# A Multimixed Strategy Improved Sparrow Search Algorithm and Its Application in TSP

Authors: Weizheng Li; Mengjian Zhang; Jing Zhang; Tao Qin; Wei Wei; Jing Yang
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

Aiming at the shortcomings of the sparrow search algorithm (SSA), such as falling into local optimum and slow convergence speed, an improved sparrow search algorithm based on multimixed strategy (MISSA) is proposed in this paper. In the initial stage, the iterative chaotic mapping is used to initialize the population in order to improve the diversity of population. In the foraging stage, the golden sine algorithm and nonlinear convergence factor strategy are introduced to optimize the discoverer-follower model, which make search process more comprehensive and extensive for the discoverer. The elite opposition-based learning strategy is used to update the optimal solution and the population obtained in each iteration to improve the self-learning ability of the algorithm. To verify the rationality of the multimixed strategy selection and efficiency of the proposed algorithm, MISSA is compared with three derived single-strategy improved algorithms, other improved SSAs, and five typical swarm intelligence algorithms using ten basic benchmark functions and CEC 2014 function. The optimization results, diversity analysis, and Wilcoxon rank-sum test results certify that the proposed MISSA has better optimization accuracy, convergence speed, and robustness than other compared methods. Moreover, the practicability and feasibility of MISSA are veriﬁed by solving the traveling salesman problem (TSP).


---

## Body

## 1. Introduction

### 1.1. Research Background

Optimization problems can be found in various fields, such as combinatorial optimization problems and engineering design problems. In general, the optimization problems and methods can be classified as shown in Figure1. A general optimization problem can mathematically be described as follows [1]:(1)minFx=f1x,f2x,…,fmx,gix≤0,i=1,2,…,p,hjx=0,j=1,2,…,q,xli≤xi≤xui,i=1,2,…,N,where x = (x1, x2, …, xn)T ∈ Ω is the feasible solution with n decision variables or real parameters, F (x) is the function with m objectives, and gi (x) and hj (x) are the ith and jth constraint functions, respectively. xli≤xi≤xui is a bound constrained over the searching space x.Figure 1
Classiﬁcation of optimization problems and methods.As shown in Figure1, optimization methods are mainly based on mathematical methods or stochastic methods. Traditional mathematical methods are susceptible to type constraints and fall into the dimension disaster when solving high-dimensional problems, as well solve the multicriteria, nonlinear, and nondifferentiable problems [2, 3] with slight effectiveness. Metaheuristic algorithm benefits from the randomness of the optimization principle and can escape the local optimal solution when dealing with high-dimensional optimization problems. Metaheuristic algorithms also have the advantages of simple structure, high efficiency, and strong robustness. Therefore, many metaheuristic algorithms were proposed for solving the high-dimensional optimization problems.In general, metaheuristic algorithms are usually classified into four categories: (1) biological evolution-based algorithms, such as genetic algorithm (GA) [4] and differential evolution (DE) [5]; (2) physics-based algorithms, such as simulated annealing (SA) [6], black hole algorithm (BHA) [7], and Runge–Kutta optimizer (RUN) [8]; (3) human social behavior-based algorithms, such as teaching-learning-based optimization (TLBO) [9], imperialist competitive algorithm (ICA) [10], and Hunger Games Search (HGS) [11]; (4) swarm-based algorithms, such as particle swarm optimization (PSO) [12], ant colony optimization (ACO) [13], whale optimization algorithm (WOA) [14], grey wolf algorithm (GWO) [15], coot optimization algorithm (COOT) [16], naked mole-rat algorithm (NMR) [17], Harris Hawk Optimization (HHO) [18], slime mould algorithm (SMA) [19], and colony predation algorithm (CPA) [20].The swarm-based algorithm simulates the behavior characteristics of a certain type of biological population in nature and performs random retrieval in the solution space to obtain the optimal solution. In addition, it is the focus of the metaheuristic algorithm due to its remarkable optimization performance and has been widely used in engineering optimization, image processing, machine learning, and other high-dimensional optimization fields.The traveling salesman problem is a typical NP-hard combinatorial optimization problem, which was first proposed by Menger in 1959. TSP has attracted the attention of the operations research, logistics science, applied mathematics, computer science, graph theory, and other disciplines. TSP has an important practical significance and is widely used in logistics and distribution [21], aircraft route arrangement [22], integrated circuit wiring [23], and other fields, which can be solved by transforming them into TSP problems. Due to the high versatility and controllable complexity of TSP, it is used as a benchmark case to test the ability of methods to deal with complex combinatorial optimization problems.


### 1.2. Related Work

The sparrow search algorithm is a new swarm-based algorithm proposed by Xue and Shen [24] in 2020, which was inspired by foraging and antipredation behavior of sparrow population. SSA has the advantages of few control parameters and high optimization precision. It has been used in the diagnosis on wheelset bearings [25], configuration of distributed generation [26], and gear fault detection [27]. But, the algorithm is prone to premature convergence in the later iterations and converges slowly when dealing with complex optimization problems.In order to enhance the optimization ability of the sparrow algorithm, Liu et al. [28] proposed an enhanced sparrow algorithm based on the merit function mechanism for improving the accuracy of the tumor diagnosis identified by the convolutional neural network. Zhang and Ding [29] proposed a chaotic sparrow search algorithm to optimize the parameters for the stochastic configuration network, which has enhanced the regression performance. Mao and Zhang [30] introduced sine-cosine algorithm and Lévy flight strategy to balance the local and global exploitation ability of SSA. Lv et al. [31] enhanced the global search ability of SSA through Tent chaotic map, generated Tent chaotic sequence based on the stagnant solution, and finally applied the algorithm to image segmentation. Tang et al. [32] introduced the logarithmic spiral strategy, adopted an adaptive stepping strategy to balance the searching abilities, and then applied the algorithm to traditional engineering problems. Ouyang et al. [33] designed an adaptive sparrow search algorithm based on the lens reverse learning strategy and sine-cosine guidance mechanism, which has better optimization ability in robot path planning. Yuan et al. [34] proposed an improved sparrow search algorithm which can track the maximum power point more accurately in the photovoltaic microgrid system.Compared with the original SSA, the above improved sparrow algorithms have many improvements in the optimization ability. However, some shortcomings are summarized as follows:(1)
Many researchers use the chaos theory to improve the diversity of the initial population but ignore the diversity of the population in subsequent iterations and the diversity analysis(2)
The existing improvement strategies have not fundamentally changed the searching mechanism of SSA. The shortcoming of insufficient self-learning ability in SSA has not been improved(3)
Some improved SSAs are only optimized for basic benchmark functions, which is not enough to explain the superiority and practicability of the improved algorithm


### 1.3. Contribution

To solve the above shortcomings, a novel multimixed strategy improved sparrow search algorithm is proposed in this paper, including four strategies: iterative chaotic mapping, golden sine algorithm, nonlinear convergence factor strategy, and elite opposition-based learning approach. The iterative chaotic map strategy is used to initialize the population in order to enhance the diversity of the population. The golden sine algorithm and nonlinear convergence factor strategies are used to improve the ability of discoverer-follower model for searching the solution space, which enhance the convergence speed and global exploration ability of the algorithm. The elite opposition-based learning strategy is used to update optimal solution and population to improve the diversity, self-learning ability, and local exploitation ability of the algorithm. In addition, we use MISSA to solve the traveling salesman problem. The innovations and main contributions of this paper are described as follows:(i)
A novel MISSA is proposed in this paper, and four improved strategies of iterative chaotic map strategy, golden sine strategy, nonlinear convergence factor strategy, and elite opposition-based learning strategy are used to enhance the performance of the proposed algorithm for solving the complex optimization problems.(ii)
Ablation experiments are designed to verify the specific impact of different strategies on population diversity and optimization performance of SSA.(iii)
Comprehensive experiments are designed and accomplished to fully verify the efficiency of the MISSA by basic benchmark functions and CEC 2014 function. Eight algorithms are compared, including SSA, other improved SSAs, WOA, GWO, COOT, and PSO.(iv)
MISSA is used to settle TSP, and the results verified the practicality of MISSA in solving the combinatorial optimization problem.


## 1.1. Research Background

Optimization problems can be found in various fields, such as combinatorial optimization problems and engineering design problems. In general, the optimization problems and methods can be classified as shown in Figure1. A general optimization problem can mathematically be described as follows [1]:(1)minFx=f1x,f2x,…,fmx,gix≤0,i=1,2,…,p,hjx=0,j=1,2,…,q,xli≤xi≤xui,i=1,2,…,N,where x = (x1, x2, …, xn)T ∈ Ω is the feasible solution with n decision variables or real parameters, F (x) is the function with m objectives, and gi (x) and hj (x) are the ith and jth constraint functions, respectively. xli≤xi≤xui is a bound constrained over the searching space x.Figure 1
Classiﬁcation of optimization problems and methods.As shown in Figure1, optimization methods are mainly based on mathematical methods or stochastic methods. Traditional mathematical methods are susceptible to type constraints and fall into the dimension disaster when solving high-dimensional problems, as well solve the multicriteria, nonlinear, and nondifferentiable problems [2, 3] with slight effectiveness. Metaheuristic algorithm benefits from the randomness of the optimization principle and can escape the local optimal solution when dealing with high-dimensional optimization problems. Metaheuristic algorithms also have the advantages of simple structure, high efficiency, and strong robustness. Therefore, many metaheuristic algorithms were proposed for solving the high-dimensional optimization problems.In general, metaheuristic algorithms are usually classified into four categories: (1) biological evolution-based algorithms, such as genetic algorithm (GA) [4] and differential evolution (DE) [5]; (2) physics-based algorithms, such as simulated annealing (SA) [6], black hole algorithm (BHA) [7], and Runge–Kutta optimizer (RUN) [8]; (3) human social behavior-based algorithms, such as teaching-learning-based optimization (TLBO) [9], imperialist competitive algorithm (ICA) [10], and Hunger Games Search (HGS) [11]; (4) swarm-based algorithms, such as particle swarm optimization (PSO) [12], ant colony optimization (ACO) [13], whale optimization algorithm (WOA) [14], grey wolf algorithm (GWO) [15], coot optimization algorithm (COOT) [16], naked mole-rat algorithm (NMR) [17], Harris Hawk Optimization (HHO) [18], slime mould algorithm (SMA) [19], and colony predation algorithm (CPA) [20].The swarm-based algorithm simulates the behavior characteristics of a certain type of biological population in nature and performs random retrieval in the solution space to obtain the optimal solution. In addition, it is the focus of the metaheuristic algorithm due to its remarkable optimization performance and has been widely used in engineering optimization, image processing, machine learning, and other high-dimensional optimization fields.The traveling salesman problem is a typical NP-hard combinatorial optimization problem, which was first proposed by Menger in 1959. TSP has attracted the attention of the operations research, logistics science, applied mathematics, computer science, graph theory, and other disciplines. TSP has an important practical significance and is widely used in logistics and distribution [21], aircraft route arrangement [22], integrated circuit wiring [23], and other fields, which can be solved by transforming them into TSP problems. Due to the high versatility and controllable complexity of TSP, it is used as a benchmark case to test the ability of methods to deal with complex combinatorial optimization problems.


## 1.2. Related Work

The sparrow search algorithm is a new swarm-based algorithm proposed by Xue and Shen [24] in 2020, which was inspired by foraging and antipredation behavior of sparrow population. SSA has the advantages of few control parameters and high optimization precision. It has been used in the diagnosis on wheelset bearings [25], configuration of distributed generation [26], and gear fault detection [27]. But, the algorithm is prone to premature convergence in the later iterations and converges slowly when dealing with complex optimization problems.In order to enhance the optimization ability of the sparrow algorithm, Liu et al. [28] proposed an enhanced sparrow algorithm based on the merit function mechanism for improving the accuracy of the tumor diagnosis identified by the convolutional neural network. Zhang and Ding [29] proposed a chaotic sparrow search algorithm to optimize the parameters for the stochastic configuration network, which has enhanced the regression performance. Mao and Zhang [30] introduced sine-cosine algorithm and Lévy flight strategy to balance the local and global exploitation ability of SSA. Lv et al. [31] enhanced the global search ability of SSA through Tent chaotic map, generated Tent chaotic sequence based on the stagnant solution, and finally applied the algorithm to image segmentation. Tang et al. [32] introduced the logarithmic spiral strategy, adopted an adaptive stepping strategy to balance the searching abilities, and then applied the algorithm to traditional engineering problems. Ouyang et al. [33] designed an adaptive sparrow search algorithm based on the lens reverse learning strategy and sine-cosine guidance mechanism, which has better optimization ability in robot path planning. Yuan et al. [34] proposed an improved sparrow search algorithm which can track the maximum power point more accurately in the photovoltaic microgrid system.Compared with the original SSA, the above improved sparrow algorithms have many improvements in the optimization ability. However, some shortcomings are summarized as follows:(1)
Many researchers use the chaos theory to improve the diversity of the initial population but ignore the diversity of the population in subsequent iterations and the diversity analysis(2)
The existing improvement strategies have not fundamentally changed the searching mechanism of SSA. The shortcoming of insufficient self-learning ability in SSA has not been improved(3)
Some improved SSAs are only optimized for basic benchmark functions, which is not enough to explain the superiority and practicability of the improved algorithm


## 1.3. Contribution

To solve the above shortcomings, a novel multimixed strategy improved sparrow search algorithm is proposed in this paper, including four strategies: iterative chaotic mapping, golden sine algorithm, nonlinear convergence factor strategy, and elite opposition-based learning approach. The iterative chaotic map strategy is used to initialize the population in order to enhance the diversity of the population. The golden sine algorithm and nonlinear convergence factor strategies are used to improve the ability of discoverer-follower model for searching the solution space, which enhance the convergence speed and global exploration ability of the algorithm. The elite opposition-based learning strategy is used to update optimal solution and population to improve the diversity, self-learning ability, and local exploitation ability of the algorithm. In addition, we use MISSA to solve the traveling salesman problem. The innovations and main contributions of this paper are described as follows:(i)
A novel MISSA is proposed in this paper, and four improved strategies of iterative chaotic map strategy, golden sine strategy, nonlinear convergence factor strategy, and elite opposition-based learning strategy are used to enhance the performance of the proposed algorithm for solving the complex optimization problems.(ii)
Ablation experiments are designed to verify the specific impact of different strategies on population diversity and optimization performance of SSA.(iii)
Comprehensive experiments are designed and accomplished to fully verify the efficiency of the MISSA by basic benchmark functions and CEC 2014 function. Eight algorithms are compared, including SSA, other improved SSAs, WOA, GWO, COOT, and PSO.(iv)
MISSA is used to settle TSP, and the results verified the practicality of MISSA in solving the combinatorial optimization problem.


## 2. The Original Sparrow Search Algorithm

The original SSA algorithm was mainly inspired by foraging and antipredation behavior of sparrow population. The population of SSA can be divided into discoverers, followers, and scouts [24]. The discoverers with high fitness value are responsible for searching for food and guiding the migration direction of the population, which accounts for 10%–20% of the entire sparrow population, and others are followers. The followers search for food in the direction pointed by the discoverers, and in order to increase the predation rate, they will constantly monitor the discoverers and compete for food with high-intake individuals. The scouts belong to the discoverers and followers, usually about 10%–20% individuals randomly within the population to detect danger and alarm the population. When predators invade, they will chirp alarming signals to make the sparrow population migrate to the safe area.The location of discoverers is updated as follows:(2)Xi,jt+1=Xi,jt⋅exp−iα⋅itermax,ifR2<ST,Xi,jt+Q⋅L,ifR2≥ST,where Xi,jt denotes the position of the ith sparrow in the jth dimension under the current tth iteration, itermax is the maximum number of iterations, α ∈ (0, 1) is a random number, and R2 ∈ [0, 1] and ST ∈ [0.5, 1] represent the alarm value and the safety threshold, respectively. Q is the migration control coefficient, which is a random number that obeys the standard normal distribution. L is a matrix of 1 × d for which each element inside is 1. If R2 is less than ST, it indicates the foraging environment is safe and the discoverers perform global search mode. If R2 is greater than or equal to ST, it indicates some individuals have detected predators and the discoverers will lead the population to migrate to the safe area quickly.The location of followers is updated as follows:(3)Xi,jt+1=Q⋅expXworstt−Xi,jti2,i>N2,Xpt+1+|Xi,jt−Xpt+1|⋅A+⋅L,other,where Xworstt represents the current global worst position and Xpt+1 denotes the optimal position of the discoverers in the (t + 1)th iteration. A is a matrix of 1 × d in which each element inside is 1 or −1, and A+=ATAAT−1.When i is greater than N/2, it indicates that the ith follower with a lower fitness value is most likely to be hungry, which needs to go to other areas for foraging.The location of scouts is updated as follows:(4)Xi,jt+1=Xbestt+β⋅|Xi,jt−Xbestt|,iffi>fg,Xi,jt+k⋅Xi,jt−Xworsttfi−fw+ε,iffi=fg,where Xbestt represents the current global optimal position. β is the step size control coefficient, which is a normal distributed random number with a mean value of 0 and variance value of 1. k ∈ [−1, 1] is a random number which represents the direction of the sparrow movement. ε is the smallest constant for avoiding from zero-division error. fi represents the fitness value of the current individual. fg and fw represent the current global optimal value and worst ﬁtness value, respectively. When fi is greater than fg, it indicates the individual is on the fringe of the population and vulnerable to predators. If fi is equal to fg, it means the individuals in the center of the population are aware of the danger of being attacked by predators and need to fly closer to the safe area.


## 3. A Multimixed Strategy Improved Sparrow Search Algorithm

Through the research on the algorithm searching mechanism, the shortcomings of SSA are mainly caused by three aspects: the method of population initialization relies on a simple random number, which cannot provide good diversity for the population; the algorithm relies on the discoverer-follower model to search the solution space, which is slow and not comprehensive enough; the algorithm lacks the learning and upgrading mechanism of the optimal individual and does not make effective use of the optimal solution during the iteration process. Based on these aspects, this paper proposes four strategies to optimize the mechanism of the algorithm. The iterative chaotic map strategy is used to initialize the population, the golden sine algorithm and nonlinear convergence factor strategies are used to improve the discoverer-follower model, and the elite opposition-based learning strategy is used to update optimal solution and population.


### 3.1. Iterative Chaotic Map Strategy

The original SSA uses a random method to generate the initial position of the sparrow population, which is likely to cause uneven distribution of individuals, resulting in poor population diversity. Chaotic motion is characterized by pseudo-randomness, ergodicity, and high sensitivity to initial conditions and parameters [35, 36].Chaotic mapping can be used to initialize the population to obtain good diversity. The iterative map has better chaotic ergodicity, hyperchaotic behavior, and high complexity. The Lyapunov exponent of the iterative map is better than one-dimensional maps such as Singer chaotic map, Tent chaotic map, and sinusoidal chaotic map [37]. The simulation results of four chaotic maps under 500 iterations are shown in Figure 2.Figure 2
Chaotic map. (a) Iterative map. (b) Singer map. (c) Tent map. (d) Sinusoidal map.
(a)(b)(c)(d)The chaotic variables generated by the Singer chaotic map have small periods and an uneven distribution phenomenon between 0.6 and 0.9. The Tent chaotic map has small periods and an unstable periodic phenomenon between 0 and 0.2, as well as the problem of easily falling into fixed points. The chaotic variables generated by the sinusoidal chaotic map have certain bimodal distribution characteristics, which are more evenly distributed in the middle of the chaotic attraction domain and densely distributed at both ends. In contrast, the distribution of iterative mapping from 0 to 1 is the most uniform. Therefore, this paper uses the iterative chaotic map strategy to initialize the population, and the equation is as follows:(5)xi+1=sinbπxi,where b ∈ (0, 1) is the control parameter, which is usually set to 0.7.


### 3.2. Golden Sine Algorithm Strategy

The current environment is safe when the alarm value (R2) is under the safety threshold (ST), and the discoverers will lead the population to forage in this area. The algorithm performs deeper local exploitation of the current solution space. Local exploitation and global exploitation are two important stages of optimization algorithms. Excessive local exploitation will inhibit the tendency of algorithm to search for global optimal solution, and too much global exploitation will reduce the optimization accuracy.The golden sine algorithm (Gold-SA) is a mathematical heuristic algorithm presented by Tanyildizi and Demir in 2017, which is inspired by the sine function and golden section coefﬁcient [38]. The mathematical relationship between the unit circle and the sine function is shown in Figure 3. The coordinates on the sine function correspond to the y-axis coordinates of the point on the unit circle centered on the origin with a radius of 1. Gold-SA improves the global exploitation ability by traversing the points on the sine function which in turn traverses the points on the unit circle.Figure 3
Correspondence diagram between the unit circle and the sine function.The golden section coefficient is widely used in the ﬁeld of art because of its strict proportionality, artistry, and harmony. Gold-SA introduces the golden section coefficient to separate the solution space in each iteration. The coefﬁcient with ﬁxed shrink step does not require gradient information. Therefore, combining the sine function with the golden section coefﬁcient can help Gold-SA comprehensively search the solution space where the optimal solution can be generated, thereby improving the local exploitation ability of algorithm. The key of Gold-SA is the golden section coefficient, which is represented byθ1 and θ2. The mathematical expression is as follows:(6)θ1=a1−h+bh,θ2=ah+b1−h,where a and b are the initial values of the golden section; according to the relationship between the period of the standard sine function and the unit circle, we set a to π and b to −π. h is the golden section coefficient. The golden section coefficient will separate the solution space every time it goes through an iteration. The position update formula is as follows:(7)Xi,jt+1=Xi,jtsinr1−r2sinr1θ1⋅Xpt−θ2⋅Xi,jt,where Xi,jt represents the position of the ith sparrow in the jth dimension under the current tth iteration. r1 and r2 are random numbers, where r1 ∈ (0, 2π) and r2 ∈ (0, π).


### 3.3. Nonlinear Convergence Factor Strategy

When the alarm value (R2) reaches or exceeds the safety threshold (ST), the discoverers will migrate with the population to avoid the predators. The algorithm performs a broader global search. The migration distance depends on the migration control coefficient Q in equation (2). In the original SSA, Q is a random number that obeys the standard normal distribution. Although the strong randomness of the migration control coefficient has the probability to help the algorithm jump out of the local optimum, the shortcoming of this strategy is also obvious. When Q cannot migrate or the migration distance is very short, the global exploitation ability and robustness of algorithm will be reduced.Therefore, this paper optimizes the coefficient and uses the nonlinear convergence factorω as the new migration control coefficient. We refer to the study of nonlinear convergence factor adjustment [39], and the parameter control strategy can be expressed as(8)ωt=1−sintitermax2⋅π2,where ω is the new migration control coefficient instead of Q in (2) and itermax is the maximum number of iterations.As shown in Figure4, at the beginning of the iteration, the value of ω is large and the changing rate is slow, keeping a large migration distance, so the discoverers can lead the population to fully explore the solution space and enhance the global optimization ability. The changing rate of ω in the middle term of the iteration is faster, which is beneficial to improving the convergence speed of the algorithm and the ability to jump out of the local optimal solution. In the later stage of the iteration, the changing rate of ω slows down and the individuals perform a local exploitation on the current solution space to enhance the local optimization ability of the algorithm.Figure 4
Variation curve of the migration control coefficient.Considering the GA strategy and the nonlinear convergence factor strategy, the discoverer location formula is updated as follows:(9)Xi,jt+1=Xi,jtsinr1−r2sinr1θ1⋅Xpt−θ2⋅Xi,jt,R2<ST,Xi,jt+ω⋅L,R2≥ST,where the meaning of each parameter is the same as in (2).


