Gaining-sharing knowledge based algorithm for solving optimization problems: a novel nature-inspired algorithm

Abstract

This paper proposes a novel nature-inspired algorithm called Gaining Sharing Knowledge based Algorithm (GSK) for solving optimization problems over continuous space. The GSK algorithm mimics the process of gaining and sharing knowledge during the human life span. It is based on two vital stages, junior gaining and sharing phase and senior gaining and sharing phase. The present work mathematically models these two phases to achieve the process of optimization. In order to verify and analyze the performance of GSK, numerical experiments on a set of 30 test problems from the CEC2017 benchmark for 10, 30, 50 and 100 dimensions. Besides, the GSK algorithm has been applied to solve the set of real world optimization problems proposed for the IEEE-CEC2011 evolutionary algorithm competition. A comparison with 10 state-of-the-art and recent metaheuristic algorithms are executed. Experimental results indicate that in terms of robustness, convergence and quality of the solution obtained, GSK is significantly better than, or at least comparable to state-of-the-art approaches with outstanding performance in solving optimization problems especially with high dimensions.

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International Journal of Machine Learning and Cybernetics
https://doi.org/10.1007/s13042-019-01053-x
ORIGINAL ARTICLE
Gaining‑sharing knowledge based algorithm forsolving optimization
problems: anovel nature‑inspired algorithm
AliWagdyMohamed1,2 · AnasA.Hadi3· AliKhaterMohamed4
Received: 12 April 2019 / Accepted: 11 December 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract
This paper proposes a novel nature-inspired algorithm called Gaining Sharing Knowledge based Algorithm (GSK) for solv-
ing optimization problems over continuous space. The GSK algorithm mimics the process of gaining and sharing knowledge
during the human life span. It is based on two vital stages, junior gaining and sharing phase and senior gaining and sharing
phase. The present work mathematically models these two phases to achieve the process of optimization. In order to verify
and analyze the performance of GSK, numerical experiments on a set of 30 test problems from the CEC2017 benchmark
for 10, 30, 50 and 100 dimensions. Besides, the GSK algorithm has been applied to solve the set of real world optimization
problems proposed for the IEEE-CEC2011 evolutionary algorithm competition. A comparison with 10 state-of-the-art and
recent metaheuristic algorithms are executed. Experimental results indicate that in terms of robustness, convergence and
quality of the solution obtained, GSK is significantly better than, or at least comparable to state-of-the-art approaches with
outstanding performance in solving optimization problems especially with high dimensions.
Keywords Evolutionary computation· Global optimization· Meta-heuristics· Nature-inspired algorithms· Population-
based algorithm
1 Introduction
Optimization is the process of finding the best combination for
a set of decision variables to solve a certain problem. Optimi-
zation arises in many fields, different disciplines and countless
applications [1]. Hard optimization problems arise in huge
number of applications in real life. In which, it is very hard
to reach the global optimum solution. Therefore, many algo-
rithms tried to tackle this kind of problems. The algorithms
could be classified as exact methods and approximate methods.
Exact methods are guaranteed to obtain an optimal solution
in a reasonable time unless the problem is classified as NP-
Hard problem, in which there is no polynomial time exists.
This leads to very high computational time. Thus, the use of
approximate methods has gained much more attention during
the last three decades. In approximate methods, the main target
is to obtain a satisfactory solution in a reasonable time.
A new kind of approximate algorithms has been devel-
oped [2] known as metaheuristic. Metaheuristic is widely
used over the last three decades because of its simplicity,
ease of implementation, ability to avoid local optima and it
deals with derivative free problems. Two important charac-
teristics of metaheuristic are exploration and exploitation.
The former one indicates the ability of the algorithm to dis-
cover new search areas, while the later one focus on finding
the best solution in a promising region of the search space.
The successful metaheuristic is the one that is able to pro-
vide the balance between exploration and exploitation.
Electronic supplementary material The online version of this
article (https ://doi.org/10.1007/s1304 2-019-01053 -x) contains
supplementary material, which is available to authorized users.
* Ali Wagdy Mohamed
aliwagdy@gmail.com
1 Operations Research Department, Faculty ofGraduate
Studies forStatisticalResearch, Cairo University,
Giza12613, Egypt
2 Wireless Intelligent Networks Center (WINC), School
ofEngineering andApplied Sciences, Nile University, Giza,
Egypt
3 College ofComputing andInformation Technology, King
Abdulaziz University, P. O. Box80200, Jeddah21589,
SaudiArabia
4 Department ofComputer Science, Faculty ofComputer
Science, October University forModern Sciences andArts
(MSA), 6thOctoberCity,Giza12451, Egypt

International Journal of Machine Learning and Cybernetics
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In the literature, it can be found different classifica-
tions for metaheuristic [3]. Nature inspired vs. non-nature
inspired, population based vs. single point search, dynamic
objective function vs. static objective function, one single
neighborhood vs. various neighborhood and memory usage
vs. memory less methods.
The literature can be divided into three main directions:
improving the current methods by controlling the parameters
of the algorithms, hybridizing different algorithms to benefit
from each one, and introducing a new algorithm.
Introducing a novel algorithm for solving optimization
problems have attracted many researchers during the last
three decades. Therefore, a new classification for the source
of inspiration is introduced through the rest of this section
and a detailed review is presented. The source of inspira-
tion for nature inspired algorithms can be classified into four
groups. As depicted in Fig.1.
• Evolutionary techniques that is inspired from biology. In
evolutionary algorithm there is an initial random popula-
tion that evolve over generations to produce new solu-
tions by means of crossover and mutation and eliminate
the worst solutions in order to improve the fitness value.
• Swarm Intelligence techniques that is inspired from the
behavior of social insects or animals, nature inspired
algorithms. In swarm intelligence, every individual has
its own intelligence and behavior, but the integration of
the individuals gives more power to solve complex prob-
lems.
• Physics based techniques that is inspired by the rules
governing a natural phenomenon.
• Human related techniques that is inspired from the
human being. Every individual does physical activities
(body activities) that affect his performance and non-
physical activities like thinking and behavior (mind
activities).
Algorithms belonging to each category are listed in
Tables1, 2, 3 and 4 respectively. Each table has the algo-
rithm, author, year and the number of citations till 20 Dec
2018.
From the above tables, Figs.2, 3, 4 and 5 are presented
to demonstrate how meta-heuristic algorithms evolved during
the last three decades. Figures2 and 3 present the number
of algorithms in each category and the share of each cate-
gory in the meta-heuristic algorithms. It is obvious that the
Swarm Intelligence algorithms is the leading category. Fig-
ure4 demonstrates the quality of each category by using the
total number of citations in each category. It can be seen that
the evolutionary algorithms are the most commonly used in
the meta-heuristic algorithms and leading the total number of
citations. Figure5 presents how the mat-heuristic algorithms
evolved over time in the last three decades. It is clear that the
evolutionary algorithms were the leader for the meta-heuristic
algorithms since 1966 and till today there are many algorithms
Nature-inspired
algorithms
Evolutionary
techniques
Swarm
intelligence
techniques
Physics-based
techniques
Human-related
techniques
Fig. 1 Classification of nature inspired algorithms
Table 1 List of evolutionary
algorithms Algorithm Author Year Citations References
Evolutionary programming Fogel etal. 1966 4418 [4]
Evolution strategy Rechenberg 1973 4397 [5]
Genetic algorithms Holland 1975 60,347 [6]
Tabu search Glover 1986 4627 [2]
Co-evolving algorithm Hillis 1990 1382 [7]
Cultural algorithm Reynolds 1994 970 [8]
Genetic Programming Koza 1994 18,989 [9]
Estimation of distribution algorithm Mühlenbein and PaaB 1996 1172 [10]
Differential evolution Storn and Price 1997 19,448 [11]
Grammatical evolution Ryan etal. 1998 642 [12]
Gene expression Ferreira 2001 2108 [13]
Quantum evolutionary algorithm Han and Kim 2002 1459 [14]
Imperialist competitive algorithm Gargari and Lucas 2007 1637 [15]
Differential search algorithm Civicioglu 2011 258 [16]
Backtracking optimization algorithm Civicioglu 2013 437 [17]
Stochastic fractal search Salimi 2014 130 [18]
Synergistic fibroblast optimization Dhivyaprabha etal. 2018 0 [19]

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Table 2 List of swarm
intelligence algorithms Algorithm Author Year Citations References
Memetic algorithm Moscato 1989 2019 [20]
Ant colony optimization Dorigo 1992 13,086 [21]
Continuous particle swarm optimization Eberhart and Kennedy 1995 13,504 [22]
Particle swarm algorithm Kennedy and Eberhart 1995 5484 [23]
Binary particle swarm optimization Kennedy and Eberhart 1997 4839 [24]
Artificial immune system Castro and Timmis 2002 2765 [25]
Clonal selection algorithm Castro and Zuben 1999 1291 [26]
Self-organizing migrating algorithm Zelinka 2000 317 [27]
Marriage in honey bees Abbass 2001 406 [28]
Artificial fish swarm algorithm LI etal. 2002 67 [29]
Bacterial foraging Passino 2002 2738 [30]
Bee dance algorithm Gordon etal. 2003 44 [31]
Bees swarm optimization heuristic algorithm Lučić and Teodorović 2003 198 [32]
Queen-bee evolution Jung 2003 131 [33]
Shuffled frog leaping algorithm Eusuff and Lansey 2003 1348 [34]
Beehive algorithm Wedde etal. 2004 258 [35]
Bee colony optimization Teodrovic, Dell’ Orco 2005 421 [36]
Cooperative bees swarm optimization Drias etal. 2005 157 [37]
Glowworm swarm optimization Krishnanand and Ghose 2005 342 [38]
Honey bee swarm optimization algorithm Karaboga 2005 5009 [39]
Virtual bee algorithm Yang 2005 349 [40]
Bees algorithms Pham etal. 2006 1296 [41]
Cat swarm optimization Chu etal. 2006 317 [42]
Invasive weed optimization Mehrabian and Lucas 2006 876 [43]
Termite swarm algorithm Roth and Wicker 2006 37 [44]
Virtual ant algorithm Yang etal. 2006 13 [45]
Artificial bee colony Karaboga and Basturk 2007 4148 [46]
Bacterial-GA foraging Chen etal. 2007 31 [47]
Good lattice swarm optimization Su etal. 2007 159 [48]
Monkey algorithm Zhao, Tang 2008 89 [49]
Accelerated PSO Yang 2008 161 [50]
Biogeography-based optimization Simon 2008 2093 [51]
Fast bacterial swarming algorithm Chu etal. 2008 71 [52]
Fish-school search Filho etal. 2008 121 [53]
Roach infestation algorithm Havens etal. 2008 82 [54]
Bumblebees algorithm Comellas and Navarro 2009 16 [55]
Cuckoo search Yang and Deb 2009 3439 [56]
Group search optimizer He etal. 2009 562 [57]
Paddy field algorithm Premaratne etal. 2009 46 [58]
Bat algorithm Yang 2010 2322 [59]
Consultant-guided search Iordache 2010 30 [60]
Eagle strategy Yang and Deb 2010 157 [61]
Firefly algorithm Yang 2010 1225 [62]
Hierarchical swarm model Chen etal. 2010 26 [63]
Termite colony optimization Hedayatzadeh etal. 2010 38 [64]
Eco-inspired evolutionary algorithm Parpinelli and Lopes 2011 23 [65]
Fruit fly optimization algorithm Pan 2011 766 [66]
Weightless swarm algorithm Ting etal. 2011 12 [67]
Artificial cooperative search algorithm Civicioglu 2012 73 [68]
Flower pollination algorithm Yang 2012 740 [69]
Japanese tree frogs calling algorithm Hern´andez and Blum 2012 31 [70]

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that are presented in the evolutionary category. During the last
two decades, the Swarm Intelligence gained more attention
from the researchers and competes strongly with the Evolu-
tionary algorithms and Physics based algorithms. The human
based algorithms start to gain attention as a new trend in the
last decade, but still cannot compete with the evolutionary
algorithms, Swarm Intelligence and physics-based algorithms.
Regarding the above discussion, it is obvious that there
are very few efforts in developing a new human-base algo-
rithm. Therefore, a novel nature inspired algorithm based on
human is presented.
The rest of this paper is organized as follows, Sect.2
presents in details the novel proposed algorithms. Numeri-
cal experiments and comparisons are presented in Sect.3.
Finally, the paper is concluded in Sect.4.
2 Proposed algorithm: gaining‑sharing
knowledge based algorithm (GSK)
Gaining-Sharing knowledge optimization algorithm (GSK)
is based on the philosophy of gaining and sharing knowl-
edge during the human life span. It is based on two vital
stages, the first stage is called beginning-intermediate or
junior gaining and sharing phase and the second stage is
called intermediate-expert or senior gaining and sharing
phase. These two phases are described in the following,
respectively.
Virtually, all individuals (persons) in a specific popula-
tion can interact together and they continuously influence
each other through cooperation and competition to be very
experienced and qualified enough to deal with real-life situ-
ations and solve complex problems. However, to be experi-
enced person, you have to gain and share knowledge from/
with others. Therefore, during the human life span, each
person in a specific population gains knowledge at early
stage (early-middle years) in which gaining knowledge
through different types of very small networks such as his/
her (family, neighbors, relatives) is more realistic than gain-
ing it from different types of larger networks such as (work,
social, friends and many others). Besides, due to limited
experience as there is very few sources of knowledge dur-
ing this stage, he/she still has a desire to share his opinion,
thoughts and skills with other different types of people that
maybe outside small networks. Actually, during this stage,
it must be taken into consideration that he is not qualified
Table 2 (continued) Algorithm Author Year Citations References
Krill herd algorithm Gandomi and Alavi 2012 787 [71]
The Great Salmon run algorithm Mozaffari 2012 35 [72]
The OptBees algorithm Maia etal. 2012 27 [73]
Wolf search algorithm Tang etal. 2012 116 [74]
Dolphin echolocation Kaveh and Farhoudi 2013 164 [75]
Egyptian vulture optimization algorithm Sur etal. 2013 24 [76]
Swallow swarm optimization algorithm Neshat etal. 2013 31 [77]
Animal migration optimization Li etal. 2014 123 [78]
Chicken swarm optimization Meng etal. 2014 142 [79]
Grey wolf optimizer Mirjalili etal. 2014 1591 [80]
Ant lion optimizer Mirjalili 2015 476 [81]
Artificial algae algorithm Uymaza etal. 2015 50 [82]
Bird swarm algorithm Meng etal. 2015 67 [83]
Dragonfly algorithm Mirjalili 2015 291 [84]
Virus colony search Li etal. 2015 48 [85]
Crow search algorithm Askarzadeh 2016 202 [86]
Dolphin swarm optimization algorithm Yong etal. 2016 2 [87]
Shark smell optimization Oveis etal. 2016 52 [88]
Whale optimization algorithm Mirjalili and Lewis 2016 538 [89]
Butterfly-inspired algorithm Qi etal. 2017 4 [90]
Grasshopper optimization algorithm Saremi etal. 2017 124 [91]
Mouth brooding fish algorithm Jahani and Chizari 2017 2 [92]
Salp swarm algorithm Mirjalili etal. 2017 102 [93]
Selfish herd optimizer Fausto etal. 2017 8 [94]
Spotted hyena optimizer Dhiman and Kumar 2017 22 [95]
Squirrel search algorithm Jain etal. 2018 3 [96]

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Table 3 List of physics-based
algorithms Algorithm Author Year Citations References
Micro-canonical annealing algorithm Creutz 1983 605 [97]
Simulated annealing Kirkpatrick etal. 1983 43,924 [98]
Stochastic diffusion search Bishop 1989 172 [99]
Self-propelled particles Vicsek etal. 1995 5083 [100]
Variable neighborhood algorithm Mladenovic and Hansen 1995 3423 [101]
Predatory search Linhares 1998 38 [102]
Photosynthetic algorithm Murase 2000 22 [103]
Harmony search Geem etal. 2001 4231 [104]
Gravitational search optimization algorithm Webster and Bernhard 2003 29 [105]
Big bang–big crunch-based optimization Erol and Eksin 2005 710 [106]
Central force optimization Formato 2007 229 [107]
Intelligent water drop Shah-Hosseini 2007 263 [108]
River formation dynamics Rabanal etal. 2007 103 [109]
Slime mold algorithm Monismith and Mayfield 2008 23 [110]
Gravitational search algorithm Rashedi etal. 2009 2927 [111]
Charged system search Kaveh and Talatahari 2010 641 [112]
Electro-magnetism optimization Cuevas etal. 2011 82 [113]
Galaxy-based search algorithm Shah-Hosseini 2011 89 [114]
Spiral optimization Tamura and Yasuda 2011 37 [115]
Black hole algorithm Hatamlou 2012 371 [116]
Curved space optimization Moghaddam etal. 2012 21 [117]
Ray optimization Kaveh and Khayatazad 2012 200 [118]
Water cycle algorithm Eskandar etal. 2012 331 [119]
Atmosphere clouds model Yan etal. 2013 12 [120]
Mine blast algorithm Sadollaha etal. 2013 226 [121]
Colliding bodies optimization Kaveh and Mahdavi 2014 234 [122]
Kinetic energy Moein and Logeswaran 2014 20 [123]
Lightning search algorithm Shareef etal. 2015 71 [124]
Weighted superposition attraction Adil and Akpinar 2015 11 [125]
A sine cosine algorithm Mirjalilia 2016 239 [126]
Multi-verse optimizer Mirjalili etal. 2016 221 [127]
Electro-search algorithm Tabari and Ahmad 2017 11 [128]
Lightning attachment procedure optimization Nematollahi etal. 2017 5 [129]
Thermal exchange optimization Kaveh and Dadras 2017 26 [130]
Find-fix-finish-exploit-analyze (F3EA) Kashan etal. 2018 0 [131]
Table 4 List of human related
algorithms Algorithm Author Year Citations References
Society and civilization Ray and Liew 2003 336 [132]
Human-inspired algorithm Zhang etal. 2009 20 [133]
League championship algorithm Kashan 2009 88 [134]
Social emotional optimization Xu etal. 2010 23 [135]
Brain storm optimization Shi 2011 221 [136]
Teaching–learning-based optimization Rao etal. 2011 791 [137]
Anarchic society optimization Shayeghi and Dadashpour 2012 26 [138]
Volleyball premier league algorithm Moghdani and Salimifard 2018 1 [139]

