HPSBA: A Modified Hybrid Framework with Convergence Analysis for Solving Wireless Sensor Network Coverage Optimization Problem

Abstract

Complex optimization (CO) problems have been solved using swarm intelligence (SI) methods. One of the CO problems is the Wireless Sensor Network (WSN) coverage optimization problem, which plays an important role in Internet of Things (IoT). A novel hybrid algorithm is proposed, named hybrid particle swarm butterfly algorithm (HPSBA), by combining their strengths of particle swarm optimization (PSO) and butterfly optimization algorithm (BOA), for solving this problem. Significantly, the value of individual scent intensity should be non-negative without consideration of the basic BOA, which is calculated with absolute value of the proposed HPSBA. Moreover, the performance of the HPSBA is comprehensively compared with the fundamental BOA, numerous potential BOA variants, and tried-and-true algorithms, for solving the twenty-six commonly used benchmark functions. The results show that HPSBA has a competitive overall performance. Finally, when compared to PSO, BOA, and MBOA, HPSBA is used to solve the node coverage optimization problem in WSN. The experimental results demonstrate that the HPSBA optimized coverage has a higher coverage rate, which effectively reduces node redundancy and extends WSN survival time.

1. Introduction

With the emergence of heuristic intelligent optimization algorithms, new methods have been provided for solving complex engineering problems. The principle is mostly to imitate the biological habits of foraging and courtship of the biological community. According to the theoretical principle of meta-heuristic optimization algorithm, it can be simply divided into four categories (See Figure 1). Typical swarm intelligence algorithms (SI-based) are: Particle swarm optimization (PSO) [1], Cuckoo search (CS) [2], Grey wolf optimizer (GWO) [3], Whale optimization algorithm (WOA) [4], Marine predators algorithm (MPA) [5], Ant colony optimization (ACO) [6], Firefly algorithm (FA) [7], Moth-flame optimization (MFO) [8], Grasshopper optimization algorithm (GOA) [9], Butterfly optimization algorithm (BOA) [10]. Evolution algorithms (Ev-based) are: Genetic algorithm (GA) [11], Differential evolution (DE) [12], Biogeography-based optimizer (BBO) [13], Genetic programming (GP) [14]. Algorithms based on physical characteristics (Phy-based) are: Simulated annealing (SA) [15], Gravitational search algorithm (GSA) [16], Harmony search (HS) [17], Sine cosine algorithm (SCA) [18], Equilibrium optimizer (EO) [19], Gradient-based optimizer (GBO) [20]. Algorithms based on human social behavior (Hu-based) are: Teaching learning based optimization (TLBO) [21], Tabu search (TS) [22], Socio evolution and learning optimization (SELO) [23], Political optimizer (PO) [24]. For a more detailed review, we can refer to the literature [25,26].

According to the biological characteristics of natural animals (such as oviparous animals or mammals, and insects) and plants, swarm intelligence optimization algorithms can be divided into three categories. Imitating animal habits such as: Particle swarm optimization (PSO) [1], Cuckoo search (CS) [2], Grey wolf optimizer (GWO) [3], Whale optimization algorithm (WOA) [4], Marine predators algorithm (MPA) [5], etc. Imitating the habits of insects such as: Ant colony optimization (ACO) [6], Firefly algorithm (FA) [7], Moth-flame optimization (MFO) [8], Grasshopper optimization algorithm (GOA) [9], Butterfly optimization algorithm (BOA) [10], etc. Imitating plant characteristics such as: Flower pollination algorithm (FPA) [27], Tree-seed algorithm (TSA) [28], etc.

PSO is a typical and widely used intelligent optimization algorithm. It has the advantage of fast convergence. Some of the recent works on hybrid PSO with other SI algorithms are as follows: hybrid PSO and DE [29], hybrid PSO and GSA [30], hybrid GA and PSO [31,32], hybrid PSO and SSA [33], etc. Butterfly optimization algorithm [10] is a novel swarm intelligent algorithm proposed by Arora and Singh, which has been used to solve the wireless sensor network node localization problem [34], and optimization training of wavelet neural network [35]. However, BOA is susceptible to local optima and suffers from premature convergence. There are several recent works on BOA as follows: improved BOA [36], modified BOA [37], hybrid BOA and PSO [38], etc.

The energy consumption of the wireless sensor network (WSN) is limited by the large number of sensor nodes [39], which is crucial to the Internet of Things (IoT). It can provide users with accurate and comprehensive real-time data by processing the detection data of sensing objects of the mutual cooperation between nodes. WSN has been widely used in military, transportation, environmental monitoring and other fields [40]. The coverage problem is one of the key tasks in the research field of the WSN, which reflects the quality of the service. Coverage ratio is an important indicator for evaluating the performance of the WSN nodes.

In a working area, sensor nodes are usually arranged at random for the initial stage. High-density nodes will result from this method, resulting in a low coverage rate that directly impacts monitoring quality [41,42]. As a result, optimizing sensor node coverage is critical to increasing WSN coverage ratio in the work area. SI optimization algorithms have recently made significant contributions to the WSN’s problem of optimizing node coverage. Wang et al. [43] proposed a resampled PSO to solve the coverage control problem in IoT. Yang et. al [44] used the improved FA to solve sensor coverage problem, which considered the target coverage and network connectivity of sensor nodes. Miao et al. [45] proposed a GWO-EH algorithm to address the WSN node coverage optimization problem. Wang et al. [46] proposed the topology optimization of coverage-oriented method for a WSN based on wolf pack algorithm (WPA). Dao et al. [47] proposed a WSN coverage optimization model based on an improved Archimedes optimization algorithm for the working area. Although the above-mentioned SI algorithms have achieved success, they are still difficult to rid of local optima when optimization problems become more challenging, which makes it necessary to find new methods.

Because of the advantages of BOA with simple structure and few adjustment parameters, and PSO with fast convergence, a novel Chaotic hybrid butterfly optimization algorithm with particle swarm optimization (HPSOBOA) [38] was proposed for solving the high-dimensional optimization problems. However, both of BOA and PSO can easily fall into a local optimum and have low convergence accuracy, and HPSOBOA also has poor results for engineering optimization problems, which makes further research necessary for them. Significantly, the value of individual scent intensity should be a non-negative in nature without consideration in the basic BOA and others. Thus, a novel hybrid particle swarm butterfly algorithm (HPSBA) is proposed on the basis of HPSOBOA. The main contributions and highlights are as follows:

• A novel hybrid particle swarm butterfly algorithm is proposed. This combination strikes a balance between exploitation and exploration. We design that the control strategy of parameter c is based on Logistic map, and the parameter $\omega$ is based on adaptive adjustment strategy of the HPSBA for improving the optimization speed, convergence accuracy and global search capability. Moreover, the individual scent intensity value is calculated with absolute value of the proposed HPSBA.
• To ensure that the proposed algorithm works, we compare the optimization results of twenty-six benchmark functions with ten intelligent optimization algorithms. According to the mean value (Mean), standard deviation (Std), Wilcoxon rank-sum (WRS) test findings, and convergence curves, the simulation results show that HPSBA has a competitive overall performance.
• The node optimization coverage problem of the WSN is solved using the proposed HPSBA. The application and advantages of the HPSBA are also discussed.

The remaining sections of this study are as follows: The mathematical model for the WSN’s node coverage optimization (NCO) problem is established in Section 2, which goes over the underlying concepts of the PSO and BOA. The proposed HPSBA is explained in detail in Section 3. The outcomes of the algorithms’ comparison experiments are presented in Section 4. In Section 5, HPSBA is applied to solve the WSN’s NCO problem. Section 6 concludes with a discussion of the next steps.

2. Basic Knowledge

2.1. Particle Swarm Optimization

There are two important characteristics of PSO algorithm [1] are the position and velocity of the particles. The position and velocity of the particles are updated as Equations (1) and  (2).

$$v_i^{t+1}=\omega ·v_i^t+c_1·ran{d}_1\times (p_{best}-x_i^t)+c_2·ran{d}_2\times (g_{best}-x_i^t)$$

(1)

$$x_i^{t+1}=x_i^t+v^{t+1}$$

(2)

where $v_i^t$ and $v_i^{t+1}$ are the velocity of the i-th particle when the iteration number is t and $t+1$, respectively. $p_{best}$ and $g_{best}$ represent the initial individual best position and global best position of the particle. $ran{d}_1$ and $ran{d}_2$ are the random number in (0, 1), and usually $c_1=c_2=2$. $\omega$ is the inertia weight coefficient.

2.2. Butterfly Optimization Algorithm

In BOA [10], each butterfly in the group has a unique sense and individual perception ability. The intensity of fragrance perception is generated between individuals. Figure 2 presents the food foraging of butterflies in the 2-D search space.