### 3.4. Elite Opposition-Based Learning Strategy

Tizhoosh [40] proposed opposition-based learning (OBL) in 2005, and it was proved that the probability of the opposite solution approaching the optimal solution was 50% higher than the current solution through experiments. The strategy can increase the population’s diversity effectively. The OBL strategy calculates the opposite solution of the current solution, finally selects a better fitness solution from the current solution and the corresponding opposite solution, and updates the individual.Definition 1 (opposite solution).
Suppose that a feasible solution of the current population in the D-dimensional solution space is expressed asX=x1,x2,…,xD, xj∈aj,bj; then, its corresponding opposite solution is X¯=x¯1,x¯2,…,x¯D, where x¯j=λaj+bj−xj, λ∈0,1 is a uniformly distributed random number.
Since OBL solves the problem in a certain space, it has the risk of falling into the local optimum and the problem of making the algorithm fall into premature convergence. The generated opposite solution may have difficulty to search for the optimal value than the current solution space [41, 42]. Elite opposition-based learning (EOBL) was proposed and successfully used to improve the characteristics of this algorithm. Sihwail et al. [43] introduced it in the HHO algorithm to improve population diversity and speed up the convergence process.
The EOBL takes advantage of the fact that elite individuals carry more effective information than ordinary individuals. First, the opposite population is formed through elite individuals with high adaptability in the population; then, elite individuals are selected as a new population from the opposite population and the current population. The EOBL strategy increases the diversity and quality of the population and improves the ability of the algorithm to jump out of the local optimum solution.Definition 2 (elite opposite solution).
Suppose that the extreme point corresponding to the general individual in the current population is elite individuals, which is expressed asXi,je=Xi,1e,Xi,2e,…,Xi,De; then, its corresponding opposite solution is Xi,je¯=Xi,1e¯,Xi,2e¯,…,Xi,De¯. We select the individuals whose fitness value is in the top 10% of the population after each iteration as elite individuals. The equation for the elite opposite solution is as follows:(10)Xi,je¯=λ×lbj+ubj−Xi,je,where λ∈0,1 is a uniformly distributed random number, Xi,je∈lbj,ubj, lbj=minXi,j, and ubj=maxXi,j. lbj and ubj are the lower and upper bounds of the dynamic boundary. Compared with the fixed boundary, the dynamic boundary can save the search experience and reduce the optimization time of the algorithm. When Xi,je¯ exceeds the boundary, it is reset by the equation as follows:(11)Xi,je¯=randlbj+ubj.


### 3.5. Pseudo-Code and Flow Chart of MISSA

The pseudo-code and flow chart of MISSA is shown in Algorithm 1 and Figure5.Figure 5
Flow chart of MISSA.Algorithm 1: Pseudo-code of MISSA.
/∗Initialization∗/(1)
Input:(2)
POP: the number of sparrows;(3)
itermax: the maximum number of iterations;(4)
PD: the proportion of discoverers in the population;(5)
SD: the proportion of scouts in the population;(6)
R2: the alarm value;(7)
ST: the safety threshold;(8)
Initialize the position ofPOP sparrows using equation (5);(9)
Rank the ﬁtness values and divide the population into discoverers and scouts;/∗Iterative search∗/(10)
while (t ≤ itermax)(11)
R2 = rand(1);(12)
fori = 1 : POP∗PD(13)
Update the discoverer’s position using equation (7);(14)
end for(15)
fori = (POP∗PD) + 1 : POP(16)
Update the follower’s position using equation (3);(17)
end for(18)
fori = 1 : POP∗SD(19)
Update the scout’s position using equation (4);(20)
end for(21)
Calculate and rank the fitness values;(22)
fori = 1 : POP∗10%(23)
Update the elite individuals’ position using equation (10);(24)
If the position exceeds the boundary, using equation (11) to reset;(25)
If the new position is better than before, update it;(26)
end for(27)
Rank the fitness values and find the current best position;(28)
t = t + 1;(29)
end while(30)
OutputXbest and fg


### 3.6. Complexity Analysis of MISSA

#### 3.6.1. Time Complexity

Assuming the population size and the solution space dimension areN and D and the maximum iteration is T, the time complexity of the original SSA includes the following: the population initialization complexity is O (1), the fitness value of calculation complexity is O (N), the complexity of exploration and exploitation phases is O (T ∗ N ∗ D), and the total time complexity of SSA is O (1) + O (N) + O (T ∗ N ∗ D) = O (T ∗ N ∗ D).MISSA does not change the structure of SSA. The chaotic mapping strategy does not increase the complexity of population initialization, and the time complexity of initializing the population is stillO (1). The Gold-SA strategy and the nonlinear convergence factor strategy only improve the position update method of the discoverers without additional operations, so the time complexity of this stage is still O (N ∗ D), and the time complexity of the EOBL strategy is O (N ∗ D). In regard to this, the total time complexity of MISSA is O (T ∗ N ∗ D), which is the same as that of SSA, indicating that the multimixed strategy used in MISSA does not affect the efficiency of the algorithm.


#### 3.6.2. Space Complexity

The storage space consumed by an algorithm can be defined as the space complexity. Assume that the ratio of elite individuals isR. Compared with SSA, only the EOBL strategy increases a certain amount of computational complexity of O (T ∗ R ∗ N ∗ D). The addition of multimixed strategies does not improve the order of magnitude of SSA. Thus, the space efficiency of the proposed strategy is effective and stable.


## 3.1. Iterative Chaotic Map Strategy

The original SSA uses a random method to generate the initial position of the sparrow population, which is likely to cause uneven distribution of individuals, resulting in poor population diversity. Chaotic motion is characterized by pseudo-randomness, ergodicity, and high sensitivity to initial conditions and parameters [35, 36].Chaotic mapping can be used to initialize the population to obtain good diversity. The iterative map has better chaotic ergodicity, hyperchaotic behavior, and high complexity. The Lyapunov exponent of the iterative map is better than one-dimensional maps such as Singer chaotic map, Tent chaotic map, and sinusoidal chaotic map [37]. The simulation results of four chaotic maps under 500 iterations are shown in Figure 2.Figure 2
Chaotic map. (a) Iterative map. (b) Singer map. (c) Tent map. (d) Sinusoidal map.
(a)(b)(c)(d)The chaotic variables generated by the Singer chaotic map have small periods and an uneven distribution phenomenon between 0.6 and 0.9. The Tent chaotic map has small periods and an unstable periodic phenomenon between 0 and 0.2, as well as the problem of easily falling into fixed points. The chaotic variables generated by the sinusoidal chaotic map have certain bimodal distribution characteristics, which are more evenly distributed in the middle of the chaotic attraction domain and densely distributed at both ends. In contrast, the distribution of iterative mapping from 0 to 1 is the most uniform. Therefore, this paper uses the iterative chaotic map strategy to initialize the population, and the equation is as follows:(5)xi+1=sinbπxi,where b ∈ (0, 1) is the control parameter, which is usually set to 0.7.


## 3.2. Golden Sine Algorithm Strategy

The current environment is safe when the alarm value (R2) is under the safety threshold (ST), and the discoverers will lead the population to forage in this area. The algorithm performs deeper local exploitation of the current solution space. Local exploitation and global exploitation are two important stages of optimization algorithms. Excessive local exploitation will inhibit the tendency of algorithm to search for global optimal solution, and too much global exploitation will reduce the optimization accuracy.The golden sine algorithm (Gold-SA) is a mathematical heuristic algorithm presented by Tanyildizi and Demir in 2017, which is inspired by the sine function and golden section coefﬁcient [38]. The mathematical relationship between the unit circle and the sine function is shown in Figure 3. The coordinates on the sine function correspond to the y-axis coordinates of the point on the unit circle centered on the origin with a radius of 1. Gold-SA improves the global exploitation ability by traversing the points on the sine function which in turn traverses the points on the unit circle.Figure 3
Correspondence diagram between the unit circle and the sine function.The golden section coefficient is widely used in the ﬁeld of art because of its strict proportionality, artistry, and harmony. Gold-SA introduces the golden section coefficient to separate the solution space in each iteration. The coefﬁcient with ﬁxed shrink step does not require gradient information. Therefore, combining the sine function with the golden section coefﬁcient can help Gold-SA comprehensively search the solution space where the optimal solution can be generated, thereby improving the local exploitation ability of algorithm. The key of Gold-SA is the golden section coefficient, which is represented byθ1 and θ2. The mathematical expression is as follows:(6)θ1=a1−h+bh,θ2=ah+b1−h,where a and b are the initial values of the golden section; according to the relationship between the period of the standard sine function and the unit circle, we set a to π and b to −π. h is the golden section coefficient. The golden section coefficient will separate the solution space every time it goes through an iteration. The position update formula is as follows:(7)Xi,jt+1=Xi,jtsinr1−r2sinr1θ1⋅Xpt−θ2⋅Xi,jt,where Xi,jt represents the position of the ith sparrow in the jth dimension under the current tth iteration. r1 and r2 are random numbers, where r1 ∈ (0, 2π) and r2 ∈ (0, π).


## 3.3. Nonlinear Convergence Factor Strategy

When the alarm value (R2) reaches or exceeds the safety threshold (ST), the discoverers will migrate with the population to avoid the predators. The algorithm performs a broader global search. The migration distance depends on the migration control coefficient Q in equation (2). In the original SSA, Q is a random number that obeys the standard normal distribution. Although the strong randomness of the migration control coefficient has the probability to help the algorithm jump out of the local optimum, the shortcoming of this strategy is also obvious. When Q cannot migrate or the migration distance is very short, the global exploitation ability and robustness of algorithm will be reduced.Therefore, this paper optimizes the coefficient and uses the nonlinear convergence factorω as the new migration control coefficient. We refer to the study of nonlinear convergence factor adjustment [39], and the parameter control strategy can be expressed as(8)ωt=1−sintitermax2⋅π2,where ω is the new migration control coefficient instead of Q in (2) and itermax is the maximum number of iterations.As shown in Figure4, at the beginning of the iteration, the value of ω is large and the changing rate is slow, keeping a large migration distance, so the discoverers can lead the population to fully explore the solution space and enhance the global optimization ability. The changing rate of ω in the middle term of the iteration is faster, which is beneficial to improving the convergence speed of the algorithm and the ability to jump out of the local optimal solution. In the later stage of the iteration, the changing rate of ω slows down and the individuals perform a local exploitation on the current solution space to enhance the local optimization ability of the algorithm.Figure 4
Variation curve of the migration control coefficient.Considering the GA strategy and the nonlinear convergence factor strategy, the discoverer location formula is updated as follows:(9)Xi,jt+1=Xi,jtsinr1−r2sinr1θ1⋅Xpt−θ2⋅Xi,jt,R2<ST,Xi,jt+ω⋅L,R2≥ST,where the meaning of each parameter is the same as in (2).


## 3.4. Elite Opposition-Based Learning Strategy

Tizhoosh [40] proposed opposition-based learning (OBL) in 2005, and it was proved that the probability of the opposite solution approaching the optimal solution was 50% higher than the current solution through experiments. The strategy can increase the population’s diversity effectively. The OBL strategy calculates the opposite solution of the current solution, finally selects a better fitness solution from the current solution and the corresponding opposite solution, and updates the individual.Definition 1 (opposite solution).
Suppose that a feasible solution of the current population in the D-dimensional solution space is expressed asX=x1,x2,…,xD, xj∈aj,bj; then, its corresponding opposite solution is X¯=x¯1,x¯2,…,x¯D, where x¯j=λaj+bj−xj, λ∈0,1 is a uniformly distributed random number.
Since OBL solves the problem in a certain space, it has the risk of falling into the local optimum and the problem of making the algorithm fall into premature convergence. The generated opposite solution may have difficulty to search for the optimal value than the current solution space [41, 42]. Elite opposition-based learning (EOBL) was proposed and successfully used to improve the characteristics of this algorithm. Sihwail et al. [43] introduced it in the HHO algorithm to improve population diversity and speed up the convergence process.
The EOBL takes advantage of the fact that elite individuals carry more effective information than ordinary individuals. First, the opposite population is formed through elite individuals with high adaptability in the population; then, elite individuals are selected as a new population from the opposite population and the current population. The EOBL strategy increases the diversity and quality of the population and improves the ability of the algorithm to jump out of the local optimum solution.Definition 2 (elite opposite solution).
Suppose that the extreme point corresponding to the general individual in the current population is elite individuals, which is expressed asXi,je=Xi,1e,Xi,2e,…,Xi,De; then, its corresponding opposite solution is Xi,je¯=Xi,1e¯,Xi,2e¯,…,Xi,De¯. We select the individuals whose fitness value is in the top 10% of the population after each iteration as elite individuals. The equation for the elite opposite solution is as follows:(10)Xi,je¯=λ×lbj+ubj−Xi,je,where λ∈0,1 is a uniformly distributed random number, Xi,je∈lbj,ubj, lbj=minXi,j, and ubj=maxXi,j. lbj and ubj are the lower and upper bounds of the dynamic boundary. Compared with the fixed boundary, the dynamic boundary can save the search experience and reduce the optimization time of the algorithm. When Xi,je¯ exceeds the boundary, it is reset by the equation as follows:(11)Xi,je¯=randlbj+ubj.


## 3.5. Pseudo-Code and Flow Chart of MISSA

The pseudo-code and flow chart of MISSA is shown in Algorithm 1 and Figure5.Figure 5
Flow chart of MISSA.Algorithm 1: Pseudo-code of MISSA.
/∗Initialization∗/(1)
Input:(2)
POP: the number of sparrows;(3)
itermax: the maximum number of iterations;(4)
PD: the proportion of discoverers in the population;(5)
SD: the proportion of scouts in the population;(6)
R2: the alarm value;(7)
ST: the safety threshold;(8)
Initialize the position ofPOP sparrows using equation (5);(9)
Rank the ﬁtness values and divide the population into discoverers and scouts;/∗Iterative search∗/(10)
while (t ≤ itermax)(11)
R2 = rand(1);(12)
fori = 1 : POP∗PD(13)
Update the discoverer’s position using equation (7);(14)
end for(15)
fori = (POP∗PD) + 1 : POP(16)
Update the follower’s position using equation (3);(17)
end for(18)
fori = 1 : POP∗SD(19)
Update the scout’s position using equation (4);(20)
end for(21)
Calculate and rank the fitness values;(22)
fori = 1 : POP∗10%(23)
Update the elite individuals’ position using equation (10);(24)
If the position exceeds the boundary, using equation (11) to reset;(25)
If the new position is better than before, update it;(26)
end for(27)
Rank the fitness values and find the current best position;(28)
t = t + 1;(29)
end while(30)
OutputXbest and fg


## 3.6. Complexity Analysis of MISSA

### 3.6.1. Time Complexity

Assuming the population size and the solution space dimension areN and D and the maximum iteration is T, the time complexity of the original SSA includes the following: the population initialization complexity is O (1), the fitness value of calculation complexity is O (N), the complexity of exploration and exploitation phases is O (T ∗ N ∗ D), and the total time complexity of SSA is O (1) + O (N) + O (T ∗ N ∗ D) = O (T ∗ N ∗ D).MISSA does not change the structure of SSA. The chaotic mapping strategy does not increase the complexity of population initialization, and the time complexity of initializing the population is stillO (1). The Gold-SA strategy and the nonlinear convergence factor strategy only improve the position update method of the discoverers without additional operations, so the time complexity of this stage is still O (N ∗ D), and the time complexity of the EOBL strategy is O (N ∗ D). In regard to this, the total time complexity of MISSA is O (T ∗ N ∗ D), which is the same as that of SSA, indicating that the multimixed strategy used in MISSA does not affect the efficiency of the algorithm.


### 3.6.2. Space Complexity

The storage space consumed by an algorithm can be defined as the space complexity. Assume that the ratio of elite individuals isR. Compared with SSA, only the EOBL strategy increases a certain amount of computational complexity of O (T ∗ R ∗ N ∗ D). The addition of multimixed strategies does not improve the order of magnitude of SSA. Thus, the space efficiency of the proposed strategy is effective and stable.


## 3.6.1. Time Complexity

Assuming the population size and the solution space dimension areN and D and the maximum iteration is T, the time complexity of the original SSA includes the following: the population initialization complexity is O (1), the fitness value of calculation complexity is O (N), the complexity of exploration and exploitation phases is O (T ∗ N ∗ D), and the total time complexity of SSA is O (1) + O (N) + O (T ∗ N ∗ D) = O (T ∗ N ∗ D).MISSA does not change the structure of SSA. The chaotic mapping strategy does not increase the complexity of population initialization, and the time complexity of initializing the population is stillO (1). The Gold-SA strategy and the nonlinear convergence factor strategy only improve the position update method of the discoverers without additional operations, so the time complexity of this stage is still O (N ∗ D), and the time complexity of the EOBL strategy is O (N ∗ D). In regard to this, the total time complexity of MISSA is O (T ∗ N ∗ D), which is the same as that of SSA, indicating that the multimixed strategy used in MISSA does not affect the efficiency of the algorithm.


## 3.6.2. Space Complexity

The storage space consumed by an algorithm can be defined as the space complexity. Assume that the ratio of elite individuals isR. Compared with SSA, only the EOBL strategy increases a certain amount of computational complexity of O (T ∗ R ∗ N ∗ D). The addition of multimixed strategies does not improve the order of magnitude of SSA. Thus, the space efficiency of the proposed strategy is effective and stable.


## 4. Experimental Simulation and Result Analysis

### 4.1. Experimental Design and Parameter Settings

To verify the rationality of the multimixed strategy selection and efficiency of the proposed algorithm, two sets of comparative experiments are designed in this paper:(1)
In order to verify the rationality of the multimixed strategy, three derivative algorithms with a single mechanism were proposed: ITSSA (using the iterative chaotic map strategy), GS-NSSA (using the Gold-SA and nonlinear convergence factor strategies for discoverers), and OBLSSA (using the EOBL strategy).(2)
In order to verify the proposed algorithm’s efficiency, MISSA has been compared to eight optimization algorithms. The comparison algorithms are SSA [24], the improved sparrow search algorithm (ISSA) [30], the chaotic sparrow search algorithm (CSSA) [31], a chaos sparrow search algorithm based on logarithmic spiral search strategy and adaptive step size strategy (CLSSA) [32], COOT [16], WOA [14], GWO [15], and PSO [12].All the experimental tests are carried out with a Windows 10 system using Intel (R) Core (TM) i5-7300HU @2.50 GHz CPU with 16 G RAM and MATLAB 2018b in this paper. The population size and maximum iteration number of each algorithm are 30 and 300, respectively. In addition, other parameters are shown in Table1. In this paper, a comparison test of 30-dimensional, 50-dimensional, and 100-dimensional function optimization is carried out on ten benchmark functions with different characteristics. The specific information of the test functions is shown in Table 2. f1–f5 are unimodal functions used to examine the local exploitation ability of the algorithm, and f6–f10 are multimodal functions used to examine the global exploration ability of the algorithm. To understand the difficulty of searching process intuitively, the two-dimensional graphs of each benchmark functions are illustrated in Figure 6.Table 1
Algorithm parameter settings.
AlgorithmParameter settingsSSA/CSSA/CLSSA /GS-NSSA/OBLSSAST = 0.6, PD = 0.7, SD = 0.2ISSAST = 0.6, PD = 0.7, SD = 0.2,θ = 0.05MISSA/ITSSAST = 0.6, PD = 0.7, SD = 0.2,b = 0.7COOTp = 0.5WOAb = 1, p = 0.5GWOafirst = 2, afinal = 0, r1, r2 ∈ (0, 1)PSOVmax = 0.9, Vmin = 0.9, C1 = C2 = 2, ω = 0.7Table 2
Benchmark functions.
NameTypeTest functionRangeOptimumSphereUNf1=∑i=1nxi2[−100, 100]0Schwefel 2.22UNf2=∑i=1nxi+∏i=1nxi[−10, 10]0Schwefel 1.2UNf3=∑i=1n∑j=1ixj2[−100, 100]0Schwefel 2.21UNf4=maxxi,1≤i≤n[−100, 100]0StepUNf5=∑i=1nxi+0.52[−100, 100]0Generalized SchwefelMUf6=∑i=1n−xisinxi[−500, 500]−418.9829 ×nRastriginMUf7=∑i=1nxi2−10cos2πxi+10[−5.12, 5.12]0AckleyMUf8=−20exp−0.21Dim∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e[−32, 32]0GriewankMUf9=14000∑i=1nxi2−∏i=1ncosxii+1[−600, 600]0PenalizedMUf10=0.1sin23πxi+∑i=1nxi−121+sin23πxi+1+xi21+sin22πxi+∑i=1nuxi,5,100,4[−50, 50]0Figure 6
2-D diagrams of the benchmark functions.


### 4.2. Ablation Experiment of the Multimixed Strategy

#### 4.2.1. Optimization Results for Different Strategies

In order to analyze the rationality of the multimixed strategy and the specific impact of each strategy on the performance of proposed algorithm, MISSA, ITSSA, GS-NSSA, and OBLSSA are compared to optimize several basic benchmark functions as in Table2. The parameters of each algorithm are shown in Table 1. To avoid the bias caused by the contingency of the algorithm, and considering the difference of the algorithms in dealing with different dimensions, three algorithms are independently run 50 times on the four test functions for dimensions 30, 50, and 100, respectively. We select the best value (Best), worst value (Worst), mean value (Mean), and standard deviation (Std) as the evaluation criterion. The Best and Worst reflect the quality of the solution, the Mean reflects the optimization accuracy of the algorithm, and Std reflects the robustness and stability of the algorithm. Table 3 shows the optimization results of the unimodal functions f1 and f2 and the multimodal functions f6 and f10 in three different dimensions.Table 3
Optimization results of different improvement strategies.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1ITSSA0.00E + 001.35E − 682.70E − 701.91E − 690.00E + 007.30E − 481.46E − 491.03E − 480.00E + 001.26E − 492.63E − 511.82E − 50GS-NSSA9.28E − 1497.15E − 421.43E − 431.01E − 421.80E − 1588.11E − 491.71E − 351.21E − 341.74E − 1595.51E − 431.10E − 447.79E − 44OBLSSA0.00E + 002.35E − 575.00E − 593.43E − 580.00E + 002.04E − 771.76E − 901.24E − 890.00E + 009.22E − 611.84E − 621.30E − 61MISSA0.00E + 008.18E − 982.20E − 991.22E − 980.00E + 005.24E − 1291.05E − 1307.40E − 1300.00E + 005.00E − 1182.00E − 1199.90E − 119f2ITSSA0.00E + 002.78E − 225.56E − 243.93E − 230.00E + 003.69E − 235.56E − 245.22E − 240.00E + 004.28E − 228.56E − 246.06E − 23GS-NSSA8.92E − 782.05E − 215.14E − 232.98E − 223.28E − 891.04E − 212.08E − 231.47E − 223.22E − 861.04E − 192.09E − 211.48E − 20OBLSSA0.00E + 001.80E − 353.92E − 372.55E − 360.00E + 001.58E − 303.17E − 322.24E − 310.00E + 002.10E − 364.19E − 382.97E − 37MISSA0.00E + 001.37E − 552.76E − 571.93E − 560.00E + 002.61E − 455.21E − 473.69E − 461.01E − 1794.57E − 641.00E − 656.47E − 65f6ITSSA−1.26E + 04−6.39E + 03−9.80E + 031.90E + 03−2.09E + 04−9.92E + 03−1.58E + 043.63E + 03−3.97E + 04−1.94E + 04−3.32E + 046.90E + 03GS-NSSA−1.26E + 04−6.60E + 03−1.14E + 041.61E + 03−2.09E + 04−1.09E + 04−1.91E + 042.24E + 03−4.19E + 04−1.89E + 04−3.82E + 045.35E + 03OBLSSA−1.16E + 04−4.54E + 03−7.69E + 032.07E + 03−2.09E + 04−7.88E + 03−1.38E + 043.97E + 03−4.13E + 04−1.31E + 04−2.98E + 047.12E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.02E + 02−2.09E + 04−1.36E + 04−1.94E + 041.72E + 03−4.19E + 04−4.02E + 04−4.14E + 043.95E + 02f10ITSSA1.14E − 042.10E − 024.02E − 036.85E − 032.04E − 042.86E − 025.31E − 038.80E − 031.33E − 043.36E − 023.36E − 036.80E − 03GS-NSSA3.92E − 083.26E − 053.45E − 061.46E − 081.13E − 096.06E − 054.32E − 061.14E − 051.26E − 103.80E − 055.99E − 069.85E − 06OBLSSA6.70E − 052.19E − 024.74E − 037.10E − 031.13E − 042.90E − 024.63E − 038.80E − 031.65E − 033.37E − 022.92E − 035.82E − 03MISSA1.40E − 102.78E − 052.92E − 061.40E − 109.93E − 106.78E − 054.96E − 061.15E − 051.16E − 091.43E − 041.09E − 052.74E − 05From a macroperspective, MISSA basically occupies all the optimal values except the functionf10 obtained by GS-NSSA in 100 dimensions. Combined with the experiment results of the three derivative algorithms, the performance of MISSA has been comprehensively improved.From a microperspective, ITSSA based on the iterative chaotic map does not fundamentally change the individual searching mechanism of sparrows and it can easily fall into the local optimal solution as SSA when dealing with complex problems. OBLSSA based on the EOBL strategy has obvious effect on the optimization of unimodal functions, indicating that the opposite solutions of the elite individuals play a dominant role in improving the local exploitation ability. The optimization effect of GS-NSSA based on the Gold-SA and the nonlinear convergence factor on multimodal functions is obvious, which shows the improved location update formula of discovers enhances the global exploration and anti-premature abilities.Overall, MISSA integrates the advantages of different single strategies, synergizes the optimization ability of each strategy, and proves the rationality of multimixed strategy selection.


