A hybrid engineering algorithm of the seeker algorithm and particle swarm optimization

Haipeng Liu; Shaomi Duan; Huilong Luo

doi:10.1515/mt-2021-2138

Article Open Access

A hybrid engineering algorithm of the seeker algorithm and particle swarm optimization

Haipeng Liu
Haipeng Liu is an associate professor in the Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, Yunnan, China. His research interests include: new energy technology, energy saving, and simulation, modeling and optimization of manufacturing process using intelligence techniques.
, Shaomi Duan
Shaomi Duan is currently a PhD Student in the Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, China. Her research interests include: modeling and optimization of manufacturing process using statistical and computational intelligence techniques; and optimization using metaheuristics.
and Huilong Luo
Huilong Luo is a professor in the Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, China. His main research interests are optimization, artificial intelligence, manufacturing processes, heat exchange, and energy saving.

Published/Copyright: July 7, 2022

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From the journal Materials Testing Volume 64 Issue 7

Abstract

A newly hybrid algorithm is proposed based on the combination of seeker optimization algorithm and particle swarm optimization. The hybrid algorithm is based on a double population evolution strategy, and the populations of individuals are evolved from the seeker optimization algorithm and the particle swarm optimization separately. The populations of individuals employ an information sharing mechanism to implement coevolution. The hybrid algorithm enhances the individuals’ diversity and averts fall into the local optimum. The hybrid algorithm is compared with particle swarm optimization, the simulated annealing and genetic algorithm, the dragonfly algorithm, the brain storming algorithm, the gravitational search algorithm, the sine cosine algorithm, the salp swarm algorithm, the multi-verse optimizer, and the seeker optimization algorithm, then 15 benchmark functions, five proportional integral differential control parameters models, and six constrained engineering optimization problems are selected for optimization experiment. According to the experimental results, the hybrid algorithm can be used in the benchmark functions, the proportional integral differential control parameters optimization, and in the optimization constrained engineering problems. The optimization ability and robustness of the hybrid algorithm are better.

Keywords: engineering optimization problems; function optimization; hybrid algorithm; particle swarm optimization; seeker optimization algorithm

1 Introduction

Recently, the modern optimization techniques have aroused great interest among the scientific and technical community in a wide variety of fields, due to their ability to solve problems with a non-linear and non-convex dependence of design parameters. Since the no free lunch (NFL) theorem, no one optimization solution can optimize overall questions [1]. Therefore, researchers pose new algorithms or enhance the current algorithms to deal with optimization problems. The current algorithms are the genetic algorithm (GA) [2], the particle swarm optimization (PSO) [3], the simulated annealing (SA) [4], the harmony search (HS) [5], the dragonfly algorithm (DA) [6], the brain storming algorithm (BSO) [7], the gravitational search algorithm (GSA) [8], the moth-flame optimization (MFO) [9, 10], the sine cosine algorithm (SCA) [11], the salp swarm algorithm (SSA) [12], the multi-verse optimizer (MVO) [13], the seeker optimization algorithm (SOA) [14], the artificial bee colony (ABC) algorithm [15], the krill herd (KH) [16], the monarch butterfly optimization (MBO) [17], the elephant herding optimization (EHO) [18], the moth search (MS) algorithm [19], the slime mould algorithm (SMA) [20], and the Harris hawks optimization (HHO) [21], the spotted hyena optimization algorithm (SHO) [22], the butterfly optimization algorithm (BOA) [23], the henry gas solubility optimization algorithm (HGS) [24], the equilibrium optimization algorithm (EO) [25], the mine blast algorithm (MBA) [26], the interior search algorithm (ISA) [27].

However, some optimization algorithms are still not very successful in optimization problems. The optimization problems include: issues with low optimization precision, being premature, having only a local optimal solution, slow convergence speed, and insufficient robustness. To better overcome the issues of optimization precision, prematurity, having only a local optimal solution, the slow convergence rate, and poor robustness, some improved algorithms have proven to be feasible optimization algorithms and have been used in practical engineering. For instance, the heuristic algorithm combined with other algorithms has been proposed according to the characteristics of some algorithms, which now is a popular strategy to improve the algorithms. These hybridizations have been shown to be effective global optimization algorithms and have been applied to solve application problems. For example, a hybrid of genetic algorithm (GA) and particle swarm optimization (PSO) was applied to the recurrent neural/fuzzy network design [28]. Comparison of grey wolf, whale, water cycle optimization algorithm, ant lion and sine cosine algorithms are used to optimize the vehicle engine connecting rod [29]. A hybrid metaheuristic differential evolution (DE) and cuckoo search (CS) algorithm was implemented to solve the unmanned combat aerial vehicles (UCAVs) path planning problem [30]. The Harris hawks optimization algorithm, salp swarm optimization algorithm, grasshopper optimization algorithm and dragonfly algorithm was applied to optimize the structural design of vehicle components [31]. A hybrid particle swarm optimization (PSO) and ant colony optimization (ACO) algorithm were implemented to solve the hierarchical classification [32]. The Harris hawks, grasshopper and multi-verse optimization algorithms are used to optimize the machining parameters in manufacturing operations [33]. The hybrid PSO and adaptive large neighborhood search (ALNS) algorithm for software and mobile was applied to the transportation in ice manufacturing industry 3.5 [34]. A new hybrid grasshopper optimization algorithm is applied to make the robust design of a robot gripper mechanism [35]. A new hybrid Taguchi-salp swarm optimization algorithm is used for the robust design of real-world engineering problems [36]. A novel hybrid Harris hawks simulated annealing algorithm and RBF-based metamodel is used to the design optimization of highway guardrails [37].

Dai et al. proposed the SOA algorithm in 2006 [38] which aims to optimize practical application problems by mimicking human behavior and information exchange. In the recent decade, the SOA algorithm has been used in many fields, such as in parameter estimation of time-delay chaotic systems problems [39], optimal reactive power dispatch [40], a challenging set of benchmark problems [41], the design of a digital filter [42], optimizing parameters of artificial neural networks [43], the optimizing model and structures of fuel cell [44], the novel human group optimizer algorithm [45], and several practical applications [46]. However, in the initial stage of dealing with optimization problems, the SOA converges faster than others; When all individuals are near the best individual for solving the optimization problem, the individuals will lose the diversity and fall into the prematurity.

The particle swarm optimization was introduced by Kennedy and Eberhart in 1995 [3], as the class of swarm intelligence techniques that are used to solve optimization problems. In the following years, the particle swarm optimization was adopted as a popular swarm intelligence (SI) algorithm. The PSO has been used in solve various mathematical, engineering, design, network, robotic, and image processing optimization problems [28, 32, 34, 47]. In the PSO, the particles explore the search space by following the personal and global best experiences. After the population initialization, the population through velocity, acceleration coefficients, and inertial weight generates new population, it is able to maintain the diversity of population, which can achieve the search of global optimal solution after several iterations. However, it also has flaws, the PSO operation is more likely to be trapped in local optima because the search process is led by only one leader.

The SOA algorithm can easily suffer from the premature convergence when solving global optimization problems. In order to overcome the deficiencies of SOA algorithm in solving a global optimization problem, in this paper, a new hybrid global optimization algorithm based on the seeker optimization algorithm (SOA) and particle swarm optimization (PSO) is proposed. This evolution strategy allows the SOA and the PSO algorithm to give full play to their respective advantages. Using the PSO algorithm enables keeping the diversity of the population, which can be a good remedy for the defect of only using the SOA algorithm optimization, avoiding falling into a local optimum due to the loss of population diversity. And the PSO algorithm converges faster; it is a good solution to make up for the shortcomings of SOA algorithm in the convergence speed, reducing the risk of falling into a local optimal solution by utilizing the individual of PSO algorithm to guide the evolution of the individual of SOA algorithm. The hybrid algorithm not only can ensure the accuracy of the algorithm, but also guarantee the speed of solving problems. Finally, compared with the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, and the SOA, the SOAPSO algorithm has been implemented and tested on a complete set of well-known benchmark functions, five proportional integral differential (PID) control parameters optimization, and six optimization constrained engineering problems taken from literature. According to the experimental results, the SOAPSO is feasible in the benchmark functions, the PID parameter optimization problems, and in the constrained engineering optimization problems. The SOAPSO can find better values for solving the questions. The improved SOA algorithm successfully overcomes its tendency to prematurely converge to local optima for problems. The SOAPSO algorithm has a better optimization performance and robustness. The algorithm also has an improvement over the original SOA algorithm. The advantages of the SOAPSO are summed up as follows.

An SOAPSO algorithm is raised to enhance the precision and robustness of the optimization process.
The hybrid strategy can improve the diversity of individuals, enhance local search, and avert premature convergence.

The rest of the article structure is as follows. Section 2 presents the SOA and the PSO algorithm. Section 3 describes the SOAPSO. Section 4 shows the algorithm optimization experiments, the results and the analyses. At last, Section 5 gives some conclusions.

2 Basic SOA and PSO algorithm

2.1 Seeker optimization algorithm

The SOA algorithm carries out in-depth research on human search behavior. It considers optimization as a search for an optimal solution by a search team in search space, taking search team as population and the site of the searcher as task method. Using “experience gradient” to determine the search direction, the search step measurement is resolved in view of using uncertain reasoning, through the scout direction and search step size to complete the searchers’ position in the search interspace update, to attain the optimization of the solution.

2.1.1 Search Direction

The forward orientation of search is defined by the experience gradient obtained from the individuals’ movement and the evaluation of other individuals’ search historical position. The egoistic direction f → i , e ( t ) , altruistic direction f → i , a ( t ) , and the preemptive direction f → i , p ( t ) of the ith individual in any dimension can be obtained.

(1) f → i , e ( t ) = p → i , best − x → i ( t )

(2) f → i , a ( t ) = g → i , best − x → i ( t )

(3) f → i , p ( t ) = x → i ( t 1 ) − x → i ( t 2 )

The searcher uses the method of a random weighted average to obtain the search orientation.

(4) f → i t = sign ω f → i , e t + φ 1 f → i , e t + φ 2 f → i , a t

with t ₁, t ₂ ∈{t, t–1, t–2}, x → i ( t 1 ) and x → i ( t 2 ) : Best advantages of { x → i ( t − 2 ) , x → i ( t − 1 ) , x → i ( t ) } separately, g _i,best: Historical optimal location in the neighborhood where the ith search factor is located, p _i,best: Optimum locality from the ith search factor to the current locality, ψ ₁ and ψ ₁: Random numbers in [0,1] nad ω is: Wight of inertia.

2.1.2 Search step size

The SOA algorithm refers to the reasoning of the fuzzy approximation ability. The SOA algorithm, through the computer language, describes some of the human natural languages that can simulate human intelligence reasoning search behavior. If the algorithm expresses a simple fuzzy rule, it adapts to the best approximation of the objective optimization problems. The greater search step length is more important. However, to the smaller fitness, step length makes the corresponding smaller. The Gaussian distribution function is adopted to describe the search step measurement.

(5) μ ( α ) = e − α 2 2 δ 2

with α and δ: parameters of a membership function.

According to Equation (5), the probability of the output variable exceeding [−3δ, 3δ] is less than 0.0111. Therefore, μ _min = 0.0111. Under normal circumstances, the optimal position of an individual has μ _max = 1.0 and the worst place is 0.0111. However, to accelerate the convergence speed and get the optimal individual to have an uncertain step size, μ _max is set as 0.9 in this paper. Select the following function as the fuzzy variable with a “small” target function value:

(6) μ i = μ max − s − I i s − I ( μ max − μ min ) , i = 1 , 2 , ⋯ , s

(7) μ i j = rand ( μ i , 1 ) , j = 1 , 2 , ⋯ , D

with μ _ij: Determined by Equations (6) and (7), I _i: Count of the sequence x _i(t) of the current individuals arranged from high to low by function value and the function rand (μ _i,1): Real number in any partition [μ _i,1].

