# Modified Atom Search Optimization Based on Immunologic Mechanism and Reinforcement Learning

Authors: Yanming Fu; Zhuohang Li; Chiwen Qu; Haiqiang Chen
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

Atom search optimization algorithm has good searching ability and has been successfully applied to calculate hydrogeological parameters and groundwater dispersion coefficient. Since the atom search optimization algorithm is only based on the atom force motion model in molecular dynamics, it has some shortcomings such as slow search speed and low precision during the later stage of iteration. A modified atom search optimization based on the immunologic mechanism and reinforcement learning is proposed to overcome the abovementioned shortcomings in this paper. The proposed algorithm introduces a vaccine operator to better utilize the dominant position in the current atom population so that the speed, accuracy, and domain search ability of the atom search optimization algorithm can be strengthened. The reinforcement learning operator is applied to dynamically adjust the vaccination probability to balance the global exploration ability and local exploitation ability. The test results of 21 benchmark functions confirm that the performance of the proposed algorithm is superior to seven contrast algorithms in search accuracy, convergence speed, and robustness. The proposed algorithm is used to optimize the permutation flow shop scheduling problem. The experimental results indicate that the proposed algorithm can achieve better optimization results than the seven comparative algorithms, so the proposed algorithm has good practical application value.


---

## Body

## 1. Introduction

Optimization has been a hot topic in scientific research and engineering application [1] such as industry, society, economy, and management. The optimization methods include traditional exact solution method, constructive algorithm, and swarm intelligence algorithm [2]. The traditional exact solution method can obtain the exact solution, but it requires the objective function to be continuous and differentiable. The algorithm of traditional exact solution method is too complex, so it is suitable for solving small-scale problems. Meanwhile, the traditional exact solution method is not good at optimizing multipeak, nonlinear, and dynamic problems. The constructive algorithm [3] can quickly obtain the solution, but the solution quality is poor and difficult to meet the actual engineering needs. The swarm intelligence algorithm simulates the swarm behavior of social organisms to optimize the given objectives and embodies the characteristics of randomness, parallel, and distribution. The swarm intelligence algorithm provides a solution to complex problems without centralized control and without providing the global mathematical model. Compared with traditional optimization methods, the swarm intelligence algorithm has simple principle and fewer parameters and does not need gradient information of the problem. At present, the common swarm intelligence algorithms include particle swarm optimization algorithm [4], differential evolution algorithm [5], cuckoo search algorithm [6], flower pollination algorithm [7], sine-cosine search algorithm [8–10], and artificial bee colony algorithm [11]. They are widely used in function optimization [12–14], combinatorial optimization [15], image segmentation [16], and flow shop scheduling [17–19].The atom search optimization (ASO) was proposed by Zhao et al. in 2018 based on the force motion model of atoms in molecular dynamics [20]. ASO is a type of physics-inspired swarm intelligence algorithm. The algorithm needs fewer parameters and has good global exploration ability in the optimization process, so it has been successfully applied to calculate hydrogeological parameters and groundwater dispersion coefficient [21, 22]. Experimental results have proved that the ASO is significantly better than particle swarm optimization and genetic algorithm [20]. Since ASO is proposed only based on the forced motion model of atoms in molecular dynamics, it has the shortcomings of low search accuracy and slow convergence speed as other swarm intelligence algorithms.The particles distribute approximately evenly in nature and constitute a complex group. The individuals of particles exchange locally acquired information and interact in group. Inspired by the above facts, the modified atom search optimization algorithm based on immunologic mechanism and reinforcement learning (MASO) is proposed to enhance the ASO. The immunologic mechanism can make better use of the dominant positions in the current atomic population to promote the speed and accuracy of the MASO. The reinforcement learning is applied to dynamically adjust the vaccination probability to balance the global exploration ability and local exploitation ability of the MASO.


## 2. Atom Search Optimization Algorithm

Matter is composed of molecules. Atoms are connected by covalent bonds to form molecules. Atoms have mass and volume. The interaction forces between atoms are shown as repulsions or attractions according to the different distances between atoms. In the repulsive region, the repulsion forces between atoms increase sharply as the distances decrease. In the attractive region, when the distances increase to a certain extent, the attraction forces reach maxima. As the distances continue to increase, the attraction forces gradually decrease to zero. When two atoms are at an equilibrium distance, the interaction force between them is zero. In molecules, atomic forces also need to consider the effects of geometric constraints and atoms’ internal motions. The geometric constraints and atoms’ internal motions are called binding force. Therefore, the constrained atomic motion equation can be simply written as follows:(1)Fi+Gi=miai,where Fi is the resultant force of the interaction on atom i, Gi is the resultant force of the constraining force on atom i, mi is the mass, and ai is the acceleration of atom i.In ASO, each atom represents a feasible solution in the search space. The mass of an atom represents the quality of a feasible solution. The better the solution is, the higher the mass of the atom will be, and vice versa. All atoms in a population attract or repel each other according to their distances, which cause lighter atoms to move toward heavier atoms. Heavier atoms have less accelerations, which allow them to exploit local space better. Lighter atoms have more accelerations, which allow them to explore a wider search space. The massmit of the atom i at the iteration t can be simply measured by the fitness function. The expression of mit is shown in equation (3):(2)Mit=e−Fitit−Fitbestt/Fitworstt−Fitbestt,(3)mit=Mit∑j=1NMjt,where N is the total number of atoms, Fitit is the fitness function value of the atom i at the iteration t, and Fitworstt and Fitbestt are the fitness function values of the worst and best atoms at iteration t, respectively.In ASO, Kbest is defined as a set of the first K atoms that have the best fitness values in the atomic population. The set is also called the K-nearest neighbors of atomi. The atom i needs to interact with its K-nearest neighbors as much as possible to strengthen exploration at the early iteration stage. The atom i needs to interact with its K neighbor as little as possible to enhance the exploitation at the late iteration stage. Therefore, K decreases as the number of iterations increases. When the total number of iterations is T, K can be calculated as follows:(4)Kt=N−N−2×tT.Since ASO needs to increase the attraction force and reduce the repulsive force with the increase of iteration times, Zhao et al. [20] proposed a simplified model of the interaction Fij. The expression of Fij is as follows:(5)Fij=−ηt2hijt13−hijt7,(6)ηt=α1−t−1T3e−20t/T,(7)hijt=hmin,rijtσt<hmin,rijtσt,hmin≤rijtσt≤hmax,hmax,rijtσt>hmax,where ηt is a depth function used to adjust the repulsion region and the attraction region. ηt is defined in equation (6), where α is the depth weight. According to equation (7), the distance function hijt is updated, where hmin=1.1 and hmax=1.4 are the upper and lower limits of h, respectively. rijt is the Euclidean distance between atom i and atom j in the iteration t. σt is the length, which is defined as follows:(8)σt=Xijt,∑ϵKbestXijtKt2,where Xij is the position component of atom i in the dimension j of the search space. The dynamically changed hmin and hmax are defined in the following equation:(9)hmin=g0+gt,hmax=u,where g is the drift component and is defined as follows:(10)gt=0.1×sinπ2×tT.The resultant force of the interaction force on atomi is expressed in the following equation, where randj is a random number that obeys the uniform distribution of 0-1:(11)Fidt=∑j∈KbestrandjFijdt.Geometric constraints in molecular dynamics theory play an important role in atomic motion. Suppose that every atom has covalent bonds with every atom in Kbest, every atom is constrained by Kbest. The constraining force of the atomi can be described by the following equation:(12)Gidt=λtXbestdt−Xidt,(13)λt=βe−20t/T,where λ is the Lagrange multiplier, β is the multiplier weight, and Xbestdt is the position component in dimension d of the best atom at iteration t. According to equations (1), (5), and (12), the updated equation of acceleration can be deduced as follows:(14)aidt=Fidtmidt+Gidtmidt=α1−t−1T3e−20t/TΣj∈Kbestrandj2×hijt13−hij7/mit×Xjdt−xidt/Xit,Xjt2+βe−20t/TXbestdt−Xidt/mit.According to the above equations, the velocity and position update equations of atomi at iteration t + 1 are equations (15) and (16), respectively:(15)vidt+1=randidvidt+aidt,(16)Xidt+1=Xidt+vidt+1.The pseudocode of ASO is given in Algorithm1.Algorithm 1:Pseudocode of the atom search optimization algorithm.
(1)
Randomly initialize a set of atomsX (solutions) and their velocity v, and FitBest=Inf.
(2)
While the stop criterion is not satisfied do
(3)
For each atomXi do
(4)
Calculate the fitness value Fit;
(5)
IfFiti<FitBest then
(6)
FitBest=Fiti;
(7)
XBest=Xi;
(8)
End If.
(9)
Calculate the massing using equations (2) and (3);
(10)
Determine its K neighbors using equation (4);
(11)
Calculate the interaction forceFi and the constraint force Gi using equations (11) and (12), respectively;
(12)
Calculate the acceleration using equation (14);
(13)
Update the velocity using equation (15);
(14)
Update the position using equation (16);
(15)
End For.
(16)
End While.
(17)
Find the best solution so farXBest


## 3. Modified Atom Search Optimization Based on Immunologic Mechanism and Reinforcement Learning

### 3.1. Chaos

Chaos is a nonlinear phenomenon widely existing in nature. It has the characteristics of randomness, ergodicity and inherent regularity [23–25]. Randomness means that the chaotic behavior has the disorderly features similar to random variable. Ergodicity means that chaos can traverse all states according to its own laws without repetition. Regularity means that chaos is generated by deterministic iteration. Therefore, the chaos strategy is used to initialize the swarm intelligence algorithm and avoid the algorithm falling into the local optimum.There are defects in random initialization of the ASO. These defects can be well solved by the chaotic strategy. In MASO, nine different chaotic maps are used to initialize the atomic population. These nine kinds of mapping will be called circularly to ensure that different problems can be properly initialized. Table1 displays the basic information about the nine maps. In this paper, the chaos strategy used to initialize population is called the chaos initialization operator.Table 1
Chaotic mapping transform kernel and parameter range.
The name of the map
Transform core
Parameter range
Circle map
x
n
+
1
=
x
n
+
b
−
a
/
2
π
sin
2
π
/
x
n
a
,
b
∈
R
Gauss/mouse map
x
n
+
1
=
e
−
α
x
n
2
+
β
α
,
β
∈
R
Sine map
x
n
+
1
=
sin
π
x
n
Logistic map
x
n
+
1
=
μ
x
n
1
−
x
n
μ
∈
0,4
Iterative map
x
n
+
1
=
sin
α
π
/
x
n
∝
∈
0,1
Singer map
x
n
+
1
=
u
7.86
x
n
−
23.31
x
n
2
+
28.75
x
n
3
−
13.302875
x
n
4
u
∈
0,4
Tent map
x
n
+
1
=
x
n
/
q
,
0
<
x
n
≤
q
1
−
x
n
/
1
−
q
,
q
<
x
n
<
1
q
∈
0,1
Sinusoidal map
x
n
+
1
=
2.3
x
n
2
sin
π
x
n
Chebyshev map
x
n
+
1
=
cos
μ
a
cos
x
n
μ
∈
0,1


### 3.2. Artificial Immune Algorithm and Vaccine Operator

Artificial immune system is an intelligent method inspired by the biological immune system. It is a new information processing system based on the theory of the human immune system. The artificial immune system provides evolutionary learning mechanisms such as noise tolerance, nonteacher learning, no negative examples, and self-organization. The artificial immune system provides a new way to solve the optimization problem of high dimension [26, 27].The artificial immune algorithm is essentially a group of operators. The artificial immune algorithm can generate and maintain the diversity of population and has the ability of self-regulation, so it is suitable for optimizing and improving the swarm intelligent algorithm.The ASO does not make full use of the dominant position in the population, so the accuracy and speed of convergence are not outstanding enough. These defects can be improved by introducing the vaccine operator [28–31]. The vaccine operator consists of three parts: vaccine extraction operator, vaccination operator, and immune detection operator.(1)
Vaccine Extraction Operator. In the artificial immune algorithm, vaccine is the prior knowledge of the solving problem. The vaccine extraction operator uses the iteration results of ASO as the prior knowledge and extracts the best current position of atoms as vaccine. The formula of vaccine extraction operator is shown below:(17)vaccine=Xit,Fitit≤Fitbest,Xit−1,Fitit>Fitbest.Fitit expressed the fitness function value of Xi in the t iteration, and Fitbest represents the fitness function value of the best position obtained by the iteration so far.(2)
Vaccination Operator. Vaccination uses prior knowledge to locally adjust the solutions so that the qualities of the candidate solutions are significantly improved. The vaccination operator will be used to inoculate atomic population. Firstly, a part of the individuals is selected from the atomic population, and then the position of the selected atomic individuals is modified by the following equation:(18)Xnewij=Vaccineij,Pv≥r,Xij,Pv<r,whereXnew is the atomic population after vaccination. r is a random number that obeys the uniform distribution of 0-1.(3)
Immunoassay Operator. After the operation of the vaccination operator, the immune detection operator compares the fitness values of the vaccinated individuals and the original individuals. If the fitness values of vaccinated individuals are not as good as the original individuals, it means that there is degradation in the process of inoculation. Then the original individuals become the parents of the next generation directly according to equation (19). Otherwise, the vaccinated individuals become the parents of next generation according to the following equation:(19)Xit+1=Xnewit,Fitit>Newfitit,Xit,Fitit≤Newfitit.The process of using the immune mechanism to enhance ASO is shown in Figure1.Figure 1
Immune mechanism improves ASO.


### 3.3. Reinforcement Learning Mechanism Updates Vaccination Probability

#### 3.3.1. Reinforcement Learning

Machine learning includes supervised learning, unsupervised learning, and reinforcement learning. Reinforcement learning takes environmental feedback as input. This method is different from the supervised learning and unsupervised learning. The supervised learning and unsupervised learning tell the Agent what behavior to take through positive and negative examples, but reinforcement learning finds the optimal behavior strategy through trial and error [32].The principle of reinforcement learning is shown in Figure2. Reinforcement learning regards learning as a test evaluation process. The Agent chooses an action for the environment, and the environment produces a reinforcement signal (award or punish) feedback to the Agent after its state changes. Then the Agent selects the next action according to the reinforcement signal and the current state of the environment. The principle of selection is to increase the probability of receiving positive reinforcement (award). The action of selection affects not only the immediate reinforcement value but also the state of the environment at the next moment and the final reinforcement value [33, 34].Figure 2
Principle of reinforcement learning.


#### 3.3.2. Adjust Vaccination Probability Dynamically

In the MASO, the vaccination probability Pv is determined according to the actual situation. The reinforcement learning method is used to determine the vaccination probability dynamically. In this paper, this method is collectively called the reinforcement learning operator.The reinforcement learning method used here is based on snap-drift neural network [35–37]. The reinforcement learning method switches between snap mode and drift mode [38]. In the snap mode, as Pv decreases, global exploration increases, thus avoiding premature convergence. In the drift mode, as Pv decreases, local exploitation increases, thus improving convergence speed. The Pv value is updated with equation (20). Agent (MASO) accepts the state (snap or drift) and reward value (Pm) at time t and then takes an action (increase Pv or decrease Pv) to convert to a new state. The Pm value is updated with the following equation:(20)Pv=max0,Pv−ωdecrease rule,μ=snap,min1,Pv+ωincrease  rule,μ=drift,(21)Pm=SeN,where Se is the number of atomic individuals updated in this iteration. Pm is the conversion probability. The value of Pm is determined by the ratio of updated individuals in the population. When Pm is less than 50%, the snap mode is used; when Pm is more than 50%, the drive mode is used. ω is the step size of Pv for each change. Figure 3 illustrates how reinforcement learning operator works.Figure 3
Principle of the reinforcement learning operator.


