# Multi-Strategy Improved Sparrow Search Algorithm and Application

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## Abstract

**:**

## 1. Introduction

## 2. Sparrow Search Algorithm

## 3. Sparrow Search Algorithm Enhancement Strategy

#### 3.1. Piecewise Chaos Mapping

#### 3.2. Gaussian Differential Variance

#### 3.3. Linear Differential Decreasing Inertia Weights

## 4. Improved Sparrow Search Algorithm

**Step 1**: The parameters are set; each parameter includes population size N, number of discoverers M, number of followers ($N-M$), number of sparrows for reconnaissance warning $(0.1-0.2)N$, dimension of the objective function $Dim$, upper and lower bounds $lb$, $ub$ of initial values, and maximum number of iterations ${iter}_{max}$;

**Step 2**: Apply the piecewise chaotic sequence in Equation (4) to initialize the population and generate N D-dimensional vectors;

**Step 3**: Calculate the fitness value ${f}_{i}$ of all individuals in the population, record the current best individual fitness value ${f}_{g}$ and the corresponding position ${X}_{b}$, and record the current worst individual fitness value ${f}_{w}$ and the corresponding position ${X}_{w}$;

**Step 5**: Therandomly selected 10%–20% individuals in the species sparrow flock are used as scouts, and the scout positions are updated by Equation (3);

**Step 6**: During the iteration of the algorithm, the diversity of individuals is generated by perturbation of the difference variables to make the algorithm converge quickly. After one complete iteration, the fitness value ${f}_{i}$ and the population average fitness value ${f}_{avg}$ are recalculated for each individual of the population, and when ${f}_{i}$ < ${f}_{avg}$, the Gaussian difference variation is performed according to Equation (5), and the pre-variation individual is replaced by the post-variation individual if it is better than the pre-variation individual;

**Step 7**: Update the historical optimal position ${X}_{b}$ and the corresponding fitness value ${f}_{g}$ of the sparrow population, and the worst position ${X}_{w}$ and the corresponding fitness value ${f}_{w}$ of the population;

**Step 8**: Determine whether the number of iterations of the algorithm reaches the maximum or the accuracy of the solution reaches the requirement, the loop ends if the requirement is reached; otherwise, return to

**Step 4**.

## 5. Simulation Experiments and Results Analysis

#### 5.1. CEC Test Functions

#### 5.2. Experimental Anvironment and Parameter Settings

#### 5.3. Comparative Analysis of Optimization Results

#### 5.4. Wilcoxon Rank-Sum Test

#### 5.5. Comparative Time Analysis

#### 5.6. Comparison of PGL-SSA with Different Improved SSA

## 6. Application of PGL-SSA for HVAC Control

#### 6.1. Fitness Function

#### 6.2. PID Parameter Tuning Simulation Experiment and Result Analysis

#### 6.2.1. Optimal Tuning of PID Parameters for First-Order Inertia Delay Systems

#### 6.2.2. Optimal Tuning of PID Parameters for Second-Order Underdamped Delay Systems

#### 6.2.3. PMSM System PID Parameter Optimization

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The distribution of 1000 iterations of six chaotic mappings, (

