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Citation: Chen, M.; Zhou, Y.; Luo, Q.
An Improved Arithmetic
Optimization Algorithm for
Numerical Optimization Problems.
Mathematics 2022,10, 2152. https://
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mathematics
Article
An Improved Arithmetic Optimization Algorithm for
Numerical Optimization Problems
Mengnan Chen 1, Yongquan Zhou 1,2,* and Qifang Luo 1,2
1College of Artificial Intelligence, Guangxi University for Nationalities, Nanning 530006, China;
2020210812000995@stu.gxmzu.edu.cn (M.C.); 20060043@gxun.edu.cn (Q.L.)
2Guangxi Key Laboratories of Hybrid Computation and IC Design Analysis, Nanning 530006, China
*Correspondence: zhouyongquan@gxun.edu.cn; Tel.: +86-136-0788-2594
Abstract:
The arithmetic optimization algorithm is a recently proposed metaheuristic algorithm. In
this paper, an improved arithmetic optimization algorithm (IAOA) based on the population control
strategy is introduced to solve numerical optimization problems. By classifying the population
and adaptively controlling the number of individuals in the subpopulation, the information of each
individual can be used effectively, which speeds up the algorithm to find the optimal value, avoids
falling into local optimum, and improves the accuracy of the solution. The performance of the
proposed IAOA algorithm is evaluated on six systems of nonlinear equations, ten integrations, and
engineering problems. The results show that the proposed algorithm outperforms other algorithms
in terms of convergence speed, convergence accuracy, stability, and robustness.
Keywords:
arithmetic optimization algorithm; population control strategy; systems of nonlinear
equations; numerical integrals; metaheuristic
MSC: 68T20
1. Introduction
In the practical application calculations of science and engineering, many mathemat-
ical problems will be involved, such as nonlinear equation systems (NESs), numerical
integration, etc. There are tremendous methods for solving NESs, including traditional
techniques and intelligent optimization algorithms. Traditional techniques to solve NESs
use gradient information [
1
], such as Newton’s method [
2
,
3
], quasi-Newton’s method [
4
],
steepest descent method, etc. Due to relying on the selection of initial points and being
prone to falling into optimal local one, these methods cannot obtain high-quality solutions
for some specific problems. The metaheuristic algorithms, however, have the characteristics
of low requirements for the initial point, a wide range of solutions, high efficiency, and
robustness. These break through the limitations of traditional methods in solving problems.
In recent years, metaheuristic algorithms have made great contributions in solving NESs
(Karr et al. [
5
]; Ouyang et al. [
6
]; Jaberipour et al. [
7
]; Pourjafari et al. [
8
]; Jia et al. [
9
];
Ren et al. [
10
]; Cai et al. [
11
]; Abdollahi et al. [
12
]; Hirsch et al. [
13
]; Sacco et al. [
14
];
Gong et al. [
15
]; Ariyaratne et al. [
16
]; Gong et al. [
17
]; Ibrahim et al. [
18
]; Liao et al. [
19
];
Ning et al. [20]; Rizk-Allah et al. [21]; Ji et al. [22]; Turgut et al. [23]).
Numerical integration is a very basic computational problem. It is well-known that,
when calculating the definite integral, the integrand is required to be easily given and
then solved by the Newton-Leibniz formula. However, this method has many limitations,
because in many practical problems, the original function of the integrand cannot be
expressed, or the calculation is too complicated, so the definite integral of the integrand
is replaced by a suitable finite sum approximation. The traditional numerical integration
methods include the trapezoidal method, rectangle method, Romberg method, Gauss
method, Simpson’s method, Newton’s method, etc. The above methods all divide the
Mathematics 2022,10, 2152. https://doi.org/10.3390/math10122152 https://www.mdpi.com/journal/mathematics
Mathematics 2022,10, 2152 2 of 27
integral interval into equal parts, and the calculation efficiency is not high. Therefore, it is
of great significance to find a new technique with a fast convergence speed, high precision,
and strong robustness for numerical integration. Zhou et al. [
24
], based on the evolutionary
strategy method, worked to solve numerical integration. Wei et al. [
25
] researched the
numerical integration method based on particle swarm optimization. Wei et al. [
26
],
based on functional networks, worked to solve numerical integration. Deng et al. [
27
]
solved the numerical integration problems based on the differential evolution algorithm.
Xiao et al. [
28
] applied the improved bat algorithm in numerical integration. The quality of
the solution obtained by the above techniques was higher than the traditional methods.
All along, engineering optimization problems have been a popular area of research.
Metaheuristic algorithms have been widely applied to engineering optimization prob-
lems due to their great practical significance, such as applied to the automatic adjust-
ment of controller coefficients (Szczepanski et al. [
29
]; Hu et al. [
30
]), applied to system
identification (Szczepanski et al. [
31
]; Liu et al. [
32
]), applied to global path planning
(Szczepanski et al. [
33
]; Brand et al. [
34
]), and applied to robotic arm scheduling (Szczepan-
ski et al. [35]; Kolakowska et al. [36]).
The Arithmetic Optimization Algorithm (AOA) [
37
] is a novel metaheuristic algo-
rithm proposed by Abualigah et al. in 2021. AOA is a mathematical model technique that
simulates the behaviors of Arithmetic operators (i.e., Multiplication, Division, Subtraction,
and Addition) and their influence on the best local solution. Some improvements and
practical applications of the algorithm have been made by scholars. Premkumar et al. [
38
]
proposed a multi-objective arithmetic optimization algorithm (MOAOA) for solving real-
world multi-objective CEC-2021-constrained optimization problems. Bansal. et al. [
39
]
used a binary arithmetic optimization algorithm for integrated features and feature selec-
tion. Agushaka et al. [
40
] introduced an advanced arithmetic optimization algorithm for
solving mechanical engineering design problems. Abualigah et al. [
41
] presented a novel
evolutionary arithmetic optimization algorithm for multilevel thresholding segmentation.
Xu et al. [
42
] hybridized an extreme learning machine and a developed version of the arith-
metic optimization algorithm for model identification of the proton exchange membrane
fuel cells. Izci et al. [
43
] introduced an improved arithmetic optimization algorithm for the
optimal design of controlled PID. Khatir et al. [
44
] proposed an improved artificial neural
network using the arithmetic optimization algorithm for damage assessments.
The basic AOA still has some drawbacks. For instance, it is easy to fall into a local
optimum due to the location update based on the optimal value, premature convergence,
and low solution accuracy, which need to be solved. Furthermore, in order to seek a more
efficient way to solve numerical problems, in this paper, an improved arithmetic opti-
mization algorithm (IAOA) based on the population control strategy is proposed to solve
numerical optimization problems. By classifying the population and adaptively controlling
the number of individuals in the subpopulation, the information of each individual can be
used effectively while increasing the population diversity. More individuals are needed in
the early iterations to perform a large-scale search that avoids falling into the local optimum.
The search around the optimal value later in the iterations by more individuals speeds up
the algorithm to find the optimal value and improves the accuracy of the solution. The
performance of the proposed IAOA algorithm is evaluated on six systems of nonlinear
equations, ten integrations, and engineering problems. The results show that the proposed
algorithm outperforms the other algorithms in terms of convergence speed, convergence
accuracy, stability, and robustness.
The main structure of this paper is as follows. Section 2reviews the relevant knowledge
for the nonlinear equation systems, integration, and basic arithmetic optimization algorithm
(AOA). Section 3introduces the proposed IAOA in detail. Section 4presents experimental
results, comparisons, and analyses. Section 5concludes the work and proposes future
research directions.
Mathematics 2022,10, 2152 3 of 27
2. Preliminaries
2.1. Nonlinear Equation Systems
Generally, a nonlinear equation system can be formulated as follows.
NES =
f1(x1,x2, . . . , xD) = 0
.
.
.
fi(x1,x2, . . . , xD) = 0
.
.
.
fn(x1,x2, . . . , xD) = 0
(1)
where xis a D-dimensional decision variable, and nis the number of equations. Some
equations are linear; the others are nonlinear. If x
*
satisfies f
i
(x
*
) = 0, then x
*
is a root of the
system of equations.
