# A Hybrid Whale Optimization with Seagull Algorithm for Global Optimization Problems

Authors: Yanhui Che; Dengxu He
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Seagull optimization algorithm (SOA) inspired by the migration and attack behavior of seagulls in nature is used to solve the global optimization problem. However, like other well-known metaheuristic algorithms, SOA has low computational accuracy and premature convergence. Therefore, in the current work, these problems are solved by proposing the modified version of SOA. This paper proposes a novel hybrid algorithm, called whale optimization with seagull algorithm (WSOA), for solving global optimization problems. The main reason is that the spiral attack prey of seagulls is very similar to the predation behavior of whale bubble net, and the WOA has strong global search ability. Therefore, firstly, this paper combines WOA’s contraction surrounding mechanism with SOA’s spiral attack behavior to improve the calculation accuracy of SOA. Secondly, the levy flight strategy is introduced into the search formula of SOA, which can effectively avoid premature convergence of algorithms and balance exploration and exploitation among algorithms more effectively. In order to evaluate the effectiveness of solving global optimization problems, 25 benchmark test functions are tested, and WSOA is compared with seven famous metaheuristic algorithms. Statistical analysis and results comparison show that WSOA has obvious advantages compared with other algorithms. Finally, four engineering examples are tested with the proposed algorithm, and the effectiveness and feasibility of WSOA are verified.


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## Body

## 1. Introduction

Over the past ten years, the metaheuristic algorithm has become very popular. For example, it has been widely used to find solutions to many complex problems in engineering and computer science. The main reasons for its popularity are flexibility, gradient-free mechanism, and avoiding falling into local optimization. In other words, metaheuristic algorithms only need to look at the input and output to consider optimization problems and do not need to calculate the derivative of search space, which makes them more flexible in solving various optimization problems.Compared with traditional optimization algorithms, the metaheuristic algorithm can solve various optimization problems more effectively. Inspired by the intelligent behaviors and evolution laws of natural swarm, this kind of algorithm has attracted the attention of scholars, so many well-known swarm intelligence algorithms have been proposed. For example, Genetic Algorithm (GA) [1] is a random search algorithm, which mainly imitates the natural selection in the biological world. The main reason for the success of GA is that the behaviors of selection, replication, and mutation are random, which can help the algorithm avoid falling into local optimum. Particle Swarm Optimization (PSO) algorithm [2] simulates the foraging behaviors of birds or fish in nature. Ant Colony Optimization (ACO) algorithm [3] is inspired by the behaviors of ants to find the shortest path to food. Grey Wolf Optimization (GWO) algorithm [4] simulates the hierarchical mechanism and predation behaviors of grey wolf population in nature. Moth-Flame Optimization (MFO) algorithm [5] is mainly derived from the simulation of the lateral positioning flight mode of moths. Whale Optimization Algorithm (WOA) [6] simulates the predatory behaviors of humpback whale populations in nature. Squirrel Search Algorithm (SqSA) [7] is mainly inspired by squirrel’s foraging behavior. The inspiration of Sea Lion Optimization (SLnO) algorithm [8] mainly comes from the hunting behavior of sea lions. Bald Eagle Search (BES) [9] is mainly inspired by vulture hunting strategy. Chimp optimization algorithm (COA) [10] is inspired by the behavior of chimpanzee’s individual intelligence and sexual motivation in hunting. Artificial Electric Field Algorithm (AEFA) [11] is mainly inspired by Coulomb’s electrostatic force law and Newton’s law of motion. Invasive Weed Optimization (IWO) [12] algorithm is mainly inspired by the process of weed invasion in nature. Artificial Bee Colony (ABC) [13] simulates the hunting honey behaviors of bees in nature.Although each metaheuristic algorithm has its own unique advantages, according to the theorem of no free lunch [14], no algorithm can solve all optimization problems [15]. The performance of the metaheuristic algorithm mainly depends on its exploration and exploitation ability. Therefore, many researchers constantly propose new algorithms and improve the original algorithm. For example, in order to improve the searching ability of grey wolves, Gupta and Deep proposed a novel random walk grey wolf optimizer in 2018 [16]. In 2019, the improved SSA tried to balance the exploration and exploitation of the algorithm to avoid the disadvantages of suboptimal solution and low accuracy [17]. In 2020, a new grey wolf optimization algorithm based on memory was proposed [18]. In 2020, a memory-guided sine and cosine algorithm was improved [19]. In 2020, an Archimedes Optimization Algorithm (AOA) was proposed to solve the optimization problem [20]. Hussien et al. proposed a new binary whale optimization algorithm for discrete optimization problems in 2020 [21]. Among them, combining two or more metaheuristic algorithms into a hybrid algorithm has attracted more and more attention in the field of optimization algorithms. For example, Gupta and Deep proposed a hybrid adaptive sine and cosine algorithm based on reverse learning in 2019 to solve the defects of the original algorithm such as low diversity and local optimal stagnation [22]. In 2019, a hybrid grey wolf optimizer with mutation operator was proposed to avoid the algorithm falling into local optimal solution [23]. Compared with the single algorithm, the hybrid algorithm can find the global optimal solution more effectively. In this paper, the contraction surrounding mechanism of WOA is combined with the spiral attack behaviors of SOA in order to obtain better performance.Inspired by seagull migration and attack behaviors in nature, Indian scholar Gaurav Dhiman proposed a novel bionic algorithm called seagull optimization algorithm (SOA) in 2019 [24]. The most important things about seagulls are their migration and attack behaviors. Migration behavior is that seagulls migrate from one place to another full of food. Attack behavior is that seagulls often attack other birds at sea during migration, and they will attack their prey in a spiral natural shape. Since the migration behaviors of seagulls are a large-scale flight, they are regarded as global search. In contrast, the attack behaviors of seagulls are considered as local search. Therefore, SOA based on simulating seagull migration and attack behaviors is also regarded as global search and local searching process.In recent years, scholars have conducted more extensive research on SOA. Jiang et al. studied a hybrid classification method based on the oppositional seagull optimization algorithm [25]. Jia et al. proposed a new hybrid seagull optimization algorithm for feature selection [26]. Dhiman et al. studied Emo SOA: a new evolutionary multitarget gull global optimization algorithm [27]. Panagant et al. studied the seagull optimization algorithm to solve the design optimization problem in the real world [28].Based on the characteristics of SOA and WOA, a hybrid whale optimization with seagull algorithm (WSOA) is proposed to improve the potential of SOA. In view of the shortcomings of SOA, such as premature convergence and low computational accuracy, this paper combines the contraction surrounding mechanism of whale optimization algorithm with the spiral attack behavior of seagull optimization algorithm to improve the computational accuracy of SOA. The core idea of this hybrid technology is to make use of the strong global search ability of WOA and the fast convergence ability of SOA and effectively accelerate the search ability of WSOA. In addition, the levy flight strategy is introduced into the search formula of SOA, which can effectively avoid premature convergence of algorithms and balance exploration and exploitation among algorithms more effectively.The main contributions of this paper can be summarized as follows: Firstly, a new hybrid whale optimization with seagull algorithm (WSOA) for optimization problems is proposed. WSOA takes advantage of the strong global search ability of WOA and the fast convergence ability of SOA. In addition, the levy flight strategy is introduced into the search formula of SOA, which can effectively avoid premature convergence of algorithms and balance exploration and exploitation among algorithms more effectively. Secondly, in order to verify the performance of WSOA, this paper verifies 25 benchmark test functions. Finally, the proposed WSOA is used to solve four constrained engineering optimization problems, which show that the proposed algorithm has strong performance.The rest of this paper is structured as follows: Section2 briefly introduces the original seagull optimization algorithm; Section 3 briefly introduces the original whale optimization algorithm; Section 4 introduces the hybrid whale optimization with seagull algorithm (WSOA); Section 5 is the experimental results and analysis; Section 6 introduces the limitations of WSOA; Section 7 is the conclusion of this paper and future work.


## 2. Seagull Optimization Algorithm (SOA)

Seagull optimization algorithm mainly simulates the migration and attack behaviors of seagulls in nature. Seagulls are a kind of seabird all over the world, with various species, which mainly feed on insects, fish, reptiles, amphibians, and earthworms. Seagulls are very clever birds that use their wisdom to find food and attack prey. For example, they use breadcrumbs to attract fish and their feet to imitate the sound of rain to lure earthworms hidden underground. Migration and attack are the most important behaviors of seagulls. Migration behaviors are defined as the source of food for seagulls. Attack behaviors are defined as the attack behaviors from the seagulls against the migrating birds at sea.During the migration process, the algorithm simulates how seagulls move from one place to another. At this stage, seagulls must meet the following conditions.In order to avoid collisions between seagulls and adjacent seagulls, variable A is added to calculate the new search agent position.(1)C⟶s=A×P⟶sx,where C⟶s represents the location of the search agent that does not collide with other search agents, P⟶s represents the current location of the search agent, x represents the current iteration times, and A represents the mobile behavior of the search agent in a given search space.(2)A=fc−x×fcMaxiterationwhere:x=0,1,2,…,Maxiteration,where fc is introduced to control the use frequency of variable A , which is linearly reduced from fc to 0, and in this paper fc is set to 2.After avoiding collisions between seagulls, the search agent moves in the direction of the best neighbor.(3)M⟶s=B×P⟶bsx−P⟶sx,where M⟶s represents the position of search agent P⟶s towards the best-fit search agent P⟶bs (i.e., the best seagull with a small fitness value). The behavior of B is randomized, which is responsible for proper balancing between exploration and exploitation. B is calculated as(4)B=2×A2×rd,where rd is a random number that lies in the range of [0, 1].Finally, the search agent can update its position relative to the best search agent.(5)D⟶s=C⟶s+M⟶s,where D⟶s represents the distance between the search agent and best-fit search agent (i.e., best seagull whose fitness value is less).In the process of attacking prey, seagulls will spiral motion behavior in the air. This behavior inx, y, and z planes is described as follows:(6)x′=r×cosk,(7)y′=r×sink,(8)z′=r×k,(9)r=u×ekv,where r is the radius of each turn of the spiral and k is a random number in the range [0, 2π]. u and v are constants to define the spiral shape, and e is the base of the natural logarithm. The updated position of search agent is calculated using equations (5)–(9).(10)P⟶sx=D⟶s×x′×y′×z′+P⟶bsx,where P⟶sx saves the best solution and updates the position of other search agents. The pseudocode for the SOA is provided in Algorithm 1.Algorithm 1: Pseudocode of the SOA.
Input: Seagull population P⟶sOutput: Optimal search agent P⟶bs(1)
Procedure SOA(2)
Initialize the parametersA, B and Maxiteration(3)
Setfc⟵2(4)
Setu⟵1(5)
Setv⟵1(6)
While (x<Maxiteration) do(7)
P⟶bs⟵ComputeFitness (P⟶s) /∗Calculate fitness values of search agent using ComputeFitness∗//∗Migration behavior∗/(8)
rd⟵Rand0,1 /∗To generate the random number in range [0, 1]∗/(9)
k⟵Rand0,2π /∗To generate the random number in range [0, 2π]∗//∗Attacking behavior∗/(10)
r⟵u×ekv /∗To generate the spiral behavior during migration∗/(11)
Calculate the distanceD⟶s using equation (5)(12)
P⟵x′×y′×z′ /∗Compute x, y, z planes using equations (6)–(9)∗/(13)
P⟶sx⟵D⟶s×P+P⟶bs(14)
x⟵x+1(15)
end while(16)
returnP⟶bs(17)
end procedure(1)
Procedure ComputeFitness (P⟶s)(2)
fori⟵1 to ndo /∗Here, n represents the dimension of a given problem∗/(3)
FITi⟵Fitness FunctionP⟶si,: /∗Calculate the best fitness of each individual∗/(4)
end for(5)
FITsbest⟵BESTFITs /∗Calculate the best fitness value using BEST function∗/(6)
returnFITsbest(7)
end procedure(1)
ProcedureBESTFITs(2)
BEST⟵FITs0(3)
fori⟵1 to ndo(4)
if (FITsi<Best) then(5)
Best⟵FITsi(6)
end if(7)
end for(8)
return Best /∗Return the best fitness value∗/(9)
end procedure


## 3. Whale Optimization Algorithm (WOA)

Whale optimization algorithm (WOA) is a new metaheuristic algorithm that was proposed by Australian scholar Mirialili and others in 2016. The main inspiration of the algorithm is to simulate the predation behaviors of humpback whale population and update the position of candidate solution through the process of whale population encircling prey, spiral updating position, and finding prey. The first stage of the algorithm is to surround prey and spiral bubble net attack (exploitation phase); the second stage is whales randomly looking for food (exploration phase). The following describes the detailed steps of the WOA.


### 3.1. Encircling Prey: Ifp<0.5 and A<1

Whales can identify the location of prey and surround them. Assuming that the optimal position in the current population is prey, other whale individuals surround the optimal individuals. Use formula (10) to update the position:(11)D⟶=C⟶.X⟶∗t−X⟶t,(12)X⟶t+1=X⟶∗t−A⟶.D⟶,where t is the current iteration number; X⟶∗ is the prey position; and A⟶ and C⟶ are coefficient vectors, and A⟶ and C⟶ can be expressed as(13)A⟶=2a⟶.r⟶1−a⟶,C⟶=2.r⟶2,where r⟶1 and r⟶2 are the random vectors between [0, 1]; is the convergence factor, which linearly decreases from 2 to 0 as the number of iterations increases; the formula is defined as(14)a⟶=2−2ttmax,where tmax is the maximum number of iterations.


### 3.2. Search for Prey: Ifp<0.5 and A≥1

In addition to bubble net predation behavior, whales can also randomly search for food, and the process of searching for food is the exploration stage of the algorithm. That is to say, whale individuals randomly search according to each other’s position, and the mathematical model can be expressed as follows:(15)D⟶=C⟶.X⟶rand−X⟶,(16)X⟶t+1=X⟶rand−A⟶.D⟶,where X⟶rand is the position vector of individual whales randomly selected from the current population.


