Journal of Robotics, Networking and Artificial Life
Vol. 5(3); December (2018), pp. 165–168
DOI: 10.2991/jrnal.2018.5.3.5; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal

Research Article

Integrated Optimization of Differential Evolution withGrasshopper Optimization Algorithm
Duangjai Jitkongchuen*, Udomlux Ampant
College of Innovative Technology and Engineering, Dhurakij Pundit University, Thailand

ARTICLE INFOABSTRACTArticle History

This paper proposes a scheme to improve the differential evolution (DE) algorithm performance with integrated thegrasshopper optimization algorithm (GOA). The grasshopper optimization algorithm mimics the behavior of grasshopper.
The characteristic of grasshoppers is slow movement in the larval stage but sudden movement in the adulthood which seemas exploration and exploitation. The grasshopper optimization algorithm concept is added to DE to guide the search process
for potential solutions. The efficiency of the DE/GOA is validated by testing on unimodal and multimodal benchmarks
optimization problems. The results prove that the DE/GOA algorithm is competitive compared to the other meta-heuristic
algorithms.

Received 24 March 2018
Accepted 6 June 2018
Keywords
Meta-heuristic
differential evolution algorithm
grasshopper optimization algorithmoptimization© 2018 The Authors. Published by Atlantis Press SARL.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Meta-heuristic algorithm techniques constitute range from simple
local search procedures to complex learning processes. The basic
concepts of meta-heuristics permit an abstract level description.
Nature-inspired meta-heuristic can be grouped in three mainclasses: Evolution-based, physics-based and swarm-based.
Evolution-based algorithms are inspired by the concepts of nature
evolution. The population in search process are randomly generated which are developed over subsequent generation. The next
generation is created by combination of the individuals in the previous generation. These provide the population to be optimized
over the course of generations. The most popular evolution-basedalgorithm is genetic algorithms (GA) [1]. GA has a process offitness-based selection and recombination to produce a successor
population for the next generation. During recombination, child
chromosomes are produced by recombined genetic material fromparent chromosomes are selected. As this process is repeated,a sequence of consecutive generations evolves and the averagefitness of the chromosomes inclines to increase until some stopping
criterion is reached. Other popular algorithms such as differential
evolution (DE) algorithms [2], the principal difference between
genetic algorithms and differential evolution is that DE use mutation as the primary search mechanism, while GA use crossover as
the probabilistic mechanism to exchange of information among
solutions to locate better solutions. Biogeography-based optimizer(BBO) [3] algorithm is applied from nature phenomenon regarding
the distribution of living creatures in various islands. In this fieldof study different ecosystems are explored for finding the relations* Corresponding author. Email: duangjai.jit@dpu.ac.th

between different species in terms of immigration, emigration, andmutation. The main inspiration of the BBO algorithm is to reach astable situation based on migration and mutation in the evolutionof ecosystems.
The second class of meta-heuristic is physics-based algorithms
that imitate the physical rules. For example, gravitational searchalgorithm (GSA) [4], GSA is a nature-inspired conceptual framework based on gravitational kinematics, a branch of physics thatmodels the motion of masses moving under the influence of gravity. Galaxy-based search algorithm [5] is inspired on the spiral armof spiral galaxies to search its surrounding. This spiral movement
is improved by chaos to escape from local optimums. Curved spaceoptimization [6] inspired by general relativity theory is used toenhance the efficiency of a simple random search, and convert it toa very robust optimization tool.
The third class of meta-heuristic is swarm-based algorithms that
imitate the social behavior of swarms, herds, flocks, or schools ofanimals. The most popular algorithms are particle swarm optimization (PSO) [7]. PSO inspired by bird flocking or fish schooling. Eachparticle will be iteratively updated to improve its solution during
movement around the search space. The result leads to the best valueof global solution. Other popular swarm-based algorithms are bat
algorithms (BA) [8]. BA is inspired by echolocation behavior of micro­
bats to avoid obstacles, find their home, and detect their prey in thedarkness. Firefly algorithm (FA) [9] applied from the flashing lightof fireflies when they want to mate or to warn others about predators.
The new meta-heuristic has been focused on this paper is differential evolution algorithm which is a global search optimization
algorithm and improved performance of it by added grasshopper
optimization algorithm (GOA) to increase convergence rate.

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The rest of the paper is structured as follows. Section 2 describes
some backgrounds of the differential evolution algorithm. In
Section 3, the proposed algorithm is presented. Section 4 showsthe experimental results and the conclusions will be discussed inSection 5.