Before using the optimization algorithm to solve the NES, first is to convert it into a
single-objective optimization problem [17] as follows.
min f(x) =
n
∑
i=1
f2
i(x),x= (x1,x2, . . . , xi, . . . , xD)(2)
Finding the minimum of an optimization problem is equivalent to finding the root of
the NES.
2.2. Numerical Integration
Definite integrals are very basic mathematical calculation problems as follows.
Zb
af(x)dx (3)
where f(x) represents the integrand function, and aand brepresent the upper and lower
bounds, respectively.
Usually, firstly, we find the original function F(x) of the integrand when finding a
definite integral and then use the Newton-Leibniz formula as follows:
Zb
af(x)dx =F(b)−F(a),(F0(x) = f(x)) (4)
However, in many cases, it is difficult to obtain the original function F(x), so the
Newton-Leibniz formula will not be able to be used.
In addition, the rest of the numerical quadrature methods are based on the quadrature
formula of equidistant node division and summation or stipulate that the equidistant nodes
remain unchanged during the whole process of calculating, as shown in Figure 1a. There
need more nodes to obtain a high accuracy. However, the best segmentation is not the
predetermined equidistant points, as shown in Figure 1b. Randomly generated subintervals
has unequal intervals according to the concave and convex changes of the function curve,
so the obtained value has a higher accuracy than the traditional methods. Based on this
idea, there is another integral method based on non-equidistant point division [
24
]. First,
generate some points randomly on the integral interval, and then, the algorithm is used
to optimize these split points. Finally, a higher accuracy value will be obtained. This not
only calculates the definite integral of the function in the usual sense but also calculates
the integral of the singular function and the integral of the oscillatory function for this
method [
27
]. The flow of the numerical integration algorithm based on unequal point
segmentation is as follows [24].
(1)
Randomly initialize the population in the search space S.
Mathematics 2022,10, 2152 4 of 27
(2)
Arrange each individual in the integral interval in ascending order. The integral
interval has n(n=D+ 2) nodes and n
−
1 segments. Calculate the distance h
i
between
two adjacent nodes and the function f(x
k
) value of each node, then calculate the
function value corresponding to the D+ 2 nodes and the function value of the middle
node of each subsection. Find the minimum value w
j
and the maximum value
W
j
(j= 1, 2,
. . .
,D+ 1) among the function values of the left endpoint, middle node,
and right endpoint of each subsection.
(3)
Calculate fitness value. F(i)1
2∑D+1
j=1hjWj−wj.
(4)
Update individuals through an optimization algorithm.
(5)
Repeat step 4 until reaching the stop condition.
(6)
Get the accuracy and integral values.
Figure 1.
Two methods of segmentation when solving numerical integrals: (
a
) equidistant division
and (b) equidistant division.
The numerical integration method based on Hermite interpolation only needs to
provide the value of the integral node functions and has high precision. However, this
method is based on equidistant segmentation. In this paper, the adaptability of unequal-
spaced partitioning and the numerical integration method based on Hermite interpolation
are combined to solve the numerical integration problem, and the formula is as follows:
Rb
af(x)dx =n
∑
k=1
hi
2[f(xk) + f(xk+1)] −
n−1
∑
i=1
25
144 hi[f(a)+ f(b)]
n−1+
n−1
∑
i=1
hi
3[f(a+hi)+ f(b−hi)]
n−1−
n−1
∑
i=1
hi
4[f(a+2hi)+ f(b−2hi)]
n−1+
n−1
∑
i=1
hi
9[f(a+3hi)+ f(b−3hi)]
n−1−
n−1
∑
i=1
hi
48 [f(a+4hi)+ f(b−4hi)]
n−1
(5)
where nis the number of random split points, h
i
is the distance between two adjacent
points, and f(x) is the integrand function. The advantage of this method is that it does not
need to calculate the derivative value and only needs to provide the node function value.
Before using the optimization algorithm to solve the integration, the first step is to convert
it into a single-objective optimization problem as follows:
minF(x) = Zb
af(x)dx −E
(6)
where Rb
af(x)dx is obtained by Equation (5), and Emeans the exact value.
Combine the optimization algorithm with Equation (5), and the whole solution process
is as follows.
(1)
Randomly initialize the population in the search space S.
Mathematics 2022,10, 2152 5 of 27
(2)
Arrange each individual in the integral interval in ascending order. The integral
interval has n(n=D + 2) nodes and n
−
1 segments. Calculate the distance h
i
between
two adjacent nodes and the function f(x
k
) value of each node and then bring them
into Equation (5).
(3)
Calculate the fitness value by Equation (6).
(4)
Update individuals through an optimization algorithm.
(5)
Repeat step 4 until reaching the stop condition.
(6)
Get the accuracy and integral values.
2.3. The Arithmetic Optimization Algorithm (AOA)
The AOA algorithm is a population-based metaheuristic algorithm to solve optimiza-
tion problems by utilizing mathematical operators (Multiplication (“
×
”), Division (“
÷
”),
Subtraction (“−”), and Addition (“+”)). The specific description is as follows.
2.3.1. Initialization Phase
Generate a candidate solution matrix randomly.
X=
x1,1 ··· · ·· x1,jx1,n−1x1,n
x2,1 ··· · ·· x2,jx2,n−1x2,n
··· ··· · ·· · ·· ··· ···
.
.
..
.
..
.
..
.
..
.
..
.
.
xN−1,1 ··· · ·· xN−1,jxN−1,n−1xN−1,n
xN,1 ··· · ·· xN,jxN,n−1xN,n
(7)
After the initialization step, calculate the Math Optimizer Accelerated (MOA) function
and use it to choose between exploration and exploitation. The function is as follows:
MOA(t) = Min +t×Max −Min
T(8)
where Max = 0.9 denotes the maximum and Min = 0.2 denotes the minimum of the function
value, MOA (t) represents the function value of the current iteration, and Tand trepresent
the maximum number of iterations and current iteration, respectively.
2.3.2. Exploration Phase
During the exploration phase, the operators (Multiplication (“
×
”) and Division (“
÷
”))
are used to explore the space randomly when the MOA > 0.5. The mathematical model is
as follows:
xi,j(t+1) = best(xj)÷(MOP +ε)×((UBj−LBj)×µ+LBj),r2<0.5
best(xj)×MOP ×((UBj−LBj)×µ+LBj),otherwise (9)
where r
2
is a random number, x
i,j
(t+ 1) represents the jth position of ith solution in the
(t+ 1)th iteration, best(x
j
) denotes the jth position in the global optimal solution,
ε
is a small
integer number that avoids the case where the denominator is zero in division, UB
j
and LB
j
represents the upper and lower bounds of each dimension, respectively, and
µ
is equal to
0.5. The Math Optimizer probability (MOP) is as follows:
MOP(t) = 1−t1
α
T1
α
(10)
where MOP(t) represents the function value for the current iteration, and
α
is a sensitive
parameter and equal to 5.
Mathematics 2022,10, 2152 6 of 27
2.3.3. Exploitation Phase
During the exploration phase, the operators (Subtraction (“
−
”) and Addition (“+”))
are used to execute the exploitation. When MOA < 0.5, the mathematical model as follows:
xi,j(t+1) = best(xj)−MOP ×((UBj−LBj)×µ+LBj),r3<0.5
best(xj) + MOP ×((U Bj−LBj)×µ+LBj),otherwise (11)
where r
3
is a random number. The pseudo-code of the AOA is as follows (Algorithm 1) [
37
].
Algorithm 1 AOA
1. Set up the initial parameters α,µ.
2. Initialize the population randomly.
3. for t= 1: T
4. Calculate the fitness function and select the best solution.
5. Update the MOA (using Equation (8)) and MOP (using Equation (10)).
6. for i= 1: N
7. for j= 1: Dim
8. Generate the random values between [0, 1] (r1,r2,r3)
9. if r1>MOA
10. if r2> 0.5
11. Update the position of the individual by Equation (9).
12. else
13. Update the position of the individual by Equation (9).
14. end
15. else
16. if r3> 0.5
17. Update the position of the individual by Equation (11).
18. else
19. Update the position of the individual by Equation (11).
20. end
21. end
22. end
23. end
24. t=t+ 1
25. end
26. Return the best solution (x).
3. Our Proposed IAOA
3.1. Motivation for Improving the AOA
In AOA, the population is updated based on the optimal global solution. Once it falls
into the optimal local one, the entire population will stagnate. There is premature coverage,
in some cases [
33
]. In addition, this algorithm does not fully utilize the information of
the individuals in the population. Therefore, to make full use of the information of the
individuals and address the weakness of AOA, the improved arithmetic optimization
algorithm (IAOA) is proposed in this paper.