### 3.4. Algorithm Flow

After summarizing the above improved methods of ASO, the pseudocode of MASO is given in Algorithm2.Algorithm 2: Pseudocode of the modified atom search optimization based on the immunologic mechanism and reinforcement learning.
(1)
Chaotic initialize a set of atomsX (solutions) and randomly initialize their velocity v, evaluate the fitness values by using the objective function, and then calculate XBest and FGbest. Determine the parameters.
(2)
While the stop criterion is not satisfied do
(3)
For each atomXi do
(4)
Calculate the massing using equations (2) and (3);
(5)
Determine its K neighbors using equation (4);
(6)
Calculate the interaction forceFi and the constraint force Gi using equations (11) and (12), respectively;
(7)
Calculate the acceleration using equation (14);
(8)
Update the velocity using equation (15);
(9)
Update the position using equation (16);
(10)
Check out, update the fitness values;
(11)
Invoke the reinforcement learning operator by equations (20) and (21);
(12)
Invoke the vaccination operator by equations (17) and (18);
(13)
Invoke the immune detection operator by equation (19);
(14)
End For.
(15)
End While
(16)
Find the best solution so farXBest


## 3.1. Chaos

Chaos is a nonlinear phenomenon widely existing in nature. It has the characteristics of randomness, ergodicity and inherent regularity [23–25]. Randomness means that the chaotic behavior has the disorderly features similar to random variable. Ergodicity means that chaos can traverse all states according to its own laws without repetition. Regularity means that chaos is generated by deterministic iteration. Therefore, the chaos strategy is used to initialize the swarm intelligence algorithm and avoid the algorithm falling into the local optimum.There are defects in random initialization of the ASO. These defects can be well solved by the chaotic strategy. In MASO, nine different chaotic maps are used to initialize the atomic population. These nine kinds of mapping will be called circularly to ensure that different problems can be properly initialized. Table1 displays the basic information about the nine maps. In this paper, the chaos strategy used to initialize population is called the chaos initialization operator.Table 1
Chaotic mapping transform kernel and parameter range.
The name of the map
Transform core
Parameter range
Circle map
x
n
+
1
=
x
n
+
b
−
a
/
2
π
sin
2
π
/
x
n
a
,
b
∈
R
Gauss/mouse map
x
n
+
1
=
e
−
α
x
n
2
+
β
α
,
β
∈
R
Sine map
x
n
+
1
=
sin
π
x
n
Logistic map
x
n
+
1
=
μ
x
n
1
−
x
n
μ
∈
0,4
Iterative map
x
n
+
1
=
sin
α
π
/
x
n
∝
∈
0,1
Singer map
x
n
+
1
=
u
7.86
x
n
−
23.31
x
n
2
+
28.75
x
n
3
−
13.302875
x
n
4
u
∈
0,4
Tent map
x
n
+
1
=
x
n
/
q
,
0
<
x
n
≤
q
1
−
x
n
/
1
−
q
,
q
<
x
n
<
1
q
∈
0,1
Sinusoidal map
x
n
+
1
=
2.3
x
n
2
sin
π
x
n
Chebyshev map
x
n
+
1
=
cos
μ
a
cos
x
n
μ
∈
0,1


## 3.2. Artificial Immune Algorithm and Vaccine Operator

Artificial immune system is an intelligent method inspired by the biological immune system. It is a new information processing system based on the theory of the human immune system. The artificial immune system provides evolutionary learning mechanisms such as noise tolerance, nonteacher learning, no negative examples, and self-organization. The artificial immune system provides a new way to solve the optimization problem of high dimension [26, 27].The artificial immune algorithm is essentially a group of operators. The artificial immune algorithm can generate and maintain the diversity of population and has the ability of self-regulation, so it is suitable for optimizing and improving the swarm intelligent algorithm.The ASO does not make full use of the dominant position in the population, so the accuracy and speed of convergence are not outstanding enough. These defects can be improved by introducing the vaccine operator [28–31]. The vaccine operator consists of three parts: vaccine extraction operator, vaccination operator, and immune detection operator.(1)
Vaccine Extraction Operator. In the artificial immune algorithm, vaccine is the prior knowledge of the solving problem. The vaccine extraction operator uses the iteration results of ASO as the prior knowledge and extracts the best current position of atoms as vaccine. The formula of vaccine extraction operator is shown below:(17)vaccine=Xit,Fitit≤Fitbest,Xit−1,Fitit>Fitbest.Fitit expressed the fitness function value of Xi in the t iteration, and Fitbest represents the fitness function value of the best position obtained by the iteration so far.(2)
Vaccination Operator. Vaccination uses prior knowledge to locally adjust the solutions so that the qualities of the candidate solutions are significantly improved. The vaccination operator will be used to inoculate atomic population. Firstly, a part of the individuals is selected from the atomic population, and then the position of the selected atomic individuals is modified by the following equation:(18)Xnewij=Vaccineij,Pv≥r,Xij,Pv<r,whereXnew is the atomic population after vaccination. r is a random number that obeys the uniform distribution of 0-1.(3)
Immunoassay Operator. After the operation of the vaccination operator, the immune detection operator compares the fitness values of the vaccinated individuals and the original individuals. If the fitness values of vaccinated individuals are not as good as the original individuals, it means that there is degradation in the process of inoculation. Then the original individuals become the parents of the next generation directly according to equation (19). Otherwise, the vaccinated individuals become the parents of next generation according to the following equation:(19)Xit+1=Xnewit,Fitit>Newfitit,Xit,Fitit≤Newfitit.The process of using the immune mechanism to enhance ASO is shown in Figure1.Figure 1
Immune mechanism improves ASO.


## 3.3. Reinforcement Learning Mechanism Updates Vaccination Probability

### 3.3.1. Reinforcement Learning

Machine learning includes supervised learning, unsupervised learning, and reinforcement learning. Reinforcement learning takes environmental feedback as input. This method is different from the supervised learning and unsupervised learning. The supervised learning and unsupervised learning tell the Agent what behavior to take through positive and negative examples, but reinforcement learning finds the optimal behavior strategy through trial and error [32].The principle of reinforcement learning is shown in Figure2. Reinforcement learning regards learning as a test evaluation process. The Agent chooses an action for the environment, and the environment produces a reinforcement signal (award or punish) feedback to the Agent after its state changes. Then the Agent selects the next action according to the reinforcement signal and the current state of the environment. The principle of selection is to increase the probability of receiving positive reinforcement (award). The action of selection affects not only the immediate reinforcement value but also the state of the environment at the next moment and the final reinforcement value [33, 34].Figure 2
Principle of reinforcement learning.


### 3.3.2. Adjust Vaccination Probability Dynamically

In the MASO, the vaccination probability Pv is determined according to the actual situation. The reinforcement learning method is used to determine the vaccination probability dynamically. In this paper, this method is collectively called the reinforcement learning operator.The reinforcement learning method used here is based on snap-drift neural network [35–37]. The reinforcement learning method switches between snap mode and drift mode [38]. In the snap mode, as Pv decreases, global exploration increases, thus avoiding premature convergence. In the drift mode, as Pv decreases, local exploitation increases, thus improving convergence speed. The Pv value is updated with equation (20). Agent (MASO) accepts the state (snap or drift) and reward value (Pm) at time t and then takes an action (increase Pv or decrease Pv) to convert to a new state. The Pm value is updated with the following equation:(20)Pv=max0,Pv−ωdecrease rule,μ=snap,min1,Pv+ωincrease  rule,μ=drift,(21)Pm=SeN,where Se is the number of atomic individuals updated in this iteration. Pm is the conversion probability. The value of Pm is determined by the ratio of updated individuals in the population. When Pm is less than 50%, the snap mode is used; when Pm is more than 50%, the drive mode is used. ω is the step size of Pv for each change. Figure 3 illustrates how reinforcement learning operator works.Figure 3
Principle of the reinforcement learning operator.


## 3.3.1. Reinforcement Learning

Machine learning includes supervised learning, unsupervised learning, and reinforcement learning. Reinforcement learning takes environmental feedback as input. This method is different from the supervised learning and unsupervised learning. The supervised learning and unsupervised learning tell the Agent what behavior to take through positive and negative examples, but reinforcement learning finds the optimal behavior strategy through trial and error [32].The principle of reinforcement learning is shown in Figure2. Reinforcement learning regards learning as a test evaluation process. The Agent chooses an action for the environment, and the environment produces a reinforcement signal (award or punish) feedback to the Agent after its state changes. Then the Agent selects the next action according to the reinforcement signal and the current state of the environment. The principle of selection is to increase the probability of receiving positive reinforcement (award). The action of selection affects not only the immediate reinforcement value but also the state of the environment at the next moment and the final reinforcement value [33, 34].Figure 2
Principle of reinforcement learning.


## 3.3.2. Adjust Vaccination Probability Dynamically

In the MASO, the vaccination probability Pv is determined according to the actual situation. The reinforcement learning method is used to determine the vaccination probability dynamically. In this paper, this method is collectively called the reinforcement learning operator.The reinforcement learning method used here is based on snap-drift neural network [35–37]. The reinforcement learning method switches between snap mode and drift mode [38]. In the snap mode, as Pv decreases, global exploration increases, thus avoiding premature convergence. In the drift mode, as Pv decreases, local exploitation increases, thus improving convergence speed. The Pv value is updated with equation (20). Agent (MASO) accepts the state (snap or drift) and reward value (Pm) at time t and then takes an action (increase Pv or decrease Pv) to convert to a new state. The Pm value is updated with the following equation:(20)Pv=max0,Pv−ωdecrease rule,μ=snap,min1,Pv+ωincrease  rule,μ=drift,(21)Pm=SeN,where Se is the number of atomic individuals updated in this iteration. Pm is the conversion probability. The value of Pm is determined by the ratio of updated individuals in the population. When Pm is less than 50%, the snap mode is used; when Pm is more than 50%, the drive mode is used. ω is the step size of Pv for each change. Figure 3 illustrates how reinforcement learning operator works.Figure 3
Principle of the reinforcement learning operator.


## 3.4. Algorithm Flow

After summarizing the above improved methods of ASO, the pseudocode of MASO is given in Algorithm2.Algorithm 2: Pseudocode of the modified atom search optimization based on the immunologic mechanism and reinforcement learning.
(1)
Chaotic initialize a set of atomsX (solutions) and randomly initialize their velocity v, evaluate the fitness values by using the objective function, and then calculate XBest and FGbest. Determine the parameters.
(2)
While the stop criterion is not satisfied do
(3)
For each atomXi do
(4)
Calculate the massing using equations (2) and (3);
(5)
Determine its K neighbors using equation (4);
(6)
Calculate the interaction forceFi and the constraint force Gi using equations (11) and (12), respectively;
(7)
Calculate the acceleration using equation (14);
(8)
Update the velocity using equation (15);
(9)
Update the position using equation (16);
(10)
Check out, update the fitness values;
(11)
Invoke the reinforcement learning operator by equations (20) and (21);
(12)
Invoke the vaccination operator by equations (17) and (18);
(13)
Invoke the immune detection operator by equation (19);
(14)
End For.
(15)
End While
(16)
Find the best solution so farXBest


## 4. Simulation Experiment

The performance of MASO will be verified by experiments in three aspects: (1) The MASO will compare with seven swarm intelligence algorithms using 21 benchmark functions [39–41] of 30 dimensions. (2) The significance of MASO improvement will be analyzed by the T test. (3) The significance of MASO improved by individual strategies will be analyzed with the Wilcoxon rank sum test.


### 4.1. Test Platform and Benchmark Functions

#### 4.1.1. Test Platform

The server model is the dawning 5000 A server. The server configures Xeon X5620 CPU (4 cores)∗2, 24 GB memory, and 300 GB SAS hard disk. The server is equipped with Rhel5.6 operating system. The programming tool is MATLAB 2012a (for Linux).


#### 4.1.2. Benchmark Functions

21 benchmark functions are selected to evaluate the algorithm performance. This set of benchmark functions has widely been used to evaluate the ability of the swarm intelligence algorithm [38]. The number, range, type, and formulation of functions are listed in Table 2. The 21 benchmark functions can be divided into three categories. F1∼F6 are the unimodal functions (U) and are applied to investigate the search convergence accuracy. F7–F17 are the complex multimodal functions (M) with multiple local extremum points. These functions are employed to test the global search capability and the ability to avoid prematurity. F18–F21 are the rotating multimodal functions (R). These functions have more extreme points, thus increasing the difficulty of searching. F18–F21 can be adopted to further detect the overall performance of the algorithm.Table 2
The benchmark functions.
No.
Range
Type
Formulation
F1
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
2
F2
[−10, 10]
U
f
x
=
∑
i
=
1
d
x
i
+
∏
i
=
1
d
x
i
F3
[−10, 10]
U
f
x
=
∑
i
=
1
d
i
x
i
2
F4
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
+
0.5
2
F5
[−1.28, 0.64]
U
f
x
=
∑
i
=
1
d
i
x
i
4
+
R
F6
[−1.28, 1.28]
U
f
x
=
∑
i
=
1
d
i
x
i
4
F7
[−32, 32]
M
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
x
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
x
i
+
20
+
e
F8
[−600, 600]
M
f
x
=
1
/
4000
⋅
∑
i
=
1
d
x
i
2
−
∏
i
=
1
d
cos
x
i
/
i
+
1
F9
[−5.12, 5.12]
M
f
x
=
10
∗
d
+
∑
i
=
1
d
x
i
2
−
10
cos
2
π
x
i
F10
[−500, 500]
M
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
x
i
sin
x
i
F11
[−10, 10]
M
f
x
=
∑
i
=
1
d
x
i
⋅
sin
x
i
+
0.1
⋅
x
i
F12
[−10, 10]
M
f
x
=
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
sin
2
3
π
x
1
F13
[−100, 100]
M
f
x
=
0.5
+
sin
2
∑
i
=
1
d
x
i
2
−
0.5
/
1
+
0.001
∑
i
=
1
d
x
i
2
2
+
x
n
−
1
1
+
sin
2
3
π
x
n
F14
[−5.12, 5.12]
M
f
x
=
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
+
10
,
y
i
=
x
i
,
x
i
<
1
/
2
round
2
x
i
/
2
,
x
i
≥
1
/
2
F15
[−5, 10]
M
f
x
=
∑
i
=
1
d
−
1
1
−
x
i
2
+
100
x
i
+
1
−
x
i
2
2
F16
[−50, 50]
M
f
x
=
π
/
d
10
sin
2
π
z
1
+
∑
i
=
1
d
−
1
z
i
−
1
2
1
+
10
sin
2
π
z
i
+
1
+
z
d
−
1
2
+
∑
i
=
1
d
u
x
i
,
10,100,4
,
z
i
=
1
+
1
/
4
x
i
+
1
,
u
x
i
,
a
,
k
,
m
=
k
x
i
−
a
m
,
x
i
>
aor
x
i
<
−
a
0
,
otherwise
F17
[−50, 50]
M
f
x
=
1
/
d
sin
2
π
x
1
+
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
x
d
−
1
2
1
+
sin
2
2
π
x
i
+
1
+
∑
i
=
1
d
u
x
i
,
5,100,4
F18
[−32, 32]
R
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
y
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
y
i
+
20
+
e
F19
[−5.12, 5.12]
R
f
x
=
10
∗
d
+
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
F20
[−600, 600]
R
f
x
=
1
/
4000
⋅
∑
i
=
1
d
y
i
2
−
∏
i
=
1
d
cos
y
i
/
i
+
1
F21
[−500, 500]
R
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
y
i
sin
y
i
U: unimodel, M: multimodel, and R: rotated.