2. THE DIFFERENTIAL EVOLUTION
ALGORITHM
The differential evolution algorithm optimizes a problem by sustaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple
procedure, and then keeping whichever candidate solution has thebest score or fitness on the optimization problem at hand.
The procedure of differential evolution algorithm starts to initialize
target vectors XiG = (x1,i , x2,i , ..., x j ,i , ..., x N ,i ) where i = 1, 2, ..., NP.
NP is the population size. N is the dimension of the population.
The superscript G identifies the Gth generation.
The target vectors are used to generate the mutant vectors
ViG = (v1,i , v2,i , ..., v j ,i , ..., v N ,i ) in next steps. The DE/current-tobest/1 mutation scheme is shown in Eq. (1).
G	   ViG = XiG + F ( X best- XiG ) + F ( XrG1 - XrG2 ) (1)where r1 and r2 are integer number chosen from the set {1, 2, ...,GNP} and must be different from index i. X bestis the best solution
in the generation G. F is used to control amplification of the differential evolution. The value of F is set in the range [0, 2]. Thetarget vectors and mutant vectors are used to generate the trialvectors U iG = (u1,i , u2,i , ..., u j ,i , ..., uN ,i ) in crossover operation toincrease the diversity of population. The trial vectors are generatedby Eq. (2).	   u = ìív j ,i , if (rand £ CR ) or ( j = jrand ) (2)j ,iî x j ,i , otherwise
where j = 1, 2, ..., N. rand is a uniform random number chosen in
the range [0, 1]. CR ∈ [0, 1] is the crossover parameter. jrand is anindex randomly chosen in the range [0, N].
The last step is the selection operation to choose a better vector for
next generation. The new generations are selected by comparing
the fitness value between the target vector and the trial vector. Theselection is shown in Eq. (3).	    X G +1 = ìíU , if ( f (U ) < f ( X )) (3)i
î X , otherwise
Gi
Gi

Gi

Gi

The whole process is repeated until the termination criteria aresatisfied or a predefined number of iterations are reached.

3. THE PROPOSED ALGORITHM
The schema of DE/current-to-best/1 in mutation operation is
updated by splitting F mutation parameter to l and F. The newmutation schema is shown in Eq. (4).
G	   ViG = XiG + l ( X best- XiG ) + F ( XrG1 - X rG2 ) (4)where l is used to balance the difference vectors between the best
vectors and the target vectors. The value of l that is generated byEq. (5) is decreased in each generation.		 
l G +1 = l G - (l G ´ c1 ) (5)Meanwhile the mutation parameter F is used to perform theamplification of the difference between two random population­members. When the procedure is trapped in the local optimum,
the value of F is increased by Eq. (6).		 
F G +1 = F G + (F G ´ c2 ) (6)The parameters c1 in Eq. (5) and c2 in Eq. (6) are uniform random
numbers. c1 ∈ [0, 0.1] is a random value and c2 ∈ [0, 1] is a random
value. The crossover parameter (CR) is regenerated at the end ofeach generation to find an optimized parameter. The new crossoverparameter is generated by Eq. (7).	    CR G +1 = CR G + (mean A (SCR ) ´ c3 ) (7)where meanA is the usual arithmetic mean. SCR is the set of crossover parameter values. C3 ∈ [0, 1] is a uniform random number.
After that, the crossover operation is rearranged. The vectors thatcreated from the target vectors and the mutant vector is called theprime vectors (U i¢). The prime vectors is used to generate the trialvectors by comparing the fitness value between the prime vectors
and the mutant vectors, otherwise the trial vectors is generated bygrasshopper optimization algorithm.
The grasshopper optimization algorithm was proposed in
Saremi et al. [10] mimics the behavior of grasshopper. The characteristic of the swarm in the larval stage is slow movement andsmall steps but is long range and suddenly movement in adulthood. These two characters seem as exploration and exploitation. The search agents are encouraged to move suddenly inexploration and conduce to move locally during exploitation.
These two functions as well as target seeking are performed bygrasshopper naturally.
The proposed algorithm takes the feature of movement of thegrasshopper to generate a new value for the trial vectors is showedin Eq. (8).
 NP
x j - xi  ubd - lbddd
	 U = c ∑cs x j - xi
+ T d (8)
 j =1
di, j 2 j ≠i(di)where ubd and lbd are the upper bound and the lower bound in-r l-r

the dth dimension. s(r ) = fe - e , di , j is the distance between
ith and jth grasshopper. T d is the value of the dth dimension in thetarget. The parameter c is decreased equivalent to the number of
iteration to balancing exploration and exploitation is calculated
as Eq. (9).		 
c = cmax - lcmax - cmin(9)Lwhere cmax and cmin are the maximum and the minimum value,respectively. l is the current iteration and L is the maximum
number of iterations.