#### 4.2.2. Diversity Analysis for Different Strategies

To distinguish the effect of different strategies on population diversity, this paper takes the Schwefel 2.22 function and performs diversity analysis on derived algorithms to visualize the diversity of agents in the process of global exploration and local exploitation. Besides, to verify the superiority of the iterative chaotic map, a comparative analysis of four chaotic mapping methods introduced in Section3.1 is carried out.The diversity [44, 45] is defined as follows:(12)Divt=∑j=1D1NDivj=1N∑j=1DDivj,Divj=1N∑i=1N∑j=1DXi,j−X¯j2X¯j=1N∑i=1NXi,j,where Div (t) represents the population diversity in the tth iteration, N represents the population size, and D is the dimension of the problem. X¯j represents the mean value of current position in the jth dimension, and Divj represents mean population diversity in the jth dimension. Thus, the exploration and exploitation percentage of the search process can be defined as follows:(13)exploration%=DivtDivmax×100%,exploitation%=Divt−DivmaxDivmax×100%,where Divmax represents the maximum diversity of the whole group’s population diversity.Figure7 shows the exploration and exploitation percentage curves of population diversity in the search space while solving the Schwefel 2.22 function. Diversity of the algorithm increases with the addition of chaotic initialization methods. Due to the small periods and unstable periodic phenomenon in Singer and Tent maps and the bipolar distribution phenomenon in sinusoidal map, the diversity curves of algorithms fluctuate greatly, which means the improved SSA is not stable. The diversity analysis is the same as the Lyapunov exponent analysis of chaotic maps in [37]. Thus, the iterative chaotic map has the best effect on improving population diversity.Figure 7
Average diversity analysis of different strategies with 20 times. (a) SSA. (b) SSA with Tent map. (c) SSA with Singer map. (d) SSA with sinusoidal map. (e) ITSSA (SSA with iterative algorithm). (f) GS-NSSA. (g) OBLSSA. (h) MISSA.
(a)(b)(c)(d)(e)(f)(g)(h)The curve of GS-NSSA indicates that the Gold-SA and the nonlinear convergence strategies sacrifice part of the stability but rapidly separate the solution space in order to improve the convergence speed. However, the EOBL strategy reconstructs the population through the opposite solution, which greatly improves the stability of population diversity. Ultimately MISSA guarantees stable and abundant population diversity. Notably, combined with optimization results in Table3, MISSA can balance the capabilities between exploration and exploitation phases to push sparrow agents to the global optimal solution.


### 4.3. Benchmark Function Test

#### 4.3.1. Optimization Results for Different Dimensions

The experimental parameter settings are consistent with the rationality experiment in Section4.2. The nine swarm intelligence algorithms performed 50 independent runs on the ten benchmark functions for low dimension (30), medium dimension (50), and high dimension (100). The specific experimental results are reported in Table 4.Table 4
Results for ten basic benchmark functions.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1SSA0.00E + 001.57E − 483.21E − 502.25E − 490.00E + 001.99E − 291.79E − 661.25E − 650.00E + 002.96E − 245.92E − 264.19E − 25ISSA4.39E − 1026.97E − 281.40E − 299.86E − 298.18E − 866.83E − 471.33E − 279.39E − 275.27E − 883.97E − 577.95E − 595.62E − 58CSSA0.00E + 001.73E − 263.45E − 282.44E − 270.00E + 005.14E − 271.04E − 287.27E − 280.00E + 006.60E − 321.48E − 339.54E − 33CLSSA0.00E + 002.00E − 52− 1.38E − 479.74E − 470.00E + 006.77E − 531.36E − 549.58E − 540.00E + 008.26E − 572.79E − 991.39E − 49MISSA0.00E + 001.57E − 487.91E − 1065.34E − 1050.00E + 006.09E − 1281.24E − 1298.71E − 1290.00E + 004.39E − 1239.55E − 1256.48E − 124COOT1.69E − 312.56E − 119.54E − 134.70E − 121.85E − 321.46E − 052.92E − 072.06E − 061.05E − 323.15E − 139.72E − 154.72E − 14WOA4.97E − 512.13E − 405.56E − 423.04E − 417.01E − 534.57E − 379.14E − 396.46E − 386.49E − 522.65E − 385.35E − 403.74E − 39GWO1.16E − 163.00E − 144.86E − 156.77E − 153.46E − 117.37E − 102.21E − 101.89E − 102.27E − 062.69E − 057.99E − 064.19E − 06PSO0.00E + 003.62E − 1047.91E − 1065.34E − 1051.44E + 006.09E + 028.19E + 012.27E + 024.15E + 011.84E + 037.72E + 025.77E + 02f2SSA0.00E + 008.21E − 221.73E − 231.17E − 220.00E + 002.22E − 204.44E − 223.14E − 210.00E + 001.60E − 253.24E − 272.27E − 26ISSA2.57E − 505.05E − 173.27E − 181.03E − 177.68E − 514.52E − 162.11E − 178.48E − 174.71E − 531.31E − 162.68E − 181.85E − 17CSSA0.00E + 008.24E − 121.65E − 131.17E − 120.00E + 002.51E − 165.06E − 183.55E − 170.00E + 006.52E − 131.62E − 149.44E − 14CLSSA0.00E + 008.15E − 502.86E − 511.45E − 500.00E + 001.44E − 442.88E − 462.04E − 450.00E + 003.23E − 416.46E − 434.57E − 42MISSA0.00E + 001.39E − 542.80E − 561.97E − 550.00E + 002.03E − 484.07E − 502.87E − 490.00E + 002.06E − 614.12E − 632.92E − 62COOT2.12E − 169.51E − 063.51E − 071.63E − 063.97E − 153.06E − 036.12E − 054.32E − 044.66E − 161.18E − 043.68E − 061.89E − 05WOA4.05E − 355.61E − 282.38E − 299.67E − 291.02E − 342.56E − 282.27E − 295.52E − 294.61E − 352.36E − 282.10E − 294.92E − 29GWO5.00E − 107.45E − 091.70E − 091.19E − 093.13E − 071.56E − 067.55E − 072.63E − 072.53E − 046.47E − 043.84E − 049.74E − 05PSO1.01E − 1671.39E − 542.80E − 561.97E − 551.32E + 003.18E + 002.01E + 004.30E − 011.13E + 012.17E + 011.54E + 011.99E + 00f3SSA0.00E + 009.96E − 352.03E − 361.42E − 350.00E + 002.43E − 714.98E − 733.44E − 720.00E + 005.96E − 591.22E − 608.52E − 60ISSA1.84E − 832.62E − 205.25E − 223.71E − 211.68E − 1091.81E − 243.75E − 262.56E − 253.17E − 881.22E − 282.46E − 301.73E − 29CSSA0.00E + 001.55E − 223.44E − 242.23E − 230.00E + 009.06E − 231.81E − 241.28E − 230.00E + 004.67E − 279.55E − 296.68E − 28CLSSA0.00E + 004.56E − 289.12E − 306.46E − 290.00E + 001.24E − 314.99E − 332.44E − 320.00E + 001.13E − 242.31E − 261.59E − 25MISSA0.00E + 003.72E − 677.43E − 695.26E − 680.00E + 009.81E − 911.96E − 921.39E − 910.00E + 005.10E − 761.16E − 777.68E − 77COOT3.01E − 326.79E − 101.36E − 119.61E − 111.44E − 302.66E − 095.31E − 113.75E − 104.16E − 348.65E − 113.37E − 121.63E − 11WOA4.79E + 041.35E + 056.04E + 041.82E + 041.46E + 053.54E + 052.37E + 054.24E + 045.74E + 051.82E + 061.14E + 063.41E + 05GWO1.86E − 043.77E − 017.86E − 021.01E − 014.24E + 003.31E + 023.99E + 015.78E + 012.77E + 021.57E + 044.23E + 033.04E + 03PSO2.92E − 2963.72E − 677.43E − 695.26E − 681.18E + 033.44E + 046.39E + 036.57E + 038.91E + 037.49E + 042.92E + 041.53E + 04f4SSA0.00E + 002.06E − 313.69E − 049.54E − 040.00E + 002.67E − 285.33E − 303.77E − 290.00E + 001.09E − 412.22E − 431.55E − 42ISSA4.83E − 582.93E − 237.61E − 299.41E − 291.95E − 559.17E − 317.39E − 172.78E − 169.81E − 467.57E − 161.87E − 171.07E − 16CSSA0.00E + 004.44E − 149.25E − 166.41E − 150.00E + 007.97E − 131.60E − 141.13E − 130.00E + 002.80E − 167.72E − 184.21E − 17CLSSA0.00E + 009.99E − 57− 4.62E − 493.24E − 480.00E + 004.67E − 489.35E − 506.61E − 490.00E + 002.51E − 484.61E − 483.54E − 49MISSA0.00E + 001.12E − 1233.2682E − 1251.7339E − 1240.00E + 005.09E − 581.11E − 597.21E − 590.00E + 001.42E − 642.95E − 662.01E − 65COOT1.06E − 482.23E − 155.74E + 007.66E + 002.29E − 173.98E − 071.27E − 085.83E − 085.96E − 161.64E − 064.64E − 082.44E − 07WOA3.80E − 039.15E + 011.63E − 364.47E − 361.32E + 019.54E + 016.75E + 012.47E + 012.60E + 019.54E + 017.70E + 011.96E + 01GWO3.95E − 161.35E − 134.24E − 221.17E − 213.57E − 021.26E − 014.92E − 022.62E − 028.04E − 011.52E + 014.91E + 002.62E + 00PSO1.27E − 012.76E − 016.00E − 691.65E − 688.08E + 009.05E + 006.01E + 001.17E + 007.54E + 001.26E + 019.98E + 001.18E + 00f5SSA5.23E − 052.06E − 034.07E − 044.09E − 043.97E − 054.53E − 041.61E − 031.92E − 031.38E − 031.85E + 003.23E − 013.45E − 01ISSA1.27E − 101.24E − 047.24E − 061.87E − 052.24E − 101.18E − 041.08E − 052.31E − 051.48E − 072.42E − 023.16E − 036.02E − 03CSSA4.87E − 114.65E − 051.47E − 066.59E − 061.64E − 086.53E − 054.72E − 061.05E − 052.57E − 114.47E − 054.05E − 059.02E − 05CLSSA4.60E − 102.96E − 051.64E − 064.47E − 061.06E − 082.57E − 052.18E − 064.42E − 063.55E − 092.94E − 041.33E − 054.22E − 05MISSA3.15E − 144.15E − 055.03E − 068.74E − 061.42E − 099.85E − 063.50E − 064.01E − 065.42E − 127.18E − 052.74E − 056.44E − 05COOT2.49E − 016.76E + 001.29E + 001.30E + 002.16E + 003.45E + 016.29E + 006.40E + 009.86E + 019.77E + 036.50E + 021.56E + 03WOA2.74E − 011.60E + 009.22E − 013.36E − 017.38E − 014.06E + 002.19E + 006.63E − 019.77E + 019.86E + 019.83E + 011.34E − 01GWO1.98E − 042.51E + 009.99E − 014.50E − 012.00E + 005.24E + 003.47E + 006.68E − 019.62E + 019.87E + 019.82E + 015.16E − 01PSO1.05E − 013.05E + 011.28E + 004.41E + 001.80E + 004.70E + 023.64E + 018.28E + 011.06E + 032.56E + 031.59E + 033.67E + 02f6SSA−1.18E + 04−4.90E + 03−8.25E + 032.23E + 03−2.06E + 04−7.41E + 03−1.49E + 044.02E + 03−4.15E + 04−1.20E + 04−3.15E + 047.89E + 03ISSA−1.22E + 04−3.92E + 03−9.72E + 032.43E + 03−2.08E + 04−6.77E + 03−1.72E + 043.80E + 03−4.19E + 04−2.14E + 04−3.64E + 045.51E + 03CSSA−1.26E + 04−5.01E + 03−8.80E + 033.11E + 01−2.08E + 04−8.01E + 03−1.72E + 043.72E + 03−4.18E + 04−1.10E + 04−3.30E + 047.80E + 03CLSSA−8.00E + 03−5.22E + 03−9.60E + 038.60E + 02−2.09E + 04−8.73E + 03−1.62E + 044.41E + 03−4.16E + 04−1.46E + 04−3.39E + 046.80E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.01E + 02−2.09E + 04−1.33E + 04−1.88E + 041.70E + 03−4.19E + 04−3.99E + 04−4.12E + 045.45E + 02COOT−8.17E + 03−4.81E + 03−6.84E + 037.60E + 02−1.42E + 04−7.36E + 03−1.04E + 041.49E + 03−2.11E + 04−1.11E + 04−1.69E + 042.82E + 03WOA−1.26E + 04−7.43E + 03−9.66E + 031.34E + 03−2.09E + 04−1.32E + 04−1.68E + 042.89E + 03−4.19E + 04−2.12E + 04−3.25E + 045.86E + 03GWO−7.17E + 03−2.78E + 03−5.66E + 031.02E + 03−1.07E + 04−4.06E + 03−8.80E + 031.59E + 03−2.00E + 04−5.69E + 03−1.44E + 043.95E + 03PSO−3.86E + 03−1.64E + 03−2.66E + 034.15E + 02−4.57E + 03−2.54E + 03−3.31E + 035.03E + 02−6.03E + 03−3.70E + 03−4.84E + 036.17E + 02f7SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 008.69E − 099.00E − 101.60E − 09MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 003.14E − 076.29E − 094.44E − 080.00E + 004.68E − 091.13E − 106.63E − 100.00E + 006.44E − 091.54E − 109.13E − 10WOA0.00E + 005.68E − 141.14E − 158.04E − 150.00E + 001.14E − 134.55E − 152.25E − 140.00E + 000.00E + 000.00E + 000.00E + 00GWO9.44E − 122.59E + 017.60E + 005.88E + 001.35E − 052.81E + 011.17E + 016.97E + 003.56E + 008.64E + 013.45E + 011.67E + 01PSO2.16E + 018.34E + 014.40E + 011.44E + 017.29E + 011.67E + 029.99E + 011.85E + 012.56E + 024.74E + 023.53E + 024.95E + 01f8SSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00ISSA8.88E − 164.44E − 151.23E − 151.07E − 158.88E − 161.96E − 138.28E − 153.05E − 148.88E − 168.88E − 168.88E − 160.00E + 00CSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00CLSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00MISSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00COOT4.44E − 151.54E − 056.83E − 073.01E − 068.88E − 167.45E − 051.49E − 061.05E − 054.44E − 158.52E − 061.77E − 071.20E − 06WOA8.88E − 161.51E − 147.13E − 153.95E − 158.88E − 162.22E − 145.93E − 153.45E − 154.44E − 151.51E − 146.93E − 152.41E − 15GWO2.50E − 094.00E − 081.18E − 087.32E − 098.94E − 077.22E − 062.35E − 061.08E − 061.83E − 045.24E − 043.18E − 048.37E − 05PSO5.78E − 024.08E + 005.00E − 018.25E − 014.65E − 012.94E + 001.21E + 005.50E − 012.32E + 004.35E + 002.93E + 003.93E − 01f9SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 007.89E − 143.42E − 151.30E − 140.00E + 008.32E − 121.73E − 131.18E − 120.00E + 007.73E − 101.55E − 111.09E − 10WOA0.00E + 003.74E − 012.95E − 029.14E − 020.00E + 003.20E − 016.39E − 034.52E − 020.00E + 001.11E − 162.22E − 181.57E − 17GWO3.11E − 153.11E − 026.10E − 039.41E − 031.29E − 023.99E − 025.31E − 031.12E − 025.57E − 027.00E − 028.65E − 032.02E − 02PSO1.60E + 022.18E + 021.87E + 021.59E + 013.10E + 023.73E + 023.40E + 021.64E + 016.69E + 027.92E + 027.25E + 023.00E + 01f10SSA8.48E − 061.96E − 022.81E − 035.68E − 033.38E − 063.58E − 024.73E − 038.95E − 032.35E − 054.16E − 025.26E − 031.05E − 02ISSA1.66E − 106.50E − 054.86E − 061.24E − 051.03E − 087.67E − 058.08E − 061.67E − 051.61E − 081.76E − 041.56E − 053.31E − 05CSSA1.66E − 102.26E − 051.73E − 061.34E − 051.52E − 098.80E − 062.47E − 063.30E − 061.89E − 081.03E − 053.69E − 065.26E − 06CLSSA3.60E − 091.17E − 051.87E − 062.73E − 062.33E − 093.86E − 053.37E − 067.89E − 061.73E − 081.79E − 055.26E − 064.00E − 06MISSA4.15E − 107.66E − 055.51E − 063.63E − 063.89E − 101.60E − 053.95E − 065.35E − 067.97E − 092.78E − 052.64E − 074.55E − 07COOT6.01E − 012.60E + 008.21E − 014.61E − 011.77E + 001.18E + 014.08E + 002.09E + 008.03E + 002.29E + 011.17E + 012.68E + 00WOA1.50E − 011.67E + 008.00E − 013.89E − 014.91E − 013.17E + 001.60E + 006.08E − 011.41E + 006.39E + 003.86E + 001.20E + 00GWO4.26E − 011.50E + 009.30E − 012.76E − 011.40E + 003.55E + 002.48E + 004.51E − 015.91E + 008.74E + 007.58E + 005.93E − 01PSO1.75E − 033.62E + 001.60E − 017.11E − 016.09E − 025.22E + 014.77E + 001.22E + 012.63E + 001.24E + 025.85E + 014.20E + 01As shown in Table4, the optimization accuracy of MISSA for ten benchmark functions is significantly better than of the other nine swarm intelligence algorithms under the same conditions.In the experiments on unimodal functions, MISSA can quickly ﬁnd the optimal values with high optimization accuracy. Forf1–f4, MISSA, CSSA, CLSSA, and SSA can all converge to the theoretical optimal value, indicating that MISSA does not reduce the search accuracy of the original SSA algorithm. Although ISSA has not converged to the theoretical value, its accuracy is more than a dozen orders of magnitude better than of other algorithms, which shows the superiority of the original SSA itself. In addition, MISSA is better than the original SSA and other improved SSAs in the evaluation of the Mean and Std, which is better than dozens of orders of magnitude in f1–f4. Besides, the trend of superiority is further improved with the increase in dimensions, showing that MISSA has better stability. For f5, although MISSA fails to find the theoretical optimal solution, the Mean, Std, and quality of its optimal solution in each dimension have been significantly better than other algorithms.In the experiments on multimodal function, MISSA demonstrates excellent global optimization ability to deal with complex problems. Forf6, only MISSA and CSSA converge to the theoretical optimal solution, but the accuracy of the evaluation criterion for MISSA increases up to one or two orders of magnitudes than of other algorithms including CSSA. For f7 and f9, the original SSA and other improved SSAs can search for the theoretical optimal solution in all dimensions and the Mean and Std are both 0, indicating the rationality and superiority of the algorithm itself. For f8, neither MISSA nor other test algorithms have found the theoretical optimal solution, but the Std of MISSA is 0, which indicates that MISSA has been optimized to the theoretical maximum accuracy. For f10, all the evaluation criteria of MISSA are optimal and the superiority becomes more significant when the dimension increases. The mathematical principle of other swarm intelligence algorithms is inferior to that of the sparrow algorithm, and it is easy to fall into local optimum when solving complex problems.Overall, the experiments verify the significant difference between the results obtained by MISSA and other compared algorithms in almost all cases. With the increase in the dimension, other swarm intelligence algorithms such as COOT, WOA, and GWO are not effective enough in solving complex optimization problems. However, the original SSA benefits from the superiority of mathematical principles, which still has the opportunity to jump out of the local optimal solution in high-dimensional optimization, but the stability of the algorithm is obviously reduced. MISSA has prominent stability and robustness, which can balance the global exploitation and local exploitation abilities to fully explore the solution space.


#### 4.3.2. Convergence Curve Analysis

Notably, to completely analyze the convergence speed and optimization accuracy of the compared algorithms, the convergence curves of the ten benchmark functions (D = 30) are shown in Figure 8. We select the semilogarithmic coordinate axis to draw the convergence curve. The y-axis and x-axis represent the fitness value and iteration, respectively.Figure 8
Convergence curves of ten benchmark functions. (a)f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)As shown in Figure8, the convergence curves clearly illustrate the advantages of MISSA in integrating the global exploration and local exploitation abilities into the search process. Under the premise of ensuring the optimization accuracy, the convergence speed of MISSA is better than that of other test algorithms.The smooth convergence curves of MISSA on benchmark functionsf1, f3, f4, f6, f7, and f9 indicate that the improved optimization mechanism is more capable to search the solution space and the algorithm does not tend to fall into local optimum when dealing with these types of benchmark functions. Especially for f7–f9, although both MISSA and other SSAs can converge to the theoretical optimal solution, MISSA has the fastest convergence speed.The steplike fluctuant convergence curves of MISSA on the benchmark functionsf2, f5, f8, and f10 indicate that MISSA can jump out of the local optimum and can be effectively implemented to avoid premature convergence. After the algorithm is temporarily trapped in a local optimum, the EOBL strategy for elite individuals helps the algorithm escape the local optimum.Combined with the optimization results in Table4, MISSA not only improves the stability and robustness but also ensures the optimization accuracy and convergence speed of the algorithm.