It can be seen from Equation (6) that it simulates the random search behavior of human beings. The step measurement of j-dimensional search interspace is determined by Equation (8):

(8) α i j = δ i j − − ln ( μ i j )

with δ _ij: Parameter of the Gaussian distribution function, which is defined by Equation (9):

(9) ω = ( i t e r max − t ) / i t e r max

(10) δ i j = ω ∗ abs ( x → min − x → max )

with ω: Weight of inertia.

As the evolutionary algebra increases, ω decreases linearly from 0.9 to 0.1. x → min and x → max are respectively the variate of the minimum value and maximum value of the function.

2.1.3 Individual location updates

After obtaining the scout direction and scout step measurement of the individual, the location update is represented by Equation (10):

(11) x i j ( t + 1 ) = x i j ( t ) + α i j ( t ) f i j ( t ) , i = 1 , 2 , ⋯ , s ; j = 1 , 2 , ⋯ , D

with i: ith searcher individual, j: Individual dimension, f _ij(t) and α _ij(t): Searchers’ search direction, respectively, as well as search step size at time t and x _ij(t) as well as x _ij(t+1): Searchers’ site at time t and (t+1), respectively.

2.2 Particle swarm optimization

The particle swarm optimization is defined by Equations (12)–(14):

(12) x i j ( t + 1 ) = x i j ( t ) + v i j ( t + 1 ) , i = 1 , 2 , ⋅ ⋅ ⋅ , s ; j = 1 , 2 , ⋯ , D

(13) ω = 0 . 9 − t . ∗ ( 0 . 4 . / G )

(14) v i j ( t + 1 ) = ω × v i j ( t ) + c 1 × r 1 × ( p i d − x i j ( t ) ) + c 2 × r 2 × ( p g d − x i j ( t ) ) , i = 1 , 2 , ⋅ ⋅ ⋅ , s ; j = 1 , 2 , ⋯ , D

with x i j ( t + 1 ) and v i j ( t + 1 ) : Position and velocity of particle x _i in the jth dimension of the (t + 1)th iteration, respectively.

The acceleration coefficients, c ₁ and c ₂ play significant role in balancing the search between cognitive (p _id) and social (p _gd) components, respectively, while r ₁ and r ₂ are the random vectors with each element having a value between 0 and 1. The personal and global best experiences are denoted as p _id and p _gd, respectively. The inertial weight ω determines the impact of the previous velocity on the current one; G is the largest optimal generation value.

3 SOAPSO algorithm

This section focuses on the rationale of the SOAPSO algorithm. The SOA and the PSO are based on the population of the global search techniques. And the SOAPSO is based on a double population evolution strategy [48, 49]. The individuals both in the SOA and the PSO employ an information sharing mechanism to implement coevolution. The strategy makes the SOAPSO enjoy the advantages of two algorithms. It can maintain the diversity of populations, and the SOAPSO algorithm has the capability to jump out of the local optimal solution. Based on the above description, the main procedure of the SOAPSO is as shown in Algorithm 1.

4 Experimental results

4.1 Setup

These algorithms used in the experiment in this paper were running under MATLAB R2016a. The computer is configured as Intel (R) Core (TM) i7-7500U CPU @2.7 GHz 2.9 GHz processor with 8 GB of memory, Windows 10 operating system.

4.2 Benchmark test function

To test the performance of SOAPSO algorithm, 15 benchmark functions [9, 14, 50], [51], [52] which have been widely used in the test are used to test. Tables 1 and 2 show the benchmark functions used in the experiment.

Table 1:

Description of unimodal benchmark functions.

Test functions	Range of search	Theoretical optimal fitness value
f 1 ( x ) = ∑ i = 1 n x i 2	[−100,100]	0
f 2 ( x ) = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|	[−10,10]	0
f 3 ( x ) = ∑ i = 1 n ∑ j = 1 i x j 2	[−100,100]	0
f 4 ( x ) = max i { \| x i \| , 1 ≤ i ≤ n }	[−100,100]	0
f 5 ( x ) = ∑ i = 1 n − 1 [ 100 ( x i + 1 − x i 2 ) 2 + ( x i − 1 ) 2 ]	[−30,30]	0
f 6 ( x ) = ∑ i = 1 n ( \| x i + 0.5 \| ) 2	[−100,100]	0
f 7 ( x ) = ∑ i = 1 n i x i 4 + r a n d m [ 0 , 1 ]	[−1.28,1.28]	0
f 8 ( x ) = ∑ i = 1 n i x i 2	[−10,10]	0
f 9 ( x ) = ∑ i = 1 n \| x i \| ( i + 1 )	[−1.28,1.28]	0

Table 2:

Description of multimodal benchmark functions.

Test functions	Range of search	Theoretical optimal fitness value
f 10 ( x ) = − ∑ i = 1 n x i sin ( \| x i \| )	[−500,500]	−418.9829 *Dimension
f 11 ( x ) = ∑ i = 1 n [ x i 2 − 10 cos ( 2 π x i ) + 10 ]	[−5.12,5.12]	0
f 12 ( x ) = 20 + e − 20 e − 0.2 1 n ∑ i = 1 n x i 2 − e 1 n ∑ i = 1 n cos ( 2 π x i )	[−32,32]	0
f 13 ( x ) = 1 4000 ∑ i = 1 n x i 2 − ∏ i = 1 n cos ( x i i ) + 1	[−600,600]	0
f 14 ( x ) = π n { 10 sin 2 ( π y 1 ) + ∑ i = 1 n − 1 ( y i − 1 ) 2 [ 1 + sin 2 ( π y i + 1 ) ] + ( y n - 1 ) 2 } + ∑ i = 1 n u ( x i , 10 , 100 , 4 ) y i = 1 + 1 4 ( x i + 1 ) , u ( x i , a , k , m ) = { k ( x i - a ) m , x i > a 0 , − a ≤ x i ≤ a ( − x i − a ) m , x i < − a	[−50,50]	0
f 15 ( x ) = 1 10 { sin 2 ( 3 π x 1 ) + ∑ i = 1 n − 1 ( x i − 1 ) 2 [ 1 + sin 2 ( 3 π x i + 1 ) ] + ( x n - 1 ) 2 [ 1 + sin 2 ( 2 π x n ) ] } + ∑ i = 1 n u ( x i , 5 , 100 , 4 )	[−50,50]	0

4.3 Parameter setting

Dai et al. performed much research for the parameters set of the SOA [38]. The parameters set of the SOAPSO is based on the practical experience to take the appropriate value. Table 3 represents the necessary parameters in the experiment. To the SOAPSO algorithm, the population number is 15, and the evolutionary algebra is 1000. The population size of other algorithms is 30, the max number of iterations of all tests is 1000.

Table 3:

Parameters sets for the algorithms.

Algorithm	Parameters and value
PSO [53]	Constant inertia: 0.9–0.4, the first and second acceleration coefficients: 1.4962.
SA_GA [54]	Select probability: 0.6, crossover probability: 0.7, the mutation scale factor: 0.05, initial temperature: 100, temperature reduction parameter: 0.98.
DA [6]	s shows the separation weight: 0–0.2, a is the alignment weight: 0–0.2, c indicates the cohesion weight: 0–0.2, f is the food factor: 0–2, e is the enemy factor: 0–0.1, w is the inertia weight: 0.9–0.4.
BSO [55]	Number of clusters: 10, probability for select one cluster: 0.8, probability for select one cluster center to be replaced: 0.2, probability for use the center: 0.4.
GSA [56]	Value of the gravitational constant at the first cosmic quantum-interval: G0 = 100, alfa = 20, proportion of an individual’s power over another final_per = 2.
SCA [11]	Random numbers: r ₁ = 0–2, r ₂ = 0∼2π, r ₃ = 0–2, r ₄ = 0–1.
SSA [12]	Random numbers: c ₁ = 0–2, c ₂ = 0–1, c ₃ = 0–1.
MVO [13]	Wormhole existence probability: WEP_Max = 1, WEP_Min = 0.2, travelling distance rate: TDR = 0–1, random numbers: r ₁ = 0–1, r ₂ = 0–1, r ₃ = 0–1.
SOA [14]	Maximum membership degree value: 0.95, minimum membership degree value: 0.0111, maximum inertia weight value: 0.8, minimum inertia weight value: 0.2.
SOAPSO	Maximum membership degree value: 0.95, minimum membership degree value: 0.0111, maximum inertia weight value: 0.9, minimum inertia weight value: 0.1, constant inertia: 0.9–0.4, first and second acceleration coefficients: 1.4962.

4.4 Algorithm performance comparison in benchmark functions

To ensure that the comparison of these algorithms is fair, the population number of algorithms is 30, and the evolutionary algebra is 1000. At the same time, for further ensuring the fairness of algorithm comparison and reducing the effect of randomness, the results of the 10 algorithms after 30 independent runs were selected for comparison.

In this section, in order to test the performance of SOAPSO algorithm, the SOAPSO algorithm has been compared with the PSO, SA-GA, DA, BSO, GSA, SCA, SSA, MVO and the SOA algorithm in low dimension and high dimension. The mean results, the standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and the rank results between the algorithms of 30 independent runs for f1 ∼ f15 are shown in low dimension in Table 4 and for f1 ∼ f15 are shown in high dimension in Table 5.

Table 4:

Comparison of performance of algorithms in benchmark function of 30 independent runs in low dimensions.

