reducer design problem
Algorithm Optimum variables Optimum cost
bmzl
1l2d1d2
AFT 3.5 0.69 17. 7.3 7.715319 3.350214 5.286654 2994.471066
SHO [99] 3.50159 0.7 17 7.3 7.8 3.35127 5.28874 2998.5507
GWO [100] 3.506690 0.7 17 7.380933 7.815726 3.357847 5.286768 3001.288
PSO [85] 3.500019 0.7 17 8.3 7.8 3.352412 5.286715 3005.763
MVO [45] 3.508502 0.7 17 7.392843 7.816034 3.358073 5.286777 3002.928
SCA [101] 3.508755 0.7 17 7.3 7.8 3.461020 5.289213 3030.563
GSA [44] 3.600000 0.7 17 8.3 7.8 3.369658 5.289224 3051.120
GA [102] 3.510253 0.7 17 8.35 7.8 3.362201 5.287723 3067.561
DE [87] 3.520124 0.7 17 8.37 7.8 3.366970 5.288719 3029.002
The best result is written in bold for the purposes of confirming the best algorithm
Table 30 Statistical results of
the AFT algorithm and other
optimization algorithms for the
speed reducer design problem
Algorithm Best Ave Worst Std
AFT 2994.471066 2994.471066 2994.471073 1.419722E-06
SHO [99] 2998.5507 2999.640 3003.889 1.93193
GWO [100] 3001.288 3005.845 3008.752 5.83794
PSO [85] 3005.763 3105.252 3211.174 79.6381
MVO [45] 3002.928 3028.841 3060.958 13.0186
SCA [101] 3030.563 3065.917 3104.779 18.0742
GSA [44] 3051.120 3170.334 3363.873 92.5726
GA [102] 3067.561 3186.523 3313.199 17.1186
DE [87] 3029.002 3295.329 3619.465 57.0235
Fig. 15 A schematic view of a rolling element bearing
Neural Computing and Applications
123

IEEE CEC-2017 benchmark functions and IEEE CEC-C06
2019 benchmark functions. Several developments were
conducted for the AFT algorithm from several aspects to
innervate its exploration and exploitation aptitudes.
Extensive comparisons with many well-studied, new and
high-performance algorithms have shown that AFT is
highly reliable and effective in getting near-optimal or
optimal solutions for most of the test functions studied. In
real-world problems, the AFT algorithm was practically
applied to solve five engineering design problems as evi-
dence of its reliability and applicability in addressing real-
life applications. In future work, a parallel optimization
algorithm could be developed by a combination of AFT
algorithm and other algorithms as potential researches to
further improve its performance level. Further expansions
of the AFT algorithm can be developed by adapting,
implementing and testing both binary and multi-objective
version of this algorithm to solve large-scale real-world
problems.
Appendix A. Unimodal, multimodal
and fixed-dimension multimodal functions
A detailed description of the unimodal benchmark func-
tions (F1–F7), multimodal benchmark functions (F8–F13 )
and fixed-dimension multimodal benchmark functions
(F14–F23 ) is tabulated in Table 33.
Table 31 Optimization results of the rolling element bearing design problem achieved by AFT and other algorithms
Algorithm Optimum variables Optimum cost
DmDbXf
ifoKDmin KDmax ef
AFT 125 21.418 11.356 0.515 0.515 0.4 0.680 0.3 0.02 0.622 85206.641
SHO [99] 125 21.407 10.932 0.515 0.515 0.4 0.7 0.3 0.02 0.6 85054.532
GWO [100] 125 21.351 10.987 0.515 0.515 0.5 0.688 0.300 0.032 0.627 84807.111
PSO [85] 125 20.753 11.173 0.515 0.515 0.5 0.615 0.3 0.051 0.6 81691.202
MVO [45] 125 21.322 10.973 0.515 0.515 0.5 0.687 0.301 0.036 0.610 84491.266
SCA [101] 125 21.148 10.969 0.515 0.515 0.5 0.7 0.3 0.027 0.629 83431.117
GSA [44] 125 20.854 11.149 0.515 0.517 0.5 0.618 0.304 0.02 0.624 82276.941
GA [102] 125 20.775 11.012 0.515 0.515 0.5 0.613 0.3 0.050 0.610 82773.982
DE [87] 125 20.871 11.166 0.515 0.516 0.5 0.619 0.301 0.050 0.614 81569.527
The best result is written in bold for the purposes of confirming the best algorithm
Table 32 Statistical results
obtained from the AFT
algorithm and others for the
rolling element bearing design
problem
Algorithm Best Ave Worst Std
AFT 85206.201105 85206.641021 85207.012231 1.290128E-03
SHO [99] 85054.532 85024.858 85853.876 0186.68
GWO [100] 84807.111 84791.613 84517.923 0137.186
PSO [85] 81691.202 50435.017 32761.546 13962.150
MVO [45] 84491.266 84353.685 84100.834 0392.431
SCA [101] 83431.117 81005.232 77992.482 1710.777
GSA [44] 82276.941 78002.107 71043.110 3119.904
GA [102] 82773.982 81198.753 80687.239 1679.367
DE [87] 81569.527 80397.998 79412.779 1756.902
Neural Computing and Applications
123