3.2. Population Control Mechanism
In the basic arithmetic optimization algorithm (AOA), the operators (Multiplication
(“
×
”), Division (“
÷
”), Subtraction (“
−
”), and Addition (“+”)) are used to wrap around
an optimal solution to search randomly in space, and it will lead to a loss of population
diversity. Therefore, it is necessary to classify for the population.
Mathematics 2022,10, 2152 7 of 27
3.2.1. The First Subpopulation
Sort the population according to the fitness value and select the first num_best individ-
uals as the first subpopulation:
num_best =round(0.1N+0.5N(1−t/T)) (12)
where Nis the number of individuals, and tand Trepresent the current iteration and
maximum iterations, respectively. Then, these individuals update their position by getting
information about each other. The mathematical model is as follows:
xbest_i(t+1) = xbest_i(t) + rand × best(x)−xbest_i(t) + xbest_j(t)
2×ω!(13)
xbest_j(t+1) = xbest_j(t) + rand × best(x)−xbest_i(t) + xbest_j(t)
2×ω!(14)
where x
best_i
(t+ 1) denotes the position of ith individual in the next iteration, the same as
x
best_j
(t+ 1), best(x) represents the global optimum that has been found through individuals
after titerations, x
best_j
is selected from the first class randomly, and
ω
means the information
acquisition rate and takes the value 1 or 2.
3.2.2. The Second Subpopulation
Select num_middle individuals from the population as the second subpopulation.
num_middle =round(0.3 ×N)(15)
These individuals fall between num_best and num_worst in the population. Then, these
individuals update their position, and the updated model is as follows:
xmid_i(t+1) = xmid_i(t) + Levy ×(best(x)−xmid_j)(16)
where x
mid_i
(t+ 1) denotes the position of ith individual in the next iteration, Levy is the
Levy distribution function [45,46], and xmid_j is selected from the second class randomly.
3.2.3. The Third Subpopulation
Select num_worst individuals from the population as the final subpopulation.
num_worst =N−(num_best +num_middl e)(17)
In the final class, the individuals update their position by the following equation:
xworst_i(t+1) = xworst_i+t
T×best(x)−xworst_j(18)
where x
worst_i
(t+ 1) denotes the position of ith individual in the next iteration, and best(x)
represents the global optimum that has been found through individuals after titerations.
At the early iteration of IAOA, there are more individuals in the first subpopulation
for speeding up the update of the global optimum. At the later iterations of the algorithm,
the number of individuals in the first subpopulation decreases, which solves the operator
crowding problem near the optimum. In addition, the number of individuals in the
third subpopulation increases, which effectively prevents the population from falling
into the local optimum. The second subpopulation utilizes the Levy flight for small-step
updates to find more promising areas. The above strategy can effectively overcome the
weaknesses of traditional AOA and improve its performance. The pseudo-code of the
IAOA in Algorithm 2 is as follows (Algorithm 2). Figure 2is the flowchart of the IAOA.
Mathematics 2022,10, 2152 8 of 27
Algorithm 2 IAOA
1. Set up the initial parameters α,µ.
2. Initialize the population randomly.
3. for t= 1: T
4. Calculate the fitness function and select the best solution.
5. Calculate the number of the first subpopulation by Equation (12).
6. Update the first subpopulation by Equations (13) and (14).
7. Calculate the number of the second subpopulation by Equation (15).
8. Update the second subpopulation by Equation (16).
9. Calculate the number of the third subpopulation by Equation (17).
10. Update the third subpopulation by Equation (18).
11. Update the MOA (using Equation (8)) and MOP (using Equation (10)).
12. for i= 1: N
13. for j= 1: Dim
14. Generate the random values between [0, 1] (r1,r2,r3)
15. if r1>MOA
16. if r2> 0.5
17. Update the position of the individual by Equation (9).
18. else
19. Update the position of the individual by Equation (9).
20. end
21. else
22. if r3> 0.5
23. Update the position of the individual by Equation (11).
24. else
25. Update the position of the individual by Equation (11).
26. end
27. end
28. end
29. end
30. t=t+ 1
31. end
32. Return the best solution (x).
Mathematics 2022,10, 2152 9 of 27
Figure 2. Flowchart of the IAOA.
4. Numerical Experiments and Analysis
4.1. Parameter Settings
Here, six groups of NESs and ten groups of integration have been used to demon-
strate the efficiency of the IAOA. The IAOA compares several popular algorithms and
two improved arithmetic optimization algorithms (The Arithmetic Optimization Algorithm
(AOA) [
37
], Sine Cosine Algorithm (SCA) [
47
], Whale Optimization Algorithm (WOA) [
48
],
Grey Wolf Optimizer (GWO) [
49
], Harris hawks optimization (HHO) [
50
], Slime mould
algorithm (SMA) [
51
], Differential evolution(DE) [
52
], Cuckoo search algorithm (CSA) [
53
],
Advanced arithmetic optimization algorithm (nAOA) [
40
], and a developed version of
Arithmetic Optimization Algorithm (dAOA) [
42
]) for tackling NES. Among them, the
parameters of these algorithms are all from the original version. These algorithms are
evaluated from four aspects: the average value, the optimal value, the worst value, and the
standard deviation. All algorithms are executed on MATLAB 2021a, running on a computer
with a Windows 10 operating system, Intel(R) Core (TM) i7-9700 CPU @ 3.00 GHz, 16 GB
of Random Access Memory (RAM), and run 30 times independently for all test problems.
The flowchart for handling issues by the IAOA is shown in Figure 3.
Mathematics 2022,10, 2152 10 of 27
Figure 3. Flowchart for handling issues.
4.2. Application in Solving NESs
Solving nonlinear problems often requires higher-precision solutions in many practical
applications. In this section, six nonlinear systems of equations are chosen to evaluate the
performance of the IAOA. The characteristics of these equations are different from each
other, where problem01 [
54
] describes the interval arithmetic problem, problem02 [
55
]
describes the multiple steady-states problem, and problem06 [56] describes the molecular
conformation. These problems come from real-world applications. For fairness, set the
population to 50 and the maximum number of iterations to 200. Tables 1–6show all the test
results of the NES. Best represents the best value, Worst represents the worst value, Mean
represents the mean value, Std represents the standard deviation, and p-value stands for
the Wilcoxon rank–sum test in Table 7. The Wilcoxon p-value test is used to verify whether
there is an obvious difference between the two sets of data.
Table 1. Comparison of the experimental results for problem01.
Variable Algorithms
AOA IAOA SCA WOA
x10.006361583402960 0.257838650825518 0.186732591196869 0.260832096649832
x20.005731653837062 0.381098185347242 0.399818814038728 0.381680691118263
x30.010586282003880 0.278742562628776 0.008959145137085 0.258353295805450
x40.002593989505334 0.200665586275865 0.227237103605413 0.215307146397956
x50.033520558095432 0.445255928027431 0.003829239926320 0.448797960971748
x60.076424218265631 0.149188813621332 0.185905381801968 0.147397359179682
x70.038862694473151 0.432010769672038 0.368813050526818 0.442390776062597
x8−0.000004007877210 0.073406152818720 0.037739989370997 0.137586270569043
x90.029054432130685 0.345966262513093 0.206476235144125 0.342058064566263
x10 0.013690425703394 0.427324518269459 0.363350844915327 0.401475021739693
f
8.45665838921712
×
10
−14.73405913551646 ×10−10
1.22078391539763
×
10
−1
9.59544885085295
×
10
−4
Mathematics 2022,10, 2152 11 of 27
Table 1. Cont.