### 3.3. Spiral Updating Position: Ifp≥0.5

Encircling prey and spiral update position methods are utilized during the exploitation phase of the WOA. The spiral update position formula is as follows:(17)X⟶t+1=D⟶′.ebl.cos2πl+X⟶∗t,where D⟶′=X⟶pt−X⟶t, which is the distance between the ith candidate solution and the best solution in the current generation. b is a constant and l∈−1,1. The specific process of the WOA is reflected in the pseudocode of Algorithm 2.Algorithm 2: Pseudocode of the WOA.
(1)
Generate the initial populationXii=1,2,…,NP(2)
Evaluate the fitness for each candidate solutions inXi(3)
X∗= the best candidate solutions(4)
while the halting criterion is not satisfied do(5)
fori=1 to NP do(6)
Updatea,A,C,l and P;(7)
for 1 to ndo(8)
ifP<0.5then(9)
if (A<1)(10)
D=C⋅X∗−Xt(11)
Xij=X∗j−A⋅D(12)
else if (A≥1)(13)
select a random individualXrand(14)
D=C⋅Xrand−Xt(15)
Xij=Xrandj−A⋅D|(16)
end if(17)
else ifP≥0.5 then(18)
D′=X∗−Xt(19)
Xij=D′⋅ebl⋅cos2πl+X∗(20)
end if(21)
end for(22)
Evaluate the fitness forXi(23)
end for(24)
end while


## 3.1. Encircling Prey: Ifp<0.5 and A<1

Whales can identify the location of prey and surround them. Assuming that the optimal position in the current population is prey, other whale individuals surround the optimal individuals. Use formula (10) to update the position:(11)D⟶=C⟶.X⟶∗t−X⟶t,(12)X⟶t+1=X⟶∗t−A⟶.D⟶,where t is the current iteration number; X⟶∗ is the prey position; and A⟶ and C⟶ are coefficient vectors, and A⟶ and C⟶ can be expressed as(13)A⟶=2a⟶.r⟶1−a⟶,C⟶=2.r⟶2,where r⟶1 and r⟶2 are the random vectors between [0, 1]; is the convergence factor, which linearly decreases from 2 to 0 as the number of iterations increases; the formula is defined as(14)a⟶=2−2ttmax,where tmax is the maximum number of iterations.


## 3.2. Search for Prey: Ifp<0.5 and A≥1

In addition to bubble net predation behavior, whales can also randomly search for food, and the process of searching for food is the exploration stage of the algorithm. That is to say, whale individuals randomly search according to each other’s position, and the mathematical model can be expressed as follows:(15)D⟶=C⟶.X⟶rand−X⟶,(16)X⟶t+1=X⟶rand−A⟶.D⟶,where X⟶rand is the position vector of individual whales randomly selected from the current population.


## 3.3. Spiral Updating Position: Ifp≥0.5

Encircling prey and spiral update position methods are utilized during the exploitation phase of the WOA. The spiral update position formula is as follows:(17)X⟶t+1=D⟶′.ebl.cos2πl+X⟶∗t,where D⟶′=X⟶pt−X⟶t, which is the distance between the ith candidate solution and the best solution in the current generation. b is a constant and l∈−1,1. The specific process of the WOA is reflected in the pseudocode of Algorithm 2.Algorithm 2: Pseudocode of the WOA.
(1)
Generate the initial populationXii=1,2,…,NP(2)
Evaluate the fitness for each candidate solutions inXi(3)
X∗= the best candidate solutions(4)
while the halting criterion is not satisfied do(5)
fori=1 to NP do(6)
Updatea,A,C,l and P;(7)
for 1 to ndo(8)
ifP<0.5then(9)
if (A<1)(10)
D=C⋅X∗−Xt(11)
Xij=X∗j−A⋅D(12)
else if (A≥1)(13)
select a random individualXrand(14)
D=C⋅Xrand−Xt(15)
Xij=Xrandj−A⋅D|(16)
end if(17)
else ifP≥0.5 then(18)
D′=X∗−Xt(19)
Xij=D′⋅ebl⋅cos2πl+X∗(20)
end if(21)
end for(22)
Evaluate the fitness forXi(23)
end for(24)
end while


## 4. Hybrid Whale Optimization With Seagull Algorithm (WSOA)

Many creatures in nature have similarities in the process of foraging. For example, whales have a unique predation behavior in foraging, that is, bubble net foraging behavior. The WOA designs shrinking encircling mechanism and upward spiral way to attack prey by simulating predation behavior of whales. Seagulls migrate from one place to another to find abundant food when they are looking for food. Once rich prey is found, they will attack prey in a spiral shape in the air. This paper proposes a hybrid whale optimization with seagull algorithm (WSOA) for global optimization problems, which mainly combines the shrinking encircling mechanism of the WOA with the spiral attack behaviors of the SOA, thus greatly improving the local searching and global searching ability of the algorithm. Levy flight is a mechanism of random walking behaviors, which can properly control local search. However, seagull optimization algorithm will converge prematurely. Therefore, this paper also considers the levy flight strategy to be introduced into the contraction surrounding mechanism of the WOA and the local searching stage of the SOA, which improves the exploitation ability and avoids the premature convergence of the algorithm. The update formula of contraction surrounding mechanism of the WOA with levy flight strategy is as follows:(18)D⟶s=levyd×levyd×P⟶bsx−P⟶sx,P⟶sx+1=P⟶bsx−A⟶×D⟶s,where d is the dimension of the current position vector, x is the current number of iterations, and levy flight [29] can be expressed as(19)levyx=0.01×r1×σr21/β,where r1 and r2 are random numbers in [0, 1], β is the constant, and σ can be expressed as(20)σ=Γ1+β×sinπβ/2Γ1+β/2×β×2β−1/21/β,where Γx=x−1!.The improvement of spiral attack method of the SOA is as follows:(21)D⟶s=C⟶s+B×levyd×P⟶bsx−P⟶sx,P⟶sx=D⟶s×x′×y′×z′+P⟶bsx,where C⟶s represents the location of the search agent that does not collide with other search agents, P⟶s represents the current location of the search agent, represents the distance between the search agent and best-fit search agent, and represents the best-fit search agent. Figure 1 describes the flow chart of WSOA. The pseudocode of the WSOA is shown in Algorithm 3.Figure 1
Flow chart of the WSOA.Algorithm 3: Pseudocode of the WSOA.
Input: Seagull population P⟶sOutput: Optimal search agent P⟶bs(1)
Procedure WSOA(2)
Initialize the SOA parameters and WOA parameters(3)
Setfc⟵2(4)
Setu⟵1(5)
Setv⟵1(6)
while (x<Maxiteration) do(7)
P⟶bs⟵ComputeFitness (P⟶s) /∗Migration behavior∗/(8)
rd⟵Rand0,1 /∗To generate the random number in range [0, 1]∗/(9)
k⟵Rand0,2π /∗To generate the random number in range [0, 2π]∗//∗Attacking behavior∗/(10)
r⟵u×ekv /∗To generate the spiral behavior during migration∗/(11)
if 1p<0.5(12)
if 2A<1(13)
update the position of the current search agent by the equation (18)(14)
else if2A≥1(15)
update the position of the current search agent by the equation (15)(16)
end if2(17)
else if1p≥0.5(18)
Calculate the distanceD⟶s using equation (15)(19)
p⟵x′×y′×z′ /∗Compute x, y, z planes using equations (6)–(9)∗/(20)
P⟶sx⟵D⟶s×P+P⟶bs(21)
x⟵x+1(22)
end if1(23)
end while(24)
returnP⟶bs(25)
end procedure(1)
Procedure ComputeFitness (P⟶s)(2)
fori⟵1to ndo /∗Here, n represents the dimension of a given problem∗/(3)
FITi⟵Fitness FunctionP⟶si,: /∗Calculate the best fitness of each individual∗/(4)
end for(5)
FITsbest⟵BESTFITs /∗Calculate the best fitness value using BEST function∗/(6)
returnFITsbest(7)
end procedure(1)
Procedure BEST (FITsbest)(2)
Best⟵FITs0(3)
fori⟵1 to ndo(4)
if (FITsi<Best) then(5)
Best⟵FITsi(6)
end if(7)
end for(8)
return Best /∗Return the best fitness value∗/(9)
end procedure


## 5. Experimental Results and Analysis

In this section, 25 benchmark functions are used to test the performance of the proposed WSOA. This section is divided into five parts. In Section5.1, the benchmark function is introduced in detail. Performance analysis of the proposed WSOA and comparison algorithm is shown in Section 5.2. In Section 5.3, statistical test of the algorithm is provided. Section 5.4 analyzes the results. Finally, the proposed WSOA is applied to solve engineering optimization problems in Section 5.5.The experiment is described in detail below. This article selects the seven following algorithms (SOA, WOA, PSO, GWO, IWO, MFO, and ABC) for comparative research. During the experiment, the population sizeN was set to be 30, the maximum number of iterations was 1000, and each benchmark function was run independently for 30 times. All experiments are implemented in MATLAB R2018b and are carried out on a computer with Intel (R) Core (TM) i5-10210U CPU @ 2.11 GHz system type as 64-bit Windows 10 operating system. In addition, in order to make the comparison more convenient, the specific parameters of these seven algorithms can be obtained according to the original paper, as shown in Table 1.Table 1
Parameter setting value of algorithm.
AlgorithmParameterValueWSOAControl parameter (A)[2, 0]fc2SOAControl parameter (A)[2, 0]fc2WOAADecreased from 2 to 0PSOInertia coefficient0.75Cognitive and social coeff.1.8 and 2GWOControl parameter (a⟶)[2, 0]IWONumber of initial populations (N0)5Maximum number of plant populations10Maximum number of seeds2Minimum number of seeds0MFOConvergence constant[−1, −2]Logarithmic spiral0.75ABCNumber of onlookers50% of the swarmNumber of employed bees50% of the swarmSwarm size20Number of scouts1


### 5.1. Benchmark Function

According to the references, we extracted 25 benchmark functions to verify the algorithm [30–32]. These benchmark functions are mainly divided into three categories: unimodal function (Table 2), multimodal function (Table 3), and fixed-dimensional multimodal function (Table 4). Unimodal function has only one optimal solution in a given interval, and these functions can test the exploitation ability of the algorithm. Multimodal function has multiple optimal solutions, among which there is only one global optimal solution, and these functions can test the exploration ability of the algorithm. The detailed information of these 25 benchmark functions is shown in Tables 2–4.Table 2
Unimodal function.
Benchmark test functionsDimRangefminf01=∑i=1nxi230[−100, 100]0f02x=∑i=1nxi+∏i=1nxi30[−10, 10]0f03x=∑i=1n∑j=1ixj230[−100, 100]0f04x=maxi|xi|,1≤i≤D30[−100, 100]0f05x=∑i=1D−1100xi+1−xi22+xi−1230[−30, 30]0f06x=∑i=1nxi+0.5230[−100, 100]0f07x=∑i=1nxi4+random0,130[−1.28, 1.28]0Table 3
Multimodal function.
Benchmark test functionsDimRangefminf08x=−∑i=1n−xisinxi30[−500, 500]−418.9829∗ Dimf09x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0f10x=−20exp−0.21n∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e30[−32, 32]0f11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+130[−600, 600]0f12x=π/D10sin2πy1+∑i=1D−1y−121+10sin2πy1+yD−12+∑i=1Duxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−am,xi>a0,−a≤xi≤ak−xi−zm,xi<a30[−50, 50]0f13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0Table 4
Fixed-dimensional multimodal function.
Benchmark test functionsDimRangefminf14x=1/500+∑j=1251/j+∑i=12xi−aij6−12[−65, 65]1f15x=∑i=111ai−x1bi2+bix2/bi2+bix+3x424[−5, 5]0.0003075f16x=4x12−2.1x14+1/3x16+x1x2−4x22+4x242[−5, 5]−1.0316285f17x=−1+cos12x12+x22/0.5x12+x22+22[−5.12, 5.12]−1f18x=1+x1+x2+1219−14x1+3x12−14x2+6x1x2+3x22×30+2x1−3x2218−32x1+12x12+48x2−36x1x2+27x222[−5, 5]3f19x=−∑i=14ciexp−∑j=16aijxj−pij26[0, 1]−3.32f20x=−∑i=15x−aix−aiT+ci−14[0, 10]−10.1532f21x=−∑i=17x−aix−aiT+ci−14[0, 10]−10.4029f22x=−∑i=110x−aix−aiT+ci−14[0, 10]−10.5364f23x=−cosx1cosx2exp−x1−π2−x2−π22[−100, 100]−1f24x=0.5+sin2x12+x22−0.5/1+0.001x12+x2222[−100, 100]−1f25x=∑i=1nxisinxi+0.1xi10[−10, 10]0