### 4.2. Experimental Results Analysis of MASO and Comparison Algorithms

In this experiment, the parameters setting of the comparison algorithms are displayed in Table3. For each benchmark function, each algorithm runs 20 times independently. The population number is 100. The evaluation times of MASO and CS are 2∗ 100 in each iteration, and evaluation times of other algorithms are 100 in each iteration. For each benchmark function, MASO and CS will iterate 2500 times and other algorithms will iterate 5000 times. So all algorithms will be compared under the evaluations of 100∗ 5000 in each run. The various performances of algorithms are measured by the four indexes of optimal value, average value, worst value, and standard deviation. The experimental statistical results are shown in Table 4. Take 500 points to draw the convergence curve. MASO and CS take one point every 50 iterations, and other algorithms take one point every 100 iterations. Figures 4–24 are the convergence curves of each function successively.Table 3
The parameters set of contrast algorithms.
Algorithm
Parameters
MASO
N = 100; depth = 50; multiplier = 0.2; Pv = 0.1; w=0.05.
ASO
N = 100; depth = 50; multiplier = 0.2.
FPA
N = 100; P = 0.8.
BAT
N = 100; A = 0.5; r = 0.5; Qmin = 0; Qmax = 2.
PSO
N = 100; c1 = 1.49445; c2 = 1.49445.
CS
N = 100; pa = 0.25.
ACOR
N = 100; nSample = 50; q = 0.5; zeta = 1.
CSA
N = 100; AP = 0.2; fl = 2.Table 4
Test statistical results of functionsf1∼f21.
Functions
Algorithms
Min
Mean
Max
STD
f
1(x)
MASO
2.99E − 45
1.82E − 40
3.5E − 39
7.82E − 40
ASO
7.33E − 27
2.64E − 26
1.05E − 25
3.08E − 26
FPA
0.006587
0.013937
0.023855
0.004844
BAT
1.09E − 05
155.0339
891.9841
257.7465
PSO
3.869551
20.12067
77.25433
16.40947
CS
0.004142
173.506
1688.332
391.8062
ACOR
6.98E − 13
2.01E − 12
3.85E − 12
7.74E − 13
CSA
1.29E − 12
8.89E − 09
7.83E − 08
1.9E − 08
f
2(x)
MASO
2.41E − 34
1.63E − 29
1.34E − 28
3.9E − 29
ASO
2.16E − 13
5.32E − 13
9.67E − 13
2.51E − 13
FPA
0.677651
1.159856
1.89272
0.340836
BAT
0.014345
42.11026
124.5488
48.56557
PSO
6.26162
9.511573
16.34641
2.636531
CS
0.024397
1.543266
8.754166
2.237715
ACOR
3.46E − 07
0.000387
0.006348
0.001411
CSA
9.9E − 07
4.2E − 05
0.000133
3.52E − 05
f
3(x)
MASO
1.5E − 47
5.73E − 42
8.19E − 41
1.87E − 41
ASO
5.73E − 26
1.73E − 25
3.92E − 25
1.03E − 25
FPA
0.000957
0.001794
0.002651
0.000518
BAT
0.000105
0.000156
0.000193
2.85E − 05
PSO
1.681533
8.073872
24.85504
6.662011
CS
0.138672
63.08822
290.7739
80.09947
ACOR
7.82E − 14
2.16E − 13
4.42E − 13
1.09E − 13
CSA
1.27E − 12
1.63E − 09
1.89E − 08
4.14E − 09
f
4(x)
MASO
0
0
0
0
ASO
0
0
0
0
FPA
0
0
0
0
BAT
4214
10180.7
16319
3108.576
PSO
57
191.7
405
90.82435
CS
9
1558.45
6328
1791.922
ACOR
0
0
0
0
CSA
0
0
0
0
f
5(x)
MASO
2.34E − 06
1.47E − 05
3.73E − 05
9.24E − 06
ASO
0.003764
0.00772
0.011135
0.002189
FPA
0.006508
0.020989
0.029649
0.005437
BAT
0.218579
0.403253
0.531159
0.080489
PSO
0.273095
0.62667
1.206052
0.285222
CS
0.94235
4.144344
13.32052
2.939838
ACOR
0.016274
0.032401
0.053185
0.009913
CSA
3.99E − 06
5.29E − 05
0.000164
4.22E − 05
f
6(x)
MASO
2.41E − 60
6.82E − 51
1.26E − 49
2.82E − 50
ASO
2.19E − 52
1.46E − 51
5.24E − 51
1.51E − 51
FPA
3.28E − 11
3.39E − 10
1.43E − 09
3.33E − 10
BAT
6.29E − 11
1.02E − 10
1.45E − 10
2.17E − 11
PSO
0.001601
0.010088
0.038612
0.010173
CS
0.000168
0.34018
2.726353
0.655415
ACOR
2.22E − 14
2.49E − 13
1.82E − 12
3.95E − 13
CSA
3.56E − 31
1.06E − 23
1.02E − 22
2.42E − 23
f
7(x)
MASO
2.22E − 14
3.83E − 13
5.36E − 12
1.18E − 12
ASO
4.35E − 14
7.41E − 14
1.47E − 13
2.97E − 14
FPA
0.17994
0.618946
1.286978
0.260178
BAT
10.99074
13.41295
19.95889
1.88331
PSO
3.213753
5.385464
7.818361
1.282747
CS
8.631765
13.19331
16.18969
1.882678
ACOR
2.3E − 07
4.2E − 07
5.56E − 07
1.07E − 07
CSA
2.84E − 07
1.95E − 05
4.98E − 05
1.54E − 05
f
8(x)
MASO
1.11E − 16
1.65E − 15
5.66E − 15
1.65E − 15
ASO
0
0
0
0
FPA
0.086888
0.136963
0.230776
0.044916
BAT
19.19178
53.02061
94.33961
21.77808
PSO
0.939324
1.178857
1.343306
0.102008
CS
0.610131
4.621765
33.27042
8.201352
ACOR
4.88E − 12
0.022291
0.247193
0.069061
CSA
3.57E − 10
4.25E − 08
4.24E − 07
9.87E − 08
f
9(x)
MASO
0
4.97E − 13
3.18E − 12
9.06E − 13
ASO
9.949591
14.5264
20.89413
3.356089
FPA
53.45329
72.74809
84.86339
8.314181
BAT
38.80531
79.29986
169.1439
35.18297
PSO
75.51275
110.7482
169.454
25.42386
CS
30.82795
51.57561
84.51111
14.55878
ACOR
183.2672
214.6805
237.272
12.35372
CSA
1.76E − 12
2.82E − 09
7.92E − 09
2.7E − 09
f
10(x)
MASO
0.000382
0.000382
0.000382
1.35E − 11
ASO
4106.132
4650.915
5626.077
521.8557
FPA
3244.947
3698.782
4219.299
237.132
BAT
5972.237
7099.381
8123.466
576.3557
PSO
3897.536
5375.416
6743.524
814.7786
CS
1848.715
2793.954
4026.17
610.2232
ACOR
1478.361
5880.244
7607.56
360.132
CSA
0.000382
0.000382
0.000382
7E −08
f
11(x)
MASO
3.44E − 33
1.83E − 05
9.49E − 05
3.11E − 05
ASO
4.23E − 14
8.76E − 14
1.97E − 13
4.72E − 14
FPA
3.227243
6.669646
9.146383
1.526006
BAT
0.834363
3.926399
9.459001
2.576305
PSO
2.748819
6.186047
10.35792
2.432746
CS
0.039447
0.854302
2.652989
0.811748
ACOR
0.034574
0.302129
5.21918
1.157372
CSA
7.64E − 08
5.4E − 06
2.31E − 05
5.86E − 06
f
12(x)
MASO
1.15E − 31
6.29E − 28
1.18E − 26
2.63E − 27
ASO
2.78E − 27
6.39E − 27
2.1E − 26
5.59E − 27
FPA
1.055809
2.341216
3.399438
0.776936
BAT
3.166887
8.757063
15.72329
3.538368
PSO
1.089431
7.019646
16.18996
3.659522
CS
9.73E − 09
0.005653
0.104223
0.02323
ACOR
1.69E − 11
9.04E − 11
4.43E − 10
9.82E − 11
CSA
1.24E − 13
5.21E − 11
4.53E − 10
1.13E − 10
f
13(x)
MASO
2.22E − 16
9.1E − 16
5.33E − 15
1.14E − 15
ASO
0.111929
0.171566
0.247934
0.04255
FPA
0.164183
0.23541
0.288289
0.035616
BAT
0.493711
0.498347
0.499585
0.001342
PSO
0.213232
0.336072
0.442089
0.05312
CS
0.470226
0.493116
0.498482
0.006452
ACOR
0.018867
0.036172
0.046726
0.006468
CSA
1.33E − 15
4.72E − 12
2.33E − 11
6.56E − 12
f
14(x)
MASO
1.78E − 15
1.44E − 14
4.09E − 14
1.04E − 14
ASO
12
18.70114
35.0005
6.254622
FPA
58.21314
75.72416
92.27162
10.98064
BAT
142.0001
225.425
320.25
43.94807
PSO
51.61432
97.72042
154.1477
27.68505
CS
22.03245
65.54915
124.1758
23.27628
ACOR
159.1282
193.0334
215.8543
15.42857
CSA
5.74E − 11
5.9E − 09
2.73E − 08
8.01E − 09
f
15(x)
MASO
9.9E − 13
7.77E −09
8.32E −08
2E −08
ASO
13.22012
14.72793
15.51416
0.63515
FPA
23.19234
26.9907
29.18676
1.618782
BAT
0.095172
18.58866
73.64575
30.2613
PSO
305.1689
574.6557
1477.057
316.5058
CS
92.87089
4700.287
44298.9
10461.76
ACOR
24.81962
25.10222
25.36799
0.158517
CSA
2.24E − 11
1.35E − 09
5.65E − 09
1.97E − 09
f
16(x)
MASO
3.12E − 32
1.12E − 29
2.02E − 28
4.51E − 29
ASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
FPA
0.795144
1.715761
2.908752
0.550818
BAT
0.519122
11.97827
28.55629
9.510974
PSO
3.03539
7.040329
11.93878
2.719544
CS
7.55E − 08
0.019757
0.366427
0.081707
ACOR
9.06E − 09
2.61E − 07
2.1E − 06
4.61E − 07
CSA
2.13E − 14
3.58E − 11
1.54E − 10
4.63E − 11
f
17(x)
MASO
1.26E − 32
2.26E − 31
1.86E − 30
4.61E − 31
ASO
6.55E − 28
2.14E − 27
5.54E − 27
1.53E − 27
FPA
0.075634
0.159202
0.383942
0.076028
BAT
28.02352
59.78051
75.72239
10.54314
PSO
3.843779
71.81837
922.9239
200.6675
CS
3.36E − 22
8.9E − 22
1.77E − 21
4.31E − 22
ACOR
3.17E − 08
8.81E − 07
6.2E − 06
1.69E − 06
CSA
9.23E − 13
2.67E − 10
8.43E − 10
2.34E − 10
f
18(x)
MASO
2.22E − 14
1.37E − 13
8.96E − 13
2.02E − 13
ASO
4.71E − 14
9.18E − 14
1.75E − 13
3.96E − 14
FPA
1.169421
1.510639
2.147189
0.28967
BAT
12.84255
14.03792
16.05812
0.884501
PSO
3.803351
5.659908
7.923137
0.986783
CS
14.20253
16.00982
17.58788
1.084051
ACOR
3.56E − 07
8.04E − 07
1.54E − 06
3.01E − 07
CSA
6.75E − 07
2.1E − 05
5.99E − 05
1.93E − 05
f
19(x)
MASO
0
5E − 13
4.66E − 12
1.05E − 12
ASO
8.954632
16.01884
19.89918
3.492605
FPA
59.36055
76.74372
100.1136
10.82332
BAT
31.84028
84.16432
198.9923
34.74113
PSO
68.84252
109.8572
159.7136
22.65396
CS
65.6753
109.6626
154.4972
25.50229
ACOR
202.1726
219.1054
230.5434
8.423366
CSA
6.31E − 12
5.12E − 09
3.79E − 08
1.01E − 08
f
20(x)
MASO
0
1.46E − 15
8.99E − 15
1.99E − 15
ASO
0
0
0
0
FPA
0.053602
0.127282
0.226275
0.05216
BAT
30.91076
58.58257
119.8216
20.73163
PSO
0.744758
1.095692
1.310903
0.116757
CS
0.868257
3.707799
19.43772
4.708682
ACOR
8.4E − 10
0.00311
0.062197
0.013908
CSA
2.6E − 13
3.35E − 08
1.31E − 07
4.47E − 08
f
21(x)
MASO
1140.688
2142.009
2596.788
363.9108
ASO
4046.881
4920.749
5321.667
391.9594
FPA
2480.206
2843.365
3076.285
173.4922
BAT
3444.441
5378.728
7371.65
998.61
PSO
1906.464
4896.28
6944.173
1242.374
CS
3032.338
4125.281
5258.021
555.419
ACOR
2691.551
3918.742
5687.103
463.1189
CSA
7233.943
8295.476
9227.791
487.6725
The best results are shown in bold.Figure 4
Convergence rates for F1.Figure 5
Convergence rates for F2.Figure 6
Convergence rates for F3.Figure 7
Convergence rates for F4.Figure 8
Convergence rates for F5.Figure 9
Convergence rates for F6.Figure 10
Convergence rates for F7.Figure 11
Convergence rates for F8.Figure 12
Convergence rates for F9.Figure 13
Convergence rates for F10.Figure 14
Convergence rates for F11.Figure 15
Convergence rates for F12.Figure 16
Convergence rates for F13.Figure 17
Convergence rates for F14.Figure 18
Convergence rates for F15.Figure 19
Convergence rates for F16.Figure 20
Convergence rates for F17.Figure 21
Convergence rates for F18.Figure 22
Convergence rates for F19.Figure 23
Convergence rates for F20.Figure 24
Convergence rates for F21.The capacities of MASO in three kinds of functions will be analyzed based on the results of Table4 and Figures 4–24.The MASO has a strong overall performance for single mode functions F1–F6. The single mode functions are relatively smooth, with only one extreme point at the origin, so the MASO is easy to converge to global optimum. In F1, F2, F3, and F5, the minimum, average, maximum, and variance of MASO are the best among the all algorithms. In F4, except BAT, CS, and PSO, the minimum, average, maximum, and variance of other algorithms get the optimums. In F6, only MASO converges to the best minimum compared with other algorithms.In complex multimodal functions F7–F17, MASO generally performs better among the all algorithms. In F9, F10, F12, F13, F14, and F17, the minimum, average, maximum, and variance of MASO are the best among the all algorithms. In F7, F11, and F16, only MASO converges to the global optimum, compared with other algorithms. In F14, the performance of MASO is worse than ASO.In the rotated multimodal functions F18–F21, the performance of MASO is not significantly superior to other comparison algorithms. There are too many extreme points in the rotating multimodal function, so MASO needs a strong global exploration ability to converge to the global optimum. On the contrary, MASO needs to consider both local development and global search, so it cannot unilaterally maximize the global search capability. This will affect the performance of MASO in optimizing rotating multimodal functions. In F18, F19, and F21, only MASO converges to the global optimum compared with other algorithms. In F20, MASO and ASO converge to the global optimum. The stability of MASO is the best in F19, but in F18, F20, and F21, the stability of MASO is not good enough.


### 4.3. Effectiveness Analysis of Improved Strategies

ASO is improved in the following three aspects to obtain MASO: (1) The atomic population is initialized by the chaotic operator to make the population location more uniform so that the MASO does not fall into local optimum easily. (2) The dominant positions in the population are used more fully by the vaccine operator to accelerate the MASO convergence. (3) The vaccination probability Pv will be determined dynamically by the reinforcement learning mechanism so that Pv adapts the problem automatically in a more scientific and reasonable way. The MASO improved from the above three aspects has high search accuracy and convergence speed. The improvement effectiveness of MASO will be analyzed by theT test and rank sum test, respectively.


#### 4.3.1. TheT Test Analysis between MASO and Comparative Algorithms

Table5 lists the results of T test, where “+” means that MASO has obvious advantages when compared with other algorithms, “=” indicates that the T test results of MASO equals other algorithms, and “≈” means that the T test results of MASO are approximately equal to other algorithms.Table 5
The comparisons ofT test for f1∼f21.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.016614
+
0
0.290266
≈
1
0.023023
+
MASO/BAT
1
3.24E − 09
+
1
0
+
0
0.14638
≈
MASO/PSO
1
9.2E − 105
+
1
1.1E − 302
+
1
1.4E − 181
+
MASO/CS
1
0.000179
+
1
1.36E − 07
+
1
1.7E − 17
+
MASO/ACOR
0
0.094316
≈
0
0.273106
≈
0
0.09149
≈
MASO/CSA
0
0.304726
≈
0
0.259636
≈
0
0.310844
≈
MASO/ASO
1
0.000526
+
0
0.481441
≈
1
0.003881
+
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.004565
+
0
0.050501
≈
1
0.049963
+
MASO/BAT
1
8.7E − 204
+
1
1.45E − 09
+
1
5.04E − 28
+
MASO/PSO
1
1.7E − 183
+
1
2.5E − 128
+
1
6.04E − 80
+
MASO/CS
1
1.47E − 09
+
1
7.87E − 73
+
1
0.001168
+
MASO/ACOR
0
0.089476
≈
1
0.029733
+
0
0.111095
≈
MASO/CSA
0
0.303911
≈
1
0.009487
+
0
0.318903
≈
MASO/ASO
N/A
N/A
=
1
4.62E − 05
+
1
0.0 03612
+
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.28E − 11
+
1
0.011808
+
1
1.6E − 30
+
MASO/BAT
1
0
+
1
4.66E − 84
+
1
0
+
MASO/PSO
1
9.1E − 295
+
1
1.1E − 188
+
1
9.5E − 210
+
MASO/CS
1
8.9E − 104
+
1
5.45E − 09
+
1
1.03E − 47
+
MASO/ACOR
1
0.001019
+
0
0.079069
≈
1
1.21E − 68
+
MASO/CSA
0
0.193249
≈
0
0.249111
≈
1
1.75E − 09
+
MASO/ASO
1
4.02E − 04
+
0
0.134598
≈
1
0.005241
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
1.96E − 56
+
1
1.91E − 25
+
1
1.21E − 07
+
MASO/BAT
1
7.53E − 91
+
1
1.15E − 40
+
1
9.57E − 87
+
MASO/PSO
1
7E − 250
+
1
2.3E − 209
+
1
1.6E − 269
+
MASO/CS
1
1.51E − 46
+
1
3.31E − 08
+
1
1.6E − 278
+
MASO/ACOR
1
0.003855
+
1
7.06E − 20
+
1
0.006418
+
MASO/CSA
1
0.003995
+
0
0.128471
≈
0
0.278777
≈
MASO/ASO
1
1.88E − 05
+
1
0.006584
+
1
0.001213
+
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.66E − 57
+
1
6.46E − 32
+
1
0.041413
+
MASO/BAT
1
0
+
1
0
+
0
0.18106
≈
MASO/PSO
1
7.1E − 227
+
1
3E − 290
+
1
5.2E − 275
+
MASO/CS
1
9.9E − 189
+
1
9.78E − 36
+
1
0.010623
+
MASO/ACOR
1
1.34E − 17
+
1
1.93E − 69
+
1
0.024626
+
MASO/CSA
1
0.001114
+
1
1.65E − 16
+
0
0.165226
≈
MASO/ASO
1
0.009734
+
1
3.09E − 11
+
1
9.92E − 53
+
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
0
0.128943
≈
1
0.048392
+
1
4.89E − 15
+
MASO/BAT
0
0.272098
≈
1
0.045067
+
1
0
+
MASO/PSO
1
6.8E − 214
+
1
2.5E − 252
+
1
2.6E − 255
+
MASO/CS
0
0.266098
≈
0
0.276366
≈
1
5.9E − 177
+
MASO/ACOR
0
0.116175
≈
1
0.049686
+
1
0.001027
+
MASO/CSA
0
0.307291
≈
0
0.319772
≈
0
0.156388
≈
MASO/ASO
1
0.008508
+
1
3.3E − 04
+
0
0.206133
+
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
5.92E − 23
+
1
0.01143
+
1
5.7E − 21
+
MASO/BAT
1
0
+
1
8.67E − 78
+
1
3.82E − 62
+
MASO/PSO
1
1.1E − 178
+
1
1.8E − 185
+
1
1.05E − 80
+
MASO/CS
1
5.5E − 54
+
1
2.14E − 05
+
1
3.74E − 28
+
MASO/ACOR
1
7.36E − 78
+
0
0.097076
≈
1
7.62E − 51
+
MASO/CSA
1
3.03E − 52
+
0
0.237911
≈
1
2.72E − 86
+
MASO/ASO
1
0.000142
+
0
0.643876
≈
1
2.28E − 36
+In F5, F7, F9, F10, F11, F12, F13, F14, F18, F19, and F21, MASO is more prominent than the most comparative algorithms. In F1, F3, F4, F6, F8, F15, F17, and F20, although performance of MASO is not outstanding, but its advantages are still better than most comparative algorithms. In F2 and F16, the effect of MASO is almost the same as the corresponding comparison algorithm. The improvement of MASO is not significant enough in F2 and F16 because most algorithms can optimize the two functions easily. Except F2 and F16, the effect of MASO is obviously better than most other algorithms, so the improvement of MASO is evidently effective. Therefore, MASO is a successful improvement to ASO.