The following is an expression of the intensity of scent that other butterflies perceive:

$$F\left(x\right)=c{I}^a$$

(3)

where $F\left(x\right)$ denotes the scent intensity function, c denotes the sensory modality, and I indicates stimulus intensity, that is, the function fitness value. a denotes the intensity factor, and the value range of parameter a is [0, 1]. The sensory modality c is calculated as follows:

$$c_{t+1}=c_t+[0.025/(c_t·T_{max})]$$

(4)

where the initial value of c is set to 0.01 in basic BOA [10]. However, the parameter c can be set to any value within $[0,\infty )$ in theory. $T_{\max }$ is the maximum number of iterations.

The switching probability $SP$ determines the global search and local search of the BOA. The position update formula is expressed as follows:

$$x_i^{t+1}=\left\{\begin{array}{c}{x}_i^t+(r^2\times g^{\ast }-x_i^t)\times F_i,SP\ge rand\hfill \\ x_i^t+(r^2\times x_j^t-x_k^t)\times F_i,SP<rand\hfill \end{array}\right.$$

(5)

where $x_i^t$ denotes the spatial position of the i-th butterfly in the t-th iteration. $g\ast$ is the best position of all butterfly individuals in the current iteration. $x_j^t$ and $x_k^t$ are the position of the j-th and k-th butterfly individual when the iteration number is t, respectively. r is a random number in (0, 1), and $F_i$ is the scent intensity value of the i-th butterfly.

2.3. Node Coverage Optimization Problem Model

For the two-dimensional point coverage problem in WSN [39], it is assumed that there are n detection points to be covered in the two-dimensional coverage area, and the coverage nodes use homogeneous sensors, that is, the sensors have the same sensing radius. Supposing the sensing radius is $r_s$, thus $r_c$ is regarded as the communication radius, and the unit of them is meter. The sensing radius is the maximum distance at which the received signal strength of a node is greater than the inherent noise, that is, the sensing range of the node. The communication radius is the maximum distance for transmitting data or signals between nodes, generally.

Assuming that the monitoring area contains n target points, the position coordinates of the i-th target point to be monitored are $(x_i,y_i)$, and the sensor nodes s position coordinates are $(x_s,y_s)$. Then, the Euclidean distance that the sensor can cover the target to be monitored can be expressed as:

$$d(i,s)=\sqrt{{(x_s-x_i)}^2+{(y_s-y_i)}^2}$$

(6)

The binary perception model [48,49] is used in this study, and the sensor node s covers the probability p that the target node i will be monitored, it can be defined as:

$$p(i,s)=\left\{\begin{array}{c}0,d(i,s)\ge r_s\hfill \\ 1,d(i,s)<r_s\hfill \end{array}\right.$$

(7)

We divide the two-dimensional plane area to be deployed along the x and y axes with the step length q, and then the length of each segment is $l=q$, and the intersection of the coverage area is $q^2$. The probability that the monitoring point set T in the coverage area is interpreted by the node set S. The node coverage rate is defined as:

$$C\mathrm{ov}=\frac{p_{cov}}{q^2}=\frac{{\displaystyle \sum _{i=1}^S}p(i,s)}{q^2}$$

(8)

Assuming that the coverage area is a square, the side length is L, and $r_s$ denotes the node sensing radius. Theoretically, the number of deployed nodes can be calculated in the coverage area. The schematic diagram of the nodes full coverage is as Figure 3.

In Figure 3, $O_1,O_2$ and $O_3$ indicate the positions of the three nodes, the triangle $O_1{O}_2{O}_3$ is an equilateral triangle where $O_{3}A=r_s$, that is, sensing radius of the sensor, $\angle A{O}_{3}B=\pi /3$, $AB=B{O}_3=O_{3}A=r_s$. According to the nature of the circle, $AB\perp O_2{O}_3,$$\angle A{O}_{3}C=1/2\angle A{O}_{3}B=\pi /6$. According to the law of cosines, the length of the line segment $O_{3}C$ can be expressed as follows:

$$L_{O_{3}C}=L_{O_{3}A}\times cos(\angle A{O}_{3}C)=r_s\times cos(\pi /6)=\frac{\sqrt{3}}{2}{r}_s$$

(9)

Thus, the number of nodes in the coverage area can be calculated by Equation (10) in theory.

$$M={\left(\frac{L}{\sqrt{3}/2·r_s+r_s}+1\right)}^2$$

(10)

According to the above analysis, the NCO problem can be simplified as a constrained optimization problem, the expression is as follows:

$$maxf\left(x\right)=Cov, \mathrm{s}.\mathrm{t}\left\{\begin{array}{c}{g}_1=\sum _{i=1}^{S}p(i,s)\ge 0,\hfill \\ g_2=\sum _{i=1}^{S}p(i,s)-q^2\ge 0,\hfill \\ g_3=d(i,s)-r_s\ge 0,\hfill \\ g_4=S-M\ge 0.\hfill \end{array}\right.$$

(11)

where $r_s$ denotes perception radius of the node, $p(i,s)$ indicates the probability of the target node i monitored and covered by the sensor node s, and $d(i,s)$ indicates the Euclidean distance between sensor node s and monitored target node i. M is the theoretical number of nodes in the coverage area, and S is the set of coverage nodes in the monitoring area.

3. Method

In this section, a novel HPSBA is proposed to improve the optimization speed, convergence accuracy and global search capability between PSO and BOA. It is possible to strike a balance between exploitation and exploration and benefits strengths of both algorithms by combining them. Significantly, the individual scent intensity value is calculated with absolute value of the proposed HPSBA. Furthermore, HPSBA is used to solve the node optimization coverage problem of the WSN for Internet of Things (IoT).

3.1. Hybrid Particle Swarm Butterfly Algorithm (HPSBA)

3.1.1. Algorithmic Population Initialization

Supposing the expression of randomly generated initial solution is defined as follow in the D-dimensional search space.

$$X_i=L_b+(U_b-L_b)·rand$$

(12)

where $X_i$ presents the spatial position of the i-th butterfly individual $(i=1,2,3,\cdots ,N)$ in the butterfly swarm, N denotes the number of initial individual solutions. $L_b$ and $U_b$ are the upper and lower bounds of the search space, and $rand$ is a random number matrix between (0, 1).

3.1.2. Algorithmic Exploration

The exploration stage of the proposed HPSBA is expressed as:

$$V_{i-1}^t=\omega V_{i-1}^t+C_1{r}_1\times (p_b-X_{i-1}^t)+C_2{r}_2\times (g_b-X_{i-1}^t)$$

(13)

$$X_i^t=X_{i-1}^t+V_{i-1}^t$$

(14)

where $\omega$ is the inertia weight coefficient. $C_1$ and $C_2$ indicate adjustment parameters, respectively. $X_i^t$ and $X_{i-1}^t$ represent the position of the i-th and $(i-1)$-th agent at t. $V_i^t$ and $V_{i-1}^t$ are the velocity of the i-th and $(i-1)$-th agent when t-th iteration, respectively. $p_b$ and $g_b$ represent the initial global best position of the agent. $r_1$ and $r_2$ are the random number in (0, 1).

3.1.3. Algorithmic Exploitation

The exploitation stage of the proposed HPSBA is expressed as:

$$X_i^{t+1}=\left\{\begin{array}{c}\hfill \omega ·X_i^t+r^2·(g_b-X_i^t)\times |F_i|,SP\ge rand\\ \hfill \omega ·X_i^t+r^2·(X_k^t-X_j^t)\times \left|F_i\right|,SP<rand\end{array}\right.$$

(15)

where $\omega$ indicates adaptive adjustment parameter. $X_i^{t+1}$ and $X_i^t$ represent the position of the i-th particle at $t+1$ and t, respectively. $X_j^t$ and $X_k^t$ are the positions of the j-th and k-th individuals randomly selected from the solution. $F_i$ is the scent intensity value of the i-th individual. Most notably, the value of individual scent intensity is a non-negative, thus we take the absolute value of $F_i$ in the proposed HPSBA.

3.1.4. The Chaotic Adjusting Strategies

Chaos theory has a lot of application research in SI algorithms, such as chaotic population initialization [38], chaotic adjusting strategies of the control parameters [50], etc. The expression of the Logistic map is defined in Ref. [51]. The chaotic sequence of the Logistic map is (0, 1). When $\mu =4$, the mapping will produce strong chaotic phenomena. In this paper, the control parameter c of the proposed HPSBA is expressed as:

$$c\left(t\right)=4·c·(1-c)$$

(16)

The inertial weight coefficient $\omega$ has a direct impact on the particle flight speed of the PSO algorithm, and can modify the algorithm’s global and local search capabilities. We adopt an adaptive adjustment strategy with chaos, and its expression is as follows:

$$\omega \left(t\right)={\omega }_u-({\omega }_u-{\omega }_l)·t/T_{max}$$

(17)

where ${\omega }_u$ = 0.9, ${\omega }_l$ = 0.2, $T_{max}$ is the maximum iteration number of the algorithm. For the control parameters c and $\omega$, $T_{max}=500$ and different $c\left(0\right)$ values are taken, and the corresponding chaotic sequences are shown in Figure 4. For selecting the best initial value of the control parameters, Schwefel 1.2 and Solomon functions are used to perform optimization tests. The optimization results are shown in

Table 1. Optimal value with different c(0) values.

c(0)	Schwefel 1.2			Solomon
	Mean	Std	Time/s	Mean	Std	Time/s
0.15	5.28E-299	0	1.15	3.85E-301	0	0.17
0.25	4.75E-300	0	1.11	1.49E-223	0	0.17
0.35	7.70E-300	0	1.12	5.73E-301	0	0.17
0.45	3.49E-299	0	1.11	1.87E-301	0	0.17
0.55	4.54E-299	0	1.13	2.40E-301	0	0.17
0.65	3.16E-299	0	1.11	8.90E-301	0	0.17
0.75	7.87E-298	0	1.11	9.75E-222	0	0.17
0.85	4.82E-299	0	1.12	4.40E-301	0	0.17
0.95	3.16E-299	0	1.12	1.93E-301	0	0.17

.