### 5.2. Performance Comparison and Analysis with Other Algorithms

In this experiment, mean, best, worst, and standard deviation (Std) are used as evaluation indexes. The experimental results of unimodal function are shown in Table5, those of multimodal function in Table 6, and those of fixed multimodal function in Table 7; bold numbers indicate the optimal solution of comparison algorithm.Table 5
Test results of unimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f1WSOA8.47E − 2196.06E − 2412.40E − 2170SOA3.82E − 282.90E − 313.66E − 277.17E − 28WOA2.71E − 1475.60E − 1737.03E − 1461.29E − 146PSO7.37E − 141.20E − 161.09E − 122.07E − 13GWO9.65E − 593.72E − 611.01E − 572.06E − 58IWO1259.167314.3674578.0546974.9639MFO1666.66691.06E − 0510000.00013790.4901ABC6.97230.007524860.457715.8377f2WSOA2.36E − 1121.68E − 1294.52E − 1119.30E − 112SOA5.31E − 183.89E − 193.05E − 176.93E − 18WOA6.02E − 1015.56E − 1161.81E − 993.30E − 100PSO0.00633536.60E − 080.189410.034578GWO6.95E − 358.01E − 362.59E − 346.01E − 35IWO0.0329180.0266140.0433990.0038925MFO34.66730.0005336610020.4653ABC0.498680.0290531.84890.45169f3WSOA2.46E − 1532.54E − 2037.37E − 1521.34E − 152SOA3.27E − 135.75E − 186.58E − 121.24E − 12WOA18928.80693503.175538099.65479210.6917PSO110.90428.0146427.2507108.2086GWO1.60E − 154.72E − 203.17E − 145.84E − 15IWO8484.03854740.426613760.18272323.4503MFO19544.2017465.481355008.133615635.4144ABC43922.930630253.818865581.93188548.9204f4WSOA7.93E − 981.37E − 1192.20E − 964.01E − 97SOA6.71E − 088.42E − 111.24E − 062.27E − 07WOA22.17140.04848576.524320.0257PSO3.63861.03818.14321.7835GWO1.44E − 141.18E − 151.14E − 132.30E − 14IWO38.099328.107549.43695.754MFO69.025152.790185.37078.6404ABC85.988279.237892.82723.3293f5WSOA4.72010.006945927.793410.3352SOA28.185827.182128.84960.52153WOA27.211926.347628.74560.63476PSO136.17587.10252495.505450.714GWO26.874425.923128.540.64642IWO118.150522.53341030.8263233.6927MFO18555.93316.8190085.351636367.0326ABC3413.6593512.244318158.44923832.0029f6WSOA0.0028810.000700950.00706740.001618SOA3.19862.65313.94670.36716WOA0.108240.0082720.32960.1094PSO9.21E − 143.62E − 167.66E − 131.73E − 13GWO0.68211.01E − 051.50620.37642IWO1148.66794.65E − 053399.7094988.8429MFO2336.72522.08E − 0510100.25014308.358ABC6.71920.005241397.637718.5232f7WSOA0.000167153.22E − 060.00111710.00020972SOA0.00105572.15E − 050.00399590.00088306WOA0.00217911.31E − 050.00927080.0023497PSO0.589780.0817891.15130.29928GWO0.000749070.000133940.00191630.0072772IWO0.0219270.0100260.0360710.0072772MFO4.16740.05647724.29126.6287ABC0.383070.236240.596550.095506Table 6
Test results of multimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f8WSOA−12273.3151−12569.3181−10662.1204457.2487SOA−5532.3042−7878.2295−4484.3954868.0981WOA−10934.7761−12569.4762−8367.041620.5229PSO−5.95E + 228−1.69E + 230−3.74E + 195InfGWO−6139.4458−7537.5931−3849.2598801.0418IWO−6762.544−7868.1968−6036.2644525.7308MFO−8563.3627−10255.3797−7319.1662750.0895ABC−10807.4564−11773.7621−9961.6702537.5941f9WSOA0000SOA1.7804024.72696.1853WOA0000PSO72.432838.8034122.379520.99GWO8.08060221.057940.2696IWO74.664238.8146137.312519.5274MFO146.883668.652248.80937.6504ABC76.470243.3295117.273615.6098f10WSOA8.88E − 168.88E − 168.88E − 160SOA19.959319.955519.96140.0014469WOA4.09E −158.88E − 167.99E −152.35E −15PSO1.55228.19E −094.50481.1664GWO1.69E −141.15E −142.93E −144.25E −15IWO19.072917.90519.51740.37948MFO16.24620.008309919.9636.3179ABC14.238310.372717.83782.128f11WSOA0000SOA0.004693300.0611940.013801WOA0.002021300.060640.011071PSO0.0417011.33E −150.207760.045345GWO0.004664800.029390.0086959IWO482.8997356.8605635.463567.1807MFO12.09092.57E −05180.961739.301ABC0.767220.0385141.55170.42278f12WSOA0.000322223.56E −050.000980510.00023705SOA0.309190.134530.828930.12059WOA0.00660890.00062080.025610.0069502PSO0.342462.16E − 131.54210.44148GWO0.0362690.013080.0986270.019866IWO20.18510.940631.98835.3077MFO8533335.4456.93E −06256000015.746738994.14ABC0.467420.0202191.10860.33933f13WSOA0.00936160.00100550.0510190.01256SOA1.95941.45422.43580.23239WOA0.246250.0261980.58630.15422PSO0.384291.15E − 163.59751.0906GWO0.488347.24E −051.12690.22795IWO0.000376246.61E −060.0110.0020065MFO13668758.950.00012905410062760.474866874.57ABC1.64280.163935.26681.2614Table 7
Test results of fixed-dimensional multimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f14WSOA0.9980.9980.9981.85E − 13SOA2.32070.9982.98211.1455WOA2.32070.9982.98211.1455PSO0.9980.9980.99800GWO0.9980.9980.9981.16E − 10IWO11.60695.928821.98849.0039MFO1.32930.9981.9920.5739ABC0.9980.9980.9981.17E − 13f15WSOA0.000568380.000307640.00157360.00036169SOA0.00123090.000663770.00138740.0001195WOA0.00105880.000307580.0131970.0023246PSO0.000706540.000307490.00106830.00029363GWO0.00241960.000307490.0203630.0060904IWO0.00138910.00030750.0203630.0036025MFO0.00226390.00037040.0203630.0049334ABC0.00125990.000625490.00191250.00031881f16WSOA−1.0316−1.0316−1.03161.07E − 08SOA−1.0316−1.0316−1.03164.41E − 07WOA−1.0316−1.0316−1.03162.29E − 10PSO−1.0316−1.0316−1.03166.71E − 16GWO−1.0316−1.0316−1.03164.23E − 09IWO−1.0316−1.0316−1.03164.51E − 09MFO−1.0316−1.0316−1.03166.78E − 16ABC−1.0316−1.0316−1.03161.16E − 15f17WSOA−1−1−10SOA−1−1−10WOA−0.97237−1−0.936250.032133PSO−0.99787−1−0.936250.01164GWO−1−1−10IWO−1−1−13.60E− 08MFO−0.9745−1−0.936250.031767ABC−0.99878−1−0.995830.0011721f18WSOA3337.28E − 05SOA333.00012.19E − 05WOA333.00023.51E − 05PSO3331.37E − 15GWO5.738414.7885IWO3333.46E − 07MFO3331.98E − 15ABC3.001633.02270.0043786f19WSOA−3.322−3.322−3.3223.31E − 05SOA−3.0507−3.121−3.01560.060884WOA−3.3218−3.3219−3.32160.00015049PSO−3.2824−3.322−3.20310.068643GWO−3.2409−3.322−3.19780.070281IWO−3.2031−3.2031−3.20312.42E − 06MFO−3.1785−3.2031−3.13270.039669ABC−3.322−3.322−3.3221.48E − 06f20WSOA−10.1532−10.1532−10.15313.45E − 05SOA−6.8513−10.138−0.351365.6292WOA−8.453−10.152−5.05512.9427PSO−5.0856−5.1008−5.05520.026312GWO−10.1527−10.1528−10.15240.0002039IWO−5.138−10.1532−2.63054.3432MFO−5.9615−10.1532−2.63053.8345ABC−10.1166−10.1489−10.08440.032264f21WSOA−10.4029−10.4029−10.40275.51E − 05SOA−5.8125−10.3992−0.372444.3645WOA−8.7562−10.4029−3.72422.5549PSO−5.9382−10.4029−2.75193.3146GWO−10.4025−10.4029−10.40150.00032137IWO−6.4519−10.4029−2.75193.594MFO−7.7868−10.4029−2.75193.3175ABC−10.3656−10.4029−10.19980.058166f22WSOA−10.5364−10.5364−10.53641.19E − 05SOA−7.3236−10.5326−0.947075.5222WOA−6.4993−10.5342−3.83523.5537PSO−10.5364−10.5364−10.53640GWO−10.536−10.5364−10.53570.00034804IWO−2.6783−2.8066−2.42170.22222MFO−6.0691−10.5364−3.83543.8688ABC−10.5247−10.5364−10.51830.010085f23WSOA−1−1−18.23E − 09SOA−0.99997−1−0.999931.92E − 05WOA−1−1−18.15E − 08PSO−1−1−10GWO−1−1−12.81E − 07IWO−1−1−11.29E − 09MFO−1−1−10ABC−0.99999−1−0.999842.90E − 05f24WSOA−1−1−10SOA−0.99806−1−0.990280.0039528WOA−0.99514−1−0.990280.004941PSO−0.99437−1−0.990280.004798GWO−0.99644−1−0.990280.0047621IWO−0.61609−0.99028−0.504050.13302MFO−0.9919−1−0.990280.0036828ABC−0.99159−0.99998−0.990240.0027293f25WSOA2.86E − 983.27E − 1338.57E − 971.56E − 97SOA1.21E − 241.43E − 353.64E − 236.65E − 24WOA0.425025.15E − 1203.62390.98594PSO2.98E − 152.29E − 281.85E − 143.89E − 15GWO0.000102048.13E − 680.000966150.00021053IWO0.0264430.00901620.0661020.011482MFO0.148012.88E− 224.44020.81067ABC0.0484450.00420350.172050.046518


#### 5.2.1. Analysis of Unimodal Function Test Results

Table5 shows the test results of 8 metaheuristic algorithms in testing unimodal functions f1 − f7. It can be seen from the table that the WSOA has better performance than other algorithms in most of the tested unimodal functions. Except for function f6, which ranks second after PSO, the WSOA ranks first among the other six unimodal test functions f1 − f5 and f7. The variances of functions f1 − f4 and f7 are all in the first place, indicating that the WSOA has better stability. Generally speaking, the WSOA can find the optimal solution faster than other algorithms, which shows that the WSOA has better convergence speed.Figures2–15 are the convergence curves and variance analysis diagram of unimodal test functions f1 − f7. The convergence curve here is drawn according to the average value of these eight metaheuristic algorithms and is the result of each algorithm running independently for 30 times. It can be seen from the figure that the WSOA converges faster than the other seven algorithms and can find the near-optimal solution faster. It can be seen from Figure 12 that the convergence speed of the WSOA in the early stage is faster than that of the other seven algorithms, and the convergence speed in the later stage is slightly inferior to that of PSO algorithm. The ANOVA chart can directly see the stability of the unimodal function tested by the algorithm; that is to say, the lower the box chart is, the more stable it is. It can be seen from the figure that the variance analysis chart except f5 is worse than SOA, WOA, and GWO, ranking fourth. In the ANOVA diagram of test functions f1 − f4, f6, and f7, the variance value of the WSOA is the smallest. Generally speaking, the WSOA has good stability in testing unimodal functions. Through the above experiments, it can be proved that the WSOA has strong convergence speed in solving unimodal test functions, which shows that the algorithm has strong global search ability.Figure 2
Convergence curve off1.Figure 3
Variance analysis diagram off1.Figure 4
Convergence curve off2.Figure 5
Variance analysis diagram off2.Figure 6
Convergence curve off3.Figure 7
Variance analysis diagram off3.Figure 8
Convergence curve off4.Figure 9
Variance analysis diagram off4.Figure 10
Convergence curve off5.Figure 11
Variance analysis diagram off5.Figure 12
Convergence curve off6.Figure 13
Variance analysis diagram off6.Figure 14
Convergence curve off7.Figure 15
Variance analysis diagram off7.


#### 5.2.2. Analysis of Multimodal Function Test Results

Table6 shows the test results of eight metaheuristic algorithms for multimodal test functions f8 − f13. It can be seen from the table that the WSOA’s test results for other multimodal test functions f8 − f12 are in the first place except that the test results of function f13 are slightly inferior to IWO and rank second. Furthermore, the WSOA can accurately find the corresponding theoretical optimal solution in the multimodal test functions f9 and f11, which fully proves that the WSOA has strong optimization ability. Although the best values of multimodal test functions f8 and f12 rank second, on the whole, the WSOA is superior to the other seven metaheuristic algorithms, and the proposed algorithm has better performance.Figures16–27 are the convergence curves and variance analysis graph of the multimodal test functions f8 − f13. From the convergence graph, it can be seen that the convergence speed and convergence accuracy of the WSOA in multimodal test function are superior to seven other metaheuristic algorithms. In the multimodal test function f13 (Figure 26), although the IWO algorithm gets smaller value than the WSOA in the later stage, the convergence speed of the WSOA is obviously better than that of the IWO algorithm in the early stage. On the whole, the WSOA has stronger convergence speed in testing multimodal functions. It can be seen from the analysis of variance that the WSOA can almost get the minimum value in the multimodal test function, which shows that it has good stability in the test of multimodal test function.Figure 16
Convergence curve off8.Figure 17
Variance analysis diagram off8.Figure 18
Convergence curve off9.Figure 19
Variance analysis diagram off9.Figure 20
Convergence curve off10.Figure 21
Variance analysis diagram off10.Figure 22
Convergence curve off11.Figure 23
Variance analysis diagram off11.Figure 24
Convergence curve off12.Figure 25
Variance analysis diagram off12.Figure 26
Convergence curve off13.Figure 27
Variance analysis diagram off13.