Variable Algorithms
GWO HHO DE CSO
x10.256851024248810 0.324317023967532 2.000000000000000 0.089951372914250
x20.383565743620699 0.303967192642514 1.948157453190990 0.309487131659014
x30.278312335483674 0.216191961411362 2.000000000000000 0.456410156556233
x40.198737300040942 0.305260974230829 1.815308511546580 0.356392775439902
x50.446311619177502 0.325255783591842 2.000000000000000 0.476086684751138
x60.145894138632280 0.223020351676054 2.000000000000000 0.078921332097133
x70.145894138632280 0.323185143014029 2.000000000000000 0.499580490394335
x8−0.007832029555062 0.327973609353822 1.915762141824520 0.197756675883883
x90.343654620394334 0.333430854648433 2.000000000000000 0.228228833675487
x10 0.425902664080806 0.324142888370713 2.000000000000000 0.470195948900759
f
1.25544451911646
×
10
−37.79220329211044 ×10−2
7.96261500819178
×
10
−2
6.61705221934444
×
10
−2
Variable Algorithms
SMA nAOA dAOA
x10.249900132290417 0.035430633051580 1.840704485033870
x20.375428314977531 0.053983062784772 1.213421005935260
x30.272448580296318 0.072735305166021 1.203555993641700
x40.199698265955405 0.021399042985613 −0.393935624266822
x50.425934189445810 0.064655913970964 −0.249476549706985
x60.057699959645613 0.012570281350831 0.459915310960444
x70.431865275874618 0.057639809639213 −0.675754718182326
x80.015005640000641 0.005520004765830 −0.895856414267328
x90.347986992756388 0.041229484511092 0.359139808282465
x10 0.415304164782275 0.079595719921909 1.529188120361250
f
4.47411205566240
×
10
−36.74563715208325 ×10−11.91503507134915
Table 2. Comparison of the experimental results for problem02.
Variable Algorithms
AOA IAOA SCA WOA
x10.040781958181860 0.042124781715274 0.000000000000000 0.041561373108785
x20.268625655728691 0.061754610138946 0.266593748985495 0.268697327813652
f
2.01752031872803
×
10
−79.24446373305873 ×10−34 8.82826387279195 ×10−5
6.92247231102962
×
10
−9
Variable Algorithms
GWO HHO DE CSO
x10.265622854930434 0.267855297066815 0.266589101862370 0.266620164671422
x20.178718146817611 0.458749279058429 0.327275026016101 0.178514261126008
f
1.13985864694418
×
10
−76.55986405733090 ×10−81.31654979128584 ×10−18
1.49504500886345
×
10
−9
Variable Algorithms
SMA nAOA dAOA
x10.021419624272050 0.000000000000000 0.236558250181286
x20.048075232460874 0.719124811309122 0.508933311549167
f
2.89316821274146
×
10
−53.07109081317222 ×10−5
3.22387407689191
×
10
−4
Mathematics 2022,10, 2152 12 of 27
Table 3. Comparison of the experimental results for problem03.
Variable Algorithms
AOA IAOA SCA WOA
x11.990744078311880 −0.947268146986263 −0.225974226141413 −1.424482905343090
x20.220001522814532 −0.785020015568289 1.245763361231140 −0.543544840817441
f
5.61739095968327
×
10
−34.02151576372412 ×10−32 7.95691890654021 ×10−4
1.06331568826728
×
10
−3
Variable Algorithms
GWO HHO DE CSO
x1−1.794053112053940 −1.495480498807310 −1.791308474954350 −0.212779003619775
x2−0.303905803005920 −0.420394691864127 0.301889327351144 −1.257141525856050
f
2.77808608355359
×
10
−56.12298193031725 ×10−51.84881969881973 ×10−9
6.26348225916795
×
10
−7
Variable Algorithms
SMA nAOA dAOA
x1−1.791387180972800 −1.475077261850100 −1.580085715978880
x2−0.302157020359872 −0.454673564762598 0.4651484d76848022
f
5.47910691165820
×
10
−82.17709293383390 ×10−4
5.12705019470938
×
10
−2
Table 4. Comparison of the experimental results for problem04.
Variable Algorithms
AOA IAOA SCA WOA
x1−0.000266868453558 −0.000000091835793 −0.120898772911816 −0.310246574315981
x2−0.000267036157051 0.000013971597535 0.491167568359585 0.467564824328878
x3−0.000267036274281 0.000030454051416 10.000000000000000 1.071469773086650
x40.000000025430197 0.000010000404353 −0.178108600809833 −0.404219784214681
x5−0.000267039311495 0.000011275918099 5.423242568753400 3.552125620609660
x6−0.000267036127224 0.000000019800029 −0.049710980654501 −1.834136698070800
x70.000000000091855 −0.000000000138437 0.445662462511328 0.286050311387620
x80.000267036101457 −0.000000454282127 −10.000000000000000 −2.931846497771810
x90.000267033832224 0.000000000736505 −0.144419405019169 −4.812450845354100
x10 0.000267043884482 −0.000002006069864 −0.518105971932846 3.756426716000660
f
1.08498006397337
×
10
−97.03339003909689 ×10−16
4.13237426374674
×
10
−1
6.47066501369328
×
10
−1
Variable Algorithms
GWO HHO DE CSO
x10.044653752694561 −0.000047703379713 0.160723693838569 −0.009650846541198
x2−0.259567674882923 0.000075691075249 0.431923139718368 0.147278561202585
x3−1.777013199398760 −0.000029713372367 0.072922517980119 −3.148557575646470
x40.042606334458592 −0.000050184914825 0.447403957744849 −0.512428980703464
x5−4.935286036663600 0.000033675529531 −0.197972459731190 −4.175819684412100
x6−8.146156623785810 0.000067989452634 1.490110445009050 −7.123183974281880
x7−0.108125274969201 0.000031288762826 0.472265426079125 1.268663892956760
x81.747052457418910 0.000048491290536 0.509493705510866 3.198230908839320
x9−0.311997778279745 0.000063892452193 1.142101578993260 −4.763105818868310
x10 8.430357427064680 −0.000123055431652 −2.110335475212350 9.463108408596410
f
7.56734706927375
×
10
−36.11971561041781 ×10−10
9.87501536049260
×
10
−12.18295386757873
Mathematics 2022,10, 2152 13 of 27
Table 4. Cont.
Variable Algorithms
SMA nAOA dAOA
x1−0.000000000028677 0.000020144848903 −0.934997016811202
x20.000014644312649 −0.000060200695401 −1.295640443505010
x30.000038790339140 −0.000020118018817 −5.634966911723890
x4−0.000000000221797 −0.000060200956330 −4.825343892476190
x50.000000055701981 −0.000020122803817 0.269511140973028
x6−0.000000030051237 −0.000020134693956 −7.253398121182340
x70.000000595936232 0.000020123341500 7.557747336452660
x8−0.000000000025333 0.000020925519435 −5.520361069927860
x90.000000799504725 0.000043615727680 −4.709534880735350
x10 0.000000000012983 0.000020120622373 8.954470788407880
f1.30095438660555 ×10−10 1.50696700666871 ×10−92.07190542503982 ×102
Table 5. Comparison of the experimental results for problem05.
Variable Algorithms
AOA IAOA SCA WOA
x10.371964486871792 0.500000000000000 0.471178994397267 0.503978268408352
x22.990337880814430 3.141592653589790 3.118271172186020 3.142976305563530
f1.89048835343036 ×10−41.85873810048745 ×10−28 3.41504906318340 ×10−52.00099014478417 ×10−7
Variable Algorithms
GWO HHO DE CSO
x10.495722089382004 0.503332577729795 0.299448692445072 0.500482294032500
x23.143566564341090 3.142753305279310 2.836927770362990 3.142098043614560
f1.12835512797232 ×10−61.16071617155615 ×10−76.25300383824133 ×10−23 2.13609775136897 ×10−8
Variable Algorithms
SMA nAOA dAOA
x10.298949061647857 0.354640044143990 2.956994389007600
x22.835691250750600 2.956994389007600 1.890717921128260
f1.05189651760469 ×10−81.59376404093113 ×10−43.65946616757579 ×10−3
Table 6. Comparison of the experimental results for problem06.