International Journal of Machine Learning and Cybernetics
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enough with very little experience in life to differentiate and
classify different types of people, that does not belong to his/
her small networks, into good and bad persons.
On the other hand, during (middle-later) years stage,
due to interacting with larger networks such as (work col-
leagues, social media friends and others), each person has
its own knowledge in different fields that can be significantly
enhanced by gaining it from others which is mainly derived
by following leaders’ success and believe on opinions of
elite persons in addition to avoiding failure persons or people
with radical concept or bad performance in all fields. In fact,
during this stage, each one has a great ability to judge, think
and classify different types of people into good, medium and
bad classes. Thus, he can easily share his knowledge and
experience in different fields with the most appropriate per-
sons with good characteristics and behaviors (i.e. to benefit
from their knowledge and experience). The mathematical
explanation of the aforementioned concept of gaining-shar-
ing knowledge is presented below.
Let
xi,i=1, 2, 3, …,N
be the persons of a specific pop-
ulation, i.e., this population contains N persons and each
person
xi
is defined by
xij
= (
xi1,xi2,…,xiD
), where D is the
number of fields of disciplines i.e. branch of knowledge
assigned to a person which determines the dimensions of
a person and
fi,i
= 1,2,…,N are their corresponding fitness
values, respectively.
Thus, it can be obviously deduced from Fig.6 that the
main idea is that during junior gaining and sharing phase
(early-middle stage) the number of dimensions of each vec-
tor will be replaced (changed) by another values using junior
gaining-sharing scheme is more than the number of updated
dimensions using senior gaining and sharing scheme i.e.
the number of updated dimensions using junior gaining and
sharing rule is more than the number of updated dimensions
using senior gaining-sharing scheme. However, during sen-
ior phase (middle-later stage), the number of updated dimen-
sions of each vector using senior gaining and sharing scheme
is more than the number of updated dimensions using junior
gaining and sharing scheme. Besides, it must be taken into
consideration that the required number of dimensions that
will be replaced using both junior and senior phases depends
on the value of knowledge rate that controls the volume of
knowledge that will be transferred during generations from
others using junior and senior gaining-sharing knowledge
schemes.
Therefore, the number of the desired number of dimen-
sions that will be updated or changed (using junior scheme)
and the other number of dimensions that will be updated
(using senior scheme) during generations must be deter-
mined for each vector at the beginning of the search. Based
on the fundamental concept of gaining-sharing knowledge,
the number of dimensions D is determined using the fol-
lowing non-linear decreasing and increasing formula
(experience equation). Note that during the generations of
optimization process, the number of dimensions that will
updated using junior scheme is decreased while the num-
ber of dimensions that will be updated by senior scheme is
decreased:
(1)
D
(juniorphase)=(problemsize)×
(
1−G
GEN )k
,
17
79
35
8
Evoluonary Swarm PhysicsHuman
Fig. 2 Number of algorithms in each category
12%
57%
25%
6%
Evoluonary Swarm Physics Human
Fig. 3 Percentage share for each category in the overall meta-heuris-
tic algorithms
122421
78545
64630
1506
0
20000
40000
60000
80000
100000
120000
140000
Evoluonary Swarm PhysicsHuman
Fig. 4 Total number of citations for each category

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where k is KNOWLEDGE rate which is a real number > 0,
G is generation number and GEN is the maximum number
of generations:
Thus, the number of gained and shared dimensions for
each vector using both schemes will be determined at initial-
ization phase. For more clarification, assume that the prob-
lem size is 100 and k is 2. Thus, this equation is decreased
in increasing rate (Table5).
Obviously, the knowledge rate k controls the experience
rate for each individual through generations using both
D(seniorphase)=problemsize −D(juniorphase).
schemes (from junior phase to senior phase). If k = 1, it is
linearly decreased and increased i.e. the number of updated
Dimensions using junior scheme is linearly decreased while
the number of updated Dimensions using senior scheme is
linearly increased through generations, otherwise it is non-
linearly decreased and increased, respectively. For k ∈ (0,1),
the number of updated dimensions using junior scheme is
decreased (non-linearly) slowly. Thus, during generation,
the junior scheme will be applied more than senior scheme
which means experience acquired in different branches of
knowledge with slow rate. On the other hand, For k > 1,
the number of updated dimensions using junior scheme is
decreased (non-linearly) rapidly. Therefore, during gen-
erations, the senior scheme will be used more than jun-
ior scheme which means experience acquired in different
branches of knowledge with fast rate.
2.1 Junior gaining‑sharing knowledge phase
In this phase, each individual tries to gain knowledge from
the closest and trusted individuals that belong to small
groups while he also tries to share knowledge with some
individual who does not belong to or, is not member in any
group due to his curiosity and desire of exploring others.
Fig. 5 Evolution of each
category over the last three
decades
0
20
40
60
80
100
120
14
0
160
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
2018
evoluonary swarm physics human
Fig. 6 Vector representa-
tion during junior and senior
gaining-sharing knowledge
phases
Vector 1during junior gaining and sharing phase (early-middle stage)
Junior dimensions (in green) Senior dimensions (in red)
11 12 1−1 1
Vector 1during senior gaining and sharing phase (middle-later stage)
Junior dimensions (in green) Senior dimensions (in red)
11 12 1−1 1
Table 5 Number of dimensions updated using junior and senior
schemes with problem size 100 and K = 2
G= G/GEN= D: number of dimensions updated
using both (junior and senior
schemes)
0 0 D(junior) = 100, D(junior) = 0
0.25 GEN 0.25 D(junior) = 57, D(senior) = 43
0.5 GEN 0.5 D(junior) = 25, D(senior) = 75
0.75 0.75 D(junior) = 7, D(senior) = 93
GEN 1 D(junior) = 0, D(senior) = 100

International Journal of Machine Learning and Cybernetics
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Thus, the updating of each individual can be computed using
junior scheme as follows:
1. Arrange all individuals in ascending order
according to their objective function value:
xbest,……,xi−1,xi,xi+1,……xwors t
2. Then, for each individual
xi
, select two different individ-
uals (the closest individuals), the nearest better (
xi−1
)and
worsen individuals (
xi+1
) than current one to constitute
the gain source of knowledge. Besides, select another
individual randomly (
xr
) to be the source of sharing
knowledge. The pseudo code of junior gaining-sharing
knowledge phase is presented in Fig.7.
Note that: in this phase, the best and worst individuals
are updated by using the closest best two individuals and the
closest worsen two individuals, respectively.
If
xi
is the global best, select the nearest forward best two
individuals as follows: (
xbest
,
xbest+1
,
xbest+2).
If
xi
is the global worst, select the nearest former worsen
two individuals as follows: (
………xworst−2
,
xworst−1
,
xworst )
.
Where
kf
is which is a real number > 0. It is the (knowl-
edge factor) that controls the total amount of gained and
shared knowledge that will be added from others to the cur-
rent individuals during generations.
Where
kr
∈[0, 1]
. It is the (knowledge ratio) that controls
the total amount of gained and shared knowledge that will be
transferred (inherited) during generations (the ratio between
the current and acquired experience).
2.2 Senior gaining‑sharing knowledge phase
This phase is concerned with utilization of available infor-
mation and appropriate knowledge from different categories
of the persons within specific population i.e. best, better and
worst persons. The utilization means the impact and effect
of others (good and bad persons) on a person. Thus, the
updating of each individual can be computed using senior
scheme as follows:
1. After sorting all individuals on ascending order accord-
ing to their objective function, they will be divided into
three category best individuals, better or middle indi-
viduals, worst individuals.
Best people
100p%(xp
−
best)
Better people
N−(
2
×
100
p%)(xm)
Worst people
100p%(xp
−
worst )
2. Then, for each individual
xi
, the proposed senior scheme
uses two random chosen vectors of the top and bottom
100p%
individuals in the current population of size NP
to form the gaining part while the third vector is selected
randomly from the middle
N−(2×100p%)
individu-
als to form the sharing part. The pseudo code of senior
gaining-sharing knowledge phase is presented in Fig.8.
where p ∈ [0,1], and p = 0.1, 10% of NP is suitable.
The pseudo code and the flow chart of GSK algorithm is
presented in Figs.9 and 10, respectively.
3 Numerical experiments andcomparisons
In this section, the computational results of GSK are dis-
cussed along with comparisons with other state-of-the-art
algorithms.
3.1 Experiments setup
The performance of the proposed GSK algorithm was tested
on 30 benchmark functions proposed in the CEC 2017 spe-
cial session on real-parameter optimization. A detailed
description of these test functions can be found in [140].
These 30 test functions can be divided into four classes:
1
For i=1:NP
2 For j=1:D
3If rand<= (Knowledge ratio)
4If ()> ( )
5=+ *[( −1-+1 )+(-)]
6else
7=+ *[( −1-+1 )+(-)]
8 End(if)
9Else =
10 End (If)
11 End (for j)
12 End (for i)
Fig. 7 Pseudo code of junior gaining-sharing knowledge phase
1
For i=1:NP
2For j=1:D
3If rand<= (Knowledge ratio)
4If ()> ( )
5=+ *[( −−−)+( − )]
6else
7=+ *[( −−−)+( − )]
8End(if)
9else =
10 End (If)
11 End (for j)
12 End (for i)
Fig. 8 Pseudo code of senior gaining-sharing knowledge phase

International Journal of Machine Learning and Cybernetics
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1. Unimodal functions f1–f3;
2. Simple multimodal functions f4–f10;
3. Hybrid functions f11–f20;
4. Composition functions f21–f30;
Besides, CEC2011 consists of 22 real-world optimization
problems for the CEC2011 special session and competition
on real world optimization problems. A detailed description
of these test functions can be found in [141].
Fig. 9 Pseudo code of GSK
algorithm
1
Be
g
in
2G=0, initialize parameters: N, ,, K and P.
3Create a random initial population ,=1,2,…..
4Evaluate ( ), ∀,=1,2,…..
5For G=1 to GEN
6 Compute the number of (Gained and Shared dimensions of both phases) using experience eq.(1);
7//Junior gaining-sharing knowledge phase //
8//Senior gaining-sharing knowledge phase//
9If ()≤ (), = , ( )= ( ) end // update each vector
10 If ( ) ≤(), = , ( ) =( ) end // update global best
11 End For…. N
12 End For…. G
14 End For…Begin
Fig. 10 The flow chart of GSK
algorithm Begin
Initialize Population size NP, knowledge factor ,
knowledge ratio , top and bottom percentage of
individuals 100 %, Knowledge rate K.
Calculate the fitness of each person of population
Identify the global best solution in the population
For each person of population, compute the number of
(Gained and Shared dimensions of both junior and
senior phases) using experience equation.
For each person of population, apply senior
gaining-sharing knowledge phasefor the
desired (D-N) dimensions.
For each person of population, apply junior
gaining-sharing knowledge phase for the
desired N dimensions.
Is termination
criteria satisfied?
Update global best solution
Update each person of population
Begin
Yes No

International Journal of Machine Learning and Cybernetics
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3.2 Parameter settings andinvolved algorithms
To evaluate the performance of algorithms, experiments
were conducted on the test suite. We adopt the solution
error measure
(f(x)−f(x∗))
, where x is the best solution
obtained by algorithms in one run and x* is well-known
global optimum of each benchmark function. Error values
and standard deviations smaller than 10–8 are taken as
zero [140]. For CEC2017, the dimensions (D) of function
are 10, 30, 50 and 100, respectively. The maximum num-
ber of function evaluations (FEs), the terminal criteria, is
set to 10,000 × D, all experiments for each function and
each algorithm run 51 times independently. For CEC2011,
the problems have different dimensions [141]. The maxi-
mum number of function evaluations (FEs), the terminal
criteria, is set to 150,000, all experiments for each function
and each algorithm run 25 times independently. GSK are
compared with 10 state-of-the-art population-based algo-
rithms in relevant literature. These algorithms are: TLBO
[137], GWO [80], SFS [18], AMO [78], DE [11], BBO
[51], ACO [21], ES [30], GA [6], PSO [24]. The control
parameters of the mentioned algorithms are given below in
Table6. Note that the control parameters of all algorithms
were directly taken from their original references. To the
best of our knowledge, this is the first study that uses all
these different types of approaches to carry out evaluation
and comparisons on up-to-date benchmark problems.
To perform comprehensive evaluation, the presentation
of the experimental results is divided into two subsec-
tions. First, the performance of the proposed algorithm
is discussed. Second, an overall performance comparison
between GSK and other 10 state-of-the-art algorithms is
provided.
The Performance assessment of the different algorithms
is based on score metric which is recently defined for the
CEC 2017 competition [140]. Thus, the evaluation method
for each algorithm is based on a score of 100 which is
based on two criteria as follows taking into account higher
weights will be given for higher dimensions:
1. 50% summation of error values for all dimensions as
follows:
SE
=0.1 ×
29
∑
i=1
ef10D+0.2 ×
29
∑
i=1
ef30D+
0.3
×
29
∑
i=1
ef50D+0.4 ×
29
∑
i=1
ef100D.
Score 1 =
(
1−SE −SEmin
SE )
×50.
Table 6 The control parameters of search algorithms
Name Specifications NP
TLBO No special parameters 50, the population size is 50 because of this algorithm has two
phases
GWO a = 2-2(g/max_g) 100
SFS Maximum diffusion number (MDN) is set to 1 50, the population size is 50 because of this algorithm has two
phases
AMO The number of animals in each group was set to 5 50, the population size is 50 because of this algorithm has two
phases
DE F = 0.5, CR = 0.9 in accordance 100
BBO Habitat modification probability = 1, immigration probability bounds
per gene = [0,1], step size for numerical integration of prob-
abilities = 1, maximum immigration and migration rates for each
island = 1, and mutation probability = 0
100
ACO Initial pheromone value = 1e−6; pheromone update constant = 20;
exploration constant = 1; global pheromone decay rate = 0.9; local
pheromone decay rate = 0.5; pheromone sensitivity = 1; visibility
sensitivity = 5;
100
ES λ = 10,
𝜎=1
, have been recommended 100
GA Roulette wheel selection, single point crossover with a crossover
probability of 1, and a mutation probability of 0.01.
100
PSO
𝜔=0.6
, c1 = c2 = 2 100
GSK P = 0.1,
kf
= 0.5,
kr
= 0.9, K = 10 100

International Journal of Machine Learning and Cybernetics
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2. 50% rank based for each problem in each dimension as
follows:
3. Combine the above two parts to find the final score as
follows:
Note that
f2
has been excluded because it shows unstable
behavior especially for higher dimensions.
Besides, to compare and analyze the solution quality from
a statistical angle of different algorithms and to check the
SR
=0.1 ×
29
∑
i=1
rank10D+0.2 ×
29
∑
i=1
rank30D+
0.3
×
29
∑
i=1
rank50D+0.4 ×
29
∑
i=1
rank100D.
Score 2 =
(
1−
SR −SRmin
SR
)
×50.
Score =Score 1 +Score 2.
behavior of the stochastic algorithms [142], the results are
compared using non-parametric statistical hypothesis tests:
multi-problem Wilcoxon signed-rank test (to check the dif-
ferences between all algorithms for all functions); at a 0.05
significance level, where R+ denotes the sum of ranks for the
test problems in which the first algorithm performs better
than the second algorithm (in the first column), and R− rep-
resents the sum of ranks for the test problems in which the
first algorithm performs worse than the second algorithm (in
the first column). Larger ranks indicate larger performance
discrepancy. As a null hypothesis, it is assumed that there is
no significance difference between the mean results of the
two samples. Whereas the alternative hypothesis is that there
is significance in the mean results of the two samples. For
CEC2017, the number of test problems is N = 29 for D = 10,
30, 50 and 100 dimensions and 5% significance level. For
CEC2011, the number of test problems is 22 and 5% signifi-
cance level. Use the p value and compare it with the signifi-
cance level. Reject the null hypothesis if the p-value is less
Table 7 Result of GSK in 10D
Function Best Median Mean Worst SD
1 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
3 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
4 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
5 1.53E+01 2.01E+01 2.03E+01 2.58E+01 2.79E+00
6 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
7 2.46E+01 3.10E+01 3.07E+01 3.74E+01 3.08E+00
8 1.45E+01 1.99E+01 2.02E+01 2.58E+01 2.92E+00
9 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
10 7.54E+02 1.10E+03 1.06E+03 1.29E+03 1.33E+02
11 0.00E+00 0.00E+00 0.00E+00 1.02E−08 0.00E+00
12 1.42E+01 7.57E+01 8.93E+01 2.69E+02 7.26E+01
13 9.19E−01 6.56E+00 6.56E+00 9.73E+00 1.41E+00
14 5.19E−03 5.42E+00 5.88E+00 1.19E+01 3.06E+00
15 2.96E−03 1.75E−01 2.22E−01 5.00E−01 2.13E−01
16 3.29E−01 9.48E−01 4.27E+00 1.18E+01 4.95E+00
17 9.76E−01 9.94E+00 1.17E+01 2.52E+01 7.09E+00
18 2.05E−02 4.30E−01 3.20E−01 5.01E−01 1.92E−01
19 2.81E−02 6.95E−02 1.55E−01 1.80E+00 3.51E−01
20 3.12E−01 3.12E−01 1.19E+00 2.03E+01 3.99E+00
21 1.00E+02 2.20E+02 1.93E+02 2.31E+02 5.07E+01
22 1.00E+02 1.00E+02 1.00E+02 1.02E+02 4.87E−01
23 3.11E+02 3.18E+02 3.18E+02 3.29E+02 3.99E+00
24 2.78E+02 3.50E+02 3.44E+02 3.57E+02 1.87E+01
25 3.98E+02 4.34E+02 4.27E+02 4.46E+02 2.05E+01
26 3.00E+02 3.00E+02 3.00E+02 3.00E+02 0.00E+00
27 3.89E+02 3.90E+02 3.89E+02 3.90E+02 2.17E−01
28 3.00E+02 3.00E+02 3.12E+02 3.97E+02 3.20E+01
29 2.39E+02 2.48E+02 2.48E+02 2.57E+02 4.76E+00
30 3.96E+02 4.65E+02 4.56E+02 5.02E+02 3.44E+01