#### 5.2.3. Analysis of Test Results of Fixed-Dimensional Multimodal Function

The fixed-dimensional multimodal test function is the same as the image of multimodal test function, but their dimensions are different, and the former’s dimension is a fixed value. Table7 shows the test results of eight metaheuristic algorithms on the fixed-dimensional multimodal test functions. It can be seen that the WSOA shows that its average value ranks first in the test of the fixed-dimensional multimodal test functions f14 − f25 and the best values of the other fixed-dimensional multimodal test functions f14 and f16 − f25 rank first except the best value of function f15. Among them, the corresponding theoretical optimum values can be found for functions f16, f17, f18, f20 − f24, and so forth.Figures28–51 are the convergence curves and variance analysis graph of the test results of optimizing the fixed-dimensional multimodal function with 8 metaheuristic algorithms. It can be seen from the convergence graph that the convergence speed of the WSOA algorithm is better than that of seven other algorithms, and the convergence accuracy is also very good, which shows the effectiveness of this algorithm. It can be seen from the analysis of variance that most of the variance values in the fixed-dimensional multimodal functions f14 − f25 are the smallest, which shows that the algorithm is stable. Generally speaking, the WSOA has better convergence speed than seven other metaheuristic algorithms.Figure 28
Convergence curve off14.Figure 29
Variance analysis diagram off14.Figure 30
Convergence curve off15.Figure 31
Variance analysis diagram off15.Figure 32
Convergence curve off16.Figure 33
Variance analysis diagram off16.Figure 34
Convergence curve off17.Figure 35
Variance analysis diagram off17.Figure 36
Convergence curve off18.Figure 37
Variance analysis diagram off18.Figure 38
Convergence curve off19.Figure 39
Variance analysis diagram off19.Figure 40
Convergence curve off20.Figure 41
Variance analysis diagram off20.Figure 42
Convergence curve off21.Figure 43
Variance analysis diagram off21.Figure 44
Convergence curve off22.Figure 45
Variance analysis diagram off22.Figure 46
Convergence curve off23.Figure 47
Variance analysis diagram off23.Figure 48
Convergence curve off24.Figure 49
Variance analysis diagram off24.Figure 50
Convergence curve off25.Figure 51
Variance analysis diagram off25.


### 5.3. Statistical Tests

In order to verify the noncontingency of the results, Wilcoxon signed-rank test [33] is carried out in this paper. Wilcoxon signed-rank test compares the performance of the proposed SOA with those of other algorithms, and it is widely used to compare the performances of two optimization algorithms [34–36]. In this experiment, the mean value obtained by running the WSOA algorithm independently for 30 times is compared with the mean values of seven other algorithms, and the Wilcoxon signed-rank test result with significance level of a = 0.05 is obtained.Table8 shows the Wilcoxon signed-rank test results between the WSOA and seven comparison algorithms. In the table, “H” is marked with “1,” which indicates that there is a significant difference between the WSOA and the comparative algorithm, and it has statistical advantage. “H” marking “0” indicates that there is no obvious difference between the WSOA and the comparison algorithm. “S” marking “+”indicates that the performance of the WSOA is better than that of the comparison algorithm. “S” marking “=” indicates that the performance of the WSOA is the same as that of the comparison algorithm. In addition, the last column of each table under the heading w/t/l represents the win, flat, and lose counts of the WSOA relative to the comparison algorithm. It can be seen from the table that the ABC algorithm is worse than the WSOA in other test functions except that the test function f23 is equal to the WSOA. The WSOA is superior to or equal to SOA, WOA, PSO, GWO, IWO, and MFO in all test functions. More specifically, in the unimodal test functions f1 − f7, the WSOA is superior to the comparison algorithm, which shows that the performance of the WSOA is superior to the comparison algorithm, and the exploitation ability of the WSOA is superior to the comparison algorithm. Among the multimodal test functions f8 − f13, the WSOA is superior to GWO, IWO, MFO, and ABC. Except that f9 and f11 are equal to the WSOA, SOA and WOA are worse than the WSOA in other test functions. In the fixed multipeak test functions f14 − f25, the WSOA is superior to or equal to the comparison algorithm. To sum up, the proposed SOA is superior to the comparative algorithm in the stability of global search capability.Table 8
Wilcoxon signed-rank test results between the WSOA and seven compared algorithms.
No.WSOA versusSOAWOAPSOGWOIWOMFOABCP valueHSP valueHSP valueHSP valueHSP valueHSPvalueHSP valueHSf11.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f21.73E − 061+3.47E − 101+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f31.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f41.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f52.60E − 061+6.05E − 071+4.69E − 081+1.03E − 061+2.02E − 081+8.15E − 111+3.02E − 111+f61.73E − 061+3.02E − 111+3.02E − 111+5.57E − 101+5.57E − 101+3.99E − 041+3.69E − 111+f71.36E − 051+1.73E − 071+3.02E − 111+5.97E − 091+3.02E − 111+3.02E − 111+3.02E − 111+f81.73E − 061+3.37E − 041+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+2.61E − 101+f91.25E − 010=1.00E + 000=1.21E − 121+1.36E − 031+1.21E − 121+1.21E − 121+1.21E − 121+f101.73E − 061+1.01E − 081+1.21E − 121+2.05E − 131+1.21E − 121+1.21E − 121+1.21E − 121+f111.25E − 010=3.34E − 010=1.21E − 121+2.79E − 031+1.21E − 121+1.21E − 121+1.21E − 121+f121.73E − 061+5.49E − 111+1.86E − 010=3.02E − 111+3.02E − 111+5.09E − 081+3.02E − 111+f131.73E − 061+4.08E − 111+9.59E − 010=5.57E − 101+2.87E − 101+1.00E − 031+3.02E − 111+f148.13E − 010=1.70E − 021+1.21E − 121+1.44E − 031+6.69E − 111+3.08E − 010=3.34E − 031+f153.52E − 061+2.52E − 010=2.34E − 010=3.56E − 041+1.63E − 021+4.12E − 061+1.47E − 071+f161.92E − 061+1.86E − 091+1.72E − 121+1.44E − 031+4.86E − 031+1.21E − 121+1.75E − 111+f171.00E+000=6.42E − 051+3.34E − 010=1.00E+000=1.21E − 121+1.40E − 041+1.21E − 121+f183.49E − 010=1.58E − 010=3.16E − 121+2.17E − 010=3.20E − 091+2.33E − 111+1.78E − 041+f194.29E − 061+1.76E − 010=1.07E − 041+2.32E − 021+3.79E − 010=1.54E − 010=1.61E − 101+f208.47E − 061+3.50E − 091+5.51E − 020=1.70E − 081+1.60E − 031+5.27E − 010=2.62E − 031+f211.73E − 061+7.39E − 111+7.86E − 031+2.67E − 091+2.51E − 021+1.83E − 010=1.03E − 061+f221.36E − 051+8.10E − 101+2.22E − 021+1.60E − 071+1.54E − 010=6.16E − 031+2.15E − 061+f231.73E − 061+9.21E − 051+1.21E − 121+3.34E − 111+9.71E − 010=1.21E − 121+2.12E − 010=f243.13E − 021+1.23E − 051+5.66E − 071+3.13E − 041+1.21E − 121+8.99E − 111+1.21E − 121+f251.73E − 061+3.16E − 101+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+w/t/l20/5/020/5/020/5/023/2/022/3/021/4/024/1/0In order to further verify the reliability of the algorithm, the Friedman rank test is carried out on the experimental results [37]. Table 9 shows the results of the Friedman rank test. It can be seen from the table that the WSOA ranks first, which shows that the performance of this algorithm is better than that of the comparison algorithm.Table 9
Friedman rank test results of the algorithm.
AlgorithmsMean rankRankWSOA2.511SOA5.216WOA3.964PSO3.843GWO3.822IWO5.817MFO4.985ABC5.878


### 5.4. Result Analysis

Through the above analysis, compared with other comparison algorithms, the proposed WSOA shows superior performance, which shows that the combination of the SOA and the WOA is very successful in solving optimization problems. The reasons for the superior performance can be summarized as follows: Firstly, the local search ability of the WSOA is effectively enhanced by using the fast search ability of contraction and encirclement mechanism of the WOA and spiral attack behavior of the SOA. Secondly, levy flight strategy is introduced, which makes use of the random flight mode characteristics of levy’s long-term small-scale search and occasionally large-scale exploration, which is beneficial to the exploration and development of balanced algorithm and avoids premature convergence. Finally, the proposed SOA retains the original migration behavior, which is helpful to improve the search ability of the algorithm. Generally speaking, the performance of the WSOA is far superior to the original SOA in testing 25 benchmark functions, which shows that the proposed algorithm has strong competitiveness.


### 5.5. Solving Engineering Optimization Problems

Engineering optimization problems are common in people’s lives, but many optimization problems have complex constraints, especially the optimization of engineering structure design. In order to verify the performance of the WSOA in solving the optimization problem of engineering structure design, the design problems such as tension/compression spring, welded beam, multiplate clutch separator, and pressure vessel are verified in this experiment.


#### 5.5.1. Tension/Compression Spring Design Problem

In real life, the optimization goal of many problems is to minimize the weight of the problem. For example, the design problem of tension/compression spring is a common example. The variables of this problem include coil diameter (x1), spring coil diameter (x2), and the number of coils (x3). The constraint conditions are minimum deflection (g1X), shear stress (g2X), impact frequency (g3X), and outer diameter limit (g4X). The spring design pattern is shown in Figure 52, and the mathematical formula can be expressed as follows:(22)consider:x⟶=x1x2x3=dDN,minimize:fX=x3+2x2x12,g2X=4x22−x1x212566x2x13−x14+15180x12−1≤0,g3X=1−140.45x1x22x3≤0,g4X=x1+x21.5−1≤0,0.05≤x1≤2,0.25≤x2≤1.3,2≤x3≤15.Figure 52
Structure of the TCS design.Table10 shows the comparison results of tension/compression spring design problems. It can be seen from the table that the WSOA proposed in this paper has better optimization performance, and the optimal function value obtained in solving spring design problems is lower than those of previous studies. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 0.01267, and the corresponding optimal solution is x⟶ = [0.0512, 0.3441, 12.0663]. It can be seen that the WSOA has better optimization accuracy in solving the design problem of tension/compression spring.Table 10
Comparative results for the tension/compression spring problem.
AlgorithmsOptimum variablesOptimum weightDDNWSOA0.05120.344112.06630.01267GDA [38]0.05169250.356810811.28350590.012875GA [1]0.051480.35166111.6322010.0127048GPSO [39]0.05170.357311.2540.0127SES [40]N/AN/AN/A0.012732SCPSO [41]0.0516880.35670511.2896870.0127576AATM [42]0.0518130.3596911.1192530.0127081UABC [43]0.0516910.35676911.2859880.012683MVO [44]0.050.31595614.226230.0144644TEO [45]0.0517750.35879211.168390.012685MGWO [46]0.0513340.34819711.808090.0126799EEGWO [47]0.0516730.3563411.31130.012685


#### 5.5.2. Welded Beam Design Problem

The goal of welding beam design problem is to minimize the manufacturing cost, which must satisfy the related constraints such as shear stress (τ), beam bending stress (θ), bar buckling load (Pc), beam end deflection (δ), normal stress (σ), and boundary. The set variables include welded beam thickness (x1), welded joint length (x2), welded beam width (x3), and beam thickness (x4). See Figure 53 for the design pattern of welded beam and the mathematical formula can be expressed as follows:(23)consider:x⟶=x1x2x3x4=hltb,minimize:fx=1.10471x12x2+0.04811x3x414+x2,s.t.g1X=τX−τmax,g2X=σX−σmax,g4X=0.125−x1≤0,g5X=δX−0.25≤0,g6X=P−PcX≤0,g7X=0.10471x12+0.04811x3x414+x2−5≤0,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,0.1≤x4≤2,where τmax is the maximum acceptable shear stress, σmax is the maximum acceptable normal stress, and P is the load. We have the following:(24)τX=τ12+2τ1τ2x22R+τ22,τ1=P2x1x2,τ2=MRJ,(25)M=PL+x22,(26)JX=22x1x2x224+x1+x322.Figure 53
The structure of welded beam design.M in equation (25) and J (X) in equation (26) represent moment of inertia and polarity, respectively. The remaining parameters are shown in equation (27) to equation (31).(27)R=x224+x1+x322.(28)σX=6PLx4x32,(29)δX=6PL3Ex33x4,(30)PcX=4.013Ex32x46/36L21−x32LE4G,(31)G=12×106psi,E=30×106psi,P=6000lb,L=14in.Table11 shows the research results of welding beam design. It can be seen from the table that the function value of the WSOA in solving the design problem of welded beam is lower than that of previous research. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 1.7519, and the corresponding optimal solution is x⟶ = [0.1919, 3.7633, 9.109, 0.2054]. Therefore, it can be shown that the WSOA has better optimization accuracy in solving the design problem of welded beams.Table 11
Comparative results for the welded beam design problem.
AlgorithmsOptimum variablesOptimum costhLTbWSOA0.19193.76339.1090.20541.7519SBM [48]0.24076.4858.23990.24972.442SA [49]0.2476.1458.2720.2492.4148SCA [50]0.2446.2388.28860.24462.3854BFOA [51]0.2053.47119.03670.20572.3868EA [52]0.24436.22018.2940.24442.3816T-cell [53]0.24446.12868.29150.24442.3811FSA [54]0.24446.12588.29390.24442.3811GSA [55]0.1821293.856979100.2023761.879952HS [56]0.24426.22318.29150.24432.3807SBO [57]0.214213.49248.557710.229451.8496


#### 5.5.3. Multiplate Disc Clutch Brake Design Problem

The purpose of multiplate disc clutch brake is to optimize the total weight of multiplate disc clutch brake, in which the variables are driving force (F), inner and outer radius (ri and r0), friction surface number (Z), and disc thickness (t). Because the problem contains eight different constraints, the feasible region in the solution space only accounts for 70%, which makes it more difficult to solve the problem. See Figure 54 for the design pattern of multiplate disc clutch brake, and the mathematical formula can be expressed as follows:(32)minimize:fx=πx22−x12x3x5+1ρ,s.t.g1x=x2−x1−ΔR≥0,g2x=Lmax−x5+1x3+δ≥0,g3x=Pmax−Prz≥0,g4x=Pmaxvsrmax−Przvsr≥0,g5x=vsrmax−vsr≥0,g6x=Tmax−T≥0,g7x=Mh−sMs≥0,g8x=T≥0,where Mh=2/3μx4x5x23−x13/x22−x12 and w=πn/30rad/s.(33)A=πx22−x12mm2,Prz=x4AN/mm2,Vsr=πRsrn30mm/s,Rsr=23x22−x12x22x12mm,T=Izπn30Mh+Mfmm,Δr=20mm,Lmax=30mm,μ=0.6,Tmax=15s,μ=0.5,s=1.5,Ms=40Nm,pmax=1Mpa,ρ=0.0000078kg/mm3,vsrmax=10m/s,δ=0.5mm,s=1.5,Tmax=15s,n=250rpm,Iz=55kg/m2,Ms=40Nm,Mf=3Nm,60≤x1≤80,90≤x2≤110,1≤x3≤3,60≤x4≤1000,2≤x5≤9,i=1,2,3,4,5.Figure 54
Multiplate disc clutch brake problem.The comparison results of design problems of multiplate disc clutch brake are shown in Table12. It can be seen from the table that the WSOA shows superiority in solving this problem, and the obtained function value is lower than those of previous studies. The experiment was run independently for 30 times. The optimal fitness value of the algorithm is fx = 0.23525, and the optimal solution is x⟶ = [69.9996, 90, 1, 600, 2]. It can be shown that the WSOA has good optimization accuracy in solving the design problem of multiplate disc clutch brake.Table 12
Comparative results for multiplate disc clutch brake problem.
Algorithmsrix1r0x2tx3Fx4Zx5Optimal costWSOA69.999690160020.23525TLBO [58]7090181030.313656WCA [59]7090191030.313656PVS [60]7090198030.31366HHO [58]69.9999999992493901100030.259768993