Variable Algorithms
AOA IAOA SCA WOA
x10.953663829653960 −0.779548045079158 11.147659127176500 1.516510183032980
x20.663112382731748 −0.779548045079158 0.900762400732728 0.694394649388567
x30.729782844271910 −0.779548045079158 0.919816117314499 10.556407054559600
f3.35330112498813 ×10−11.00553388370096 ×10−20 2.75666643131973 8.65817545834561
Variable Algorithms
GWO HHO DE CSO
x10.781303537791760 −0.782460718139219 −0.779277448448367 −0.765447632695953
x20.777872878718449 −0.789339702437282 −0.779700789186745 −0.784775197498564
x30.779780469890485 −0.766810453292313 −0.780020611467694 −0.735052686517780
f5.49159538279891 ×10−41.00882211687459 ×10−26.71295836563811 ×10−62.92512803990831 ×10−1
Variable Algorithms
SMA nAOA dAOA
x1−0.779731780102931 −0.437772635064718 −1.056395480177350
x2−0.779371556451744 −7.659741643877890 6.893981344148980
x3−0.779303513685515 −2.620897335617900 −1.876924860155790
f1.03517116885362 ×10−51.49720612584788 2.61017698945353 ×104
Mathematics 2022,10, 2152 14 of 27
Table 7. Statistical results for the NES.
Algorithms Systems of Nonlinear Equations
problem01 problem02 problem03 problem04 problem05 problem06
AOA best 7.02711 ×10−11.20198 ×10−88.30574 ×10−12 2.99534 ×10−10 5.32587 ×10−61.60969 ×10−8
worst 9.05980 ×10−17.47231 ×10−79.55457 ×10−33.58264 ×10−95.96026 ×10−41.00599 ×10
mean 8.45666 ×10−12.01752 ×10−73.18486 ×10−41.08498 ×10−91.89049 ×10−43.35330 ×10−1
std 4.40686 ×10−21.78065 ×10−71.74442 ×10−38.49280 ×10−10 1.40374 ×10−41.83668
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
IAOA best 1.05462 ×10−10 0.00000 4.93038 ×10−32 2.97972 ×10−19 0.00000 1.81191 ×10−30
worst 1.25230 ×10−93.08149 ×10−33 2.09541 ×10−31 5.52546 ×10−15 5.57614 ×10−27 2.98754 ×10−19
mean 4.73406 ×10−10 9.24446 ×10−34 7.27231 ×10−32 7.03339 ×10−16 1.85874 ×10−28 1.00553 ×10−20
std 2.84371 ×10−10 1.43626 ×10−33 4.02152 ×10−32 1.22291 ×10−15 1.01806 ×10−27 5.45273 ×10−20
SCA best 4.64629 ×10−21.20156 ×10−88.29788 ×10−67.08592 ×10−47.53679 ×10−91.19890 ×10−1
worst 2.98744 ×10−18.60445 ×10−43.13588 ×10−32.83503 2.00649 ×10−43.29896 ×10
mean 1.22078 ×10−18.82826 ×10−55.47683 ×10−44.13237 ×10−13.41505 ×10−52.75667
std 5.72692 ×10−22.61875 ×10−47.59630 ×10−46.58494 ×10−14.69615 ×10−56.25475
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
WOA best 1.87873 ×10−46.72146 ×10−14 6.18945 ×10−13 4.04945 ×10−62.16928 ×10−11 1.76476 ×10−5
worst 5.56233 ×10−31.30541 ×10−74.48907 ×10−24.99725 4.78904 ×10−67.91148 ×10
mean 9.59545 ×10−46.92247 ×10−94.26773 ×10−36.47067 ×10−12.00099 ×10−78.65818
std 1.06419 ×10−32.49080 ×10−81.24385 ×10−21.07197 8.71177 ×10−72.24136 ×10
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
GWO best 2.65480 ×10−62.31886 ×10−12 1.77817 ×10−81.01688 ×10−62.21126 ×10−99.05730 ×10−5
worst 6.59898 ×10−31.73256 ×10−69.94266 ×10−25.57604 ×10−21.70979 ×10−51.58625 ×10−3
mean 1.25544 ×10−31.13986 ×10−73.33932 ×10−37.56735 ×10−31.12836 ×10−65.49160 ×10−4
std 2.25868 ×10−34.16137 ×10−71.81481 ×10−21.36923 ×10−23.33417 ×10−63.69947 ×10−4
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
HHO best 2.03768 ×10−28.99794 ×10−31 4.93038 ×10−32 1.21192 ×10−11 7.70372 ×10−34 3.83242 ×10−5
worst 1.33302 ×10−11.91904 ×10−65.78702 ×10−41.00491 ×10−93.34700 ×10−67.08247 ×10−2
mean 7.79220 ×10−26.55986 ×10−84.12782 ×10−56.11972 ×10−10 1.16072 ×10−71.00882 ×10−2
std 2.90524 ×10−23.50117 ×10−71.19896 ×10−42.78236 ×10−10 6.10656 ×10−71.45023 ×10−2
p-value 3.01986 ×10−11 1.01490 ×10−11 5.56066 ×10−83.01986 ×10−11 1.30542 ×10−10 3.01230 ×10−11
DE best 6.05782 ×10−38.15969 ×10−28 2.49399 ×10−20 2.59514 ×10−12.59615 ×10−31 4.23182 ×10−11
worst 9.69921 ×10−11.19322 ×10−17 5.91181 ×10−72.58615 6.37964 ×10−22 1.17012 ×10−4
mean 7.96262 ×10−21.31655 ×10−18 3.33313 ×10−89.87502 ×10−16.25300 ×10−23 6.71296 ×10−6
std 2.40157 ×10−12.91169 ×10−18 1.26981 ×10−76.21653 ×10−11.66035 ×10−22 2.15862 ×10−5
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 6.22236 ×10−11 3.01230 ×10−11
CSO best 2.82411 ×10−27.30711 ×10−11 2.92752 ×10−96.03864 ×10−12.67109 ×10−10 2.27267 ×10−2
worst 1.34962 ×10−17.15408 ×10−92.57784 ×10−64.34942 1.32416 ×10−71.31894
mean 6.61705 ×10−21.49505 ×10−96.53698 ×10−72.18295 2.13610 ×10−82.92513 ×10−1
std 2.71383 ×10−21.66707 ×10−95.69101 ×10−71.05318 3.36401 ×10−83.41112 ×10−1
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
SMA best 5.18988 ×10−41.26496 ×10−72.37253 ×10−11 2.08208 ×10−11 6.22359 ×10−11 3.95601 ×10−7
worst 1.17331 ×10−22.46549 ×10−45.80093 ×10−72.89907 ×10−10 5.94920 ×10−84.75099 ×10−5
mean 4.47411 ×10−32.89317 ×10−55.98652 ×10−81.30095 ×10−10 1.05190 ×10−81.03517 ×10−5
std 3.00476 ×10−35.64857 ×10−51.28713 ×10−77.25135 ×10−11 1.30068 ×10−81.04158 ×10−5
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
nAOA best 4.73537 ×10−11.16733 ×10−93.11364 ×10−12 3.28064 ×10−10 2.13953 ×10−57.56334 ×10−8
worst 7.39125 ×10−19.06936 ×10−48.22290 ×10−12.69391 ×10−94.30978 ×10−44.49162 ×10
mean 6.74564 ×10−13.07109 ×10−52.77064 ×10−21.50697 ×10−91.59376 ×10−41.49721
std 5.68300 ×10−21.65502 ×10−41.50077 ×10−16.31248 ×10−10 7.06193 ×10−58.20053
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
dAOA best 2.01052 ×10−18.99368 ×10−92.54429 ×10−43.09426 ×10−10 5.69606 ×10−68.50407 ×10−4
worst 6.87872 1.28121 ×10−34.68145 ×10−19.87499 ×1021.56431 ×10−23.78263 ×105
mean 1.91504 3.22387 ×10−46.56368 ×10−22.07191 ×1023.65947 ×10−32.61018 ×104
std 2.16147 3.20053 ×10−41.21675 ×10−12.92259 ×1025.26309 ×10−38.07193 ×104
p-value 3.01986 ×10−11 1.01490 ×10−11 1.07516 ×10−11 3.01986 ×10−11 1.49399 ×10−11 3.01230 ×10−11
Mathematics 2022,10, 2152 15 of 27
Problem 01. The description of the system is as follows [54]:
x1−0.25428722 −0.18324757x4x3x9=0
x2−0.37842197 −0.16275449x1x10 x6=0
x3−0.27162577 −0.16955071x1x2x10 =0
x4−0.19807914 −0.15585316x7x1x6=0
x5−0.44166728 −0.19950920x7x6x3=0
x6−0.14654113 −0.18922793x8x5x10 =0
x7−0.42937161 −0.21180486x2x5x8=0
x8−0.07056438 −0.17081208x1x7x6=0
x9−0.34504906 −0.19612740x10 x6x8=0
x10 −0.42651102 −0.21466544x4x8x1=0
(19)
There are ten equations in the system, where
xi∈[−
2, 2
]
,i= 1,
. . .