#### 4.3.3. Wilcoxon Rank-Sum Test

To further verify the difference from the perspective of statistics between MISSA and the comparison algorithms, the Wilcoxon rank-sum test is carried out at theP = 5% significance level in this paper [46]. Table 5 shows the corresponding results of the Wilcoxon rank-sum test on the benchmark functions (D = 30), in which the value of significance level is set to 0.05. There is a significant advantage between MISSA and the comparison algorithm if the P value is less than 0.05. On the contrary, the optimization performance of MISSA is equal to or worse than the comparison algorithm. NaN means the results of the comparison algorithms are close, and the significance judgment cannot be made. The sign “+/−” means that the MISSA is “better than/equivalent or inferior to” other algorithms in performance.Table 5
Results of the Wilcoxon rank-sum test.
FunctionSSAISSACSSACLSSACOOTWOAGWOPSOf16.43E − 111.12E − 188.49E − 024.81E − 011.12E − 181.12E − 181.12E − 181.12E − 18f28.16E − 061.52E − 171.66E − 024.00E − 031.52E − 173.31E − 151.52E − 171.52E − 17f33.36E − 061.34E − 181.98E − 027.11E − 021.34E − 181.34E − 181.34E − 181.34E − 18f41.33E − 053.41E − 171.17E − 013.21E − 023.41E − 173.41E − 173.41E − 173.41E − 17f57.07E − 18NaN4.04E − 056.96E − 037.07E − 187.07E − 187.07E − 187.07E − 18f62.54E − 161.93E − 111.77E − 106.98E − 187.07E − 187.68E − 157.07E − 187.07E − 18f7NaNNaNNaNNaN1.73E − 073.27E − 013.31E − 203.31E − 20f8NaN4.28E − 02NaNNaN8.24E − 172.32E − 158.24E − 178.24E − 17f9NaNN/ANaNNaN6.50E − 052.31E − 023.31E − 203.31E − 20f101.54E − 172.50E − 034.63E − 024.46E − 027.07E − 187.07E − 187.07E − 187.07E − 18NaN/+/−3/7/03/7/03/6/13/6/10/10/00/9/10/10/00/10/0For theP value in Table 5, MISSA has a significant difference with other eight algorithms. For f7–f9, due to the superiority of the SSA, SSAs can converge to the theoretical optimal value, so the difference is not obvious and the optimization ability is equivalent.


### 4.4. CEC 2014 Function Test

Compared to the basic benchmark functions, the CEC 2014 [47] is more complicated and can evaluate the effectiveness and robustness of the proposed algorithm better. The CEC 2014 consists of unimodal functions (F1–F3), simple multimodal functions (F4–F16), hybrid functions (F17–F22), and composition functions (F23–F30). According to the evaluation criteria of CEC 2014, the number of iterations is 10,000 × dim, the dimension is 30, the search range is [100, 100], and other parameter settings are the same as in Section 4.1.Table6 illustrates that MISSA has a better optimization eﬀect on CEC 2014. From the Wilcoxon rank-sum test, it is seen that MISSA is signiﬁcantly diﬀerent from other algorithms, showing better advantages, and some composition functions are similar to SSAs because of the superiority of the sparrow search algorithm itself which is analyzed in benchmark functions f7–f9. Generally, MISSA has high effectiveness and robustness and is more suitable for some complex optimization problems than other algorithms.Table 6
Results of each algorithm in CEC 2014.
FCriteriaSSAISSACSSACLSSACOOTWOAGWOPSOMISSAF1Best1.592E + 082.567E + 068.560E + 063.745E + 062.400E + 055.548E + 068.723E + 062.585E + 079.413E + 04Worst5.321E + 081.321E + 072.380E + 072.012E + 078.529E + 052.382E + 071.308E + 086.025E + 096.025E + 05Mean1.601E + 087.609E + 061.399E + 071.255E + 075.334E + 051.217E + 074.544E + 078.539E + 083.239E + 05Std1.760E + 083.806E + 065.323E + 065.995E + 062.033E + 055.486E + 063.635E + 071.961E + 091.776E + 05P2.827E − 031.402E − 021.827E − 041.827E − 042.202E − 031.827E − 041.827E − 041.827E − 04F2Best4.636E + 024.473E + 022.172E + 043.446E + 042.807E + 033.581E + 044.116E + 071.330E + 102.025E + 02Worst2.450E + 042.220E + 042.221E + 101.092E + 053.197E + 041.102E + 053.098E + 092.225E + 108.635E + 03Mean1.107E + 046.900E + 031.784E + 108.130E + 041.286E + 046.618E + 041.240E + 091.681E + 101.771E + 03Std8.510E + 037.633E + 036.515E + 092.312E + 041.035E + 043.005E + 049.874E + 083.162E + 092.787E + 03P2.827E − 031.402E − 021.827E − 041.827E − 042.202E − 031.827E − 041.827E − 041.827E − 04F3Best7.248E + 041.796E + 056.625E + 046.504E + 046.012E + 044.606E + 041.845E + 043.142E + 023.195E + 03Worst9.039E + 042.869E + 058.757E + 048.525E + 048.190E + 041.264E + 056.590E + 049.248E + 039.734E + 03Mean8.405E + 042.408E + 058.303E + 047.839E + 047.289E + 047.405E + 043.928E + 045.238E + 035.071E + 03Std5.338E + 033.171E + 047.377E + 036.719E + 037.671E + 033.116E + 041.268E + 043.596E + 031.952E + 03P1.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 046.776E − 01F4Best1.246E + 038.594E + 021.064E + 034.056E + 035.858E + 025.583E + 025.651E + 024.694E + 024.222E + 02Worst3.336E + 032.445E + 038.607E + 031.021E + 041.062E + 037.400E + 027.834E + 024.774E + 025.259E + 02Mean1.838E + 031.770E + 036.405E + 036.818E + 037.996E + 026.711E + 026.650E + 024.717E + 024.774E + 02Std6.265E + 025.105E + 022.253E + 031.927E + 031.578E + 025.354E + 017.403E + 012.878E + 002.131E + 01P1.817E − 041.817E − 041.817E − 041.817E − 041.817E − 041.817E − 041.817E − 048.747E − 03F5Best5.200E + 025.200E + 025.200E + 025.200E + 025.200E + 025.203E + 025.209E + 025.200E + 025.200E + 02Worst5.204E + 025.205E + 025.202E + 025.202E + 025.208E + 025.207E + 025.211E + 025.209E + 025.200E + 02Mean5.200E + 025.201E + 025.201E + 025.200E + 025.203E + 025.205E + 025.210E + 025.205E + 025.200E + 02Std1.153E − 011.587E − 015.872E − 025.831E − 022.534E − 011.138E − 014.143E − 023.427E − 011.618E − 03P4.309E − 041.315E − 031.620E − 013.281E − 042.461E − 041.827E − 041.827E − 042.461E − 04F6Best6.360E + 026.384E + 026.398E + 026.384E + 026.367E + 026.328E + 026.102E + 026.295E + 026.173E + 02Worst6.454E + 026.466E + 026.486E + 026.456E + 026.438E + 026.420E + 026.182E + 026.421E + 026.264E + 02Mean6.425E + 026.419E + 026.421E + 026.432E + 026.390E + 026.370E + 026.145E + 026.352E + 026.214E + 02Std2.816E + 002.521E + 002.621E + 002.168E + 002.203E + 003.012E + 002.642E + 004.779E + 003.423E + 00P1.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 047.685E − 041.827E − 04F7Best7.305E + 027.476E + 027.241E + 028.942E + 027.000E + 027.013E + 027.040E + 027.016E + 027.000E + 02Worst8.443E + 028.490E + 021.110E + 031.050E + 037.000E + 027.061E + 027.643E + 027.021E + 027.021E + 02Mean7.919E + 027.898E + 029.265E + 029.743E + 027.000E + 027.022E + 027.239E + 027.019E + 027.017E + 02Std3.405E + 013.439E + 011.467E + 025.949E + 018.745E − 031.401E + 001.837E + 012.292E − 016.519E − 01P1.827E − 041.827E − 041.827E − 041.827E − 042.827E − 038.796E − 011.827E − 043.195E − 01F8Best9.640E + 029.291E + 029.277E + 029.383E + 029.572E + 029.170E + 028.681E + 029.015E + 028.627E + 02Worst1.025E + 039.887E + 021.016E + 031.007E + 031.024E + 031.094E + 039.615E + 029.731E + 029.473E + 02Mean9.860E + 029.675E + 029.743E + 029.710E + 029.750E + 029.908E + 028.951E + 029.389E + 029.046E + 02Std1.838E + 011.669E + 012.236E + 012.075E + 012.076E + 015.343E + 012.629E + 012.009E + 012.765E + 01P1.827E − 043.298E − 043.298E − 042.461E − 041.827E − 041.008E − 033.847E − 011.726E − 02F9Best1.100E + 031.093E + 031.124E + 031.117E + 039.995E + 021.065E + 031.001E + 031.021E + 039.914E + 02Worst1.130E + 031.161E + 031.188E + 031.183E + 031.039E + 031.228E + 031.139E + 031.097E + 031.131E + 03Mean1.117E + 031.132E + 031.160E + 031.144E + 031.015E + 031.133E + 031.031E + 031.063E + 031.095E + 03Std1.069E + 012.137E + 012.410E + 012.383E + 011.339E + 015.343E + 015.097E + 012.220E + 013.958E + 01P1.041E − 011.133E − 024.396E − 041.706E − 032.827E − 031.859E − 014.362E − 021.133E − 02F10Best4.708E + 033.946E + 035.095E + 034.458E + 033.416E + 034.487E + 033.473E + 033.617E + 032.752E + 03Worst5.737E + 037.185E + 037.089E + 036.473E + 034.740E + 036.788E + 036.226E + 035.298E + 034.153E + 03Mean5.153E + 035.484E + 035.926E + 035.397E + 034.164E + 035.735E + 035.049E + 034.569E + 033.598E + 03Std3.121E + 028.421E + 026.559E + 026.528E + 024.148E + 026.061E + 028.830E + 025.878E + 024.754E + 02P1.827E − 044.396E − 041.827E − 041.827E − 041.402E − 021.827E − 041.315E − 034.586E − 03F11Best4.599E + 035.689E + 034.942E + 035.895E + 034.321E + 035.239E + 034.910E + 034.251E + 032.714E + 03Worst7.325E + 037.937E + 038.411E + 037.682E + 035.879E + 037.681E + 037.039E + 039.197E + 034.860E + 03Mean6.390E + 036.570E + 036.584E + 036.748E + 034.954E + 036.531E + 035.671E + 035.882E + 033.875E + 03Std8.529E + 027.063E + 021.036E + 036.091E + 025.479E + 028.841E + 027.064E + 021.360E + 036.634E + 02P3.298E − 041.827E − 041.827E − 041.827E − 044.586E − 031.827E − 041.827E − 047.685E − 04F12Best1.201E + 031.201E + 031.201E + 031.201E + 031.201E + 031.201E + 031.200E + 031.201E + 031.201E + 03Worst1.203E + 031.203E + 031.203E + 031.202E + 031.202E + 031.203E + 031.203E + 031.202E + 031.202E + 03Mean1.202E + 031.202E + 031.202E + 031.202E + 031.202E + 031.202E + 031.203E + 031.202E + 031.201E + 03Std7.197E − 015.079E − 016.170E − 013.856E − 014.297E − 016.000E − 018.211E − 014.977E − 013.077E − 01P1.399E − 011.388E − 021.207E − 017.910E − 011.613E − 014.362E − 022.786E − 033.840E − 01F13Best1.303E + 031.303E + 031.306E + 031.306E + 031.300E + 031.300E + 031.300E + 031.300E + 031.300E + 03Worst1.305E + 031.305E + 031.308E + 031.308E + 031.301E + 031.301E + 031.303E + 031.301E + 031.303E + 03Mean1.304E + 031.304E + 031.307E + 031.307E + 031.300E + 031.301E + 031.301E + 031.301E + 031.301E + 03Std5.492E − 018.018E − 017.393E − 016.089E − 011.261E − 011.122E − 018.971E − 011.260E − 017.857E − 01P1.827E − 041.827E − 041.827E − 041.827E − 045.795E − 033.447E − 012.413E − 014.390E − 02F14Best1.467E + 031.475E + 031.422E + 031.420E + 031.400E + 031.400E + 031.400E + 031.400E + 031.400E + 03Worst1.507E + 031.515E + 031.491E + 031.510E + 031.401E + 031.400E + 031.416E + 031.401E + 031.438E + 03Mean1.491E + 031.493E + 031.457E + 031.464E + 031.400E + 031.400E + 031.404E + 031.400E + 031.418E + 03Std1.373E + 011.367E + 012.338E + 012.428E + 011.708E − 013.211E − 025.397E + 003.465E − 011.235E + 01P1.827E − 041.827E − 045.828E − 045.828E − 041.133E − 025.795E − 033.121E − 021.133E − 02F15Best4.278E + 034.537E + 032.272E + 032.263E + 031.571E + 031.555E + 031.508E + 031.540E + 031.518E + 03Worst1.537E + 042.229E + 046.568E + 031.394E + 042.222E + 031.691E + 031.885E + 031.570E + 031.533E + 03Mean7.556E + 038.870E + 034.873E + 034.617E + 031.746E + 031.624E + 031.610E + 031.552E + 031.523E + 03Std3.669E + 035.508E + 031.524E + 033.448E + 032.256E + 024.273E + 011.200E + 021.035E + 014.466E + 00P1.817E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 042.827E − 031.827E − 04F16Best1.613E + 031.613E + 031.613E + 031.613E + 031.612E + 031.612E + 031.614E + 031.612E + 031.612E + 03Worst1.614E + 031.614E + 031.614E + 031.614E + 031.613E + 031.613E + 031.613E + 031.613E + 031.613E + 03Mean1.613E + 031.613E + 031.613E + 031.613E + 031.612E + 031.613E + 031.613E + 031.613E + 031.613E + 03Std3.827E − 012.789E − 012.316E − 013.621E − 013.050E − 013.664E − 016.362E − 015.379E − 014.230E − 01P2.409E − 041.786E − 041.688E − 042.786E − 031.726E − 023.447E − 013.298E − 043.440E − 01F17Best8.051E + 056.594E + 054.092E + 053.200E + 053.505E + 046.989E + 052.133E + 051.236E + 057.154E + 03Worst9.578E + 063.698E + 067.083E + 064.444E + 062.988E + 057.565E + 061.656E + 069.602E + 055.006E + 04Mean3.533E + 062.071E + 062.808E + 061.874E + 061.515E + 053.003E + 067.017E + 054.927E + 052.645E + 04Std2.477E + 061.136E + 062.262E + 061.442E + 069.963E + 042.469E + 065.702E + 053.004E + 051.383E + 04P1.827E − 041.827E − 041.827E − 041.827E − 045.828E − 041.827E − 041.827E − 041.827E − 04F18Best3.365E + 038.957E + 037.399E + 046.574E + 082.092E + 042.067E + 044.002E + 032.193E + 031.885E + 03Worst1.290E + 082.893E + 073.872E + 094.259E + 091.574E + 056.860E + 057.632E + 075.631E + 077.298E + 03Mean2.185E + 076.500E + 061.435E + 092.713E + 094.205E + 041.514E + 051.738E + 071.035E + 073.291E + 03Std4.097E + 079.788E + 061.311E + 091.395E + 094.222E + 032.037E + 052.603E + 071.943E + 071.761E + 03P4.586E − 031.827E − 041.827E − 041.827E − 044.708E − 031.827E − 044.396E − 041.133E − 02F19Best2.066E + 032.052E + 032.060E + 032.065E + 031.912E + 031.928E + 031.912E + 031.959E + 031.918E + 03Worst2.331E + 032.199E + 032.332E + 032.364E + 031.971E + 032.094E + 032.002E + 032.038E + 031.946E + 03Mean2.160E + 032.141E + 032.146E + 032.216E + 031.920E + 031.994E + 031.946E + 032.005E + 031.927E + 03Std8.977E + 014.816E + 011.006E + 021.008E + 021.819E + 015.239E + 013.136E + 012.448E + 018.322E + 00P1.827E − 041.827E − 041.827E − 041.827E − 042.827E − 035.828E − 042.123E − 011.827E − 04F20Best9.172E + 041.226E + 053.271E + 046.443E + 045.353E + 032.061E + 041.262E + 042.486E + 043.490E + 03Worst3.861E + 052.749E + 065.141E + 054.471E + 052.929E + 041.165E + 053.594E + 043.048E + 051.041E + 04Mean2.281E + 057.579E + 051.972E + 051.969E + 051.854E + 046.012E + 042.709E + 041.308E + 056.484E + 03Std1.031E + 059.890E + 051.451E + 051.151E + 058.090E + 032.632E + 047.251E + 038.808E + 042.063E + 03P1.827E − 041.827E − 041.827E − 041.827E − 041.315E − 031.827E − 041.827E − 041.827E − 04F21Best1.357E + 061.514E + 062.552E + 052.722E + 054.970E + 041.156E + 066.811E + 041.346E + 062.009E + 04Worst5.335E + 073.639E + 072.150E + 074.371E + 074.151E + 051.029E + 071.846E + 061.521E + 071.177E + 05Mean1.192E + 071.237E + 079.264E + 061.412E + 072.425E + 056.436E + 065.982E + 056.043E + 067.626E + 04Std1.498E + 071.137E + 078.219E + 061.505E + 071.483E + 053.172E + 065.877E + 054.241E + 063.785E + 04P1.817E − 041.817E − 041.817E − 041.817E − 041.398E − 021.817E − 045.777E − 031.817E − 04F22Best3.098E + 033.245E + 033.235E + 033.248E + 032.376E + 033.003E + 032.389E + 032.873E + 032.821E + 03Worst4.849E + 035.707E + 034.393E + 035.464E + 033.132E + 033.499E + 032.869E + 033.998E + 033.658E + 03Mean3.824E + 034.160E + 033.877E + 033.745E + 032.768E + 033.264E + 032.602E + 033.380E + 033.260E + 03Std5.866E + 027.812E + 023.807E + 026.163E + 021.998E + 021.645E + 021.427E + 023.380E + 022.635E + 02P1.726E − 024.586E − 032.202E − 031.402E − 027.685E − 049.698E − 012.461E − 043.447E − 02F23Best2.500E + 032.500E + 032.500E + 032.500E + 032.615E + 032.640E + 032.626E + 032.614E + 032.500E + 03Worst2.500E + 032.500E + 032.500E + 032.500E + 032.616E + 032.678E + 032.662E + 032.614E + 032.500E + 03Mean2.500E + 032.500E + 032.500E + 032.500E + 032.615E + 032.657E + 032.641E + 032.614E + 032.500E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.123E − 011.380E + 011.208E + 011.228E − 030.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F24Best2.600E + 032.600E + 032.600E + 032.600E + 032.625E + 032.605E + 032.600E + 032.621E + 032.600E + 03Worst2.600E + 032.600E + 032.600E + 032.600E + 032.631E + 032.617E + 032.600E + 032.651E + 032.600E + 03Mean2.600E + 032.600E + 032.600E + 032.600E + 032.628E + 032.611E + 032.600E + 032.633E + 032.600E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.989E + 005.162E + 005.381E − 031.063E + 010.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F25Best2.700E + 032.700E + 032.700E + 032.700E + 032.700E + 032.716E + 032.706E + 032.711E + 032.700E + 03Worst2.700E + 032.700E + 032.700E + 032.700E + 032.718E + 032.759E + 032.719E + 032.731E + 032.700E + 03Mean2.700E + 032.700E + 032.700E + 032.700E + 032.713E + 032.727E + 032.711E + 032.719E + 032.700E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.821E + 001.472E + 014.320E + 005.773E + 000.000E + 00PNaNNaNNaNNaN1.576E − 045.511E − 056.340E − 056.386E − 05F26Best2.702E + 032.703E + 032.701E + 032.707E + 032.718E + 032.800E + 032.800E + 032.800E + 032.700E + 03Worst2.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.701E + 03Mean2.790E + 032.771E + 032.773E + 032.783E + 032.792E + 032.800E + 032.761E + 032.800E + 032.700E + 03Std3.089E + 014.648E + 014.427E + 013.524E + 012.598E + 010.000E + 005.082E + 011.902E − 021.288E − 01P8.745E − 051.317E − 042.438E − 041.107E − 048.745E − 056.386E − 055.828E − 041.727E − 04F27Best2.900E + 032.900E + 032.900E + 032.900E + 033.111E + 033.177E + 033.192E + 033.106E + 032.900E + 03Worst2.900E + 032.900E + 032.900E + 032.900E + 033.835E + 034.170E + 033.699E + 034.477E + 032.900E + 03Mean2.900E + 032.900E + 032.900E + 032.900E + 033.592E + 033.785E + 033.417E + 033.787E + 032.900E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.978E + 023.861E + 021.412E + 025.854E + 020.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F28Best3.000E + 033.000E + 033.000E + 033.000E + 033.796E + 034.711E + 033.689E + 036.271E + 033.000E + 03Worst3.000E + 033.000E + 033.000E + 033.000E + 035.108E + 035.966E + 034.747E + 038.157E + 033.000E + 03Mean3.000E + 033.000E + 033.000E + 033.000E + 034.432E + 035.351E + 034.009E + 037.026E + 033.000E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.163E + 023.647E + 023.494E + 026.857E + 020.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F29Best3.100E + 033.100E + 033.100E + 033.100E + 035.558E + 032.146E + 069.872E + 034.580E + 033.100E + 03Worst3.100E + 033.100E + 033.100E + 033.100E + 031.644E + 072.111E + 071.683E + 064.343E + 053.100E + 03Mean3.100E + 033.100E + 033.100E + 033.100E + 036.121E + 061.269E + 072.989E + 051.071E + 053.100E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 006.805E + 065.525E + 065.341E + 051.387E + 050.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F30Best3.200E + 033.200E + 033.200E + 033.200E + 031.068E + 041.172E + 051.817E + 045.782E + 033.200E + 03Worst3.200E + 033.200E + 033.200E + 033.200E + 032.423E + 041.124E + 061.441E + 055.947E + 043.200E + 03Mean3.200E + 033.200E + 033.200E + 033.200E + 031.539E + 044.686E + 055.813E + 041.648E + 043.200E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.192E + 032.990E + 053.914E + 041.642E + 040.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05NaN/+/−7/21/27/23/07/21/27/21/20/29/10/25/50/27/30/26/4


## 4.1. Experimental Design and Parameter Settings

To verify the rationality of the multimixed strategy selection and efficiency of the proposed algorithm, two sets of comparative experiments are designed in this paper:(1)
In order to verify the rationality of the multimixed strategy, three derivative algorithms with a single mechanism were proposed: ITSSA (using the iterative chaotic map strategy), GS-NSSA (using the Gold-SA and nonlinear convergence factor strategies for discoverers), and OBLSSA (using the EOBL strategy).(2)
In order to verify the proposed algorithm’s efficiency, MISSA has been compared to eight optimization algorithms. The comparison algorithms are SSA [24], the improved sparrow search algorithm (ISSA) [30], the chaotic sparrow search algorithm (CSSA) [31], a chaos sparrow search algorithm based on logarithmic spiral search strategy and adaptive step size strategy (CLSSA) [32], COOT [16], WOA [14], GWO [15], and PSO [12].All the experimental tests are carried out with a Windows 10 system using Intel (R) Core (TM) i5-7300HU @2.50 GHz CPU with 16 G RAM and MATLAB 2018b in this paper. The population size and maximum iteration number of each algorithm are 30 and 300, respectively. In addition, other parameters are shown in Table1. In this paper, a comparison test of 30-dimensional, 50-dimensional, and 100-dimensional function optimization is carried out on ten benchmark functions with different characteristics. The specific information of the test functions is shown in Table 2. f1–f5 are unimodal functions used to examine the local exploitation ability of the algorithm, and f6–f10 are multimodal functions used to examine the global exploration ability of the algorithm. To understand the difficulty of searching process intuitively, the two-dimensional graphs of each benchmark functions are illustrated in Figure 6.Table 1
Algorithm parameter settings.
AlgorithmParameter settingsSSA/CSSA/CLSSA /GS-NSSA/OBLSSAST = 0.6, PD = 0.7, SD = 0.2ISSAST = 0.6, PD = 0.7, SD = 0.2,θ = 0.05MISSA/ITSSAST = 0.6, PD = 0.7, SD = 0.2,b = 0.7COOTp = 0.5WOAb = 1, p = 0.5GWOafirst = 2, afinal = 0, r1, r2 ∈ (0, 1)PSOVmax = 0.9, Vmin = 0.9, C1 = C2 = 2, ω = 0.7Table 2
Benchmark functions.
NameTypeTest functionRangeOptimumSphereUNf1=∑i=1nxi2[−100, 100]0Schwefel 2.22UNf2=∑i=1nxi+∏i=1nxi[−10, 10]0Schwefel 1.2UNf3=∑i=1n∑j=1ixj2[−100, 100]0Schwefel 2.21UNf4=maxxi,1≤i≤n[−100, 100]0StepUNf5=∑i=1nxi+0.52[−100, 100]0Generalized SchwefelMUf6=∑i=1n−xisinxi[−500, 500]−418.9829 ×nRastriginMUf7=∑i=1nxi2−10cos2πxi+10[−5.12, 5.12]0AckleyMUf8=−20exp−0.21Dim∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e[−32, 32]0GriewankMUf9=14000∑i=1nxi2−∏i=1ncosxii+1[−600, 600]0PenalizedMUf10=0.1sin23πxi+∑i=1nxi−121+sin23πxi+1+xi21+sin22πxi+∑i=1nuxi,5,100,4[−50, 50]0Figure 6
2-D diagrams of the benchmark functions.