#### 5.5.4. Pressure Vessel Design Problem

The optimization purpose of pressure vessel problem is to minimize the total cost. Constraints include material cost, forming cost, and welding cost. The variables of this problem are shell thickness (Ts), head thickness (Th), inner diameter (R), and cylindrical section length (L) [61]. See Figure 55 for the design of pressure vessel, and the mathematical formula can be expressed as follows:(34)consider:Z=z1,z2,z3,z4=h,l,t,b,minimize:fx=0.6224x1x3x4+1.7781x2x23+3.1661x12x4+19.84x12x3,subject tog1x=−x1+0.0193x3≤0,g2x=−x3+0.00954z3≤0,g3x=−πx32x4−43πx33+1,296,00≤0,g4x=x4−240≤0,where 0≤x1, x2≤99, 0≤x3, and x4≤200.Figure 55
Pressure vessel design and its features.Table13 shows the research results of the WSOA and other algorithms on pressure vessel design. It can be seen from the table that the results obtained by the WSOA algorithm are better than those obtained by other algorithms, in which the optimal fitness value is fx = 5964.6114 and the optimal solution is x⟶ = [0.8056, 0.4081, 41.7401, 181.1285]. It is shown that the algorithm has better optimization performance and accuracy in solving the design problem of pressure vessels.Table 13
Comparative results for the pressure vessel design problem.
AlgorithmsOptimum variablesOptimum costTsThRLWSOA0.80560.408141.7401181.12855964.6114GWO [4]0.81250.434542.089181176.7587316051.5639MFO [5]0.81250.437542.098445176.6365966059.7143CPSO [62]0.81250.437542.091266176.74656061.0777HAIS-GA [63]0.81250.437542.0931176.70316060.367ES [64]0.81250.437542.098087176.6405186059.745CDE [65]0.81250.437542.098411176.637696059.734BA [66]0.81250.437542.098445176.6365956059.7143WOA [6]0.81250.437542.0982699176.6389986059.741DTS-GA [67]0.81250.437542.097398176.6540476059.9463GeneAS [68]0.93750.548.329112.6796410.3811WEO [69]0.81250.437542.098444176.6366226059.71HHO [70]0.817583830.407292742.09174576176.71963526000.46259


## 5.1. Benchmark Function

According to the references, we extracted 25 benchmark functions to verify the algorithm [30–32]. These benchmark functions are mainly divided into three categories: unimodal function (Table 2), multimodal function (Table 3), and fixed-dimensional multimodal function (Table 4). Unimodal function has only one optimal solution in a given interval, and these functions can test the exploitation ability of the algorithm. Multimodal function has multiple optimal solutions, among which there is only one global optimal solution, and these functions can test the exploration ability of the algorithm. The detailed information of these 25 benchmark functions is shown in Tables 2–4.Table 2
Unimodal function.
Benchmark test functionsDimRangefminf01=∑i=1nxi230[−100, 100]0f02x=∑i=1nxi+∏i=1nxi30[−10, 10]0f03x=∑i=1n∑j=1ixj230[−100, 100]0f04x=maxi|xi|,1≤i≤D30[−100, 100]0f05x=∑i=1D−1100xi+1−xi22+xi−1230[−30, 30]0f06x=∑i=1nxi+0.5230[−100, 100]0f07x=∑i=1nxi4+random0,130[−1.28, 1.28]0Table 3
Multimodal function.
Benchmark test functionsDimRangefminf08x=−∑i=1n−xisinxi30[−500, 500]−418.9829∗ Dimf09x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0f10x=−20exp−0.21n∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e30[−32, 32]0f11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+130[−600, 600]0f12x=π/D10sin2πy1+∑i=1D−1y−121+10sin2πy1+yD−12+∑i=1Duxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−am,xi>a0,−a≤xi≤ak−xi−zm,xi<a30[−50, 50]0f13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0Table 4
Fixed-dimensional multimodal function.
Benchmark test functionsDimRangefminf14x=1/500+∑j=1251/j+∑i=12xi−aij6−12[−65, 65]1f15x=∑i=111ai−x1bi2+bix2/bi2+bix+3x424[−5, 5]0.0003075f16x=4x12−2.1x14+1/3x16+x1x2−4x22+4x242[−5, 5]−1.0316285f17x=−1+cos12x12+x22/0.5x12+x22+22[−5.12, 5.12]−1f18x=1+x1+x2+1219−14x1+3x12−14x2+6x1x2+3x22×30+2x1−3x2218−32x1+12x12+48x2−36x1x2+27x222[−5, 5]3f19x=−∑i=14ciexp−∑j=16aijxj−pij26[0, 1]−3.32f20x=−∑i=15x−aix−aiT+ci−14[0, 10]−10.1532f21x=−∑i=17x−aix−aiT+ci−14[0, 10]−10.4029f22x=−∑i=110x−aix−aiT+ci−14[0, 10]−10.5364f23x=−cosx1cosx2exp−x1−π2−x2−π22[−100, 100]−1f24x=0.5+sin2x12+x22−0.5/1+0.001x12+x2222[−100, 100]−1f25x=∑i=1nxisinxi+0.1xi10[−10, 10]0


## 5.2. Performance Comparison and Analysis with Other Algorithms

In this experiment, mean, best, worst, and standard deviation (Std) are used as evaluation indexes. The experimental results of unimodal function are shown in Table5, those of multimodal function in Table 6, and those of fixed multimodal function in Table 7; bold numbers indicate the optimal solution of comparison algorithm.Table 5
Test results of unimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f1WSOA8.47E − 2196.06E − 2412.40E − 2170SOA3.82E − 282.90E − 313.66E − 277.17E − 28WOA2.71E − 1475.60E − 1737.03E − 1461.29E − 146PSO7.37E − 141.20E − 161.09E − 122.07E − 13GWO9.65E − 593.72E − 611.01E − 572.06E − 58IWO1259.167314.3674578.0546974.9639MFO1666.66691.06E − 0510000.00013790.4901ABC6.97230.007524860.457715.8377f2WSOA2.36E − 1121.68E − 1294.52E − 1119.30E − 112SOA5.31E − 183.89E − 193.05E − 176.93E − 18WOA6.02E − 1015.56E − 1161.81E − 993.30E − 100PSO0.00633536.60E − 080.189410.034578GWO6.95E − 358.01E − 362.59E − 346.01E − 35IWO0.0329180.0266140.0433990.0038925MFO34.66730.0005336610020.4653ABC0.498680.0290531.84890.45169f3WSOA2.46E − 1532.54E − 2037.37E − 1521.34E − 152SOA3.27E − 135.75E − 186.58E − 121.24E − 12WOA18928.80693503.175538099.65479210.6917PSO110.90428.0146427.2507108.2086GWO1.60E − 154.72E − 203.17E − 145.84E − 15IWO8484.03854740.426613760.18272323.4503MFO19544.2017465.481355008.133615635.4144ABC43922.930630253.818865581.93188548.9204f4WSOA7.93E − 981.37E − 1192.20E − 964.01E − 97SOA6.71E − 088.42E − 111.24E − 062.27E − 07WOA22.17140.04848576.524320.0257PSO3.63861.03818.14321.7835GWO1.44E − 141.18E − 151.14E − 132.30E − 14IWO38.099328.107549.43695.754MFO69.025152.790185.37078.6404ABC85.988279.237892.82723.3293f5WSOA4.72010.006945927.793410.3352SOA28.185827.182128.84960.52153WOA27.211926.347628.74560.63476PSO136.17587.10252495.505450.714GWO26.874425.923128.540.64642IWO118.150522.53341030.8263233.6927MFO18555.93316.8190085.351636367.0326ABC3413.6593512.244318158.44923832.0029f6WSOA0.0028810.000700950.00706740.001618SOA3.19862.65313.94670.36716WOA0.108240.0082720.32960.1094PSO9.21E − 143.62E − 167.66E − 131.73E − 13GWO0.68211.01E − 051.50620.37642IWO1148.66794.65E − 053399.7094988.8429MFO2336.72522.08E − 0510100.25014308.358ABC6.71920.005241397.637718.5232f7WSOA0.000167153.22E − 060.00111710.00020972SOA0.00105572.15E − 050.00399590.00088306WOA0.00217911.31E − 050.00927080.0023497PSO0.589780.0817891.15130.29928GWO0.000749070.000133940.00191630.0072772IWO0.0219270.0100260.0360710.0072772MFO4.16740.05647724.29126.6287ABC0.383070.236240.596550.095506Table 6
Test results of multimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f8WSOA−12273.3151−12569.3181−10662.1204457.2487SOA−5532.3042−7878.2295−4484.3954868.0981WOA−10934.7761−12569.4762−8367.041620.5229PSO−5.95E + 228−1.69E + 230−3.74E + 195InfGWO−6139.4458−7537.5931−3849.2598801.0418IWO−6762.544−7868.1968−6036.2644525.7308MFO−8563.3627−10255.3797−7319.1662750.0895ABC−10807.4564−11773.7621−9961.6702537.5941f9WSOA0000SOA1.7804024.72696.1853WOA0000PSO72.432838.8034122.379520.99GWO8.08060221.057940.2696IWO74.664238.8146137.312519.5274MFO146.883668.652248.80937.6504ABC76.470243.3295117.273615.6098f10WSOA8.88E − 168.88E − 168.88E − 160SOA19.959319.955519.96140.0014469WOA4.09E −158.88E − 167.99E −152.35E −15PSO1.55228.19E −094.50481.1664GWO1.69E −141.15E −142.93E −144.25E −15IWO19.072917.90519.51740.37948MFO16.24620.008309919.9636.3179ABC14.238310.372717.83782.128f11WSOA0000SOA0.004693300.0611940.013801WOA0.002021300.060640.011071PSO0.0417011.33E −150.207760.045345GWO0.004664800.029390.0086959IWO482.8997356.8605635.463567.1807MFO12.09092.57E −05180.961739.301ABC0.767220.0385141.55170.42278f12WSOA0.000322223.56E −050.000980510.00023705SOA0.309190.134530.828930.12059WOA0.00660890.00062080.025610.0069502PSO0.342462.16E − 131.54210.44148GWO0.0362690.013080.0986270.019866IWO20.18510.940631.98835.3077MFO8533335.4456.93E −06256000015.746738994.14ABC0.467420.0202191.10860.33933f13WSOA0.00936160.00100550.0510190.01256SOA1.95941.45422.43580.23239WOA0.246250.0261980.58630.15422PSO0.384291.15E − 163.59751.0906GWO0.488347.24E −051.12690.22795IWO0.000376246.61E −060.0110.0020065MFO13668758.950.00012905410062760.474866874.57ABC1.64280.163935.26681.2614Table 7
Test results of fixed-dimensional multimodal test function.
FunctionsAlgorithmsMeanBestWorstStd.f14WSOA0.9980.9980.9981.85E − 13SOA2.32070.9982.98211.1455WOA2.32070.9982.98211.1455PSO0.9980.9980.99800GWO0.9980.9980.9981.16E − 10IWO11.60695.928821.98849.0039MFO1.32930.9981.9920.5739ABC0.9980.9980.9981.17E − 13f15WSOA0.000568380.000307640.00157360.00036169SOA0.00123090.000663770.00138740.0001195WOA0.00105880.000307580.0131970.0023246PSO0.000706540.000307490.00106830.00029363GWO0.00241960.000307490.0203630.0060904IWO0.00138910.00030750.0203630.0036025MFO0.00226390.00037040.0203630.0049334ABC0.00125990.000625490.00191250.00031881f16WSOA−1.0316−1.0316−1.03161.07E − 08SOA−1.0316−1.0316−1.03164.41E − 07WOA−1.0316−1.0316−1.03162.29E − 10PSO−1.0316−1.0316−1.03166.71E − 16GWO−1.0316−1.0316−1.03164.23E − 09IWO−1.0316−1.0316−1.03164.51E − 09MFO−1.0316−1.0316−1.03166.78E − 16ABC−1.0316−1.0316−1.03161.16E − 15f17WSOA−1−1−10SOA−1−1−10WOA−0.97237−1−0.936250.032133PSO−0.99787−1−0.936250.01164GWO−1−1−10IWO−1−1−13.60E− 08MFO−0.9745−1−0.936250.031767ABC−0.99878−1−0.995830.0011721f18WSOA3337.28E − 05SOA333.00012.19E − 05WOA333.00023.51E − 05PSO3331.37E − 15GWO5.738414.7885IWO3333.46E − 07MFO3331.98E − 15ABC3.001633.02270.0043786f19WSOA−3.322−3.322−3.3223.31E − 05SOA−3.0507−3.121−3.01560.060884WOA−3.3218−3.3219−3.32160.00015049PSO−3.2824−3.322−3.20310.068643GWO−3.2409−3.322−3.19780.070281IWO−3.2031−3.2031−3.20312.42E − 06MFO−3.1785−3.2031−3.13270.039669ABC−3.322−3.322−3.3221.48E − 06f20WSOA−10.1532−10.1532−10.15313.45E − 05SOA−6.8513−10.138−0.351365.6292WOA−8.453−10.152−5.05512.9427PSO−5.0856−5.1008−5.05520.026312GWO−10.1527−10.1528−10.15240.0002039IWO−5.138−10.1532−2.63054.3432MFO−5.9615−10.1532−2.63053.8345ABC−10.1166−10.1489−10.08440.032264f21WSOA−10.4029−10.4029−10.40275.51E − 05SOA−5.8125−10.3992−0.372444.3645WOA−8.7562−10.4029−3.72422.5549PSO−5.9382−10.4029−2.75193.3146GWO−10.4025−10.4029−10.40150.00032137IWO−6.4519−10.4029−2.75193.594MFO−7.7868−10.4029−2.75193.3175ABC−10.3656−10.4029−10.19980.058166f22WSOA−10.5364−10.5364−10.53641.19E − 05SOA−7.3236−10.5326−0.947075.5222WOA−6.4993−10.5342−3.83523.5537PSO−10.5364−10.5364−10.53640GWO−10.536−10.5364−10.53570.00034804IWO−2.6783−2.8066−2.42170.22222MFO−6.0691−10.5364−3.83543.8688ABC−10.5247−10.5364−10.51830.010085f23WSOA−1−1−18.23E − 09SOA−0.99997−1−0.999931.92E − 05WOA−1−1−18.15E − 08PSO−1−1−10GWO−1−1−12.81E − 07IWO−1−1−11.29E − 09MFO−1−1−10ABC−0.99999−1−0.999842.90E − 05f24WSOA−1−1−10SOA−0.99806−1−0.990280.0039528WOA−0.99514−1−0.990280.004941PSO−0.99437−1−0.990280.004798GWO−0.99644−1−0.990280.0047621IWO−0.61609−0.99028−0.504050.13302MFO−0.9919−1−0.990280.0036828ABC−0.99159−0.99998−0.990240.0027293f25WSOA2.86E − 983.27E − 1338.57E − 971.56E − 97SOA1.21E − 241.43E − 353.64E − 236.65E − 24WOA0.425025.15E − 1203.62390.98594PSO2.98E − 152.29E − 281.85E − 143.89E − 15GWO0.000102048.13E − 680.000966150.00021053IWO0.0264430.00901620.0661020.011482MFO0.148012.88E− 224.44020.81067ABC0.0484450.00420350.172050.046518