International Journal of Machine Learning and Cybernetics
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than or equal the significance level (5%). All the p values
in this paper were computed using SPSS (version 20.00).
3.3 Experimental results anddiscussions
3.3.1 Results oftheproposed approach (GSK) onCEC2017
The statistical results of the GSK on the CEC2017 bench-
marks with 10, 30, 50 and 100 dimensions are summarized
in Tables7, 8, 9 and 10, respectively. It includes the obtained
best, median, mean, worst values and the standard deviations
of error from optimum solution of the proposed GSK over
51 runs for all 29 benchmark functions.
Generally, from Tables7, 8, 9 and 10 it can be clearly
seen that GSK succeeded at solving, at least once, six
problems in 10D, four problems in 30D, three problems
in 50D, and one problem in 100D. GSK has outstanding
performance on unimodal problems (f1–f3), GSK is able to
find the global optimal solution consistently over 51 runs
in 10 and 30 dimensions. In 50 dimensions, the optimum
is detected in 1 case, while the mean error ranges from
1.09E + 03 to 3.85E + 03 and the standard deviation ranges
from 1.24E + 03 to 1.51E + 03. In 100 dimensions, the mean
error ranges from 5.80E + 03 to 1.15E + 05 and the standard
deviation ranges from 4.63E + 03 to 2.15E + 04.
As for the simple multimodal functions (f4–f10), in 10
dimensions the optimum is detected continuously over 51
runs in 3 cases, while the mean error ranges from 0.00E + 00
to 1.06E + 03 and the standard deviation ranges from
0.00E + 00 to 1.33E + 02. In 30 dimensions, the optimum
is detected continuously over 51 runs in 2 cases, while the
mean error ranges from 0.00E + 00 to 6.69E + 03 and the
standard deviation ranges from 0.00E + 00 to 3.54E + 02.
In 50 dimensions, the optimum is detected in 2 case, while
the mean error ranges from 3.78E−06 to 1.30E + 04 and the
standard deviation ranges from 3.52E−06 to 4.52E + 02. In
100 dimensions, the mean error ranges from 1.72E−03 to
2.95E + 04 and the standard deviation ranges from 6.46E−03
Table 8 Result of GSK in 30D
Function Best Median Mean Worst SD
1 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
3 0.00E+00 5.66E−08 7.07E−07 8.54E−06 1.82E−06
4 9.28E−03 4.00E+00 1.11E+01 7.25E+01 2.17E+01
5 1.36E+02 1.62E+02 1.60E+02 1.73E+02 8.87E+00
6 0.00E+00 5.47E−07 1.52E−06 8.42E−06 2.36E−06
7 1.64E+02 1.87E+02 1.87E+02 1.99E+02 8.40E+00
8 1.24E+02 1.56E+02 1.55E+02 1.73E+02 1.11E+01
9 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
10 5.84E+03 6.68E+03 6.69E+03 7.16E+03 3.54E+02
11 6.00E−01 8.57E+00 3.30E+01 9.85E+01 3.83E+01
12 1.03E+03 5.32E+03 6.62E+03 1.91E+04 4.59E+03
13 4.61E+01 9.11E+01 9.83E+01 1.82E+02 3.42E+01
14 4.72E+01 5.73E+01 5.69E+01 6.65E+01 5.49E+00
15 2.13E+00 9.17E+00 1.45E+01 7.48E+01 1.49E+01
16 4.35E+02 7.76E+02 7.96E+02 1.15E+03 1.94E+02
17 6.28E+01 2.04E+02 1.89E+02 3.77E+02 9.44E+01
18 2.25E+01 3.69E+01 3.68E+01 4.80E+01 5.42E+00
19 3.42E+00 1.13E+01 1.29E+01 2.22E+01 6.02E+00
20 1.34E+00 5.51E+01 1.08E+02 4.51E+02 1.14E+02
21 3.20E+02 3.49E+02 3.46E+02 3.55E+02 8.30E+00
22 1.00E+02 1.00E+02 1.00E+02 1.00E+02 0.00E+00
23 3.50E+02 4.86E+02 4.70E+02 5.07E+02 4.49E+01
24 5.29E+02 5.70E+02 5.68E+02 5.86E+02 1.45E+01
25 3.87E+02 3.87E+02 3.87E+02 3.87E+02 2.11E−01
26 6.93E+02 9.56E+02 9.87E+02 2.12E+03 2.49E+02
27 4.80E+02 4.92E+02 4.93E+02 5.14E+02 8.02E+00
28 3.00E+02 3.00E+02 3.21E+02 4.04E+02 4.22E+01
29 4.23E+02 5.92E+02 5.77E+02 7.69E+02 1.01E+02
30 1.94E+03 2.09E+03 2.08E+03 2.36E+03 9.27E+01
Table 9 Result of GSK in 50D
Function Best Median Mean Worst SD
1 1.71E+01 4.73E+02 1.09E+03 4.61E+03 1.24E+03
3 1.61E+03 3.79E+03 3.85E+03 6.73E+03 1.51E+03
4 1.33E−02 7.21E+01 8.33E+01 1.46E+02 5.00E+01
5 2.65E+02 3.25E+02 3.20E+02 3.45E+02 1.79E+01
6 3.11E−07 2.80E−06 3.78E−06 1.64E−05 3.52E−06
7 3.29E+02 3.74E+02 3.70E+02 3.86E+02 1.41E+01
8 2.90E+02 3.28E+02 3.24E+02 3.38E+02 1.36E+01
9 0.00E+00 0.00E+00 1.07E−02 8.95E−02 2.79E−02
10 1.21E+04 1.28E+04 1.30E+04 1.37E+04 4.50E+02
11 2.34E+01 2.96E+01 3.45E+01 1.45E+02 2.32E+01
12 2.16E+03 7.71E+03 9.46E+03 3.17E+04 7.01E+03
13 7.41E+01 6.24E+02 1.49E+03 8.98E+03 2.16E+03
14 5.76E+01 1.28E+02 1.24E+02 1.42E+02 1.87E+01
15 2.52E+01 3.62E+01 4.20E+01 1.00E+02 1.68E+01
16 1.30E+02 2.01E+03 1.83E+03 2.70E+03 6.59E+02
17 7.75E+02 1.39E+03 1.35E+03 1.63E+03 1.90E+02
18 1.78E+02 5.01E+02 5.98E+02 1.45E+03 3.37E+02
19 1.84E+01 2.92E+01 3.05E+01 5.13E+01 9.59E+00
20 1.17E+03 1.40E+03 1.37E+03 1.62E+03 1.28E+02
21 4.90E+02 5.25E+02 5.21E+02 5.46E+02 1.31E+01
22 1.00E+02 1.31E+04 1.10E+04 1.35E+04 4.85E+03
23 4.20E+02 4.46E+02 5.42E+02 7.49E+02 1.39E+02
24 5.01E+02 5.24E+02 6.34E+02 8.11E+02 1.39E+02
25 4.60E+02 5.64E+02 5.56E+02 6.11E+02 4.62E+01
26 1.06E+03 1.29E+03 1.27E+03 1.42E+03 9.18E+01
27 5.18E+02 5.64E+02 5.92E+02 9.16E+02 8.29E+01
28 4.59E+02 4.97E+02 4.94E+02 5.70E+02 2.24E+01
29 3.28E+02 3.55E+02 3.60E+02 4.09E+02 2.23E+01
30 5.79E+05 5.81E+05 5.96E+05 6.52E+05 2.24E+04

International Journal of Machine Learning and Cybernetics
1 3
to 4.44E + 02. Considering hybrid functions (f11–f20), GSK
is able to obtain good solutions for all test problems, except
for f10 and f12 in which the performance was reduced when
D increased.
Finally, in regards to composition functions
(f21–f30),which considered the most difficult problems in the
benchmark suite as they are highly multimodal, non-separa-
ble and possess different properties around huge number of
local optima [140].the optimum is detected in 1 case, while
the mean error ranges from 1.00E + 02 to 4.56E + 02, from
1.00E + 02 to 2.08E + 03, from 3.60E + 02 to 5.96E + 05,
and from 5.53E + 02to 3.00E + 04, while the standard devia-
tion ranges from 0.00E + 00 to 5.07E + 01, from 0.00E + 00
to 2.49E + 02, from 1.31E + 01 to 2.24E + 04, and from
1.58E + 01 to 4.57E + 02 in 10, 30, 50 and 100 dimensions,
respectively. Thus, GSK is often trapped in local optimum
which is still not far away from the optimum in all functions.
Therefore, it can be concluded that in all functions for all
the four dimensionalities, the differences between mean and
median are small even in the cases when the final results are
far away from the optimum, regardless of the dimensions.
That implies the GSK is a robust algorithm. Finally, due to
insignificant difference between the results in four dimen-
sions, it can be concluded that the performance of the GSK
algorithm slightly diminishes, and it is still more stable and
robust against the curse of dimensionality i.e. it is overall
steady as the dimensions of the problems increases.
3.3.2 Comparison againststate‑of‑the‑art algorithms
The statistical results of the comparisons on the bench-
marks with 10, 30, 50 and 100 dimensions are summarized
in Tables11, 12, 13, 14, 15, 16, 17, and 18 respectively. It
includes the obtained best and the standard deviations of
error from optimum solution of GSK and other ten state-of-
the-art algorithms over 51 runs for all 29 benchmark func-
tions. The best results are marked in bold for all problems.
Ranking of the algorithms using score metric on the CEC
Table 10 Result of GSK in 100D
Function Best Median Mean Worst SD
1 3.96E−01 5.38E+03 5.80E+03 2.11E+04 4.63E+03
3 6.01E+04 1.15E+05 1.15E+05 1.48E+05 2.15E+04
4 8.46E+01 2.17E+02 2.05E+02 2.89E+02 4.57E+01
5 7.26E+01 7.70E+02 5.31E+02 8.14E+02 3.39E+02
6 1.25E−05 6.83E−05 1.72E−03 3.14E−02 6.46E−03
7 8.33E+02 8.76E+02 8.75E+02 9.16E+02 1.83E+01
8 5.90E+01 7.36E+02 4.92E+02 8.18E+02 3.41E+02
9 5.45E+00 7.69E+00 8.46E+00 1.72E+01 3.41E+00
10 2.87E+04 2.94E+04 2.95E+04 3.04E+04 4.44E+02
11 1.61E+02 2.65E+02 2.78E+02 4.64E+02 6.85E+01
12 2.25E+04 6.53E+04 8.34E+04 3.23E+05 7.52E+04
13 5.17E+01 2.85E+03 3.20E+03 9.31E+03 2.64E+03
14 3.30E+02 2.53E+03 4.64E+03 1.69E+04 4.47E+03
15 3.20E+01 4.23E+02 7.33E+02 5.20E+03 1.09E+03
16 2.87E+02 8.25E+02 2.27E+03 7.04E+03 2.61E+03
17 2.14E+03 4.12E+03 3.91E+03 4.59E+03 6.68E+02
18 1.93E+04 4.43E+04 5.73E+04 1.87E+05 3.63E+04
19 5.04E+01 8.41E+02 1.00E+03 3.25E+03 8.30E+02
20 3.97E+03 4.52E+03 4.46E+03 4.80E+03 2.22E+02
21 2.83E+02 3.29E+02 6.07E+02 1.02E+03 3.37E+02
22 2.86E+04 3.01E+04 3.00E+04 3.09E+04 4.57E+02
23 5.86E+02 6.11E+02 6.11E+02 6.49E+02 1.58E+01
24 8.89E+02 9.33E+02 9.32E+02 9.69E+02 1.70E+01
25 7.61E+02 8.20E+02 8.21E+02 8.99E+02 4.34E+01
26 3.33E+03 3.63E+03 3.66E+03 3.96E+03 1.78E+02
27 6.20E+02 6.46E+02 6.57E+02 7.30E+02 3.07E+01
28 4.99E+02 5.57E+02 5.53E+02 6.09E+02 3.25E+01
29 9.12E+02 1.21E+03 1.21E+03 1.56E+03 1.74E+02
30 2.41E+03 3.02E+03 2.99E+03 3.72E+03 2.73E+02

International Journal of Machine Learning and Cybernetics
1 3
2017 functions is given in Table19. The multi-problem Wil-
coxon signed-rank GSK and others in 10D, 30D, 50D and
100D are summarized in Table20.
Firstly, regarding evolutionary and physical based algo-
rithms in four dimensions, it can be observed that SFS and
DE algorithms can be good at different functions in some or
all dimensions. However, GA and ES perform poorly on all
the functions in all dimensions. Generally, GSK, SFS and
DE do significantly better than the others on most functions
in different dimensions.
On the other hand, regarding swarm intelligence-based
algorithms, it can be obviously shown that AMO is competi-
tive with GSK on some functions in different dimensions.
However, GSK is superior to the others in most functions
in all dimensions. Actually, it can be obviously seen that
the performance of all compared algorithms shows com-
plete and/or significant deterioration with the growth of the
search-space dimensionality from 10D to 100D while the
performance of the GSK algorithm slightly diminishes, and
it is still more stable, efficient and robust against the curse
of dimensionality.
Secondly, the performance of GSK and other competi-
tive algorithms on the functions of different dimensions is
discussed. Table19 clearly shows that GSK gets the first
ranking among all algorithms, followed by DE and AMO in
second and third place, respectively. However, GWO, ACO
and ES are the poorest algorithms, respectively. The score
of all algorithms on the CEC 2017 functions is shown in
Fig.11. Table20 summarizes the statistical analysis results
of applying multiple-problem Wilcoxon’s test between GSK
and other compared algorithms for 10D, 30D, 50D and 100D
problems.
From Table20, we can see that GSK obtains higher R+
values than R− in all the cases with exception to DE and
AMO in 10D. Precisely, we can draw the following con-
clusions: GSK outperforms TLBO, GA, ES, GWO, PSO,
BBO and ACO significantly in all dimensions with excep-
tion to DE, SFS and AMO in 10, 30 and 50 dimensions
and SFS and AMO 100 dimensions. Thus, according to the
Wilcoxon’s test at α = 0.05, the significance difference can
be observed in 28 cases out of 40, which means that GSK is
Table 11 Experimental results of TLBO, SFS, DE, GA, ES and GSK over 51 independent runs on 29 test functions of 10 variables with 100,000
FES
Function TLBO SFS DE GA ES GSK
11.96E+03 ± 2.52E+03 5.60E+03 ± 3.26E+03 0.00E+00 ± 0.00E+00 1.14E+06 ± 2.18E+05 7.20E+09 ± 2.05E+09 0.00E+00 ± 0.00E+00
30.00E+00 ± 0.00E+00 2.05E−02 ± 1.17E−02 0.00E+00 ± 0.00E+00 1.24E+04 ± 7.87E+03 2.47E+04 ± 7.25E+03 0.00E+00 ± 0.00E+00
41.79E−01 ± 3.45E−01 9.37E−01 ± 6.74E−01 0.00E+00 ± 0.00E+00 1.17E+01 ± 3.29E+00 5.28E+02 ± 1.34E+02 0.00E+00 ± 0.00E+00
58.24E+00 ± 3.38E+00 8.68E+00 ± 3.30E+00 2.48E+01 ± 4.13E+00 4.90E+01 ± 1.22E+01 9.40E+01 ± 1.27E+01 2.03E+01 ± 2.79E+00
63.76E−02 ± 1.28E−01 5.35E−03 ± 1.59E−03 0.00E+00 ± 0.00E+00 1.24E+01 ± 4.54E+00 5.60E+01 ± 9.08E+00 0.00E+00 ± 0.00E+00
71.76E+01 ± 3.25E+00 2.33E+01 ± 3.60E+00 3.43E+01 ± 4.84E+00 6.87E+01 ± 1.46E+01 2.70E+02 ± 3.87E+01 3.07E+01 ± 3.08E+00
86.84E+00 ± 2.75E+00 7.42E+00 ± 2.76E+00 2.37E+01 ± 3.76E+00 3.54E+01 ± 8.94E+00 8.98E+01 ± 9.39E+00 2.02E+01 ± 2.92E+00
92.64E−01 ± 4.22E−01 6.68E−06 ± 4.34E−06 0.00E+00 ± 0.00E+00 2.39E+01 ± 1.75E+01 1.78E+03 ± 3.34E+02 0.00E+00 ± 0.00E+00
10 4.95E+02 ± 2.92E+02 3.63E+02 ± 1.91E+02 4.94E+02 ± 2.99E+02 7.52E+02 ± 2.36E+02 1.82E+03 ± 2.12E+02 1.06E+03 ± 1.33E+02
11 7.06E+00 ± 5.17E+00 4.61E+00 ± 1.25E+00 4.33E−02 ± 1.95E−01 8.21E+01 ± 7.89E+01 1.30E+03 ± 6.80E+02 0.00E+00 ± 0.00E+00
12 1.28E+04 ± 9.07E+03 5.04E+03 ± 2.11E+03 8.06E+00 ± 1.89E+01 1.12E+06 ± 1.78E+06 2.97E+08 ± 1.77E+08 8.93E+01 ± 7.26E+01
13 1.70E+03 ± 1.91E+03 4.55E+01 ± 9.83E+00 6.56E+00 ± 1.86E+00 2.62E+04 ± 2.19E+04 3.26E+06 ± 3.13E+06 6.56E+00 ± 1.41E+00
14 3.40E+01 ± 7.43E+00 2.20E+01 ± 3.82E+00 4.30E−02 ± 1.96E−01 1.48E+04 ± 6.43E+03 4.46E+03 ± 4.50E+03 5.88E+00 ± 3.06E+00
15 4.75E+01 ± 2.09E+01 1.00E+01 ± 2.14E+00 3.60E−02 ± 1.03E−01 1.38E+04 ± 1.26E+04 2.16E+04 ± 2.26E+04 2.22E−01 ± 2.13E−01
16 1.44E+01 ± 3.48E+01 4.17E+00 ± 3.16E+00 2.93E+00 ± 4.65E+00 3.00E+01 ± 4.18E+01 5.25E+02 ± 1.47E+02 4.27E+00 ± 4.95E+00
17 2.76E+01 ± 8.83E+00 2.34E+01 ± 5.66E+00 4.82E+00 ± 6.77E+00 4.12E+01 ± 1.29E+01 2.48E+02 ± 5.75E+01 1.17E+01 ± 7.09E+00
18 4.55E+03 ± 3.67E+03 5.22E+01 ± 1.05E+01 8.18E−02 ± 1.73E−01 8.30E+04 ± 9.59E+04 7.61E+06 ± 8.08E+06 3.20E−01 ± 1.92E−01
19 2.94E+01 ± 1.83E+01 5.84E+00 ± 8.94E−01 6.27E−03 ± 1.11E−02 6.30E+04 ± 4.21E+04 9.69E+04 ± 1.08E+05 1.55E−01 ± 3.51E−01
20 1.75E+01 ± 1.04E+01 1.21E+01 ± 3.31E+00 1.10E−01 ± 1.63E−01 1.02E+02 ± 5.01E+01 2.41E+02 ± 6.43E+01 1.19E+00 ± 3.99E+00
21 1.62E+02 ± 5.26E+01 1.00E+02 ± 5.06E−02 1.80E+02 ± 6.22E+01 1.54E+02 ± 1.49E+01 1.73E+02 ± 5.57E+00 1.93E+02 ± 5.07E+01
22 9.98E+01 ± 1.07E+01 9.24E+01 ± 3.00E+01 9.48E+01 ± 2.30E+01 1.25E+02 ± 2.17E+01 9.06E+02 ± 2.61E+02 1.00E+02 ± 4.87E−01
23 3.10E+02 ± 4.31E+00 3.03E+02 ± 4.35E+01 3.17E+02 ± 5.20E+00 3.63E+02 ± 1.51E+01 4.05E+02 ± 1.01E+01 3.18E+02 ± 3.99E+00
24 3.23E+02 ± 5.62E+01 2.18E+02 ± 1.18E+02 3.43E+02 ± 4.97E+01 3.90E+02 ± 3.40E+01 4.37E+02 ± 3.31E+01 3.44E+02 ± 1.87E+01
25 4.26E+02 ± 2.23E+01 4.21E+02 ± 2.30E+01 4.11E+02 ± 2.10E+01 4.30E+02 ± 2.52E+01 9.17E+02 ± 1.74E+02 4.27E+02 ± 2.05E+01
26 3.31E+02 ± 5.60E+01 2.92E+02 ± 4.40E+01 3.00E+02 ± 0.00E+00 4.19E+02 ± 8.88E+01 1.44E+03 ± 2.85E+02 3.00E+02 ± 0.00E+00
27 3.93E+02 ± 2.62E+00 3.92E+02 ± 1.85E+00 3.90E+02 ± 2.63E−01 4.10E+02 ± 9.10E+00 4.37E+02 ± 7.98E+00 3.89E+02 ± 2.17E−01
28 4.01E+02 ± 1.26E+02 3.06E+02 ± 3.97E+01 3.63E+02 ± 1.21E+02 5.82E+02 ± 1.56E+02 8.11E+02 ± 9.22E+01 3.12E+02 ± 3.20E+01
29 2.63E+02 ± 1.46E+01 2.59E+02 ± 1.18E+01 2.40E+02 ± 4.73E+00 3.04E+02 ± 4.09E+01 5.69E+02 ± 6.52E+01 2.48E+02 ± 4.76E+00
30 1.25E+05 ± 2.98E+05 2.03E+03 ± 1.63E+03 1.64E+04 ± 1.14E+05 3.73E+05 ± 3.78E+05 8.83E+05 ± 0.00E+00 4.56E+02 ± 3.44E+01