Test functions	Result	Algorithms
Test functions	Result	PSO	SA_GA	DA	BSO	GSA	SCA	SSA	MVO	SOA	SOAPSO
f ₁ (D = 100)	Mean	0.0031	3.02E+04	1.67E+04	70.588780	6.21E+02	6.42E+03	2.4258997	40.269355	1.0524957	1.93039e-04
	Std.	6.31E-04	6.69E+03	7.16E+03	14.271695	2.71E+02	5.62E+03	1.6157099	6.8812839	0.2051906	3.06900e-05
	Worst	0.0044	4.05E+04	3.09E+04	1.25E+02	1.15E+03	1.89E+04	6.2348128	60.394282	1.4502675	2.67295e-04
	Best	0.0022	1.71E+04	4.70E+02	49.150576	2.50E+02	96.291688	0.3206193	27.376332	0.6215966	1.39244e-04
	Rank	2	10	9	6	8	7	3	5	4	1
f ₂ (D = 100)	Mean	0.2192801	1.19E+02	68.015914	N/A	7.1387617	1.3795842	22.960614	7.34E+20	13.002085	0.28414018
	Std.	0.0260935	13.168468	23.977776	N/A	3.2135126	1.4335667	6.7732128	3.92E+21	2.0008395	0.04094720
	Worst	0.2684631	1.42E+02	1.27E+02	N/A	14.185711	5.8449387	36.961794	2.15E+22	18.305798	0.35504890
	Best	0.1732089	92.513923	28.540106	N/A	2.0654838	0.0310204	11.525488	5.43E+02	9.4729280	0.21052655
	Rank	2	8	7	10	4	1	6	9	5	3
f ₃ (D = 100)	Mean	2.27E+03	3.77E+05	1.90E+05	1.63E+04	8.80E+03	1.84E+05	3.58E+04	4.24E+04	9.87E+03	2.3896e+03
	Std.	1.30E+03	8.95E+04	7.28E+04	5.18E+03	1.84E+03	3.48E+04	1.67E+04	6.09E+03	3.47E+03	1.8910e+03
	Worst	5.58E+03	5.79E+05	3.44E+05	3.40E+04	1.40E+04	2.78E+05	1.01E+05	5.30E+04	1.78E+04	7.0512e+03
	Best	8.43E+02	2.55E+05	6.87E+04	9.50E+03	6.26E+03	1.24E+05	1.36E+04	3.11E+04	3.32E+03	1.6448e+02
	Rank	2	10	8	5	4	9	6	7	3	1
f ₄ (D = 100)	Mean	1.4624169	89.626879	54.046551	23.239192	15.671857	85.539422	26.50225	49.557442	21.743821	1.53637621
	Std.	0.2035038	3.4544225	5.8105757	3.3497291	1.5605923	3.5980134	3.0835448	6.2884298	5.6809434	0.75124538
	Worst	2.0084420	95.051439	60.342117	31.391629	20.036975	90.947406	34.066627	62.165472	29.187985	3.95113076
	Best	1.0929345	83.496263	38.891864	18.846933	10.958917	77.909721	22.210338	37.299715	2.9045088	0.17953942
	Rank	2	10	8	5	4	9	6	7	3	1
f ₅ (D = 100)	Mean	2.51E+02	5.78E+07	1.34E+07	1.57E+04	1.78E+04	6.43E+07	4.70E+03	2.44E+03	7.09E+02	1.1032e+02
	Std.	59.018749	2.62E+07	6.99E+06	4.80E+03	1.58E+04	4.67E+07	7.23E+03	1.96E+03	1.67E+02	41.4146145
	Worst	3.61E+02	1.25E+08	2.75E+07	2.66E+04	7.03E+04	2.11E+08	3.41E+04	9.44E+03	1.01E+03	2.8092e+02
	Best	1.24E+02	1.92E+07	1.73E+06	6.88E+03	1.99E+03	7.16E+06	9.05E+02	7.75E+02	2.79E+02	91.3081407
	Rank	2	10	8	7	6	9	5	4	3	1
f ₆ (D = 100)	Mean	0.0119146	3.07E+04	1.80E+04	71.311851	7.00E+02	6.13E+03	4.0344913	40.089213	1.1365348	0.00261787
	Std.	0.0020781	8.32E+03	7.81E+03	19.030272	3.89E+02	4.65E+03	3.5377887	5.9162672	0.1751582	5.63575e-04
	Worst	0.0163601	5.23E+04	3.33E+04	1.32E+02	1.84E+03	1.99E+04	16.516267	52.047400	1.5083506	0.00392564
	Best	0.0086605	1.36E+04	4.09E+03	46.048287	2.17E+02	6.73E+02	0.6398690	28.916720	0.7992740	0.00159638
	Rank	2	10	9	6	7	8	3	5	4	1
f ₇ (D = 100)	Mean	0.2211229	77.487242	20.638807	13.058642	2.3136998	73.024146	1.4115482	0.3355664	3.4645595	0.06687563
	Std.	0.0419681	49.238469	16.771470	3.8383051	1.1506156	46.772681	0.3068424	0.0733789	0.9443482	0.01908711
	Worst	0.3406667	2.23E+02	85.271478	25.176691	6.3399231	2.08E+02	2.2269363	0.4938711	6.5532009	0.11685566
	Best	0.1180966	19.170953	3.9670294	6.6128823	0.8371509	11.108763	0.9996030	0.2176315	2.0472780	0.04060105
	Rank	2	10	7	8	4	9	5	3	6	1
f ₈ (D = 100)	Mean	0.0624146	1.29E+04	6.68E+03	1.38E+03	1.32E+02	1.52E+03	1.28E+02	1.05E+02	1.39E+02	0.05831202
	Std.	0.0164020	3.57E+03	3.65E+03	3.91E+02	74.414138	1.16E+03	49.510547	43.821833	33.925549	0.01512488
	Worst	0.1011455	2.33E+04	1.61E+04	2.28E+03	3.78E+02	5.07E+03	2.51E+02	2.04E+02	2.27E+02	0.09834756
	Best	0.0291246	7.55E+03	1.59E+03	6.76E+02	31.981022	2.47E+02	43.759004	34.296414	79.288776	0.03366305
	Rank	1	10	9	8	3	7	5	4	6	2
f ₉ (D = 100)	Mean	2.08E-26	1.78E+03	8.14E-04	3.91E-05	3.67E-12	35.077327	9.57E-07	1.21E-06	1.47E-05	1.32212e-08
	Std.	3.99E-26	9.54E+03	0.0028250	8.18E-05	7.66E-12	80.100469	6.27E-07	5.23E-07	1.28E-05	8.86567e-09
	Worst	1.72E-25	5.23E+04	0.0154537	3.88E-04	3.81E-11	4.17E+02	2.59E-06	3.03E-06	5.47E-05	3.86518e-08
	Best	2.07E-29	0.0028833	2.59E-06	6.77E-07	4.23E-16	0.0268174	2.02E-07	4.55E-07	6.32E-07	4.92666e-10
	Rank	1	9	8	7	2	10	4	5	6	3
f ₁₀ (D = 100)	Mean	-4.75E+03	−2.47E+04	−1.07E+04	−1.81E+04	−4.73E+03	−7.26E+03	−2.36E+04	−2.41E+04	−2.32E+04	−2.5735e+04
	Std.	6.89E+02	8.85E+02	1.32E+03	1.52E+03	8.70E+02	6.96E+02	1.49E+03	1.53E+03	3.40E+03	4.3087e+03
	Worst	−3.56E+03	−2.32E+04	−7.30E+03	−1.49E+04	−3.60E+03	−6.21E+03	−1.99E+04	−2.14E+04	−1.87E+04	−1.8817e+04
	Best	−6.44E+03	−2.66E+04	−1.28E+04	−2.11E+04	−7.35E+03	−9.46E+03	−2.59E+04	−2.66E+04	−3.17E+04	−3.4341e+04
	Rank	10	3	7	6	9	8	5	3	2	1
f ₁₁ (D = 100)	Mean	33.896488	4.26E+02	7.64E+02	5.00E+02	1.36E+02	2.06E+02	1.62E+02	6.41E+02	4.15E+02	26.0813111
	Std.	6.2350851	48.908770	1.19E+02	44.678043	18.532649	92.249647	44.581629	69.443553	47.133647	25.7528475
	Worst	49.898365	5.40E+02	1.04E+03	5.82E+02	1.73E+02	4.32E+02	2.67E+02	8.01E+02	5.05E+02	1.0811e+02
	Best	23.655297	3.34E+02	5.60E+02	3.93E+02	88.161003	67.314606	84.812071	5.01E+02	3.30E+02	0.09947806
	Rank	2	7	10	8	5	3	4	9	6	1
f ₁₂ (D = 100)	Mean	0.0220555	15.347972	12.879210	4.7554512	3.1378392	18.309269	7.5175875	6.5910720	2.4523845	0.00599338
	Std.	0.0032932	0.7816167	2.1899705	0.3262885	0.6404753	4.7183569	1.2414366	6.0322718	0.3106583	6.57922e-04
	Worst	0.0297039	16.669138	16.049443	5.5752180	4.6781816	20.678970	9.4909679	20.171725	3.2763189	0.00777180
	Best	0.0161767	13.796287	7.8089273	4.2290358	2.0521342	6.872015	4.8882702	3.2031355	1.9634230	0.00486383
	Rank	2	10	9	7	4	8	5	6	3	1
f ₁₃ (D = 100)	Mean	9.1115292	2.75E+02	1.36E+02	1.24E+02	98.761595	53.145123	0.7064324	1.3775687	0.5354221	0.00381629
	Std.	1.8646352	66.687333	71.679953	34.409820	11.527169	38.195663	0.2200883	0.0626040	0.3343659	0.01426213
	Worst	14.163886	4.19E+02	2.75E+02	2.05E+02	1.24E+02	1.45E+02	1.0776799	1.5175488	1.0812126	0.06113412
	Best	6.1937260	1.38E+02	7.041402	76.125105	81.688324	1.7219609	0.2641328	1.2241138	0.0806019	2.16274e-05
	Rank	6	10	7	8	9	5	3	4	2	1
f ₁₄ (D = 100)	Mean	0.0167129	4.42E+07	5.63E+06	14.732482	4.5497727	1.58E+08	17.291593	11.656751	20.231983	1.47161804
	Std.	0.0241246	5.11E+07	1.06E+07	3.4008233	1.2139614	1.35E+08	4.5189004	4.2010589	8.8688013	1.10331377
	Worst	0.0623283	2.13E+08	5.56E+07	22.801357	6.8218450	6.52E+08	25.115105	25.911307	48.892258	3.33398551
	Best	4.84E-05	3.88E+06	1.33E+05	9.5966961	2.1673113	6.76E+06	7.5727656	6.7785747	8.9451796	1.35789e-04
	Rank	1	9	8	7	3	10	5	4	6	2
f ₁₅ (D = 100)	Mean	0.0010357	1.44E+08	2.33E+07	1.17E+02	1.31E+02	2.67E+08	1.77E+02	1.21E+02	1.30E+02	0.13034585
	Std.	0.0028003	1.01E+08	1.59E+07	29.055176	64.254053	1.53E+08	15.845079	30.717732	62.979168	0.12750016
	Worst	0.0113607	4.26E+08	6.87E+07	1.67E+02	4.50E+02	5.83E+08	2.15E+02	1.80E+02	2.18E+02	0.46282091
	Best	1.90E-04	2.31E+07	1.83E+06	55.987388	76.410368	3.11E+07	1.45E+02	51.265613	1.7998260	0.00680983
	Rank	1	9	8	5	6	10	7	4	3	2
Average Rank		2.533333333	9	8.133333333	6.866666667	5.2	7.533333333	4.8	5.266666667	4.133333333	1.466666667
Overall Rank		2	9	8	7	5	10	3	4	6	1

The bold values in the table indicate the best results.

Table 5:

Comparison of performance of algorithms in benchmark function of 30 independent runs in high dimensions.