,n, and n= 10.
The aim was to obtain a higher precision solution x(x
1
,
. . .
,x
n
) through the proposed
optimization method, and the results are recorded in Table 1. The IAOA is better than
others compared with several algorithms. The WOA ranks second, and the rest obtain
competitive results. The convergence curve for this problem shows in Figure 4a.
Figure 4. Cont.
Mathematics 2022,10, 2152 16 of 27
Figure 4. Convergence curve for tackling the NES (problem01–06 (a–f)).
Problem 02. The description of the system is as follows [55]:
(1−R)D
10(1+β1)−x1·exp10x1
1+10x1
γ−x1=0
(1−R)D
10 −β1x1−(1+β2)x2·exp10x2
1+10x2
γ+x1−(1+β2)x2=0
(20)
There are two equations in system, where
xi∈[
0, 1
]
,i= 1,
. . .
,n, and n= 2. In Table 2,
the experimental results for this problem proved that the proposed IAOA outperforms the
other methods. The DE ranks second, and the rest obtain competitive results. The AOA,
WOA, GWO, HHO, and CSO are in the third echelon. Furthermore, the rest are in the
fourth echelon. The convergence curve for this problem is shown in Figure 4b.
Problem 03. The description of the system is as follows [13]:
sinx3
1−3x1x2
2−1=0
cos3x2
1x2−x3
2+1=0
(21)
There are two equations in the system, where
xi∈[−
2, 2
]
,i= 1,
. . .
,n, and n= 2. The
simulation results for this problem are shown in Table 3. It revealed that the IAOA is better
than the other algorithms. The DE, CSO, and SMA are in the second echelon. The rest are
in the third echelon. The convergence curve for this problem is shown in Figure 4c.
Problem 04. The description of the system is as follows [54]:
x2+2x6+x9+2x10 −10−5=0
x3+x8−3·10−5=0
x1+x3+2x5+2x8+x9+x10 −5·10−5=0
x4+2x7−10−5=0
0.5140437 ·10−7x5−x2
1=0
0.1006932 ·10−6x6−2x2
2=0
0.7816278 ·10−15x7−x2
4=0
0.1496236 ·10−6x8−x1x3=0
0.6194411 ·10−7x9−x1x2=0
0.2089296 ·10−14x10 −x1x2
2=0
(22)
There are ten equations in the system:
xi∈[−
10, 10
]
,i= 1,
. . .
,n, and n= 10. Table 4
shows that the IAOA outperforms the others, and AOA, HHO, SMA, and nAOA obtain
the competitive results. The convergence curve for this problem is shown in Figure 4d.
Mathematics 2022,10, 2152 17 of 27
Problem 05. The description of the system is as follows [17]:
0.5 sin(x1x2)−0.25
πx2−0.5x1=0
1−0.25
π[exp(2x1)−e]+e
πx2−2ex1=0
(23)
There are two equations in the system, where
x1∈[
0.25, 1
]
and
x2∈[
1.5, 2
π]
. In
Table 5, the IAOA obtained the optimal solution, DE obtained the suboptimal solution,
and the rest of the algorithms obtained competitive results. The convergence curve for this
problem is shown in Figure 4e.
Problem 06. The description of the system is as follows [56]:
β11 +β12x2
2+β13x2
3+β14x2x3+β15x2
2x2
3=0
β21 +β22x2
3+β23x2
1+β24x3x1+β25x2
3x2
1=0
β31 +β32x2
1+β33x2
2+β34x1x2+β35x2
1x2
2=0
(24)
There are three equations in the system, where the details about
βij
can be found in
the literature [
56
]:
xi∈[−
20, 20
]
,i= 1,
. . .
,n, and n= 3. In Table 6, the proposed IAOA
outperforms the other algorithms; the GWO, SMA, and DE get competitive results. The
convergence curve for this problem is shown in Figure 4f.
The statistical results show that the IAOA outperforms all algorithms on the remaining
problems in Table 7. These demonstrate that the IAOA has stronger ability and higher
stability than the other methods when solving a nonlinear system of equations. In Figure 4,
IAOA’s convergence speed is slower than the others before the 110th iteration, but after
that, the IAOA still maintains a high convergence speed and achieves the optimum at the
200th iteration for problem01; for problem02 and problem03, the IAOA has the fastest speed
throughout the whole process and reaches the optimum at the 120th iteration and before
120 iterations, respectively; for problem04, the IAOA is slower than the other algorithms
before 70 iterations; however it continues to converge after that and obtains the optimal
value after 200 iterations; for problem05, there is a close convergence rate for the IAOA and
DE, but a better value is obtained by the IAOA; for problem06, it has a slower convergence
speed than the others before 20 iterations, but after that, the fastest convergence rate is
obtained by the IAOA. All the experimental results prove that the algorithm proposed in
this paper has the characteristics that include a fast convergence speed, high convergence
accuracy, high solution quality, good stability, and strong robustness when dealing with
nonlinear systems of equations. The p-values of almost all test functions in the table are
less than 0.05, indicating that the IAOA is significantly different from the other algorithms.
4.3. Numerical Integration
The performance of the proposed new method is evaluated in this section using
the ten numerical integration problems in Table 8, where F08 is a singular integral and
F10 is an oscillatory integral. The IAOA compared with the traditional methods and
population-based algorithms in tackling these cases. Tables 9–12 show the best integral
values obtained by solving ten problems in 30 independent runs, where the R-method,
T-method, S-method, H-method, G32, and 2n
×
L5 represent the traditional methods
(rectangle method, trapezoid method, Simpson method, Hermite interpolation method,
the 32-point Gaussian formula, and the 5-point Gauss-Roberto-Legendre formula). The
rest are swarm intelligence algorithms applied to solve numerical integration problems
(evolutionary strategy method [
24
], particle swarm optimization [
25
], differential evolution
algorithm [
27
], and improved bat algorithm [
28
]). The population size and the maximum
number of iterations are set to 30 and 200 during the process, respectively. In Table 9, for
F01, the solution accuracy of the IAOA is higher than the other methods, and then, the
S-method, FN, ES, DEBA, PSO, and DE obtain close results; for F02, the IAOA achieves
the best result, and the FN, ES, DEBA, PSO, and DE are in the second echelon; for F03, the
Mathematics 2022,10, 2152 18 of 27
IAOA achieves the better result compared to the FN, ES, and PSO. The MBFES, DEBA, and
DE rank third. In Table 10, for F04, the IAOA gets a perfect result, and the FN, ES, DEBA,
PSO, and DE obtain similar values; for F05, the IAOA ranks first, and the FN, ES, DEBA,
PSO, and DE rank second; for F06, the IAOA, FN, and DE achieve competitive results.
For F07–F09, the IAOA obtains the best value, and the FN, ES, and DEBA rank second
in Table 11. The traditional methods (R-method, T-method, and S-method) fail to solve
F10; therefore, G32 and 2n
×
L5 are utilized to tackle this problem. In Table 12, the IAOA
and DEBA obtain similar values and ranks first. Tables 13 and 14 are statistical results for
the numerical integration (F01–F10) are obtained by swarm intelligence algorithms. For
F01–F09, the IAOA is better than the other algorithms across all the assessment criteria
(the best value, the worst value, mean value, and standard deviation). However, for F10,
the IAOA achieves the only optimal result in the best value, and the rest rank second, in
which the DEBA obtains the best results. From Figure 5, the method proposed in this paper
has the fastest convergence speed and convergence accuracy for all the problems except
F10. The above experimental results prove that the IAOA has fast convergence speed, high
solution accuracy, and strong robustness. These enable the IAOA to handle numerical
integration problems; therefore, it is a worthwhile direction to apply the IAOA to solve the
integration solution problems in practical engineering applications.