International Journal of Machine Learning and Cybernetics
1 3
significantly better than 7 algorithms out of 10 algorithms
on 29 test functions at α = 0.05.
Alternatively, to be more precise, it is obvious from
Table20 that GSK is inferior to, equal to, superior to other
algorithms in 71, 14, 205 out of the total 290 cases in 10D,
55, 7, 228 out of the total 290 cases in 30D, 56, 1, 233 out
of the total 290 cases in 50D, 51, 0, 239 out of the total 290
cases in 100D, respectively. In summary, GSK is inferior to,
equal to, superior to other algorithms in 233, 22, 905 cases,
respectively out of total 1160 cases.
Thus, it can be concluded that the performance of GSK is
almost better than the performance of compared algorithms
in 78% of all cases, respectively, and it is just outperformed
by other compared algorithms in 20% of all problems in all
dimensions. Furthermore, it can be obviously deduced from
Fig.12 that the superiority of the GSK algorithm against
the compared algorithms increases as the dimensions of the
problems increases from 10 to 100 dimensions.
From the above results, comparisons and discussion
through this section, the proposed GSK algorithm is of bet-
ter searching quality, efficiency and robustness for solving
small, moderate and high dimensions unconstrained global
optimization problems. It is clear that the proposed GSK
algorithm perform well, and it has shown its outstanding
superiority with separable, non-separable, unimodal, mul-
timodal, hybrid and composition functions with shifts in
dimensionality, rotation, multiplicative noise in fitness and
composition of functions.
Consequently, its performance is not influenced by all
these obstacles. Contrarily, it greatly keeps the balance the
local optimization speed and the global optimization diver-
sity in challenging optimization environment with invariant
performance. Besides, its performance is superior and com-
petitive with the performance of the state-of-the-art well-
known algorithms. Finally, it can be concluded that the pro-
posed junior and senior phases help to maintain effectively
the balance between the global exploration and local exploi-
tation abilities for searching process of the GSK. Besides,
GSK is very simple and easy to implement and program in
many programming languages.
Furthermore, in order to analyze the convergence
behavior of GSK and other state-of-the-art algorithms, the
Table 12 Experimental results of GWO, AMO, PSO, BBO, ACO and GSK over 51 independent runs on 29 test functions of 10 variables with
100,000 FES
Function GWO AMO PSO BBO ACO GSK
11.50E+08 ± 6.50E+07 9.84E+00 ± 2.69E+01 2.30E+03 ± 3.05E+03 1.12E+06 ± 2.26E+05 1.57E+10 ± 3.75E+09 0.00E+00 ± 0.00E+00
35.28E+02 ± 7.98E+02 2.90E−08 ± 9.16E−08 0.00E+00 ± 0.00E+00 8.35E+02 ± 8.07E+02 6.32E+03 ± 1.74E+03 0.00E+00 ± 0.00E+00
41.84E+01 ± 1.62E+01 1.39E+00 ± 6.29E−01 2.82E+00 ± 1.21E+00 6.20E+00 ± 1.72E+00 1.83E+02 ± 4.26E+01 0.00E+00 ± 0.00E+00
53.01E+01 ± 4.85E+00 6.36E+00 ± 1.49E+00 1.59E+01 ± 7.08E+00 7.96E+00 ± 2.77E+00 6.86E+01 ± 8.53E+00 2.03E+01 ± 2.79E+00
67.81E+00 ± 1.09E+00 0.00E+00 ± 0.00E+00 8.27E−02 ± 3.36E−01 9.19E−01 ± 0.00E+00 3.57E+01 ± 5.40E+00 0.00E+00 ± 0.00E+00
74.35E+01 ± 5.14E+00 1.82E+01 ± 1.42E+00 1.72E+01 ± 4.46E+00 2.31E+01 ± 3.82E+00 1.23E+02 ± 1.43E+01 3.07E+01 ± 3.08E+00
82.38E+01 ± 4.44E+00 6.84E+00 ± 1.49E+00 1.23E+01 ± 5.33E+00 8.66E+00 ± 3.05E+00 4.85E+01 ± 6.66E+00 2.02E+01 ± 2.92E+00
91.10E+01 ± 3.79E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 3.15E−01 ± 8.16E−02 5.56E+02 ± 1.53E+02 0.00E+00 ± 0.00E+00
10 8.53E+02 ± 2.50E+02 3.65E+02 ± 1.01E+02 6.30E+02 ± 2.64E+02 2.57E+02 ± 1.55E+02 1.45E+03 ± 1.34E+02 1.06E+03 ± 1.33E+02
11 4.03E+01 ± 9.91E+00 2.76E+00 ± 7.90E−01 1.10E+01 ± 7.27E+00 8.92E+00 ± 4.26E+00 1.96E+02 ± 6.99E+01 0.00E+00 ± 0.00E+00
12 2.58E+06 ± 3.10E+06 1.13E+04 ± 6.49E+03 1.33E+04 ± 1.24E+04 2.55E+05 ± 1.60E+05 9.79E+08 ± 9.01E+08 8.93E+01 ± 7.26E+01
13 1.16E+04 ± 8.11E+03 2.67E+01 ± 8.99E+00 6.45E+03 ± 5.72E+03 8.09E+04 ± 7.12E+04 7.02E+06 ± 2.36E+07 6.56E+00 ± 1.41E+00
14 5.91E+02 ± 1.21E+03 3.83E+00 ± 1.67E+00 4.57E+01 ± 1.93E+01 1.20E+04 ± 1.01E+04 4.33E+02 ± 1.10E+02 5.88E+00 ± 3.06E+00
15 8.04E+02 ± 1.12E+03 1.75E+00 ± 5.40E−01 5.62E+01 ± 5.89E+01 1.95E+04 ± 1.94E+04 3.74E+03 ± 2.37E+03 2.22E−01 ± 2.13E−01
16 8.52E+01 ± 9.41E+01 1.13E+00 ± 2.64E−01 2.05E+02 ± 1.19E+02 1.06E+02 ± 8.75E+01 2.65E+02 ± 5.71E+01 4.27E+00 ± 4.95E+00
17 5.42E+01 ± 8.45E+00 4.92E+00 ± 2.53E+00 4.56E+01 ± 2.30E+01 2.74E+01 ± 3.49E+01 1.14E+02 ± 2.31E+01 1.17E+01 ± 7.09E+00
18 3.68E+04 ± 2.11E+04 3.13E+01 ± 8.02E+00 5.10E+03 ± 5.76E+03 1.45E+05 ± 1.01E+05 6.67E+07 ± 1.65E+08 3.20E−01 ± 1.92E−01
19 1.75E+03 ± 3.81E+03 1.11E+00 ± 4.28E−01 9.57E+01 ± 2.95E+02 4.34E+04 ± 4.02E+04 1.24E+04 ± 1.31E+04 1.55E−01 ± 3.51E−01
20 7.76E+01 ± 3.83E+01 6.19E−04 ± 1.74E−03 5.51E+01 ± 5.42E+01 1.29E+01 ± 5.91E+00 1.20E+02 ± 2.77E+01 1.19E+00 ± 3.99E+00
21 2.03E+02 ± 4.93E+01 1.37E+02 ± 5.02E+01 1.79E+02 ± 5.65E+01 2.00E+02 ± 3.07E+01 1.37E+02 ± 1.29E+01 1.93E+02 ± 5.07E+01
22 1.25E+02 ± 6.03E+00 9.91E+01 ± 7.18E+00 9.38E+01 ± 2.55E+01 1.06E+02 ± 7.47E+00 2.57E+02 ± 6.14E+01 1.00E+02 ± 4.87E−01
23 3.33E+02 ± 3.86E+00 3.08E+02 ± 1.61E+00 3.28E+02 ± 1.24E+01 3.13E+02 ± 4.97E+00 3.78E+02 ± 9.61E+00 3.18E+02 ± 3.99E+00
24 3.63E+02 ± 4.56E+00 2.85E+02 ± 8.09E+01 3.24E+02 ± 8.33E+01 3.36E+02 ± 2.66E+01 2.93E+02 ± 3.63E+01 3.44E+02 ± 1.87E+01
25 4.42E+02 ± 1.56E+01 4.18E+02 ± 2.23E+01 4.25E+02 ± 2.29E+01 4.34E+02 ± 2.26E+01 5.76E+02 ± 4.23E+01 4.27E+02 ± 2.05E+01
26 4.09E+02 ± 1.47E+02 3.00E+02 ± 0.00E+00 2.74E+02 ± 7.63E+01 3.40E+02 ± 1.51E+02 7.47E+02 ± 8.18E+01 3.00E+02 ± 0.00E+00
27 3.96E+02 ± 1.15E+00 3.91E+02 ± 2.38E+00 4.03E+02 ± 1.97E+01 3.97E+02 ± 4.39E+00 4.40E+02 ± 8.57E+00 3.89E+02 ± 2.17E−01
28 5.39E+02 ± 9.99E+01 2.99E+02 ± 3.99E+00 4.54E+02 ± 1.57E+02 5.48E+02 ± 9.68E+01 5.85E+02 ± 4.64E+01 3.12E+02 ± 3.20E+01
29 2.94E+02 ± 3.04E+01 2.64E+02 ± 8.17E+00 3.05E+02 ± 4.50E+01 2.65E+02 ± 1.56E+01 3.90E+02 ± 3.41E+01 2.48E+02 ± 4.76E+00
30 4.84E+05 ± 7.31E+05 7.42E+03 ± 5.17E+03 2.00E+05 ± 3.79E+05 4.63E+05 ± 5.39E+05 2.23E+07 ± 2.32E+07 4.56E+02 ± 3.44E+01