## 4.2. Ablation Experiment of the Multimixed Strategy

### 4.2.1. Optimization Results for Different Strategies

In order to analyze the rationality of the multimixed strategy and the specific impact of each strategy on the performance of proposed algorithm, MISSA, ITSSA, GS-NSSA, and OBLSSA are compared to optimize several basic benchmark functions as in Table2. The parameters of each algorithm are shown in Table 1. To avoid the bias caused by the contingency of the algorithm, and considering the difference of the algorithms in dealing with different dimensions, three algorithms are independently run 50 times on the four test functions for dimensions 30, 50, and 100, respectively. We select the best value (Best), worst value (Worst), mean value (Mean), and standard deviation (Std) as the evaluation criterion. The Best and Worst reflect the quality of the solution, the Mean reflects the optimization accuracy of the algorithm, and Std reflects the robustness and stability of the algorithm. Table 3 shows the optimization results of the unimodal functions f1 and f2 and the multimodal functions f6 and f10 in three different dimensions.Table 3
Optimization results of different improvement strategies.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1ITSSA0.00E + 001.35E − 682.70E − 701.91E − 690.00E + 007.30E − 481.46E − 491.03E − 480.00E + 001.26E − 492.63E − 511.82E − 50GS-NSSA9.28E − 1497.15E − 421.43E − 431.01E − 421.80E − 1588.11E − 491.71E − 351.21E − 341.74E − 1595.51E − 431.10E − 447.79E − 44OBLSSA0.00E + 002.35E − 575.00E − 593.43E − 580.00E + 002.04E − 771.76E − 901.24E − 890.00E + 009.22E − 611.84E − 621.30E − 61MISSA0.00E + 008.18E − 982.20E − 991.22E − 980.00E + 005.24E − 1291.05E − 1307.40E − 1300.00E + 005.00E − 1182.00E − 1199.90E − 119f2ITSSA0.00E + 002.78E − 225.56E − 243.93E − 230.00E + 003.69E − 235.56E − 245.22E − 240.00E + 004.28E − 228.56E − 246.06E − 23GS-NSSA8.92E − 782.05E − 215.14E − 232.98E − 223.28E − 891.04E − 212.08E − 231.47E − 223.22E − 861.04E − 192.09E − 211.48E − 20OBLSSA0.00E + 001.80E − 353.92E − 372.55E − 360.00E + 001.58E − 303.17E − 322.24E − 310.00E + 002.10E − 364.19E − 382.97E − 37MISSA0.00E + 001.37E − 552.76E − 571.93E − 560.00E + 002.61E − 455.21E − 473.69E − 461.01E − 1794.57E − 641.00E − 656.47E − 65f6ITSSA−1.26E + 04−6.39E + 03−9.80E + 031.90E + 03−2.09E + 04−9.92E + 03−1.58E + 043.63E + 03−3.97E + 04−1.94E + 04−3.32E + 046.90E + 03GS-NSSA−1.26E + 04−6.60E + 03−1.14E + 041.61E + 03−2.09E + 04−1.09E + 04−1.91E + 042.24E + 03−4.19E + 04−1.89E + 04−3.82E + 045.35E + 03OBLSSA−1.16E + 04−4.54E + 03−7.69E + 032.07E + 03−2.09E + 04−7.88E + 03−1.38E + 043.97E + 03−4.13E + 04−1.31E + 04−2.98E + 047.12E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.02E + 02−2.09E + 04−1.36E + 04−1.94E + 041.72E + 03−4.19E + 04−4.02E + 04−4.14E + 043.95E + 02f10ITSSA1.14E − 042.10E − 024.02E − 036.85E − 032.04E − 042.86E − 025.31E − 038.80E − 031.33E − 043.36E − 023.36E − 036.80E − 03GS-NSSA3.92E − 083.26E − 053.45E − 061.46E − 081.13E − 096.06E − 054.32E − 061.14E − 051.26E − 103.80E − 055.99E − 069.85E − 06OBLSSA6.70E − 052.19E − 024.74E − 037.10E − 031.13E − 042.90E − 024.63E − 038.80E − 031.65E − 033.37E − 022.92E − 035.82E − 03MISSA1.40E − 102.78E − 052.92E − 061.40E − 109.93E − 106.78E − 054.96E − 061.15E − 051.16E − 091.43E − 041.09E − 052.74E − 05From a macroperspective, MISSA basically occupies all the optimal values except the functionf10 obtained by GS-NSSA in 100 dimensions. Combined with the experiment results of the three derivative algorithms, the performance of MISSA has been comprehensively improved.From a microperspective, ITSSA based on the iterative chaotic map does not fundamentally change the individual searching mechanism of sparrows and it can easily fall into the local optimal solution as SSA when dealing with complex problems. OBLSSA based on the EOBL strategy has obvious effect on the optimization of unimodal functions, indicating that the opposite solutions of the elite individuals play a dominant role in improving the local exploitation ability. The optimization effect of GS-NSSA based on the Gold-SA and the nonlinear convergence factor on multimodal functions is obvious, which shows the improved location update formula of discovers enhances the global exploration and anti-premature abilities.Overall, MISSA integrates the advantages of different single strategies, synergizes the optimization ability of each strategy, and proves the rationality of multimixed strategy selection.


### 4.2.2. Diversity Analysis for Different Strategies

To distinguish the effect of different strategies on population diversity, this paper takes the Schwefel 2.22 function and performs diversity analysis on derived algorithms to visualize the diversity of agents in the process of global exploration and local exploitation. Besides, to verify the superiority of the iterative chaotic map, a comparative analysis of four chaotic mapping methods introduced in Section3.1 is carried out.The diversity [44, 45] is defined as follows:(12)Divt=∑j=1D1NDivj=1N∑j=1DDivj,Divj=1N∑i=1N∑j=1DXi,j−X¯j2X¯j=1N∑i=1NXi,j,where Div (t) represents the population diversity in the tth iteration, N represents the population size, and D is the dimension of the problem. X¯j represents the mean value of current position in the jth dimension, and Divj represents mean population diversity in the jth dimension. Thus, the exploration and exploitation percentage of the search process can be defined as follows:(13)exploration%=DivtDivmax×100%,exploitation%=Divt−DivmaxDivmax×100%,where Divmax represents the maximum diversity of the whole group’s population diversity.Figure7 shows the exploration and exploitation percentage curves of population diversity in the search space while solving the Schwefel 2.22 function. Diversity of the algorithm increases with the addition of chaotic initialization methods. Due to the small periods and unstable periodic phenomenon in Singer and Tent maps and the bipolar distribution phenomenon in sinusoidal map, the diversity curves of algorithms fluctuate greatly, which means the improved SSA is not stable. The diversity analysis is the same as the Lyapunov exponent analysis of chaotic maps in [37]. Thus, the iterative chaotic map has the best effect on improving population diversity.Figure 7
Average diversity analysis of different strategies with 20 times. (a) SSA. (b) SSA with Tent map. (c) SSA with Singer map. (d) SSA with sinusoidal map. (e) ITSSA (SSA with iterative algorithm). (f) GS-NSSA. (g) OBLSSA. (h) MISSA.
(a)(b)(c)(d)(e)(f)(g)(h)The curve of GS-NSSA indicates that the Gold-SA and the nonlinear convergence strategies sacrifice part of the stability but rapidly separate the solution space in order to improve the convergence speed. However, the EOBL strategy reconstructs the population through the opposite solution, which greatly improves the stability of population diversity. Ultimately MISSA guarantees stable and abundant population diversity. Notably, combined with optimization results in Table3, MISSA can balance the capabilities between exploration and exploitation phases to push sparrow agents to the global optimal solution.


## 4.2.1. Optimization Results for Different Strategies

In order to analyze the rationality of the multimixed strategy and the specific impact of each strategy on the performance of proposed algorithm, MISSA, ITSSA, GS-NSSA, and OBLSSA are compared to optimize several basic benchmark functions as in Table2. The parameters of each algorithm are shown in Table 1. To avoid the bias caused by the contingency of the algorithm, and considering the difference of the algorithms in dealing with different dimensions, three algorithms are independently run 50 times on the four test functions for dimensions 30, 50, and 100, respectively. We select the best value (Best), worst value (Worst), mean value (Mean), and standard deviation (Std) as the evaluation criterion. The Best and Worst reflect the quality of the solution, the Mean reflects the optimization accuracy of the algorithm, and Std reflects the robustness and stability of the algorithm. Table 3 shows the optimization results of the unimodal functions f1 and f2 and the multimodal functions f6 and f10 in three different dimensions.Table 3
Optimization results of different improvement strategies.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1ITSSA0.00E + 001.35E − 682.70E − 701.91E − 690.00E + 007.30E − 481.46E − 491.03E − 480.00E + 001.26E − 492.63E − 511.82E − 50GS-NSSA9.28E − 1497.15E − 421.43E − 431.01E − 421.80E − 1588.11E − 491.71E − 351.21E − 341.74E − 1595.51E − 431.10E − 447.79E − 44OBLSSA0.00E + 002.35E − 575.00E − 593.43E − 580.00E + 002.04E − 771.76E − 901.24E − 890.00E + 009.22E − 611.84E − 621.30E − 61MISSA0.00E + 008.18E − 982.20E − 991.22E − 980.00E + 005.24E − 1291.05E − 1307.40E − 1300.00E + 005.00E − 1182.00E − 1199.90E − 119f2ITSSA0.00E + 002.78E − 225.56E − 243.93E − 230.00E + 003.69E − 235.56E − 245.22E − 240.00E + 004.28E − 228.56E − 246.06E − 23GS-NSSA8.92E − 782.05E − 215.14E − 232.98E − 223.28E − 891.04E − 212.08E − 231.47E − 223.22E − 861.04E − 192.09E − 211.48E − 20OBLSSA0.00E + 001.80E − 353.92E − 372.55E − 360.00E + 001.58E − 303.17E − 322.24E − 310.00E + 002.10E − 364.19E − 382.97E − 37MISSA0.00E + 001.37E − 552.76E − 571.93E − 560.00E + 002.61E − 455.21E − 473.69E − 461.01E − 1794.57E − 641.00E − 656.47E − 65f6ITSSA−1.26E + 04−6.39E + 03−9.80E + 031.90E + 03−2.09E + 04−9.92E + 03−1.58E + 043.63E + 03−3.97E + 04−1.94E + 04−3.32E + 046.90E + 03GS-NSSA−1.26E + 04−6.60E + 03−1.14E + 041.61E + 03−2.09E + 04−1.09E + 04−1.91E + 042.24E + 03−4.19E + 04−1.89E + 04−3.82E + 045.35E + 03OBLSSA−1.16E + 04−4.54E + 03−7.69E + 032.07E + 03−2.09E + 04−7.88E + 03−1.38E + 043.97E + 03−4.13E + 04−1.31E + 04−2.98E + 047.12E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.02E + 02−2.09E + 04−1.36E + 04−1.94E + 041.72E + 03−4.19E + 04−4.02E + 04−4.14E + 043.95E + 02f10ITSSA1.14E − 042.10E − 024.02E − 036.85E − 032.04E − 042.86E − 025.31E − 038.80E − 031.33E − 043.36E − 023.36E − 036.80E − 03GS-NSSA3.92E − 083.26E − 053.45E − 061.46E − 081.13E − 096.06E − 054.32E − 061.14E − 051.26E − 103.80E − 055.99E − 069.85E − 06OBLSSA6.70E − 052.19E − 024.74E − 037.10E − 031.13E − 042.90E − 024.63E − 038.80E − 031.65E − 033.37E − 022.92E − 035.82E − 03MISSA1.40E − 102.78E − 052.92E − 061.40E − 109.93E − 106.78E − 054.96E − 061.15E − 051.16E − 091.43E − 041.09E − 052.74E − 05From a macroperspective, MISSA basically occupies all the optimal values except the functionf10 obtained by GS-NSSA in 100 dimensions. Combined with the experiment results of the three derivative algorithms, the performance of MISSA has been comprehensively improved.From a microperspective, ITSSA based on the iterative chaotic map does not fundamentally change the individual searching mechanism of sparrows and it can easily fall into the local optimal solution as SSA when dealing with complex problems. OBLSSA based on the EOBL strategy has obvious effect on the optimization of unimodal functions, indicating that the opposite solutions of the elite individuals play a dominant role in improving the local exploitation ability. The optimization effect of GS-NSSA based on the Gold-SA and the nonlinear convergence factor on multimodal functions is obvious, which shows the improved location update formula of discovers enhances the global exploration and anti-premature abilities.Overall, MISSA integrates the advantages of different single strategies, synergizes the optimization ability of each strategy, and proves the rationality of multimixed strategy selection.


## 4.2.2. Diversity Analysis for Different Strategies

To distinguish the effect of different strategies on population diversity, this paper takes the Schwefel 2.22 function and performs diversity analysis on derived algorithms to visualize the diversity of agents in the process of global exploration and local exploitation. Besides, to verify the superiority of the iterative chaotic map, a comparative analysis of four chaotic mapping methods introduced in Section3.1 is carried out.The diversity [44, 45] is defined as follows:(12)Divt=∑j=1D1NDivj=1N∑j=1DDivj,Divj=1N∑i=1N∑j=1DXi,j−X¯j2X¯j=1N∑i=1NXi,j,where Div (t) represents the population diversity in the tth iteration, N represents the population size, and D is the dimension of the problem. X¯j represents the mean value of current position in the jth dimension, and Divj represents mean population diversity in the jth dimension. Thus, the exploration and exploitation percentage of the search process can be defined as follows:(13)exploration%=DivtDivmax×100%,exploitation%=Divt−DivmaxDivmax×100%,where Divmax represents the maximum diversity of the whole group’s population diversity.Figure7 shows the exploration and exploitation percentage curves of population diversity in the search space while solving the Schwefel 2.22 function. Diversity of the algorithm increases with the addition of chaotic initialization methods. Due to the small periods and unstable periodic phenomenon in Singer and Tent maps and the bipolar distribution phenomenon in sinusoidal map, the diversity curves of algorithms fluctuate greatly, which means the improved SSA is not stable. The diversity analysis is the same as the Lyapunov exponent analysis of chaotic maps in [37]. Thus, the iterative chaotic map has the best effect on improving population diversity.Figure 7
Average diversity analysis of different strategies with 20 times. (a) SSA. (b) SSA with Tent map. (c) SSA with Singer map. (d) SSA with sinusoidal map. (e) ITSSA (SSA with iterative algorithm). (f) GS-NSSA. (g) OBLSSA. (h) MISSA.
(a)(b)(c)(d)(e)(f)(g)(h)The curve of GS-NSSA indicates that the Gold-SA and the nonlinear convergence strategies sacrifice part of the stability but rapidly separate the solution space in order to improve the convergence speed. However, the EOBL strategy reconstructs the population through the opposite solution, which greatly improves the stability of population diversity. Ultimately MISSA guarantees stable and abundant population diversity. Notably, combined with optimization results in Table3, MISSA can balance the capabilities between exploration and exploitation phases to push sparrow agents to the global optimal solution.