Test functions	Result	Algorithm
Test functions	Result	PSO	SA_GA	DA	BSO	GSA	SCA	SSA	MVO	SOA	SOAPSO
f ₁ (D = 1000)	Mean	9.7477e+03	2.5732e+06	3.6829e+05	1.5118e+05	9.7519e+04	3.5105e+05	1.9108e+05	3.5977e+05	8.3814e+03	39.2393
	Std.	1.6106e+03	5.6744e+04	1.8383e+05	1.6112e+04	4.9089e+03	1.3646e+05	9.6344e+03	1.6121e+04	1.5983e+03	1.4424e+2
	Worst	1.4005e+04	2.6799e+06	7.8629e+05	1.8873e+05	1.0536e+05	7.7196e+05	2.1341e+05	3.9287e+05	1.2907e+04	8.0148e+2
	Best	6.9495e+03	2.4536e+06	1.0420e+05	1.1957e+05	8.7230e+04	4.0211e+04	1.7058e+05	3.3044e+05	4.7822e+03	2.1465002
	Rank	3	10	6	7	5	4	8	9	2	1
f ₂ (D = 1000)	Mean	Inf	Inf	1.0431e+03	N/A	1.801e+281	Inf	1.0772e+03	1.344e+273	7.7812e+278	57.1333
	Std.	N/A	N/A	3.3461e+02	N/A	Inf	N/A	20.2070	Inf	Inf	4.1107
	Worst	Inf	Inf	1.6602e+03	N/A	3.795e+282	Inf	1.1116e+03	4.031e+274	2.334e+280	66.4263
	Best	2.9025e+02	Inf	3.0660e+02	N/A	4.608e+244	Inf	1.0358e+03	2.315e+209	2.1280e+03	51.2417
	Rank	2	8	3	10	7	8	4	6	5	1
f ₃ (D = 1000)	Mean	3.3245e+06	3.3451e+07	2.3366e+07	1.8797e+06	1.9611e+06	2.4026e+07	4.1724e+06	6.1928e+06	2.3906e+06	1.2672e+6
	Std.	1.4068e+06	1.1343e+07	5.9873e+06	4.7421e+05	8.1529e+05	4.7320e+06	1.9370e+06	4.3229e+05	7.6981e+05	6.3085e+5
	Worst	7.9869e+06	8.2745e+07	3.7730e+07	3.2090e+06	4.8408e+06	3.4515e+07	8.4958e+06	7.3742e+06	3.9890e+06	3.1987e+6
	Best	1.6748e+06	1.9532e+07	1.1474e+07	1.1610e+06	9.5585e+05	1.3982e+07	1.5071e+06	5.5268e+06	2.0598e+05	5.4842e+4
	Rank	6	9	8	4	3	10	5	7	2	1
f ₄ (D = 1000)	Mean	18.3429	99.5440076	73.9886785	42.9599576	28.7444424	99.5223	43.9089	97.4126	37.5552471	23.9116482
	Std.	1.3182	0.19176900	14.4216872	3.44320734	1.79619392	0.1313	2.9961	0.7405	8.75145918	17.9559077
	Worst	21.4724	99.7898020	88.5995335	50.5377515	33.6662360	99.7171	53.5153	98.7713	44.9098123	48.1995287
	Best	16.3774	98.958778	19.5323743	36.5946922	25.8833818	99.1965	39.4689	95.6086	12.0877478	0.51588439
	Rank	3	9	4	6	5	10	7	8	2	1
f ₅ (D = 1000)	Mean	2.7249e+05	1.1187e+10	4.8447e+08	8.5617e+07	1.6750e+07	3.2676e+09	7.5113e+07	6.7902e+08	1.0308e+08	6.2151e+4
	Std.	1.4792e+05	3.3483e+08	3.8528e+08	1.4661e+07	1.7268e+06	7.0048e+08	8.8633e+06	7.3930e+07	2.5353e+07	7.6347e+4
	Worst	7.5049e+05	1.2027e+10	1.5016e+09	1.2280e+08	2.2405e+07	4.6597e+09	1.0506e+08	8.4443e+08	1.4810e+08	2.8829e+5
	Best	1.2499e+05	1.0569e+10	1.8947e+07	5.7480e+07	1.4932e+07	2.0841e+09	6.0279e+07	5.4843e+08	6.2501e+07	1.4431e+3
	Rank	2	10	4	5	3	9	6	8	7	1
f ₆ (D = 1000)	Mean	1.0284e+04	2.5635e+06	2.9250e+05	1.5572e+05	9.8733e+04	4.1315e+05	1.9105e+05	3.5579e+05	7.9675e+03	3.5788e+2
	Std.	1.3817e+03	4.0795e+04	1.2442e+05	1.8394e+04	4.9212e+03	1.2787e+05	1.0328e+04	1.7687e+04	1.41579e+03	3.2306e+2
	Worst	1.3566e+04	2.6317e+06	5.6448e+05	1.9830e+05	1.0908e+05	7.3580e+05	2.1256e+05	3.9770e+05	1.1528e+04	1.5098e+3
	Best	6.1887e+03	2.4668e+06	6.1537e+04	1.1722e+05	9.0772e+04	1.6278e+05	1.6527e+05	3.1703e+05	4.8067e+03	1.5114e+2
	Rank	3	10	4	6	5	7	8	9	2	1
f ₇ (D = 1000)	Mean	1.1474e+02	1.7871e+05	7.9249e+03	6.5609e+03	5.2986e+03	4.7647e+04	1.1040e+03	8.7236e+03	2.0441e+04	1.0252e+3
	Std.	22.2657388	6.0811e+03	5.9118e+03	2.1474e+03	5.8169e+02	1.1048e+04	129.0645	771.5517	2.6651e+03	4.8520e+2
	Worst	1.7347e+02	1.9064e+05	1.9723e+04	1.3399e+04	6.9318e+03	6.9571e+04	1.3689e+03	1.1079e+04	2.6136e+04	2.1401e+3
	Best	80.9849331	1.6720e+05	667.2428	3.2310e+03	4.3702e+03	2.6553e+04	878.7155	7.2739e+03	1.4371e+04	80.264704
	Rank	2	10	3	5	6	9	4	7	8	1
f ₈ (D = 1000)	Mean	4.7291e+04	1.2547e+07	1.9203e+06	1.3736e+06	3.8862e+05	1.8012e+06	8.6050e+05	1.5454e+06	1.6177e+06	4.5990e+3
	Std.	6.6212e+03	3.5539e+05	9.6039e+05	1.6383e+05	2.2329e+04	5.5949e+05	5.1965e+04	6.4511e+04	1.1284e+05	3.3693e+3
	Worst	6.0802e+04	1.3042e+07	3.8303e+06	1.6356e+06	4.3797 e+05	2.6519e+06	9.8685e+05	1.6892e+06	1.8715e+06	1.2994e+4
	Best	3.6574e+04	1.1381e+07	6.5489e+05	1.0579e+06	3.5017e+05	4.5453e+05	7.8606e+05	1.4470e+06	1.3452e+06	1.1801e+3
	Rank	2	10	5	7	3	4	6	9	8	1
f ₉ (D = 1000)	Mean	9.7670e-09	1.5616e+92	4.8316e+27	N/A	7.6578e-05	9.1116e+83	1.6993e-06	1.0103e+56	1.4012e+12	4.0119e-08
	Std.	3.3014e-08	7.3622e+92	2.6457e+28	N/A	2.5646e-04	4.6589e+84	1.2518e-06	5.5321e+56	5.4965e+12	4.4393e-08
	Worst	1.5129e-07	4.0431e+93	1.4491e+29	N/A	0.00128117	2.5521e+85	5.3454e-06	3.0301e+57	2.6100e+13	2.3807e-07
	Best	5.4973e-16	1.1184e+7	4.8301e-04	N/A	1.9892e-09	9.2943e+69	4.1424e-07	5.6083e+38	0.024180374	2.7889e-09
	Rank	1	7	5	10	2	9	4	8	6	3
f ₁₀ (D = 1000)	Mean	−1.7007e+4	−5.8435e+4	−3.1555e+4	−6.5837e+4	−1.4744e+4	−2.2849e+4	−1.1867e+5	−1.3394e+5	−1.2726e+5	−1.4785e+5
	Std.	2.5847e+3	3.4160e+3	4.8833e+3	4.3367e+3	2.3262e+3	1.3757e+3	6.9059e+3	5.9809e+3	2.7008e+4	4.6972e+4
	Worst	−1.0338e+4	−5.2541e+4	−2.3985e+4	−5.4523e+4	−1.1005e+4	−2.0040e+4	−1.0778e+5	−1.1839e+5	−9.0577e+4	−9.5352e+4
	Best	−2.1550e+4	−6.7056e04	−4.2025e+4	−7.3504e+4	−1.8442e+4	−2.6647e+4	−1.3612e+5	−1.4670e+5	−1.9652e+5	−2.6675e+5
	Rank	9	6	7	5	10	8	4	3	2	1
f ₁₁ (D = 1000)	Mean	2.8217e+03	1.5245e+04	9.3813e+03	9.4292e+03	5.7869e+03	1.8362e+03	6.3367e+03	1.3788e+04	9.9711e+03	1.6919e+3
	Std.	1.7027e+02	1.3944e+02	1.1877e+03	1.9719e+02	1.5766e+02	874.1338	155.2772	300.8269	3.1150e+02	5.4168e+2
	Worst	3.2546e+03	1.5581e+04	1.1288e+04	9.7810e+03	6.0413e+03	3.8740e+03	6.6892e+03	1.4298e+04	1.0555e+04	3.1932e+3
	Best	2.5372e+03	1.4955e+04	6.9656e+03	9.1231e+03	5.4142e+03	502.6460	6.1002e+03	1.3250e+04	9.3790e+03	1.1473e+3
	Rank	3	10	6	7	4	1	5	9	8	2
f ₁₂ (D = 1000)	Mean	4.49472157	20.7976705	15.4922	14.3714165	10.3286273	18.8639	14.4206	20.9497	10.3504556	1.83801020
	Std.	0.17456627	0.02616263	1.8936	0.40087062	0.15377404	4.0427	0.2008	0.0222	0.57923796	1.62755230
	Worst	5.06374703	20.8506176	18.6432	15.2249329	10.6339573	20.8454	14.7842	20.9935	11.1238556	5.60320124
	Best	4.24420457	20.7492288	11.3346	13.4575332	9.93588986	8.4914	13.9782	20.8992	9.19276609	0.40833143
	Rank	2	9	6	7	5	3	8	10	4	1
f ₁₃ (D = 1000)	Mean	1.0210e+03	2.3100e+04	2.9176e+03	2.2307e+03	1.3996e+04	3.1433e+03	1.7159e+03	3.2496e+03	11.8192945	0.36210206
	Std.	32.8774099	5.1266e+02	1.4869e+03	2.8098e+02	2.1571e+02	1.0529e+03	91.5869	169.9815	11.8731051	0.96226594
	Worst	1.1068e+03	2.3908e+04	6.5186e+03	2.7103e+03	1.4470e+04	5.8108e+03	1.9131e+03	3.5731e+03	53.8148847	4.60350838
	Best	9.5388e+02	2.1976e+04	725.5926	1.5753e+03	1.3539e+04	1.1471e+03	1.5693e+03	2.8639e+03	2.94729201	0.00835621
	Rank	4	10	3	7	9	5	6	8	2	1
f ₁₄ (D = 1000)	Mean	3.58238444	2.6235e+10	9.4922e+08	3.6287e+06	3.8689e+04	9.5072e+09	3.1511e+06	9.0076e+08	1.0956e+07	29.7193724
	Std.	0.46770694	9.4178e+08	8.9381e+08	1.2754e+06	2.4728e+04	2.0642e+09	1.3752e+06	1.0468e+08	6.6012e+06	10.8640272
	Worst	4.62169611	2.7753e+10	4.1170e+09	7.3906e+06	1.3000e+05	1.4117e+10	7.7277e+06	1.1163e+09	2.4899e+07	47.2595636
	Best	2.74775928	2.4014e+10	2.3930e+07	1.5174e+06	5.6569e+03	5.1822e+09	1.5134e+06	7.1553e+08	1.9753e+05	0.51065701
	Rank	2	10	7	6	3	9	5	8	4	1
f ₁₅ (D = 1000)	Mean	1.4226e+04	4.9243e+10	2.0082e+09	5.5104e+07	6.1049e+06	1.6105e+10	7.2399e+07	2.2732e+09	7.41299e+07	1.1273e+3
	Std.	2.2790e+04	1.6004e+09	1.4799e+09	1.6028e+07	1.2137e+06	3.5103e+09	1.0071e+07	2.8365e+08	3.8466e+07	6.6367e+2
	Worst	7.5633e+04	5.3105e+10	6.5263e+09	8.3955e+07	8.3946e+06	2.3878e+10	8.9785e+07	2.9667e+09	2.1183e+08	2.8810e+3
	Best	4.0719e+02	4.6279e+10	3.7708e+08	3.3109e+07	4.2910e+06	8.3903e+09	5.4157e+07	1.8130e+09	2.4155e+07	86.1454
	Rank	2	10	7	5	3	9	6	8	4	1
Average Rank		3.066666667	9.2	5.2	6.466666667	4.866666667	7	5.733333333	7.8	4.4	1.2
Overall Rank		2	10	5	7	4	8	6	9	3	1

The bold values in the table indicate the best results.

4.4.1 Algorithm performance comparison in low dimension functions

For the low-dimensional case in the benchmark functions f1–f15, based on Table 4, except f2, f8, f9, f14, and f15, the optimal value of the SOAPSO algorithm is better than the others. To f8, f14, and f15, the PSO algorithm gives the better results, but the SOAPSO has already reached the theoretical optimal value. To f9, the optimal value of the SOAPSO has reached the theoretical best value, although the optimal fitness value of the SOAPSO is worse than the PSO and the GSA. To f2 function, the optimal value of the SOAPSO is worse than the PSO and the SCA algorithm.

For the low-dimensional case in the benchmark functions f1–f15, based on Table 4, except f2, f3, f4, f9, f10, f14, and f15, the worst value of the SOAPSO algorithm is better than the others. To f2, f3, f4, f14, and f15, the PSO algorithm gives the better results. To f9, the worst fitness value of the SOAPSO is worse than the PSO and the GSA. To f10 function, the optimal value of the SOAPSO is worse than the SA_GA, SSA and the MVO algorithm.