International Journal of Machine Learning and Cybernetics
1 3
convergence characteristics in terms of the best fitness value
of the median run of all algorithms for some functions with
dimensions 10,30, 50 and 100 is illustrated in the supple-
mental file (Fig. S1). It is clear that the convergence speed of
the GSK algorithm is fast at the early stage of the optimiza-
tion process for all functions with different shapes, complex-
ity, and dimensions. Furthermore, the convergence speed is
dramatically decreased, and its improvement is found to be
significant in the middle and later stages of the optimization
process.
Additionally, the convergent figure suggests that the GSK
algorithm can reach the global solution or better solution in
most problems in a fewer number of generations less than
the maximum predetermined number of generations. In gen-
eral, GSK is scalable enough and can balance greatly the
exploration and exploitation abilities until the maximum
FEs is reached. Therefore, the proposed GSK algorithm is
proven to be an effective and powerful approach for solving
unconstrained global optimization problems within limited
number of function evaluations which is a very important
issue when dealing with real-world problems. Finally, it can
be obviously deduced from Fig. S1 that the GSK converges
faster than other compared algorithms in most cases espe-
cially with high dimensions.
3.3.3 Algorithm complexity
The algorithm complexity of all algorithms on 10, 30, 50
dimensions are shown in Tables21, 22, 23, respectively. All
experiments were implemented and executed using MAT-
LAB R2014a running on a PC with core i7-4790 (3.60GHz)
CPU and 12GB RAM running win 10 OS. In order to eval-
uate the computational complexity of the compared algo-
rithms we follow the guidelines described in [40].
T0 is the time in seconds needed to run the following
program:
Table 13 Experimental results of TLBO, SFS, DE, GA, ES and GSK over 51 independent runs on 29 test functions of 30 variables with 300,000
FES
Function TLBO SFS DE GA ES GSK
13.36E+03 ± 3.19E+03 3.17E+03 ± 3.76E+03 0.00E+00 ± 0.00E+00 6.06E+06 ± 1.62E+06 8.32E+10 ± 1.11E+10 0.00E+00 ± 0.00E+00
31.03E−04 ± 2.38E−04 5.36E+01 ± 2.72E+01 1.36E+02 ± 1.36E+02 6.81E+04 ± 2.02E+04 1.95E+05 ± 3.27E+04 7.07E−07 ± 1.82E−06
45.63E+01 ± 3.51E+01 5.91E+01 ± 4.04E+01 5.92E+01 ± 1.81E+00 1.51E+02 ± 4.09E+01 2.45E+04 ± 6.13E+03 1.11E+01 ± 2.17E+01
59.06E+01 ± 2.00E+01 6.76E+01 ± 1.34E+01 1.79E+02 ± 1.27E+01 2.26E+02 ± 2.93E+01 5.41E+02 ± 3.34E+01 1.60E+02 ± 8.87E+00
68.45E+00 ± 4.18E+00 1.18E−02 ± 2.04E−02 0.00E+00 ± 0.00E+00 3.86E+01 ± 9.95E+00 1.11E+02 ± 6.94E+00 1.52E−06 ± 2.36E−06
71.48E+02 ± 3.18E+01 1.01E+02 ± 2.19E+01 2.12E+02 ± 9.99E+00 3.24E+02 ± 5.43E+01 2.09E+03 ± 1.89E+02 1.87E+02 ± 8.40E+00
87.06E+01 ± 1.44E+01 7.50E+01 ± 1.96E+01 1.80E+02 ± 1.14E+01 2.43E+02 ± 3.07E+01 4.88E+02 ± 2.69E+01 1.55E+02 ± 1.11E+01
92.07E+02 ± 1.29E+02 2.47E+01 ± 6.83E+01 0.00E+00 ± 0.00E+00 1.12E+03 ± 1.42E+03 2.24E+04 ± 1.93E+03 0.00E+00 ± 0.00E+00
10 6.01E+03 ± 1.17E+03 2.35E+03 ± 5.20E+02 6.61E+03 ± 4.36E+02 4.48E+03 ± 8.44E+02 7.41E+03 ± 2.37E+02 6.69E+03 ± 3.54E+02
11 1.36E+02 ± 4.75E+01 4.57E+01 ± 2.85E+01 7.49E+01 ± 3.10E+01 1.33E+03 ± 1.01E+03 1.65E+04 ± 4.77E+03 3.30E+01 ± 3.83E+01
12 4.44E+04 ± 6.48E+04 2.75E+04 ± 1.38E+04 7.55E+03 ± 7.17E+03 4.29E+06 ± 3.34E+06 1.32E+10 ± 2.92E+09 6.62E+03 ± 4.59E+03
13 1.40E+04 ± 1.41E+04 9.39E+02 ± 2.79E+02 8.24E+01 ± 9.84E+00 2.54E+06 ± 2.22E+06 8.70E+09 ± 3.21E+09 9.83E+01 ± 3.42E+01
14 2.99E+03 ± 2.58E+03 8.72E+01 ± 9.94E+00 6.33E+01 ± 4.72E+00 9.85E+05 ± 9.57E+05 5.46E+06 ± 3.38E+06 5.69E+01 ± 5.49E+00
15 5.17E+03 ± 6.89E+03 1.52E+02 ± 2.55E+01 3.91E+01 ± 5.79E+00 7.21E+05 ± 2.53E+05 1.56E+09 ± 7.25E+08 1.45E+01 ± 1.49E+01
16 5.67E+02 ± 2.18E+02 4.41E+02 ± 1.58E+02 7.78E+02 ± 4.11E+02 1.17E+03 ± 3.00E+02 4.10E+03 ± 4.79E+02 7.96E+02 ± 1.94E+02
17 2.00E+02 ± 9.15E+01 1.05E+02 ± 6.29E+01 1.02E+02 ± 5.03E+01 6.43E+02 ± 2.04E+02 2.33E+03 ± 4.24E+02 1.89E+02 ± 9.44E+01
18 2.05E+05 ± 1.40E+05 3.32E+02 ± 7.40E+01 3.83E+01 ± 4.10E+00 3.77E+06 ± 4.84E+06 8.61E+07 ± 3.76E+07 3.68E+01 ± 5.42E+00
19 5.57E+03 ± 6.13E+03 5.83E+01 ± 9.10E+00 1.89E+01 ± 5.75E+00 8.32E+05 ± 3.31E+05 2.09E+09 ± 7.48E+08 1.29E+01 ± 6.02E+00
20 2.20E+02 ± 7.33E+01 1.30E+02 ± 6.05E+01 6.03E+01 ± 6.50E+01 4.10E+02 ± 1.11E+02 1.24E+03 ± 1.42E+02 1.08E+02 ± 1.14E+02
21 2.70E+02 ± 1.83E+01 2.56E+02 ± 2.71E+01 3.66E+02 ± 1.34E+01 4.57E+02 ± 3.28E+01 6.91E+02 ± 3.09E+01 3.46E+02 ± 8.30E+00
22 2.08E+02 ± 7.55E+02 1.00E+02 ± 5.50E−04 1.81E+03 ± 2.95E+03 5.16E+03 ± 1.60E+03 7.96E+03 ± 2.24E+02 1.00E+02 ± 0.00E+00
23 4.42E+02 ± 2.62E+01 4.08E+02 ± 1.35E+01 5.26E+02 ± 9.51E+00 6.77E+02 ± 3.70E+01 1.10E+03 ± 5.19E+01 4.70E+02 ± 4.49E+01
24 4.97E+02 ± 2.37E+01 4.93E+02 ± 2.27E+01 5.96E+02 ± 7.08E+00 8.14E+02 ± 6.40E+01 1.03E+03 ± 4.56E+01 5.68E+02 ± 1.45E+01
25 4.08E+02 ± 2.28E+01 3.86E+02 ± 2.68E+00 3.87E+02 ± 2.19E−02 5.61E+02 ± 9.50E+01 9.40E+03 ± 1.72E+03 3.87E+02 ± 2.11E−01
26 2.10E+03 ± 1.08E+03 1.32E+03 ± 8.12E+02 2.58E+03 ± 2.61E+02 3.61E+03 ± 8.15E+02 9.14E+03 ± 6.85E+02 9.87E+02 ± 2.49E+02
27 5.34E+02 ± 2.02E+01 5.16E+02 ± 1.25E+01 4.96E+02 ± 7.01E+00 6.13E+02 ± 4.04E+01 8.91E+02 ± 5.91E+01 4.93E+02 ± 8.02E+00
28 3.85E+02 ± 5.63E+01 3.61E+02 ± 4.99E+01 3.20E+02 ± 4.41E+01 5.66E+02 ± 4.52E+01 5.67E+03 ± 5.95E+02 3.21E+02 ± 4.22E+01
29 8.37E+02 ± 1.80E+02 5.53E+02 ± 1.04E+02 5.42E+02 ± 9.62E+01 9.07E+02 ± 1.78E+02 2.94E+03 ± 1.10E+02 5.77E+02 ± 1.01E+02
30 5.24E+03 ± 2.76E+03 8.22E+03 ± 3.40E+03 2.00E+03 ± 5.62E+01 3.45E+05 ± 1.43E+05 1.19E+09 ± 4.18E+08 2.08E+03 ± 9.27E+01

International Journal of Machine Learning and Cybernetics
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for i=1:1000000
x=x + x;
x=x/2;
x=x*x;
x=sqrt(x);
x=log(x);
x=exp(x);
x=x/(x+2);
end
T1 is the time in seconds to execute 200,000 evaluations
of benchmark function f18 by itself with D dimensions, and
T2 is the mean time of executing the compared algorithms
5 times for 200,000 valuations for f18 in D dimensions.
Finally, the algorithm complexity is shown in T2, T1, and
(T2
− T1)/T0.
Actually, it can be easily noticed from these tables that
the complexity of GSK algorithm should be considered with
some care, since the measured values of run-time are very
small relative to the remaining algorithms. Thus, regard-
ing implementation and design of algorithms, it can be
derived that it is easily implemented more than compared
algorithms.
3.3.4 Results oftheproposed approach (GSK) onCEC2011
The statistical results of the GSK on the CEC2011 bench-
marks are summarized in Table24, respectively. It includes
the obtained best, median, mean, worst values and the stand-
ard deviations of objective function value of the proposed
GSK over 25 runs for all 22 benchmark functions. Generally,
from Table24 it can be clearly seen that GSK is able to find
the global optimal solution consistently in 3 test functions
over 25 runs. With respect to first test function, although
the optimal solutions are not consistently found, the best
result achieved is very close to the global optimal solution
which can be verified by the very small standard deviation.
Additionally, regarding the remaining test functions, the
Table 14 Experimental results of GWO, AMO, PSO, BBO, ACO and GSK over 51 independent runs on 29 test functions of 30 variables with
300,000 FES
Function GWO AMO PSO BBO ACO GSK
13.68E+09 ± 7.80E+08 5.34E+00 ± 7.57E+00 3.43E+03 ± 3.97E+03 4.20E+06 ± 1.87E+05 8.95E+10 ± 2.59E+10 0.00E+00 ± 0.00E+00
32.82E+04 ± 6.90E+03 4.56E+03 ± 1.48E+03 1.48E+02 ± 5.90E+01 4.39E+04 ± 2.58E+04 2.08E+07 ± 1.23E+08 7.07E−07 ± 1.82E−06
42.58E+02 ± 3.54E+01 4.60E+00 ± 1.07E+01 8.59E+01 ± 3.22E+01 9.99E+01 ± 2.27E+01 9.61E+03 ± 1.72E+03 1.11E+01 ± 2.17E+01
52.02E+02 ± 1.63E+01 5.41E+01 ± 6.68E+00 1.14E+02 ± 2.67E+01 4.21E+01 ± 1.06E+01 4.11E+02 ± 2.27E+01 1.60E+02 ± 8.87E+00
62.60E+01 ± 2.54E+00 0.00E+00 ± 0.00E+00 3.09E+00 ± 4.01E+00 8.98E−01 ± 4.07E−02 8.31E+01 ± 6.11E+00 1.52E−06 ± 2.36E−06
72.81E+02 ± 1.90E+01 9.15E+01 ± 6.90E+00 9.71E+01 ± 1.66E+01 1.14E+02 ± 1.46E+01 9.44E+02 ± 9.83E+01 1.87E+02 ± 8.40E+00
81.87E+02 ± 1.53E+01 5.42E+01 ± 5.76E+00 9.73E+01 ± 2.11E+01 4.31E+01 ± 1.00E+01 3.63E+02 ± 1.97E+01 1.55E+02 ± 1.11E+01
91.42E+03 ± 3.61E+02 1.05E−01 ± 2.07E−01 1.01E+03 ± 1.15E+03 1.09E+02 ± 8.42E+01 1.39E+04 ± 1.46E+03 0.00E+00 ± 0.00E+00
10 6.21E+03 ± 4.87E+02 3.62E+03 ± 2.66E+02 3.20E+03 ± 5.52E+02 2.29E+03 ± 4.48E+02 7.38E+03 ± 2.24E+02 6.69E+03 ± 3.54E+02
11 5.22E+02 ± 3.70E+02 4.91E+01 ± 2.47E+01 1.14E+02 ± 3.55E+01 1.43E+03 ± 1.40E+03 5.54E+03 ± 1.53E+03 3.30E+01 ± 3.83E+01
12 2.96E+08 ± 8.71E+07 6.07E+04 ± 6.88E+04 1.43E+05 ± 9.98E+04 3.52E+06 ± 2.19E+06 1.89E+10 ± 4.59E+09 6.62E+03 ± 4.59E+03
13 1.10E+08 ± 5.32E+07 6.78E+03 ± 3.23E+03 1.49E+04 ± 1.63E+04 1.51E+06 ± 4.77E+05 1.23E+10 ± 7.17E+09 9.83E+01 ± 3.42E+01
14 1.31E+05 ± 2.24E+05 2.22E+03 ± 1.39E+03 1.05E+04 ± 8.32E+03 8.23E+05 ± 8.54E+05 9.44E+06 ± 2.25E+07 5.69E+01 ± 5.49E+00
15 2.94E+06 ± 5.30E+06 4.22E+02 ± 5.61E+02 7.96E+03 ± 8.62E+03 7.01E+05 ± 3.26E+05 3.56E+09 ± 2.14E+09 1.45E+01 ± 1.49E+01
16 1.25E+03 ± 2.51E+02 5.52E+02 ± 1.19E+02 8.59E+02 ± 2.38E+02 8.95E+02 ± 3.11E+02 2.90E+03 ± 2.10E+02 7.96E+02 ± 1.94E+02
17 3.96E+02 ± 1.36E+02 8.49E+01 ± 1.77E+01 3.57E+02 ± 1.64E+02 3.81E+02 ± 2.16E+02 1.28E+03 ± 1.66E+02 1.89E+02 ± 9.44E+01
18 1.07E+06 ± 7.22E+05 1.43E+05 ± 5.21E+04 1.84E+05 ± 1.36E+05 1.95E+06 ± 1.96E+06 1.78E+08 ± 1.13E+08 3.68E+01 ± 5.42E+00
19 5.71E+06 ± 2.88E+06 1.32E+03 ± 1.53E+03 7.77E+03 ± 1.12E+04 8.37E+05 ± 3.40E+05 3.91E+09 ± 2.60E+09 1.29E+01 ± 6.02E+00
20 4.57E+02 ± 1.33E+02 1.42E+02 ± 4.76E+01 3.78E+02 ± 1.36E+02 4.33E+02 ± 1.85E+02 8.37E+02 ± 1.02E+02 1.08E+02 ± 1.14E+02
21 3.83E+02 ± 1.68E+01 2.53E+02 ± 6.95E+00 3.06E+02 ± 2.35E+01 2.49E+02 ± 1.04E+01 5.90E+02 ± 1.72E+01 3.46E+02 ± 8.30E+00
22 3.91E+03 ± 2.80E+03 1.00E+02 ± 0.00E+00 9.62E+02 ± 1.61E+03 1.58E+03 ± 1.49E+03 5.74E+03 ± 4.29E+02 1.00E+02 ± 0.00E+00
23 5.67E+02 ± 1.89E+01 3.98E+02 ± 9.02E+00 5.08E+02 ± 5.60E+01 4.02E+02 ± 1.24E+01 9.40E+02 ± 4.40E+01 4.70E+02 ± 4.49E+01
24 6.33E+02 ± 1.27E+01 4.64E+02 ± 8.44E+00 5.73E+02 ± 6.32E+01 4.67E+02 ± 1.33E+01 1.06E+03 ± 5.58E+01 5.68E+02 ± 1.45E+01
25 5.02E+02 ± 3.08E+01 3.87E+02 ± 1.07E+00 3.90E+02 ± 8.76E+00 3.93E+02 ± 9.81E+00 3.49E+03 ± 6.51E+02 3.87E+02 ± 2.11E−01
26 3.10E+03 ± 1.47E+02 1.49E+03 ± 1.95E+02 1.27E+03 ± 1.39E+03 1.63E+03 ± 1.56E+02 7.10E+03 ± 4.27E+02 9.87E+02 ± 2.49E+02
27 5.63E+02 ± 2.37E+01 5.16E+02 ± 4.81E+00 5.43E+02 ± 3.02E+01 5.26E+02 ± 7.66E+00 1.14E+03 ± 7.37E+01 4.93E+02 ± 8.02E+00
28 6.63E+02 ± 5.98E+01 3.14E+02 ± 3.61E+01 4.14E+02 ± 2.58E+01 4.33E+02 ± 2.47E+01 3.51E+03 ± 4.77E+02 3.21E+02 ± 4.22E+01
29 1.08E+03 ± 1.62E+02 5.33E+02 ± 2.46E+01 7.56E+02 ± 1.79E+02 7.18E+02 ± 1.45E+02 2.59E+03 ± 2.20E+02 5.77E+02 ± 1.01E+02
30 2.50E+07 ± 8.05E+06 4.71E+03 ± 7.99E+02 5.95E+03 ± 2.82E+03 3.31E+05 ± 1.23E+05 3.16E+09 ± 9.12E+08 2.08E+03 ± 9.27E+01