**a**) logistic mapping, (

**b**) tent mapping, (

**c**) iterative mapping, (

**d**) intermittency mapping, (

**e**) chebyshev mapping, and (

**f**) piecewise mapping.

**Figure 2.**Iteration curves of 6 chaotic mapping improved SSA algorithms for different types of test functions, (

**a**) for high-dimensional single-peaked function, (

**b**) for high-dimensional multi-peaked function.

**Figure 4.**Different algorithms convergence curves of F1–F11 with 30 dimensions. (

**a**) F1 Convergence curve. (

**b**) F2 Convergence curve. (

**c**) F3 Convergence curve. (

**d**) F4 Convergence curve. (

**e**) F5 Convergence curve. (

**f**) F6 Convergence curve. (

**g**) F7 Convergence curve. (

**h**) F8 Convergence curve. (

**i**) F9 Convergence curve. (

**j**) F10 Convergence curve. (

**k**) F11 Convergence curve.

**Figure 5.**Different algorithms convergence curves of F1–F11 with 100 dimensions. (

**a**) F1 Convergence curve. (

**b**) F2 Convergence curve. (

**c**) F3 Convergence curve. (

**d**) F4 Convergence curve. (

**e**) F5 Convergence curve. (

**f**) F6 Convergence curve. (

**g**) F7 Convergence curve. (

**h**) F8 Convergence curve. (

**i**) F9 Convergence curve. (

**j**) F10 Convergence curve. (

**k**) F11 Convergence curve.

**Figure 6.**Convergence curves of fixed-dimensional peak functions for different algorithms. (

**a**) F12 Convergence curve. (

**b**) F13 Convergence curve. (

**c**) F14 Convergence curve. (

**d**) F15 Convergence curve. (

**e**) F16 Convergence curve. (

**f**) F17 Convergence curve. (

**g**) F18 Convergence curve. (

**h**) F19 Convergence curve. (

**i**) F20 Convergence curve. (

**j**) F21 Convergence curve.

**Figure 8.**Comparison of running time of different algorithms in each dimension. (

**a**) 30D mean running time, (

**b**) 100D mean running time, (

**c**) Fixed dimension test function running time.

**Figure 10.**Convergence curves and step response curves of different algorithms for first-order inertial delay systems. (

**a**) Convergence curve. (

**b**) Step response curve.

**Figure 11.**Convergence curves and step response curves of different algorithms for second-order inertial delay systems. (

**a**) Convergence curve. (

**b**) step response curve.

**Figure 12.**Convergence curves and step response curves of different algorithms for PMSM systems. (

**a**) Convergence curve. (

**b**) Step response curve.

Type | Title | Function | Interval | Dimension | Min |
---|---|---|---|---|---|

Single peak | Sphere | ${F}_{1}\left(x\right)={\sum}_{i=1}^{n}{x}_{i}^{2}$ | $[-100,100]$ | 30/100 | 0 |

Schwefel 2.22 | ${F}_{2}\left(x\right)={\sum}_{i=1}^{n}\left|{x}_{i}\right|+{\prod}_{i=1}^{n}\left|{x}_{i}\right|$ | $[-10,10]$ | 30/100 | 0 | |

Quadric | ${F}_{3}\left(x\right)={\sum}_{i=1}^{n}{\left({\sum}_{j=1}^{i}{\chi}_{j}\right)}^{2}$ | $[-100,100]$ | 30/100 | 0 | |

Schwefel 2.21 | ${F}_{4}={max}_{i}\left\{\left|{x}_{i}\right|,1\u2a7di\u2a7dn\right\}$ | $[-100,100]$ | 30/100 | 0 | |

Step | ${F}_{5}\left(x\right)={\sum}_{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | $[-100,100]$ | 30/100 | 0 | |

Multi-peak | Schwefel 2.26 | ${F}_{6}\left(x\right)={\sum}_{i=1}^{n}-\left({x}_{i}sin\left(\sqrt{\left|{x}_{i}\right|}\right)\right)$ | $[-500,500]$ | 30/100 | $-418.9829d$ |

Rastrigin | ${F}_{7}\left(x\right)={\sum}_{i=1}^{n}\left[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}\right)+10\right]$ | $[-5.12,5.12]$ | 30/100 | 0 | |

Ackley | $\begin{array}{cc}& \hfill {F}_{8}\left(\mathbf{x}\right)=-20exp\left(-0.2\sqrt{\frac{1}{D}\sum _{i=1}^{D}{x}_{i}^{2}}\right)\hfill \\ & \hfill -exp\left(\frac{1}{D}\sum _{i=1}^{D}cos\left(2\pi {x}_{i}\right)\right)+20+\mathrm{e}\hfill \end{array}$ | $[-32,32]$ | 30/100 | 0 | |