D. Jitkongchuen and U. Ampant / Journal of Robotics, Networking and Artificial Life 5(3) 165–168

4. THE EXPERIMENTAL RESULTS
The proposed algorithm has been evaluated performance with thirteen benchmark functions [11]. The test functions are unimodal( f1 - f7) and multimodal ( f8 - f13). The details of benchmark functions
are shown in Table 1. The mutation parameters set l, F = 0.5, thecrossover parameter set CR = 0.9 and population size set NP = 100.
The proposed algorithm was run 30 times on each benchmark
functions. The average results are summarized in Table 2. The performance of the proposed algorithm is compared with two EAs:
DE and fast evolutionary programming (FEP) and compared withswarm-based algorithm: PSO. The results of DE, FEP and PSO
were taken from the results reported in Saremi et al. [12].
According to the results of Table 2, DE/GOA is able to provide very
competitive results. This algorithm outperforms all others in f1, f2,f3, and f7. Both DE/GOA and DE were successful to find the optimal
solution in f4 and f6. In the function f5, only DE could solve the optimum. Therefore, these results show the performance of DE/GOA
in terms of exploiting the optimum.
The multimodal functions have many local optima with thenumber increasing exponentially with dimension. This makes themsuitable for benchmarking the exploration ability of an algorithm.

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The experimental showed that DE/GOA is able to provide very
competitive results on the multimodal benchmark functions
as well.
The experimental results show that the global search algorithms which added grasshopper optimization algorithm could
Table 2 | The experimental results
Functions

DE/GOADEFEP

PSO

f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13

2.35E−220
4.28E−11
3.50E−226028.98017
02.11E−05−8831.53
6.76E−10
4.44E−1602.87E−09
3.01E−08

8.20E−14
1.50E−09
6.80E−11000
4.63E−03−11080.1
69.20
9.70E−0807.90E−15
5.10E−14

5.70E−048.10E−03
1.60E−02
3.00E−01
5.06
01.42E−01−12554.5
4.60E−02
1.80E−02
1.60E−02
9.20E−06
1.60E−04

1.36E−04
4.21E−02
70.12562
1.086481
96.71832
1.02E−04
1.23E−01−4841.29
46.70423
2.76E−02
9.22E−03
6.92E−03
6.68E−03

Table 1 | The benchmark functions
FunctionsRange

f1 (x ) = ∑ i=1 xi2N[−100, 100]

f 2 (x ) = ∑ i=1 xi + ∏ i=1 xi[−100, 100]()[−10, 10]

{ x , 1 ≤ i ≤ N}[−100, 100]

f 5 (x ) = ∑ i=1 [100 (xi+1 - xi2 )2 + (xi - 1)2 ][−100, 100]NN

f 3 (x ) = ∑ i=1 ∑ j=1 x j
N

f 4 (x ) = max ii

2i

N -1

(f 6 (x ) = ∑ i=1  xi + 0.5N)

2[−30, 30]

f 7 (x ) = ∑ i=1 ixi4 + random [0, 1)N

f 8 (x ) = ∑ i=1 - xi sin
N[−100, 100]

( | x |)[−1.28, 1.28]if 9 (x ) = ∑ i=1  xi2 - 10 cos(2p xi) + 10 [−500, 500]

NN11
f10 (x ) = −20 exp  −0.2
x 2 − exp  ∑ i =1 cos 2p xi + 20 + e∑
i =1 i NN[−5.12, 5.12]N

f11 (x ) =N1
æx öN
å xi2 - Õ i=1 cos çè ii ÷ø + 1
4000 i =1[−32, 32]f12 (x ) =n-1np{10 sin(p y1) + å i =1 ( yi - 1)2[1 + 10 sin2 (p yi +1 )] + ( yn - 1)2 } + å i =1 u (xi , 10, 100, 4)n[−600, 600]

ìk(xi - a)m , xi > aïïu (xi , a, k , m) = í0, - a < xi < aïmïîk(- xi - a) , xi < -ayi = 1 +xi + 1
4[−50, 50]

f13 (x ) = 0.1 {sin2 (3p x1) + å i =1 (x1 - 1)2[1 + sin2 (3p xi + 1)] + (xn - 1)2[1 + sin2 (2p xn)]} + å i =1 u (x1 , 5, 100, 4)
nn[−50, 50]

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increase convergence rate because the characteristic of GOA
is both exploration and exploitation. Meanwhile the results ofmultimodal functions are not perfect that may be due to themultimodal functions have many increasing number of local
optimum.

5. CONCLUSION
This work proposed to improve the DE algorithm performancewith integrated the GOA. Thirteen benchmark functions wereemployed the performance in terms of exploitation and exploration. The results showed that DE/GOA was able to provide competitive results compared to well known heuristics such as DE, FEP
and PSO.

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Authors Introduction
Dr. Duangjai Jitkongchuen
She graduated Ph.D. in Information
Technology from King Mongkut’s
Institute of Technology Ladkrabang(KMITL) in 2014 and now she
is working as Assistant Professor
at Dhurakij Pundit University,
College of Innovative Technologyand Engineering as a deputy deanadministration. Her research area isabout machine learning, data mining
and bio-inspired algorithms.

Ms. Udomlux Ampant
She graduated Master of Business
Administration (Marketing) Dhurakij
Pundit University. She is an lecturer in
the College of Innovative Technology
and Engineering in Dhurakij PunditUniversity.