For the low-dimensional case in the benchmark functions f1–f15, based on Table 4, except f2, f3, f4, f9, f10, f11, f14, and f15 the standard deviation results of the SOAPSO algorithm are better than the others. The standard deviation results of the SOAPSO to f2, f4, f14, and f15 are only worse than the PSO algorithm. The results of the SOAPSO to f3, f9, and f11 are worse than the PSO, and the GSA. The result of the SOAPSO to f10 is worse than the PSO, DA, BSO, SSA, MVO, and the SOA algorithm.

For the low-dimensional case in the benchmark functions f1–f15, based on Table 4, except f2, f3, f4, f9, f14 and f15 the mean values of the SOAPSO are better than the others. To f9, the mean test results of the SOAPSO have reached the theoretical best value, although the mean test result of the SOAPSO is worse than the PSO and GSA. The results of the SOAPSO to f2, f3, f4, f14 and f15 are worse than the PSO algorithm.

According to the optimal fitness value mean rank and all rank results from Table 4, the SOAPSO can find solutions, and has strong optimization ability and strong robustness to the low-dimensional benchmark function.

4.4.2 Algorithm performance comparison in high dimension functions

In order to test the optimization ability of the algorithms in high-dimensional space, this paper selects 1000 dimensions for tests [57]. In all the tests, the max number of iterations is 1000, and the set of other parameters is the same. The mean results, the standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and the rank results between the algorithms of 30 independent runs for f1 ∼ f15 are shown in Table 5.

For the high-dimensional case in the benchmark functions f1–f15, based on Table 5, except f9 and f11, the optimal value of the SOAPSO algorithm is better than the others. To f9, the PSO and the GSA algorithm give the better results, but the SOAPSO has already reached the theoretical optimal value. To f11 function, the optimal value of the SOAPSO is worse than the SCA algorithm.

For the high -dimensional case in the benchmark functions f1–f15, based on Table 5, except f4, f7, f9, f12, and f14, the worst value of the SOAPSO algorithm is better than the others. To f4 function, the optimal value of the SOAPSO is worse than the PSO, GSA and the SOA algorithm. To f7 function, the optimal value of the SOAPSO is worse than the PSA and the SSA algorithm. To f9, the worst fitness value of the SOAPSO is worse than the PSO, but the SOAPSO has already reached the theoretical optimal value. To f12 and f14, the PSO algorithm gives the better results.

For the high -dimensional case in the benchmark functions f1–f15, based on Table 5, except f3, f7, f9, f10, f11, f12, and f14, the standard deviation results of the SOAPSO algorithm are better than the others. The standard deviation results of the SOAPSO to f9, and f14 are worse than the PSO algorithm. The results of the SOAPSO to f3 are worse than the SSA and the BSO. The result of the SOAPSO to f7 is worse than the PSO, and the SSA. The results of the SOAPSO to f10 are worse than the that of others. The result of the SOAPSO to f11 is worse than the PSO, SA_GA, BSO, GSA, SSA, MVO and the SOA. The results of the SOAPSO to f12 are worse than the PSO, SA_GA, BSO, GSA, SCA, SSA, and the SOA algorithm.

For the high-dimensional case in the benchmark functions f1–f15, based on Table 5, except f4, f7, f9 and f14 the mean values of the SOAPSO are better than the others. To f9, the mean test results of the SOAPSO have reached the theoretical best value, although the mean test result of the SOAPSO is worse than the PSO. The results of the SOAPSO to f4, f7, and f14 are worse than the PSO algorithm.

According to the optimal fitness value mean rank and all rank results from Table 5, the SOAPSO can find solutions, and has strong optimization ability and strong robustness to the high-dimensional benchmark function.

The same as before, the average rank based on these 15 functions’ ranking is calculated. Then, the average rank and obtain the overall rank are ranked. From the average rank of each algorithm, it shows that the SOAPSO is very robust and efficient.

4.4.3 Search history of the SOAPSO

Figure 1 shows the graph of the optimized function f1, the initial population’s positions, the population’s positions search history, and the convergence curves; the search history behaviors of the search seekers are marked with red mark +. Based on Figure 1, for the benchmark function f1, the convergence curve of the SOAPSO algorithm is the fast. From the population’s positions search history of the SOAPSO algorithm, the search seekers of the SOAPSO extensively move towards promising search regions in the search space; the search seekers searched the given search space by the moment in the change search step size and different search directions; this gives the way to increase local search, escape local optima, and avoid premature convergence.

Algorithm 1:

SOAPSO.

Figure 1:

Graphs for f1, a) graph of test function f1, b) initial population, iteration = 0, best fitness = 604.6074, c) SOA, search history, best fitness = 1.2588e-09, d) PSO, search history, best fitness = 1.4698e-10 e) SOAPSO, search history, best fitness = 1.7443e-11 f) convergence curve of f1 sphere.

Similarly, Figure 2 shows the graph of the optimized function f10, the initial population’s positions, the population’s positions search history, and the convergence curves; the search history behaviors of the search seekers are marked with red mark +. Based on Figure 2, for the benchmark function f10, the convergence curve of the SOAPSO algorithm is the fast. From the population’s positions search history of the SOAPSO algorithm, the search seekers of the SOAPSO extensively move towards promising search regions in the search space; the search seekers searched the given search space by the moment in the change search step size and different search directions; this gives the way to increase local search, escape local optima, and avoid premature convergence.

Figure 2:

Graphs for f10, a) graph of test function f10, b) initial population, iteration = 0, best fitness = −613.7265 c) SOA, search history, best fitness = −837.9657, d) PSO, search history, best fitness = −617.9003, e) SOAPSO, search history, best fitness = −837.9658, f) convergence curve of f10 Schwefel.

4.4.4 Convergence of algorithm analysis in benchmark functions

Figure 3 shows the fitness curves of the best values for the benchmark functions f1–f15 (D = 1000). As seen from Figure 3, in all of these functions only except f3, f8, f11, f14, and f15, the convergence of the SOAPSO is faster, and the accuracy of the SOAPSO is better.

Figure 3:
Fitness curves of the test global minimum values for f1–f15 (D = 1000), a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel 2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2.

Figure 3:

Fitness curves of the test global minimum values for f1–f15 (D = 1000), a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel 2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2.

4.4.5 The ANOVA of algorithm analysis in benchmark functions

Figure 4 is the ANOVA for the benchmark functions f1–f15 (D = 1000). As seen from Figure 4, the SOAPSO is the most robust. The SOAPSO algorithm showed better robustness and an improved SOA. Therefore, the SOAPSO is a feasible solution in the optimization of benchmark functions.

Figure 4:
ANOVA tests of the global minimum values for f1–f15 (D = 1000), a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel 2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2.

Figure 4:

ANOVA tests of the global minimum values for f1–f15 (D = 1000), a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel 2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2.

4.4.6 Complexity analysis

The calculational complexity of the SOA is O (N.D.M), N represents the total individual count, D represents the dimension count, M represents the maximum count of algebras. The computational complexity of the first phase of the SOA stage is O (N.D.M). The hybrid strategy is introduced to calculate the O (N.D.M) value. So, the overall complexity of the SOAPSO is O (N.D.M + N.D.M). Based on the principle of the Big-O representation [58], if the count of algebras is high (M ≫ N, D), the calculational complexity is O (N.D.M). Therefore, the overall calculational complexity of the SOAPSO is almost the same as the basic SOA.

4.4.7 Statistical testing of algorithms in benchmark functions

Using the Wilcoxon’s rank-sum test [59] can discover the important differentia between the two algorithms. This test gives the value p < 0.05. Table 6 is the results of statistical testing. N/A represents the best algorithm. From Table 6, the SOAPSO is suitable for 15 functions only except f9 and f11. Therefore, the SOAPSO is better than the other algorithms.

Table 6:

The p-values of the Wilcoxon rank-sum test.

Test functions (D = 1000)	p test values of various algorithms
Test functions (D = 1000)	PSO	SA_GA	DA	BSO	GSA	SCA	SSA	MVO	SOA	SOAPSO
f ₁	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₂	2.8646e-11	1.2118e-12	3.0199e-11	Inf	3.0199e-11	1.2118e-12	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₃	5.0723e-10	3.0199e-11	3.0199e-11	1.3250e-04	4.2175e-04	3.0199e-11	5.5727e-10	3.0199e-11	2.1959e-07	N/A
f ₄	0.6627	3.0199e-11	1.4643e-10	9.5139e-06	0.7618	3.0199e-11	1.0277e-06	3.0199e-11	0.0042	N/A
f ₅	3.4971e-09	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₆	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₇	5.5727e-10	3.0199e-11	3.8249e-09	3.0199e-11	3.0199e-11	3.0199e-11	0.4376	3.0199e-11	3.0199e-11	N/A
f ₈	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₉	N/A	3.0199e-11	3.0199e-11	Inf	1.3289e-10	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	2.6015e-08
f ₁₀	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	0.0392	0.8883	0.1188	N/A
f ₁₁	1.2493e-05	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A	3.0199e-11	3.0199e-11	3.0199e-11	0.9941
f ₁₂	7.0430e-07	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₁₃	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	7.3891e-11	N/A
f ₁₄	8.4848e-09	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A
f ₁₅	0.1958	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	3.0199e-11	N/A

4.4.8 Run time comparison of algorithms in benchmark functions

In the subsection, the running time of the algorithm for each function is recorded under the same conditions: the population number is 30, the evolution algebra is 1000, and 30 independent runs of the above 15 benchmark functions f1–f15 (d = 1000). Then, the running time of the 15 functions is added to obtain the sum of the 30 independent running times of each algorithm for the 15 functions listed in this paper, and the ranking of the total time, as shown in Table 7. As seen from Table 7, the PSO algorithm has the most minor program running time, followed by the SCA algorithm, which has more minor program running time. The SOAPSO algorithm ranks sixth, which has a relatively longer program running time. The program running time of the SOAPSO algorithm is shorter than the SA_GA, DA, BSO, and the GSA algorithm. At the bottom of the list is the DA and the BSO algorithm; the DA takes the most running time; the BSO algorithm can’t find the solution when optimizing function f2 and f9 (d = 1000).

Table 7:

Run time comparison of 30 independent runs for benchmark functions f1–f15 (D = 1000).