### 5.2.1. Analysis of Unimodal Function Test Results

Table5 shows the test results of 8 metaheuristic algorithms in testing unimodal functions f1 − f7. It can be seen from the table that the WSOA has better performance than other algorithms in most of the tested unimodal functions. Except for function f6, which ranks second after PSO, the WSOA ranks first among the other six unimodal test functions f1 − f5 and f7. The variances of functions f1 − f4 and f7 are all in the first place, indicating that the WSOA has better stability. Generally speaking, the WSOA can find the optimal solution faster than other algorithms, which shows that the WSOA has better convergence speed.Figures2–15 are the convergence curves and variance analysis diagram of unimodal test functions f1 − f7. The convergence curve here is drawn according to the average value of these eight metaheuristic algorithms and is the result of each algorithm running independently for 30 times. It can be seen from the figure that the WSOA converges faster than the other seven algorithms and can find the near-optimal solution faster. It can be seen from Figure 12 that the convergence speed of the WSOA in the early stage is faster than that of the other seven algorithms, and the convergence speed in the later stage is slightly inferior to that of PSO algorithm. The ANOVA chart can directly see the stability of the unimodal function tested by the algorithm; that is to say, the lower the box chart is, the more stable it is. It can be seen from the figure that the variance analysis chart except f5 is worse than SOA, WOA, and GWO, ranking fourth. In the ANOVA diagram of test functions f1 − f4, f6, and f7, the variance value of the WSOA is the smallest. Generally speaking, the WSOA has good stability in testing unimodal functions. Through the above experiments, it can be proved that the WSOA has strong convergence speed in solving unimodal test functions, which shows that the algorithm has strong global search ability.Figure 2
Convergence curve off1.Figure 3
Variance analysis diagram off1.Figure 4
Convergence curve off2.Figure 5
Variance analysis diagram off2.Figure 6
Convergence curve off3.Figure 7
Variance analysis diagram off3.Figure 8
Convergence curve off4.Figure 9
Variance analysis diagram off4.Figure 10
Convergence curve off5.Figure 11
Variance analysis diagram off5.Figure 12
Convergence curve off6.Figure 13
Variance analysis diagram off6.Figure 14
Convergence curve off7.Figure 15
Variance analysis diagram off7.


### 5.2.2. Analysis of Multimodal Function Test Results

Table6 shows the test results of eight metaheuristic algorithms for multimodal test functions f8 − f13. It can be seen from the table that the WSOA’s test results for other multimodal test functions f8 − f12 are in the first place except that the test results of function f13 are slightly inferior to IWO and rank second. Furthermore, the WSOA can accurately find the corresponding theoretical optimal solution in the multimodal test functions f9 and f11, which fully proves that the WSOA has strong optimization ability. Although the best values of multimodal test functions f8 and f12 rank second, on the whole, the WSOA is superior to the other seven metaheuristic algorithms, and the proposed algorithm has better performance.Figures16–27 are the convergence curves and variance analysis graph of the multimodal test functions f8 − f13. From the convergence graph, it can be seen that the convergence speed and convergence accuracy of the WSOA in multimodal test function are superior to seven other metaheuristic algorithms. In the multimodal test function f13 (Figure 26), although the IWO algorithm gets smaller value than the WSOA in the later stage, the convergence speed of the WSOA is obviously better than that of the IWO algorithm in the early stage. On the whole, the WSOA has stronger convergence speed in testing multimodal functions. It can be seen from the analysis of variance that the WSOA can almost get the minimum value in the multimodal test function, which shows that it has good stability in the test of multimodal test function.Figure 16
Convergence curve off8.Figure 17
Variance analysis diagram off8.Figure 18
Convergence curve off9.Figure 19
Variance analysis diagram off9.Figure 20
Convergence curve off10.Figure 21
Variance analysis diagram off10.Figure 22
Convergence curve off11.Figure 23
Variance analysis diagram off11.Figure 24
Convergence curve off12.Figure 25
Variance analysis diagram off12.Figure 26
Convergence curve off13.Figure 27
Variance analysis diagram off13.


### 5.2.3. Analysis of Test Results of Fixed-Dimensional Multimodal Function

The fixed-dimensional multimodal test function is the same as the image of multimodal test function, but their dimensions are different, and the former’s dimension is a fixed value. Table7 shows the test results of eight metaheuristic algorithms on the fixed-dimensional multimodal test functions. It can be seen that the WSOA shows that its average value ranks first in the test of the fixed-dimensional multimodal test functions f14 − f25 and the best values of the other fixed-dimensional multimodal test functions f14 and f16 − f25 rank first except the best value of function f15. Among them, the corresponding theoretical optimum values can be found for functions f16, f17, f18, f20 − f24, and so forth.Figures28–51 are the convergence curves and variance analysis graph of the test results of optimizing the fixed-dimensional multimodal function with 8 metaheuristic algorithms. It can be seen from the convergence graph that the convergence speed of the WSOA algorithm is better than that of seven other algorithms, and the convergence accuracy is also very good, which shows the effectiveness of this algorithm. It can be seen from the analysis of variance that most of the variance values in the fixed-dimensional multimodal functions f14 − f25 are the smallest, which shows that the algorithm is stable. Generally speaking, the WSOA has better convergence speed than seven other metaheuristic algorithms.Figure 28
Convergence curve off14.Figure 29
Variance analysis diagram off14.Figure 30
Convergence curve off15.Figure 31
Variance analysis diagram off15.Figure 32
Convergence curve off16.Figure 33
Variance analysis diagram off16.Figure 34
Convergence curve off17.Figure 35
Variance analysis diagram off17.Figure 36
Convergence curve off18.Figure 37
Variance analysis diagram off18.Figure 38
Convergence curve off19.Figure 39
Variance analysis diagram off19.Figure 40
Convergence curve off20.Figure 41
Variance analysis diagram off20.Figure 42
Convergence curve off21.Figure 43
Variance analysis diagram off21.Figure 44
Convergence curve off22.Figure 45
Variance analysis diagram off22.Figure 46
Convergence curve off23.Figure 47
Variance analysis diagram off23.Figure 48
Convergence curve off24.Figure 49
Variance analysis diagram off24.Figure 50
Convergence curve off25.Figure 51
Variance analysis diagram off25.


## 5.2.1. Analysis of Unimodal Function Test Results

Table5 shows the test results of 8 metaheuristic algorithms in testing unimodal functions f1 − f7. It can be seen from the table that the WSOA has better performance than other algorithms in most of the tested unimodal functions. Except for function f6, which ranks second after PSO, the WSOA ranks first among the other six unimodal test functions f1 − f5 and f7. The variances of functions f1 − f4 and f7 are all in the first place, indicating that the WSOA has better stability. Generally speaking, the WSOA can find the optimal solution faster than other algorithms, which shows that the WSOA has better convergence speed.Figures2–15 are the convergence curves and variance analysis diagram of unimodal test functions f1 − f7. The convergence curve here is drawn according to the average value of these eight metaheuristic algorithms and is the result of each algorithm running independently for 30 times. It can be seen from the figure that the WSOA converges faster than the other seven algorithms and can find the near-optimal solution faster. It can be seen from Figure 12 that the convergence speed of the WSOA in the early stage is faster than that of the other seven algorithms, and the convergence speed in the later stage is slightly inferior to that of PSO algorithm. The ANOVA chart can directly see the stability of the unimodal function tested by the algorithm; that is to say, the lower the box chart is, the more stable it is. It can be seen from the figure that the variance analysis chart except f5 is worse than SOA, WOA, and GWO, ranking fourth. In the ANOVA diagram of test functions f1 − f4, f6, and f7, the variance value of the WSOA is the smallest. Generally speaking, the WSOA has good stability in testing unimodal functions. Through the above experiments, it can be proved that the WSOA has strong convergence speed in solving unimodal test functions, which shows that the algorithm has strong global search ability.Figure 2
Convergence curve off1.Figure 3
Variance analysis diagram off1.Figure 4
Convergence curve off2.Figure 5
Variance analysis diagram off2.Figure 6
Convergence curve off3.Figure 7
Variance analysis diagram off3.Figure 8
Convergence curve off4.Figure 9
Variance analysis diagram off4.Figure 10
Convergence curve off5.Figure 11
Variance analysis diagram off5.Figure 12
Convergence curve off6.Figure 13
Variance analysis diagram off6.Figure 14
Convergence curve off7.Figure 15
Variance analysis diagram off7.


## 5.2.2. Analysis of Multimodal Function Test Results

Table6 shows the test results of eight metaheuristic algorithms for multimodal test functions f8 − f13. It can be seen from the table that the WSOA’s test results for other multimodal test functions f8 − f12 are in the first place except that the test results of function f13 are slightly inferior to IWO and rank second. Furthermore, the WSOA can accurately find the corresponding theoretical optimal solution in the multimodal test functions f9 and f11, which fully proves that the WSOA has strong optimization ability. Although the best values of multimodal test functions f8 and f12 rank second, on the whole, the WSOA is superior to the other seven metaheuristic algorithms, and the proposed algorithm has better performance.Figures16–27 are the convergence curves and variance analysis graph of the multimodal test functions f8 − f13. From the convergence graph, it can be seen that the convergence speed and convergence accuracy of the WSOA in multimodal test function are superior to seven other metaheuristic algorithms. In the multimodal test function f13 (Figure 26), although the IWO algorithm gets smaller value than the WSOA in the later stage, the convergence speed of the WSOA is obviously better than that of the IWO algorithm in the early stage. On the whole, the WSOA has stronger convergence speed in testing multimodal functions. It can be seen from the analysis of variance that the WSOA can almost get the minimum value in the multimodal test function, which shows that it has good stability in the test of multimodal test function.Figure 16
Convergence curve off8.Figure 17
Variance analysis diagram off8.Figure 18
Convergence curve off9.Figure 19
Variance analysis diagram off9.Figure 20
Convergence curve off10.Figure 21
Variance analysis diagram off10.Figure 22
Convergence curve off11.Figure 23
Variance analysis diagram off11.Figure 24
Convergence curve off12.Figure 25
Variance analysis diagram off12.Figure 26
Convergence curve off13.Figure 27
Variance analysis diagram off13.


## 5.2.3. Analysis of Test Results of Fixed-Dimensional Multimodal Function

The fixed-dimensional multimodal test function is the same as the image of multimodal test function, but their dimensions are different, and the former’s dimension is a fixed value. Table7 shows the test results of eight metaheuristic algorithms on the fixed-dimensional multimodal test functions. It can be seen that the WSOA shows that its average value ranks first in the test of the fixed-dimensional multimodal test functions f14 − f25 and the best values of the other fixed-dimensional multimodal test functions f14 and f16 − f25 rank first except the best value of function f15. Among them, the corresponding theoretical optimum values can be found for functions f16, f17, f18, f20 − f24, and so forth.Figures28–51 are the convergence curves and variance analysis graph of the test results of optimizing the fixed-dimensional multimodal function with 8 metaheuristic algorithms. It can be seen from the convergence graph that the convergence speed of the WSOA algorithm is better than that of seven other algorithms, and the convergence accuracy is also very good, which shows the effectiveness of this algorithm. It can be seen from the analysis of variance that most of the variance values in the fixed-dimensional multimodal functions f14 − f25 are the smallest, which shows that the algorithm is stable. Generally speaking, the WSOA has better convergence speed than seven other metaheuristic algorithms.Figure 28
Convergence curve off14.Figure 29
Variance analysis diagram off14.Figure 30
Convergence curve off15.Figure 31
Variance analysis diagram off15.Figure 32
Convergence curve off16.Figure 33
Variance analysis diagram off16.Figure 34
Convergence curve off17.Figure 35
Variance analysis diagram off17.Figure 36
Convergence curve off18.Figure 37
Variance analysis diagram off18.Figure 38
Convergence curve off19.Figure 39
Variance analysis diagram off19.Figure 40
Convergence curve off20.Figure 41
Variance analysis diagram off20.Figure 42
Convergence curve off21.Figure 43
Variance analysis diagram off21.Figure 44
Convergence curve off22.Figure 45
Variance analysis diagram off22.Figure 46
Convergence curve off23.Figure 47
Variance analysis diagram off23.Figure 48
Convergence curve off24.Figure 49
Variance analysis diagram off24.Figure 50
Convergence curve off25.Figure 51
Variance analysis diagram off25.