Griewank | ${F}_{9}\left(x\right)=\frac{1}{4000}{\sum}_{i=1}^{n}{x}_{i}^{2}-{\prod}_{i=1}^{n}cos\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | $[-600,600]$ | 30/100 | 0 | |

Penalized | $\begin{array}{cc}& \hfill {F}_{10}\left(x\right)=\frac{\pi}{D}\{10{sin}^{2}\left(\pi {y}_{i}\right)+\sum _{i-1}^{D-1}{\left({y}_{i}-1\right)}^{2}\hfill \\ & \hfill \left[1+10{sin}^{2}\left(\pi {y}_{i+1}\right)\right]+\left({y}_{D}-1\right)\hfill \\ & \hfill +\sum _{i-1}^{D}u\left({x}_{i},10,100,4\right){y}_{i}=1+\frac{{x}_{i}+1}{4}\hfill \\ & \hfill u\left({x}_{i},a,k,m\right)=\{\begin{array}{c}\hfill k{\left({x}_{i}-a\right)}^{m}{x}_{i}>a\hfill \\ \hfill 0-a<{x}_{i}<a\hfill \\ \hfill k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}<a\hfill \end{array}\hfill \end{array}$ | $[-50,50]$ | 30/100 | 0 | |

Penalized2 | $\begin{array}{cc}& \hfill {F}_{11}\left(x\right)=0.1\{{sin}^{2}\left(3\pi {x}_{i}\right)+\sum _{i=1}^{D}{\left({x}_{i}-1\right)}^{2}\hfill \\ & \hfill \left[1+{sin}^{2}\left(3\pi {x}_{i}\right)\right]+{\left({x}_{D}-1\right)}^{2}\hfill \\ & \hfill [1+{sin}^{2}\left(2\pi {x}_{D}\right)]\}+\sum _{i-1}^{D}u\left({x}_{i},5,100,4\right)\hfill \end{array}$ | $[-50,50]$ | 30/100 | 0 | |

Fixed dimensional multi-peak | Foxholes | ${F}_{12}\left(x\right)={\left({\displaystyle \frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+{\sum}_{i=1}^{2}{\left({x}_{i}-{a}_{ij}\right)}^{6}}}\right)}^{-1}$ | $[-65.5360,65.5360]$ | 2 | 0.998004 |

Kowalik | ${F}_{13}\left(x\right)={\sum}_{i=1}^{11}{\left({\displaystyle {a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}}\right)}^{-1}$ | $[-5,5]$ | 4 | 0.0003075 | |

Six Hump Camel Back | ${F}_{14}\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+1/3{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+{x}_{2}^{4}$ | $[-5,5]$ | 2 | $-1.03163$ | |

Branin | $\begin{array}{c}\hfill {F}_{15}\left(x\right)={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+\\ \hfill 10\left(1-\frac{1}{8\pi}\right)cos{x}_{1}+10\end{array}$ | $[-5,5]$ | 2 | 0.398 | |

Goldstein Price | $\begin{array}{cc}& {F}_{16}\left(x\right)=[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\hfill \\ & \left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)]\hfill \\ & \times [30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\hfill \\ & \times \left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)]\hfill \end{array}$ | $[-5,5]$ | 2 | 3 | |

Hartman 3 | ${F}_{17}\left(x\right)=-{\sum}_{i=1}^{4}{c}_{i}exp\left(-{\sum}_{j=1}^{3}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)$ | $[0,1]$ | 3 | −3.86 | |

Hartman 6 | ${F}_{18}\left(x\right)=-{\sum}_{i=1}^{4}{c}_{i}exp\left(-{\sum}_{j=1}^{6}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)$ | $[0,1]$ | 6 | −3.32 | |

Langermann 5 | ${F}_{19}\left(x\right)=-{\sum}_{i=1}^{5}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | $[0,10]$ | 4 | −10.1532 | |

Langermann 7 | ${F}_{20}\left(x\right)=-{\sum}_{i=1}^{7}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | $[0,10]$ | 4 | −10.4029 | |