Functions (D = 1000)	Run time of algorithms
Functions (D = 1000)	PSO	SA_GA	DA	BSO	GSA	SCA	SSA	MVO	SOA	SOAPSO
f ₁	100.719730	880.039134	17573.656518	377.919485	1238.641454	123.011947	111.919349	291.672145	140.420268	212.995084
f ₂	68.229668	848.231807	16987.875185	Inf	1319.453189	158.694010	108.328681	120.413171	181.073442	226.594775
f ₃	2083.966419	59198.227940	20326.621973	2078.700365	3001.030713	1813.227548	1793.574307	1978.830114	6117.125815	7629.600897
f ₄	55.090061	763.100004	17128.983039	247.633844	1182.766535	148.610507	90.994624	282.693611	128.181110	219.457919
f ₅	59.726643	907.492580	18250.985859	258.984598	1190.101594	133.724941	109.915102	347.492206	134.319506	257.103444
f ₆	60.136433	768.869660	18085.212404	268.858467	1321.559282	127.490008	93.639898	283.733081	117.076822	200.372646
f ₇	171.086743	2893.833975	17934.262519	371.589100	1345.984756	203.877433	164.750798	366.515217	429.653999	585.376849
f ₈	56.468720	824.393464	17824.021441	279.127729	1361.309649	131.849271	92.755506	296.880364	126.427399	434.676154
f ₉	119.475367	2814.347692	18457.107878	Inf	1338.479782	202.890012	164.340304	270.227397	379.676861	727.189193
f ₁₀	96.665578	1331.028369	17637.361668	287.377356	1207.890758	162.894968	142.796763	163.620252	222.037061	525.320156
f ₁₁	67.987998	1166.055564	18049.793773	280.069048	1200.353798	147.172828	104.299900	321.039558	199.011258	431.311811
f ₁₂	85.468484	1298.972295	17955.135260	346.407765	1504.250466	168.363508	133.421811	421.893238	230.803220	372.640589
f ₁₃	105.918299	1489.631271	18681.850475	282.811171	1217.562709	147.627982	112.427627	330.277401	204.931102	349.125084
f ₁₄	227.567551	5817.019363	18773.786856	430.793297	1364.117264	381.158993	260.063965	485.116503	656.703311	957.626150
f ₁₅	252.124024	5953.028083	21243.332251	433.075547	1367.575562	320.156020	278.872164	494.213399	674.918798	958.778772
The total time	3610.631718	86954.2712	274909.9871	Inf	21161.07751	4370.749976	3762.100799	6454.617657	9942.359972	14088.16952
Overall rank	1	8	9	10	7	3	2	4	5	6

To learn more trait about the program running time of the 10 algorithms in the 15 functions, a bar chart in Figure 5 was made for the total time of each algorithm after 30 independent runs. From Figure 5, to the running time, the SOAPSO is less than the SA_GA, DA, and the GSA; the PSO is the least; the DA is the most. The running time of the BSO algorithm is not plotted, because the BSO algorithm can’t find the solution when optimizing function f2 and f9 (d = 1000).

Figure 5:

Total time of 30 independent runs of 10 algorithms on 15 benchmark functions.

4.4.9 Exploration and exploitation in benchmark functions

According to the literature [60], [61], [62], Equations (15)–(18) represent the exploration and development capability of an algorithm.

(15) D i v j = 1 n ∑ i = 1 n median ( x j ) − x i j

(16) D i v = 1 D ∑ i = 1 D D i v j

(17) X pl% = D i v D i v max × 100

(18) X pt% = | D i v − D i v max | D i v max × 100

with median x ^j: Median of dimension j in whole swarm, x _i ^j: Dimension j of the swam individual I, n: Size of swarm, Div_j: Average for all the individuals, Div: Diversity of swarm in an iteration and Div _max: Maximum diversity in all iterations, Xpl% and Xpt%: Exploration and exploitation percentages for an iteration, respectively.

Figure 6 shows the exploration and exploitation abilities of the SOAPSO as the number of iterations increases in the benchmark functions f1–f15. As observed from the plotted curves shown in Figure 6, the SOAPSO maintain good balance between the exploration and exploitation ratios as the number of iterations increases.

Figure 6:
Exploration and exploitation abilities of the SOAPSO in benchmark functions, a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2 .

Figure 6:

Exploration and exploitation abilities of the SOAPSO in benchmark functions, a) f1 Sphere, b) f2 Schwefel 2.22, c) f3 Rotated, d) f4 Schwefel2.21, e) f5 Rosenbrock, f) f6 Step, g) f7 Quartic, h) f8 SumSquares, i) f9 SumPower, j) f10 Schwefel, k) f11 Rastrigin, l) f12 Ackley, m) f13 Griewank, n) f14 Penalized1, o) f15 Penalized2 .

4.4.10 Performance profiles of algorithms in benchmark functions

The average fitness was selected as the capability index. The algorithmic capability is expressed in performance profiles, which is calculated by Equations (19) and (20).

(19) r f , g = μ f , g / min { μ f , g : g ∈ G }

(20) ρ g ( T ) = size { f ∈ F : r f , g ≤ T } / n f

with g: An algorithm, G : Algorithms set, f: A function, F : Function set, n _g: Count of algorithms in the experiment, n _f: Number of functions in the experiment, μ _f,g: Average fitness after the algorithm g solving function f, r _f,g: Capability ratio, ρ _g: Algorithmic capability and T: Factor of the best probability [63].

Figure 7 shows the capability ratios of the average value for the 10 algorithms on the benchmark functions f1–f15 (D = 1000). The consequences are revealed by a log scale 2. As shown in Figure 7, the SOAPSO has the highest probability. When τ = 1, the SOAPSO is about 0.73, which is better than that of the others. When τ = 9, the SOAPSO is the winner on the given test functions is about 1, the PSO is 0.53, SA_GA is 0.13, DA is 0.27, BSO is 0.33, GSA is 0.27, SCA is 0.2, SSA is 0.33, MVO is 0.27, and the SOA is 0.33. When τ = 15, the SOAPSO is the winner on the given test functions is about 1, the PSO is 0.6, SA_GA is 0.2, DA is 0.27, BSO is 0.33, GSA is 0.27, SCA is 0.2, SSA is 0.4, MVO is 0.33, and the SOA is 0.33. Regarding the performance curve, the SOAPSO is the best; the SOAPSO can achieve 100% when τ ≥ 9. Thus, the performance of the SOAPSO is better than that of the other algorithms.

Figure 7:

Performance profile of 10 algorithms on 15 benchmark functions.

4.5 Algorithm performance comparison in PID parameter optimization

In this section, to test the performance of the SOAPSO algorithm, five test control system functions are used optimizing PID parameters by algorithm. Equations (21)–(25) showed the test control system functions optimizing PID parameters used in the experiment.

(21) g 1 ( s ) = 2.6 ( 2.7 s + 1 ) ( 0.3 s + 1 )

(22) g 2 ( s ) = 981.966306 804.882485 s 2 + 459.744086 s + 2.32887523

(23) g 3 ( s ) = 5 ( 2.7 s + 1 ) e − 0.5 s

(24) g 4 ( s ) = 3 ( 2 s + 1 ) e − 3 s

(25) g 5 ( s ) = 1 ( s + 1 ) 8

In order to test the performance of the SOAPSO algorithm, the SOAPSO algorithm has been compared with the PSO, SA-GA, DA, BSO, GSA, SCA, SSA, MVO and the SOA algorithm in the PID controller parameter optimization. The mean results, the standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and the rank results between the algorithms of 30 independent runs for g1 ∼ g5 are shown in Table 8. To g1 ∼ g4, the population size of the algorithms is 20, the max number of iterations of all tests is 20, the step response time set 10s. To g5, the population size of algorithms is 50, the max number of iterations of all tests is 50, the step response time set 50s. Figure 8 shows the process diagram for optimizing test control system PID parameters by the SOAPSO. Figure 9 shows the optimization PID parameters model structure of the test control system. Figure 10 is shows the fitness curves. Figure 11 is shows the ANOVA tests of the global minimum values and Figure 12 shows the unit step functions PID controller parameter optimization for g1 to g5.

Table 8:

Comparison of performance of algorithms in PID controller parameter optimization of 30 independent runs.

Test functions	Result	Algorithms
Test functions	Result	PSO	SA_GA	DA	BSO	GSA	SCA	SSA	MVO	SOA	SOAPSO
g₁	Mean	0.2267	0.3169	0.1462	0.4320	0.4571	0.0918	0.0651	0.2501	0.1917	0.0500
	Std.	0.0877	0.0649	0.0803	0.1403	0.1569	0.0263	0.0513	0.0532	0.11226	0.00235
	Worst	0.4205	0.4321	0.3062	0.8831	0.8622	0.1139	0.3329	0.2668	0.42650	0.05830
	Best	0.0485	0.1002	0.0503	0.2630	0.2732	0.0483	0.0479	0.0513	0.05774	0.04785
	Rank	4	8	5	9	10	3	2	6	7	1
g₂	Mean	0.35251	0.5053	0.2967	0.6804	0.6494	0.1007	0.2699	0.3953	0.28875	0.10254
	Std.	0.20955	0.1059	0.2134	0.1516	0.1232	8.7707e-4	0.2122	0.1963	0.18233	0.00700
	Worst	0.54032	0.6176	0.5411	1.0432	0.9683	0.1034	0.5448	0.5343	0.53616	0.13865
	Best	0.1000003	0.1555	0.1000	0.3982	0.3806	0.1000	0.0996	0.1000	0.09991	0.09979
	Rank	7	8	4	10	9	4	1	4	3	2
g₃	Mean	58.4757	62.4599	58.1507	62.4345	60.7787	24.8454	30.9646	59.5805	42.1538	0.49115
	Std.	7.75976	0.1216	9.5559	0.8076	5.3034	21.5239	29.6325	7.6556	27.9025	0.13170
	Worst	62.5971	62.6105	62.4997	63.0899	64.1487	62.5282	62.5897	62.5030	62.5479	0.82873
	Best	36.0409	62.0356	14.3698	58.3042	42.7711	0.4898	0.3392	32.6095	0.39301	0.32158
	Rank	7	10	5	9	8	4	2	6	3	1
g₄	Mean	1.8481e+2	2.7179e+2	1.3533e+2	2.7181e+2	2.7665e+2	29.0458	41.7597	1.0848e+2	2.6269e+2	10.0080
	Std.	59.6434	0.62334	71.5086	5.5362	10.3088	11.9839	40.9625	56.6750	44.8106	1.01517
	Worst	3.1486e+2	2.7391e+2	1.9761e+2	2.7955e+2	3.1639e+2	50.000	1.4962e+2	1.9618e+2	2.7470e+2	12.98208
	Best	32.5445	2.71191	24.9192	2.4361e+2	2.7139e+2	14.5588	9.10102	20.0492	26.5763	8.74922
	Rank	8	1	6	9	10	4	3	5	7	2
g₅	Mean	1.7713e+2	55.3556	81.651133	41.441442	2.3413e+2	85.196656	64.196288	35.721213	46.10528	34.642843
	Std.	4.2182e+2	36.00807	79.815329	11.386779	2.1754e+2	1.0050e+2	41.708210	1.411226	26.992197	0.0536118
	Worst	2.1298e+3	163.6571	3.6475e+2	90.501277	1.1508e+3	4.2034e+2	2.1021e+2	41.285971	1.8468e+2	34.920259
	Best	34.625063	34.6294	34.638061	35.045432	58.321733	34.867448	34.625153	34.643162	34.745734	34.62476
	Rank	2	4	5	9	10	8	3	6	7	1
Average Rank		5.6	6.2	5	9.2	9.4	4.6	2.2	5.4	5.4	1.4
Overall Rank		7	8	4	9	10	3	2	5	5	1

The bold values in the table indicate the best results.

Figure 8:

A process diagram for optimizing test control system PID parameters by the SOAPSO.

Figure 9:

Optimization PID parameters model structure of test control system.

Figure 10:

Convergence curves for PID controller parameter optimization, a) g1, b) g2, c) g3, d) g4, e) g5.

Figure 11:

ANOVA tests for PID controller parameter optimization, a) g1, b) g2, c) g3, d) g4, e) g5.

Figure 12:

Unit step functions PID controller parameter optimization, a) g1, b) g2, c) g3, d) g4, e) g5.

As can be seen from Table 8, the SOAPSO algorithm has the fastest global convergence speed and the highest convergence precision in all of these functions only except g2 and g4, for the benchmark function g2, the SOAPSO algorithm has already reached the theoretical optimal value. From the evolutionary process of fitness value, it can be seen that the SOAPSO algorithm has a strong ability to find the optimal solutions.

For the PID controller parameter optimization, the solution of SOAPSO algorithm is the smallest value to the mean results, the standard deviation (Std.) results, and the worst fitness value. The SOAPSO algorithm can converge within the maximum number of iterations, and it has a faster global convergence speed in many functions and higher convergence precision. Moreover, as seen from Figure 11, the SOAPSO is the most robust in these algorithms. In the last, as seen from Figure 12, by the SOAPSO algorithm to optimization the unit step functions PID controller parameter for g1–g5, the unit step functions tend to stabilize very quickly and accurately. Therefore, the SOAPSO is also an effective and feasible solution for optimization problems in control system functions optimizing PID parameters by algorithm.