As seen from Figure 4, when $c\left(0\right)=0.25$ and $c\left(0\right)=0.75$, according to the property of Logistic map, the control strategy of parameter c falls into a fixed point. That is, the initial value of parameter c cannot be set 0.25 or 0.75 in this study.

According to Figure 4 and Table 1, when $c\left(0\right)=0.35$, HPSBA can obtain the best value for the Schwefel 1.2 function (Uni-modal). For the Solomon function (Multi-modal), although the optimal value is not the best with $c\left(0\right)=0.35$, its search value is in the same order of magnitude as the best search value. They take the same time to search the optimal value. In summary, the initial value of parameter c is set to 0.35 in the following experiments.

Figure 5 shows the optimization process of the proposed algorithm. From Figure 5, the new fitness value ($F_{new}$) and new agent positions are obtained in the exploration stage, which is the local optimum. Then, according to the parameters value in exploration, the proposed HPSBA will get the global optimum in the theory. Finally, the best fitness and agent position are output.

3.2. Complexity Analysis of the HPSBA

To better understand computational complexity the proposed HPSBA, the time and space complexity of HPSBA are given in this section.

3.2.1. Time Complexity

Assuming that the population size is n, the search space dimension is d, and the maximum iteration is $T_{max}$. The complexity of the HPSBA includes: the population initialization complexity is $O\left(nd\right)$, the fitness value calculation complexity is $O\left(nd\right)$, the global and local search location update complexity is $O(n^{2}logn)$, the fitness value sorting complexity is $O\left(n^2\right)$, and the control parameter update complexity of the algorithm is $O\left(nd\right)$. The HPSBA’s total time complexity can be shown as follows by looking at all of the aforementioned components:

$$O\left(HPSBA\right)=O\left(nd\right)+O\left(T_{max}\right)O\left(nd+n^2{log}_n+n^2+nd\right)$$

(18)

The time complexity of BOA is:

$$O\left(BOA\right)=O\left(nd\right)+O\left(T_{max}\right)O\left(n^2{log}_n+n+nd\right)$$

(19)

3.2.2. Space Complexity

The space complexity of an algorithm is regarded as the storage space consumed by the algorithm. The population size is n and the dimension is d. The hybrid algorithm is used to calculate the space complexity. The total space complexity of the proposed HPSBA is $O\left(nd\right)$. The butterfly optimization algorithm uses n search agents to calculate the space complexity, and the total space complexity of the BOA is $O\left(nd\right)$. Therefore, the total space complexity of the basic BOA is the same as the total space complexity of the HPSBA. Therefore, the proposed algorithm has a reliable and effective space efficiency.

3.3. The Pseudo-Code and Flowchart of the HPSBA

The pseudo-code of the HPSBA is presented in Algorithm 1.

Algorithm 1: Pseudo-code of HPSBA

In addition, the flowchart of the proposed HPSBA is shown as Figure 6.

3.4. Convergence Analysis of the HPSBA

Theorem 1. 

The population position vector sequence $\left\{X\right(t),t\ge 0\}$ of the proposed hybrid HPSBA method is a finite homogeneous Markov process.

Proof. 

The search space of any optimization algorithm is limited, so the population position vector sequence $\left\{X\left(t\right),t\ge 0\right\}$ of the hybrid particle swarm butterfly algorithm is also limited. In addition, the position vector of the population in the optimization process is determined by the odor behavior $F_i\left(t\right)$ and the flight speed $V_i\left(t\right)$. It can be seen that $X(t+1)$ is only related to $X\left(t\right)$, namely $\left\{X\left(t\right),t\ge 0\right\}$ is a Markov chain. Individuals gradually approach the optimal position base on the search space’s fitness value, that is, when $f\left(x_{t+1}\right)>f\left(x_t\right)$, the movement of the population is adjusted, which only has to do with time t. In summary, the population position vector sequence $\left\{X\left(t\right),t\ge 0\right\}$ of the HPSBA is a finite homogeneous Markov process. □

The essence of the HPSBA belongs to the category of random search algorithm, so the convergence criterion of random optimization algorithm [52] is used to prove the convergence of the hybrid algorithm HPSBA.

3.4.1. Convergence Criterion

For the problem $<Y,f>$, there is a random optimization algorithm Z [53]. The result of the k-th iteration is $x_k$, and the result of the next iteration is $x_{k+1}=D(x_k,\zeta )$. Y represents the space of potential solutions, f denotes the fitness function, and $\zeta$ is the solution searched in Z iteration of the algorithm. The search’s lower bound of the Lebesgue measure space [52] is defined as follows:

$$\sigma =inf\left\{t\right|U(x\in Y|f\left(x\right)<t)>0\}$$

(20)

where $U\left(X\right)$ denotes the Lebesgue measure on the set X, and the optimum can be defined as:

$$R_{\xi ,M}=\left\{\begin{array}{c}\hfill \{x\in Y|f\left(x\right)<\sigma +\xi \},\sigma limited\\ \hfill \{x\in Y|f\left(x\right)<-ϵ\},\sigma =-\infty \end{array}\right.$$

(21)

where $\xi$ denotes greater than zero, and $ϵ$ is a sufficiently large positive number. If the algorithm can find a point in $R_{\xi ,M}$, that is, the algorithm may have reached an acceptable global optimal point or an approximate global optimal point.

Condition 1:$f\left(Z\right(x,\zeta \left)\right)\le f\left(x\right)$, and if $\zeta \in Y,f\left(Z\right(x,\zeta \left)\right)\le f\left(\zeta \right)$.

Condition 2: If $\forall B\in Y$, s.t. $U\left(B\right)>0$, then

$$\prod _{k=0}^{\infty }(1-U_k\left(B\right))=0$$

(22)

where $U_k\left(B\right)$ denotes the measure of probability of algorithm Z searching solution on set B at the k-th iteration.

Theorem 2. 

(Conditions necessary and sufficient for global convergence) Supposing f is measurable, the measurable space Y denotes a measurable subset of $R_n$, and algorithm Z fulfills both Conditions 1 and 2, and ${\left\{x_k\right\}}_{k=0}^{\infty }$ is the algorithm Z used to generate the solution sequence. So there is a probability measure:

$$\underset{k\to \infty }{lim}P(x_k\in R_{\xi ,M})=1$$

(23)

Thus, the algorithm Z converges globally. $P(x_k,R_{\xi ,M})$ is the probability measure of the solution $x_k$ in $R_{\xi ,M}$ of the iterative search step of the algorithm.

3.4.2. Convergence Analysis

Lemma 1. 

Condition 1 is met by HPSBA, which states that the hybrid algorithm’s direction of population optimization is monotonic.

Lemma 2. 

The HPSBA population state space’s general state has a transition probability of one to the optimal state, that is, ${\displaystyle \underset{t\to \infty }{lim}}{P}^{\left(t\right)}({\zeta }_i\to {\zeta }_j)=1$.

Proof. 

Assuming the population state $\zeta \left(j\right)$ is the optimal solution, if the algorithm converges, after infinite state transitions, the probability of its state space from the general state to the optimal state should be 1. Due to

$$\underset{t\to \infty }{lim}{P}^{\left(t\right)}({\zeta }_i\to {\zeta }_j)=\sum _{k=1}^N{P}^{\left(t\right)}({\zeta }_{ik}\to {\zeta }_{jk})$$

(24)

Each iteration of the HPSBA population state is based on the transfer of individual odor to the optimal state, that is, the position of the worst individual state of the population is updated. Therefore

$$\underset{t\to \infty }{lim}{P}^{\left(t\right)}({\zeta }_i\to {\zeta }_j)=1$$

(25)

□

Lemma 3. 

HPSBA satisfies Condition 2.

Theorem 3. 

HPSBA converges to the global optimum, namely ${\displaystyle \underset{t\to \infty }{lim}}P\{X\left(t\right)\in G|X\left(0\right)={\Phi }_0\}=1$.

Proof. 