Table 8. Details of the integrations F01–F10.
Integrations Details Range
F01 f(x) = x2[0, 2]
F02 f(x) = x4[0, 2]
F03 f(x) = √1+x2[0, 2]
F04 f(x) = 1
1+x[0, 2]
F05 f(x) = sin x[0, 2]
F06 f(x) = ex[0, 2]
F07 f(x) = q1+ (cos x)2[0, 48]
F08 f(x) =
e−x, 0 ≤x<1
e−x/2, 1 ≤x<2
e−x/3, 2 ≤x≤3
[0, 3]
F09 f(x) = e−x2[0, 1]
F10 f(x) = xcos xsin xmx,(m=10, 20, 30)[0, 2π]
Table 9. Comparison of the experimental results for F01–F03.
Methods Integrations
F01 F02 F03
R-method 2.000 2.000 2.828
T-method 4.000 16.000 3.236
S-method 2.667 6.667 2.964
H-method 2.830 7.066 3.048
FN [26] 2.667 6.3995 2.95789
MBFES [24] 2.659 6.338 2.956
ES [24] 2.666 6.398 2.9577
DEBA [28] 2.66698573 6.401201 2.958169
PSO [25] 2.666 6.398 2.9578
DE [27] 2.667 6.3995 2.958
AOA 2.61006134 6.20147125 2.94004382
IAOA 2.66661710 6.40000000 2.95788286
Exact 2.66666667 6.40000000 2.95788572
Mathematics 2022,10, 2152 19 of 27
Table 10. Comparison of the experimental results for F04–F06.
Methods Integrations
F04 F05 F06
R-method 1.000 1.683 5.437
T-method 1.333 0.909 8.389
S-method 1.111 1.425 6.421
H-method 1.112 1.452 6.691
FN [26] 1.0986 1.416 6.389
MBFES [24] 1.090 1.419 6.390
ES [24] 1.098 1.416 6.388
DEBA [28] 1.098754 1.416082 6.388921
PSO [25] 1.0985 1.416 6.3887
DE [27] 1.099 1.416 6.389
AOA 1.08923818 1.40101546 6.29531692
IAOA 1.09861229 1.41613957 6.38901606
Exact 1.09861229 1.41614684 6.38905610
Table 11. Comparison of the experimental results for F07–F09.
Methods Integrations
F07 F08 F09
R-method 52.13975183 1.51349542 0.77782078
T-method 62.43737140 1.61179305 0.74621972
S-method 117.61490334 2.48720505 0.74683657
H-method 58.99776108 1.56164258 0.75403569
FN [26] 58.4705 1.54604 0.746823
MBFES [24] 58.48828 1.5455 0.74652
ES [24] 58.47065 1.5459805 0.74683
DEBA [28] 58.470505372351 1.5460388345767 0.7468269544604
PSO 56.80139775 1.52897330 0.74328459
DE 56.04598085 1.52425900 0.74202909
AOA 56.17497970 1.52641514 0.74223182
IAOA 58.47046915 1.54603603 0.74682413
Exact 58.47046915 1.54603603 0.74682413
Table 12. Comparison of the experimental results for F10.
Methods Integrations
F10 (m = 10) F10 (m = 20) F10 (m = 30)
G32 −0.6340207 −1.2092524 −1.5822272
2n ×L5 −0.55875940 −0.27789620 −0.18508448
H-method −0.21043575 0.17309499 −0.02945756
MBFES [24]−0.68134052 −0.37280425 −0.17305621
ES [24]−0.65034080 −0.30583435 −0.23556815
DEBA −0.63466518 −0.31494663 −0.20967248
PSO −1.50150183 −1.33949737 −1.10170197
DE [27]−0.63982173 −0.31035906 −0.21438251
AOA −3.07253909 −0.56489050 −0.42642997
IAOA −0.63466518 −0.31494663 −0.20967248
Exact −0.63466518 −0.31494663 −0.20967248
Mathematics 2022,10, 2152 20 of 27
Table 13. Statistical results for the numerical integrations (F01–F06).
Algorithms Integrations
F01 F02 F03 F04 F05 F06
AOA best 5.660532 ×10−21.985287 ×10−11.784189 ×10−29.374106 ×10−31.513137 ×10−29.373918 ×10−2
worst 6.785842 ×10−22.466178 ×10−12.112411 ×10−21.103594 ×10−21.827849 ×10−21.105054 ×10−1
mean 6.196485 ×10−22.238141 ×10−11.970905 ×10−21.041648 ×10−21.679104 ×10−21.013200 ×10−1
std 2.473863 ×10−31.277362 ×10−26.790772 ×10−44.381854 ×10−47.886715 ×10−43.985235 ×10−3
IAOA best 4.956295 ×10−50.000000 2.855397 ×10−60.000000 7.267277 ×10−64.004088 ×10−5
worst 1.070986 ×10−49.632589 ×10−61.471988 ×10−57.241931 ×10−63.035345 ×10−51.136393 ×10−4
mean 7.267766 ×10−59.617999 ×10−76.357033 ×10−61.274560 ×10−61.595556 ×10−57.989662 ×10−5
std 1.561025 ×10−52.672207 ×10−62.828416 ×10−61.942626 ×10−65.989208 ×10−62.032255 ×10−5
PSO [25] best 3.966996 ×10−21.282142 ×10−11.263049 ×10−26.772669 ×10−31.115352 ×10−26.495427 ×10−2
worst 5.467546 ×10−21.880821 ×10−11.614274 ×10−29.112184 ×10−31.385859 ×10−29.718717 ×10−2
mean 4.406724 ×10−21.593799 ×10−11.405265 ×10−27.745239 ×10−31.208230 ×10−27.327404 ×10−2
std 3.262431 ×10−31.528260 ×10−29.707823 ×10−46.532329 ×10−47.146743 ×10−46.698801 ×10−3
DE [27] best 5.444535 ×10−21.776272 ×10−11.740389 ×10−29.410606 ×10−31.537737 ×10−29.229490 ×10−2
worst 6.223208 ×10−21.992612 ×10−11.943564 ×10−21.043440 ×10−21.668422 ×10−21.003285 ×10−1
mean 5.887766 ×10−21.887098 ×10−11.881844 ×10−21.003350 ×10−21.606658 ×10−29.665791 ×10−2
std 1.717478 ×10−35.056921 ×10−34.230737 ×10−42.412656 ×10−43.636407 ×10−41.886442 ×10−3
DEBA [28] best 5.858312 ×10−21.958779 ×10−11.797733 ×10−29.632554 ×10−31.541447 ×10−29.078063 ×10−2
worst 6.805128 ×10−22.566962 ×10−12.194973 ×10−21.144459 ×10−21.824156 ×10−21.096576 ×10−1
mean 6.306158 ×10−22.287206 ×10−12.005007 ×10−21.048558 ×10−21.700868 ×10−21.008133 ×10−1
std 2.059708 ×10−31.384008 ×10−28.428458 ×10−44.319549 ×10−47.193521 ×10−44.457879 ×10−3
ES [24] best 3.634854 ×10−21.053634 ×10−11.178783 ×10−26.152581 ×10−39.742411 ×10−36.028495 ×10−2
worst 3.704455 ×10−21.076016 ×10−11.197536 ×10−26.272540 ×10−39.921388 ×10−36.120127 ×10−2
mean 3.662145 ×10−21.064150 ×10−11.189432 ×10−26.206519 ×10−39.813727 ×10−36.070549 ×10−2
std 1.618502 ×10−44.726931 ×10−44.687831 ×10−52.718416 ×10−54.560503 ×10−52.303572 ×10−4
Table 14. Statistical results for numerical integrations (F07–F10).