## 5.3. Statistical Tests

In order to verify the noncontingency of the results, Wilcoxon signed-rank test [33] is carried out in this paper. Wilcoxon signed-rank test compares the performance of the proposed SOA with those of other algorithms, and it is widely used to compare the performances of two optimization algorithms [34–36]. In this experiment, the mean value obtained by running the WSOA algorithm independently for 30 times is compared with the mean values of seven other algorithms, and the Wilcoxon signed-rank test result with significance level of a = 0.05 is obtained.Table8 shows the Wilcoxon signed-rank test results between the WSOA and seven comparison algorithms. In the table, “H” is marked with “1,” which indicates that there is a significant difference between the WSOA and the comparative algorithm, and it has statistical advantage. “H” marking “0” indicates that there is no obvious difference between the WSOA and the comparison algorithm. “S” marking “+”indicates that the performance of the WSOA is better than that of the comparison algorithm. “S” marking “=” indicates that the performance of the WSOA is the same as that of the comparison algorithm. In addition, the last column of each table under the heading w/t/l represents the win, flat, and lose counts of the WSOA relative to the comparison algorithm. It can be seen from the table that the ABC algorithm is worse than the WSOA in other test functions except that the test function f23 is equal to the WSOA. The WSOA is superior to or equal to SOA, WOA, PSO, GWO, IWO, and MFO in all test functions. More specifically, in the unimodal test functions f1 − f7, the WSOA is superior to the comparison algorithm, which shows that the performance of the WSOA is superior to the comparison algorithm, and the exploitation ability of the WSOA is superior to the comparison algorithm. Among the multimodal test functions f8 − f13, the WSOA is superior to GWO, IWO, MFO, and ABC. Except that f9 and f11 are equal to the WSOA, SOA and WOA are worse than the WSOA in other test functions. In the fixed multipeak test functions f14 − f25, the WSOA is superior to or equal to the comparison algorithm. To sum up, the proposed SOA is superior to the comparative algorithm in the stability of global search capability.Table 8
Wilcoxon signed-rank test results between the WSOA and seven compared algorithms.
No.WSOA versusSOAWOAPSOGWOIWOMFOABCP valueHSP valueHSP valueHSP valueHSP valueHSPvalueHSP valueHSf11.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f21.73E − 061+3.47E − 101+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f31.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f41.73E − 061+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+f52.60E − 061+6.05E − 071+4.69E − 081+1.03E − 061+2.02E − 081+8.15E − 111+3.02E − 111+f61.73E − 061+3.02E − 111+3.02E − 111+5.57E − 101+5.57E − 101+3.99E − 041+3.69E − 111+f71.36E − 051+1.73E − 071+3.02E − 111+5.97E − 091+3.02E − 111+3.02E − 111+3.02E − 111+f81.73E − 061+3.37E − 041+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+2.61E − 101+f91.25E − 010=1.00E + 000=1.21E − 121+1.36E − 031+1.21E − 121+1.21E − 121+1.21E − 121+f101.73E − 061+1.01E − 081+1.21E − 121+2.05E − 131+1.21E − 121+1.21E − 121+1.21E − 121+f111.25E − 010=3.34E − 010=1.21E − 121+2.79E − 031+1.21E − 121+1.21E − 121+1.21E − 121+f121.73E − 061+5.49E − 111+1.86E − 010=3.02E − 111+3.02E − 111+5.09E − 081+3.02E − 111+f131.73E − 061+4.08E − 111+9.59E − 010=5.57E − 101+2.87E − 101+1.00E − 031+3.02E − 111+f148.13E − 010=1.70E − 021+1.21E − 121+1.44E − 031+6.69E − 111+3.08E − 010=3.34E − 031+f153.52E − 061+2.52E − 010=2.34E − 010=3.56E − 041+1.63E − 021+4.12E − 061+1.47E − 071+f161.92E − 061+1.86E − 091+1.72E − 121+1.44E − 031+4.86E − 031+1.21E − 121+1.75E − 111+f171.00E+000=6.42E − 051+3.34E − 010=1.00E+000=1.21E − 121+1.40E − 041+1.21E − 121+f183.49E − 010=1.58E − 010=3.16E − 121+2.17E − 010=3.20E − 091+2.33E − 111+1.78E − 041+f194.29E − 061+1.76E − 010=1.07E − 041+2.32E − 021+3.79E − 010=1.54E − 010=1.61E − 101+f208.47E − 061+3.50E − 091+5.51E − 020=1.70E − 081+1.60E − 031+5.27E − 010=2.62E − 031+f211.73E − 061+7.39E − 111+7.86E − 031+2.67E − 091+2.51E − 021+1.83E − 010=1.03E − 061+f221.36E − 051+8.10E − 101+2.22E − 021+1.60E − 071+1.54E − 010=6.16E − 031+2.15E − 061+f231.73E − 061+9.21E − 051+1.21E − 121+3.34E − 111+9.71E − 010=1.21E − 121+2.12E − 010=f243.13E − 021+1.23E − 051+5.66E − 071+3.13E − 041+1.21E − 121+8.99E − 111+1.21E − 121+f251.73E − 061+3.16E − 101+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+3.02E − 111+w/t/l20/5/020/5/020/5/023/2/022/3/021/4/024/1/0In order to further verify the reliability of the algorithm, the Friedman rank test is carried out on the experimental results [37]. Table 9 shows the results of the Friedman rank test. It can be seen from the table that the WSOA ranks first, which shows that the performance of this algorithm is better than that of the comparison algorithm.Table 9
Friedman rank test results of the algorithm.
AlgorithmsMean rankRankWSOA2.511SOA5.216WOA3.964PSO3.843GWO3.822IWO5.817MFO4.985ABC5.878


## 5.4. Result Analysis

Through the above analysis, compared with other comparison algorithms, the proposed WSOA shows superior performance, which shows that the combination of the SOA and the WOA is very successful in solving optimization problems. The reasons for the superior performance can be summarized as follows: Firstly, the local search ability of the WSOA is effectively enhanced by using the fast search ability of contraction and encirclement mechanism of the WOA and spiral attack behavior of the SOA. Secondly, levy flight strategy is introduced, which makes use of the random flight mode characteristics of levy’s long-term small-scale search and occasionally large-scale exploration, which is beneficial to the exploration and development of balanced algorithm and avoids premature convergence. Finally, the proposed SOA retains the original migration behavior, which is helpful to improve the search ability of the algorithm. Generally speaking, the performance of the WSOA is far superior to the original SOA in testing 25 benchmark functions, which shows that the proposed algorithm has strong competitiveness.


## 5.5. Solving Engineering Optimization Problems

Engineering optimization problems are common in people’s lives, but many optimization problems have complex constraints, especially the optimization of engineering structure design. In order to verify the performance of the WSOA in solving the optimization problem of engineering structure design, the design problems such as tension/compression spring, welded beam, multiplate clutch separator, and pressure vessel are verified in this experiment.


### 5.5.1. Tension/Compression Spring Design Problem

In real life, the optimization goal of many problems is to minimize the weight of the problem. For example, the design problem of tension/compression spring is a common example. The variables of this problem include coil diameter (x1), spring coil diameter (x2), and the number of coils (x3). The constraint conditions are minimum deflection (g1X), shear stress (g2X), impact frequency (g3X), and outer diameter limit (g4X). The spring design pattern is shown in Figure 52, and the mathematical formula can be expressed as follows:(22)consider:x⟶=x1x2x3=dDN,minimize:fX=x3+2x2x12,g2X=4x22−x1x212566x2x13−x14+15180x12−1≤0,g3X=1−140.45x1x22x3≤0,g4X=x1+x21.5−1≤0,0.05≤x1≤2,0.25≤x2≤1.3,2≤x3≤15.Figure 52
Structure of the TCS design.Table10 shows the comparison results of tension/compression spring design problems. It can be seen from the table that the WSOA proposed in this paper has better optimization performance, and the optimal function value obtained in solving spring design problems is lower than those of previous studies. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 0.01267, and the corresponding optimal solution is x⟶ = [0.0512, 0.3441, 12.0663]. It can be seen that the WSOA has better optimization accuracy in solving the design problem of tension/compression spring.Table 10
Comparative results for the tension/compression spring problem.
AlgorithmsOptimum variablesOptimum weightDDNWSOA0.05120.344112.06630.01267GDA [38]0.05169250.356810811.28350590.012875GA [1]0.051480.35166111.6322010.0127048GPSO [39]0.05170.357311.2540.0127SES [40]N/AN/AN/A0.012732SCPSO [41]0.0516880.35670511.2896870.0127576AATM [42]0.0518130.3596911.1192530.0127081UABC [43]0.0516910.35676911.2859880.012683MVO [44]0.050.31595614.226230.0144644TEO [45]0.0517750.35879211.168390.012685MGWO [46]0.0513340.34819711.808090.0126799EEGWO [47]0.0516730.3563411.31130.012685


### 5.5.2. Welded Beam Design Problem

The goal of welding beam design problem is to minimize the manufacturing cost, which must satisfy the related constraints such as shear stress (τ), beam bending stress (θ), bar buckling load (Pc), beam end deflection (δ), normal stress (σ), and boundary. The set variables include welded beam thickness (x1), welded joint length (x2), welded beam width (x3), and beam thickness (x4). See Figure 53 for the design pattern of welded beam and the mathematical formula can be expressed as follows:(23)consider:x⟶=x1x2x3x4=hltb,minimize:fx=1.10471x12x2+0.04811x3x414+x2,s.t.g1X=τX−τmax,g2X=σX−σmax,g4X=0.125−x1≤0,g5X=δX−0.25≤0,g6X=P−PcX≤0,g7X=0.10471x12+0.04811x3x414+x2−5≤0,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,0.1≤x4≤2,where τmax is the maximum acceptable shear stress, σmax is the maximum acceptable normal stress, and P is the load. We have the following:(24)τX=τ12+2τ1τ2x22R+τ22,τ1=P2x1x2,τ2=MRJ,(25)M=PL+x22,(26)JX=22x1x2x224+x1+x322.Figure 53
The structure of welded beam design.M in equation (25) and J (X) in equation (26) represent moment of inertia and polarity, respectively. The remaining parameters are shown in equation (27) to equation (31).(27)R=x224+x1+x322.(28)σX=6PLx4x32,(29)δX=6PL3Ex33x4,(30)PcX=4.013Ex32x46/36L21−x32LE4G,(31)G=12×106psi,E=30×106psi,P=6000lb,L=14in.Table11 shows the research results of welding beam design. It can be seen from the table that the function value of the WSOA in solving the design problem of welded beam is lower than that of previous research. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 1.7519, and the corresponding optimal solution is x⟶ = [0.1919, 3.7633, 9.109, 0.2054]. Therefore, it can be shown that the WSOA has better optimization accuracy in solving the design problem of welded beams.Table 11
Comparative results for the welded beam design problem.
AlgorithmsOptimum variablesOptimum costhLTbWSOA0.19193.76339.1090.20541.7519SBM [48]0.24076.4858.23990.24972.442SA [49]0.2476.1458.2720.2492.4148SCA [50]0.2446.2388.28860.24462.3854BFOA [51]0.2053.47119.03670.20572.3868EA [52]0.24436.22018.2940.24442.3816T-cell [53]0.24446.12868.29150.24442.3811FSA [54]0.24446.12588.29390.24442.3811GSA [55]0.1821293.856979100.2023761.879952HS [56]0.24426.22318.29150.24432.3807SBO [57]0.214213.49248.557710.229451.8496


### 5.5.3. Multiplate Disc Clutch Brake Design Problem

The purpose of multiplate disc clutch brake is to optimize the total weight of multiplate disc clutch brake, in which the variables are driving force (F), inner and outer radius (ri and r0), friction surface number (Z), and disc thickness (t). Because the problem contains eight different constraints, the feasible region in the solution space only accounts for 70%, which makes it more difficult to solve the problem. See Figure 54 for the design pattern of multiplate disc clutch brake, and the mathematical formula can be expressed as follows:(32)minimize:fx=πx22−x12x3x5+1ρ,s.t.g1x=x2−x1−ΔR≥0,g2x=Lmax−x5+1x3+δ≥0,g3x=Pmax−Prz≥0,g4x=Pmaxvsrmax−Przvsr≥0,g5x=vsrmax−vsr≥0,g6x=Tmax−T≥0,g7x=Mh−sMs≥0,g8x=T≥0,where Mh=2/3μx4x5x23−x13/x22−x12 and w=πn/30rad/s.(33)A=πx22−x12mm2,Prz=x4AN/mm2,Vsr=πRsrn30mm/s,Rsr=23x22−x12x22x12mm,T=Izπn30Mh+Mfmm,Δr=20mm,Lmax=30mm,μ=0.6,Tmax=15s,μ=0.5,s=1.5,Ms=40Nm,pmax=1Mpa,ρ=0.0000078kg/mm3,vsrmax=10m/s,δ=0.5mm,s=1.5,Tmax=15s,n=250rpm,Iz=55kg/m2,Ms=40Nm,Mf=3Nm,60≤x1≤80,90≤x2≤110,1≤x3≤3,60≤x4≤1000,2≤x5≤9,i=1,2,3,4,5.Figure 54
Multiplate disc clutch brake problem.The comparison results of design problems of multiplate disc clutch brake are shown in Table12. It can be seen from the table that the WSOA shows superiority in solving this problem, and the obtained function value is lower than those of previous studies. The experiment was run independently for 30 times. The optimal fitness value of the algorithm is fx = 0.23525, and the optimal solution is x⟶ = [69.9996, 90, 1, 600, 2]. It can be shown that the WSOA has good optimization accuracy in solving the design problem of multiplate disc clutch brake.Table 12
Comparative results for multiplate disc clutch brake problem.
Algorithmsrix1r0x2tx3Fx4Zx5Optimal costWSOA69.999690160020.23525TLBO [58]7090181030.313656WCA [59]7090191030.313656PVS [60]7090198030.31366HHO [58]69.9999999992493901100030.259768993