Table 33 Characteristics of the unimodal, multimodal and fixed-dimension multimodal functions used in this work
F Function formulation fmin Class Dim Range
F1Pn
i¼1x2
i0U10xi2[-100,100]
F2Pn
i¼1jxij?Qn
i¼1jxij0U10xi2[-10,10]
F3Pn
i¼1Pi
j1xj

20U10xi2[-100,100]
F4maxijxij;1infg 0U10xi2[-100,100]
F5Pn1
i¼1100 xiþ1x2
i

2þxi1ðÞ
2
hi 0U10xi2[-30,30]
F6Pn
i¼1xiþ0:5½ðÞ
20U10xi2[-100,100]
F7Pn
i¼1ix4
iþrandom½0;1Þ0U10xi2[-1.28,1.28]
F8Pn
i¼1xisin ffiffiffiffiffiffi
jxij
p
 -418E?5M 10 xi2[-500,500]
F9Pn
i¼1x2
i10 cos 2pxi
ðÞþ10
 0M10xi2[-5.12,5.12]
F10 20 exp ð0:2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nPn
i¼1x2
i
qÞexp 1
nPn
i¼1cos 2pxi
ðÞ

þ20 þe0M10xi2[-32,32]
F11 1
4000 Pn
i¼1x2
iQn
i¼1cos xiffii
p

þ10M10xi2[-600,600]
F12 p
n10 sin py1
ðÞþ
Pn1
i¼1ðyi1Þ21þ10 sin2ðpyiþ1Þ

þðyn1Þ2
no
þPn
i¼1uðxi;10;100;4Þ
yi¼1þxiþ1
4uðxi;a;k;mÞ¼
kðxiaÞmxi[a
0a\xi\a
kðxiaÞmxi\a
8
<
:
0M10xi2[-50,50]
F13 0:1 sin2ð3px1ÞþPn
i¼1xi1ðÞ
21þsin2ð3pxiþ1Þ

þðxn1Þ21þsin2ð2pxnÞ

no
þPn
i¼1uðxi;5;100;4Þ0M10xi2[-50,50]
F14 1
500 þP25
j¼1
1
jþP2
i¼1xiaij
ðÞ
6

11F2xi2
65:536;65:536½
F15 P11
i¼1aix1b2
iþbix2
ðÞ
b2
iþbix3þx4

20.0003 F 4 xi25;5½
F16 4x2
12:1x4
1þ1
3x6
1þx1x24x2
2þ4x4
2-1.0316 F 2 xi25;5½
F17 x25:1
4p2x2
1þ5
px16

2þ10 1 1
8p

cos x1þ10 0.398 F 2 xi25;0½&10;15½
F18 1þx1þx2þ1ðÞ
219 14x1þ3x2
114x2þ6x1x2þ3x2
2

hi 3F2xi22;2
½
30 þ2x13x2
ðÞ
218 32x1þ12x2
1þ48x236x1x2þ27x2
2

hi
F19 P4
i¼1ciexp P3
j¼1aij xjpij

2
 -3.86 F 3 xi20;1½
F20 P4
i¼1ciexp P6
j¼1aij xjpij

2
 -3.32 F 6 xi20;1½
F21 P5
i¼1ðXaiÞXai
ðÞ
Tþci

1-10.1532 F 4 xi20;10½
F22 P7
i¼1ðXaiÞXai
ðÞ
Tþci

1-10.4028 F 4 xi20;10½
F23 P10
i¼1ðXaiÞXai
ðÞ
Tþci

1-10.5363 F 4 xi20:10½
U: Unimodal, M: Multimodal, F: fixed-dimension multimodal
Neural Computing and Applications
123