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differences between mean and median are small even in the
cases when the final results are far away from the optimum,
regardless of the dimensions. That implies the GSK is a
robust algorithm.
Finally, due to the exact optimum of most of these prob-
lems are not available, it is difficult to gauge the absolute
performance of the algorithm at this stage. However, relative
performance will be evaluated in next subsection when the
performance of GSK will be compared with the performance
of other algorithms.
3.3.5 Comparison againststate‑of‑the‑art algorithms
The statistical results of the comparisons on the bench-
marks are summarized in Tables25 and 26, respectively.
It includes the obtained best and the standard deviations of
the objective function value of GSK and other ten state-of-
the-art algorithms over 25 runs for all 22 benchmark func-
tions. The best results are marked in bold and the second
best results are underlined for all problems. Ranking of the
algorithms using Friedman test [142] is given in Table27,
where the algorithm with smallest ranking is the better. The
null hypothesis for this test is that “there is no difference
among the performance of all algorithms” and that alterna-
tive hypothesis states “there is a difference among the per-
formance of all algorithms”. The multi-problem Wilcoxon
signed-rank GSK and others are summarized in Table28.
Firstly, regarding evolutionary and physical based algo-
rithms in four dimensions, it can be observed that SFS and
TLBO algorithms can be good at different functions and DE
shows moderate performance compared with others. How-
ever, GA and ES perform poorly on most of the functions.
Generally, GSK, SFS and TLBO do significantly better than
the others on most functions.
On the other hand, regarding swarm intelligence-based
algorithms, it can be obviously shown that AMO and BBO
are competitive with GSK on some functions. However,
GSK is superior to the others in most functions. Actually, it
can be obviously seen that the performance of all compared
algorithms shows complete and/or significant deteriora-
tion on the problems with the growth of the search-space
dimensionality. Besides, GWO shows moderate performance
Table 15 Experimental results of TLBO, SFS, DE, GA, ES and GSK over 51 independent runs on 29 test functions of 50 variables with 500,000
FES
Function TLBO SFS DE GA ES GSK
12.56E+03 ± 3.07E+03 3.65E+03 ± 5.38E+03 7.43E−01 ± 1.81E+00 1.93E+07 ± 4.99E+06 2.01E+11 ± 1.43E+10 1.09E+03 ± 1.24E+03
31.40E+03 ± 1.03E+03 4.44E+03 ± 1.28E+03 9.29E+04 ± 1.68E+04 8.00E+04 ± 1.78E+04 3.90E+05 ± 5.44E+04 3.85E+03 ± 1.51E+03
41.03E+02 ± 4.48E+01 1.10E+02 ± 5.13E+01 8.03E+01 ± 4.94E+01 3.03E+02 ± 8.39E+01 6.57E+04 ± 1.15E+04 8.33E+01 ± 5.00E+01
51.87E+02 ± 3.14E+01 2.13E+02 ± 4.68E+01 3.50E+02 ± 1.40E+01 4.47E+02 ± 5.07E+01 1.01E+03 ± 4.69E+01 3.20E+02 ± 1.79E+01
62.17E+01 ± 5.12E+00 1.01E−01 ± 1.99E−01 2.60E−07 ± 1.09E−06 4.11E+01 ± 4.95E+00 1.28E+02 ± 6.97E+00 3.78E−06 ± 3.52E−06
73.84E+02 ± 6.37E+01 2.56E+02 ± 5.48E+01 4.07E+02 ± 1.09E+01 5.87E+02 ± 6.93E+01 4.46E+03 ± 2.60E+02 3.70E+02 ± 1.41E+01
82.02E+02 ± 2.96E+01 2.03E+02 ± 4.35E+01 3.53E+02 ± 1.42E+01 4.53E+02 ± 5.51E+01 1.02E+03 ± 4.95E+01 3.24E+02 ± 1.36E+01
92.68E+03 ± 1.47E+03 2.90E+02 ± 4.52E+02 3.99E−02 ± 1.26E−01 3.96E+03 ± 2.17E+03 6.83E+04 ± 6.85E+03 1.07E−02 ± 2.79E−02
10 1.02E+04 ± 2.67E+03 4.68E+03 ± 6.67E+02 1.30E+04 ± 7.47E+02 8.76E+03 ± 7.41E+02 1.38E+04 ± 3.75E+02 1.30E+04 ± 4.50E+02
11 2.16E+02 ± 6.69E+01 1.28E+02 ± 3.60E+01 1.43E+02 ± 2.27E+01 6.84E+03 ± 4.28E+03 4.70E+04 ± 9.97E+03 3.45E+01 ± 2.32E+01
12 5.90E+05 ± 1.31E+06 3.19E+05 ± 2.42E+05 6.19E+04 ± 3.75E+04 1.65E+07 ± 8.55E+06 8.68E+10 ± 1.41E+10 9.46E+03 ± 7.01E+03
13 5.23E+03 ± 3.91E+03 3.50E+03 ± 3.20E+03 5.33E+02 ± 1.39E+03 2.17E+06 ± 5.56E+05 4.08E+10 ± 7.67E+09 1.49E+03 ± 2.16E+03
14 4.92E+04 ± 4.29E+04 1.68E+02 ± 1.53E+01 1.25E+02 ± 9.35E+00 2.22E+06 ± 1.10E+06 4.58E+07 ± 1.66E+07 1.24E+02 ± 1.87E+01
15 6.65E+03 ± 5.53E+03 3.00E+02 ± 4.76E+01 1.08E+02 ± 1.00E+01 1.59E+06 ± 3.13E+05 1.42E+10 ± 3.99E+09 4.20E+01 ± 1.68E+01
16 1.17E+03 ± 3.09E+02 1.18E+03 ± 3.37E+02 2.51E+03 ± 6.16E+02 2.67E+03 ± 5.68E+02 8.06E+03 ± 7.76E+02 1.83E+03 ± 6.59E+02
17 9.96E+02 ± 2.23E+02 7.49E+02 ± 1.96E+02 1.19E+03 ± 4.77E+02 1.26E+03 ± 3.48E+02 5.07E+04 ± 4.10E+04 1.35E+03 ± 1.90E+02
18 5.45E+05 ± 3.28E+05 4.53E+02 ± 1.55E+02 7.67E+02 ± 1.31E+03 2.72E+06 ± 2.05E+06 2.27E+08 ± 8.65E+07 5.98E+02 ± 3.37E+02
19 1.32E+04 ± 7.59E+03 1.22E+02 ± 2.92E+01 6.24E+01 ± 6.16E+00 5.69E+05 ± 2.18E+05 5.22E+09 ± 1.46E+09 3.05E+01 ± 9.59E+00
20 5.81E+02 ± 2.76E+02 5.01E+02 ± 2.20E+02 9.98E+02 ± 5.60E+02 1.52E+03 ± 2.68E+02 2.76E+03 ± 1.81E+02 1.37E+03 ± 1.28E+02
21 3.91E+02 ± 3.70E+01 3.47E+02 ± 3.36E+01 5.52E+02 ± 1.28E+01 7.20E+02 ± 6.23E+01 1.24E+03 ± 6.41E+01 5.21E+02 ± 1.31E+01
22 7.62E+03 ± 5.62E+03 2.44E+03 ± 2.74E+03 1.32E+04 ± 3.25E+02 1.01E+04 ± 7.97E+02 1.44E+04 ± 3.51E+02 1.10E+04 ± 4.85E+03
23 7.01E+02 ± 6.69E+01 5.98E+02 ± 4.60E+01 7.69E+02 ± 2.12E+01 9.94E+02 ± 4.44E+01 1.99E+03 ± 7.95E+01 5.42E+02 ± 1.39E+02
24 7.25E+02 ± 5.00E+01 6.91E+02 ± 3.95E+01 8.48E+02 ± 1.38E+01 1.27E+03 ± 9.86E+01 1.76E+03 ± 5.64E+01 6.34E+02 ± 1.39E+02
25 5.71E+02 ± 3.42E+01 5.60E+02 ± 2.79E+01 4.96E+02 ± 3.14E+01 6.90E+02 ± 6.08E+01 3.90E+04 ± 5.17E+03 5.56E+02 ± 4.62E+01
26 5.28E+03 ± 2.03E+03 1.60E+03 ± 2.23E+03 4.40E+03 ± 1.82E+02 5.46E+03 ± 5.13E+02 1.86E+04 ± 1.02E+03 1.27E+03 ± 9.18E+01
27 8.98E+02 ± 1.36E+02 6.29E+02 ± 5.61E+01 5.45E+02 ± 4.05E+01 1.17E+03 ± 1.25E+02 1.86E+03 ± 1.42E+02 5.92E+02 ± 8.29E+01
28 5.04E+02 ± 2.40E+01 5.08E+02 ± 3.05E+01 4.67E+02 ± 1.85E+01 1.40E+03 ± 4.65E+02 1.23E+04 ± 8.03E+02 4.94E+02 ± 2.24E+01
29 1.54E+03 ± 3.86E+02 8.61E+02 ± 2.33E+02 1.18E+03 ± 5.28E+02 1.53E+03 ± 3.29E+02 1.69E+04 ± 1.99E+03 3.60E+02 ± 2.23E+01
30 9.35E+05 ± 1.49E+05 9.88E+05 ± 1.73E+05 5.91E+05 ± 2.40E+04 3.09E+06 ± 3.38E+06 8.50E+09 ± 1.80E+09 5.96E+05 ± 2.24E+04

International Journal of Machine Learning and Cybernetics
1 3
compared with others. Generally, GSK, AMO and BBO do
significantly better than the others on most functions.
Secondly, the performance of GSK and other competi-
tive algorithms is discussed. Table27 clearly shows that
AMO gets the first ranking among all algorithms, followed
by SFS and GSK in second and third place, respectively.
However, PSO, ES and ACO are the poorest algorithms,
respectively. The ranking of all algorithms on the CEC
2017 functions is shown in Fig.13. Table28 summarizes
the statistical analysis results of applying multiple-prob-
lem Wilcoxon’s test between GSK and other compared
algorithms on CEC2011 problems.
From Table28, we can see that GSK obtains higher R+
values than R− in all the cases with exception to AMO
which means that GSK is better than 9 algorithms out
of 10 algorithms on 22 test functions. Precisely, we can
draw the following conclusions: GSK outperforms GA,
DE, GWO, PSO, ES and ACO significantly while GSK is
insignificantly better than SFS, TLBO and BBO.
From the above results, comparisons and discussion
through this section, the proposed GSK algorithm is of
better searching quality, efficiency and robustness for
solving small, moderate and high dimensions real-world
unconstrained global optimization problems.
Furthermore, in order to analyze the convergence behav-
ior of GSK and other state-of-the-art algorithms, the conver-
gence characteristics in terms of the best fitness value of the
median run of all algorithms for all functions is illustrated
in the supplemental file (Fig. S2). It is clear that the conver-
gence speed of the GSK algorithm is fast at the early stage
of the optimization process for all functions with different
shapes, complexity, and dimensions. Furthermore, the con-
vergence speed is dramatically decreased, and its improve-
ment is found to be significant in the middle and later stages
of the optimization process.
Additionally, the convergent figure suggests that the GSK
algorithm can reach the global solution or better solution in
most problems in a fewer number of generations less than
the maximum predetermined number of generations. In
Table 16 Experimental results of GWO, AMO, PSO, BBO, ACO and GSK over 51 independent runs on 29 test functions of 50 variables with
500,000 FES
Function GWO AMO PSO BBO ACO GSK
11.27E+10 ± 2.94E+09 1.30E+03 ± 1.27E+03 3.87E+03 ± 6.33E+03 6.10E+06 ± 7.65E+05 1.94E+11 ± 5.32E+10 1.09E+03 ± 1.24E+03
37.54E+04 ± 1.25E+04 3.15E+04 ± 4.71E+03 1.90E+03 ± 3.35E+02 1.23E+05 ± 4.22E+04 1.31E+06 ± 4.92E+06 3.85E+03 ± 1.51E+03
49.59E+02 ± 2.18E+02 7.53E+01 ± 4.77E+01 1.57E+02 ± 4.67E+01 1.43E+02 ± 5.04E+01 3.28E+04 ± 3.36E+03 8.33E+01 ± 5.00E+01
54.22E+02 ± 2.51E+01 1.34E+02 ± 1.30E+01 2.31E+02 ± 4.15E+01 8.37E+01 ± 1.39E+01 8.46E+02 ± 2.80E+01 3.20E+02 ± 1.79E+01
63.84E+01 ± 4.11E+00 2.10E−04 ± 1.22E−03 1.55E+01 ± 1.18E+01 8.96E−01 ± 3.65E−02 1.05E+02 ± 4.46E+00 3.78E−06 ± 3.52E−06
75.98E+02 ± 3.47E+01 2.00E+02 ± 1.64E+01 1.99E+02 ± 2.83E+01 2.30E+02 ± 2.38E+01 2.05E+03 ± 1.71E+02 3.70E+02 ± 1.41E+01
84.29E+02 ± 1.87E+01 1.34E+02 ± 1.35E+01 2.33E+02 ± 3.73E+01 8.71E+01 ± 1.82E+01 8.06E+02 ± 2.61E+01 3.24E+02 ± 1.36E+01
97.94E+03 ± 2.50E+03 3.94E+00 ± 4.75E+00 6.53E+03 ± 2.49E+03 5.69E+02 ± 3.49E+02 4.52E+04 ± 5.03E+03 1.07E−02 ± 2.79E−02
10 1.19E+04 ± 7.11E+02 6.96E+03 ± 4.28E+02 5.56E+03 ± 8.76E+02 4.11E+03 ± 7.06E+02 1.36E+04 ± 3.20E+02 1.30E+04 ± 4.50E+02
11 2.49E+03 ± 9.73E+02 9.62E+01 ± 1.51E+01 1.63E+02 ± 3.71E+01 3.99E+03 ± 3.26E+03 2.00E+04 ± 3.07E+03 3.45E+01 ± 2.32E+01
12 3.10E+09 ± 1.12E+09 4.74E+05 ± 2.23E+05 1.87E+06 ± 1.32E+06 1.10E+07 ± 4.55E+06 1.11E+11 ± 1.89E+10 9.46E+03 ± 7.01E+03
13 8.03E+08 ± 2.05E+08 1.45E+03 ± 1.00E+03 3.93E+03 ± 4.85E+03 2.12E+06 ± 7.10E+05 6.60E+10 ± 1.62E+10 1.49E+03 ± 2.16E+03
14 8.55E+05 ± 4.68E+05 3.45E+04 ± 1.68E+04 5.32E+04 ± 3.68E+04 3.70E+06 ± 2.67E+06 8.63E+07 ± 3.34E+07 1.24E+02 ± 1.87E+01
15 7.83E+07 ± 3.16E+07 2.24E+03 ± 1.63E+03 5.16E+03 ± 4.69E+03 1.49E+06 ± 4.43E+05 2.04E+10 ± 7.77E+09 4.20E+01 ± 1.68E+01
16 2.36E+03 ± 4.13E+02 1.05E+03 ± 1.69E+02 1.50E+03 ± 3.84E+02 1.70E+03 ± 3.68E+02 5.80E+03 ± 3.21E+02 1.83E+03 ± 6.59E+02
17 1.77E+03 ± 2.72E+02 7.37E+02 ± 1.11E+02 1.14E+03 ± 2.81E+02 1.20E+03 ± 3.13E+02 5.14E+03 ± 6.89E+02 1.35E+03 ± 1.90E+02
18 4.27E+06 ± 4.18E+06 6.43E+05 ± 2.78E+05 1.02E+06 ± 5.61E+05 8.01E+06 ± 5.70E+06 5.13E+08 ± 5.72E+08 5.98E+02 ± 3.37E+02
19 5.60E+07 ± 2.37E+07 1.01E+04 ± 3.71E+03 1.36E+04 ± 7.22E+03 6.51E+05 ± 1.96E+05 9.17E+09 ± 2.48E+09 3.05E+01 ± 9.59E+00
20 1.23E+03 ± 2.25E+02 5.02E+02 ± 9.84E+01 9.06E+02 ± 2.68E+02 1.16E+03 ± 3.44E+02 2.05E+03 ± 1.01E+02 1.37E+03 ± 1.28E+02
21 6.12E+02 ± 2.69E+01 3.27E+02 ± 1.58E+01 4.33E+02 ± 4.96E+01 2.96E+02 ± 1.58E+01 1.04E+03 ± 3.00E+01 5.21E+02 ± 1.31E+01
22 1.21E+04 ± 1.91E+03 5.65E+03 ± 3.18E+03 6.02E+03 ± 2.58E+03 4.92E+03 ± 8.04E+02 1.38E+04 ± 3.66E+02 1.10E+04 ± 4.85E+03
23 8.99E+02 ± 3.04E+01 5.60E+02 ± 1.49E+01 8.06E+02 ± 1.03E+02 5.48E+02 ± 2.41E+01 1.74E+03 ± 8.83E+01 5.42E+02 ± 1.39E+02
24 9.54E+02 ± 3.00E+01 6.13E+02 ± 1.50E+01 8.83E+02 ± 8.98E+01 5.88E+02 ± 1.77E+01 1.96E+03 ± 1.08E+02 6.34E+02 ± 1.39E+02
25 1.31E+03 ± 2.68E+02 5.62E+02 ± 2.87E+01 5.57E+02 ± 2.37E+01 5.75E+02 ± 2.68E+01 1.65E+04 ± 1.61E+03 5.56E+02 ± 4.62E+01
26 5.87E+03 ± 2.68E+02 2.54E+03 ± 2.28E+02 1.94E+03 ± 2.23E+03 2.30E+03 ± 1.77E+02 1.57E+04 ± 8.19E+02 1.27E+03 ± 9.18E+01
27 9.15E+02 ± 8.20E+01 6.00E+02 ± 2.82E+01 7.51E+02 ± 1.09E+02 7.48E+02 ± 7.01E+01 2.80E+03 ± 1.80E+02 5.92E+02 ± 8.29E+01
28 1.41E+03 ± 2.99E+02 5.02E+02 ± 2.01E+01 5.12E+02 ± 3.21E+01 5.26E+02 ± 1.78E+01 1.03E+04 ± 6.44E+02 4.94E+02 ± 2.24E+01
29 2.35E+03 ± 3.11E+02 6.58E+02 ± 1.05E+02 1.27E+03 ± 2.56E+02 1.12E+03 ± 2.31E+02 8.78E+03 ± 1.30E+03 3.60E+02 ± 2.23E+01
30 2.12E+08 ± 5.37E+07 8.16E+05 ± 5.64E+04 9.06E+05 ± 1.24E+05 1.83E+06 ± 3.95E+05 1.31E+10 ± 3.91E+09 5.96E+05 ± 2.24E+04

