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

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## Abstract

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## 1. Introduction

- 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.

## 2. Basic Knowledge

#### 2.1. Particle Swarm Optimization

#### 2.2. Butterfly Optimization Algorithm

#### 2.3. Node Coverage Optimization Problem Model

## 3. Method

#### 3.1. Hybrid Particle Swarm Butterfly Algorithm (HPSBA)

#### 3.1.1. Algorithmic Population Initialization

#### 3.1.2. Algorithmic Exploration

#### 3.1.3. Algorithmic Exploitation

#### 3.1.4. The Chaotic Adjusting Strategies

#### 3.2. Complexity Analysis of the HPSBA

#### 3.2.1. Time Complexity

#### 3.2.2. Space Complexity

#### 3.3. The Pseudo-Code and Flowchart of the HPSBA

Algorithm 1: Pseudo-code of HPSBA |

#### 3.4. Convergence Analysis of the HPSBA

**Theorem 1.**

**Proof.**

#### 3.4.1. Convergence Criterion

**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

**Theorem 2.**

#### 3.4.2. Convergence Analysis

**Lemma 1.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Theorem 3.**

**Proof.**

## 4. Analyses and Results of Numerical Optimization

#### 4.1. Parameter Setting of Comparison Algorithms

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

#### 4.2.1. Analysis of the Numerical Results

#### 4.2.2. Convergence Behavior Analysis

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

#### 4.3.1. Analysis of the Numerical Results

#### 4.3.2. Boxplot Results Analysis

## 5. Nodes Coverage Optimization in WSN

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

- 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

#### 5.2.2. The Effect of the Number of Iterations on Coverage

#### 5.2.3. Node Obstacle Avoidance Coverage Based on HPSBA

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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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 |

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 | ${F}_{12}={({x}_{1}-1)}^{2}+\sum _{i=2}^{Dim}i{(2{x}_{i}^{2}-{x}_{i-1})}^{2}$ | [−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 | ${F}_{17}={\displaystyle \sum _{i=1}^{Dim}}\left({y}_{i}^{2}-10cos\left(2\pi {y}_{i}\right)+10\right),{y}_{i}=\left\{\begin{array}{c}\hfill {x}_{i},\left|{x}_{i}\right|<0.5,\\ \hfill round\left(2{x}_{i}\right)/2,\left|{x}_{i}\right|<0.5\end{array}\right.$ | [−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 | ${F}_{20}={\displaystyle \sum _{i=1}^{Dim}}\left|{x}_{i}\xb7sin\left({x}_{i}\right)+0.1{x}_{i}\right|$ | [−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 | ${F}_{23}={sin}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum _{i=1}^{Dim-1}}{({x}_{i}-1)}^{2}[1+{sin}^{2}\left(3\pi {x}_{i+1}\right)]+\left|{x}_{Dim}-1\right|\xb7[1+{sin}^{2}\left(2\pi {x}_{Dim}\right)]$ | [−2,2] | 30/100 | 0 | M |

Weierstrass | ${F}_{24}={\displaystyle \sum _{i=1}^{Dim}}\left({\displaystyle \sum _{k=0}^{{k}_{max}}}\left[{a}^{k}cos\left(2\pi {b}^{k}({x}_{i}+0.5)\right)\right]\right)-Dim{\displaystyle \sum _{k=0}^{{k}_{max}}}\left[{a}^{k}cos\left(2\pi {b}^{k}\xb70.5\right)\right],a=0.5,b=3,{k}_{max}=20$ | [−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 | ${F}_{26}={\displaystyle \sum _{i=1}^{Dim}}\left[{x}_{i}^{2}+2{x}_{i+1}^{2}-0.3\xb7cos\left(3\pi {x}_{i}\right)-0.4\xb7cos\left(4\pi {x}_{i+1}\right)+0.7\right]$ | [−5,5] | 30/100 | 0 | M |

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 | $N=30,a=0.1,c\left(0\right)=0.35,SP=0.6,\mu =4,{\omega}_{u}=0.9,{\omega}_{l}=0.2,{C}_{1}={C}_{2}=2$ |

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 |

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 |

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 | ${r}_{s}/3$ | ${r}_{s}/3$ |

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 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Wang, D.; Yang, M.; Tan, W.; Yang, J.
HPSBA: A Modified Hybrid Framework with Convergence Analysis for Solving Wireless Sensor Network Coverage Optimization Problem. *Axioms* **2022**, *11*, 675.
https://doi.org/10.3390/axioms11120675

**AMA Style**

Zhang M, Wang D, Yang M, Tan W, Yang J.
HPSBA: A Modified Hybrid Framework with Convergence Analysis for Solving Wireless Sensor Network Coverage Optimization Problem. *Axioms*. 2022; 11(12):675.
https://doi.org/10.3390/axioms11120675

**Chicago/Turabian Style**

Zhang, Mengjian, Deguang Wang, Ming Yang, Wei Tan, and Jing Yang.
2022. "HPSBA: A Modified Hybrid Framework with Convergence Analysis for Solving Wireless Sensor Network Coverage Optimization Problem" *Axioms* 11, no. 12: 675.
https://doi.org/10.3390/axioms11120675