### 5.5.4. Pressure Vessel Design Problem

The optimization purpose of pressure vessel problem is to minimize the total cost. Constraints include material cost, forming cost, and welding cost. The variables of this problem are shell thickness (Ts), head thickness (Th), inner diameter (R), and cylindrical section length (L) [61]. See Figure 55 for the design of pressure vessel, and the mathematical formula can be expressed as follows:(34)consider:Z=z1,z2,z3,z4=h,l,t,b,minimize:fx=0.6224x1x3x4+1.7781x2x23+3.1661x12x4+19.84x12x3,subject tog1x=−x1+0.0193x3≤0,g2x=−x3+0.00954z3≤0,g3x=−πx32x4−43πx33+1,296,00≤0,g4x=x4−240≤0,where 0≤x1, x2≤99, 0≤x3, and x4≤200.Figure 55
Pressure vessel design and its features.Table13 shows the research results of the WSOA and other algorithms on pressure vessel design. It can be seen from the table that the results obtained by the WSOA algorithm are better than those obtained by other algorithms, in which the optimal fitness value is fx = 5964.6114 and the optimal solution is x⟶ = [0.8056, 0.4081, 41.7401, 181.1285]. It is shown that the algorithm has better optimization performance and accuracy in solving the design problem of pressure vessels.Table 13
Comparative results for the pressure vessel design problem.
AlgorithmsOptimum variablesOptimum costTsThRLWSOA0.80560.408141.7401181.12855964.6114GWO [4]0.81250.434542.089181176.7587316051.5639MFO [5]0.81250.437542.098445176.6365966059.7143CPSO [62]0.81250.437542.091266176.74656061.0777HAIS-GA [63]0.81250.437542.0931176.70316060.367ES [64]0.81250.437542.098087176.6405186059.745CDE [65]0.81250.437542.098411176.637696059.734BA [66]0.81250.437542.098445176.6365956059.7143WOA [6]0.81250.437542.0982699176.6389986059.741DTS-GA [67]0.81250.437542.097398176.6540476059.9463GeneAS [68]0.93750.548.329112.6796410.3811WEO [69]0.81250.437542.098444176.6366226059.71HHO [70]0.817583830.407292742.09174576176.71963526000.46259


## 5.5.1. Tension/Compression Spring Design Problem

In real life, the optimization goal of many problems is to minimize the weight of the problem. For example, the design problem of tension/compression spring is a common example. The variables of this problem include coil diameter (x1), spring coil diameter (x2), and the number of coils (x3). The constraint conditions are minimum deflection (g1X), shear stress (g2X), impact frequency (g3X), and outer diameter limit (g4X). The spring design pattern is shown in Figure 52, and the mathematical formula can be expressed as follows:(22)consider:x⟶=x1x2x3=dDN,minimize:fX=x3+2x2x12,g2X=4x22−x1x212566x2x13−x14+15180x12−1≤0,g3X=1−140.45x1x22x3≤0,g4X=x1+x21.5−1≤0,0.05≤x1≤2,0.25≤x2≤1.3,2≤x3≤15.Figure 52
Structure of the TCS design.Table10 shows the comparison results of tension/compression spring design problems. It can be seen from the table that the WSOA proposed in this paper has better optimization performance, and the optimal function value obtained in solving spring design problems is lower than those of previous studies. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 0.01267, and the corresponding optimal solution is x⟶ = [0.0512, 0.3441, 12.0663]. It can be seen that the WSOA has better optimization accuracy in solving the design problem of tension/compression spring.Table 10
Comparative results for the tension/compression spring problem.
AlgorithmsOptimum variablesOptimum weightDDNWSOA0.05120.344112.06630.01267GDA [38]0.05169250.356810811.28350590.012875GA [1]0.051480.35166111.6322010.0127048GPSO [39]0.05170.357311.2540.0127SES [40]N/AN/AN/A0.012732SCPSO [41]0.0516880.35670511.2896870.0127576AATM [42]0.0518130.3596911.1192530.0127081UABC [43]0.0516910.35676911.2859880.012683MVO [44]0.050.31595614.226230.0144644TEO [45]0.0517750.35879211.168390.012685MGWO [46]0.0513340.34819711.808090.0126799EEGWO [47]0.0516730.3563411.31130.012685


## 5.5.2. Welded Beam Design Problem

The goal of welding beam design problem is to minimize the manufacturing cost, which must satisfy the related constraints such as shear stress (τ), beam bending stress (θ), bar buckling load (Pc), beam end deflection (δ), normal stress (σ), and boundary. The set variables include welded beam thickness (x1), welded joint length (x2), welded beam width (x3), and beam thickness (x4). See Figure 53 for the design pattern of welded beam and the mathematical formula can be expressed as follows:(23)consider:x⟶=x1x2x3x4=hltb,minimize:fx=1.10471x12x2+0.04811x3x414+x2,s.t.g1X=τX−τmax,g2X=σX−σmax,g4X=0.125−x1≤0,g5X=δX−0.25≤0,g6X=P−PcX≤0,g7X=0.10471x12+0.04811x3x414+x2−5≤0,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,0.1≤x4≤2,where τmax is the maximum acceptable shear stress, σmax is the maximum acceptable normal stress, and P is the load. We have the following:(24)τX=τ12+2τ1τ2x22R+τ22,τ1=P2x1x2,τ2=MRJ,(25)M=PL+x22,(26)JX=22x1x2x224+x1+x322.Figure 53
The structure of welded beam design.M in equation (25) and J (X) in equation (26) represent moment of inertia and polarity, respectively. The remaining parameters are shown in equation (27) to equation (31).(27)R=x224+x1+x322.(28)σX=6PLx4x32,(29)δX=6PL3Ex33x4,(30)PcX=4.013Ex32x46/36L21−x32LE4G,(31)G=12×106psi,E=30×106psi,P=6000lb,L=14in.Table11 shows the research results of welding beam design. It can be seen from the table that the function value of the WSOA in solving the design problem of welded beam is lower than that of previous research. The algorithm of this experiment is the result of running independently for 30 times. The optimal fitness value of the WSOA is fx = 1.7519, and the corresponding optimal solution is x⟶ = [0.1919, 3.7633, 9.109, 0.2054]. Therefore, it can be shown that the WSOA has better optimization accuracy in solving the design problem of welded beams.Table 11
Comparative results for the welded beam design problem.
AlgorithmsOptimum variablesOptimum costhLTbWSOA0.19193.76339.1090.20541.7519SBM [48]0.24076.4858.23990.24972.442SA [49]0.2476.1458.2720.2492.4148SCA [50]0.2446.2388.28860.24462.3854BFOA [51]0.2053.47119.03670.20572.3868EA [52]0.24436.22018.2940.24442.3816T-cell [53]0.24446.12868.29150.24442.3811FSA [54]0.24446.12588.29390.24442.3811GSA [55]0.1821293.856979100.2023761.879952HS [56]0.24426.22318.29150.24432.3807SBO [57]0.214213.49248.557710.229451.8496


## 5.5.3. Multiplate Disc Clutch Brake Design Problem

The purpose of multiplate disc clutch brake is to optimize the total weight of multiplate disc clutch brake, in which the variables are driving force (F), inner and outer radius (ri and r0), friction surface number (Z), and disc thickness (t). Because the problem contains eight different constraints, the feasible region in the solution space only accounts for 70%, which makes it more difficult to solve the problem. See Figure 54 for the design pattern of multiplate disc clutch brake, and the mathematical formula can be expressed as follows:(32)minimize:fx=πx22−x12x3x5+1ρ,s.t.g1x=x2−x1−ΔR≥0,g2x=Lmax−x5+1x3+δ≥0,g3x=Pmax−Prz≥0,g4x=Pmaxvsrmax−Przvsr≥0,g5x=vsrmax−vsr≥0,g6x=Tmax−T≥0,g7x=Mh−sMs≥0,g8x=T≥0,where Mh=2/3μx4x5x23−x13/x22−x12 and w=πn/30rad/s.(33)A=πx22−x12mm2,Prz=x4AN/mm2,Vsr=πRsrn30mm/s,Rsr=23x22−x12x22x12mm,T=Izπn30Mh+Mfmm,Δr=20mm,Lmax=30mm,μ=0.6,Tmax=15s,μ=0.5,s=1.5,Ms=40Nm,pmax=1Mpa,ρ=0.0000078kg/mm3,vsrmax=10m/s,δ=0.5mm,s=1.5,Tmax=15s,n=250rpm,Iz=55kg/m2,Ms=40Nm,Mf=3Nm,60≤x1≤80,90≤x2≤110,1≤x3≤3,60≤x4≤1000,2≤x5≤9,i=1,2,3,4,5.Figure 54
Multiplate disc clutch brake problem.The comparison results of design problems of multiplate disc clutch brake are shown in Table12. It can be seen from the table that the WSOA shows superiority in solving this problem, and the obtained function value is lower than those of previous studies. The experiment was run independently for 30 times. The optimal fitness value of the algorithm is fx = 0.23525, and the optimal solution is x⟶ = [69.9996, 90, 1, 600, 2]. It can be shown that the WSOA has good optimization accuracy in solving the design problem of multiplate disc clutch brake.Table 12
Comparative results for multiplate disc clutch brake problem.
Algorithmsrix1r0x2tx3Fx4Zx5Optimal costWSOA69.999690160020.23525TLBO [58]7090181030.313656WCA [59]7090191030.313656PVS [60]7090198030.31366HHO [58]69.9999999992493901100030.259768993


## 5.5.4. Pressure Vessel Design Problem

The optimization purpose of pressure vessel problem is to minimize the total cost. Constraints include material cost, forming cost, and welding cost. The variables of this problem are shell thickness (Ts), head thickness (Th), inner diameter (R), and cylindrical section length (L) [61]. See Figure 55 for the design of pressure vessel, and the mathematical formula can be expressed as follows:(34)consider:Z=z1,z2,z3,z4=h,l,t,b,minimize:fx=0.6224x1x3x4+1.7781x2x23+3.1661x12x4+19.84x12x3,subject tog1x=−x1+0.0193x3≤0,g2x=−x3+0.00954z3≤0,g3x=−πx32x4−43πx33+1,296,00≤0,g4x=x4−240≤0,where 0≤x1, x2≤99, 0≤x3, and x4≤200.Figure 55
Pressure vessel design and its features.Table13 shows the research results of the WSOA and other algorithms on pressure vessel design. It can be seen from the table that the results obtained by the WSOA algorithm are better than those obtained by other algorithms, in which the optimal fitness value is fx = 5964.6114 and the optimal solution is x⟶ = [0.8056, 0.4081, 41.7401, 181.1285]. It is shown that the algorithm has better optimization performance and accuracy in solving the design problem of pressure vessels.Table 13
Comparative results for the pressure vessel design problem.
AlgorithmsOptimum variablesOptimum costTsThRLWSOA0.80560.408141.7401181.12855964.6114GWO [4]0.81250.434542.089181176.7587316051.5639MFO [5]0.81250.437542.098445176.6365966059.7143CPSO [62]0.81250.437542.091266176.74656061.0777HAIS-GA [63]0.81250.437542.0931176.70316060.367ES [64]0.81250.437542.098087176.6405186059.745CDE [65]0.81250.437542.098411176.637696059.734BA [66]0.81250.437542.098445176.6365956059.7143WOA [6]0.81250.437542.0982699176.6389986059.741DTS-GA [67]0.81250.437542.097398176.6540476059.9463GeneAS [68]0.93750.548.329112.6796410.3811WEO [69]0.81250.437542.098444176.6366226059.71HHO [70]0.817583830.407292742.09174576176.71963526000.46259


## 6. Research Limitations

Although the above experimental results show that the WSOA has better performance than most comparative algorithms, it also has some shortcomings. There is a room for improvement in the performance of the WSOA. For example, in the unimodal test function, it is found that the number of theoretical optimum values obtained by the WSOA is small, which shows that the exploitation ability of the WSOA can be further improved in the test unimodal function. After testing the multimodal test function, it is found that the number of theoretical optimal values obtained is small, which indicates that the exploration and exploitation ability of the WSOA under multiple optimal solutions needs to be further improved. Moreover, the WSOA has not been further verified in a higher dimension. Finally, because the WSOA combines the WOA with the SOA, the structure of this algorithm is more complex than the original algorithm, so the flexibility of the WSOA is worse than that of the original algorithm.


## 7. Conclusion and Future Work

In this paper, an improved seagull optimization algorithm (WSOA) is proposed to solve the global optimization problem. The combination of contraction and encirclement mechanism of the WOA and spiral attack behavior of the SOA can improve the calculation accuracy of the SOA and balance the exploitation and exploration capabilities more effectively. In addition, introducing levy flight strategy into the search formula can avoid premature convergence of the SOA. In the aspect of performance evaluation, 25 benchmark test functions are used for verification. Comparing the performance of the WSOA with the performances of seven famous metaheuristic algorithms, the results show that the algorithm has strong competitiveness. Statistical test analysis and results comparison show that the improved SOA has better performance than the SOA. In this paper, four engineering examples are also used to test the performance of the proposed algorithm, which proves that the WSOA has better performance.For the WSOA, further research is needed in future work. First of all, the WSOA can be applied to target allocation problem and more constrained engineering examples. Second, the WSOA is applied to solve real-life problems, such as medical data.



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*Source: 6639671-2021-01-29.xml*