#### 4.3.2. Wilcoxon Rank Sum Test for Improved Strategies

MASO is obtained by using the chaotic initialization operator, vaccine operator, and reinforcement learning operator to ameliorate ASO. The ASO enhanced by the chaotic initialization operator is named ASO_chaos. The ASO enhanced by the vaccine operator is named ASO_vaccine. The ASO enhanced by the reinforcement learning operator is named ASO_rl.In this experiment, each algorithm runs 20 times independently in each benchmark function. The population number is 100. The evaluation times of ASO_vaccine and ASO_rl are2∗ 100 in each iteration, and evaluation times of ASO and ASO_chaos are 100 in each iteration. For each benchmark function, ASO_vaccine and ASO_rl will iterate 2500 times, and ASO and ASO_chaos will iterate 5000 times. So the four algorithms will be compared under the evaluation of 100∗ 5000 in each run.Table6 lists the statistical results with different strategies. Among the three improved methods, the improvement effect of ASO_chaos is relatively general, and its stability is not good too. ASO_vaccine and ASO_rl are slightly improved compared with the ASO, but the improvement effect is not significant. Generally, the minimums of ASO_vaccine and ASO_rl are at most several orders of magnitude lower than the minimum of ASO. Only the MASO algorithm, that is, enhanced by three operators, can achieve remarkable results.Table 6
Test statistical results with different strategies.
Functions
Algorithms
Min
Mean
Max
STD
f
1(x)
ASO
7.33E − 27
2.64E − 26
1.05E − 25
3.08E − 26
ASO_chaos
2.69E − 27
38220.86
134747.6
49900.47
ASO_vaccine
9.34E − 27
3.35E − 26
6.9E − 26
2.51E − 26
ASO_rl
5.59E − 27
4.05E − 26
9.19E − 26
3.97E − 26
f
2(x)
ASO
2.16E − 13
5.32E − 13
9.67E − 13
2.51E − 13
ASO_chaos
2.99E − 13
1.58E + 18
2.84E + 19
6.35E + 18
ASO_vaccine
5.68E − 13
7.88E − 13
1.01E − 12
1.6E − 13
ASO_rl
3.18E − 13
1.25E − 12
2.35E − 12
7.73E − 13
f
3(x)
ASO
5.73E − 26
1.73E − 25
3.92E − 25
1.03E − 25
ASO_chaos
3.79E − 26
5651.508
17487.91
7310.399
ASO_vaccine
8.75E − 26
2.01E − 25
5.32E − 25
1.86E − 25
ASO_rl
1.23E − 25
3.62E − 25
5.3E − 25
1.63E − 25
f
4(x)
ASO
0
0
0
0
ASO_chaos
0
42588.7
116706
53955.46
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
5(x)
ASO
0.003764
0.00772
0.011135
0.002189
ASO_chaos
0.003324
72.91083
301.5848
100.3357
ASO_vaccine
0.003663
0.005553
0.008711
0.00202
ASO_rl
0.024793
0.032927
0.046203
0.010069
f
6(x)
ASO
2.19E − 52
1.46E − 51
5.24E − 51
1.51E − 51
ASO_chaos
2.34E − 53
94.61799
345.6983
134.9649
ASO_vaccine
1.02E − 51
3.62E − 51
8.42E − 51
3.45E − 51
ASO_rl
5.22E − 52
9.87E − 51
3.93E − 50
1.67E − 50
f
7(x)
ASO
4.35E − 14
7.41E − 14
1.47E − 13
2.97E − 14
ASO_chaos
5.42E − 14
8.462759
21.3434
10.63432
ASO_vaccine
6.84E − 14
9.25E − 14
1.29E − 13
2.22E − 14
ASO_rl
7.55E − 14
1.02E − 13
1.82E − 13
4.54E − 14
f
8(x)
ASO
0
0
0
0
ASO_chaos
0
344.9339
956.8887
437.1594
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
9(x)
ASO
9.949591
14.5264
20.89413
3.356089
ASO_chaos
7.959672
230.112
648.2703
274.7404
ASO_vaccine
11.93951
16.11833
20.89413
4.065958
ASO_rl
7.959672
12.33749
14.92438
2.68823
f
10(x)
ASO
4106.132
4650.915
5626.077
521.8557
ASO_chaos
4312.996
5089.873
5718.976
566.3842
ASO_vaccine
4500.827
5160.146
5625.985
471.8367
ASO_rl
3928.42
4852.266
5823.46
748.2354
f
11(x)
ASO
4.23E − 14
8.76E − 14
1.97E − 13
4.72E − 14
ASO_chaos
2.55E − 14
37.10606
116.9176
47.21336
ASO_vaccine
7.52E − 14
1.12E − 13
1.6E − 13
3.61E − 14
ASO_rl
7.19E − 14
1.3E − 13
2.7E − 13
8.45E − 14
f
12(x)
ASO
2.78E − 27
6.39E − 27
2.1E − 26
5.59E − 27
ASO_chaos
2.46E − 27
172.4654
562.9697
222.5449
ASO_vaccine
2.06E − 27
1.86E − 26
6.71E − 26
2.73E − 26
ASO_rl
3.9E − 27
1.11E − 26
1.95E − 26
7.21E − 27
f
13(x)
ASO
0.111929
0.171566
0.247934
0.04255
ASO_chaos
0.006224
0.221173
0.5
0.239022
ASO_vaccine
0.18622
0.268834
0.353136
0.072799
ASO_rl
0.094156
0.109595
0.122451
0.010779
f
14(x)
ASO
12
18.70114
35.0005
6.254622
ASO_chaos
9
226.4739
622.2764
263.4471
ASO_vaccine
16.08234
23.04624
27.001
4.654738
ASO_rl
13
23
30
6.892024
f
15(x)
ASO
13.22012
14.72793
15.51416
0.63515
ASO_chaos
14.05125
1691798
5141408
2180894
ASO_vaccine
15.5044
15.89732
16.17919
0.26884
ASO_rl
15.80763
16.45844
17.09231
0.562882
f
16(x)
ASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
ASO_chaos
2.17E − 29
2.2E + 09
8.17E + 09
2.87E + 09
ASO_vaccine
3.97E − 29
1.23E − 28
1.94E − 28
6.21E − 29
ASO_rl
3.35E − 29
1.49E − 28
4.22E − 28
1.59E − 28
MASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
f
17(x)
ASO
6.55E − 28
2.14E − 27
5.54E − 27
1.53E − 27
ASO_chaos
6.04E − 28
1.47E + 09
5.17E + 09
1.96E + 09
ASO_vaccine
7.69E − 28
2.92E − 27
7.94E − 27
2.89E − 27
ASO_rl
7.23E − 28
2.19E − 27
5.26E − 27
1.84E − 27
f
18(x)
ASO
4.71E − 14
9.18E − 14
1.75E − 13
3.96E − 14
ASO_chaos
5.06E − 14
8.510808
21.42481
10.69466
ASO_vaccine
6.13E − 14
1.59E − 13
4.34E − 13
1.55E − 13
ASO_rl
1.04E − 13
1.34E − 13
1.82E − 13
2.95E − 14
f
19(x)
ASO
8.954632
16.01884
19.89918
3.492605
ASO_chaos
9.949591
226.8982
594.1118
268.2596
ASO_vaccine
11.9395
22.28707
31.83867
7.629455
ASO_rl
9.949591
14.5264
19.89917
4.750859
f
20(x)
ASO
0
0
0
0
ASO_chaos
0
375.431
1145.34
482.6689
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
21(x)
ASO
4046.881
4920.749
5321.667
391.9594
ASO_chaos
4561.110
5235.886
3421.026
339.6557
ASO_vaccine
4817.261
5210.066
5626.202
326.6599
ASO_rl
4540.61
5634.291
6731.833
945.2253Table7 displays the statistical results of the Wilcoxon rank sum test, where “+” means that MASO has obvious advantages when compared with other algorithms and “≈” means that the Wilcoxon rank sum test results of MASO approximately equal to other algorithms. In F5, F9, F10, F13, F14, F19, and F21, ASO_chaos, ASO_vaccine, and ASO_rl are more effective than the ASO. The ASO_chaos has better performance than ASO in F15. The improved effect of ASO_chaos, ASO_vaccine, and ASO_rl is not significant in functions F1–F4, F6–F8, F11–F12, F16–F18, and F20. The effect of MASO is significantly better than ASO and comparative algorithms in most benchmark functions. Therefore, the remarkable effect of MASO can only be achieved by using three operators together.Table 7
Test statistical results of the Wilcoxon rank sum test.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.314619
≈
0
0.324488
≈
0
0.328083
≈
ASO_vaccine/ASO
0
0.488314
≈
0
0.641407
≈
0
0.195942
≈
ASO_rl/ASO
0
0.701861
≈
0
0.925725
≈
0
0.920272
≈
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.244594
≈
0
0.062368
≈
0
0.701051
≈
ASO_vaccine/ASO
0
0.553674
≈
1
0.030511
+
0
0.315137
≈
ASO_rl/ASO
0
0.893699
≈
1
0.000289
+
0
0.706156
≈
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.321172
≈
0
0.307772
≈
1
0.000331
+
ASO_vaccine/ASO
0
0.41769
≈
0
0.457487
≈
1
0.004432
+
ASO_rl/ASO
0
0.801112
≈
0
0.636002
≈
1
0.03666
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
8.67E − 08
+
0
0.301404
≈
0
0.321245
≈
ASO_vaccine/ASO
1
7.45E − 07
+
0
0.205611
≈
0
0.231387
≈
ASO_rl/ASO
1
1.61E − 07
+
0
0.898369
≈
0
0.914785
≈
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
2.53E − 18
+
1
0.018586
+
1
0.02315
+
ASO_vaccine/ASO
1
8.18E − 05
+
0
0.064058
≈
0
0.479393
≈
ASO_rl/ASO
1
0.000381
+
1
0.000728
+
0
0.075423
≈
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.449624
≈
0
0.385929
≈
0
0.311159
≈
ASO_vaccine/ASO
0
0.462677
≈
0
0.458521
≈
0
0.555353
≈
ASO_rl/ASO
0
0.769354
≈
0
0.753607
≈
0
0.8118
≈
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
0.000516
+
0
0.389344
≈
1
4.02E − 07
+
ASO_vaccine/ASO
1
0.003634
+
0
0.412808
≈
1
1.92E − 08
+
ASO_rl/ASO
1
0.013797
+
0
0.636028
≈
1
6.05E − 07
+


## 4.1. Test Platform and Benchmark Functions

### 4.1.1. Test Platform

The server model is the dawning 5000 A server. The server configures Xeon X5620 CPU (4 cores)∗2, 24 GB memory, and 300 GB SAS hard disk. The server is equipped with Rhel5.6 operating system. The programming tool is MATLAB 2012a (for Linux).


### 4.1.2. Benchmark Functions

21 benchmark functions are selected to evaluate the algorithm performance. This set of benchmark functions has widely been used to evaluate the ability of the swarm intelligence algorithm [38]. The number, range, type, and formulation of functions are listed in Table 2. The 21 benchmark functions can be divided into three categories. F1∼F6 are the unimodal functions (U) and are applied to investigate the search convergence accuracy. F7–F17 are the complex multimodal functions (M) with multiple local extremum points. These functions are employed to test the global search capability and the ability to avoid prematurity. F18–F21 are the rotating multimodal functions (R). These functions have more extreme points, thus increasing the difficulty of searching. F18–F21 can be adopted to further detect the overall performance of the algorithm.Table 2
The benchmark functions.
No.
Range
Type
Formulation
F1
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
2
F2
[−10, 10]
U
f
x
=
∑
i
=
1
d
x
i
+
∏
i
=
1
d
x
i
F3
[−10, 10]
U
f
x
=
∑
i
=
1
d
i
x
i
2
F4
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
+
0.5
2
F5
[−1.28, 0.64]
U
f
x
=
∑
i
=
1
d
i
x
i
4
+
R
F6
[−1.28, 1.28]
U
f
x
=
∑
i
=
1
d
i
x
i
4
F7
[−32, 32]
M
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
x
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
x
i
+
20
+
e
F8
[−600, 600]
M
f
x
=
1
/
4000
⋅
∑
i
=
1
d
x
i
2
−
∏
i
=
1
d
cos
x
i
/
i
+
1
F9
[−5.12, 5.12]
M
f
x
=
10
∗
d
+
∑
i
=
1
d
x
i
2
−
10
cos
2
π
x
i
F10
[−500, 500]
M
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
x
i
sin
x
i
F11
[−10, 10]
M
f
x
=
∑
i
=
1
d
x
i
⋅
sin
x
i
+
0.1
⋅
x
i
F12
[−10, 10]
M
f
x
=
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
sin
2
3
π
x
1
F13
[−100, 100]
M
f
x
=
0.5
+
sin
2
∑
i
=
1
d
x
i
2
−
0.5
/
1
+
0.001
∑
i
=
1
d
x
i
2
2
+
x
n
−
1
1
+
sin
2
3
π
x
n
F14
[−5.12, 5.12]
M
f
x
=
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
+
10
,
y
i
=
x
i
,
x
i
<
1
/
2
round
2
x
i
/
2
,
x
i
≥
1
/
2
F15
[−5, 10]
M
f
x
=
∑
i
=
1
d
−
1
1
−
x
i
2
+
100
x
i
+
1
−
x
i
2
2
F16
[−50, 50]
M
f
x
=
π
/
d
10
sin
2
π
z
1
+
∑
i
=
1
d
−
1
z
i
−
1
2
1
+
10
sin
2
π
z
i
+
1
+
z
d
−
1
2
+
∑
i
=
1
d
u
x
i
,
10,100,4
,
z
i
=
1
+
1
/
4
x
i
+
1
,
u
x
i
,
a
,
k
,
m
=
k
x
i
−
a
m
,
x
i
>
aor
x
i
<
−
a
0
,
otherwise
F17
[−50, 50]
M
f
x
=
1
/
d
sin
2
π
x
1
+
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
x
d
−
1
2
1
+
sin
2
2
π
x
i
+
1
+
∑
i
=
1
d
u
x
i
,
5,100,4
F18
[−32, 32]
R
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
y
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
y
i
+
20
+
e
F19
[−5.12, 5.12]
R
f
x
=
10
∗
d
+
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
F20
[−600, 600]
R
f
x
=
1
/
4000
⋅
∑
i
=
1
d
y
i
2
−
∏
i
=
1
d
cos
y
i
/
i
+
1
F21
[−500, 500]
R
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
y
i
sin
y
i
U: unimodel, M: multimodel, and R: rotated.


## 4.1.1. Test Platform

The server model is the dawning 5000 A server. The server configures Xeon X5620 CPU (4 cores)∗2, 24 GB memory, and 300 GB SAS hard disk. The server is equipped with Rhel5.6 operating system. The programming tool is MATLAB 2012a (for Linux).