Langermann 10 | ${F}_{21}\left(x\right)=-{\sum}_{i=1}^{10}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | $[0,10]$ | 4 | −10.5364 |

Algorithm | Parameters |
---|---|

PSO | ${W}_{1}=0.9;{C}_{1},{C}_{2}=2;{V}_{min}=-5,{V}_{max}=5$ |

GWO | $\alpha $ decreases linearly from 2 to 0; ${r}_{1},{r}_{2}\in [0,1]$ |

SSA | $M=0.7N;ST=0.6;SD=0.2N$ |

PGL-SSA | $M=0.7N;ST=0.6;SD=0.2N$ |

Function | Algorithm | $\mathit{Dim}=30$ | $\mathit{Dim}=100$ | ||||
---|---|---|---|---|---|---|---|

Mean | Std | Best | Mean | Std | Best | ||

PSO | 1.88 × ${10}^{0}$ | 4.27 × ${10}^{-1}$ | 1.28 × ${10}^{0}$ | 1.10 × ${10}^{2}$ | 6.59 × ${10}^{0}$ | 1.02 × ${10}^{2}$ | |

F1 | GWO | 2.35 × ${10}^{-33}$ | 2.53 × ${10}^{-33}$ | $4.46\times {10}^{-34}$ | 3.05 × ${10}^{-15}$ | 6.03 × ${10}^{-16}$ | 2.36 × ${10}^{-15}$ |

SSA | 1.45 × ${10}^{-55}$ | 2.91 × ${10}^{-55}$ | 0.0 | 4.57 × ${10}^{-82}$ | 6.46 × ${10}^{-82}$ | 0.0 | |

PGL-SSA | $\mathbf{1}.\mathbf{72}\times {\mathbf{10}}^{-\mathbf{222}}$ | 0.0 | 0.0 | $\mathbf{2}.\mathbf{75}\times {\mathbf{10}}^{-\mathbf{203}}$ | 0.0 | 0.0 | |

PSO | 5.65 × ${10}^{0}$ | 9.29 × ${10}^{-1}$ | 4.64 × ${10}^{0}$ | 2.19 × ${10}^{2}$ | 5.75 × ${10}^{1}$ | 1.61 × ${10}^{2}$ | |

F2 | GWO | 6.62 × ${10}^{-20}$ | 3.65 × ${10}^{-20}$ | 2.49 × ${10}^{-20}$ | 1.56 × ${10}^{-9}$ | 6.59 × ${10}^{-10}$ | 9.02 × ${10}^{-10}$ |

SSA | 2.90 × ${10}^{-44}$ | 5.81 × ${10}^{-44}$ | 0.0 | 9.61 × ${10}^{-31}$ | 9.61 × ${10}^{-31}$ | 5.65 × ${10}^{-59}$ | |

PGL-SSA | $\mathbf{4}.\mathbf{79}\times {\mathbf{10}}^{-\mathbf{263}}$ | 0.0 | 0.0 | $\mathbf{3}.\mathbf{70}\times {\mathbf{10}}^{-\mathbf{294}}$ | 0.0 | 0.0 | |

PSO | 8.29 × ${10}^{1}$ | 1.43 × ${10}^{1}$ | 6.69 × ${10}^{1}$ | 9.91 × ${10}^{3}$ | 2.53 × ${10}^{3}$ | 7.37 × ${10}^{3}$ | |

F3 | GWO | 2.69 × ${10}^{-9}$ | 3.59 × ${10}^{-9}$ | 1.15 × ${10}^{-11}$ | 1.13 × ${10}^{2}$ | 5.62 × ${10}^{1}$ | 5.71 × ${10}^{1}$ |

SSA | 1.07 × ${10}^{-88}$ | 2.15 × ${10}^{-88}$ | 0.0 | 0.0 | 0.0 | 0.0 | |

PGL-SSA | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

PSO | 2.06 × ${10}^{0}$ | 1.81 × ${10}^{-1}$ | 1.82 × ${10}^{0}$ | 1.08 × ${10}^{1}$ | 1.32 × ${10}^{0}$ | 9.48 × ${10}^{0}$ | |