Since HPSBA satisfies Condition 1 and Condition 2, in each iteration of the algorithm, the individual will choose to update the retention mechanism of the optimal individual. That is, when the iteration has a tendency to be infinite, ${lim}_{n\to \infty }P(x_k,R_{\xi ,M})$ = 1, and ${\left\{x_k\right\}}_{k=0}^{\infty }$ is the solution sequence generated by HPSBA iteration. It can be concluded from Theorem 2 that HPSBA is a globally convergent algorithm. □

4. Analyses and Results of Numerical Optimization

Twenty-six benchmark functions are used to test the proposed hybrid algorithm performance, which are listed in

Table 2. Twenty-six benchmark functions.

Name	Formula	Search Range	Dim	${\mathit{f}}_{\mathbf{min}}$	Category
Sphere	$F_1={\displaystyle \sum _{i=1}^{Dim}}{x}_i^2$	[−100,100]	30/100	0	U
Schwefel 2.22	$F_2={\displaystyle \sum _{i=1}^{Dim}}\left|x_i\right|+{\displaystyle \prod _{i=1}^{Dim}}\left|x_i\right|$	[−10,10]	30/100	0	U
Schwefel 1.2	$F_3={\displaystyle \sum _{i=1}^{Dim}}{\left({\displaystyle \sum _{j=1}^i}{x}_j\right)}^2$	[−10,10]	30/100	0	U
Schwefel 2.21	$F_4=max\left\{\left|x_i\right|,1\le i\le Dim\right\}$	[−10,10]	30/100	0	U
Step	$F_5={\displaystyle \sum _{i=1}^{Dim}}{\left(x_i+0.5\right)}^2$	[−10,10]	30/100	0	U
Quartic	$F_6={\displaystyle \sum _{i=1}^{Dim}}i{x}_i^4+rand(0,1)$	[−1.28,1.28]	30/100	0	U
Exponential	$F_7=exp\left(0.5{\displaystyle \sum _{i=1}^{Dim}}{x}_i\right)$	[−10,10]	30/100	0	U
Sum Power	$F_8={\displaystyle \sum _{i=1}^{Dim}}{\left|x_i\right|}^{(i+1)}$	[−1,1]	30/100	0	U
Sum Square	$F_9={\displaystyle \sum _{i=1}^{Dim}}i{x}_i^2$	[−10,10]	30/100	0	U
Rosenbrock	$F_{10}={\displaystyle \sum _{i=1}^{Dim}}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right)$	[−10,10]	30/100	0	U
Zakharov	$F_{11}={\displaystyle \sum _{i=1}^{Dim}}{x}_i^2+{\left({\displaystyle \sum _{i=1}^{Dim}}0.5i{x}_i\right)}^2+{\left({\displaystyle \sum _{i=1}^{Dim}}0.5i{x}_i\right)}^4$	[−5.12,5.12]	30/100	0	U
Trid	$F_{12}={(x_1-1)}^2+\sum _{i=2}^{Dim}i{(2x_i^2-x_{i-1})}^2$	[−5,5]	30/100	0	U
Elliptic	$F_{13}={\displaystyle \sum _{i=1}^{Dim}}{\left({10}^6\right)}^{\frac{i-1}{D-1}}{x}_i^2$	[−100,100]	30/100	0	U
Cigar	$F_{14}=x_1^2+{10}^6{\displaystyle \sum _{i=2}^{Dim}}{x}_i^2$	[−100,100]	30/100	0	U
Tablet	$F_{15}={10}^6{x}_1^2+{\displaystyle \sum _{i=2}^{Dim}}{x}_i^6$	[−10,10]	30/100	0	U
Rastrigin	$F_{16}={\displaystyle \sum _{i=1}^{Dim}}\left(x_i^2-10cos\left(2\pi x_i\right)+10\right)$	[−5.12,5.12]	30/100	0	M
NCRastrigin	$F_{17}={\displaystyle \sum _{i=1}^{Dim}}\left(y_i^2-10cos\left(2\pi y_i\right)+10\right),y_i=\left\{\begin{array}{c}\hfill x_i,\left|x_i\right|<0.5,\\ \hfill round\left(2x_i\right)/2,\left|x_i\right|<0.5\end{array}\right.$	[−5.12,5.12]	30/100	0	M
Ackley	$F_{18}=-20exp\left(-0.2\sqrt{\frac{1}{Dim}{\displaystyle \sum _{i=1}^{Dim}}{x}_i^2}\right)-exp\left(\frac{1}{Dim}{\displaystyle \sum _{i=1}^{Dim}}cos\left(2\pi x_i\right)\right)+20+e$	[−20,20]	30/100	0	M
Griewank	$F_{19}=\frac{1}{4000}{\displaystyle \sum _{i=1}^{Dim}}{x}_i^2-{\displaystyle \prod _{i=1}^{Dim}}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	[−600,600]	30/100	0	M
Alpine	$F_{20}={\displaystyle \sum _{i=1}^{Dim}}\left|x_i·sin\left(x_i\right)+0.1x_i\right|$	[−10,10]	30/100	0	M
Penalized 1	$\begin{array}{l}{F}_{21}=\frac{\pi }{Dim}\left\{{\displaystyle \sum _{i=1}^{Dim-1}}{(y_i-1)}^2[1+10{sin}^2\left(\pi y_{i+1}\right)]+{\left(y_{Dim-1}\right)}^2+10{sin}^2\left(\pi y_1\right)\right\}+{\displaystyle \sum _{i=1}^{Dim}}u(x_i,10,100,4),\\ y_i=1+\frac{x_i+1}{4},u_{y_i,a,k,m}=\left\{\begin{array}{c}\hfill k{(x_i-a)}^m,x_i>a,\\ \hfill 0,-a\le x_i\le a,\\ \hfill k{(-x_i-a)}^m,x_i<a\end{array}\right.\end{array}$	[−10,10]	30/100	0	M
Penalized 2	$\begin{array}{l}{F}_{22}=\frac{1}{10}\left\{{sin}^2\left(\pi x_1\right)+\sum _{i=1}^{Dim-1}{(x_i-1)}^2\left[1+{sin}^2\left(3\pi x_{i+1}\right)\right]+{\left(x_{Dim-1}\right)}^2\left(1+{sin}^2\left(2\pi x_{i+1}\right)\right)\right\}+\\ {\displaystyle \sum _{i=1}^{Dim}}u(x_i,5,100,4)\end{array}$	[−5,5]	30/100	0	M
Levy	$F_{23}={sin}^2\left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^{Dim-1}}{(x_i-1)}^2[1+{sin}^2\left(3\pi x_{i+1}\right)]+\left|x_{Dim}-1\right|·[1+{sin}^2\left(2\pi x_{Dim}\right)]$	[−2,2]	30/100	0	M
Weierstrass	$F_{24}={\displaystyle \sum _{i=1}^{Dim}}\left({\displaystyle \sum _{k=0}^{k_{max}}}\left[a^{k}cos\left(2\pi b^k(x_i+0.5)\right)\right]\right)-Dim{\displaystyle \sum _{k=0}^{k_{max}}}\left[a^{k}cos\left(2\pi b^k·0.5\right)\right],a=0.5,b=3,k_{max}=20$	[−1,1]	30/100	0	M
Solomon	$F_{25}=1-cos\left(2\pi \sqrt{{\displaystyle \sum _{i=1}^{Dim}}{x}_i^2}\right)+0.1\sqrt{{\displaystyle \sum _{i=1}^{Dim}}{x}_i^2}$	[−20,20]	30/100	0	M
Bohachevsky	$F_{26}={\displaystyle \sum _{i=1}^{Dim}}\left[x_i^2+2x_{i+1}^2-0.3·cos\left(3\pi x_i\right)-0.4·cos\left(4\pi x_{i+1}\right)+0.7\right]$	[−5,5]	30/100	0	M

. There are two categories of the benchmark functions. F1–F15 are unimodal functions, which notes the category as U. F16–F26 are multimodal functions, which notes the category as M.

4.1. Parameter Setting of Comparison Algorithms

To verify the performance of HPSBA for solving numerical optimization problems, ten algorithms are employed as competitors for the experiment. The proposed algorithm is compared with five standard algorithms (they are PSO [1], GWO [3], BOA [10], EO [19], MPA [5]), and four BOA variants (they are LBOA [34], CBOA [36], HPSOBOA [38], and IBOA [35]), and SOGWO [54].

The comparison algorithm parameter settings for numerical optimization experiments are shown in

Table 3. Parameter settings of the comparison algorithms.