## 4.1.2. Benchmark Functions

21 benchmark functions are selected to evaluate the algorithm performance. This set of benchmark functions has widely been used to evaluate the ability of the swarm intelligence algorithm [38]. The number, range, type, and formulation of functions are listed in Table 2. The 21 benchmark functions can be divided into three categories. F1∼F6 are the unimodal functions (U) and are applied to investigate the search convergence accuracy. F7–F17 are the complex multimodal functions (M) with multiple local extremum points. These functions are employed to test the global search capability and the ability to avoid prematurity. F18–F21 are the rotating multimodal functions (R). These functions have more extreme points, thus increasing the difficulty of searching. F18–F21 can be adopted to further detect the overall performance of the algorithm.Table 2
The benchmark functions.
No.
Range
Type
Formulation
F1
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
2
F2
[−10, 10]
U
f
x
=
∑
i
=
1
d
x
i
+
∏
i
=
1
d
x
i
F3
[−10, 10]
U
f
x
=
∑
i
=
1
d
i
x
i
2
F4
[−100, 100]
U
f
x
=
∑
i
=
1
d
x
i
+
0.5
2
F5
[−1.28, 0.64]
U
f
x
=
∑
i
=
1
d
i
x
i
4
+
R
F6
[−1.28, 1.28]
U
f
x
=
∑
i
=
1
d
i
x
i
4
F7
[−32, 32]
M
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
x
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
x
i
+
20
+
e
F8
[−600, 600]
M
f
x
=
1
/
4000
⋅
∑
i
=
1
d
x
i
2
−
∏
i
=
1
d
cos
x
i
/
i
+
1
F9
[−5.12, 5.12]
M
f
x
=
10
∗
d
+
∑
i
=
1
d
x
i
2
−
10
cos
2
π
x
i
F10
[−500, 500]
M
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
x
i
sin
x
i
F11
[−10, 10]
M
f
x
=
∑
i
=
1
d
x
i
⋅
sin
x
i
+
0.1
⋅
x
i
F12
[−10, 10]
M
f
x
=
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
sin
2
3
π
x
1
F13
[−100, 100]
M
f
x
=
0.5
+
sin
2
∑
i
=
1
d
x
i
2
−
0.5
/
1
+
0.001
∑
i
=
1
d
x
i
2
2
+
x
n
−
1
1
+
sin
2
3
π
x
n
F14
[−5.12, 5.12]
M
f
x
=
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
+
10
,
y
i
=
x
i
,
x
i
<
1
/
2
round
2
x
i
/
2
,
x
i
≥
1
/
2
F15
[−5, 10]
M
f
x
=
∑
i
=
1
d
−
1
1
−
x
i
2
+
100
x
i
+
1
−
x
i
2
2
F16
[−50, 50]
M
f
x
=
π
/
d
10
sin
2
π
z
1
+
∑
i
=
1
d
−
1
z
i
−
1
2
1
+
10
sin
2
π
z
i
+
1
+
z
d
−
1
2
+
∑
i
=
1
d
u
x
i
,
10,100,4
,
z
i
=
1
+
1
/
4
x
i
+
1
,
u
x
i
,
a
,
k
,
m
=
k
x
i
−
a
m
,
x
i
>
aor
x
i
<
−
a
0
,
otherwise
F17
[−50, 50]
M
f
x
=
1
/
d
sin
2
π
x
1
+
∑
i
=
1
d
−
1
x
i
−
1
2
1
+
sin
2
3
π
x
i
+
1
+
x
d
−
1
2
1
+
sin
2
2
π
x
i
+
1
+
∑
i
=
1
d
u
x
i
,
5,100,4
F18
[−32, 32]
R
f
x
=
−
20
exp
−
0.2
1
/
d
∑
i
=
1
d
y
i
2
−
exp
1
/
d
∑
i
=
1
d
cos
2
π
y
i
+
20
+
e
F19
[−5.12, 5.12]
R
f
x
=
10
∗
d
+
∑
i
=
1
d
y
i
2
−
10
cos
2
π
y
i
F20
[−600, 600]
R
f
x
=
1
/
4000
⋅
∑
i
=
1
d
y
i
2
−
∏
i
=
1
d
cos
y
i
/
i
+
1
F21
[−500, 500]
R
f
x
=
418.9829
∗
d
+
∑
i
=
1
d
−
y
i
sin
y
i
U: unimodel, M: multimodel, and R: rotated.


## 4.2. Experimental Results Analysis of MASO and Comparison Algorithms

In this experiment, the parameters setting of the comparison algorithms are displayed in Table3. For each benchmark function, each algorithm runs 20 times independently. The population number is 100. The evaluation times of MASO and CS are 2∗ 100 in each iteration, and evaluation times of other algorithms are 100 in each iteration. For each benchmark function, MASO and CS will iterate 2500 times and other algorithms will iterate 5000 times. So all algorithms will be compared under the evaluations of 100∗ 5000 in each run. The various performances of algorithms are measured by the four indexes of optimal value, average value, worst value, and standard deviation. The experimental statistical results are shown in Table 4. Take 500 points to draw the convergence curve. MASO and CS take one point every 50 iterations, and other algorithms take one point every 100 iterations. Figures 4–24 are the convergence curves of each function successively.Table 3
The parameters set of contrast algorithms.
Algorithm
Parameters
MASO
N = 100; depth = 50; multiplier = 0.2; Pv = 0.1; w=0.05.
ASO
N = 100; depth = 50; multiplier = 0.2.
FPA
N = 100; P = 0.8.
BAT
N = 100; A = 0.5; r = 0.5; Qmin = 0; Qmax = 2.
PSO
N = 100; c1 = 1.49445; c2 = 1.49445.
CS
N = 100; pa = 0.25.
ACOR
N = 100; nSample = 50; q = 0.5; zeta = 1.
CSA
N = 100; AP = 0.2; fl = 2.Table 4
Test statistical results of functionsf1∼f21.
Functions
Algorithms
Min
Mean
Max
STD
f
1(x)
MASO
2.99E − 45
1.82E − 40
3.5E − 39
7.82E − 40
ASO
7.33E − 27
2.64E − 26
1.05E − 25
3.08E − 26
FPA
0.006587
0.013937
0.023855
0.004844
BAT
1.09E − 05
155.0339
891.9841
257.7465
PSO
3.869551
20.12067
77.25433
16.40947
CS
0.004142
173.506
1688.332
391.8062
ACOR
6.98E − 13
2.01E − 12
3.85E − 12
7.74E − 13
CSA
1.29E − 12
8.89E − 09
7.83E − 08
1.9E − 08
f
2(x)
MASO
2.41E − 34
1.63E − 29
1.34E − 28
3.9E − 29
ASO
2.16E − 13
5.32E − 13
9.67E − 13
2.51E − 13
FPA
0.677651
1.159856
1.89272
0.340836
BAT
0.014345
42.11026
124.5488
48.56557
PSO
6.26162
9.511573
16.34641
2.636531
CS
0.024397
1.543266
8.754166
2.237715
ACOR
3.46E − 07
0.000387
0.006348
0.001411
CSA
9.9E − 07
4.2E − 05
0.000133
3.52E − 05
f
3(x)
MASO
1.5E − 47
5.73E − 42
8.19E − 41
1.87E − 41
ASO
5.73E − 26
1.73E − 25
3.92E − 25
1.03E − 25
FPA
0.000957
0.001794
0.002651
0.000518
BAT
0.000105
0.000156
0.000193
2.85E − 05
PSO
1.681533
8.073872
24.85504
6.662011
CS
0.138672
63.08822
290.7739
80.09947
ACOR
7.82E − 14
2.16E − 13
4.42E − 13
1.09E − 13
CSA
1.27E − 12
1.63E − 09
1.89E − 08
4.14E − 09
f
4(x)
MASO
0
0
0
0
ASO
0
0
0
0
FPA
0
0
0
0
BAT
4214
10180.7
16319
3108.576
PSO
57
191.7
405
90.82435
CS
9
1558.45
6328
1791.922
ACOR
0
0
0
0
CSA
0
0
0
0
f
5(x)
MASO
2.34E − 06
1.47E − 05
3.73E − 05
9.24E − 06
ASO
0.003764
0.00772
0.011135
0.002189
FPA
0.006508
0.020989
0.029649
0.005437
BAT
0.218579
0.403253
0.531159
0.080489
PSO
0.273095
0.62667
1.206052
0.285222
CS
0.94235
4.144344
13.32052
2.939838
ACOR
0.016274
0.032401
0.053185
0.009913
CSA
3.99E − 06
5.29E − 05
0.000164
4.22E − 05
f
6(x)
MASO
2.41E − 60
6.82E − 51
1.26E − 49
2.82E − 50
ASO
2.19E − 52
1.46E − 51
5.24E − 51
1.51E − 51
FPA
3.28E − 11
3.39E − 10
1.43E − 09
3.33E − 10
BAT
6.29E − 11
1.02E − 10
1.45E − 10
2.17E − 11
PSO
0.001601
0.010088
0.038612
0.010173
CS
0.000168
0.34018
2.726353
0.655415
ACOR
2.22E − 14
2.49E − 13
1.82E − 12
3.95E − 13
CSA
3.56E − 31
1.06E − 23
1.02E − 22
2.42E − 23
f
7(x)
MASO
2.22E − 14
3.83E − 13
5.36E − 12
1.18E − 12
ASO
4.35E − 14
7.41E − 14
1.47E − 13
2.97E − 14
FPA
0.17994
0.618946
1.286978
0.260178
BAT
10.99074
13.41295
19.95889
1.88331
PSO
3.213753
5.385464
7.818361
1.282747
CS
8.631765
13.19331
16.18969
1.882678
ACOR
2.3E − 07
4.2E − 07
5.56E − 07
1.07E − 07
CSA
2.84E − 07
1.95E − 05
4.98E − 05
1.54E − 05
f
8(x)
MASO
1.11E − 16
1.65E − 15
5.66E − 15
1.65E − 15
ASO
0
0
0
0
FPA
0.086888
0.136963
0.230776
0.044916
BAT
19.19178
53.02061
94.33961
21.77808
PSO
0.939324
1.178857
1.343306
0.102008
CS
0.610131
4.621765
33.27042
8.201352
ACOR
4.88E − 12
0.022291
0.247193
0.069061
CSA
3.57E − 10
4.25E − 08
4.24E − 07
9.87E − 08
f
9(x)
MASO
0
4.97E − 13
3.18E − 12
9.06E − 13
ASO
9.949591
14.5264
20.89413
3.356089
FPA
53.45329
72.74809
84.86339
8.314181
BAT
38.80531
79.29986
169.1439
35.18297
PSO
75.51275
110.7482
169.454
25.42386
CS
30.82795
51.57561
84.51111
14.55878
ACOR
183.2672
214.6805
237.272
12.35372
CSA
1.76E − 12
2.82E − 09
7.92E − 09
2.7E − 09
f
10(x)
MASO
0.000382
0.000382
0.000382
1.35E − 11
ASO
4106.132
4650.915
5626.077
521.8557
FPA
3244.947
3698.782
4219.299
237.132
BAT
5972.237
7099.381
8123.466
576.3557
PSO
3897.536
5375.416
6743.524
814.7786
CS
1848.715
2793.954
4026.17
610.2232
ACOR
1478.361
5880.244
7607.56
360.132
CSA
0.000382
0.000382
0.000382
7E −08
f
11(x)
MASO
3.44E − 33
1.83E − 05
9.49E − 05
3.11E − 05
ASO
4.23E − 14
8.76E − 14
1.97E − 13
4.72E − 14
FPA
3.227243
6.669646
9.146383
1.526006
BAT
0.834363
3.926399
9.459001
2.576305
PSO
2.748819
6.186047
10.35792
2.432746
CS
0.039447
0.854302
2.652989
0.811748
ACOR
0.034574
0.302129
5.21918
1.157372
CSA
7.64E − 08
5.4E − 06
2.31E − 05
5.86E − 06
f
12(x)
MASO
1.15E − 31
6.29E − 28
1.18E − 26
2.63E − 27
ASO
2.78E − 27
6.39E − 27
2.1E − 26
5.59E − 27
FPA
1.055809
2.341216
3.399438
0.776936
BAT
3.166887
8.757063
15.72329
3.538368
PSO
1.089431
7.019646
16.18996
3.659522
CS
9.73E − 09
0.005653
0.104223
0.02323
ACOR
1.69E − 11
9.04E − 11
4.43E − 10
9.82E − 11
CSA
1.24E − 13
5.21E − 11
4.53E − 10
1.13E − 10
f
13(x)
MASO
2.22E − 16
9.1E − 16
5.33E − 15
1.14E − 15
ASO
0.111929
0.171566
0.247934
0.04255
FPA
0.164183
0.23541
0.288289
0.035616
BAT
0.493711
0.498347
0.499585
0.001342
PSO
0.213232
0.336072
0.442089
0.05312
CS
0.470226
0.493116
0.498482
0.006452
ACOR
0.018867
0.036172
0.046726
0.006468
CSA
1.33E − 15
4.72E − 12
2.33E − 11
6.56E − 12
f
14(x)
MASO
1.78E − 15
1.44E − 14
4.09E − 14
1.04E − 14
ASO
12
18.70114
35.0005
6.254622
FPA
58.21314
75.72416
92.27162
10.98064
BAT
142.0001
225.425
320.25
43.94807
PSO
51.61432
97.72042
154.1477
27.68505
CS
22.03245
65.54915
124.1758
23.27628
ACOR
159.1282
193.0334
215.8543
15.42857
CSA
5.74E − 11
5.9E − 09
2.73E − 08
8.01E − 09
f
15(x)
MASO
9.9E − 13
7.77E −09
8.32E −08
2E −08
ASO
13.22012
14.72793
15.51416
0.63515
FPA
23.19234
26.9907
29.18676
1.618782
BAT
0.095172
18.58866
73.64575
30.2613
PSO
305.1689
574.6557
1477.057
316.5058
CS
92.87089
4700.287
44298.9
10461.76
ACOR
24.81962
25.10222
25.36799
0.158517
CSA
2.24E − 11
1.35E − 09
5.65E − 09
1.97E − 09
f
16(x)
MASO
3.12E − 32
1.12E − 29
2.02E − 28
4.51E − 29
ASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
FPA
0.795144
1.715761
2.908752
0.550818
BAT
0.519122
11.97827
28.55629
9.510974
PSO
3.03539
7.040329
11.93878
2.719544
CS
7.55E − 08
0.019757
0.366427
0.081707
ACOR
9.06E − 09
2.61E − 07
2.1E − 06
4.61E − 07
CSA
2.13E − 14
3.58E − 11
1.54E − 10
4.63E − 11
f
17(x)
MASO
1.26E − 32
2.26E − 31
1.86E − 30
4.61E − 31
ASO
6.55E − 28
2.14E − 27
5.54E − 27
1.53E − 27
FPA
0.075634
0.159202
0.383942
0.076028
BAT
28.02352
59.78051
75.72239
10.54314
PSO
3.843779
71.81837
922.9239
200.6675
CS
3.36E − 22
8.9E − 22
1.77E − 21
4.31E − 22
ACOR
3.17E − 08
8.81E − 07
6.2E − 06
1.69E − 06
CSA
9.23E − 13
2.67E − 10
8.43E − 10
2.34E − 10
f
18(x)
MASO
2.22E − 14
1.37E − 13
8.96E − 13
2.02E − 13
ASO
4.71E − 14
9.18E − 14
1.75E − 13
3.96E − 14
FPA
1.169421
1.510639
2.147189
0.28967
BAT
12.84255
14.03792
16.05812
0.884501
PSO
3.803351
5.659908
7.923137
0.986783
CS
14.20253
16.00982
17.58788
1.084051
ACOR
3.56E − 07
8.04E − 07
1.54E − 06
3.01E − 07
CSA
6.75E − 07
2.1E − 05
5.99E − 05
1.93E − 05
f
19(x)
MASO
0
5E − 13
4.66E − 12
1.05E − 12
ASO
8.954632
16.01884
19.89918
3.492605
FPA
59.36055
76.74372
100.1136
10.82332
BAT
31.84028
84.16432
198.9923
34.74113
PSO
68.84252
109.8572
159.7136
22.65396
CS
65.6753
109.6626
154.4972
25.50229
ACOR
202.1726
219.1054
230.5434
8.423366
CSA
6.31E − 12
5.12E − 09
3.79E − 08
1.01E − 08
f
20(x)
MASO
0
1.46E − 15
8.99E − 15
1.99E − 15
ASO
0
0
0
0
FPA
0.053602
0.127282
0.226275
0.05216
BAT
30.91076
58.58257
119.8216
20.73163
PSO
0.744758
1.095692
1.310903
0.116757
CS
0.868257
3.707799
19.43772
4.708682
ACOR
8.4E − 10
0.00311
0.062197
0.013908
CSA
2.6E − 13
3.35E − 08
1.31E − 07
4.47E − 08
f
21(x)
MASO
1140.688
2142.009
2596.788
363.9108
ASO
4046.881
4920.749
5321.667
391.9594
FPA
2480.206
2843.365
3076.285
173.4922
BAT
3444.441
5378.728
7371.65
998.61
PSO
1906.464
4896.28
6944.173
1242.374
CS
3032.338
4125.281
5258.021
555.419
ACOR
2691.551
3918.742
5687.103
463.1189
CSA
7233.943
8295.476
9227.791
487.6725
The best results are shown in bold.Figure 4
Convergence rates for F1.Figure 5
Convergence rates for F2.Figure 6
Convergence rates for F3.Figure 7
Convergence rates for F4.Figure 8
Convergence rates for F5.Figure 9
Convergence rates for F6.Figure 10
Convergence rates for F7.Figure 11
Convergence rates for F8.Figure 12
Convergence rates for F9.Figure 13
Convergence rates for F10.Figure 14
Convergence rates for F11.Figure 15
Convergence rates for F12.Figure 16
Convergence rates for F13.Figure 17
Convergence rates for F14.Figure 18
Convergence rates for F15.Figure 19
Convergence rates for F16.Figure 20
Convergence rates for F17.Figure 21
Convergence rates for F18.Figure 22
Convergence rates for F19.Figure 23
Convergence rates for F20.Figure 24
Convergence rates for F21.The capacities of MASO in three kinds of functions will be analyzed based on the results of Table4 and Figures 4–24.The MASO has a strong overall performance for single mode functions F1–F6. The single mode functions are relatively smooth, with only one extreme point at the origin, so the MASO is easy to converge to global optimum. In F1, F2, F3, and F5, the minimum, average, maximum, and variance of MASO are the best among the all algorithms. In F4, except BAT, CS, and PSO, the minimum, average, maximum, and variance of other algorithms get the optimums. In F6, only MASO converges to the best minimum compared with other algorithms.In complex multimodal functions F7–F17, MASO generally performs better among the all algorithms. In F9, F10, F12, F13, F14, and F17, the minimum, average, maximum, and variance of MASO are the best among the all algorithms. In F7, F11, and F16, only MASO converges to the global optimum, compared with other algorithms. In F14, the performance of MASO is worse than ASO.In the rotated multimodal functions F18–F21, the performance of MASO is not significantly superior to other comparison algorithms. There are too many extreme points in the rotating multimodal function, so MASO needs a strong global exploration ability to converge to the global optimum. On the contrary, MASO needs to consider both local development and global search, so it cannot unilaterally maximize the global search capability. This will affect the performance of MASO in optimizing rotating multimodal functions. In F18, F19, and F21, only MASO converges to the global optimum compared with other algorithms. In F20, MASO and ASO converge to the global optimum. The stability of MASO is the best in F19, but in F18, F20, and F21, the stability of MASO is not good enough.