F4 | GWO | 2.86 × ${10}^{-8}$ | 2.36 × ${10}^{-8}$ | 7.67 × ${10}^{-9}$ | 1.03 × ${10}^{-1}$ | 5.25 × ${10}^{-2}$ | 5.07 × ${10}^{-2}$ |

SSA | 1.45 × ${10}^{-58}$ | 2.91 × ${10}^{-58}$ | 0.0 | 3.04 × ${10}^{-50}$ | 3.04 × ${10}^{-50}$ | 0.0 | |

PGL-SSA | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

PSO | 1.73 × ${10}^{0}$ | 6.97 × ${10}^{-1}$ | 6.85 × ${10}^{-1}$ | 8.71 × ${10}^{1}$ | 4.47 × ${10}^{0}$ | 8.27 × ${10}^{1}$ | |

F5 | GWO | 4.54 × ${10}^{-1}$ | 3.36 × ${10}^{-1}$ | 6.63 × ${10}^{-5}$ | 8.57 × ${10}^{0}$ | 5.39 × ${10}^{-2}$ | 8.51 × ${10}^{0}$ |

SSA | 2.40 × ${10}^{-7}$ | 4.48 × ${10}^{-7}$ | 5.38 × ${10}^{-11}$ | 8.27 × ${10}^{-8}$ | 6.22 × ${10}^{-8}$ | 2.05 × ${10}^{-8}$ | |

PGL-SSA | $\mathbf{2}.\mathbf{19}\times {\mathbf{10}}^{-\mathbf{9}}$ | $\mathbf{2}.\mathbf{22}\times {\mathbf{10}}^{-\mathbf{9}}$ | $\mathbf{5}.\mathbf{56}\times {\mathbf{10}}^{-\mathbf{13}}$ | $\mathbf{4}.\mathbf{48}\times {\mathbf{10}}^{-\mathbf{9}}$ | $\mathbf{4}.\mathbf{26}\times {\mathbf{10}}^{-\mathbf{11}}$ | $\mathbf{4}.\mathbf{44}\times {\mathbf{10}}^{-\mathbf{9}}$ | |

PSO | 5.50 × ${10}^{3}$ | 3.56 × ${10}^{2}$ | 5.05 × ${10}^{3}$ | 3.18 × ${10}^{4}$ | 3.38 × ${10}^{3}$ | 2.84 × ${10}^{4}$ | |

F6 | GWO | 5.93 × ${10}^{3}$ | 2.95 × ${10}^{2}$ | 5.47 × ${10}^{3}$ | 2.52 × ${10}^{4}$ | 5.93 × ${10}^{2}$ | 2.46 × ${10}^{4}$ |

SSA | 2.90 × ${10}^{3}$ | 2.54 × ${10}^{3}$ | 6.64 × ${10}^{0}$ | 1.16 × ${10}^{2}$ | 1.16 × ${10}^{2}$ | 1.96 × ${10}^{-1}$ | |

PGL-SSA | $\mathbf{5}.\mathbf{04}\times {\mathbf{10}}^{-\mathbf{1}}$ | $\mathbf{3}.\mathbf{92}\times {\mathbf{10}}^{-\mathbf{1}}$ | $\mathbf{1}.\mathbf{50}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{1}.\mathbf{27}\times {\mathbf{10}}^{\mathbf{0}}$ | $\mathbf{1}.\mathbf{22}\times {\mathbf{10}}^{\mathbf{0}}$ | $\mathbf{4}.\mathbf{97}\times {\mathbf{10}}^{-\mathbf{2}}$ | |

PSO | 1.43 × ${10}^{2}$ | 1.59 × ${10}^{1}$ | 1.26 × ${10}^{2}$ | 6.76 × ${10}^{2}$ | 4.42 × ${10}^{0}$ | 6.72 × ${10}^{2}$ | |