4.6 Algorithm performance comparison in constrained engineering problems

Six constrained engineering problems are used to test the capability of the SOAPSO further. These constrained engineering problems are very popular in the literature. The penalty function is used to calculate the constrained problem. The parameters set for all of the heuristic algorithms still adopt the parameter setting Table 3 of Section 4.3.

4.6.1 Welded beam design

This is a least fabrication cost problem, which has four parameters and seven constraints. The formulations of the problem are Equations (26)–(43). The parameters of the structural system are shown in Figure 13 [9]. Some of the works come from these kinds of literature: GSA [8], MFO [9], MVO [13], CPSO [64], and HS [65]. For the problem in this paper, the SOAPSO is compared to the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, and the SOA, and provide the best-obtained values in Table 9.

(26) Consider x → = [ x 1 , x 2 , x 3 , x 4 ] = [ h , l , t , b ] ,

(27) Minimize f ( x → ) = 1.10471 x 1 2 x 2 + 0.04811 x 3 x 4 ( 14 + x 2 ) ,

(28) Subject to g 1 ( x → ) = τ ( x → ) − τ max ≤ 0 ,

(29) g 2 ( x → ) = σ ( x → ) − σ max ≤ 0 ,

(30) g 3 ( x → ) = x 1 − x 4 ≤ 0 ,

(31) g 4 ( x → ) = 1.10471 x 1 2 + 0.04811 x 3 x 4 ( 14 + x 2 ) − 5 ≤ 0 ,

(32) g 5 ( x → ) = 0.125 − x 1 ≤ 0 ,

(33) g 6 ( x → ) = δ ( x → ) − δ max ≤ 0 ,

(34) g 7 ( x → ) = P − P c ( x → ) ≤ 0 ,

(35) Variable range 0.1 ≤ x 1 ≤ 2 ,

(36) 0.1 ≤ x 2 ≤ 10 ,

(37) 0.1 ≤ x 3 ≤ 10 ,

(38) 0.1 ≤ x 4 ≤ 2 ,

(39) with τ ( x → ) = ( τ ′ ) 2 + 2 τ ′ τ ″ x 2 2 R + ( τ ″ ) 2 ,

(40) τ ′ = P 2 x 1 x 2 , τ ″ = M P J , M = P ( L + x 2 2 ) ,

(41) R = x 2 2 4 + ( x 1 + x 3 2 ) 2 , J = 2 { 2 x 1 x 2 [ x 2 2 4 + ( x 1 + x 3 2 ) 2 ] } ,

(42) σ ( x → ) = 6 PL x 4 x 3 2 , δ ( x → ) = 6 PL 3 E x 4 x 3 3 ,

(43) P c ( x → ) = 4.013 E x 3 2 x 4 6 36 L 2 ( 1 − x 3 2 L E 4 G ) ,

and P = 6000 lbs = 2721.6 kg, L = 14 in = 0.3556 m, E = 30 × 10⁶ psi = 206,850 MPa, G = 12 × 10⁶ psi = 82,740 MPa, τ _max = 136,000 psi = 93.772 MPa, σ _max = 30,000 psi = 206.85 MPa, δ _{ma x} = 0.25 in = 0.0625 m.

Figure 13:

Design parameters of the welded beam design problem.

Table 9:

Comparison results of the welded beam design problem.

Algorithm	Optimal values for variables				Optimal cost	Rank
Algorithm	h	l	t	b	Optimal cost	Rank
GSA [8]	0.182129	3.856979	10.0000	0.202376	1.87995	14
MFO [9]	0.2057	3.4703	9.0364	0.2057	1.72452	8
MVO [13]	0.205463	3.473193	9.044502	0.205695	1.72645	9
CPSO [64]	0.202369	3.544214	9.048210	0.205723	1.72802	10
HS [65]	0.2442	6.2231	8.2915	0.2443	2.3807	15
PSO	0.20437461682	3.27746206207	9.03907307954	0.20573458497	1.69700648019	2
SA-GA	0.26572876298	2.77789863579	7.63164040030	0.28853829376	1.99412873170	13
DA	0.204403919934271	3.270476762038852	9.060189939938688	0.205612356418801	1.698792090915354	5
BSO	0.200986698000924	3.340190407114191	9.034684361089031	0.205817980582575	1.700320898804440	7
GSA	0.12743403146	5.89076184871	8.05262845397	0.25908004232	2.10212926568	12
SCA	0.20112344041	3.23948182622	9.40574225336	0.20795790595	1.76704865429	11
SSA	0.202070956658453	3.319454357077143	9.036787303432986	0.205728837179598	1.698832459557541	6
MVO	0.20397627841	3.28970350716	9.03536739179	0.20582407425	1.69811381975	4
SOA	0.19348578918	3.489546622637	9.027709656861	0.20615302629	1.69714450048	3
SOAPSO	0.205737406556505	3.253602499355056	9.036942735165496	0.205751419536403	1.695542183515181	1

The bold values in the table indicate the best results.

In Table 9, the SOAPSO algorithm is better than the GSA, MFO, MVO, CPSO, and the HS algorithm in other kinds of literature. The SOAPSO is also better than the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, and the SOA.

4.6.2 Pressure vessel design

This is also the least fabrication cost problem of four parameters and four constraints. The formulations of the problem are Equations (44)–(53). The parameters of the structural system are shown in Figure 14 [9]. Some of the works come from the literature: MFO [9], ES [66], DE [67], ACO [68], and GA [69]. For the problem the SOAPSO is compared to the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, and the SOA, and provides the best-obtained values in Table 10.

(44) Consider x → = [ x 1 , x 2 , x 3 ] = [ T , s T , h R , L ] ,

(45) Minimize f ( x → ) = 0.6224 x 1 x 3 x 4 + 1.7781 x 2 x 3 2 + 3.1661 x 1 2 x 4 + 19.84 x 1 2 x 3 ,

(46) Subject to g 1 ( x → ) = − x 1 + 0.0193 x 3 ≤ 0 ,

(47) g 2 ( x → ) = − x 2 + 0.00954 x 3 ≤ 0 ,

(48) g 3 ( x → ) = − π x 3 2 x 4 − 4 3 π x 3 3 + 1296000 ≤ 0 ,

(49) g 4 ( x → ) = x 4 − 240 ≤ 0 ,

(50) Variable range 0 ≤ x 1 ≤ 99 ,

(51) 0 ≤ x 2 ≤ 99 ,

(52) 10 ≤ x 3 ≤ 200 ,

(53) 10 ≤ x 4 ≤ 200 ,

Figure 14:

Pressure vessel design problem.

Table 10:

Comparison of results for the pressure vessel design.

Algorithm	Optimal values for variables				Optimal cost	Rank
Algorithm	T _s	T _h	R	L	Optimal cost	Rank
MFO [9]	0.8125	0.4375	42.098445	176.636596	6059.7143	11
ES [66]	0.8125	0.4375	42.098087	176.640518	6059.7456	13
DE [67]	0.8125	0.4375	42.098411	176.637690	6059.7340	12
ACO [68]	0.8125	0.4375	42.103624	176.572656	6059.0888	10
GA [69]	0.8125	0.4375	42.097398	176.654050	6059.9463	14
PSO	0.93627266112	0.41391783346	47.19019859907	123.06285131625	6317.0167340514	15
SA-GA	0.83804097369	0.41223740796	45.10610463950	142.64078515697	5931.2868373440	8
DA	0.7424534886963	0.3694874063445	40.3200553665492	200.0000000000000	5735.104525550256	2
BSO	0.7577708260759	0.3768722559688	41.0753218111850	189.7409640585777	5764.131209400236	5
GSA	0.89533101776	0.43654377356	47.89640596198	115.96279725902	6057.9309555313	9
SCA	0.71165237901	0.39215740603	40.39056304889	200.00000000000	5903.0036698882	7
SSA	0.7453325684133	0.3714831435280	40.4653356671543	197.9814104958969	5740.503728086691	4
MVO	0.75462696023	0.37830685291	40.94839768196	191.64503059607	5764.4347452930	6
SOA	0.76961590364	41.5284631287	0.388196715944	183.84147207932	5735.1355906012	3
SOAPSO	0.7430438520196	0.3704103258374	40.3197048517771	200.0000000000000	5734.983040625856	1

The bold values in the table indicate the best results.

For the problem, the SOAPSO algorithm is better than the MFO, ES, DE, ACO, and the GA algorithm in other kinds of literature. The SOAPSO is also better than the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO and the SOA.

4.6.3 Cantilever beam design problem

The promble is determined by five parameters, and is only applied to the scope of variables of constraints. The formulations of the problem are Equations (54)–(57). The parameters of the structural system are shown in Figure 15 [9]. Some of the works come from these kinds of literature: MFO [9] CS [70], GCA [71], MMA [71], and SOS [72]. For the problem, the SOAPSO is compared to the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, and the SOA, and provides the best-obtained values in Table 11.

(54) Consider x → = [ x 1 , x 2 , x 3 , x 4 , x 5 ] ,

(55) Minimize f ( x → ) = 0.0624 ( x 1 + x 2 + x 3 + x 4 + x 5 ) ,

(56) Subject to g ( x → ) = 61 x 1 3 + 37 x 2 3 + 19 x 3 3 + 7 x 4 3 + 1 x 5 3 − 1 ≤ 0 ,

(57) Variable range 0.01 ≤ x 1 , x 2 , x 3 , x 4 , x 5 ≤ 100 ,

Figure 15:

Cantilever beam design problem.

Table 11:

Comparison of results for the cantilever beam design.

Algorithm	Optimal values for variables					Optimum weight	Rank
Algorithm	x ₁	x ₂	x ₃	x ₄	x ₅	Optimum weight	Rank
MFO [9]	5.9848717732	5.3167269243	4.4973325858	3.5136164677	2.1616202934	1.339988086	8
CS [70]	6.0089	5.3049	4.5023	3.5077	2.1504	1.33999	9
GCA [71]	6.0100	5.3000	4.4900	3.4900	2.1500	1.3400	10
MMA [71]	6.0100	5.3000	4.4900	3.4900	2.1500	1.3400	11
SOS [72]	6.01878	5.30344	4.49587	3.49896	2.15564	1.33996	4
PSO	6.007219438	5.311747232	4.505611438	3.4904346887	2.158626706	1.339963522	5
SA-GA	6.251285023	5.460509756	4.149903306	3.8032391760	1.974102742	1.350285757	14
DA	6.0563407804	5.3241124746	4.4954051989	3.50655517270	2.0949566737	1.3401929632	13
BSO	6.0357867996	5.2975225360	4.4867307802	3.49489797255	2.1588279511	1.3399679864	6
GSA	6.020285873	5.305304583	4.512114944	3.4939372220	2.142187864	1.339969652	7
SCA	5.801308754	5.589807963	4.497563735	3.4994713866	2.262668613	1.351011196	15
SSA	6.0161915790	5.3091446860	4.4940346160	3.50135576164	2.1527729288	1.3399513774	1
MVO	6.017944991	5.336576175	4.493102726	3.4797461041	2.146292918	1.340024388	12
SOA	6.014092415	5.315583298	4.484154000	3.5033360363	2.156331174	1.339957455	3
SOAPSO	6.0197181873	5.3045384437	4.4909167501	3.50400187111	2.1543547842	1.3399528868	2

The bold values in the table indicate the best results.

In Table 11, the SOAPSO algorithm proves to be better than the MFO, CS, GCA, MMA, and the SOS algorithm in other kinds of literature. The SOAPSO is also better than the PSO, SA_GA, DA, BSO, GSA, SCA, MVO, and the SOA. There is not much of a difference between the optimal value of the SOAPSO algorithm and that of SSA algorithm.