## 4.3. Effectiveness Analysis of Improved Strategies

ASO is improved in the following three aspects to obtain MASO: (1) The atomic population is initialized by the chaotic operator to make the population location more uniform so that the MASO does not fall into local optimum easily. (2) The dominant positions in the population are used more fully by the vaccine operator to accelerate the MASO convergence. (3) The vaccination probability Pv will be determined dynamically by the reinforcement learning mechanism so that Pv adapts the problem automatically in a more scientific and reasonable way. The MASO improved from the above three aspects has high search accuracy and convergence speed. The improvement effectiveness of MASO will be analyzed by theT test and rank sum test, respectively.


### 4.3.1. TheT Test Analysis between MASO and Comparative Algorithms

Table5 lists the results of T test, where “+” means that MASO has obvious advantages when compared with other algorithms, “=” indicates that the T test results of MASO equals other algorithms, and “≈” means that the T test results of MASO are approximately equal to other algorithms.Table 5
The comparisons ofT test for f1∼f21.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.016614
+
0
0.290266
≈
1
0.023023
+
MASO/BAT
1
3.24E − 09
+
1
0
+
0
0.14638
≈
MASO/PSO
1
9.2E − 105
+
1
1.1E − 302
+
1
1.4E − 181
+
MASO/CS
1
0.000179
+
1
1.36E − 07
+
1
1.7E − 17
+
MASO/ACOR
0
0.094316
≈
0
0.273106
≈
0
0.09149
≈
MASO/CSA
0
0.304726
≈
0
0.259636
≈
0
0.310844
≈
MASO/ASO
1
0.000526
+
0
0.481441
≈
1
0.003881
+
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.004565
+
0
0.050501
≈
1
0.049963
+
MASO/BAT
1
8.7E − 204
+
1
1.45E − 09
+
1
5.04E − 28
+
MASO/PSO
1
1.7E − 183
+
1
2.5E − 128
+
1
6.04E − 80
+
MASO/CS
1
1.47E − 09
+
1
7.87E − 73
+
1
0.001168
+
MASO/ACOR
0
0.089476
≈
1
0.029733
+
0
0.111095
≈
MASO/CSA
0
0.303911
≈
1
0.009487
+
0
0.318903
≈
MASO/ASO
N/A
N/A
=
1
4.62E − 05
+
1
0.0 03612
+
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.28E − 11
+
1
0.011808
+
1
1.6E − 30
+
MASO/BAT
1
0
+
1
4.66E − 84
+
1
0
+
MASO/PSO
1
9.1E − 295
+
1
1.1E − 188
+
1
9.5E − 210
+
MASO/CS
1
8.9E − 104
+
1
5.45E − 09
+
1
1.03E − 47
+
MASO/ACOR
1
0.001019
+
0
0.079069
≈
1
1.21E − 68
+
MASO/CSA
0
0.193249
≈
0
0.249111
≈
1
1.75E − 09
+
MASO/ASO
1
4.02E − 04
+
0
0.134598
≈
1
0.005241
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
1.96E − 56
+
1
1.91E − 25
+
1
1.21E − 07
+
MASO/BAT
1
7.53E − 91
+
1
1.15E − 40
+
1
9.57E − 87
+
MASO/PSO
1
7E − 250
+
1
2.3E − 209
+
1
1.6E − 269
+
MASO/CS
1
1.51E − 46
+
1
3.31E − 08
+
1
1.6E − 278
+
MASO/ACOR
1
0.003855
+
1
7.06E − 20
+
1
0.006418
+
MASO/CSA
1
0.003995
+
0
0.128471
≈
0
0.278777
≈
MASO/ASO
1
1.88E − 05
+
1
0.006584
+
1
0.001213
+
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.66E − 57
+
1
6.46E − 32
+
1
0.041413
+
MASO/BAT
1
0
+
1
0
+
0
0.18106
≈
MASO/PSO
1
7.1E − 227
+
1
3E − 290
+
1
5.2E − 275
+
MASO/CS
1
9.9E − 189
+
1
9.78E − 36
+
1
0.010623
+
MASO/ACOR
1
1.34E − 17
+
1
1.93E − 69
+
1
0.024626
+
MASO/CSA
1
0.001114
+
1
1.65E − 16
+
0
0.165226
≈
MASO/ASO
1
0.009734
+
1
3.09E − 11
+
1
9.92E − 53
+
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
0
0.128943
≈
1
0.048392
+
1
4.89E − 15
+
MASO/BAT
0
0.272098
≈
1
0.045067
+
1
0
+
MASO/PSO
1
6.8E − 214
+
1
2.5E − 252
+
1
2.6E − 255
+
MASO/CS
0
0.266098
≈
0
0.276366
≈
1
5.9E − 177
+
MASO/ACOR
0
0.116175
≈
1
0.049686
+
1
0.001027
+
MASO/CSA
0
0.307291
≈
0
0.319772
≈
0
0.156388
≈
MASO/ASO
1
0.008508
+
1
3.3E − 04
+
0
0.206133
+
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
5.92E − 23
+
1
0.01143
+
1
5.7E − 21
+
MASO/BAT
1
0
+
1
8.67E − 78
+
1
3.82E − 62
+
MASO/PSO
1
1.1E − 178
+
1
1.8E − 185
+
1
1.05E − 80
+
MASO/CS
1
5.5E − 54
+
1
2.14E − 05
+
1
3.74E − 28
+
MASO/ACOR
1
7.36E − 78
+
0
0.097076
≈
1
7.62E − 51
+
MASO/CSA
1
3.03E − 52
+
0
0.237911
≈
1
2.72E − 86
+
MASO/ASO
1
0.000142
+
0
0.643876
≈
1
2.28E − 36
+In F5, F7, F9, F10, F11, F12, F13, F14, F18, F19, and F21, MASO is more prominent than the most comparative algorithms. In F1, F3, F4, F6, F8, F15, F17, and F20, although performance of MASO is not outstanding, but its advantages are still better than most comparative algorithms. In F2 and F16, the effect of MASO is almost the same as the corresponding comparison algorithm. The improvement of MASO is not significant enough in F2 and F16 because most algorithms can optimize the two functions easily. Except F2 and F16, the effect of MASO is obviously better than most other algorithms, so the improvement of MASO is evidently effective. Therefore, MASO is a successful improvement to ASO.


### 4.3.2. Wilcoxon Rank Sum Test for Improved Strategies

MASO is obtained by using the chaotic initialization operator, vaccine operator, and reinforcement learning operator to ameliorate ASO. The ASO enhanced by the chaotic initialization operator is named ASO_chaos. The ASO enhanced by the vaccine operator is named ASO_vaccine. The ASO enhanced by the reinforcement learning operator is named ASO_rl.In this experiment, each algorithm runs 20 times independently in each benchmark function. The population number is 100. The evaluation times of ASO_vaccine and ASO_rl are2∗ 100 in each iteration, and evaluation times of ASO and ASO_chaos are 100 in each iteration. For each benchmark function, ASO_vaccine and ASO_rl will iterate 2500 times, and ASO and ASO_chaos will iterate 5000 times. So the four algorithms will be compared under the evaluation of 100∗ 5000 in each run.Table6 lists the statistical results with different strategies. Among the three improved methods, the improvement effect of ASO_chaos is relatively general, and its stability is not good too. ASO_vaccine and ASO_rl are slightly improved compared with the ASO, but the improvement effect is not significant. Generally, the minimums of ASO_vaccine and ASO_rl are at most several orders of magnitude lower than the minimum of ASO. Only the MASO algorithm, that is, enhanced by three operators, can achieve remarkable results.Table 6
Test statistical results with different strategies.
Functions
Algorithms
Min
Mean
Max
STD
f
1(x)
ASO
7.33E − 27
2.64E − 26
1.05E − 25
3.08E − 26
ASO_chaos
2.69E − 27
38220.86
134747.6
49900.47
ASO_vaccine
9.34E − 27
3.35E − 26
6.9E − 26
2.51E − 26
ASO_rl
5.59E − 27
4.05E − 26
9.19E − 26
3.97E − 26
f
2(x)
ASO
2.16E − 13
5.32E − 13
9.67E − 13
2.51E − 13
ASO_chaos
2.99E − 13
1.58E + 18
2.84E + 19
6.35E + 18
ASO_vaccine
5.68E − 13
7.88E − 13
1.01E − 12
1.6E − 13
ASO_rl
3.18E − 13
1.25E − 12
2.35E − 12
7.73E − 13
f
3(x)
ASO
5.73E − 26
1.73E − 25
3.92E − 25
1.03E − 25
ASO_chaos
3.79E − 26
5651.508
17487.91
7310.399
ASO_vaccine
8.75E − 26
2.01E − 25
5.32E − 25
1.86E − 25
ASO_rl
1.23E − 25
3.62E − 25
5.3E − 25
1.63E − 25
f
4(x)
ASO
0
0
0
0
ASO_chaos
0
42588.7
116706
53955.46
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
5(x)
ASO
0.003764
0.00772
0.011135
0.002189
ASO_chaos
0.003324
72.91083
301.5848
100.3357
ASO_vaccine
0.003663
0.005553
0.008711
0.00202
ASO_rl
0.024793
0.032927
0.046203
0.010069
f
6(x)
ASO
2.19E − 52
1.46E − 51
5.24E − 51
1.51E − 51
ASO_chaos
2.34E − 53
94.61799
345.6983
134.9649
ASO_vaccine
1.02E − 51
3.62E − 51
8.42E − 51
3.45E − 51
ASO_rl
5.22E − 52
9.87E − 51
3.93E − 50
1.67E − 50
f
7(x)
ASO
4.35E − 14
7.41E − 14
1.47E − 13
2.97E − 14
ASO_chaos
5.42E − 14
8.462759
21.3434
10.63432
ASO_vaccine
6.84E − 14
9.25E − 14
1.29E − 13
2.22E − 14
ASO_rl
7.55E − 14
1.02E − 13
1.82E − 13
4.54E − 14
f
8(x)
ASO
0
0
0
0
ASO_chaos
0
344.9339
956.8887
437.1594
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
9(x)
ASO
9.949591
14.5264
20.89413
3.356089
ASO_chaos
7.959672
230.112
648.2703
274.7404
ASO_vaccine
11.93951
16.11833
20.89413
4.065958
ASO_rl
7.959672
12.33749
14.92438
2.68823
f
10(x)
ASO
4106.132
4650.915
5626.077
521.8557
ASO_chaos
4312.996
5089.873
5718.976
566.3842
ASO_vaccine
4500.827
5160.146
5625.985
471.8367
ASO_rl
3928.42
4852.266
5823.46
748.2354
f
11(x)
ASO
4.23E − 14
8.76E − 14
1.97E − 13
4.72E − 14
ASO_chaos
2.55E − 14
37.10606
116.9176
47.21336
ASO_vaccine
7.52E − 14
1.12E − 13
1.6E − 13
3.61E − 14
ASO_rl
7.19E − 14
1.3E − 13
2.7E − 13
8.45E − 14
f
12(x)
ASO
2.78E − 27
6.39E − 27
2.1E − 26
5.59E − 27
ASO_chaos
2.46E − 27
172.4654
562.9697
222.5449
ASO_vaccine
2.06E − 27
1.86E − 26
6.71E − 26
2.73E − 26
ASO_rl
3.9E − 27
1.11E − 26
1.95E − 26
7.21E − 27
f
13(x)
ASO
0.111929
0.171566
0.247934
0.04255
ASO_chaos
0.006224
0.221173
0.5
0.239022
ASO_vaccine
0.18622
0.268834
0.353136
0.072799
ASO_rl
0.094156
0.109595
0.122451
0.010779
f
14(x)
ASO
12
18.70114
35.0005
6.254622
ASO_chaos
9
226.4739
622.2764
263.4471
ASO_vaccine
16.08234
23.04624
27.001
4.654738
ASO_rl
13
23
30
6.892024
f
15(x)
ASO
13.22012
14.72793
15.51416
0.63515
ASO_chaos
14.05125
1691798
5141408
2180894
ASO_vaccine
15.5044
15.89732
16.17919
0.26884
ASO_rl
15.80763
16.45844
17.09231
0.562882
f
16(x)
ASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
ASO_chaos
2.17E − 29
2.2E + 09
8.17E + 09
2.87E + 09
ASO_vaccine
3.97E − 29
1.23E − 28
1.94E − 28
6.21E − 29
ASO_rl
3.35E − 29
1.49E − 28
4.22E − 28
1.59E − 28
MASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
f
17(x)
ASO
6.55E − 28
2.14E − 27
5.54E − 27
1.53E − 27
ASO_chaos
6.04E − 28
1.47E + 09
5.17E + 09
1.96E + 09
ASO_vaccine
7.69E − 28
2.92E − 27
7.94E − 27
2.89E − 27
ASO_rl
7.23E − 28
2.19E − 27
5.26E − 27
1.84E − 27
f
18(x)
ASO
4.71E − 14
9.18E − 14
1.75E − 13
3.96E − 14
ASO_chaos
5.06E − 14
8.510808
21.42481
10.69466
ASO_vaccine
6.13E − 14
1.59E − 13
4.34E − 13
1.55E − 13
ASO_rl
1.04E − 13
1.34E − 13
1.82E − 13
2.95E − 14
f
19(x)
ASO
8.954632
16.01884
19.89918
3.492605
ASO_chaos
9.949591
226.8982
594.1118
268.2596
ASO_vaccine
11.9395
22.28707
31.83867
7.629455
ASO_rl
9.949591
14.5264
19.89917
4.750859
f
20(x)
ASO
0
0
0
0
ASO_chaos
0
375.431
1145.34
482.6689
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
21(x)
ASO
4046.881
4920.749
5321.667
391.9594
ASO_chaos
4561.110
5235.886
3421.026
339.6557
ASO_vaccine
4817.261
5210.066
5626.202
326.6599
ASO_rl
4540.61
5634.291
6731.833
945.2253Table7 displays the statistical results of the Wilcoxon rank sum test, where “+” means that MASO has obvious advantages when compared with other algorithms and “≈” means that the Wilcoxon rank sum test results of MASO approximately equal to other algorithms. In F5, F9, F10, F13, F14, F19, and F21, ASO_chaos, ASO_vaccine, and ASO_rl are more effective than the ASO. The ASO_chaos has better performance than ASO in F15. The improved effect of ASO_chaos, ASO_vaccine, and ASO_rl is not significant in functions F1–F4, F6–F8, F11–F12, F16–F18, and F20. The effect of MASO is significantly better than ASO and comparative algorithms in most benchmark functions. Therefore, the remarkable effect of MASO can only be achieved by using three operators together.Table 7
Test statistical results of the Wilcoxon rank sum test.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.314619
≈
0
0.324488
≈
0
0.328083
≈
ASO_vaccine/ASO
0
0.488314
≈
0
0.641407
≈
0
0.195942
≈
ASO_rl/ASO
0
0.701861
≈
0
0.925725
≈
0
0.920272
≈
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.244594
≈
0
0.062368
≈
0
0.701051
≈
ASO_vaccine/ASO
0
0.553674
≈
1
0.030511
+
0
0.315137
≈
ASO_rl/ASO
0
0.893699
≈
1
0.000289
+
0
0.706156
≈
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.321172
≈
0
0.307772
≈
1
0.000331
+
ASO_vaccine/ASO
0
0.41769
≈
0
0.457487
≈
1
0.004432
+
ASO_rl/ASO
0
0.801112
≈
0
0.636002
≈
1
0.03666
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
8.67E − 08
+
0
0.301404
≈
0
0.321245
≈
ASO_vaccine/ASO
1
7.45E − 07
+
0
0.205611
≈
0
0.231387
≈
ASO_rl/ASO
1
1.61E − 07
+
0
0.898369
≈
0
0.914785
≈
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
2.53E − 18
+
1
0.018586
+
1
0.02315
+
ASO_vaccine/ASO
1
8.18E − 05
+
0
0.064058
≈
0
0.479393
≈
ASO_rl/ASO
1
0.000381
+
1
0.000728
+
0
0.075423
≈
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.449624
≈
0
0.385929
≈
0
0.311159
≈
ASO_vaccine/ASO
0
0.462677
≈
0
0.458521
≈
0
0.555353
≈
ASO_rl/ASO
0
0.769354
≈
0
0.753607
≈
0
0.8118
≈
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
0.000516
+
0
0.389344
≈
1
4.02E − 07
+
ASO_vaccine/ASO
1
0.003634
+
0
0.412808
≈
1
1.92E − 08
+
ASO_rl/ASO
1
0.013797
+
0
0.636028
≈
1
6.05E − 07
+