International Journal of Machine Learning and Cybernetics
1 3
general, GSK is scalable enough and can balance greatly
the exploration and exploitation abilities until the maximum
FEs is reached. Therefore, the proposed GSK algorithm is
proven to be an effective and powerful approach for solving
unconstrained global optimization problems within limited
number of function evaluations which is a very important
issue when dealing with real-world problems.
4 Conclusion
The gaining and sharing knowledge (GSK) algorithm is pro-
posed as a novel metaheuristic for solving optimization prob-
lems. The GSK is a population based stochastic algorithm
that mimics the process of gaining and sharing knowledge
during the human life span. It is based on two vital stages,
junior gaining-sharing phase and senior gaining-sharing
phase. In the proposed GSK algorithm, each candidate solu-
tion is presented as a person in the entire population of peo-
ple. This person has different level of knowledge in various
disciplines and fields by utilization of both junior and senior
phases, where each discipline represents a specific dimen-
sion of the optimized problem.
During Junior phase, the person gains and shares knowl-
edge from/with small private and social networks while he
cooperates and competes with different types of larger net-
works of people with various talents, experience, character-
istics in senior phase which are the source of gaining and
sharing knowledge.
The mathematical expressions of this process is formu-
lated. In order to test the effectiveness of GSK, it is applied
to solve the CEC-2017 real-parameter benchmark optimiza-
tion problems. Experimental results are compared with state-
of-the-art algorithms which were 4 evolutionary algorithms
like SFS, DE, GA and ES and 1 human related algorithm
Table 17 Experimental results of TLBO, SFS, DE, GA, ES and GSK over 51 independent runs on 29 test functions of 100 variables with
1,000,000 FES
Function TLBO SFS DE GA ES GSK
17.13E+03 ± 8.54E+03 8.57E+03 ± 1.21E+04 1.10E+04 ± 1.62E+04 2.03E+08 ± 1.62E+08 5.41E+11 ± 2.51E+10 5.80E+03 ± 4.63E+03
36.30E+04 ± 1.28E+04 3.00E+04 ± 6.89E+03 4.23E+05 ± 2.81E+04 2.07E+05 ± 3.03E+04 9.06E+05 ± 1.15E+05 1.15E+05 ± 2.15E+04
42.70E+02 ± 5.22E+01 2.72E+02 ± 3.84E+01 2.16E+02 ± 2.36E+01 4.76E+02 ± 6.04E+01 2.10E+05 ± 2.61E+04 2.05E+02 ± 4.57E+01
55.90E+02 ± 5.32E+01 6.36E+02 ± 9.34E+01 8.19E+02 ± 1.77E+01 1.14E+03 ± 7.63E+01 2.37E+03 ± 6.77E+01 5.31E+02 ± 3.39E+02
64.20E+01 ± 3.90E+00 1.76E+00 ± 1.94E+00 2.57E−03 ± 4.31E−03 4.24E+01 ± 4.85E+00 1.47E+02 ± 5.47E+00 1.72E−03 ± 6.46E−03
71.35E+03 ± 2.09E+02 9.78E+02 ± 1.76E+02 9.39E+02 ± 1.95E+01 1.60E+03 ± 1.19E+02 1.16E+04 ± 5.35E+02 8.75E+02 ± 1.83E+01
86.31E+02 ± 5.78E+01 6.27E+02 ± 9.44E+01 8.18E+02 ± 2.02E+01 1.08E+03 ± 9.95E+01 2.48E+03 ± 8.28E+01 4.92E+02 ± 3.41E+02
92.82E+04 ± 9.68E+03 1.18E+04 ± 3.78E+03 3.86E+00 ± 4.56E+00 3.11E+04 ± 7.46E+03 1.78E+05 ± 8.70E+03 8.46E+00 ± 3.41E+00
10 2.30E+04 ± 6.19E+03 1.21E+04 ± 1.33E+03 3.01E+04 ± 3.96E+02 2.28E+04 ± 1.51E+03 3.14E+04 ± 4.56E+02 2.95E+04 ± 4.44E+02
11 1.16E+03 ± 2.76E+02 6.43E+02 ± 8.04E+01 6.05E+02 ± 8.70E+01 6.12E+04 ± 2.52E+04 3.92E+05 ± 5.99E+04 2.78E+02 ± 6.85E+01
12 1.46E+06 ± 9.86E+05 2.11E+06 ± 9.27E+05 2.72E+05 ± 1.28E+05 8.67E+07 ± 3.86E+07 2.74E+11 ± 2.49E+10 8.34E+04 ± 7.52E+04
13 8.96E+03 ± 4.36E+03 6.42E+03 ± 6.05E+03 5.32E+03 ± 4.39E+03 2.75E+06 ± 2.82E+06 6.11E+10 ± 9.99E+09 3.20E+03 ± 2.64E+03
14 1.70E+05 ± 1.96E+05 3.37E+02 ± 4.29E+01 7.71E+03 ± 8.97E+03 9.93E+06 ± 5.46E+06 1.13E+08 ± 5.11E+07 4.64E+03 ± 4.47E+03
15 2.41E+03 ± 2.13E+03 3.12E+03 ± 3.95E+03 8.91E+03 ± 7.19E+03 1.78E+06 ± 4.52E+05 2.95E+10 ± 5.17E+09 7.33E+02 ± 1.09E+03
16 3.36E+03 ± 6.41E+02 3.35E+03 ± 6.00E+02 7.65E+03 ± 3.37E+02 6.22E+03 ± 7.26E+02 2.46E+04 ± 2.85E+03 2.27E+03 ± 2.61E+03
17 3.17E+03 ± 5.95E+02 2.30E+03 ± 4.85E+02 4.71E+03 ± 4.85E+02 3.70E+03 ± 4.74E+02 1.05E+07 ± 4.84E+06 3.91E+03 ± 6.68E+02
18 4.27E+05 ± 2.51E+05 3.18E+04 ± 2.32E+04 1.21E+05 ± 6.33E+04 8.40E+06 ± 4.08E+06 3.32E+08 ± 9.55E+07 5.73E+04 ± 3.63E+04
19 2.36E+03 ± 2.03E+03 3.39E+03 ± 4.82E+03 8.95E+03 ± 1.05E+04 1.30E+06 ± 2.09E+05 3.10E+10 ± 4.10E+09 1.00E+03 ± 8.30E+02
20 2.34E+03 ± 8.22E+02 2.04E+03 ± 4.66E+02 4.15E+03 ± 8.41E+02 3.92E+03 ± 4.42E+02 6.51E+03 ± 2.21E+02 4.46E+03 ± 2.22E+02
21 8.80E+02 ± 8.83E+01 7.12E+02 ± 7.84E+01 1.04E+03 ± 2.54E+01 1.48E+03 ± 1.20E+02 2.90E+03 ± 1.12E+02 6.07E+02 ± 3.37E+02
22 2.49E+04 ± 8.31E+03 1.30E+04 ± 5.94E+03 3.02E+04 ± 7.15E+02 2.41E+04 ± 1.07E+03 3.19E+04 ± 4.29E+02 3.00E+04 ± 4.57E+02
23 1.35E+03 ± 1.05E+02 9.75E+02 ± 7.94E+01 8.60E+02 ± 2.99E+02 1.54E+03 ± 1.02E+02 3.54E+03 ± 8.56E+01 6.11E+02 ± 1.58E+01
24 1.99E+03 ± 1.96E+02 1.54E+03 ± 1.00E+02 1.63E+03 ± 1.49E+02 2.11E+03 ± 1.33E+02 5.46E+03 ± 1.17E+02 9.32E+02 ± 1.70E+01
25 8.28E+02 ± 6.11E+01 7.90E+02 ± 4.99E+01 7.31E+02 ± 5.56E+01 1.63E+03 ± 1.63E+02 1.04E+05 ± 1.09E+04 8.21E+02 ± 4.34E+01
26 2.01E+04 ± 5.23E+03 1.37E+04 ± 6.56E+03 1.08E+04 ± 1.37E+03 1.58E+04 ± 1.46E+03 5.10E+04 ± 1.66E+03 3.66E+03 ± 1.78E+02
27 1.42E+03 ± 2.02E+02 8.81E+02 ± 7.99E+01 6.16E+02 ± 2.15E+01 1.27E+03 ± 1.20E+02 6.72E+03 ± 6.63E+02 6.57E+02 ± 3.07E+01
28 6.30E+02 ± 2.91E+01 6.13E+02 ± 3.09E+01 5.59E+02 ± 3.52E+01 2.60E+03 ± 1.62E+03 4.88E+04 ± 2.86E+03 5.53E+02 ± 3.25E+01
29 4.34E+03 ± 6.73E+02 3.09E+03 ± 4.26E+02 4.74E+03 ± 1.19E+03 4.37E+03 ± 4.85E+02 2.49E+06 ± 1.36E+06 1.21E+03 ± 1.74E+02
30 2.15E+04 ± 2.11E+04 1.22E+04 ± 6.93E+03 4.68E+03 ± 3.05E+03 3.17E+06 ± 3.02E+05 4.85E+10 ± 9.66E+09 2.99E+03 ± 2.73E+02

International Journal of Machine Learning and Cybernetics
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TLBO and 5 swarm intelligence algorithms like GWO,
ACO, BBO, AMO and PSO. In order to evaluate the perfor-
mance of each algorithm, a score metric which is recently
defined for the CEC 2017 competition is used. It takes into
account the error values for all dimensions and the rank for
each problem in each dimension.
GSK gets the first ranking among all algorithms, fol-
lowed by DE and AMO in second and third place, respec-
tively. Furthermore, in order to statistically analyze the
performance of GSK, non-parametric tests (the Wilcoxon’s
test) are used with the significance level of 0.05.
As a summary of results, the performance of the GSK
algorithm was statistically superior to and competitive
with other recent and well-known state-of-the-art algo-
rithms in the majority of functions and for different dimen-
sions especially in high dimensions.
Table 18 Experimental results of GWO, AMO, PSO, BBO, ACO and GSK over 51 independent runs on 29 test functions of 100 variables with
1,000,000 FES
Function GWO AMO PSO BBO ACO GSK
15.72E+10 ± 6.63E+09 2.51E+03 ± 1.89E+03 1.08E+04 ± 1.42E+04 1.28E+07 ± 5.83E+05 4.99E+11 ± 1.38E+11 5.80E+03 ± 4.63E+03
32.08E+05 ± 1.74E+04 1.75E+05 ± 1.41E+04 3.17E+04 ± 5.19E+03 4.21E+05 ± 8.31E+04 1.32E+11 ± 2.81E+11 1.15E+05 ± 2.15E+04
44.68E+03 ± 8.31E+02 1.41E+02 ± 6.04E+01 3.03E+02 ± 3.91E+01 2.87E+02 ± 4.43E+01 1.28E+05 ± 1.09E+04 2.05E+02 ± 4.57E+01
51.09E+03 ± 4.01E+01 4.41E+02 ± 4.10E+01 6.02E+02 ± 6.13E+01 2.24E+02 ± 3.29E+01 1.98E+03 ± 3.89E+01 5.31E+02 ± 3.39E+02
65.82E+01 ± 3.57E+00 6.28E−02 ± 8.93E−02 3.64E+01 ± 8.16E+00 8.79E−01 ± 4.51E−02 1.35E+02 ± 3.70E+00 1.72E−03 ± 6.46E−03
71.69E+03 ± 7.81E+01 5.86E+02 ± 4.46E+01 5.71E+02 ± 1.17E+02 6.31E+02 ± 5.66E+01 8.86E+03 ± 1.95E+02 8.75E+02 ± 1.83E+01
81.08E+03 ± 3.93E+01 4.33E+02 ± 4.24E+01 6.01E+02 ± 7.76E+01 2.28E+02 ± 3.48E+01 2.00E+03 ± 2.18E+01 4.92E+02 ± 3.41E+02
93.79E+04 ± 4.59E+03 1.83E+03 ± 1.30E+03 1.81E+04 ± 3.09E+03 2.25E+03 ± 9.01E+02 1.25E+05 ± 7.21E+03 8.46E+00 ± 3.41E+00
10 2.84E+04 ± 8.64E+02 1.81E+04 ± 7.17E+02 1.32E+04 ± 1.28E+03 1.10E+04 ± 9.31E+02 3.01E+04 ± 7.44E+02 2.95E+04 ± 4.44E+02
11 4.19E+04 ± 9.04E+03 6.14E+02 ± 7.28E+01 1.12E+03 ± 2.00E+02 6.48E+04 ± 1.92E+04 1.17E+07 ± 3.79E+07 2.78E+02 ± 6.85E+01
12 1.55E+10 ± 2.53E+09 1.34E+06 ± 5.48E+05 1.11E+07 ± 5.28E+06 4.10E+07 ± 1.51E+07 3.32E+11 ± 3.41E+10 8.34E+04 ± 7.52E+04
13 2.43E+09 ± 6.66E+08 2.37E+03 ± 1.03E+03 4.22E+03 ± 3.95E+03 2.20E+06 ± 3.17E+05 8.01E+10 ± 1.35E+10 3.20E+03 ± 2.64E+03
14 6.79E+06 ± 2.94E+06 8.50E+05 ± 3.54E+05 5.24E+05 ± 2.42E+05 1.54E+07 ± 7.21E+06 2.74E+08 ± 2.30E+08 4.64E+03 ± 4.47E+03
15 7.11E+08 ± 1.87E+08 5.84E+02 ± 3.50E+02 2.17E+03 ± 1.59E+03 1.37E+06 ± 2.99E+05 3.65E+10 ± 5.65E+09 7.33E+02 ± 1.09E+03
16 7.91E+03 ± 7.06E+02 3.28E+03 ± 2.77E+02 3.31E+03 ± 5.55E+02 3.44E+03 ± 7.75E+02 1.88E+04 ± 1.35E+03 2.27E+03 ± 2.61E+03
17 6.14E+03 ± 4.78E+02 2.40E+03 ± 2.29E+02 2.93E+03 ± 5.54E+02 3.20E+03 ± 7.82E+02 4.34E+06 ± 2.67E+06 3.91E+03 ± 6.68E+02
18 9.63E+06 ± 3.30E+06 1.60E+06 ± 4.67E+05 1.92E+06 ± 8.57E+05 6.60E+06 ± 3.01E+06 5.51E+08 ± 1.57E+08 5.73E+04 ± 3.63E+04
19 6.23E+08 ± 1.19E+08 1.24E+03 ± 9.31E+02 1.93E+03 ± 2.77E+03 1.39E+06 ± 2.01E+05 3.35E+10 ± 8.97E+09 1.00E+03 ± 8.30E+02
20 4.30E+03 ± 4.60E+02 2.30E+03 ± 2.63E+02 2.75E+03 ± 4.15E+02 2.74E+03 ± 4.91E+02 5.42E+03 ± 1.66E+02 4.46E+03 ± 2.22E+02
21 1.30E+03 ± 5.09E+01 6.23E+02 ± 2.31E+01 9.70E+02 ± 1.34E+02 4.78E+02 ± 3.30E+01 2.62E+03 ± 6.65E+01 6.07E+02 ± 3.37E+02
22 2.95E+04 ± 9.92E+02 1.94E+04 ± 7.43E+02 1.55E+04 ± 1.75E+03 1.24E+04 ± 1.12E+03 3.19E+04 ± 5.87E+02 3.00E+04 ± 4.57E+02
23 1.66E+03 ± 5.24E+01 8.22E+02 ± 2.46E+01 1.49E+03 ± 1.49E+02 7.11E+02 ± 3.06E+01 3.91E+03 ± 1.67E+02 6.11E+02 ± 1.58E+01
24 2.19E+03 ± 8.62E+01 1.23E+03 ± 4.73E+01 1.60E+03 ± 1.56E+02 1.20E+03 ± 4.17E+01 7.41E+03 ± 6.15E+02 9.32E+02 ± 1.70E+01
25 3.84E+03 ± 5.24E+02 8.31E+02 ± 5.21E+01 8.12E+02 ± 7.29E+01 8.03E+02 ± 7.33E+01 4.90E+04 ± 5.90E+03 8.21E+02 ± 4.34E+01
26 1.65E+04 ± 6.27E+02 7.91E+03 ± 6.30E+02 9.73E+03 ± 5.69E+03 6.26E+03 ± 4.03E+02 5.23E+04 ± 2.10E+03 3.66E+03 ± 1.78E+02
27 1.60E+03 ± 1.20E+02 8.49E+02 ± 5.06E+01 9.31E+02 ± 1.14E+02 8.28E+02 ± 6.41E+01 7.92E+03 ± 2.58E+02 6.57E+02 ± 3.07E+01
28 5.23E+03 ± 9.72E+02 5.60E+02 ± 3.26E+01 6.55E+02 ± 4.30E+01 6.44E+02 ± 4.08E+01 3.82E+04 ± 1.51E+03 5.53E+02 ± 3.25E+01
29 7.49E+03 ± 5.12E+02 2.74E+03 ± 3.04E+02 3.66E+03 ± 5.24E+02 3.29E+03 ± 5.09E+02 1.19E+05 ± 5.53E+04 1.21E+03 ± 1.74E+02
30 1.95E+09 ± 4.30E+08 5.25E+03 ± 1.28E+03 8.19E+03 ± 3.68E+03 2.63E+06 ± 4.88E+05 6.75E+10 ± 1.47E+10 2.99E+03 ± 2.73E+02
Table 19 Ranking of algorithms using score metric on the CEC 2017
functions
Algorithm Score1 Score2 Score Ranking
GSK 50 47.16667 97.17 1
DE 27.10 33.27 60.37 2
AMO 7.49 50 57.49 3
SFS 12.88 40.01 52.89 4
TLBO 10.38 28.22 38.60 5
PSO 2.53 29.44 31.97 6
BBO 0.36 26.90 27.26 7
GA 0.11 18.06 18.17 8
GWO 4.58E−04 17.01 17.01 9
ACO 2.73E−05 13.75 13.76 10
ES 3.25E−05 13.42 13.42 11

International Journal of Machine Learning and Cybernetics
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Besides, the GSK algorithm has been applied to solve
the set of real world optimization problems proposed for
the IEEE-CEC2011 evolutionary algorithm competition.
Generally, GSK, AMO and BBO do significantly better
than the others on most functions. Virtually, it is easily
implemented and has been proven to be a reliable approach
for real parameter optimization.
Table 20 Results of multiple-
problem Wilcoxon’s test
between GSK and other
algorithms for D = 10, 30,50
and 100
D Algorithms R+R−p value + ≈ − Dec.
10 GSK vs TLBO 298.5 107.5 0.030 19 1 9 +
GSK vs SFS 245 190 0.552 17 0 12 ≈
GSK vs DE 81 172 0.140 7 7 15 ≈
GSK vs GA 403 32 0.000 27 0 2 +
GSK vs ES 405 1 0.000 27 1 1 +
GSK vs GWO 416 19 0.000 28 0 1 +
GSK vs AMO 155 196 0.603 12 3 14 ≈
GSK vs PSO 294.5 83.5 0.011 18 2 9 +
GSK vs BBO 368.5 66.5 0.001 23 0 6 +
GSK vs ACO 424.5 10.5 0.000 27 0 2 +
30 GSK vs TLBO 340 95 0.005 21 0 8 +
GSK vs SFS 237 169 0.439 17 1 11 ≈
GSK vs DE 238 113 0.112 17 3 9 ≈
GSK vs GA 417 18 0.000 28 0 1 +
GSK vs ES 406 0 0.000 28 1 0 +
GSK vs GWO 420 15 0.000 28 0 1 +
GSK vs AMO 228 150 0.350 14 2 13 ≈
GSK vs PSO 382 53 0.000 24 0 5 +
GSK vs BBO 366 69 0.001 22 0 7 +
GSK vs ACO 435 0 0.000 29 0 0 +
50 GSK vs TLBO 310 125 0.045 20 0 9 +
GSK vs SFS 251 184 0.469 19 0 10 ≈
GSK vs DE 268 138 0.139 18 1 10 ≈
GSK vs GA 400 35 0.000 26 0 3 +
GSK vs ES 435 0 0.000 29 0 0 +
GSK vs GWO 414.5 20.5 0.000 27 0 2 +
GSK vs AMO 264 171 0.315 17 0 12 ≈
GSK vs PSO 304 131 0.061 19 0 10 ≈
GSK vs BBO 317 118 0.031 19 0 10 +
GSK vs ACO 435 0 0.000 29 0 0 +
100 GSK vs TLBO 339 96 0.009 24 0 5 +
GSK vs SFS 274 161 0.222 21 0 8 ≈
GSK vs DE 410 25 0.000 25 0 4 +
GSK vs GA 396 39 0.000 25 0 4 +
GSK vs ES 435 0 0.000 29 0 0 +
GSK vs GWO 420 15 0.000 26 0 3 +
GSK vs AMO 274 161 0.222 18 0 11 ≈
GSK vs PSO 321 114 0.025 22 0 7 +
GSK vs BBO 336 99 0.010 20 0 9 +
GSK vs ACO 435 0 0.000 29 0 0 +