Algorithms	Parameter Settings
PSO	$N=30,c_1=c_2=2,v_{max}=1,v_{min}=-1,\omega =0.7$
GWO	$N=30,a_{first}=2,a_{final}=0$
BOA	$N=30,a=0.1,c\left(0\right)=0.01,SP=0.6$
EO	$N=25,a_1=2,a_2=1,GP=0.5,\lambda \in (0,1)$
MPA	$N=30,p=0.5,FADs=0.2$
LBOA	$N=30,a=0.1,c\left(0\right)=0.01,p=0.6,\gamma =1.5$
CBOA	$N=30,a\left(0\right)=0.1,c\left(0\right)=0.01,p=0.6,r\left(0\right)=0.33,\mu =4$
HPSOBOA	$N=30,a_{first}=0.1,a_{final}=0.3,c_0=0.01,p=0.6,x_0=0.315,\rho =0.295,c_1=c_2=0.5$
IBOA	$N=30,a=0.1,c\left(0\right)=0.01,andpisdynamic.$
SOGWO	$N=50,a_{first}=2,a_{final}=0$
HPSBA	$N=30,a=0.1,c\left(0\right)=0.35,SP=0.6,\mu =4,{\omega }_u=0.9,{\omega }_l=0.2,C_1=C_2=2$

. All of the experimental series were carried out using MATLAB 2018a and an Intel(R) Core (TM) i5-10210U CPU @2.11G with 8G RAM in this study. Furthermore, to better set the number of nodes in the node optimization coverage experiment, the dimensions of the function are respectively set to 30 and 100 in the numerical optimization problems, that is, the setting range of the number of nodes in 2-D monitoring area.

4.2. Comparison Results of HPSBA with Others (Dim = 30)

In order to ensure the reasonableness and fairness of the comparison results, the dimension of the test function in the numerical optimization experiment is set to 30, and the maximum number of iterations is set to 500. For the same test function, each algorithm is independently run 30 times. The mean (Mean), standard deviation (Std) are calculated according to the statistical value.

Table 4. Comparison results of eleven algorithms: Dim = 30.

Functions		PSO	GWO	BOA	EO	MPA	LBOA	CBOA	HPSOBOA	IBOA	SOGWO	HPSBA
F1	Mean	1.16E-01	1.87E-27	7.70E-11	3.35E-40	5.72E-23	3.54E-12	2.20E-30	3.59E-152	7.47E-15	3.46E-33	3.29E-252
	Std	3.89E-02	3.11E-27	6.78E-12	1.70E-39	5.83E-23	3.57E-12	4.29E-30	7.89E-153	1.72E-15	7.28E-33	0.00E+00
F2	Mean	6.35E-01	9.50E-17	2.35E-08	6.78E-24	2.53E-13	1.24E-09	3.92E-19	5.04E-60	4.97E+12	8.25E-20	2.52E-134
	Std	2.05E-01	7.46E-17	6.49E-09	6.18E-24	2.68E-13	2.10E-09	6.67E-19	2.06E-59	1.22E+13	9.41E-20	6.87E-134
F3	Mean	4.16E+00	7.10E-08	5.31E-11	2.38E-11	2.71E-06	2.77E-12	1.11E-30	4.05E-153	9.32E-15	1.97E-09	7.60E-300
	Std	9.64E-01	1.69E-07	5.83E-12	1.04E-10	7.11E-06	2.71E-12	3.47E-30	1.13E-153	1.11E-15	8.79E-09	0.00E+00
F4	Mean	3.36E-01	7.05E-08	2.65E-08	2.10E-11	3.39E-10	2.48E-09	1.48E-19	1.05E-77	8.05E-12	1.82E-09	6.02E-152
	Std	4.82E-02	4.84E-08	2.94E-09	4.31E-11	2.12E-10	3.13E-09	2.49E-19	7.23E-79	1.06E-12	1.47E-09	1.69E-152
F5	Mean	7.12E-02	5.40E-01	5.23E+00	7.61E-06	3.42E-08	3.40E+00	4.55E+00	4.12E-02	3.36E+00	3.46E-01	5.38E+00
	Std	3.20E-02	3.28E-01	6.84E-01	6.22E-06	1.73E-08	6.50E-01	5.83E-01	2.40E-02	7.86E-01	2.66E-01	6.36E-01
F6	Mean	2.60E-01	1.79E-03	1.99E-03	1.36E-03	1.28E-03	1.92E-03	1.17E-04	2.31E-04	3.10E-04	1.20E-03	9.21E-05
	Std	8.64E-02	9.30E-04	5.51E-04	9.12E-04	7.43E-04	9.45E-04	1.17E-04	3.77E-04	2.58E-04	5.60E-04	9.73E-05
F7	Mean	0.00E+00	3.19E-58	4.94E-11	7.18E-66	7.18E-66	6.36E-21	3.84E-19	1.53E-62	7.09E-14	8.16E-61	8.51E-16
	Std	0.00E+00	1.20E-57	1.34E-10	1.02E-78	1.40E-69	2.34E-20	1.16E-18	6.12E-63	3.06E-13	4.36E-60	4.27E-15
F8	Mean	7.23E-07	1.75E-95	8.88E-14	1.97E-134	1.98E-60	8.27E-16	1.46E-36	1.02E-156	4.45E-19	2.00E-116	1.70E-307
	Std	1.17E-06	9.25E-95	5.52E-14	9.42E-134	4.89E-60	9.08E-16	6.48E-36	8.09E-158	2.44E-19	9.70E-116	0.00E+00
F9	Mean	7.51E-01	2.35E-28	6.94E-11	1.36E-41	4.83E-24	3.11E-12	8.60E-31	2.02E-152	9.61E-15	2.03E-34	3.19E-263
	Std	2.87E-01	4.02E-28	8.22E-12	3.56E-41	6.23E-24	4.11E-12	1.77E-30	2.89E-153	1.34E-15	2.28E-34	0.00E+00
F10	Mean	5.99E+01	2.72E+01	2.89E+01	2.53E+01	2.51E+01	2.88E+01	2.89E+01	2.71E+01	2.89E+01	2.68E+01	2.89E+01
	Std	3.87E+01	8.55E-01	2.70E-02	1.54E-01	3.89E-01	3.25E-02	3.73E-02	6.30E+00	3.53E-02	8.00E-01	3.37E-02
F11	Mean	1.47E+00	3.17E-28	6.67E-11	3.42E-41	1.23E-23	3.52E-12	2.81E-30	6.89E-153	8.32E-15	1.18E-33	1.28E-252
	Std	8.03E-01	5.15E-28	7.26E-12	1.26E-40	2.24E-23	3.22E-12	7.66E-30	1.17E-153	1.52E-15	2.59E-33	0.00E+00
F12	Mean	4.22E+00	6.67E-01	9.74E-01	6.67E-01	6.67E-01	9.18E-01	9.76E-01	1.00E+00	9.35E-01	6.67E-01	6.67E-01
	Std	1.82E+00	3.76E-05	8.43E-03	3.08E-10	3.87E-08	2.48E-02	9.00E-03	1.25E-05	1.81E-02	4.37E-06	1.86E-04
F13	Mean	6.08E-31	0.00E+00	2.80E-21	0.00E+00	3.55E-174	5.49E-26	1.77E-34	2.30E-148	8.87E-31	0.00E+00	2.44E-302
	Std	2.42E-30	0.00E+00	8.91E-21	0.00E+00	0.00E+00	1.14E-25	5.23E-34	1.14E-147	4.33E-30	0.00E+00	0.00E+00
F14	Mean	1.16E-24	2.82E-205	1.92E-17	7.30E-207	1.34E-61	3.31E-18	3.49E-31	1.95E-147	5.73E-23	4.59E-228	6.12E-296
	Std	2.26E-24	0.00E+00	2.08E-17	0.00E+00	7.35E-61	4.01E-18	7.94E-31	4.53E-147	1.22E-22	0.00E+00	0.00E+00
F15	Mean	3.02E-30	6.90E-261	4.54E-19	8.38E-255	8.23E-94	1.65E-19	1.09E-34	1.92E-153	3.69E-22	1.06E-313	3.61E-304
	Std	1.65E-29	0.00E+00	8.61E-19	0.00E+00	3.33E-93	3.98E-19	5.42E-34	6.79E-153	7.04E-22	0.00E+00	0.00E+00
F16	Mean	2.37E+02	4.02E+00	6.54E+01	1.89E-15	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	2.27E+00	0.00E+00
	Std	5.65E+01	3.88E+00	9.09E+01	1.04E-14	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	3.20E+00	0.00E+00
F17	Mean	2.76E+02	8.31E+00	1.24E+02	2.33E-01	3.96E-07	0.00E+00	0.00E+00	0.00E+00	0.00E+00	8.87E+00	0.00E+00
	Std	7.80E+01	4.39E+00	7.02E+01	6.26E-01	2.17E-06	0.00E+00	0.00E+00	0.00E+00	0.00E+00	5.03E+00	0.00E+00
F18	Mean	2.47E-01	9.05E-14	2.75E-08	8.47E-15	8.53E-13	2.47E-09	8.88E-16	8.88E-16	7.10E-12	4.14E-14	8.88E-16
	Std	7.83E-02	1.67E-14	2.47E-09	1.80E-15	5.41E-13	1.38E-09	0.00E+00	0.00E+00	7.10E-13	2.89E-15	0.00E+00
F19	Mean	3.41E+01	3.23E-03	9.73E-12	0.00E+00	0.00E+00	1.79E-13	0.00E+00	0.00E+00	4.18E-16	2.70E-03	0.00E+00
	Std	5.57E+00	8.81E-03	1.06E-11	0.00E+00	0.00E+00	3.96E-13	0.00E+00	0.00E+00	1.90E-15	5.90E-03	0.00E+00
F20	Mean	1.53E-01	4.74E-04	3.47E-09	6.29E-09	6.61E-14	6.42E-14	1.00E-19	9.42E-60	6.67E-12	3.52E-04	7.05E-136
	Std	9.79E-02	7.66E-04	7.60E-09	3.44E-08	4.58E-14	1.86E-13	1.28E-19	2.99E-59	7.69E-13	5.84E-04	3.83E-135
F21	Mean	7.09E+00	5.03E-02	5.39E-01	3.46E-03	7.59E-05	2.90E-01	4.78E-01	2.47E-03	1.48E+00	3.38E-02	5.49E-01
	Std	3.04E+00	2.12E-02	1.58E-01	1.89E-02	4.15E-04	9.21E-02	1.36E-01	2.64E-03	2.31E-01	1.49E-02	1.37E-01
F22	Mean	8.06E-03	7.07E-01	3.40E+00	2.18E-02	3.45E-03	2.37E+00	3.00E+00	4.07E+00	2.63E+00	5.15E-01	3.42E+00
	Std	4.77E-03	2.10E-01	4.83E-01	4.72E-02	1.65E-02	6.28E-01	5.74E-01	2.15E+00	5.90E-01	1.88E-01	5.79E-01
F23	Mean	3.88E-01	1.67E+00	1.18E+01	1.52E-01	1.38E-01	9.31E+00	1.01E+01	8.89E-01	1.06E+01	1.25E+00	1.09E+01
	Std	2.43E-01	1.02E+00	2.10E+00	3.22E-01	1.89E-01	2.67E+00	2.71E+00	1.18E+00	1.94E+00	8.19E-01	3.23E+00
F24	Mean	5.70E+00	4.93E+00	1.23E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	4.61E+00	0.00E+00
	Std	2.76E+00	2.04E+00	2.35E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.67E+00	0.00E+00
F25	Mean	7.11E-01	2.79E-01	7.66E-01	9.95E-02	9.95E-02	2.99E-02	1.68E-32	6.43E-02	9.95E-02	2.89E-01	4.07E-301
	Std	3.41E-01	1.49E-01	2.18E-01	2.08E-12	7.05E-17	4.64E-02	3.64E-32	4.99E-02	1.24E-06	1.46E-01	0.00E+00
F26	Mean	1.24E+00	0.00E+00	8.02E-11	0.00E+00	0.00E+00	4.45E-12	0.00E+00	0.00E+00	9.79E-15	0.00E+00	0.00E+00
	Std	5.92E-01	0.00E+00	8.59E-12	0.00E+00	0.00E+00	5.45E-12	0.00E+00	0.00E+00	1.33E-15	0.00E+00	0.00E+00
+/-/≈		1/25/0	0/24/2	0/26/0	1/20/5	4/18/4	0/23/3	0/20/6	0/20/6	0/23/3	1/22/3	∼
Rank		7	6	8	3	2	4	3	3	5	4	1