## 4.3.1. TheT Test Analysis between MASO and Comparative Algorithms

Table5 lists the results of T test, where “+” means that MASO has obvious advantages when compared with other algorithms, “=” indicates that the T test results of MASO equals other algorithms, and “≈” means that the T test results of MASO are approximately equal to other algorithms.Table 5
The comparisons ofT test for f1∼f21.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.016614
+
0
0.290266
≈
1
0.023023
+
MASO/BAT
1
3.24E − 09
+
1
0
+
0
0.14638
≈
MASO/PSO
1
9.2E − 105
+
1
1.1E − 302
+
1
1.4E − 181
+
MASO/CS
1
0.000179
+
1
1.36E − 07
+
1
1.7E − 17
+
MASO/ACOR
0
0.094316
≈
0
0.273106
≈
0
0.09149
≈
MASO/CSA
0
0.304726
≈
0
0.259636
≈
0
0.310844
≈
MASO/ASO
1
0.000526
+
0
0.481441
≈
1
0.003881
+
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
0.004565
+
0
0.050501
≈
1
0.049963
+
MASO/BAT
1
8.7E − 204
+
1
1.45E − 09
+
1
5.04E − 28
+
MASO/PSO
1
1.7E − 183
+
1
2.5E − 128
+
1
6.04E − 80
+
MASO/CS
1
1.47E − 09
+
1
7.87E − 73
+
1
0.001168
+
MASO/ACOR
0
0.089476
≈
1
0.029733
+
0
0.111095
≈
MASO/CSA
0
0.303911
≈
1
0.009487
+
0
0.318903
≈
MASO/ASO
N/A
N/A
=
1
4.62E − 05
+
1
0.0 03612
+
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.28E − 11
+
1
0.011808
+
1
1.6E − 30
+
MASO/BAT
1
0
+
1
4.66E − 84
+
1
0
+
MASO/PSO
1
9.1E − 295
+
1
1.1E − 188
+
1
9.5E − 210
+
MASO/CS
1
8.9E − 104
+
1
5.45E − 09
+
1
1.03E − 47
+
MASO/ACOR
1
0.001019
+
0
0.079069
≈
1
1.21E − 68
+
MASO/CSA
0
0.193249
≈
0
0.249111
≈
1
1.75E − 09
+
MASO/ASO
1
4.02E − 04
+
0
0.134598
≈
1
0.005241
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
1.96E − 56
+
1
1.91E − 25
+
1
1.21E − 07
+
MASO/BAT
1
7.53E − 91
+
1
1.15E − 40
+
1
9.57E − 87
+
MASO/PSO
1
7E − 250
+
1
2.3E − 209
+
1
1.6E − 269
+
MASO/CS
1
1.51E − 46
+
1
3.31E − 08
+
1
1.6E − 278
+
MASO/ACOR
1
0.003855
+
1
7.06E − 20
+
1
0.006418
+
MASO/CSA
1
0.003995
+
0
0.128471
≈
0
0.278777
≈
MASO/ASO
1
1.88E − 05
+
1
0.006584
+
1
0.001213
+
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
2.66E − 57
+
1
6.46E − 32
+
1
0.041413
+
MASO/BAT
1
0
+
1
0
+
0
0.18106
≈
MASO/PSO
1
7.1E − 227
+
1
3E − 290
+
1
5.2E − 275
+
MASO/CS
1
9.9E − 189
+
1
9.78E − 36
+
1
0.010623
+
MASO/ACOR
1
1.34E − 17
+
1
1.93E − 69
+
1
0.024626
+
MASO/CSA
1
0.001114
+
1
1.65E − 16
+
0
0.165226
≈
MASO/ASO
1
0.009734
+
1
3.09E − 11
+
1
9.92E − 53
+
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
0
0.128943
≈
1
0.048392
+
1
4.89E − 15
+
MASO/BAT
0
0.272098
≈
1
0.045067
+
1
0
+
MASO/PSO
1
6.8E − 214
+
1
2.5E − 252
+
1
2.6E − 255
+
MASO/CS
0
0.266098
≈
0
0.276366
≈
1
5.9E − 177
+
MASO/ACOR
0
0.116175
≈
1
0.049686
+
1
0.001027
+
MASO/CSA
0
0.307291
≈
0
0.319772
≈
0
0.156388
≈
MASO/ASO
1
0.008508
+
1
3.3E − 04
+
0
0.206133
+
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
MASO/FPA
1
5.92E − 23
+
1
0.01143
+
1
5.7E − 21
+
MASO/BAT
1
0
+
1
8.67E − 78
+
1
3.82E − 62
+
MASO/PSO
1
1.1E − 178
+
1
1.8E − 185
+
1
1.05E − 80
+
MASO/CS
1
5.5E − 54
+
1
2.14E − 05
+
1
3.74E − 28
+
MASO/ACOR
1
7.36E − 78
+
0
0.097076
≈
1
7.62E − 51
+
MASO/CSA
1
3.03E − 52
+
0
0.237911
≈
1
2.72E − 86
+
MASO/ASO
1
0.000142
+
0
0.643876
≈
1
2.28E − 36
+In F5, F7, F9, F10, F11, F12, F13, F14, F18, F19, and F21, MASO is more prominent than the most comparative algorithms. In F1, F3, F4, F6, F8, F15, F17, and F20, although performance of MASO is not outstanding, but its advantages are still better than most comparative algorithms. In F2 and F16, the effect of MASO is almost the same as the corresponding comparison algorithm. The improvement of MASO is not significant enough in F2 and F16 because most algorithms can optimize the two functions easily. Except F2 and F16, the effect of MASO is obviously better than most other algorithms, so the improvement of MASO is evidently effective. Therefore, MASO is a successful improvement to ASO.


## 4.3.2. Wilcoxon Rank Sum Test for Improved Strategies

MASO is obtained by using the chaotic initialization operator, vaccine operator, and reinforcement learning operator to ameliorate ASO. The ASO enhanced by the chaotic initialization operator is named ASO_chaos. The ASO enhanced by the vaccine operator is named ASO_vaccine. The ASO enhanced by the reinforcement learning operator is named ASO_rl.In this experiment, each algorithm runs 20 times independently in each benchmark function. The population number is 100. The evaluation times of ASO_vaccine and ASO_rl are2∗ 100 in each iteration, and evaluation times of ASO and ASO_chaos are 100 in each iteration. For each benchmark function, ASO_vaccine and ASO_rl will iterate 2500 times, and ASO and ASO_chaos will iterate 5000 times. So the four algorithms will be compared under the evaluation of 100∗ 5000 in each run.Table6 lists the statistical results with different strategies. Among the three improved methods, the improvement effect of ASO_chaos is relatively general, and its stability is not good too. ASO_vaccine and ASO_rl are slightly improved compared with the ASO, but the improvement effect is not significant. Generally, the minimums of ASO_vaccine and ASO_rl are at most several orders of magnitude lower than the minimum of ASO. Only the MASO algorithm, that is, enhanced by three operators, can achieve remarkable results.Table 6
Test statistical results with different strategies.
Functions
Algorithms
Min
Mean
Max
STD
f
1(x)
ASO
7.33E − 27
2.64E − 26
1.05E − 25
3.08E − 26
ASO_chaos
2.69E − 27
38220.86
134747.6
49900.47
ASO_vaccine
9.34E − 27
3.35E − 26
6.9E − 26
2.51E − 26
ASO_rl
5.59E − 27
4.05E − 26
9.19E − 26
3.97E − 26
f
2(x)
ASO
2.16E − 13
5.32E − 13
9.67E − 13
2.51E − 13
ASO_chaos
2.99E − 13
1.58E + 18
2.84E + 19
6.35E + 18
ASO_vaccine
5.68E − 13
7.88E − 13
1.01E − 12
1.6E − 13
ASO_rl
3.18E − 13
1.25E − 12
2.35E − 12
7.73E − 13
f
3(x)
ASO
5.73E − 26
1.73E − 25
3.92E − 25
1.03E − 25
ASO_chaos
3.79E − 26
5651.508
17487.91
7310.399
ASO_vaccine
8.75E − 26
2.01E − 25
5.32E − 25
1.86E − 25
ASO_rl
1.23E − 25
3.62E − 25
5.3E − 25
1.63E − 25
f
4(x)
ASO
0
0
0
0
ASO_chaos
0
42588.7
116706
53955.46
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
5(x)
ASO
0.003764
0.00772
0.011135
0.002189
ASO_chaos
0.003324
72.91083
301.5848
100.3357
ASO_vaccine
0.003663
0.005553
0.008711
0.00202
ASO_rl
0.024793
0.032927
0.046203
0.010069
f
6(x)
ASO
2.19E − 52
1.46E − 51
5.24E − 51
1.51E − 51
ASO_chaos
2.34E − 53
94.61799
345.6983
134.9649
ASO_vaccine
1.02E − 51
3.62E − 51
8.42E − 51
3.45E − 51
ASO_rl
5.22E − 52
9.87E − 51
3.93E − 50
1.67E − 50
f
7(x)
ASO
4.35E − 14
7.41E − 14
1.47E − 13
2.97E − 14
ASO_chaos
5.42E − 14
8.462759
21.3434
10.63432
ASO_vaccine
6.84E − 14
9.25E − 14
1.29E − 13
2.22E − 14
ASO_rl
7.55E − 14
1.02E − 13
1.82E − 13
4.54E − 14
f
8(x)
ASO
0
0
0
0
ASO_chaos
0
344.9339
956.8887
437.1594
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
9(x)
ASO
9.949591
14.5264
20.89413
3.356089
ASO_chaos
7.959672
230.112
648.2703
274.7404
ASO_vaccine
11.93951
16.11833
20.89413
4.065958
ASO_rl
7.959672
12.33749
14.92438
2.68823
f
10(x)
ASO
4106.132
4650.915
5626.077
521.8557
ASO_chaos
4312.996
5089.873
5718.976
566.3842
ASO_vaccine
4500.827
5160.146
5625.985
471.8367
ASO_rl
3928.42
4852.266
5823.46
748.2354
f
11(x)
ASO
4.23E − 14
8.76E − 14
1.97E − 13
4.72E − 14
ASO_chaos
2.55E − 14
37.10606
116.9176
47.21336
ASO_vaccine
7.52E − 14
1.12E − 13
1.6E − 13
3.61E − 14
ASO_rl
7.19E − 14
1.3E − 13
2.7E − 13
8.45E − 14
f
12(x)
ASO
2.78E − 27
6.39E − 27
2.1E − 26
5.59E − 27
ASO_chaos
2.46E − 27
172.4654
562.9697
222.5449
ASO_vaccine
2.06E − 27
1.86E − 26
6.71E − 26
2.73E − 26
ASO_rl
3.9E − 27
1.11E − 26
1.95E − 26
7.21E − 27
f
13(x)
ASO
0.111929
0.171566
0.247934
0.04255
ASO_chaos
0.006224
0.221173
0.5
0.239022
ASO_vaccine
0.18622
0.268834
0.353136
0.072799
ASO_rl
0.094156
0.109595
0.122451
0.010779
f
14(x)
ASO
12
18.70114
35.0005
6.254622
ASO_chaos
9
226.4739
622.2764
263.4471
ASO_vaccine
16.08234
23.04624
27.001
4.654738
ASO_rl
13
23
30
6.892024
f
15(x)
ASO
13.22012
14.72793
15.51416
0.63515
ASO_chaos
14.05125
1691798
5141408
2180894
ASO_vaccine
15.5044
15.89732
16.17919
0.26884
ASO_rl
15.80763
16.45844
17.09231
0.562882
f
16(x)
ASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
ASO_chaos
2.17E − 29
2.2E + 09
8.17E + 09
2.87E + 09
ASO_vaccine
3.97E − 29
1.23E − 28
1.94E − 28
6.21E − 29
ASO_rl
3.35E − 29
1.49E − 28
4.22E − 28
1.59E − 28
MASO
3.53E − 29
8.02E − 29
1.31E − 28
3.4E − 29
f
17(x)
ASO
6.55E − 28
2.14E − 27
5.54E − 27
1.53E − 27
ASO_chaos
6.04E − 28
1.47E + 09
5.17E + 09
1.96E + 09
ASO_vaccine
7.69E − 28
2.92E − 27
7.94E − 27
2.89E − 27
ASO_rl
7.23E − 28
2.19E − 27
5.26E − 27
1.84E − 27
f
18(x)
ASO
4.71E − 14
9.18E − 14
1.75E − 13
3.96E − 14
ASO_chaos
5.06E − 14
8.510808
21.42481
10.69466
ASO_vaccine
6.13E − 14
1.59E − 13
4.34E − 13
1.55E − 13
ASO_rl
1.04E − 13
1.34E − 13
1.82E − 13
2.95E − 14
f
19(x)
ASO
8.954632
16.01884
19.89918
3.492605
ASO_chaos
9.949591
226.8982
594.1118
268.2596
ASO_vaccine
11.9395
22.28707
31.83867
7.629455
ASO_rl
9.949591
14.5264
19.89917
4.750859
f
20(x)
ASO
0
0
0
0
ASO_chaos
0
375.431
1145.34
482.6689
ASO_vaccine
0
0
0
0
ASO_rl
0
0
0
0
f
21(x)
ASO
4046.881
4920.749
5321.667
391.9594
ASO_chaos
4561.110
5235.886
3421.026
339.6557
ASO_vaccine
4817.261
5210.066
5626.202
326.6599
ASO_rl
4540.61
5634.291
6731.833
945.2253Table7 displays the statistical results of the Wilcoxon rank sum test, where “+” means that MASO has obvious advantages when compared with other algorithms and “≈” means that the Wilcoxon rank sum test results of MASO approximately equal to other algorithms. In F5, F9, F10, F13, F14, F19, and F21, ASO_chaos, ASO_vaccine, and ASO_rl are more effective than the ASO. The ASO_chaos has better performance than ASO in F15. The improved effect of ASO_chaos, ASO_vaccine, and ASO_rl is not significant in functions F1–F4, F6–F8, F11–F12, F16–F18, and F20. The effect of MASO is significantly better than ASO and comparative algorithms in most benchmark functions. Therefore, the remarkable effect of MASO can only be achieved by using three operators together.Table 7
Test statistical results of the Wilcoxon rank sum test.
Algorithm
f
1(x)
f
2(x)
f
3(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.314619
≈
0
0.324488
≈
0
0.328083
≈
ASO_vaccine/ASO
0
0.488314
≈
0
0.641407
≈
0
0.195942
≈
ASO_rl/ASO
0
0.701861
≈
0
0.925725
≈
0
0.920272
≈
f
4(x)
f
5(x)
f
6(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.244594
≈
0
0.062368
≈
0
0.701051
≈
ASO_vaccine/ASO
0
0.553674
≈
1
0.030511
+
0
0.315137
≈
ASO_rl/ASO
0
0.893699
≈
1
0.000289
+
0
0.706156
≈
f
7(x)
f
8(x)
f
9(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.321172
≈
0
0.307772
≈
1
0.000331
+
ASO_vaccine/ASO
0
0.41769
≈
0
0.457487
≈
1
0.004432
+
ASO_rl/ASO
0
0.801112
≈
0
0.636002
≈
1
0.03666
+
f
10(x)
f
11(x)
f
12(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
8.67E − 08
+
0
0.301404
≈
0
0.321245
≈
ASO_vaccine/ASO
1
7.45E − 07
+
0
0.205611
≈
0
0.231387
≈
ASO_rl/ASO
1
1.61E − 07
+
0
0.898369
≈
0
0.914785
≈
f
13(x)
f
14(x)
f
15(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
2.53E − 18
+
1
0.018586
+
1
0.02315
+
ASO_vaccine/ASO
1
8.18E − 05
+
0
0.064058
≈
0
0.479393
≈
ASO_rl/ASO
1
0.000381
+
1
0.000728
+
0
0.075423
≈
f
16(x)
f
17(x)
f
18(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
0
0.449624
≈
0
0.385929
≈
0
0.311159
≈
ASO_vaccine/ASO
0
0.462677
≈
0
0.458521
≈
0
0.555353
≈
ASO_rl/ASO
0
0.769354
≈
0
0.753607
≈
0
0.8118
≈
f
19(x)
f
20(x)
f
21(x)
H
P
Sig
H
P
Sig
H
P
Sig
ASO_chaos/ASO
1
0.000516
+
0
0.389344
≈
1
4.02E − 07
+
ASO_vaccine/ASO
1
0.003634
+
0
0.412808
≈
1
1.92E − 08
+
ASO_rl/ASO
1
0.013797
+
0
0.636028
≈
1
6.05E − 07
+


## 5. Application of MASO in Optimization of Permutation Flow Shop Scheduling Problem

In order to verify the optimization capability of MASO for practical engineering problems, MASO will be used to optimize the permutation flow shop scheduling problem.