International Journal of Machine Learning and Cybernetics
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Fig. 11 The score of all
algorithms on the CEC 2017
functions
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
100.00
GSKDE AMO SFS TLBO PSO BBO GA GWOACO ES
97.17
60.37 57.49
52.89
38.61
31.97
27.26
18.17 17.01 13.76 13.42
0
50
100
150
200
250
inferior equalsuperior
D=10 D=30 D=50 D=100
Fig. 12 Statistical comparison results of GSK against other state-of-
the-art algorithms with the growth of the dimensionality
Table 21 Algorithm complexity results for D = 10
Alg. T0T1T2(T2 − T1)/T0
GSK 0.0398 0.5934 1.916 33.2311558
PSO 9.7927 231.138191
TLBO 4.85 106.949749
GWO 2.787 55.1155779
SFS 4.6507 101.942211
AMO 7.2405 167.012563
ES 8.2422 192.180905
DE 4.0051 85.721105
GA 54.1597 1345.88693
ACO 23.3552 571.904523
BBO 26.9809 663.002513
Table 22 Algorithm complexity results for D = 30
Alg. T0T1T2(T2 − T1)/T0
GSK 0.0398 0.7577 2.3803 40.7674945
PSO 9.969 231.438349
TLBO 5.159 110.584077
GWO 3.3747 65.7524191
SFS 5.0192 107.071515
AMO 17.1161 411.013726
ES 16.7547 401.933324
DE 6.8723 113.652632
GA 51.9629 1286.56146
ACO 67.0559 1665.78257
BBO 36.5998 900.553927
Table 23 Algorithm complexity results for D = 50
Alg. T0T1T2(T2 − T1)/T0
GSK 0.0398 1.0077 3.0954 52.4547739
PSO 10.3271 234.155779
TLBO 5.5209 113.396985
GWO 4.2553 81.5979899
SFS 5.6214 115.922111
AMO 26.8281 648.753769
ES 27.0341 653.929648
DE 9.9907 225.703517
GA 51.8108 1276.4598
ACO 111.4702 2775.4397
BBO 46.9975 1155.52261

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Several current and future works can be developed from
this study. Firstly, Current research efforts focus on how to
modify the GSK algorithm for handling constrained and
multi-objective optimization problems as well as solving
Table 24 Result of GSK on
CEC2011 Function Best Median Mean Worst SD
1 0.00E+00 1.85E−21 3.28E+00 1.49E+01 5.21E+00
2− 1.35E+01 − 1.14E+01 − 1.13E+01 − 9.25E+00 1.03E+00
3 1.15E−05 1.15E−05 1.15E−05 1.15E−05 9.12E−13
4 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
5− 2.39E+01 − 2.08E+01 − 2.06E+01 − 1.87E+01 1.21E+00
6− 1.16E+01 − 6.85E+00 − 6.94E+00 0.00E+00 2.48E+00
7 1.59E+00 1.78E+00 1.78E+00 1.97E+00 1.08E−01
8 2.20E+02 2.20E+02 2.20E+02 2.20E+02 0.00E+00
9 1.29E+03 2.07E+03 2.11E+03 3.09E+03 5.02E+02
10 − 2.18E+01 − 2.16E+01 − 2.16E+01 − 2.14E+01 1.19E−01
11 5.09E+04 5.24E+04 5.24E+04 5.39E+04 6.88E+02
12 1.07E+06 1.07E+06 1.07E+06 1.08E+06 1.73E+03
13 1.54E+04 1.54E+04 1.54E+04 1.55E+04 2.44E+00
14 1.82E+04 1.84E+04 1.84E+04 1.86E+04 1.22E+02
15 3.28E+04 3.28E+04 3.28E+04 3.28E+04 1.55E+01
16 1.32E+05 1.35E+05 1.35E+05 1.39E+05 2.22E+03
17 1.97E+06 2.03E+06 2.09E+06 2.36E+06 1.20E+05
18 1.17E+06 1.27E+06 1.27E+06 1.57E+06 7.56E+04
19 1.75E+06 2.03E+06 2.00E+06 2.25E+06 1.36E+05
20 1.12E+06 1.29E+06 1.29E+06 1.46E+06 9.20E+04
21 1.34E+01 1.61E+01 1.70E+01 2.49E+01 3.11E+00
22 8.61E+00 1.28E+01 1.29E+01 2.09E+01 2.93E+00
Table 25 Experimental results of TLBO, SFS, DE, GA, ES and GSK over 25 independent runs on 22 test functions with 150,000 FES
Func. TLBO SFS DE GA ES GSK
16.60E+00 ± 6.78E+00 4.69E+00 ± 4.49E+00 4.30E+00 ± 5.38E+00 1.95E+01 ± 3.01E+00 2.84E+01 ± 1.31E+00 3.28E+00 ± 5.21E+00
2− 2.08E+01 ± 2.15E+00 − 2.66E+01 ± 1.28E+00 − 1.31E+01 ± 4.60E+00 − 9.87E+00 ± 2.20E+00 − 2.54E+00 ± 3.38E−01 − 1.13E+01 ± 1.03E+00
31.15E−05 ± 4.08E−17 1.15E−05 ± 4.87E−10 1.15E−05 ± 1.86E−15 1.15E−05 ± 9.88E−10 1.15E−05 ± 0.00E+00 1.15E−05 ± 9.12E−13
40.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00
5− 2.83E+01 ± 3.70E+00 − 3.37E+01 ± 1.61E+00 − 1.95E+01 ± 9.34E−01 − 1.38E+01 ± 1.94E+00 − 2.76E+01 ± 3.23E+00 − 2.06E+01 ± 1.21E+00
6− 1.94E+01 ± 1.70E+00 − 2.74E+01 ± 1.86E+00 − 1.41E+01 ± 1.53E+00 − 3.35E+00 ± 3.99E+00 − 1.99E+01 ± 6.06E+00 − 6.94E+00 ± 2.48E+00
71.16E+00 ± 2.69E−01 1.36E+00 ± 1.39E−01 1.75E+00 ± 9.79E−02 1.13E+00 ± 1.61E−01 2.44E+00 ± 1.98E−01 1.78E+00 ± 1.08E−01
82.20E+02 ± 0.00E+00 2.20E+02 ± 0.00E+00 2.20E+02 ± 0.00E+00 2.20E+02 ± 0.00E+00 5.55E+02 ± 3.15E+02 2.20E+02 ± 0.00E+00
93.99E+03 ± 2.88E+03 2.27E+04 ± 6.55E+03 2.37E+04 ± 2.35E+03 1.32E+05 ± 3.76E+04 2.00E+06 ± 6.38E+04 2.11E+03 ± 5.02E+02
10 − 1.89E+01 ± 2.43E+00 − 2.15E+01 ± 1.31E−01 − 1.41E+01 ± 2.68E+00 − 9.61E+00 ± 1.67E+00 − 8.90E+00 ± 3.55E−01 − 2.16E+01 ± 1.19E−01
11 3.79E+06 ± 9.41E+05 4.92E+05 ± 2.14E+05 1.26E+05 ± 2.43E+04 5.52E+05 ± 8.33E+05 2.91E+08 ± 2.42E+07 5.24E+04 ± 6.88E+02
12 1.53E+06 ± 2.23E+05 1.30E+06 ± 6.45E+04 1.14E+06 ± 9.67E+03 1.05E+07 ± 7.73E+05 1.45E+07 ± 6.02E+05 1.07E+06 ± 1.73E+03
13 1.55E+04 ± 1.24E+01 1.54E+04 ± 1.44E+00 1.54E+04 ± 1.18E+01 1.55E+04 ± 2.23E+01 1.81E+04 ± 4.44E+03 1.54E+04 ± 2.44E+00
14 1.93E+04 ± 9.32E+01 1.88E+04 ± 8.04E+01 1.84E+04 ± 1.66E+02 1.98E+04 ± 7.01E+02 1.88E+04 ± 3.71E−12 1.84E+04 ± 1.22E+02
15 3.29E+04 ± 7.14E+01 3.30E+04 ± 2.35E+01 3.29E+04 ± 3.03E+01 3.30E+04 ± 7.93E+01 2.23E+05 ± 8.85E+04 3.28E+04 ± 1.55E+01
16 1.36E+05 ± 3.40E+03 1.37E+05 ± 2.16E+03 1.37E+05 ± 2.91E+03 1.50E+05 ± 7.18E+03 1.46E+05 ± 5.07E+03 1.35E+05 ± 2.22E+03
17 2.05E+06 ± 2.09E+05 2.21E+06 ± 2.60E+05 2.26E+06 ± 2.34E+05 8.78E+08 ± 1.10E+09 9.93E+09 ± 1.45E+09 2.09E+06 ± 1.20E+05
18 1.14E+06 ± 8.67E+04 1.05E+06 ± 4.38E+04 1.88E+06 ± 3.06E+05 1.32E+06 ± 2.70E+05 5.04E+07 ± 6.79E+06 1.27E+06 ± 7.56E+04
19 1.42E+06 ± 1.58E+05 1.51E+06 ± 1.93E+05 2.52E+06 ± 2.58E+05 2.24E+06 ± 2.48E+06 4.93E+07 ± 2.98E+06 2.00E+06 ± 1.36E+05
20 1.12E+06 ± 8.06E+04 1.05E+06 ± 4.64E+04 1.77E+06 ± 2.78E+05 1.37E+06 ± 1.09E+06 4.82E+07 ± 6.01E+06 1.29E+06 ± 9.20E+04
21 1.55E+01 ± 2.02E+00 1.77E+01 ± 3.13E+00 1.77E+01 ± 3.62E+00 3.15E+01 ± 6.07E+00 8.75E+01 ± 1.54E+01 1.70E+01 ± 3.11E+00
22 2.12E+01 ± 2.41E+00 2.00E+01 ± 3.09E+00 1.32E+01 ± 2.78E+00 3.50E+01 ± 8.81E+00 5.99E+01 ± 6.17E+00 1.29E+01 ± 2.93E+00

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practical engineering optimization problems and real-
world applications.
Secondly, concerning the improvement of GSK, it
would be very interesting to propose another adaptive GSK
version such that each individual has its parameter val-
ues which changed adaptively during generations. Future
research studies may focus on applying the algorithm to
solve high dimensions or large-scale global optimization
problems. Another possible direction is developing binary,
discrete versions of GSK to solve mixed integer optimi-
zation problems. Finally, hybridizing the GSK algorithm
with other powerful metaheuristics are thought to be prom-
ising direction. Furthermore, it will be greatly beneficial
as a future direction to investigate a complete parameter
tune-free adaptive GSK by combining novel adaptive
Table 26 Experimental results of GWO, AMO, PSO, BBO, ACO and GSK over 25 independent runs on 22 test functions with 150,000 FES
Func. GWO AMO PSO B BO ACO GSK
11.54E+01 ± 4.91E+00 4.85E−01 ± 1.17E+00 2.63E+01 ± 0.00E+00 2.05E+01 ± 3.40E+00 3.05E+01 ± 8.23E−01 3.28E+00 ± 5.21E+00
2− 1.48E+01 ± 1.10E+00 − 1.98E+01 ± 1.03E+00 − 3.79E+00 ± 5.52E−02 − 1.85E+01 ± 1.85E+00 − 6.16E+00 ± 8.28E−01 − 1.13E+01 ± 1.03E+00
31.15E−05 ± 4.61E−10 1.15E−05 ± 7.25E−14 1.15E−05 ± 1.46E−12 1.15E−05 ± 3.80E−09 1.15E−05 ± 0.00E+00 1.15E−05 ± 9.12E−13
40.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00
5− 2.56E+01 ± 2.56E+00 − 3.32E+01 ± 7.33E−01 − 1.98E+01 ± 9.91E−01 − 3.30E+01 ± 7.34E−01 − 1.48E+01 ± 1.21E+00 − 2.06E+01 ± 1.21E+00
6− 1.86E+01 ± 1.63E+00 − 2.70E+01 ± 9.45E−01 − 1.06E+01 ± 1.03E+00 − 2.79E+01 ± 7.95E−01 − 1.04E+01 ± 1.42E+00 − 6.94E+00 ± 2.48E+00
71.72E+00 ± 1.15E−01 1.37E+00 ± 9.38E−02 1.53E+00 ± 1.21E−01 1.44E+00 ± 8.64E−02 1.89E+00 ± 9.89E−02 1.78E+00 ± 1.08E−01
82.33E+02 ± 1.17E+01 2.20E+02 ± 0.00E+00 2.21E+02 ± 2.84E+00 2.20E+02 ± 0.00E+00 2.39E+03 ± 1.32E+03 2.20E+02 ± 0.00E+00
91.09E+05 ± 1.45E+04 1.15E+03 ± 3.24E+02 1.85E+06 ± 1.54E+05 7.83E+04 ± 2.42E+04 1.03E+06 ± 2.42E+04 2.11E+03 ± 5.02E+02
10 − 1.45E+01 ± 2.50E+00 − 2.13E+01 ± 9.12E−02 − 9.31E+00 ± 8.20E−01 − 1.52E+01 ± 1.78E+00 − 9.04E+00 ± 1.49E−01 − 2.16E+01 ± 1.19E−01
11 4.71E+07 ± 5.09E+07 5.26E+04 ± 5.02E+02 5.36E+06 ± 3.85E+05 5.66E+04 ± 8.38E+02 9.25E+06 ± 2.73E+06 5.24E+04 ± 6.88E+02
12 1.29E+07 ± 7.39E+05 1.07E+06 ± 1.27E+03 1.42E+07 ± 9.63E+05 1.09E+06 ± 1.93E+04 8.10E+06 ± 2.33E+05 1.07E+06 ± 1.73E+03
13 1.55E+04 ± 2.10E+01 1.54E+04 ± 2.59E+00 1.56E+04 ± 5.13E+01 1.55E+04 ± 2.49E+01 1.29E+05 ± 9.70E+04 1.54E+04 ± 2.44E+00
14 1.93E+04 ± 2.34E+02 1.92E+04 ± 1.48E+02 1.97E+04 ± 2.14E+02 1.94E+04 ± 3.21E+02 4.15E+05 ± 4.79E+05 1.84E+04 ± 1.22E+02
15 3.32E+04 ± 1.70E+02 3.30E+04 ± 2.09E+01 1.26E+05 ± 5.23E+04 3.31E+04 ± 6.90E+01 4.13E+06 ± 3.30E+06 3.28E+04 ± 1.55E+01
16 1.43E+05 ± 6.64E+03 1.37E+05 ± 1.70E+03 4.76E+06 ± 4.24E+06 1.42E+05 ± 4.91E+03 7.43E+07 ± 1.77E+07 1.35E+05 ± 2.22E+03
17 6.33E+09 ± 1.12E+09 2.02E+06 ± 1.63E+05 1.24E+10 ± 2.39E+09 2.70E+06 ± 1.98E+06 1.78E+10 ± 3.18E+09 2.09E+06 ± 1.20E+05
18 5.95E+06 ± 1.51E+06 1.04E+06 ± 7.56E+04 1.29E+08 ± 1.15E+07 9.70E+05 ± 1.47E+04 1.53E+08 ± 1.89E+07 1.27E+06 ± 7.56E+04
19 6.84E+06 ± 1.82E+06 1.56E+06 ± 1.29E+05 1.34E+08 ± 1.95E+07 1.53E+06 ± 2.56E+05 1.47E+08 ± 2.71E+07 2.00E+06 ± 1.36E+05
20 6.03E+06 ± 1.58E+06 1.03E+06 ± 5.94E+04 1.37E+08 ± 1.98E+07 9.75E+05 ± 1.54E+04 1.59E+08 ± 1.44E+07 1.29E+06 ± 9.20E+04
21 3.59E+01 ± 3.52E+00 1.85E+01 ± 1.39E+00 6.11E+01 ± 5.34E+00 2.24E+01 ± 3.32E+00 8.61E+01 ± 2.33E+01 1.70E+01 ± 3.11E+00
22 3.10E+01 ± 3.78E+00 2.22E+01 ± 1.81E+00 5.09E+01 ± 4.65E+00 2.52E+01 ± 2.77E+00 8.85E+01 ± 1.26E+01 1.29E+01 ± 2.93E+00
Table 27 Average ranks for all algorithms across all problems using
CEC2011
Algorithm Mean Rank Ranking
AMO 3.27 1
SFS 3.48 2
GSK 3.80 3
TLBO 3.98 4
BBO 4.75 5
DE 5.05 6
GA 7.02 7
GWO 7.18 8
PSO 8.77 9
ES 8.89 10
ACO 9.82 11
Table 28 Results of multiple-
problem Wilcoxon’s test
between GSK and other
algorithms using CEC2011
Algorithms R+R−p value + ≈ − Dec.
GSK vs. TLBO 104 86 0.717 10 3 9 ≈
GSK vs. SFS 100 71 0.528 11 4 7 ≈
GSK vs. DE 139 14 0.000 14 5 3 +
GSK vs. GA 189 1 0.000 18 3 1 +
GSK vs. ES 203 7 0.000 18 2 2 +
GSK vs. GWO 199 11 0.000 16 2 4 +
GSK vs. AMO 53 100 0.266 7 5 10 ≈
GSK vs. PSO 205 5 0.000 18 2 2 +
GSK vs. BBO 120 70 0.314 12 3 7 ≈
GSK vs. ACO 208 2 0.000 19 2 1 +

International Journal of Machine Learning and Cybernetics
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population reduction and increment method. The Matlab
source code of the proposed GSK algorithm can be down-
loaded from https ://sites .googl e.com/view/optim izati on-
proje ct/files optim izati on-proje ct/files .
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