## 4.3. Benchmark Function Test

### 4.3.1. Optimization Results for Different Dimensions

The experimental parameter settings are consistent with the rationality experiment in Section4.2. The nine swarm intelligence algorithms performed 50 independent runs on the ten benchmark functions for low dimension (30), medium dimension (50), and high dimension (100). The specific experimental results are reported in Table 4.Table 4
Results for ten basic benchmark functions.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1SSA0.00E + 001.57E − 483.21E − 502.25E − 490.00E + 001.99E − 291.79E − 661.25E − 650.00E + 002.96E − 245.92E − 264.19E − 25ISSA4.39E − 1026.97E − 281.40E − 299.86E − 298.18E − 866.83E − 471.33E − 279.39E − 275.27E − 883.97E − 577.95E − 595.62E − 58CSSA0.00E + 001.73E − 263.45E − 282.44E − 270.00E + 005.14E − 271.04E − 287.27E − 280.00E + 006.60E − 321.48E − 339.54E − 33CLSSA0.00E + 002.00E − 52− 1.38E − 479.74E − 470.00E + 006.77E − 531.36E − 549.58E − 540.00E + 008.26E − 572.79E − 991.39E − 49MISSA0.00E + 001.57E − 487.91E − 1065.34E − 1050.00E + 006.09E − 1281.24E − 1298.71E − 1290.00E + 004.39E − 1239.55E − 1256.48E − 124COOT1.69E − 312.56E − 119.54E − 134.70E − 121.85E − 321.46E − 052.92E − 072.06E − 061.05E − 323.15E − 139.72E − 154.72E − 14WOA4.97E − 512.13E − 405.56E − 423.04E − 417.01E − 534.57E − 379.14E − 396.46E − 386.49E − 522.65E − 385.35E − 403.74E − 39GWO1.16E − 163.00E − 144.86E − 156.77E − 153.46E − 117.37E − 102.21E − 101.89E − 102.27E − 062.69E − 057.99E − 064.19E − 06PSO0.00E + 003.62E − 1047.91E − 1065.34E − 1051.44E + 006.09E + 028.19E + 012.27E + 024.15E + 011.84E + 037.72E + 025.77E + 02f2SSA0.00E + 008.21E − 221.73E − 231.17E − 220.00E + 002.22E − 204.44E − 223.14E − 210.00E + 001.60E − 253.24E − 272.27E − 26ISSA2.57E − 505.05E − 173.27E − 181.03E − 177.68E − 514.52E − 162.11E − 178.48E − 174.71E − 531.31E − 162.68E − 181.85E − 17CSSA0.00E + 008.24E − 121.65E − 131.17E − 120.00E + 002.51E − 165.06E − 183.55E − 170.00E + 006.52E − 131.62E − 149.44E − 14CLSSA0.00E + 008.15E − 502.86E − 511.45E − 500.00E + 001.44E − 442.88E − 462.04E − 450.00E + 003.23E − 416.46E − 434.57E − 42MISSA0.00E + 001.39E − 542.80E − 561.97E − 550.00E + 002.03E − 484.07E − 502.87E − 490.00E + 002.06E − 614.12E − 632.92E − 62COOT2.12E − 169.51E − 063.51E − 071.63E − 063.97E − 153.06E − 036.12E − 054.32E − 044.66E − 161.18E − 043.68E − 061.89E − 05WOA4.05E − 355.61E − 282.38E − 299.67E − 291.02E − 342.56E − 282.27E − 295.52E − 294.61E − 352.36E − 282.10E − 294.92E − 29GWO5.00E − 107.45E − 091.70E − 091.19E − 093.13E − 071.56E − 067.55E − 072.63E − 072.53E − 046.47E − 043.84E − 049.74E − 05PSO1.01E − 1671.39E − 542.80E − 561.97E − 551.32E + 003.18E + 002.01E + 004.30E − 011.13E + 012.17E + 011.54E + 011.99E + 00f3SSA0.00E + 009.96E − 352.03E − 361.42E − 350.00E + 002.43E − 714.98E − 733.44E − 720.00E + 005.96E − 591.22E − 608.52E − 60ISSA1.84E − 832.62E − 205.25E − 223.71E − 211.68E − 1091.81E − 243.75E − 262.56E − 253.17E − 881.22E − 282.46E − 301.73E − 29CSSA0.00E + 001.55E − 223.44E − 242.23E − 230.00E + 009.06E − 231.81E − 241.28E − 230.00E + 004.67E − 279.55E − 296.68E − 28CLSSA0.00E + 004.56E − 289.12E − 306.46E − 290.00E + 001.24E − 314.99E − 332.44E − 320.00E + 001.13E − 242.31E − 261.59E − 25MISSA0.00E + 003.72E − 677.43E − 695.26E − 680.00E + 009.81E − 911.96E − 921.39E − 910.00E + 005.10E − 761.16E − 777.68E − 77COOT3.01E − 326.79E − 101.36E − 119.61E − 111.44E − 302.66E − 095.31E − 113.75E − 104.16E − 348.65E − 113.37E − 121.63E − 11WOA4.79E + 041.35E + 056.04E + 041.82E + 041.46E + 053.54E + 052.37E + 054.24E + 045.74E + 051.82E + 061.14E + 063.41E + 05GWO1.86E − 043.77E − 017.86E − 021.01E − 014.24E + 003.31E + 023.99E + 015.78E + 012.77E + 021.57E + 044.23E + 033.04E + 03PSO2.92E − 2963.72E − 677.43E − 695.26E − 681.18E + 033.44E + 046.39E + 036.57E + 038.91E + 037.49E + 042.92E + 041.53E + 04f4SSA0.00E + 002.06E − 313.69E − 049.54E − 040.00E + 002.67E − 285.33E − 303.77E − 290.00E + 001.09E − 412.22E − 431.55E − 42ISSA4.83E − 582.93E − 237.61E − 299.41E − 291.95E − 559.17E − 317.39E − 172.78E − 169.81E − 467.57E − 161.87E − 171.07E − 16CSSA0.00E + 004.44E − 149.25E − 166.41E − 150.00E + 007.97E − 131.60E − 141.13E − 130.00E + 002.80E − 167.72E − 184.21E − 17CLSSA0.00E + 009.99E − 57− 4.62E − 493.24E − 480.00E + 004.67E − 489.35E − 506.61E − 490.00E + 002.51E − 484.61E − 483.54E − 49MISSA0.00E + 001.12E − 1233.2682E − 1251.7339E − 1240.00E + 005.09E − 581.11E − 597.21E − 590.00E + 001.42E − 642.95E − 662.01E − 65COOT1.06E − 482.23E − 155.74E + 007.66E + 002.29E − 173.98E − 071.27E − 085.83E − 085.96E − 161.64E − 064.64E − 082.44E − 07WOA3.80E − 039.15E + 011.63E − 364.47E − 361.32E + 019.54E + 016.75E + 012.47E + 012.60E + 019.54E + 017.70E + 011.96E + 01GWO3.95E − 161.35E − 134.24E − 221.17E − 213.57E − 021.26E − 014.92E − 022.62E − 028.04E − 011.52E + 014.91E + 002.62E + 00PSO1.27E − 012.76E − 016.00E − 691.65E − 688.08E + 009.05E + 006.01E + 001.17E + 007.54E + 001.26E + 019.98E + 001.18E + 00f5SSA5.23E − 052.06E − 034.07E − 044.09E − 043.97E − 054.53E − 041.61E − 031.92E − 031.38E − 031.85E + 003.23E − 013.45E − 01ISSA1.27E − 101.24E − 047.24E − 061.87E − 052.24E − 101.18E − 041.08E − 052.31E − 051.48E − 072.42E − 023.16E − 036.02E − 03CSSA4.87E − 114.65E − 051.47E − 066.59E − 061.64E − 086.53E − 054.72E − 061.05E − 052.57E − 114.47E − 054.05E − 059.02E − 05CLSSA4.60E − 102.96E − 051.64E − 064.47E − 061.06E − 082.57E − 052.18E − 064.42E − 063.55E − 092.94E − 041.33E − 054.22E − 05MISSA3.15E − 144.15E − 055.03E − 068.74E − 061.42E − 099.85E − 063.50E − 064.01E − 065.42E − 127.18E − 052.74E − 056.44E − 05COOT2.49E − 016.76E + 001.29E + 001.30E + 002.16E + 003.45E + 016.29E + 006.40E + 009.86E + 019.77E + 036.50E + 021.56E + 03WOA2.74E − 011.60E + 009.22E − 013.36E − 017.38E − 014.06E + 002.19E + 006.63E − 019.77E + 019.86E + 019.83E + 011.34E − 01GWO1.98E − 042.51E + 009.99E − 014.50E − 012.00E + 005.24E + 003.47E + 006.68E − 019.62E + 019.87E + 019.82E + 015.16E − 01PSO1.05E − 013.05E + 011.28E + 004.41E + 001.80E + 004.70E + 023.64E + 018.28E + 011.06E + 032.56E + 031.59E + 033.67E + 02f6SSA−1.18E + 04−4.90E + 03−8.25E + 032.23E + 03−2.06E + 04−7.41E + 03−1.49E + 044.02E + 03−4.15E + 04−1.20E + 04−3.15E + 047.89E + 03ISSA−1.22E + 04−3.92E + 03−9.72E + 032.43E + 03−2.08E + 04−6.77E + 03−1.72E + 043.80E + 03−4.19E + 04−2.14E + 04−3.64E + 045.51E + 03CSSA−1.26E + 04−5.01E + 03−8.80E + 033.11E + 01−2.08E + 04−8.01E + 03−1.72E + 043.72E + 03−4.18E + 04−1.10E + 04−3.30E + 047.80E + 03CLSSA−8.00E + 03−5.22E + 03−9.60E + 038.60E + 02−2.09E + 04−8.73E + 03−1.62E + 044.41E + 03−4.16E + 04−1.46E + 04−3.39E + 046.80E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.01E + 02−2.09E + 04−1.33E + 04−1.88E + 041.70E + 03−4.19E + 04−3.99E + 04−4.12E + 045.45E + 02COOT−8.17E + 03−4.81E + 03−6.84E + 037.60E + 02−1.42E + 04−7.36E + 03−1.04E + 041.49E + 03−2.11E + 04−1.11E + 04−1.69E + 042.82E + 03WOA−1.26E + 04−7.43E + 03−9.66E + 031.34E + 03−2.09E + 04−1.32E + 04−1.68E + 042.89E + 03−4.19E + 04−2.12E + 04−3.25E + 045.86E + 03GWO−7.17E + 03−2.78E + 03−5.66E + 031.02E + 03−1.07E + 04−4.06E + 03−8.80E + 031.59E + 03−2.00E + 04−5.69E + 03−1.44E + 043.95E + 03PSO−3.86E + 03−1.64E + 03−2.66E + 034.15E + 02−4.57E + 03−2.54E + 03−3.31E + 035.03E + 02−6.03E + 03−3.70E + 03−4.84E + 036.17E + 02f7SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 008.69E − 099.00E − 101.60E − 09MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 003.14E − 076.29E − 094.44E − 080.00E + 004.68E − 091.13E − 106.63E − 100.00E + 006.44E − 091.54E − 109.13E − 10WOA0.00E + 005.68E − 141.14E − 158.04E − 150.00E + 001.14E − 134.55E − 152.25E − 140.00E + 000.00E + 000.00E + 000.00E + 00GWO9.44E − 122.59E + 017.60E + 005.88E + 001.35E − 052.81E + 011.17E + 016.97E + 003.56E + 008.64E + 013.45E + 011.67E + 01PSO2.16E + 018.34E + 014.40E + 011.44E + 017.29E + 011.67E + 029.99E + 011.85E + 012.56E + 024.74E + 023.53E + 024.95E + 01f8SSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00ISSA8.88E − 164.44E − 151.23E − 151.07E − 158.88E − 161.96E − 138.28E − 153.05E − 148.88E − 168.88E − 168.88E − 160.00E + 00CSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00CLSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00MISSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00COOT4.44E − 151.54E − 056.83E − 073.01E − 068.88E − 167.45E − 051.49E − 061.05E − 054.44E − 158.52E − 061.77E − 071.20E − 06WOA8.88E − 161.51E − 147.13E − 153.95E − 158.88E − 162.22E − 145.93E − 153.45E − 154.44E − 151.51E − 146.93E − 152.41E − 15GWO2.50E − 094.00E − 081.18E − 087.32E − 098.94E − 077.22E − 062.35E − 061.08E − 061.83E − 045.24E − 043.18E − 048.37E − 05PSO5.78E − 024.08E + 005.00E − 018.25E − 014.65E − 012.94E + 001.21E + 005.50E − 012.32E + 004.35E + 002.93E + 003.93E − 01f9SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 007.89E − 143.42E − 151.30E − 140.00E + 008.32E − 121.73E − 131.18E − 120.00E + 007.73E − 101.55E − 111.09E − 10WOA0.00E + 003.74E − 012.95E − 029.14E − 020.00E + 003.20E − 016.39E − 034.52E − 020.00E + 001.11E − 162.22E − 181.57E − 17GWO3.11E − 153.11E − 026.10E − 039.41E − 031.29E − 023.99E − 025.31E − 031.12E − 025.57E − 027.00E − 028.65E − 032.02E − 02PSO1.60E + 022.18E + 021.87E + 021.59E + 013.10E + 023.73E + 023.40E + 021.64E + 016.69E + 027.92E + 027.25E + 023.00E + 01f10SSA8.48E − 061.96E − 022.81E − 035.68E − 033.38E − 063.58E − 024.73E − 038.95E − 032.35E − 054.16E − 025.26E − 031.05E − 02ISSA1.66E − 106.50E − 054.86E − 061.24E − 051.03E − 087.67E − 058.08E − 061.67E − 051.61E − 081.76E − 041.56E − 053.31E − 05CSSA1.66E − 102.26E − 051.73E − 061.34E − 051.52E − 098.80E − 062.47E − 063.30E − 061.89E − 081.03E − 053.69E − 065.26E − 06CLSSA3.60E − 091.17E − 051.87E − 062.73E − 062.33E − 093.86E − 053.37E − 067.89E − 061.73E − 081.79E − 055.26E − 064.00E − 06MISSA4.15E − 107.66E − 055.51E − 063.63E − 063.89E − 101.60E − 053.95E − 065.35E − 067.97E − 092.78E − 052.64E − 074.55E − 07COOT6.01E − 012.60E + 008.21E − 014.61E − 011.77E + 001.18E + 014.08E + 002.09E + 008.03E + 002.29E + 011.17E + 012.68E + 00WOA1.50E − 011.67E + 008.00E − 013.89E − 014.91E − 013.17E + 001.60E + 006.08E − 011.41E + 006.39E + 003.86E + 001.20E + 00GWO4.26E − 011.50E + 009.30E − 012.76E − 011.40E + 003.55E + 002.48E + 004.51E − 015.91E + 008.74E + 007.58E + 005.93E − 01PSO1.75E − 033.62E + 001.60E − 017.11E − 016.09E − 025.22E + 014.77E + 001.22E + 012.63E + 001.24E + 025.85E + 014.20E + 01As shown in Table4, the optimization accuracy of MISSA for ten benchmark functions is significantly better than of the other nine swarm intelligence algorithms under the same conditions.In the experiments on unimodal functions, MISSA can quickly ﬁnd the optimal values with high optimization accuracy. Forf1–f4, MISSA, CSSA, CLSSA, and SSA can all converge to the theoretical optimal value, indicating that MISSA does not reduce the search accuracy of the original SSA algorithm. Although ISSA has not converged to the theoretical value, its accuracy is more than a dozen orders of magnitude better than of other algorithms, which shows the superiority of the original SSA itself. In addition, MISSA is better than the original SSA and other improved SSAs in the evaluation of the Mean and Std, which is better than dozens of orders of magnitude in f1–f4. Besides, the trend of superiority is further improved with the increase in dimensions, showing that MISSA has better stability. For f5, although MISSA fails to find the theoretical optimal solution, the Mean, Std, and quality of its optimal solution in each dimension have been significantly better than other algorithms.In the experiments on multimodal function, MISSA demonstrates excellent global optimization ability to deal with complex problems. Forf6, only MISSA and CSSA converge to the theoretical optimal solution, but the accuracy of the evaluation criterion for MISSA increases up to one or two orders of magnitudes than of other algorithms including CSSA. For f7 and f9, the original SSA and other improved SSAs can search for the theoretical optimal solution in all dimensions and the Mean and Std are both 0, indicating the rationality and superiority of the algorithm itself. For f8, neither MISSA nor other test algorithms have found the theoretical optimal solution, but the Std of MISSA is 0, which indicates that MISSA has been optimized to the theoretical maximum accuracy. For f10, all the evaluation criteria of MISSA are optimal and the superiority becomes more significant when the dimension increases. The mathematical principle of other swarm intelligence algorithms is inferior to that of the sparrow algorithm, and it is easy to fall into local optimum when solving complex problems.Overall, the experiments verify the significant difference between the results obtained by MISSA and other compared algorithms in almost all cases. With the increase in the dimension, other swarm intelligence algorithms such as COOT, WOA, and GWO are not effective enough in solving complex optimization problems. However, the original SSA benefits from the superiority of mathematical principles, which still has the opportunity to jump out of the local optimal solution in high-dimensional optimization, but the stability of the algorithm is obviously reduced. MISSA has prominent stability and robustness, which can balance the global exploitation and local exploitation abilities to fully explore the solution space.


### 4.3.2. Convergence Curve Analysis

Notably, to completely analyze the convergence speed and optimization accuracy of the compared algorithms, the convergence curves of the ten benchmark functions (D = 30) are shown in Figure 8. We select the semilogarithmic coordinate axis to draw the convergence curve. The y-axis and x-axis represent the fitness value and iteration, respectively.Figure 8
Convergence curves of ten benchmark functions. (a)f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)As shown in Figure8, the convergence curves clearly illustrate the advantages of MISSA in integrating the global exploration and local exploitation abilities into the search process. Under the premise of ensuring the optimization accuracy, the convergence speed of MISSA is better than that of other test algorithms.The smooth convergence curves of MISSA on benchmark functionsf1, f3, f4, f6, f7, and f9 indicate that the improved optimization mechanism is more capable to search the solution space and the algorithm does not tend to fall into local optimum when dealing with these types of benchmark functions. Especially for f7–f9, although both MISSA and other SSAs can converge to the theoretical optimal solution, MISSA has the fastest convergence speed.The steplike fluctuant convergence curves of MISSA on the benchmark functionsf2, f5, f8, and f10 indicate that MISSA can jump out of the local optimum and can be effectively implemented to avoid premature convergence. After the algorithm is temporarily trapped in a local optimum, the EOBL strategy for elite individuals helps the algorithm escape the local optimum.Combined with the optimization results in Table4, MISSA not only improves the stability and robustness but also ensures the optimization accuracy and convergence speed of the algorithm.


### 4.3.3. Wilcoxon Rank-Sum Test

To further verify the difference from the perspective of statistics between MISSA and the comparison algorithms, the Wilcoxon rank-sum test is carried out at theP = 5% significance level in this paper [46]. Table 5 shows the corresponding results of the Wilcoxon rank-sum test on the benchmark functions (D = 30), in which the value of significance level is set to 0.05. There is a significant advantage between MISSA and the comparison algorithm if the P value is less than 0.05. On the contrary, the optimization performance of MISSA is equal to or worse than the comparison algorithm. NaN means the results of the comparison algorithms are close, and the significance judgment cannot be made. The sign “+/−” means that the MISSA is “better than/equivalent or inferior to” other algorithms in performance.Table 5
Results of the Wilcoxon rank-sum test.
FunctionSSAISSACSSACLSSACOOTWOAGWOPSOf16.43E − 111.12E − 188.49E − 024.81E − 011.12E − 181.12E − 181.12E − 181.12E − 18f28.16E − 061.52E − 171.66E − 024.00E − 031.52E − 173.31E − 151.52E − 171.52E − 17f33.36E − 061.34E − 181.98E − 027.11E − 021.34E − 181.34E − 181.34E − 181.34E − 18f41.33E − 053.41E − 171.17E − 013.21E − 023.41E − 173.41E − 173.41E − 173.41E − 17f57.07E − 18NaN4.04E − 056.96E − 037.07E − 187.07E − 187.07E − 187.07E − 18f62.54E − 161.93E − 111.77E − 106.98E − 187.07E − 187.68E − 157.07E − 187.07E − 18f7NaNNaNNaNNaN1.73E − 073.27E − 013.31E − 203.31E − 20f8NaN4.28E − 02NaNNaN8.24E − 172.32E − 158.24E − 178.24E − 17f9NaNN/ANaNNaN6.50E − 052.31E − 023.31E − 203.31E − 20f101.54E − 172.50E − 034.63E − 024.46E − 027.07E − 187.07E − 187.07E − 187.07E − 18NaN/+/−3/7/03/7/03/6/13/6/10/10/00/9/10/10/00/10/0For theP value in Table 5, MISSA has a significant difference with other eight algorithms. For f7–f9, due to the superiority of the SSA, SSAs can converge to the theoretical optimal value, so the difference is not obvious and the optimization ability is equivalent.


## 4.3.1. Optimization Results for Different Dimensions

The experimental parameter settings are consistent with the rationality experiment in Section4.2. The nine swarm intelligence algorithms performed 50 independent runs on the ten benchmark functions for low dimension (30), medium dimension (50), and high dimension (100). The specific experimental results are reported in Table 4.Table 4
Results for ten basic benchmark functions.
FDimD = 30D = 50D = 100AlgorithmBestWorstMeanStdBestWorstMeanStdBestWorstMeanStdf1SSA0.00E + 001.57E − 483.21E − 502.25E − 490.00E + 001.99E − 291.79E − 661.25E − 650.00E + 002.96E − 245.92E − 264.19E − 25ISSA4.39E − 1026.97E − 281.40E − 299.86E − 298.18E − 866.83E − 471.33E − 279.39E − 275.27E − 883.97E − 577.95E − 595.62E − 58CSSA0.00E + 001.73E − 263.45E − 282.44E − 270.00E + 005.14E − 271.04E − 287.27E − 280.00E + 006.60E − 321.48E − 339.54E − 33CLSSA0.00E + 002.00E − 52− 1.38E − 479.74E − 470.00E + 006.77E − 531.36E − 549.58E − 540.00E + 008.26E − 572.79E − 991.39E − 49MISSA0.00E + 001.57E − 487.91E − 1065.34E − 1050.00E + 006.09E − 1281.24E − 1298.71E − 1290.00E + 004.39E − 1239.55E − 1256.48E − 124COOT1.69E − 312.56E − 119.54E − 134.70E − 121.85E − 321.46E − 052.92E − 072.06E − 061.05E − 323.15E − 139.72E − 154.72E − 14WOA4.97E − 512.13E − 405.56E − 423.04E − 417.01E − 534.57E − 379.14E − 396.46E − 386.49E − 522.65E − 385.35E − 403.74E − 39GWO1.16E − 163.00E − 144.86E − 156.77E − 153.46E − 117.37E − 102.21E − 101.89E − 102.27E − 062.69E − 057.99E − 064.19E − 06PSO0.00E + 003.62E − 1047.91E − 1065.34E − 1051.44E + 006.09E + 028.19E + 012.27E + 024.15E + 011.84E + 037.72E + 025.77E + 02f2SSA0.00E + 008.21E − 221.73E − 231.17E − 220.00E + 002.22E − 204.44E − 223.14E − 210.00E + 001.60E − 253.24E − 272.27E − 26ISSA2.57E − 505.05E − 173.27E − 181.03E − 177.68E − 514.52E − 162.11E − 178.48E − 174.71E − 531.31E − 162.68E − 181.85E − 17CSSA0.00E + 008.24E − 121.65E − 131.17E − 120.00E + 002.51E − 165.06E − 183.55E − 170.00E + 006.52E − 131.62E − 149.44E − 14CLSSA0.00E + 008.15E − 502.86E − 511.45E − 500.00E + 001.44E − 442.88E − 462.04E − 450.00E + 003.23E − 416.46E − 434.57E − 42MISSA0.00E + 001.39E − 542.80E − 561.97E − 550.00E + 002.03E − 484.07E − 502.87E − 490.00E + 002.06E − 614.12E − 632.92E − 62COOT2.12E − 169.51E − 063.51E − 071.63E − 063.97E − 153.06E − 036.12E − 054.32E − 044.66E − 161.18E − 043.68E − 061.89E − 05WOA4.05E − 355.61E − 282.38E − 299.67E − 291.02E − 342.56E − 282.27E − 295.52E − 294.61E − 352.36E − 282.10E − 294.92E − 29GWO5.00E − 107.45E − 091.70E − 091.19E − 093.13E − 071.56E − 067.55E − 072.63E − 072.53E − 046.47E − 043.84E − 049.74E − 05PSO1.01E − 1671.39E − 542.80E − 561.97E − 551.32E + 003.18E + 002.01E + 004.30E − 011.13E + 012.17E + 011.54E + 011.99E + 00f3SSA0.00E + 009.96E − 352.03E − 361.42E − 350.00E + 002.43E − 714.98E − 733.44E − 720.00E + 005.96E − 591.22E − 608.52E − 60ISSA1.84E − 832.62E − 205.25E − 223.71E − 211.68E − 1091.81E − 243.75E − 262.56E − 253.17E − 881.22E − 282.46E − 301.73E − 29CSSA0.00E + 001.55E − 223.44E − 242.23E − 230.00E + 009.06E − 231.81E − 241.28E − 230.00E + 004.67E − 279.55E − 296.68E − 28CLSSA0.00E + 004.56E − 289.12E − 306.46E − 290.00E + 001.24E − 314.99E − 332.44E − 320.00E + 001.13E − 242.31E − 261.59E − 25MISSA0.00E + 003.72E − 677.43E − 695.26E − 680.00E + 009.81E − 911.96E − 921.39E − 910.00E + 005.10E − 761.16E − 777.68E − 77COOT3.01E − 326.79E − 101.36E − 119.61E − 111.44E − 302.66E − 095.31E − 113.75E − 104.16E − 348.65E − 113.37E − 121.63E − 11WOA4.79E + 041.35E + 056.04E + 041.82E + 041.46E + 053.54E + 052.37E + 054.24E + 045.74E + 051.82E + 061.14E + 063.41E + 05GWO1.86E − 043.77E − 017.86E − 021.01E − 014.24E + 003.31E + 023.99E + 015.78E + 012.77E + 021.57E + 044.23E + 033.04E + 03PSO2.92E − 2963.72E − 677.43E − 695.26E − 681.18E + 033.44E + 046.39E + 036.57E + 038.91E + 037.49E + 042.92E + 041.53E + 04f4SSA0.00E + 002.06E − 313.69E − 049.54E − 040.00E + 002.67E − 285.33E − 303.77E − 290.00E + 001.09E − 412.22E − 431.55E − 42ISSA4.83E − 582.93E − 237.61E − 299.41E − 291.95E − 559.17E − 317.39E − 172.78E − 169.81E − 467.57E − 161.87E − 171.07E − 16CSSA0.00E + 004.44E − 149.25E − 166.41E − 150.00E + 007.97E − 131.60E − 141.13E − 130.00E + 002.80E − 167.72E − 184.21E − 17CLSSA0.00E + 009.99E − 57− 4.62E − 493.24E − 480.00E + 004.67E − 489.35E − 506.61E − 490.00E + 002.51E − 484.61E − 483.54E − 49MISSA0.00E + 001.12E − 1233.2682E − 1251.7339E − 1240.00E + 005.09E − 581.11E − 597.21E − 590.00E + 001.42E − 642.95E − 662.01E − 65COOT1.06E − 482.23E − 155.74E + 007.66E + 002.29E − 173.98E − 071.27E − 085.83E − 085.96E − 161.64E − 064.64E − 082.44E − 07WOA3.80E − 039.15E + 011.63E − 364.47E − 361.32E + 019.54E + 016.75E + 012.47E + 012.60E + 019.54E + 017.70E + 011.96E + 01GWO3.95E − 161.35E − 134.24E − 221.17E − 213.57E − 021.26E − 014.92E − 022.62E − 028.04E − 011.52E + 014.91E + 002.62E + 00PSO1.27E − 012.76E − 016.00E − 691.65E − 688.08E + 009.05E + 006.01E + 001.17E + 007.54E + 001.26E + 019.98E + 001.18E + 00f5SSA5.23E − 052.06E − 034.07E − 044.09E − 043.97E − 054.53E − 041.61E − 031.92E − 031.38E − 031.85E + 003.23E − 013.45E − 01ISSA1.27E − 101.24E − 047.24E − 061.87E − 052.24E − 101.18E − 041.08E − 052.31E − 051.48E − 072.42E − 023.16E − 036.02E − 03CSSA4.87E − 114.65E − 051.47E − 066.59E − 061.64E − 086.53E − 054.72E − 061.05E − 052.57E − 114.47E − 054.05E − 059.02E − 05CLSSA4.60E − 102.96E − 051.64E − 064.47E − 061.06E − 082.57E − 052.18E − 064.42E − 063.55E − 092.94E − 041.33E − 054.22E − 05MISSA3.15E − 144.15E − 055.03E − 068.74E − 061.42E − 099.85E − 063.50E − 064.01E − 065.42E − 127.18E − 052.74E − 056.44E − 05COOT2.49E − 016.76E + 001.29E + 001.30E + 002.16E + 003.45E + 016.29E + 006.40E + 009.86E + 019.77E + 036.50E + 021.56E + 03WOA2.74E − 011.60E + 009.22E − 013.36E − 017.38E − 014.06E + 002.19E + 006.63E − 019.77E + 019.86E + 019.83E + 011.34E − 01GWO1.98E − 042.51E + 009.99E − 014.50E − 012.00E + 005.24E + 003.47E + 006.68E − 019.62E + 019.87E + 019.82E + 015.16E − 01PSO1.05E − 013.05E + 011.28E + 004.41E + 001.80E + 004.70E + 023.64E + 018.28E + 011.06E + 032.56E + 031.59E + 033.67E + 02f6SSA−1.18E + 04−4.90E + 03−8.25E + 032.23E + 03−2.06E + 04−7.41E + 03−1.49E + 044.02E + 03−4.15E + 04−1.20E + 04−3.15E + 047.89E + 03ISSA−1.22E + 04−3.92E + 03−9.72E + 032.43E + 03−2.08E + 04−6.77E + 03−1.72E + 043.80E + 03−4.19E + 04−2.14E + 04−3.64E + 045.51E + 03CSSA−1.26E + 04−5.01E + 03−8.80E + 033.11E + 01−2.08E + 04−8.01E + 03−1.72E + 043.72E + 03−4.18E + 04−1.10E + 04−3.30E + 047.80E + 03CLSSA−8.00E + 03−5.22E + 03−9.60E + 038.60E + 02−2.09E + 04−8.73E + 03−1.62E + 044.41E + 03−4.16E + 04−1.46E + 04−3.39E + 046.80E + 03MISSA−1.26E + 04−1.04E + 04−1.22E + 044.01E + 02−2.09E + 04−1.33E + 04−1.88E + 041.70E + 03−4.19E + 04−3.99E + 04−4.12E + 045.45E + 02COOT−8.17E + 03−4.81E + 03−6.84E + 037.60E + 02−1.42E + 04−7.36E + 03−1.04E + 041.49E + 03−2.11E + 04−1.11E + 04−1.69E + 042.82E + 03WOA−1.26E + 04−7.43E + 03−9.66E + 031.34E + 03−2.09E + 04−1.32E + 04−1.68E + 042.89E + 03−4.19E + 04−2.12E + 04−3.25E + 045.86E + 03GWO−7.17E + 03−2.78E + 03−5.66E + 031.02E + 03−1.07E + 04−4.06E + 03−8.80E + 031.59E + 03−2.00E + 04−5.69E + 03−1.44E + 043.95E + 03PSO−3.86E + 03−1.64E + 03−2.66E + 034.15E + 02−4.57E + 03−2.54E + 03−3.31E + 035.03E + 02−6.03E + 03−3.70E + 03−4.84E + 036.17E + 02f7SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 008.69E − 099.00E − 101.60E − 09MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 003.14E − 076.29E − 094.44E − 080.00E + 004.68E − 091.13E − 106.63E − 100.00E + 006.44E − 091.54E − 109.13E − 10WOA0.00E + 005.68E − 141.14E − 158.04E − 150.00E + 001.14E − 134.55E − 152.25E − 140.00E + 000.00E + 000.00E + 000.00E + 00GWO9.44E − 122.59E + 017.60E + 005.88E + 001.35E − 052.81E + 011.17E + 016.97E + 003.56E + 008.64E + 013.45E + 011.67E + 01PSO2.16E + 018.34E + 014.40E + 011.44E + 017.29E + 011.67E + 029.99E + 011.85E + 012.56E + 024.74E + 023.53E + 024.95E + 01f8SSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00ISSA8.88E − 164.44E − 151.23E − 151.07E − 158.88E − 161.96E − 138.28E − 153.05E − 148.88E − 168.88E − 168.88E − 160.00E + 00CSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00CLSSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00MISSA8.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 008.88E − 168.88E − 168.88E − 160.00E + 00COOT4.44E − 151.54E − 056.83E − 073.01E − 068.88E − 167.45E − 051.49E − 061.05E − 054.44E − 158.52E − 061.77E − 071.20E − 06WOA8.88E − 161.51E − 147.13E − 153.95E − 158.88E − 162.22E − 145.93E − 153.45E − 154.44E − 151.51E − 146.93E − 152.41E − 15GWO2.50E − 094.00E − 081.18E − 087.32E − 098.94E − 077.22E − 062.35E − 061.08E − 061.83E − 045.24E − 043.18E − 048.37E − 05PSO5.78E − 024.08E + 005.00E − 018.25E − 014.65E − 012.94E + 001.21E + 005.50E − 012.32E + 004.35E + 002.93E + 003.93E − 01f9SSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00ISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00CLSSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00MISSA0.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 000.00E + 00COOT0.00E + 007.89E − 143.42E − 151.30E − 140.00E + 008.32E − 121.73E − 131.18E − 120.00E + 007.73E − 101.55E − 111.09E − 10WOA0.00E + 003.74E − 012.95E − 029.14E − 020.00E + 003.20E − 016.39E − 034.52E − 020.00E + 001.11E − 162.22E − 181.57E − 17GWO3.11E − 153.11E − 026.10E − 039.41E − 031.29E − 023.99E − 025.31E − 031.12E − 025.57E − 027.00E − 028.65E − 032.02E − 02PSO1.60E + 022.18E + 021.87E + 021.59E + 013.10E + 023.73E + 023.40E + 021.64E + 016.69E + 027.92E + 027.25E + 023.00E + 01f10SSA8.48E − 061.96E − 022.81E − 035.68E − 033.38E − 063.58E − 024.73E − 038.95E − 032.35E − 054.16E − 025.26E − 031.05E − 02ISSA1.66E − 106.50E − 054.86E − 061.24E − 051.03E − 087.67E − 058.08E − 061.67E − 051.61E − 081.76E − 041.56E − 053.31E − 05CSSA1.66E − 102.26E − 051.73E − 061.34E − 051.52E − 098.80E − 062.47E − 063.30E − 061.89E − 081.03E − 053.69E − 065.26E − 06CLSSA3.60E − 091.17E − 051.87E − 062.73E − 062.33E − 093.86E − 053.37E − 067.89E − 061.73E − 081.79E − 055.26E − 064.00E − 06MISSA4.15E − 107.66E − 055.51E − 063.63E − 063.89E − 101.60E − 053.95E − 065.35E − 067.97E − 092.78E − 052.64E − 074.55E − 07COOT6.01E − 012.60E + 008.21E − 014.61E − 011.77E + 001.18E + 014.08E + 002.09E + 008.03E + 002.29E + 011.17E + 012.68E + 00WOA1.50E − 011.67E + 008.00E − 013.89E − 014.91E − 013.17E + 001.60E + 006.08E − 011.41E + 006.39E + 003.86E + 001.20E + 00GWO4.26E − 011.50E + 009.30E − 012.76E − 011.40E + 003.55E + 002.48E + 004.51E − 015.91E + 008.74E + 007.58E + 005.93E − 01PSO1.75E − 033.62E + 001.60E − 017.11E − 016.09E − 025.22E + 014.77E + 001.22E + 012.63E + 001.24E + 025.85E + 014.20E + 01As shown in Table4, the optimization accuracy of MISSA for ten benchmark functions is significantly better than of the other nine swarm intelligence algorithms under the same conditions.In the experiments on unimodal functions, MISSA can quickly ﬁnd the optimal values with high optimization accuracy. Forf1–f4, MISSA, CSSA, CLSSA, and SSA can all converge to the theoretical optimal value, indicating that MISSA does not reduce the search accuracy of the original SSA algorithm. Although ISSA has not converged to the theoretical value, its accuracy is more than a dozen orders of magnitude better than of other algorithms, which shows the superiority of the original SSA itself. In addition, MISSA is better than the original SSA and other improved SSAs in the evaluation of the Mean and Std, which is better than dozens of orders of magnitude in f1–f4. Besides, the trend of superiority is further improved with the increase in dimensions, showing that MISSA has better stability. For f5, although MISSA fails to find the theoretical optimal solution, the Mean, Std, and quality of its optimal solution in each dimension have been significantly better than other algorithms.In the experiments on multimodal function, MISSA demonstrates excellent global optimization ability to deal with complex problems. Forf6, only MISSA and CSSA converge to the theoretical optimal solution, but the accuracy of the evaluation criterion for MISSA increases up to one or two orders of magnitudes than of other algorithms including CSSA. For f7 and f9, the original SSA and other improved SSAs can search for the theoretical optimal solution in all dimensions and the Mean and Std are both 0, indicating the rationality and superiority of the algorithm itself. For f8, neither MISSA nor other test algorithms have found the theoretical optimal solution, but the Std of MISSA is 0, which indicates that MISSA has been optimized to the theoretical maximum accuracy. For f10, all the evaluation criteria of MISSA are optimal and the superiority becomes more significant when the dimension increases. The mathematical principle of other swarm intelligence algorithms is inferior to that of the sparrow algorithm, and it is easy to fall into local optimum when solving complex problems.Overall, the experiments verify the significant difference between the results obtained by MISSA and other compared algorithms in almost all cases. With the increase in the dimension, other swarm intelligence algorithms such as COOT, WOA, and GWO are not effective enough in solving complex optimization problems. However, the original SSA benefits from the superiority of mathematical principles, which still has the opportunity to jump out of the local optimal solution in high-dimensional optimization, but the stability of the algorithm is obviously reduced. MISSA has prominent stability and robustness, which can balance the global exploitation and local exploitation abilities to fully explore the solution space.