### 5.1. Permutation Flow Shop Scheduling Problem

The flow shop scheduling problem studies the process ofn jobs on m machines. According to different constraints, flow shop scheduling can be divided into the following different problems: permutation flow shop scheduling problem, no-wait flow shop scheduling problem, no-idle flow shop scheduling problem, blocking flow shop scheduling problem, flow shop scheduling problem with limited buffers, lot-streaming flow shop scheduling problem, and hybrid flow shop scheduling problem [42].The solution of the permutation flow shop scheduling problem is to find a scheduling sequence so that a certain production index can reach the optimal value. The optimization objective is the minimum makespan. The minimum makespan equals to the completion time of the last workpiece on the last machine.


### 5.2. Model Assumptions

Some assumptions are made to establish the mathematical model of permutation flow shop scheduling problem:(1)
N workpieces are processed on M machines. Each workpiece must be processed in sequence on different machines without interruption.(2)
There is an infinite buffer area between machines.(3)
In each process, a workpiece can only be processed on one machine without interruption and a machine can only process one workpiece at the same time.(4)
The processing time of the workpiece is known, and the preparation time of the workpiece and the start time of the machine are all included in the processing time.The mathematical models of permutation flow shop scheduling problems are shown in equations (22), (23), and (24) [43].(1)
Objective function:(22)objective=MinMaxCj,s,j=1,…,n.(2)
Constraint conditions:(23)Cj,k=Sj,k+Pj,k,j=1,…,n;k=1,…,s,(24)Cj,k≤Sj,k+1,j=1,…,n;k=1,…,s.The number of workpieces isn, and a single workpiece is marked as j (j = 1, …, n). The number of machines is s, and a single machine is marked k (k = 1, …, s). Sj,k is the start time of workpiece j in machine k. Pj,k is the processing time of workpiece j on machine k. Cj,k is the makespan of workpiece j in machine k.


### 5.3. Engineering Sample

An engineering example of permutation flow shop scheduling problem is given below. Suppose three jobs A, B, and C are processed on three machines in the order of A, B, and C, respectively. The starting time of the first job on the first machine is 0. The processing time of all jobs on each machine is listed in Table8. The Gantt chart and the makespan are shown in Figure 25.Table 8
The processing time of the workpiece on each machine.
Workpieces
A
B
C
Machines
1
3
5
1
2
4
2
2
3
1
1
3Figure 25
Gantt chart of permutation flow shop scheduling with makespan.


### 5.4. Experimental Design

The configuration data about the size and structure of the problem need to be obtained to systematically evaluate the performance of algorithm. The actual data in industry are difficult to obtain, so the test data will be generated randomly. These test data will represent the industrial data. The test is divided into nine problem combinations. Each problem can be identified in the form of “number of jobs-number of machines.” For example, problems with 20 jobs and 4 machines can be expressed as “20-4.” The factors of level and scope are given in Table9.Table 9
The factors of level and range.
Factors
Level
Number of level
Number of jobs
10, 20, and 50
3
Number of machines
2, 4, and 8
3
The processing time of each workpiece on each machine
Discrete uniform [1, 5]
1The setting of experimental parameters is the same as in Section3.2. But for each test problem, MASO is compared with ASO, FPA, BAT, PSO, CS, ACOR, and CSA by running ten times separately.


### 5.5. Analysis of Experimental Results

The above eight algorithms have been tested ten times on nine combinations of problems, and the test results are listed in Table10. The minimum makespan and average are two important performance evaluation indexes. The minimum makespan represents the optimization ability of the algorithm. The average represents the stability of the algorithm.Table 10
Permutation flow shop scheduling test results.
Problem
MASO
ASO
FPA
BAT
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
10-2
28
33.7
39
26
32.9
41
28
33.5
38
29
34.7
38
10-4
36
39.1
44
36
40.7
46
38
41.5
46
39
41.2
43
10-8
50
55.2
63
51
55.8
60
52
56.1
61
50
56.8
60
20-2
56
63.7
71
58
65.3
75
59
64.2
71
57
64.2
75
20-4
64
72.6
76
66
74.7
83
62
70
78
68
73.9
80
20-8
81
87.6
95
85
89.3
97
84
88.4
93
80
88.4
92
50-2
147
158.3
180
150
159.1
169
145
157
167
150
157.6
172
50-4
153
164.2
180
155
164.4
186
154
167.2
184
157
166.6
176
50-8
171
180.4
191
173
178.6
185
180
186.6
194
181
185.2
190
PSO
CS
ACOR
CSA
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
10-2
30
35.3
40
30
35.4
39
29
34.6
44
28
33.5
40
10-4
35
40.6
46
35
43.5
50
35
39.1
42
34
40.2
44
10-8
51
57.2
64
51
57.1
62
50
54.2
57
50
55.5
61
20-2
55
66.4
74
57
61.3
67
62
65.5
69
56
62
70
20-4
69
74.8
83
68
71.1
80
64
69
77
64
71.9
83
20-8
85
89.7
97
88
92.8
98
82
87.9
92
85
91.5
100
50-2
143
158.3
166
153
160.6
171
157
159.9
167
147
157.1
173
50-4
162
167.5
178
160
167.4
191
149
161.2
172
161
168.3
173
50-8
183
189.8
200
180
194.9
201
177
182.9
189
184
188.7
199In the “10-2” problem, the minimum makespan obtained by MASO is better than BAT, PSO, and ACOR and is equal to FPA and CS but is worse than the ASO. The stability of MASO is in the middle of all the test algorithms. Generally, the effect is not ideal. In “10-4,” the search effect of MASO is general, but the average is the best, and the stability is good. In the “10-8” problem, the search ability of MASO is the best, but it is equal to CSA, ACOR, and BAT, respectively. And the average of MASO is the best in all algorithms. In summary, MASO has almost the same performance as other algorithms under the problem scale of 10 jobs because the scale of the problem is too small to reflect the good search capability of MASO.In the “20-2” problem, MASO has the better performance. The minimum makespan of MASO is equal to CSA and is slightly inferior to PSO. In the “20-4” problem, the minimum makespan of MASO is equal to ACOR and CSA and is slightly inferior to FPA. The minimum makespan of MASO ranks second among all the algorithms. In the “20-8” problem, the optimization capability of MASO is only inferior to BAT, ranking second among all algorithms, but its average is the best. In summary, the overall performance of MASO is better than all other algorithms under the scale of 20 workpieces because MASO has strong optimization accuracy and can coordinate the balance between global exploration and local development. In addition to MASO, ACOR performs well on the scale of the problem, so ACOR is also suitable for optimizing this complex permutation flow shop scheduling problem.In the problem of “50-2,” the search effect of MASO is the best, but the average of MASO is worse. In the “50-4” problem, the search effect of MASO is only slightly worse than ACOR. In the “50-8” problem, the search ability of MASO is the best and far better than other algorithms.Therefore, MASO works better on the problem of higher complexity, especially on the problem of 20 and 50 workpieces. The reason is that the immune operator and reinforcement learning operator strengthen the utilization of the dominant position in the atom population, so the search speed and accuracy of MASO are promoted. Experiments confirm that MASO has advantages to optimize engineering problem.


## 5.1. Permutation Flow Shop Scheduling Problem

The flow shop scheduling problem studies the process ofn jobs on m machines. According to different constraints, flow shop scheduling can be divided into the following different problems: permutation flow shop scheduling problem, no-wait flow shop scheduling problem, no-idle flow shop scheduling problem, blocking flow shop scheduling problem, flow shop scheduling problem with limited buffers, lot-streaming flow shop scheduling problem, and hybrid flow shop scheduling problem [42].The solution of the permutation flow shop scheduling problem is to find a scheduling sequence so that a certain production index can reach the optimal value. The optimization objective is the minimum makespan. The minimum makespan equals to the completion time of the last workpiece on the last machine.


## 5.2. Model Assumptions

Some assumptions are made to establish the mathematical model of permutation flow shop scheduling problem:(1)
N workpieces are processed on M machines. Each workpiece must be processed in sequence on different machines without interruption.(2)
There is an infinite buffer area between machines.(3)
In each process, a workpiece can only be processed on one machine without interruption and a machine can only process one workpiece at the same time.(4)
The processing time of the workpiece is known, and the preparation time of the workpiece and the start time of the machine are all included in the processing time.The mathematical models of permutation flow shop scheduling problems are shown in equations (22), (23), and (24) [43].(1)
Objective function:(22)objective=MinMaxCj,s,j=1,…,n.(2)
Constraint conditions:(23)Cj,k=Sj,k+Pj,k,j=1,…,n;k=1,…,s,(24)Cj,k≤Sj,k+1,j=1,…,n;k=1,…,s.The number of workpieces isn, and a single workpiece is marked as j (j = 1, …, n). The number of machines is s, and a single machine is marked k (k = 1, …, s). Sj,k is the start time of workpiece j in machine k. Pj,k is the processing time of workpiece j on machine k. Cj,k is the makespan of workpiece j in machine k.


## 5.3. Engineering Sample

An engineering example of permutation flow shop scheduling problem is given below. Suppose three jobs A, B, and C are processed on three machines in the order of A, B, and C, respectively. The starting time of the first job on the first machine is 0. The processing time of all jobs on each machine is listed in Table8. The Gantt chart and the makespan are shown in Figure 25.Table 8
The processing time of the workpiece on each machine.
Workpieces
A
B
C
Machines
1
3
5
1
2
4
2
2
3
1
1
3Figure 25
Gantt chart of permutation flow shop scheduling with makespan.


## 5.4. Experimental Design

The configuration data about the size and structure of the problem need to be obtained to systematically evaluate the performance of algorithm. The actual data in industry are difficult to obtain, so the test data will be generated randomly. These test data will represent the industrial data. The test is divided into nine problem combinations. Each problem can be identified in the form of “number of jobs-number of machines.” For example, problems with 20 jobs and 4 machines can be expressed as “20-4.” The factors of level and scope are given in Table9.Table 9
The factors of level and range.
Factors
Level
Number of level
Number of jobs
10, 20, and 50
3
Number of machines
2, 4, and 8
3
The processing time of each workpiece on each machine
Discrete uniform [1, 5]
1The setting of experimental parameters is the same as in Section3.2. But for each test problem, MASO is compared with ASO, FPA, BAT, PSO, CS, ACOR, and CSA by running ten times separately.


## 5.5. Analysis of Experimental Results

The above eight algorithms have been tested ten times on nine combinations of problems, and the test results are listed in Table10. The minimum makespan and average are two important performance evaluation indexes. The minimum makespan represents the optimization ability of the algorithm. The average represents the stability of the algorithm.Table 10
Permutation flow shop scheduling test results.
Problem
MASO
ASO
FPA
BAT
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
10-2
28
33.7
39
26
32.9
41
28
33.5
38
29
34.7
38
10-4
36
39.1
44
36
40.7
46
38
41.5
46
39
41.2
43
10-8
50
55.2
63
51
55.8
60
52
56.1
61
50
56.8
60
20-2
56
63.7
71
58
65.3
75
59
64.2
71
57
64.2
75
20-4
64
72.6
76
66
74.7
83
62
70
78
68
73.9
80
20-8
81
87.6
95
85
89.3
97
84
88.4
93
80
88.4
92
50-2
147
158.3
180
150
159.1
169
145
157
167
150
157.6
172
50-4
153
164.2
180
155
164.4
186
154
167.2
184
157
166.6
176
50-8
171
180.4
191
173
178.6
185
180
186.6
194
181
185.2
190
PSO
CS
ACOR
CSA
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
Min
Mean
Max
10-2
30
35.3
40
30
35.4
39
29
34.6
44
28
33.5
40
10-4
35
40.6
46
35
43.5
50
35
39.1
42
34
40.2
44
10-8
51
57.2
64
51
57.1
62
50
54.2
57
50
55.5
61
20-2
55
66.4
74
57
61.3
67
62
65.5
69
56
62
70
20-4
69
74.8
83
68
71.1
80
64
69
77
64
71.9
83
20-8
85
89.7
97
88
92.8
98
82
87.9
92
85
91.5
100
50-2
143
158.3
166
153
160.6
171
157
159.9
167
147
157.1
173
50-4
162
167.5
178
160
167.4
191
149
161.2
172
161
168.3
173
50-8
183
189.8
200
180
194.9
201
177
182.9
189
184
188.7
199In the “10-2” problem, the minimum makespan obtained by MASO is better than BAT, PSO, and ACOR and is equal to FPA and CS but is worse than the ASO. The stability of MASO is in the middle of all the test algorithms. Generally, the effect is not ideal. In “10-4,” the search effect of MASO is general, but the average is the best, and the stability is good. In the “10-8” problem, the search ability of MASO is the best, but it is equal to CSA, ACOR, and BAT, respectively. And the average of MASO is the best in all algorithms. In summary, MASO has almost the same performance as other algorithms under the problem scale of 10 jobs because the scale of the problem is too small to reflect the good search capability of MASO.In the “20-2” problem, MASO has the better performance. The minimum makespan of MASO is equal to CSA and is slightly inferior to PSO. In the “20-4” problem, the minimum makespan of MASO is equal to ACOR and CSA and is slightly inferior to FPA. The minimum makespan of MASO ranks second among all the algorithms. In the “20-8” problem, the optimization capability of MASO is only inferior to BAT, ranking second among all algorithms, but its average is the best. In summary, the overall performance of MASO is better than all other algorithms under the scale of 20 workpieces because MASO has strong optimization accuracy and can coordinate the balance between global exploration and local development. In addition to MASO, ACOR performs well on the scale of the problem, so ACOR is also suitable for optimizing this complex permutation flow shop scheduling problem.In the problem of “50-2,” the search effect of MASO is the best, but the average of MASO is worse. In the “50-4” problem, the search effect of MASO is only slightly worse than ACOR. In the “50-8” problem, the search ability of MASO is the best and far better than other algorithms.Therefore, MASO works better on the problem of higher complexity, especially on the problem of 20 and 50 workpieces. The reason is that the immune operator and reinforcement learning operator strengthen the utilization of the dominant position in the atom population, so the search speed and accuracy of MASO are promoted. Experiments confirm that MASO has advantages to optimize engineering problem.


## 6. Conclusion

The MASO is proposed to improve ASO in this paper. The chaotic operator, immune operator, and reinforcement learning operator are introduced to enhance ASO. The chaotic operator overcomes the disadvantages of nonuniform initialization in ASO. The immune operator and reinforcement learning operator can make better use of the dominant position in the atom population and employs adaptive parameters to avoid human interference factors. The experimental results in 21 benchmark functions indicate that MASO is better than the seven comparison algorithms in global search ability and convergence speed.T test proves that the performance of MASO is significantly better than the seven comparison algorithms on most benchmark functions. The Wilcoxon rank sum test illustrates that the excellent performance of MASO is obtained by the interaction of three operators. When MASO is applied to optimize the permutation flow shop scheduling problem, the search accuracy of MASO is better than the seven contrast algorithms. MASO does not require the objective function to be convex, continuous, or derivable, so it has strong advantages in the field of numerical optimization. Therefore, MASO can be applied in machine learning, engineering optimization, municipal traffic management, and other optimization fields.



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*Source: 4568906-2020-01-14.xml*