F7 | GWO | 3.42 × ${10}^{0}$ | 4.21 × ${10}^{0}$ | 1.13 × ${10}^{-13}$ | 3.90 × ${10}^{0}$ | 3.90 × ${10}^{0}$ | 1.90 × ${10}^{-11}$ |

SSA | 2.47 × ${10}^{-224}$ | 3.75 × ${10}^{-225}$ | 0.0 | 1.95 × ${10}^{-211}$ | 4.34 × ${10}^{-212}$ | 0.0 | |

PGL-SSA | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

PSO | 3.05 × ${10}^{0}$ | 3.43 × ${10}^{-1}$ | 2.55 × ${10}^{0}$ | 7.50 × ${10}^{0}$ | 2.98 × ${10}^{-1}$ | 7.21 × ${10}^{0}$ | |

F8 | GWO | 4.09 × ${10}^{-14}$ | 1.74 × ${10}^{-15}$ | 3.95 × ${10}^{-14}$ | 1.13 × ${10}^{-8}$ | 3.76 × ${10}^{-9}$ | 7.60 × ${10}^{-9}$ |

SSA | $4.44\times {10}^{-16}$ | 0.0 | $4.44\times {10}^{-16}$ | $4.44\times {10}^{-16}$ | 0.0 | $4.44\times {10}^{-16}$ | |

PGL-SSA | $\mathbf{4}.\mathbf{44}\times {\mathbf{10}}^{-\mathbf{16}}$ | 0.0 | $\mathbf{4}.\mathbf{44}\times {\mathbf{10}}^{-\mathbf{16}}$ | $\mathbf{4}.\mathbf{44}\times {\mathbf{10}}^{-\mathbf{16}}$ | 0.0 | $\mathbf{4}.\mathbf{44}\times {\mathbf{10}}^{-\mathbf{16}}$ | |

PSO | 1.67 × ${10}^{-1}$ | 2.27 × ${10}^{-2}$ | 1.24 × ${10}^{-1}$ | 1.03 × ${10}^{0}$ | 6.45 × ${10}^{-3}$ | 1.02 × ${10}^{0}$ | |

F9 | GWO | 2.31 × ${10}^{-3}$ | 4.63 × ${10}^{-3}$ | 0.0 | 7.16 × ${10}^{-15}$ | 1.66 × ${10}^{-16}$ | 6.99 × ${10}^{-15}$ |

SSA | 5.14 × ${10}^{-199}$ | 4.32 × ${10}^{-200}$ | 0.0 | 4.47 × ${10}^{-207}$ | 1.25 × ${10}^{-208}$ | 0.0 | |

PGL-SSA | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

PSO | 3.38 × ${10}^{0}$ | 1.46 × ${10}^{0}$ | 1.68 × ${10}^{0}$ | 1.19 × ${10}^{1}$ | 2.77 × ${10}^{0}$ | 7.80 × ${10}^{0}$ | |

F10 | GWO | 3.29 × ${10}^{-2}$ | 1.64 × ${10}^{-2}$ | 2.03 × ${10}^{-2}$ | 1.85 × ${10}^{-1}$ | 2.45 × ${10}^{-2}$ | 1.43 × ${10}^{-1}$ |

SSA | 1.05 × ${10}^{-8}$ | 1.70 × ${10}^{-8}$ | 1.09 × ${10}^{-9}$ | 2.54 × ${10}^{-9}$ | 3.74 × ${10}^{-9}$ | 4.10 × ${10}^{-10}$ | |

PGL-SSA | $\mathbf{3}.\mathbf{83}\times {\mathbf{10}}^{-\mathbf{9}}$ | $\mathbf{4}.\mathbf{40}\times {\mathbf{10}}^{-\mathbf{9}}$ | $\mathbf{9}.\mathbf{87}\times {\mathbf{10}}^{-\mathbf{14}}$ | $\mathbf{8}.\mathbf{05}\times {\mathbf{10}}^{-\mathbf{10}}$ | $\mathbf{6}.\mathbf{94}\times {\mathbf{10}}^{-\mathbf{10}}$ | $\mathbf{8}.\mathbf{41}\times {\mathbf{10}}^{-\mathbf{11}}$ | |