shows the comparison results of eleven algorithms with Dim = 30. Results of Wilcoxon rank-sum test calculated at a significance level of $\alpha =0.05$ are also listed in Table 4 and

Table 5. Comparison results of eleven algorithms: Dim = 100.

Functions		PSO	GWO	BOA	EO	MPA	LBOA	CBOA	HPSOBOA	IBOA	SOGWO	HPSBA
F1	Mean	8.45E+00	1.48E-12	8.75E-11	4.14E-29	1.77E-19	6.12E-12	6.86E-30	3.31E-152	9.05E-15	6.44E-15	7.12E-299
	Std	7.28E-01	1.83E-12	9.69E-12	5.38E-29	2.10E-19	7.39E-12	1.51E-29	1.35E-152	1.87E-15	8.86E-15	0.00E+00
F2	Mean	1.82E+01	4.04E-08	4.71E+50	1.85E-17	1.43E-11	1.52E+50	3.74E-18	1.48E+36	8.10E+49	1.60E-09	4.35E+50
	Std	2.23E+00	1.35E-08	1.92E+51	1.36E-17	1.15E-11	6.27E+50	6.73E-18	2.80E+35	3.95E+50	6.14E-10	1.42E+51
F3	Mean	2.80E+02	4.54E+00	6.08E-11	3.15E-01	1.04E-01	3.40E-12	4.42E-31	1.48E-152	9.74E-15	1.39E+00	1.06E-299
	Std	7.38E+01	4.04E+00	5.91E-12	1.11E+00	1.50E-01	3.36E-12	8.91E-31	1.35E-152	1.11E-15	1.98E+00	0.00E+00
F4	Mean	1.77E+00	5.79E-02	2.98E-08	2.64E-01	2.40E-08	2.86E-09	1.20E-19	1.08E-77	8.47E-12	1.60E-02	7.43E-152
	Std	1.71E-01	6.77E-02	2.71E-09	1.45E+00	1.16E-08	2.88E-09	1.58E-19	5.19E-79	1.17E-12	1.52E-02	2.49E-152
F5	Mean	5.47E+00	9.20E+00	2.21E+01	2.94E+00	2.56E+00	2.03E+01	2.12E+01	1.48E-01	2.07E+01	7.67E+00	2.24E+01
	Std	9.43E-01	9.87E-01	1.06E+00	5.46E-01	7.71E-01	1.60E+00	1.18E+00	1.27E-01	1.06E+00	9.06E-01	8.81E-01
F6	Mean	6.02E+01	7.14E-03	2.11E-03	2.44E-03	1.87E-03	2.01E-03	1.03E-04	1.13E-04	2.77E-04	4.90E-03	6.85E-05
	Std	1.38E+01	2.79E-03	8.96E-04	1.41E-03	9.55E-04	1.18E-03	7.42E-05	1.28E-04	2.49E-04	1.70E-03	6.09E-05
F7	Mean	0.00E+00	9.84E-135	1.71E-23	7.16E-218	1.70E-202	2.74E-28	4.05E-32	1.93E-207	1.92E-20	2.96E-155	4.15E-27
	Std	0.00E+00	5.39E-134	8.68E-23	0.00E+00	0.00E+00	1.39E-27	1.12E-31	0.00E+00	8.47E-20	1.62E-154	2.27E-26
F8	Mean	9.97E-02	1.28E-66	7.32E-14	1.16E-129	2.23E-60	9.96E-16	4.86E-37	9.98E-157	4.00E-19	2.38E-64	1.82E-306
	Std	2.20E-01	4.84E-66	6.04E-14	6.03E-129	4.85E-60	1.62E-15	1.49E-36	7.46E-158	2.59E-19	1.30E-63	0.00E+00
F9	Mean	2.82E+02	6.63E-13	8.60E-11	1.94E-29	8.17E-20	3.97E-12	1.13E-30	4.23E-152	9.06E-15	1.38E-15	4.72E-299
	Std	5.17E+01	5.34E-13	8.92E-12	2.56E-29	5.83E-20	3.66E-12	2.60E-30	1.02E-152	1.95E-15	1.01E-15	0.00E+00
F10	Mean	1.24E+03	9.79E+01	9.89E+01	9.66E+01	9.69E+01	9.88E+01	9.89E+01	9.07E+01	9.89E+01	9.77E+01	9.89E+01
	Std	2.28E+02	5.73E-01	2.93E-02	1.08E+00	8.66E-01	3.76E-02	4.76E-02	2.30E+01	3.48E-02	7.22E-01	3.49E-02
F11	Mean	3.65E+02	7.60E-13	8.13E-11	3.57E-29	3.77E-20	4.82E-12	3.26E-30	9.41E-153	1.00E-14	1.93E-15	4.59E-299
	Std	7.04E+01	6.45E-13	6.27E-12	1.09E-28	3.04E-20	3.99E-12	1.45E-29	5.44E-153	1.66E-15	1.49E-15	0.00E+00
F12	Mean	6.01E+02	6.67E-01	9.98E-01	6.67E-01	6.67E-01	9.95E-01	9.98E-01	1.00E+00	9.96E-01	6.67E-01	9.99E-01
	Std	1.55E+02	3.47E-05	8.04E-04	3.93E-08	1.41E-06	9.93E-04	5.10E-04	8.11E-05	8.74E-04	5.49E-06	4.19E-04
F13	Mean	1.06E-33	0.00E+00	5.13E-22	0.00E+00	3.61E-169	3.15E-25	3.19E-34	1.33E-150	4.62E-31	0.00E+00	1.53E-302
	Std	5.50E-33	0.00E+00	1.69E-21	0.00E+00	0.00E+00	8.18E-25	1.63E-33	3.20E-150	1.56E-30	0.00E+00	0.00E+00
F14	Mean	6.38E-24	1.24E-205	3.84E-17	2.66E-201	1.86E-63	4.88E-18	3.01E-31	6.40E-150	9.63E-23	2.87E-186	3.05E-298
	Std	1.76E-23	0.00E+00	6.24E-17	0.00E+00	1.02E-62	7.45E-18	6.15E-31	1.34E-149	2.26E-22	0.00E+00	0.00E+00
F15	Mean	1.88E-32	8.39E-261	1.23E-19	3.97E-253	9.63E-92	3.93E-19	1.55E-34	1.71E-153	3.26E-22	6.24E-310	1.69E-303
	Std	5.59E-32	0.00E+00	2.55E-19	0.00E+00	4.94E-91	9.39E-19	8.12E-34	5.23E-153	4.34E-22	0.00E+00	0.00E+00
F16	Mean	4.87E+02	9.55E+00	1.77E-06	3.79E-15	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	9.04E+00	0.00E+00
	Std	6.30E+01	8.37E+00	9.70E-06	2.08E-14	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	5.80E+00	0.00E+00
F17	Mean	4.34E+02	2.34E+01	7.57E+01	1.00E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00	5.92E-17	1.32E+01	0.00E+00
	Std	6.22E+01	2.09E+01	2.31E+02	3.05E-01	0.00E+00	0.00E+00	0.00E+00	0.00E+00	3.24E-16	9.07E+00	0.00E+00
F18	Mean	2.52E+00	8.15E-08	3.08E-08	3.27E-14	2.87E-11	2.76E-09	8.88E-16	8.88E-16	7.78E-12	4.06E-09	8.88E-16
	Std	1.91E-01	3.11E-08	2.55E-09	6.08E-15	1.33E-11	2.31E-09	0.00E+00	0.00E+00	9.15E-13	1.29E-09	0.00E+00
F19	Mean	1.38E+02	4.01E-03	6.69E-11	0.00E+00	0.00E+00	1.98E-12	0.00E+00	0.00E+00	9.25E-15	1.83E-03	0.00E+00
	Std	1.42E+01	1.15E-02	2.72E-11	0.00E+00	0.00E+00	1.80E-12	0.00E+00	0.00E+00	4.81E-15	5.61E-03	0.00E+00
F20	Mean	1.33E+01	3.92E-03	2.01E-09	3.66E-18	3.01E-12	3.26E-11	1.05E-19	1.15E-57	7.60E-12	2.49E-03	1.27E-151
	Std	3.35E+00	2.70E-03	1.85E-09	2.04E-18	2.42E-12	5.92E-11	1.32E-19	6.30E-57	9.27E-13	1.50E-03	3.56E-152
F21	Mean	1.13E-01	2.05E-01	9.83E-01	2.83E-02	3.74E-02	7.49E-01	9.33E-01	1.49E-03	7.70E-01	1.51E-01	1.09E+00
	Std	8.19E-02	4.33E-02	8.89E-02	8.01E-03	9.89E-03	1.14E-01	1.20E-01	6.36E-04	1.10E-01	4.35E-02	7.82E-02
F22	Mean	1.02E+00	5.66E+00	9.99E+00	5.16E+00	6.05E+00	9.99E+00	9.98E+00	9.70E+00	9.94E+00	4.96E+00	9.99E+00
	Std	1.98E-01	4.20E-01	5.25E-03	1.35E+00	2.96E+00	4.55E-03	4.85E-03	6.11E-01	1.34E-01	4.19E-01	2.48E-03
F23	Mean	2.35E+01	1.81E+01	6.84E+01	3.96E+00	4.54E+00	6.05E+01	6.85E+01	1.94E+00	6.54E+01	1.37E+01	6.69E+01
	Std	7.20E+00	4.80E+00	4.20E+00	1.59E+00	1.69E+00	6.39E+00	4.94E+00	8.69E-01	5.49E+00	3.09E+00	5.65E+00
F24	Mean	5.16E+01	1.67E+01	2.66E+00	3.20E-05	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.33E+01	0.00E+00
	Std	9.49E+00	1.08E+01	3.18E+00	1.75E-04	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	4.05E+00	0.00E+00
F25	Mean	3.97E+00	6.80E-01	4.00E-01	2.29E-01	2.49E-01	3.33E-02	1.09E-32	8.86E-02	9.95E-02	6.47E-01	7.39E-301
	Std	1.06E+00	2.51E-01	3.16E-03	1.50E-01	1.52E-01	4.77E-02	3.81E-32	3.58E-02	3.00E-06	2.53E-01	0.00E+00
F26	Mean	6.60E+01	1.01E-13	8.60E-11	0.00E+00	0.00E+00	5.22E-12	0.00E+00	0.00E+00	7.87E-15	2.96E-16	0.00E+00
	Std	6.71E+00	1.50E-13	8.49E-12	0.00E+00	0.00E+00	6.98E-12	0.00E+00	0.00E+00	1.50E-15	4.86E-16	0.00E+00
+/-/≈		1/25/0	1/25/0	0/26/0	2/21/3	0/21/5	0/23/3	1/19/6	4/16/6	0/24/2	1/25/0	∼
Rank		7	7	8	4	4	5	3	2	6	7	1