4.6.4 Piston lever problem

This is a locating the piston components problem, which has four variables and four constraints. The formulations of the problem are the Equations (58)–(66). The parameters of the structural system are shown in Figure 16 [73]. Some of these works come from the kinds of literature: DE [74], GA [74], HPSO [74], CS [75], SNS [73]. And the SOAPSO is compared to the PSO, SA_GA, GSA, SCA, MVO, and the SOA, and provides the best-obtained values for variables and the best obtained values in Table 12.

In Table 12, except for the SNS, the SOAPSO algorithm proves to be better than the DE, GA, HPSO, and the CS algorithm in other kinds of literature. The SOAPSO is also better than the PSO, SA-GA, SCA, GSA, MVO, and the SOA. The result of the SOAPSO has reached the theoretical best solution, although the optimum of the SOAPSO is worse than that of the SNS algorithm.

(58) Consider x → = [ x 1 , x 2 , x 3 , x 4 ] = [ H , B , D , X ] ,

(59) Minimize f ( x → ) = 1 4 π x 3 2 ( L 2 − L 1 ) ,

(60) Subject to g 1 ( x → ) = Q L cos θ − R × F ≤ 0 ,

(61) g 2 ( x → ) = Q ( L − x 4 ) − M max ≤ 0 ,

(62) g 3 ( x → ) = 1.2 ( L 2 − L 1 ) − L 1 ≤ 0 ,

(63) g 4 ( x → ) = x 3 2 − x 2 ≤ 0 ,

(64) with R = | − x 4 ( x 4 sin θ + x 1 ) + x 1 ( x 2 − x 4 cos θ ) | ( x 4 − x 2 ) 2 + x 1 2 ,

(65) F = π P x 3 2 4 ,

(66) L 1 = ( x 4 − x 2 ) 2 + x 1 2 ,

with θ = 45°, Q = 10,000lbs = 4536 kg, L = 240in = 6.096 m, M _max = 1.8 × 10⁶lbs/in = 315 N/m, P = 1500psi = 10.3425 MPa.

Table 12:

Comparison of results of the piston lever.

Algorithm	Optimal values for variables				Optimal value	Rank
Algorithm	x ₁(H)	x ₂(B)	x ₃(D)	x ₄(X)	Optimal value	Rank
DE [74]	129.4	2.43	119.80	4.75	159	9
GA [74]	250.0	3.96	60.03	5.91	161	11
HPSO [74]	135.5	2.48	116.62	4.75	161	11
CS [75]	0.050	2.043	120.000	4.085	8.427	5
SNS [73]	0.050	2.042	120.000	4.083	8.412698349	1
PSO	0.0500000000000	2.2004458009526	4.3429316143480	110.6668235984532	10.24259065154840	8
SA-GA	0.0500000000000	2.0417619164866	4.0830416789885	119.9999715756912	8.413739285922011	4
GSA	215.9646294854686	344.6817103997157	03.2179597335794	60.1907279614403	329.0280579311141	12
SCA	0.0589732446248	2.0456822965266	4.0848813955281	120.0000000000000	8.520961983238227	7
MVO	0.0500000000000	2.0502088605801	4.0908333358488	119.9640112340799	8.479423852552021	6
SOA	0.7649083155587	2.0351385415011	4.0554693386457	120.0000000000000	8.413476646923973	3
SOAPSO	0.0500000000000	2.0414808420678	4.0830580681750	120.0000000000000	8.412928462010163	2

The bold values in the table indicate the best results.

(67) L 2 = ( x 4 sin θ + x 1 ) 2 + ( x 2 − x 4 cos θ ) 2 ,

(68) variable range 0.05 ≤ x 1 , x 2 , x 4 ≤ 500 ,

(69) 0.05 ≤ x 3 ≤ 120 .

4.6.5 Tubular column design

This is also a minimum cost promble of two parameters and six constraints. The formulations of the problem are Equations (70)–(79). The parameters of the structural system are shown in Figure 17 [73]. Some of these works come from the kinds of literature: CS [75], ISA [76], FA [77], ASO [78], SNS [73]. And the SOAPSO is compared to the PSO, SA_GA, GSA, SCA, MVO, and the SOA, and provides the best-obtained values for variables and the best-obtained values in Table 13.

(70) Consider x → = [ x 1 , x 2 ] = [ d , t ] ,

(71) Minimize f ( x → ) = 9.8 x 1 x 2 + 2 x 1 ,

(72) Subject to g 1 ( x → ) = P π x 1 x 2 σ y − 1 ≤ 0 ,

(73) g 2 ( x → ) = 8 P L 2 π 3 E x 1 x 2 ( x 1 2 + x 2 2 ) − 1 ≤ 0 ,

(74) g 3 ( x → ) = 2.0 x 1 − 1 ≤ 0 ,

(75) g 4 ( x → ) = x 1 14 − 1 ≤ 0 ,

(76) g 5 ( x → ) = 0.2 x 2 − 1 ≤ 0 ,

(77) g 6 ( x → ) = x 2 8 − 1 ≤ 0 ,

(78) variable range 2 ≤ x 1 ≤ 14 ,

(79) 0.2 ≤ x 2 ≤ 0.8 .

with σ _y = 500 kgf/cm² = 49.03325 MPa, E = 0.85 × 10⁶ kgf/cm² = 83356.525 MPa.

Figure 16:

Cantilever beam design.

Table 13:

Comparison results of the Tubular column design.

Algorithm	Optimal values for variables		Optimal cost	Rank
Algorithm	x ₁(d)	x ₂(t)	Optimal cost	Rank
CS [75]	5.45139	0.29196	26.53217	10
ISA [76]	5.45115623	0.29196547	26.5313	8
FA [77]	N/A	N/A	26.4994969	5
ASO [78]	N/A	N/A	26.53137828	9
SNS [73]	26.5313	26.5313	26.4994969	5
PSO	5.45241248	0.29161380	26.48643791	3
SA-GA	5.48259506	0.28999485	26.54702124	11
GSA	5.46443535	0.29261817	26.59900262	12
SCA	5.45179801	0.29199765	26.50433586	7
MVO	5.45225365	0.29161486	26.48627240	1
SOA	5.45386190	0.29153427	26.48664773	4
SOAPSO	5.452336447668137	0.291608418803226	26.48643423	2

The bold values in the table indicate the best results.

In Table 13, the SOAPSO algorithm proves to be better than the CS, ISA, FA, ASO, and the SNS algorithms in other kinds of literature. Except for the MVO, the SOAPSO is also better than the PSO, SA-GA, GSA, SCA, and the SOA. Although the optimum of the SOAPSO is worse than that of the MVO, the result of the SOAPSO has reached the theoretical best solution.

4.6.6 Reinforced concrete beam design

This is an optimization problem of designing a reinforced concrete beam, which has three variables and two constraints. The formulations of the problem are Equations (80)–(86). Figure 18 is the schematic diagram [73]. Some of these works come from the kinds of literature: GHN-EP [74], FA [79], CS [75], ASO [78], SNS [73]. And the SOAPSO is compared to the PSO, SA_GA, GSA, SCA, MVO, and the SOA, and provides the best-obtained values for variables and the best-obtained values in Table 14.

(80) Consider x → = [ x 1 , x 2 , x 3 ] = [ A s , b , h ] ,

(81) Minimize f ( x → ) = 2.9 x 1 + 0.6 x 2 x 3 ,

(82) Subject to g 1 ( x → ) = x 2 x 3 − 4 ≤ 0 ,

(83) g 2 ( x → ) = 180 + 7.375 x 1 2 x 3 − x 1 x 2 ≤ 0 ,

(84) Variable range x 1 ∈ { 6 , 6.16 , 6.32 , 6.6 , 7 , 7.11 , 7.2 , 7.8 , 7.9 , 8 , 8.4 } ,

(85) x 2 ∈ { 28 , 29 , 30 , … , 40 } ,

(86) 5 ≤ x 3 ≤ 10 .

Figure 17:

Tubular column design problem.

Figure 18:

Reinforced concrete beam design problem.

Table 14:

Comparison results for reinforced concrete beam design.

Algorithm	Optimal values for variables			Optimal value	Rank
Algorithm	x ₁(A _s)	x ₂(b)	x ₃(h)	Optimal value	Rank
GHN-EP [74]	6.32	34	8.637180	362.00648	12
FA [79]	6.32	34	8.5000	359.2080	6
CS [75]	6.32	34	8.5000	359.2080	6
ASO [78]	6.32	34	8.5000	359.2080	6
SNS [73]	6.32	34	8.5000	359.2080	6
PSO	0.21973978	0.48774465	8.49953948	359.20330245	1
SA-GA	0.22271591	0.49673391	8.49953948	359.20330245	1
GSA	0.27021862	0.46197772	8.52039835	359.62412635	11
SCA	0.23315802	0.50078494	8.49841820	359.23116167	10
MVO	0.22893546	0.48404040	8.49958944	359.20335774	5
SOA	0.21878243	0.50307295	8.49954126	359.20330246	3
SOAPSO	0.20313921	0.49336095	8.49953944	359.20330245	1

The bold values in the table indicate the best results.

In Table 14, the SOAPSO algorithm proves to be better than the GHN-EP, FA, CS, ASO, and the SNS algorithm in other kinds of literature. Except for the PSO, SA_GA, and the SOA, the SOAPSO is also better than the GSA, SCA, and the MVO. Although the optimum of the SOAPSO is same that of the PSO and the SA_GA algorithm, the result of the SOAPSO has reached the theoretical best solution.

In brief, the SOAPSO algorithm proves to be better than the other algorithms in most actual studies. The SOAPSO can better resolve these practical problems than that of the other algorithms.

5 Conclusions

A SOAPSO algorithm is presented based on the hybrid of seeker optimization algorithm and particle swarm optimization. The SOAPSO algorithm is tested to the SOAPSO from different perspectives, the benchmark functions optimization, the PID control parameters optimization problems, and the constrained engineering problems.

According to the comparative analysis of the experiments, the following conclusions can be drawn:

The hybrid algorithm of seeker optimization algorithm and particle swarm optimization tend to generate in any seeker, increase the diversity of seeker, increases the search space, and avoid premature convergence.
Among the PSO, SA_GA, DA, BSO, GSA, SCA, SSA, MVO, SOA, and the SOAPSO algorithm, the SOAPSO optimization benchmark function has higher optimization capability.
The SOAPSO optimization benchmark functions have almost the same calculational complexity as the SOA.
The running time of the SOAPSO optimization benchmark function is relatively high. Among the 10 algorithms compared, the running time is better than that of the SA-GA, DA, BSO, and the GSA algorithm.
The SOAPSO maintain good balance between the exploration and exploitation as the number of iterations increases.
The SOAPSO can solve real challenging problems, such as: the PID control parameters optimization problems and the classical constrained engineering optimization problems.
Further improving and application can be incorporated into future studies.

Corresponding author: Shaomi Duan, Kunming University of Science and Technology, Kunming, Yunnan, China, E-mail: 529345484@qq.com

Funding source: National Natural Science Foundation of China http://dx.doi.org/10.13039/501100001809

Award Identifier / Grant number: 51766005

Award Identifier / Grant number: 52166001

About the authors

Haipeng Liu

Haipeng Liu is an associate professor in the Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, Yunnan, China. His research interests include: new energy technology, energy saving, and simulation, modeling and optimization of manufacturing process using intelligence techniques.

Shaomi Duan

Shaomi Duan is currently a PhD Student in the Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, China. Her research interests include: modeling and optimization of manufacturing process using statistical and computational intelligence techniques; and optimization using metaheuristics.

Huilong Luo

Huilong Luo is a professor in the Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, China. His main research interests are optimization, artificial intelligence, manufacturing processes, heat exchange, and energy saving.

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This work was supported by National Natural Science Foundation of China (http://dx.doi.org/10.13039/501100001809) Grant 51766005, 52166001.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Published Online: 2022-07-07

Published in Print: 2022-07-26

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Keywords for this article

engineering optimization problems; function optimization; hybrid algorithm; particle swarm optimization; seeker optimization algorithm

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