Algorithms Integrations
F07 F08 F09 F10 (m = 10) F10 (m = 20) F10 (m = 30)
AOA best 2.295489 1.962088 ×10−24.592313 ×10−32.437873 2.499438 ×10−12.167574 ×10−1
worst 2.524012 2.400262 ×10−25.421672 ×10−33.611012 3.429053 3.115022
mean 2.424997 2.226327 ×10−25.031127 ×10−33.225836 1.617425 9.721188 ×10−1
std 5.634089 ×10−21.017542 ×10−32.167135 ×10−42.620454 ×10−19.081448 ×10−17.417795 ×10−1
IAOA best 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
worst 4.285648 ×10−49.665730 ×10−67.650313 ×10−94.941453 ×10−48.932970 ×10−44.121824 ×10−4
mean 5.817808 ×10−51.079836 ×10−61.094646 ×10−96.843408 ×10−59.159354 ×10−56.487479 ×10−5
std 9.331558 ×10−52.377176 ×10−62.051844 ×10−91.219906 ×10−41.972260 ×10−49.370544 ×10−5
PSO [25] best 1.093717 1.499542 ×10−23.212480 ×10−35.688245 ×10−11.024550 8.920294 ×10−1
worst 2.077297 2.010782 ×10−24.674802 ×10−31.599995 1.485451 1.953066
mean 1.669071 1.706272 ×10−23.539538 ×10−38.668366 ×10−11.219538 1.489201
std 2.419795 ×10−11.205259 ×10−33.409595 ×10−42.759571 ×10−11.216184 ×10−12.065585 ×10−1
DE [27] best 2.255785 2.091958 ×10−24.575317 ×10−32.543013 3.461794 3.889322
worst 2.522405 2.254710 ×10−25.009106 ×10−33.236645 4.684467 5.201887
mean 2.424488 2.177702 ×10−24.795040 ×10−33.015091 4.242609 4.687029
std 5.766110 ×10−24.602533 ×10−41.146454 ×10−41.967397 ×10−12.313007 ×10−12.923496 ×10−1
DEBA [28] best 2.361570 ×10−12.057410 ×10−24.776881 ×10−36.043389 ×10−14 1.208677 ×10−13 5.319404 ×10−13
worst 2.468831 2.474051 ×10−25.441200 ×10−36.043389 ×10−14 1.208677 ×10−13 5.319404 ×10−13
mean 1.163514 2.294436 ×10−25.157892 ×10−36.043389 ×10−14 1.208677 ×10−13 5.319404 ×10−13
std 6.919695 ×10−19.765442 ×10−41.475304 ×10−43.851264 ×10−29 7.702528 ×10−29 3.081011 ×10−28
ES [24] best 1.298269 1.319474 ×10−23.051746 ×10−31.460773 1.634373 1.152204
worst 1.321623 1.341748 ×10−23.121709 ×10−31.665912 2.355153 2.380726
mean 1.308546 1.331615 ×10−23.081151 ×10−31.568781 1.869004 1.719830
std 5.523404 ×10−35.640941 ×10−51.521690 ×10−54.627499 ×10−21.831224 ×10−12.898513 ×10−1
Mathematics 2022,10, 2152 21 of 27
Figure 5. Cont.
Mathematics 2022,10, 2152 22 of 27
Figure 5. Convergence curve for the numerical integrations (F01–F10 (a–l)).
4.4. Sovling Engineering Problem
Compared with three-dimensional motion, planar motion restricts the robot to a single
plane and is simpler to calculate. However, most robot mechanisms can simplify plane
mechanisms or planes for tackling. Now, the robotic arm plays an increasingly important
role, which has also attracted the extensive attention of researchers. Improving the working
efficiency of the robotic arm under the premise of low energy consumption is a challenging
problem facing the industrial field [
57
]. The kinematics of the robotic arm mainly include
Mathematics 2022,10, 2152 23 of 27
forward kinematics and inverse kinematics. One is the pose of the end effector determined
according to the rotation angle of each joint based on the base coordinates; the other is
taking the end joint as the starting point and, finally, back-to-base coordinates. The inverse
kinematics problem is essentially a nonlinear equation problem. The tasks performed by
the robotic arm are usually described by its base coordinate system in practical applications.
Therefore, the inverse kinematics solution is particularly important in the field of the
control. The robotic arm model [
58
] is shown in Figure 6a, and the mathematical model
in coordinates is shown in Figure 6b. The nonlinear equation system for this model is
as follows.
10, 000 ×((a×sin(A2)−b×sin(A2+B2) + c×sin(A2+B2+C2)−X)2) = 0
10, 000 ×((h−a×cos(A2)−b×cos(A2+B2) + c×cos(A2+B2+C2)−Y)2) = 0
|A2−A1|+|B2−B1|+|C2−C1|=0
(25)
where a= 16.5 cm; b= 7.9 cm; c= 5.3 cm; and h= 7.4 cm (
A1
= 150
◦
,
B1
= 132.7026
◦
, and
C1
= 127.0177
◦
) are the initial angles of the three joints; (X= 10 cm, Y= 10 cm) is the
coordinate of the end effector; and (
A2
,
B2
, and
C2
) are the aims required to obtain three
joint angles in the final stage. The first two equations in the nonlinear equation system find
the three joint angles when the end effector reaches the target position (X,Y), and the third
equation ensures that the change of the joint angle is the smallest to meet the requirements
for saving energy.
Figure 6. (a) The model of a robotic arm, and (b) a mathematical model for a robotic arm.
Tables 15–18 demonstrate that the IAOA obtains the closest results to the initial angle
compared with the PSO, GA and PSSA in solving the inverse kinematics problem of the
robotic arm. This shows that the method proposed in this paper allows the robotic arm to
consume less energy during movement. In Table 19,frepresents the fitness value obtain
by Equation (25) and is the difference between the final angle and initial angle of the
joint. Obviously, the IAOA achieves the best results for both evaluations. Therefore, it is
a great significance to the stability, operation efficiency, operation accuracy, and energy
consumption of the robotic arm trajectory control. A new method is provided for the
inverse motion solution, which makes up for the deficiency of the traditional method.
Mathematics 2022,10, 2152 24 of 27
Table 15. The results obtained by the IAOA for the engineering problem.
Algorithm Joint Angles
A2B2C2
IAOA initial angle 150 132.7026 127.0177
Result 145.7291 139.0180 123.9864
Table 16. The results obtained by the PSO for the engineering problem.
Algorithm Joint Angles
A2B2C2
PSO initial angle 150 132.7026 127.0177
result 139.6534 68.2235 96.4886
Table 17. The results obtained by the GA for the engineering problem.
Algorithm Joint Angles
A2B2C2
GA initial angle 150 132.7026 127.0177
result 129.8653 118.9625 52.6691
Table 18. The results obtained by the PSSA for the engineering problem.
Algorithm Joint Angles
A2B2C2
PSSA [58] initial angle 150 132.7026 127.0177
result 147.1015 92.5371 89.5116
Table 19. Comparison of the experimental results for the IAOA, PSO, GA, and PSSA.
Objective Funtions Algorithms
IAOA PSO GA PSSA
f
1.3618
×
10
3.0608 ×1063.2329 ×1062.0199 ×105
|A2−A1|+|B2−B1|+|C2−C1|13.6176 105.3548 118.2234 80.5701
5. Conclusions and Future Works
In this paper, the shortcomings are analyzed of the traditional AOA so that an im-
proved AOA based on a population control strategy is proposed to overcome the weakness.
The algorithm can find the best global value faster by classifying the population and adap-
tively controlling the number of individuals in each subpopulation. This method effectively
enhances the information sharing strength between individuals, can better search the space,
avoids falling into the local optimum, accelerates the convergence process, and improves
the optimization accuracy. The AOA, IAOA, and some other algorithms are compared
based on solving 6 nonlinear systems of equations, 10 numerical integrations, and an engi-
neering problem. The experimental results show that the IAOA can solve these problems
well and outperform the other algorithms. In the future, the IAOA can be used to solve
more nonlinear problems in practical engineering applications. Secondly, it can try to
solve multiple integrals. Finally, the algorithm can be further improved and enhanced in
its performance.
Author Contributions:
Conceptualization and methodology, M.C. and Y.Z.; software, M.C.; writing—
original draft preparation, M.C.; writing—review and editing, Y.Z. and Q.L.; and funding acquisition,
Y.Z. All authors have read and agreed to the published version of the manuscript.
Mathematics 2022,10, 2152 25 of 27
Funding:
This research was funded by the National Natural Science Foundation of China, Grant No.
U21A20464 and 62066005.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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