. The second last row indicates the number of success (+), failure (–), and approximate (≈) of the compared algorithms with respect to HPSBA. The last row shows the rank of the compared algorithms.

4.2.1. Analysis of the Numerical Results

As can be seen from Table 4, for the benchmark functions F1, F2, F3, F4, F6, F8, F9, F11, F14, F20, and F25, the proposed algorithm outperforms all comparative algorithms. For the F16, F17, F19, F24, and F26, HPSBA achieved the theoretical optimal value. For the F18, HPSBA, HPSOBOA and CBOA have the superior results over the other algorithms. For the F5, F21, F22, and F23, MPA have the best results. For the F7, PSO achieved the theoretical optimal value. For the F10, EO has the best result. For the F12, although the Means of GWO, EO, SOGWO, and HPSBA are the same, the Std of EO is the smallest. For the F13, GWO, EO, and SOGWO achieved the theoretical optimal value.

Although the performance of the PSO algorithm is poor, the optimization time is the shortest, which shows that the optimization speed of the algorithm is relatively strong. The conclusions that can be drawn from the results presented in Table 4 are that the rankings of the comparison algorithms are HPSBA > MPA > EO = HPSOBOA = CBOA > LBOA = SOGWO > IBOA > GWO > PSO > BOA.

4.2.2. Convergence Behavior Analysis

Figure 7 shows the 2D search space for twenty-six benchmark functions in three dimensional visualization. Figure 8 shows the convergence curves of the comparison algorithms for the functions F1 to F4 (unimodal functions), F16 to F19 and F24 (multimodal functions) when Dim = 30. As can be seen from Figure 8, the comparison curves confirm the superiority of HPSBA over the PSO, GWO, BOA, EO, MPA and other comparison algorithms for functions F1 to F4. In addition, for F16, F17, and F19, HPSBA, HPSOBOA, IBOA, MPA and EO can obtain the best value of the functions in theory. For F24, there are two comparison algorithms can obtain the optimal value, called HPSBA and EO algorithm from the Figure 8. From the curves of the proposed HPSBA, the performance of the algorithm needs to be further improved, especially in terms of convergence speed.

4.3. Comparison Results of HPSBA with Others (Dim = 100)

Table 5 shows the comparison results of eleven algorithms with Dim = 100. Results of the Wilcoxon rank-sum test calculated at a significance level of $\alpha =0.05$ are listed in Table 5. Where the number of success (+), failure (–), and approximate (≈) of the compared algorithms with respect to HPSBA are listed in the second last row. The last row shows the rank of the compared algorithms.

4.3.1. Analysis of the Numerical Results

As can be seen from Table 5, for the benchmark functions F1, F3, F4, F6, F8, F9, F11, F14, F15, F20, and F25, the proposed algorithm outperforms all comparative algorithms. For the F16, F17, F19, F24, and F26, HPSBA achieved the theoretical optimal value. For the F2, CBOA has the best result. For the F18, HPSBA, HPSOBOA and CBOA have the superior results over the other algorithms. For the F5, F10, F21, and F23, HPSOBOA has the best results. For the F7 and F12, EO algorithm has the best results. For the F13, GWO, EO, and SOGWO achieved the theoretical optimal value. For the F22, PSO algorithm has the best result.

It can be seen from Table 5 that with the increase of the problem dimensions, the optimization speed of PSO algorithm decreases to be some extent. This shows that the difficulty of solving a problem increases with its complexity. The conclusion that can be drawn from the results presented in Table 5 are that the rankings of the comparison algorithms is HPSBA > HPSOBOA > CBOA > EO = MPA > LBOA > IBOA > PSO = SOGWO = GWO > BOA.

4.3.2. Boxplot Results Analysis

To better explain the stability of the comparison algorithms for solving the high dimensional optimization problems. Figure 9 presents the boxplot results of the eleven algorithms on the four test functions (F3, F8, F20, and F25). 30 independent runs of each algorithm are conducted for the same test function.

5. Nodes Coverage Optimization in WSN

Through the above experiments, it can be known that HPSBA performs better in numerical optimization problems, but for practical problems, the effectiveness of the proposed algorithm remains to be verified. In this section, we apply the proposed HPSBA without parameter $\omega$ in Equation (14) to the NOC problem of the WSN. The model description and objective function of the problem are detailed in Section 2.3.