Appendix B. IEEE CEC-2017 test suite
A description of the IEEE CEC-2017 benchmark test
functions is shown in Table 34.
Table 34 Characteristics of the IEEE CEC-2017 benchmark test functions
Function No. Function fmin Range Dim Class
C-2017-f1 Shifted and Rotated Bent Cigar function 100 [-100,100] 10 U
C-2017-f3 Shifted and Rotated Zakharov function 300 [-100,100] 10 U
C-2017-f4 Shifted and Rotated Rosenbrock’s function 400 [-100,100] 10 M
C-2017-f5 Shifted and Rotated Rastrigin’s function 500 [-100,100] 10 M
C-2017-f6 Shifted and Rotated Expanded Scaffer’s function 600 [-100,100] 10 M
C-2017-f7 Shifted and Rotated Lunacek Bi-Rastrigin function 700 [-100,100] 10 M
C-2017-f8 Shifted and Rotated Non-Continuous Rastrigin’s function 800 [-100,100] 10 M
C-2017-f9 Shifted and Rotated Levy Function 900 [-100,100] 10 M
C-2017-f10 Shifted and Rotated Schwefel’s Function 1000 [-100,100] 10 M
C-2017-f11 Hybrid function of Zakharov, Rosenbrock and Rastrigin’s 1100 [-100,100] 10 H
C-2017-f12 Hybrid function of High Conditioned Elliptic, Modified Schwefel and
Bent Cigar
1200 [-100,100] 10 H
C-2017-f13 Hybrid function of Bent Ciagr, Rosenbrock and Lunache Bi-Rastrigin 1300 [-100,100] 10 H
C-2017-f14 Hybrid function of Eliptic, Ackley, Schaffer and Rastrigin 1400 [-100,100] 10 H
C-2017-f15 Hybrid function of Bent Cigar, HGBat, Rastrigin and Rosenbrock 1500 [-100,100] 10 H
C-2017-f16 Hybrid function of Expanded Schaffer, HGBat, Rosenbrock and
Modified Schwefel
1600 [-100,100] 10 H
C-2017-f17 Hybrid function of Katsuura, Ackley, Expanded Griewank plus
Rosenbrock, Modified Schwefel and Rastrigin
1700 [-100,100] 10 H
C-2017-f18 Hybrid function of high conditioned Elliptic, Ackley, Rastrigin,
HGBat and Discus
1800 [-100,100] 10 H
C-2017-f19 Hybrid function of Bent Cigar, Rastrigin, Expanded Grienwank plus
Rosenbrock, Weierstrass and expanded Schaffer
1900 [-100,100] 10 H
C-2017-f20 Hybrid function of Happycat, Katsuura, Ackley, Rastrigin, Modified
Schwefel and Schaffer
2000 [-100,100] 10 H
C-2017-f21 Composition function of Rosenbrock, High Conditioned Elliptic and
Rastrigin
2100 [-100,100] 10 C
C-2017-f22 Composition function of Rastrigin’s, Griewank’s and Modifed
Schwefel’s
2200 [-100,100] 10 C
C-2017-f23 Composition function of Rosenbrock, Ackley, Modified Schwefel and
Rastrigin
2300 [-100,100] 10 C
C-2017-f24 Composition function of Ackley, High Conditioned Elliptic,
Girewank and Rastrigin
2400 [-100,100] 10 C
C-2017-f25 Composition function of Rastrigin, Happycat, Ackley, Discus and
Rosenbrock
2500 [-100,100] 10 C
C-2017-f26 Composition function of Expanded Scaffer, Modified Schwefel,
Griewank, Rosenbrock and Rastrigin
2600 [-100,100] 10 C
C-2017-f27 Composition function of HGBat, Rastrigin, Modified Schwefel, Bent-
Cigar, High Conditioned Elliptic and Expanded Scaffer
2700 [-100,100] 10 C
C-2017-f28 Composition function of Ackley, Griewank, Discus, Rosenbrock,
HappyCat, Expanded Scaffer
2800 [-100,100] 10 C
C-2017-f29 Composition function of shifted and rotated Rastrigin, Expanded
Scaffer and Lunacek Bi-Rastrigin
2900 [-100,100] 10 C
C-2017-f30 Composition function of shifted and rotated Rastrigin, Non-
Continuous Rastrigin and Levy function
3000 [-100,100] 10 C
U: Unimodal, M: Multimodal, H: Hybrid, C: Composition
Neural Computing and Applications
123

Appendix C. IEEE CEC-C06 2019 benchmark
test functions
A description of the IEEE CEC-C06 2019 benchmark
functions is given in Table 35.
Declarations
Conflict of Interest The authors declare that there is no conflict of
interest regarding the publication of this paper.
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Table 35 Characteristics of the
IEEE CEC-C06 2019
benchmark functions
Function Functions Dim Range fmin
CEC01 Storn’s Chebyshev polynomial fitting problem 9 [-8192, 8192] 1
CEC02 Inverse Hilbert matrix problem 16 [-16384, 16384] 1
CEC03 Lennard–Jones minimum energy cluster 18 [-4, 4] 1
CEC04 Rastrigin’s function 10 [-100, 100] 1
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CEC06 Weierstrass function 10 [-100, 100] 1
CEC07 Modified Schwefel’s function 10 [-100, 100] 1
CEC08 Expanded schaffer’s F6function 10 [-100, 100] 1
CEC09 Happy CAT function 10 [-100, 100] 1
CEC10 Ackley function 10 [-100, 100] 1
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