## 4.3.2. Convergence Curve Analysis

Notably, to completely analyze the convergence speed and optimization accuracy of the compared algorithms, the convergence curves of the ten benchmark functions (D = 30) are shown in Figure 8. We select the semilogarithmic coordinate axis to draw the convergence curve. The y-axis and x-axis represent the fitness value and iteration, respectively.Figure 8
Convergence curves of ten benchmark functions. (a)f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)As shown in Figure8, the convergence curves clearly illustrate the advantages of MISSA in integrating the global exploration and local exploitation abilities into the search process. Under the premise of ensuring the optimization accuracy, the convergence speed of MISSA is better than that of other test algorithms.The smooth convergence curves of MISSA on benchmark functionsf1, f3, f4, f6, f7, and f9 indicate that the improved optimization mechanism is more capable to search the solution space and the algorithm does not tend to fall into local optimum when dealing with these types of benchmark functions. Especially for f7–f9, although both MISSA and other SSAs can converge to the theoretical optimal solution, MISSA has the fastest convergence speed.The steplike fluctuant convergence curves of MISSA on the benchmark functionsf2, f5, f8, and f10 indicate that MISSA can jump out of the local optimum and can be effectively implemented to avoid premature convergence. After the algorithm is temporarily trapped in a local optimum, the EOBL strategy for elite individuals helps the algorithm escape the local optimum.Combined with the optimization results in Table4, MISSA not only improves the stability and robustness but also ensures the optimization accuracy and convergence speed of the algorithm.


## 4.3.3. Wilcoxon Rank-Sum Test

To further verify the difference from the perspective of statistics between MISSA and the comparison algorithms, the Wilcoxon rank-sum test is carried out at theP = 5% significance level in this paper [46]. Table 5 shows the corresponding results of the Wilcoxon rank-sum test on the benchmark functions (D = 30), in which the value of significance level is set to 0.05. There is a significant advantage between MISSA and the comparison algorithm if the P value is less than 0.05. On the contrary, the optimization performance of MISSA is equal to or worse than the comparison algorithm. NaN means the results of the comparison algorithms are close, and the significance judgment cannot be made. The sign “+/−” means that the MISSA is “better than/equivalent or inferior to” other algorithms in performance.Table 5
Results of the Wilcoxon rank-sum test.
FunctionSSAISSACSSACLSSACOOTWOAGWOPSOf16.43E − 111.12E − 188.49E − 024.81E − 011.12E − 181.12E − 181.12E − 181.12E − 18f28.16E − 061.52E − 171.66E − 024.00E − 031.52E − 173.31E − 151.52E − 171.52E − 17f33.36E − 061.34E − 181.98E − 027.11E − 021.34E − 181.34E − 181.34E − 181.34E − 18f41.33E − 053.41E − 171.17E − 013.21E − 023.41E − 173.41E − 173.41E − 173.41E − 17f57.07E − 18NaN4.04E − 056.96E − 037.07E − 187.07E − 187.07E − 187.07E − 18f62.54E − 161.93E − 111.77E − 106.98E − 187.07E − 187.68E − 157.07E − 187.07E − 18f7NaNNaNNaNNaN1.73E − 073.27E − 013.31E − 203.31E − 20f8NaN4.28E − 02NaNNaN8.24E − 172.32E − 158.24E − 178.24E − 17f9NaNN/ANaNNaN6.50E − 052.31E − 023.31E − 203.31E − 20f101.54E − 172.50E − 034.63E − 024.46E − 027.07E − 187.07E − 187.07E − 187.07E − 18NaN/+/−3/7/03/7/03/6/13/6/10/10/00/9/10/10/00/10/0For theP value in Table 5, MISSA has a significant difference with other eight algorithms. For f7–f9, due to the superiority of the SSA, SSAs can converge to the theoretical optimal value, so the difference is not obvious and the optimization ability is equivalent.


## 4.4. CEC 2014 Function Test

Compared to the basic benchmark functions, the CEC 2014 [47] is more complicated and can evaluate the effectiveness and robustness of the proposed algorithm better. The CEC 2014 consists of unimodal functions (F1–F3), simple multimodal functions (F4–F16), hybrid functions (F17–F22), and composition functions (F23–F30). According to the evaluation criteria of CEC 2014, the number of iterations is 10,000 × dim, the dimension is 30, the search range is [100, 100], and other parameter settings are the same as in Section 4.1.Table6 illustrates that MISSA has a better optimization eﬀect on CEC 2014. From the Wilcoxon rank-sum test, it is seen that MISSA is signiﬁcantly diﬀerent from other algorithms, showing better advantages, and some composition functions are similar to SSAs because of the superiority of the sparrow search algorithm itself which is analyzed in benchmark functions f7–f9. Generally, MISSA has high effectiveness and robustness and is more suitable for some complex optimization problems than other algorithms.Table 6
Results of each algorithm in CEC 2014.
FCriteriaSSAISSACSSACLSSACOOTWOAGWOPSOMISSAF1Best1.592E + 082.567E + 068.560E + 063.745E + 062.400E + 055.548E + 068.723E + 062.585E + 079.413E + 04Worst5.321E + 081.321E + 072.380E + 072.012E + 078.529E + 052.382E + 071.308E + 086.025E + 096.025E + 05Mean1.601E + 087.609E + 061.399E + 071.255E + 075.334E + 051.217E + 074.544E + 078.539E + 083.239E + 05Std1.760E + 083.806E + 065.323E + 065.995E + 062.033E + 055.486E + 063.635E + 071.961E + 091.776E + 05P2.827E − 031.402E − 021.827E − 041.827E − 042.202E − 031.827E − 041.827E − 041.827E − 04F2Best4.636E + 024.473E + 022.172E + 043.446E + 042.807E + 033.581E + 044.116E + 071.330E + 102.025E + 02Worst2.450E + 042.220E + 042.221E + 101.092E + 053.197E + 041.102E + 053.098E + 092.225E + 108.635E + 03Mean1.107E + 046.900E + 031.784E + 108.130E + 041.286E + 046.618E + 041.240E + 091.681E + 101.771E + 03Std8.510E + 037.633E + 036.515E + 092.312E + 041.035E + 043.005E + 049.874E + 083.162E + 092.787E + 03P2.827E − 031.402E − 021.827E − 041.827E − 042.202E − 031.827E − 041.827E − 041.827E − 04F3Best7.248E + 041.796E + 056.625E + 046.504E + 046.012E + 044.606E + 041.845E + 043.142E + 023.195E + 03Worst9.039E + 042.869E + 058.757E + 048.525E + 048.190E + 041.264E + 056.590E + 049.248E + 039.734E + 03Mean8.405E + 042.408E + 058.303E + 047.839E + 047.289E + 047.405E + 043.928E + 045.238E + 035.071E + 03Std5.338E + 033.171E + 047.377E + 036.719E + 037.671E + 033.116E + 041.268E + 043.596E + 031.952E + 03P1.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 046.776E − 01F4Best1.246E + 038.594E + 021.064E + 034.056E + 035.858E + 025.583E + 025.651E + 024.694E + 024.222E + 02Worst3.336E + 032.445E + 038.607E + 031.021E + 041.062E + 037.400E + 027.834E + 024.774E + 025.259E + 02Mean1.838E + 031.770E + 036.405E + 036.818E + 037.996E + 026.711E + 026.650E + 024.717E + 024.774E + 02Std6.265E + 025.105E + 022.253E + 031.927E + 031.578E + 025.354E + 017.403E + 012.878E + 002.131E + 01P1.817E − 041.817E − 041.817E − 041.817E − 041.817E − 041.817E − 041.817E − 048.747E − 03F5Best5.200E + 025.200E + 025.200E + 025.200E + 025.200E + 025.203E + 025.209E + 025.200E + 025.200E + 02Worst5.204E + 025.205E + 025.202E + 025.202E + 025.208E + 025.207E + 025.211E + 025.209E + 025.200E + 02Mean5.200E + 025.201E + 025.201E + 025.200E + 025.203E + 025.205E + 025.210E + 025.205E + 025.200E + 02Std1.153E − 011.587E − 015.872E − 025.831E − 022.534E − 011.138E − 014.143E − 023.427E − 011.618E − 03P4.309E − 041.315E − 031.620E − 013.281E − 042.461E − 041.827E − 041.827E − 042.461E − 04F6Best6.360E + 026.384E + 026.398E + 026.384E + 026.367E + 026.328E + 026.102E + 026.295E + 026.173E + 02Worst6.454E + 026.466E + 026.486E + 026.456E + 026.438E + 026.420E + 026.182E + 026.421E + 026.264E + 02Mean6.425E + 026.419E + 026.421E + 026.432E + 026.390E + 026.370E + 026.145E + 026.352E + 026.214E + 02Std2.816E + 002.521E + 002.621E + 002.168E + 002.203E + 003.012E + 002.642E + 004.779E + 003.423E + 00P1.827E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 047.685E − 041.827E − 04F7Best7.305E + 027.476E + 027.241E + 028.942E + 027.000E + 027.013E + 027.040E + 027.016E + 027.000E + 02Worst8.443E + 028.490E + 021.110E + 031.050E + 037.000E + 027.061E + 027.643E + 027.021E + 027.021E + 02Mean7.919E + 027.898E + 029.265E + 029.743E + 027.000E + 027.022E + 027.239E + 027.019E + 027.017E + 02Std3.405E + 013.439E + 011.467E + 025.949E + 018.745E − 031.401E + 001.837E + 012.292E − 016.519E − 01P1.827E − 041.827E − 041.827E − 041.827E − 042.827E − 038.796E − 011.827E − 043.195E − 01F8Best9.640E + 029.291E + 029.277E + 029.383E + 029.572E + 029.170E + 028.681E + 029.015E + 028.627E + 02Worst1.025E + 039.887E + 021.016E + 031.007E + 031.024E + 031.094E + 039.615E + 029.731E + 029.473E + 02Mean9.860E + 029.675E + 029.743E + 029.710E + 029.750E + 029.908E + 028.951E + 029.389E + 029.046E + 02Std1.838E + 011.669E + 012.236E + 012.075E + 012.076E + 015.343E + 012.629E + 012.009E + 012.765E + 01P1.827E − 043.298E − 043.298E − 042.461E − 041.827E − 041.008E − 033.847E − 011.726E − 02F9Best1.100E + 031.093E + 031.124E + 031.117E + 039.995E + 021.065E + 031.001E + 031.021E + 039.914E + 02Worst1.130E + 031.161E + 031.188E + 031.183E + 031.039E + 031.228E + 031.139E + 031.097E + 031.131E + 03Mean1.117E + 031.132E + 031.160E + 031.144E + 031.015E + 031.133E + 031.031E + 031.063E + 031.095E + 03Std1.069E + 012.137E + 012.410E + 012.383E + 011.339E + 015.343E + 015.097E + 012.220E + 013.958E + 01P1.041E − 011.133E − 024.396E − 041.706E − 032.827E − 031.859E − 014.362E − 021.133E − 02F10Best4.708E + 033.946E + 035.095E + 034.458E + 033.416E + 034.487E + 033.473E + 033.617E + 032.752E + 03Worst5.737E + 037.185E + 037.089E + 036.473E + 034.740E + 036.788E + 036.226E + 035.298E + 034.153E + 03Mean5.153E + 035.484E + 035.926E + 035.397E + 034.164E + 035.735E + 035.049E + 034.569E + 033.598E + 03Std3.121E + 028.421E + 026.559E + 026.528E + 024.148E + 026.061E + 028.830E + 025.878E + 024.754E + 02P1.827E − 044.396E − 041.827E − 041.827E − 041.402E − 021.827E − 041.315E − 034.586E − 03F11Best4.599E + 035.689E + 034.942E + 035.895E + 034.321E + 035.239E + 034.910E + 034.251E + 032.714E + 03Worst7.325E + 037.937E + 038.411E + 037.682E + 035.879E + 037.681E + 037.039E + 039.197E + 034.860E + 03Mean6.390E + 036.570E + 036.584E + 036.748E + 034.954E + 036.531E + 035.671E + 035.882E + 033.875E + 03Std8.529E + 027.063E + 021.036E + 036.091E + 025.479E + 028.841E + 027.064E + 021.360E + 036.634E + 02P3.298E − 041.827E − 041.827E − 041.827E − 044.586E − 031.827E − 041.827E − 047.685E − 04F12Best1.201E + 031.201E + 031.201E + 031.201E + 031.201E + 031.201E + 031.200E + 031.201E + 031.201E + 03Worst1.203E + 031.203E + 031.203E + 031.202E + 031.202E + 031.203E + 031.203E + 031.202E + 031.202E + 03Mean1.202E + 031.202E + 031.202E + 031.202E + 031.202E + 031.202E + 031.203E + 031.202E + 031.201E + 03Std7.197E − 015.079E − 016.170E − 013.856E − 014.297E − 016.000E − 018.211E − 014.977E − 013.077E − 01P1.399E − 011.388E − 021.207E − 017.910E − 011.613E − 014.362E − 022.786E − 033.840E − 01F13Best1.303E + 031.303E + 031.306E + 031.306E + 031.300E + 031.300E + 031.300E + 031.300E + 031.300E + 03Worst1.305E + 031.305E + 031.308E + 031.308E + 031.301E + 031.301E + 031.303E + 031.301E + 031.303E + 03Mean1.304E + 031.304E + 031.307E + 031.307E + 031.300E + 031.301E + 031.301E + 031.301E + 031.301E + 03Std5.492E − 018.018E − 017.393E − 016.089E − 011.261E − 011.122E − 018.971E − 011.260E − 017.857E − 01P1.827E − 041.827E − 041.827E − 041.827E − 045.795E − 033.447E − 012.413E − 014.390E − 02F14Best1.467E + 031.475E + 031.422E + 031.420E + 031.400E + 031.400E + 031.400E + 031.400E + 031.400E + 03Worst1.507E + 031.515E + 031.491E + 031.510E + 031.401E + 031.400E + 031.416E + 031.401E + 031.438E + 03Mean1.491E + 031.493E + 031.457E + 031.464E + 031.400E + 031.400E + 031.404E + 031.400E + 031.418E + 03Std1.373E + 011.367E + 012.338E + 012.428E + 011.708E − 013.211E − 025.397E + 003.465E − 011.235E + 01P1.827E − 041.827E − 045.828E − 045.828E − 041.133E − 025.795E − 033.121E − 021.133E − 02F15Best4.278E + 034.537E + 032.272E + 032.263E + 031.571E + 031.555E + 031.508E + 031.540E + 031.518E + 03Worst1.537E + 042.229E + 046.568E + 031.394E + 042.222E + 031.691E + 031.885E + 031.570E + 031.533E + 03Mean7.556E + 038.870E + 034.873E + 034.617E + 031.746E + 031.624E + 031.610E + 031.552E + 031.523E + 03Std3.669E + 035.508E + 031.524E + 033.448E + 032.256E + 024.273E + 011.200E + 021.035E + 014.466E + 00P1.817E − 041.827E − 041.827E − 041.827E − 041.827E − 041.827E − 042.827E − 031.827E − 04F16Best1.613E + 031.613E + 031.613E + 031.613E + 031.612E + 031.612E + 031.614E + 031.612E + 031.612E + 03Worst1.614E + 031.614E + 031.614E + 031.614E + 031.613E + 031.613E + 031.613E + 031.613E + 031.613E + 03Mean1.613E + 031.613E + 031.613E + 031.613E + 031.612E + 031.613E + 031.613E + 031.613E + 031.613E + 03Std3.827E − 012.789E − 012.316E − 013.621E − 013.050E − 013.664E − 016.362E − 015.379E − 014.230E − 01P2.409E − 041.786E − 041.688E − 042.786E − 031.726E − 023.447E − 013.298E − 043.440E − 01F17Best8.051E + 056.594E + 054.092E + 053.200E + 053.505E + 046.989E + 052.133E + 051.236E + 057.154E + 03Worst9.578E + 063.698E + 067.083E + 064.444E + 062.988E + 057.565E + 061.656E + 069.602E + 055.006E + 04Mean3.533E + 062.071E + 062.808E + 061.874E + 061.515E + 053.003E + 067.017E + 054.927E + 052.645E + 04Std2.477E + 061.136E + 062.262E + 061.442E + 069.963E + 042.469E + 065.702E + 053.004E + 051.383E + 04P1.827E − 041.827E − 041.827E − 041.827E − 045.828E − 041.827E − 041.827E − 041.827E − 04F18Best3.365E + 038.957E + 037.399E + 046.574E + 082.092E + 042.067E + 044.002E + 032.193E + 031.885E + 03Worst1.290E + 082.893E + 073.872E + 094.259E + 091.574E + 056.860E + 057.632E + 075.631E + 077.298E + 03Mean2.185E + 076.500E + 061.435E + 092.713E + 094.205E + 041.514E + 051.738E + 071.035E + 073.291E + 03Std4.097E + 079.788E + 061.311E + 091.395E + 094.222E + 032.037E + 052.603E + 071.943E + 071.761E + 03P4.586E − 031.827E − 041.827E − 041.827E − 044.708E − 031.827E − 044.396E − 041.133E − 02F19Best2.066E + 032.052E + 032.060E + 032.065E + 031.912E + 031.928E + 031.912E + 031.959E + 031.918E + 03Worst2.331E + 032.199E + 032.332E + 032.364E + 031.971E + 032.094E + 032.002E + 032.038E + 031.946E + 03Mean2.160E + 032.141E + 032.146E + 032.216E + 031.920E + 031.994E + 031.946E + 032.005E + 031.927E + 03Std8.977E + 014.816E + 011.006E + 021.008E + 021.819E + 015.239E + 013.136E + 012.448E + 018.322E + 00P1.827E − 041.827E − 041.827E − 041.827E − 042.827E − 035.828E − 042.123E − 011.827E − 04F20Best9.172E + 041.226E + 053.271E + 046.443E + 045.353E + 032.061E + 041.262E + 042.486E + 043.490E + 03Worst3.861E + 052.749E + 065.141E + 054.471E + 052.929E + 041.165E + 053.594E + 043.048E + 051.041E + 04Mean2.281E + 057.579E + 051.972E + 051.969E + 051.854E + 046.012E + 042.709E + 041.308E + 056.484E + 03Std1.031E + 059.890E + 051.451E + 051.151E + 058.090E + 032.632E + 047.251E + 038.808E + 042.063E + 03P1.827E − 041.827E − 041.827E − 041.827E − 041.315E − 031.827E − 041.827E − 041.827E − 04F21Best1.357E + 061.514E + 062.552E + 052.722E + 054.970E + 041.156E + 066.811E + 041.346E + 062.009E + 04Worst5.335E + 073.639E + 072.150E + 074.371E + 074.151E + 051.029E + 071.846E + 061.521E + 071.177E + 05Mean1.192E + 071.237E + 079.264E + 061.412E + 072.425E + 056.436E + 065.982E + 056.043E + 067.626E + 04Std1.498E + 071.137E + 078.219E + 061.505E + 071.483E + 053.172E + 065.877E + 054.241E + 063.785E + 04P1.817E − 041.817E − 041.817E − 041.817E − 041.398E − 021.817E − 045.777E − 031.817E − 04F22Best3.098E + 033.245E + 033.235E + 033.248E + 032.376E + 033.003E + 032.389E + 032.873E + 032.821E + 03Worst4.849E + 035.707E + 034.393E + 035.464E + 033.132E + 033.499E + 032.869E + 033.998E + 033.658E + 03Mean3.824E + 034.160E + 033.877E + 033.745E + 032.768E + 033.264E + 032.602E + 033.380E + 033.260E + 03Std5.866E + 027.812E + 023.807E + 026.163E + 021.998E + 021.645E + 021.427E + 023.380E + 022.635E + 02P1.726E − 024.586E − 032.202E − 031.402E − 027.685E − 049.698E − 012.461E − 043.447E − 02F23Best2.500E + 032.500E + 032.500E + 032.500E + 032.615E + 032.640E + 032.626E + 032.614E + 032.500E + 03Worst2.500E + 032.500E + 032.500E + 032.500E + 032.616E + 032.678E + 032.662E + 032.614E + 032.500E + 03Mean2.500E + 032.500E + 032.500E + 032.500E + 032.615E + 032.657E + 032.641E + 032.614E + 032.500E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.123E − 011.380E + 011.208E + 011.228E − 030.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F24Best2.600E + 032.600E + 032.600E + 032.600E + 032.625E + 032.605E + 032.600E + 032.621E + 032.600E + 03Worst2.600E + 032.600E + 032.600E + 032.600E + 032.631E + 032.617E + 032.600E + 032.651E + 032.600E + 03Mean2.600E + 032.600E + 032.600E + 032.600E + 032.628E + 032.611E + 032.600E + 032.633E + 032.600E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.989E + 005.162E + 005.381E − 031.063E + 010.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F25Best2.700E + 032.700E + 032.700E + 032.700E + 032.700E + 032.716E + 032.706E + 032.711E + 032.700E + 03Worst2.700E + 032.700E + 032.700E + 032.700E + 032.718E + 032.759E + 032.719E + 032.731E + 032.700E + 03Mean2.700E + 032.700E + 032.700E + 032.700E + 032.713E + 032.727E + 032.711E + 032.719E + 032.700E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.821E + 001.472E + 014.320E + 005.773E + 000.000E + 00PNaNNaNNaNNaN1.576E − 045.511E − 056.340E − 056.386E − 05F26Best2.702E + 032.703E + 032.701E + 032.707E + 032.718E + 032.800E + 032.800E + 032.800E + 032.700E + 03Worst2.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.800E + 032.701E + 03Mean2.790E + 032.771E + 032.773E + 032.783E + 032.792E + 032.800E + 032.761E + 032.800E + 032.700E + 03Std3.089E + 014.648E + 014.427E + 013.524E + 012.598E + 010.000E + 005.082E + 011.902E − 021.288E − 01P8.745E − 051.317E − 042.438E − 041.107E − 048.745E − 056.386E − 055.828E − 041.727E − 04F27Best2.900E + 032.900E + 032.900E + 032.900E + 033.111E + 033.177E + 033.192E + 033.106E + 032.900E + 03Worst2.900E + 032.900E + 032.900E + 032.900E + 033.835E + 034.170E + 033.699E + 034.477E + 032.900E + 03Mean2.900E + 032.900E + 032.900E + 032.900E + 033.592E + 033.785E + 033.417E + 033.787E + 032.900E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 001.978E + 023.861E + 021.412E + 025.854E + 020.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F28Best3.000E + 033.000E + 033.000E + 033.000E + 033.796E + 034.711E + 033.689E + 036.271E + 033.000E + 03Worst3.000E + 033.000E + 033.000E + 033.000E + 035.108E + 035.966E + 034.747E + 038.157E + 033.000E + 03Mean3.000E + 033.000E + 033.000E + 033.000E + 034.432E + 035.351E + 034.009E + 037.026E + 033.000E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.163E + 023.647E + 023.494E + 026.857E + 020.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F29Best3.100E + 033.100E + 033.100E + 033.100E + 035.558E + 032.146E + 069.872E + 034.580E + 033.100E + 03Worst3.100E + 033.100E + 033.100E + 033.100E + 031.644E + 072.111E + 071.683E + 064.343E + 053.100E + 03Mean3.100E + 033.100E + 033.100E + 033.100E + 036.121E + 061.269E + 072.989E + 051.071E + 053.100E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 006.805E + 065.525E + 065.341E + 051.387E + 050.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05F30Best3.200E + 033.200E + 033.200E + 033.200E + 031.068E + 041.172E + 051.817E + 045.782E + 033.200E + 03Worst3.200E + 033.200E + 033.200E + 033.200E + 032.423E + 041.124E + 061.441E + 055.947E + 043.200E + 03Mean3.200E + 033.200E + 033.200E + 033.200E + 031.539E + 044.686E + 055.813E + 041.648E + 043.200E + 03Std0.000E + 000.000E + 000.000E + 000.000E + 004.192E + 032.990E + 053.914E + 041.642E + 040.000E + 00PNaNNaNNaNNaN6.386E − 056.386E − 056.386E − 056.386E − 05NaN/+/−7/21/27/23/07/21/27/21/20/29/10/25/50/27/30/26/4