5.1. Parameter Setting and Pseudo Code of Node Coverage Using HPSBA

To confirm the performance of the HPSBA for solving the NOC problem, there are two group experiments are designed as follows: (1) To study the performance of the HPSBA in node optimization coverage problem, three comparison algorithms, PSO, BOA, and MBOA, are employed as competitors. (2) The HPSBA algorithm is applied to the problem of optimal coverage of nodes with obstacles.

Equation (10) can be used to determine the number of sensor nodes needed to cover the theoretical area. The node coverage area, the number of sensor nodes, and the parameter settings of the simulation experiment are shown in

Table 6. Node coverage parameter settings.

Parameters	Setting Values
Side length of coverage area/m	100 × 100	100 × 100
Number of nodes	45	40, 45, 50
Perception radius/m	10	10
Communication radius $r_c$/m	20	20
Maximum iterations ($T_{max}$)	100, 150, 200	150
Boundary threshold/m	$r_s/3$	$r_s/3$

.

The simulations of node optimization coverage of the WSN are conducted using PSO, BOA, MBOA, and HPSBA. The population and number of iterations of comparison algorithms are respectively set 30 and 150. Additionally, the following is a discussion of the parameters for the comparison methods:

• PSO based node optimization coverage: Each target node runs the PSO to become a deployed node. Parameters that are considered for coverage are: inertial weight = 0.7, cognitive and social scaling parameters $c_1=c_2=2$.
• BOA based node optimization coverage: Each target node runs the BOA to become a deployed node. Parameters that are considered for coverage are: probability switch weight $SP=0.8$, cognitive and social scaling parameters $c=1$, and $a=0.1$.
• MBOA based node optimization coverage: Each target node runs the MBOA to become a deployed node. Parameters that are considered for coverage are: probability switch weight $SP=0.5$, cognitive and social scaling parameters $c\left(0\right)=r_1=0.35$ with chaotic adjust strategy, and $a=0.1$.
• HPSBA based node optimization coverage: Each target node runs the proposed HPSBA to become a deployed node. Parameters that are considered for coverage are set as follows: initial value of inertial weight = 0.9, probability switch weight $SP=0.6$, cognitive and social scaling parameters $c_1=c_2=2,a=0.1$ and $c\left(0\right)=0.35$.

5.2. Results Analyses of Coverage Optimization Problem

5.2.1. The Effect of the Number of Nodes on Coverage

To further test the optimization performance of the HPSBA for coverage optimization problem, the coverage rates of various algorithms in the monitoring area under different numbers of sensor nodes are compared. Deploy sensor nodes in a 100 m × 100 m square monitoring area, with that the sensing radius is set 10 m, the communication radius is set 20 m, and the maximum iterations is set 150. Comparison algorithms are used when sensor nodes numbers are 40, 45, and 50, respectively. When nodes are 45 with 150 iterations, the coverage simulation results are shown in Figure 10. Other parameters remain unchanged, and the variation trend of the coverage rate with the number of nodes under different coverage strategies is shown in Figure 10a.

As seen from Figure 10 that the initial random coverage is shown in Figure 10a with 45 sensor nodes. As the number of iterations of HPSBA reaches 150 times, the coverage position of the nodes is shown in Figure 10b. According to the HPSBA to optimize the sensor node coordinate positions before and after coverage, the minimum spanning tree Prim algorithm [55] is used to draw the node communication network in the coverage area, as shown in Figure 10c,d.

As seen from Figure 10c,d, the uniformity of the communication distance between the initially deployed nodes is poor, the sink node is located in the center of the coverage area, and the data transmission distance between the nodes is longer. Energy consumption is large, and the optimized communication distance between nodes is more uniform, there are multiple convergence nodes, and the location is located at the boundary, thereby enhancing the reliability of the network, thereby reducing the energy consumption of node data transmission, and extending the network life.

The coverage curves of node coverage based on BOA, MBOA, PSO and HPSBA node are shown in Figure 11. The coverage effect of the BOA is poor. HPSBA is in the leading position after 10 iterations. The final coverage rates of the four algorithms after 150 iterations are: 83.65%, 83.90%, 94.12%, and 96.54%.

In the case of different numbers of nodes, Figure 12 provides an illustration of the calculated fitness average values. The proposed HPSBA coverage strategy is better than BOA, MBOA and PSO in the WSN node coverage with different number of nodes. Moreover, it is evident that the most effective simulation results are obtained with the proposed HPSBA, and the coverage rate reaches 93.15% in nodes = 40, 96.54% in nodes $N=45$, and 98.42% in nodes $N=50$. Compared to the standard BOA, the coverage rate is increased by the percentage points of 11.49, 12.89 and 12.41 in $N=40$, $N=45$ and $N=50$, respectively. It is noteworthy that as the coverage optimization problem (80-dimension when $N=40$, 90-dimension when $N=45$, 100-dimension when $N=50$) grows in size, HPSBA has more outstanding advantages over the basic BOA.

5.2.2. The Effect of the Number of Iterations on Coverage

To verify the influence of the number of algorithm iterations on the coverage rate, we set different iteration numbers, 100, 150, and 200, respectively. The optimized coverage results with 45 sensor nodes of the four comparison algorithms are shown in

Table 7. Node coverage rate of different iterations.

Item	${\mathit{T}}_{\mathit{Max}}$ = 100		${\mathit{T}}_{\mathit{Max}}$ = 150		${\mathit{T}}_{\mathit{Max}}$ = 200
	Cov/%	Time/s	Cov/%	Time/s	Cov/%	Time/s
BOA	84.79	12.21	83.65	16.55	86.42	40.98
MBOA	85.11	10.93	83.90	16.04	85.82	21.10
PSO	92.20	11.94	94.12	24.4	94.32	51.92
HPSBA	93.28	11.09	96.54	16.55	96.32	21.31

.

As seen from Table 7, it is evident that the most effective simulation results are obtained with the proposed HPSBA, and the coverage rate reaches 93.28% in $T_{max}=100$, 96.54% in $T_{max}=150$, and 96.32% in $T_{max}=200$. Compared to the standard BOA, the coverage rate is increased by the percentage points of 8.49, 12.89, and 9.90 in $T_{max}$ = 100, 150 and 200, respectively. It is noteworthy that as the coverage optimization problem (80-dimension when $N=40$, 90-dimension when $N=45$, 100-dimension when $N=50$) grows in size, the proposed HPSBA demonstrates a superior advantage to the basic BOA. Furthermore, the proposed HPSBA compared with MBOA, the coverage rate is increased by the percentage points of 8.17, 12.64. HPSBA compared with PSO, the coverage rate is increased by the percentage points of 1.08, 2.42, and 2.00. Overall, when it comes to the WSN node coverage optimization problem, HPSBA outperforms the other three competitors.

5.2.3. Node Obstacle Avoidance Coverage Based on HPSBA

The proposed HPSBA has a positive effect on the sensor network’s coverage of two-dimensional nodes when compared to the previous experimental results. To verify the effectiveness of the HPSBA node obstacle avoidance coverage problem, a node coverage experiment is designed under obstacles: the node coverage area is a two-dimensional plane of 100 m × 100 m, the number of sensor nodes is 40, the sensing radius $r_s$ is set 10 m, communication radius $r_c=20$ m, the number of iterations is 200, and the obstacle is a rectangular area of 20 m × 20 m. The optimization results of the coverage of obstacle avoidance nodes based on HPSBA and the coverage curve of 200 iterations are depicted in Figure 13.

As seen from Figure 13, HPSBA has a better application effect in the coverage of obstacle avoidance nodes. After 200 iterations, the node coverage in the set coverage area increased from 77.01% to 98.67%, an increase point of 21.66, and the time consumption is 21.34 s.

6. Conclusions

Aiming at the uneven distribution of nodes and low coverage in the random coverage of sensor networks, a hybrid particle swarm butterfly algorithm (HPSBA) is proposed for deploying WSN nodes. HPSBA improves the convergence speed through Logistic map and adaptive adjustment strategies, and the algorithm’s accuracy at convergence is also improved. Through the optimization experiment of twenty-six benchmark functions with ten comparison swarm intelligence algorithms, the optimization results show that HPSBA’s optimization ability and convergence accuracy are improved, and the stability is enhanced.

For the WSN node coverage problem, HPSBA can effectively coordinate the global exploration and local development capabilities of the algorithm. Compared with other algorithms, HPSBA effectively improves the coverage of WSN nodes while using fewer nodes, thus reducing it. The configuration cost of the network is reduced. However, the energy of nodes is usually taken into account in the real node coverage. More advanced algorithms [56] will be considered in the future work.

We will concentrate on the following tasks in future work: (i) In order to guarantee optimization precision, we will further develop HPSBA in light of the high complexity of its main framework. (ii) The proposed HPSBA will be further applied to solve multi-objective optimization problem, such as energy, distance, and uniformity for WSN in a three-dimensional environment, etc, between nodes.