## 5. Inductive Analysis of Strategies

MISSA improves the original SSA from three aspects: initialization method, discoverer-follower model, and optimal solution. After the experiments and analysis in Section4, we generalize the impact influence of four strategies, as shown in Table 7; the sign “↑/↓/—” means “enhance/reduce/no significant change.”Table 7
Impact influence of the strategy.
AspectStrategyDiversitySearching abilityAbundanceStabilityExplorationExploitationPopulation initializationIterative chaotic map↑↑——Discoverer-follower modelGold-SA with nonlinear convergence factor↑↓↑—Optimal solutionEOBL↑↑—↑From Table7, the abundance of population diversity has been improved after the addition of different strategies. The chaotic map enhances the diversity of the population and slightly improves the algorithm’s ability to deal with unimodal functions. The Gold-SA with nonlinear convergence factor strategies can quickly search the solution space and improve the convergence speed of the algorithm and the ability to deal with multimodal functions, but it reduces the stability of population diversity. The EOBL has a strong ability to handle unimodal functions by opposite solution and population reconstruction of elite individuals. However, when EOBL is used alone, the algorithm does not improve the ability to handle multimodal functions, because the reverse learning of the current optimal solution amplifies the probability of the algorithm falling into a local optimum.Notably, the EOBL improves the stability of population diversity, which was reduced by Gold-SA with the nonlinear convergence factor, and Gold-SA with nonlinear convergence factor can help EOBL to escape local optimum, which shows that there is a complementary relationship between them when strategies are mixed.


## 6. Experimental Simulation and Result Analysis

In order to verify the efficiency of the proposed MISSA in dealing with combinatorial optimization problems, Section6 applies the proposed MISSA to the traveling salesman problem, the tourism route planning problems of 31 provincial capital cities in China, the standard TSPLIB instance is selected for testing, and the results are compared with SSA [24], ISSA [30], GWO [15], and PSO [12].


### 6.1. The Traveling Salesman Problem

The traveling salesman problem is a well-known problem in combinatorial optimization filed, which can be described as follows: the traveling salesman intends to sell goods inn cities, selecting one of the cities as the starting point and visiting all the remaining cities without repetition, and finally returns to the starting point to select the shortest path among all routes [48].In the graph theory, TSP can be described as finding a Hamilton cycle with the lowest weight in a complete weighted graph. LetG = (V, E) be a complete weighted graph, where V = {1, 2, …, n} represents the vertex set and E represents the edge set. Each edge has a nonnegative weight m (e) on e = (I n, j) ∈ E and finds the Hamiltonian cycle C of G, in which the total weight of C satisfies MC=min∑ECme.The matrix W of distance between cities is as follows:(14)W=d1,1⋯d1,n⋮⋱⋮dn,1⋯dn,n,where d(i,j) represents the distance from the ith city to the jth city, d(i, j) ≥ 0, i, j ∈ v. The mathematical model of the objective function and constraint function of TSP are established as follows:(15)f=min∑∑di,jxi,j,s.t.∑j≠1xi,j=1,i∈V,∑i≠jxi,j=1,j∈V,∑i.j∈Sxi,j=S−1,S⊆V,xi,j∈0,1,i,j∈V,xi,j=1,i,j∈optimal path,0,other,where |S| represents the number of vertices in the set S. The constraint functions show that there is only one outgoing edge and one incoming edge for each vertex and no subloops will be generated.


### 6.2. Practical Problem Test

Suppose a tourist wants to travel around China passing through 31 provincial capital cities, an appropriate travel route to reduce economic costs is an important issue. The tourism scheme can be abstracted into a typical TSP problem, and the proposed MISSA, ISSA, SSA, GWO, and PSO are used for solving and comparative analysis. The city coordinates are shown in Table8. To ensure the consistency of the comparative experiments, the population size and maximum iteration number of each algorithm are 30 and 300, respectively. In addition, other parameters are as shown in Table 1.Table 8
City coordinates.
OrderCoordinate1(1304, 2312)2(3639, 1315)3(4177, 2244)4(3712, 1399)5(3488, 1535)6(3326, 1556)7(3238, 1229)8(4196, 1004)9(4312, 790)10(4386, 570)11(3007, 1970)12(2562, 1756)13(2788, 1491)14(2381, 1676)15(1332, 695)16(3715, 1678)17(3918, 2179)18(4061, 2370)19(3780, 2212)20(3676, 2578)21(4029, 2838)22(4263, 2931)23(3429, 1908)24(3507, 2367)25(3394, 2643)26(3439, 3201)27(2935, 3240)28(3140, 3550)29(2545, 2357)30(2778, 2826)31(2370, 2975)Figure9(a) shows the optimal travel route searched by MISSA, which does not show a crossed route and is consistent with the theoretical optimal solution. Figure 9(b) shows the convergence curves of the four algorithms. MISSA converges to the theoretical optimal solution in the 10th iteration, and the optimization accuracy and convergence speed are better than of compared algorithms.Figure 9
Provincial capitals tourism route planning. (a) Optimal route of MISSA. (b) Convergence curves.
(a)(b)To further verify the practicality and feasibility of the proposed MISSA, the algorithm is applied to five standard TSPLIB instances [49] in Table 9. We select the Euclidean distance as the calculation standard, and four algorithms are run 20 times on the five TSPLIB instances independently. The theoretical optimal (TO) value is the current optimal solution given by TSPLIB. The relative error (RE) is defined as follows:(16)RE=Mean−TOTO.Table 9
TSPLIB instances.
TSPLIBInstanceburma14bayg29dantzig42eil51eil76D1429425176TO3.09E + 019.07E + 036.99E + 024.26E + 025.38E + 02MISSABest3.09E + 019.07E + 036.88E + 024.26E + 025.38E + 02RE0.00%0.00%0.14%0.40%0.89%Mean3.09E + 019.07E + 037.00E + 024.28E + 025.43E + 02ISSABest3.09E + 019.13E + 036.99E + 024.35E + 025.52E + 02RE0.91%1.21%1.57%5.15%6.32%Mean3.12E + 019.18E + 037.10E + 024.48E + 025.72E + 02SSABest3.09E + 019.23E + 037.01E + 024.38E + 025.62E + 02RE1.17%2.21%2.72%6.09%8.00%Mean3.12E + 019.27E + 037.18E + 024.52E + 025.81E + 02GWOBest3.09E + 019.24E + 037.21E + 024.83E + 026.35E + 02RE1.69%2.09%4.43%12.96%17.89%Mean3.14E + 019.26E + 037.31E + 025.06E + 026.74E + 02PSOBest3.09E + 019.23E + 037.26E + 026.38E + 025.82E + 02RE2.01%3.09%5.01%16.57%21.61%Mean3.15E + 019.35E + 037.34E + 024.97E + 026.54E + 02Table9 illustrates that the solution precision and dimension of MISSA have been greatly improved in TSPLIB instances. For the instances tested, MISSA can converge to the theoretical optimal value with a lower relative error. For dantzig42, the Best of MISSA is better than the theoretical optimal value. Notably, as the dimensionality and dense distribution of cities increase, the requirements for algorithms to handle complex problems increase explosively. The relative error of the compared algorithms increases rapidly in eil51 and eil76, especially for GWO and PSO, which indicates the algorithms are falling into the local optimum. However, MISSA still maintains excellent optimization ability, indicating that the multimixed strategy is useful. The chaotic map provides good population diversity, the improved discoverer-follower model rapidly separates the solution space, and the EOBL mechanism upgrades the optimal solution and reconstructs the population. The multimixed strategy helps MISSA solve complex optimization problems with high dimensions. Figure 10 shows the optimal routes obtained by MISSA for burma14, bayg29, dantzig42, eil51, and eil76. None of the optimal routes obtained by MISSA have a crossed route.Figure 10
Optimal routes of MISSA. (a) burma14. (b) bayg29. (c) dantzig42. (d) eil51. (e) eil76.
(a)(b)(c)(d)(e)


## 6.1. The Traveling Salesman Problem

The traveling salesman problem is a well-known problem in combinatorial optimization filed, which can be described as follows: the traveling salesman intends to sell goods inn cities, selecting one of the cities as the starting point and visiting all the remaining cities without repetition, and finally returns to the starting point to select the shortest path among all routes [48].In the graph theory, TSP can be described as finding a Hamilton cycle with the lowest weight in a complete weighted graph. LetG = (V, E) be a complete weighted graph, where V = {1, 2, …, n} represents the vertex set and E represents the edge set. Each edge has a nonnegative weight m (e) on e = (I n, j) ∈ E and finds the Hamiltonian cycle C of G, in which the total weight of C satisfies MC=min∑ECme.The matrix W of distance between cities is as follows:(14)W=d1,1⋯d1,n⋮⋱⋮dn,1⋯dn,n,where d(i,j) represents the distance from the ith city to the jth city, d(i, j) ≥ 0, i, j ∈ v. The mathematical model of the objective function and constraint function of TSP are established as follows:(15)f=min∑∑di,jxi,j,s.t.∑j≠1xi,j=1,i∈V,∑i≠jxi,j=1,j∈V,∑i.j∈Sxi,j=S−1,S⊆V,xi,j∈0,1,i,j∈V,xi,j=1,i,j∈optimal path,0,other,where |S| represents the number of vertices in the set S. The constraint functions show that there is only one outgoing edge and one incoming edge for each vertex and no subloops will be generated.


## 6.2. Practical Problem Test

Suppose a tourist wants to travel around China passing through 31 provincial capital cities, an appropriate travel route to reduce economic costs is an important issue. The tourism scheme can be abstracted into a typical TSP problem, and the proposed MISSA, ISSA, SSA, GWO, and PSO are used for solving and comparative analysis. The city coordinates are shown in Table8. To ensure the consistency of the comparative experiments, the population size and maximum iteration number of each algorithm are 30 and 300, respectively. In addition, other parameters are as shown in Table 1.Table 8
City coordinates.
OrderCoordinate1(1304, 2312)2(3639, 1315)3(4177, 2244)4(3712, 1399)5(3488, 1535)6(3326, 1556)7(3238, 1229)8(4196, 1004)9(4312, 790)10(4386, 570)11(3007, 1970)12(2562, 1756)13(2788, 1491)14(2381, 1676)15(1332, 695)16(3715, 1678)17(3918, 2179)18(4061, 2370)19(3780, 2212)20(3676, 2578)21(4029, 2838)22(4263, 2931)23(3429, 1908)24(3507, 2367)25(3394, 2643)26(3439, 3201)27(2935, 3240)28(3140, 3550)29(2545, 2357)30(2778, 2826)31(2370, 2975)Figure9(a) shows the optimal travel route searched by MISSA, which does not show a crossed route and is consistent with the theoretical optimal solution. Figure 9(b) shows the convergence curves of the four algorithms. MISSA converges to the theoretical optimal solution in the 10th iteration, and the optimization accuracy and convergence speed are better than of compared algorithms.Figure 9
Provincial capitals tourism route planning. (a) Optimal route of MISSA. (b) Convergence curves.
(a)(b)To further verify the practicality and feasibility of the proposed MISSA, the algorithm is applied to five standard TSPLIB instances [49] in Table 9. We select the Euclidean distance as the calculation standard, and four algorithms are run 20 times on the five TSPLIB instances independently. The theoretical optimal (TO) value is the current optimal solution given by TSPLIB. The relative error (RE) is defined as follows:(16)RE=Mean−TOTO.Table 9
TSPLIB instances.
TSPLIBInstanceburma14bayg29dantzig42eil51eil76D1429425176TO3.09E + 019.07E + 036.99E + 024.26E + 025.38E + 02MISSABest3.09E + 019.07E + 036.88E + 024.26E + 025.38E + 02RE0.00%0.00%0.14%0.40%0.89%Mean3.09E + 019.07E + 037.00E + 024.28E + 025.43E + 02ISSABest3.09E + 019.13E + 036.99E + 024.35E + 025.52E + 02RE0.91%1.21%1.57%5.15%6.32%Mean3.12E + 019.18E + 037.10E + 024.48E + 025.72E + 02SSABest3.09E + 019.23E + 037.01E + 024.38E + 025.62E + 02RE1.17%2.21%2.72%6.09%8.00%Mean3.12E + 019.27E + 037.18E + 024.52E + 025.81E + 02GWOBest3.09E + 019.24E + 037.21E + 024.83E + 026.35E + 02RE1.69%2.09%4.43%12.96%17.89%Mean3.14E + 019.26E + 037.31E + 025.06E + 026.74E + 02PSOBest3.09E + 019.23E + 037.26E + 026.38E + 025.82E + 02RE2.01%3.09%5.01%16.57%21.61%Mean3.15E + 019.35E + 037.34E + 024.97E + 026.54E + 02Table9 illustrates that the solution precision and dimension of MISSA have been greatly improved in TSPLIB instances. For the instances tested, MISSA can converge to the theoretical optimal value with a lower relative error. For dantzig42, the Best of MISSA is better than the theoretical optimal value. Notably, as the dimensionality and dense distribution of cities increase, the requirements for algorithms to handle complex problems increase explosively. The relative error of the compared algorithms increases rapidly in eil51 and eil76, especially for GWO and PSO, which indicates the algorithms are falling into the local optimum. However, MISSA still maintains excellent optimization ability, indicating that the multimixed strategy is useful. The chaotic map provides good population diversity, the improved discoverer-follower model rapidly separates the solution space, and the EOBL mechanism upgrades the optimal solution and reconstructs the population. The multimixed strategy helps MISSA solve complex optimization problems with high dimensions. Figure 10 shows the optimal routes obtained by MISSA for burma14, bayg29, dantzig42, eil51, and eil76. None of the optimal routes obtained by MISSA have a crossed route.Figure 10
Optimal routes of MISSA. (a) burma14. (b) bayg29. (c) dantzig42. (d) eil51. (e) eil76.
(a)(b)(c)(d)(e)


## 7. Conclusion

A multimixed strategy improved sparrow search algorithm is proposed in this paper, overcoming the shortcomings of the original SSA, and we apply it to TSP. The original SSA is improved from three aspects: population initialization method, individuals’ location update principle, and optimal solution mutation strategy. The iterative chaotic mapping is used to improve the diversity of the initial population, making the population to be distributed more uniformly. The Gold-SA and nonlinear convergence factor strategies are introduced to enhance the discoverer-follower model, which make the discoverer’s searching process more comprehensive and extensive. Besides, the optimal solution and population are updated by the EOBL strategy, which enable the algorithm to increase self-learning ability.Compared with three derived algorithms through four basic benchmark functions and diversity analysis, the rationality of the multimixed strategy selection is proved. Through the optimization and Wilcoxon rank-sum test of ten basic benchmark functions and CEC 2014 function, MISSA is certified to have better optimization accuracy, convergence speed, and robustness than SSA, ISSA, CSSA, CLSSA, WOA, GWO, COOT, and PSO. Finally, the practicability of MISSA is veriﬁed by TSP. Experimental results demonstrate the feasibility of MISSA to solve complex combinatorial optimization problems.In the future, the application of MISSA in TSP will be further studied in generalized fields. Further research will be done on the multimixed strategy to stabilize population diversity while improving global exploration ability. The convergence of MISSA will be researched theoretically, and it will be used to solve large-scale combinatorial optimization problems.



---

*Source: 8171164-2022-05-18.xml*