PSO | 7.70 × ${10}^{-1}$ | 3.71 × ${10}^{-1}$ | 4.43 × ${10}^{-1}$ | 3.05 × ${10}^{2}$ | 1.25 × ${10}^{2}$ | 2.13 × ${10}^{2}$ | |

F11 | GWO | 3.26 × ${10}^{-1}$ | 2.92 × ${10}^{-1}$ | 6.23 × ${10}^{-5}$ | 5.92 × ${10}^{0}$ | 3.76 × ${10}^{-1}$ | 5.49 × ${10}^{0}$ |

SSA | 6.10 × ${10}^{-8}$ | 3.95 × ${10}^{-8}$ | 1.19 × ${10}^{-8}$ | 1.02 × ${10}^{-7}$ | 9.70 × ${10}^{-8}$ | 1.06 × ${10}^{-8}$ | |

PGL-SSA | $\mathbf{2}.\mathbf{39}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{2}.\mathbf{07}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{5}.\mathbf{29}\times {\mathbf{10}}^{-\mathbf{10}}$ | $\mathbf{3}.\mathbf{57}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{2}.\mathbf{94}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{1}.\mathbf{65}\times {\mathbf{10}}^{-\mathbf{9}}$ | |

Dim = 2 | |||||||

PSO | 2.48 × ${10}^{1}$ | 4.95 × ${10}^{-1}$ | 1.99 × ${10}^{0}$ | ||||

F12 | GWO | 1.99 × ${10}^{0}$ | 9.92 × ${10}^{-1}$ | 9.98 × ${10}^{-1}$ | |||

SSA | 7.82 × ${10}^{0}$ | 4.84 × ${10}^{0}$ | 2.98 × ${10}^{0}$ | ||||

PGL-SSA | $\mathbf{9}.\mathbf{98}\times {\mathbf{10}}^{-\mathbf{1}}$ | $\mathbf{4}.\mathbf{35}\times {\mathbf{10}}^{-\mathbf{12}}$ | $\mathbf{9}.\mathbf{98}\times {\mathbf{10}}^{-\mathbf{1}}$ | ||||

Dim = 4 | |||||||

PSO | 1.27 × ${10}^{-3}$ | 5.22 × ${10}^{-4}$ | 7.50 × ${10}^{-4}$ | ||||

F13 | GWO | 3.15 × ${10}^{-4}$ | $\mathbf{1}.\mathbf{23}\times {\mathbf{10}}^{-\mathbf{8}}$ | 3.08 × ${10}^{-4}$ | |||

SSA | 3.41 × ${10}^{-4}$ | 2.66 × ${10}^{-6}$ | 3.38 × ${10}^{-4}$ | ||||

PGL-SSA | $\mathbf{3}.\mathbf{07}\times {\mathbf{10}}^{-\mathbf{4}}$ | 6.55 × ${10}^{-6}$ | $\mathbf{3}.\mathbf{07}\times {\mathbf{10}}^{-\mathbf{4}}$ | ||||

Dim = 2 | |||||||

PSO | $-1.03\times {10}^{0}$ | 4.06 × ${10}^{-5}$ | $-1.03\times {10}^{0}$ | ||||

F14 | GWO | $-1.03\times {10}^{0}$ | $\mathbf{2}.\mathbf{45}\times {\mathbf{10}}^{-\mathbf{9}}$ | $-1.03\times {10}^{0}$ | |||

SSA | $-1.03\times {10}^{0}$ | 4.28 × ${10}^{-5}$ | $-1.03\times {10}^{0}$ | ||||

PGL-SSA | $-1.03\times {10}^{0}$ | 2.10 × ${10}^{-5}$ | $-1.03\times {10}^{0}$ | ||||

Dim = 2 | |||||||

PSO | 3.97 × ${10}^{-1}$ | 2.15 × ${10}^{-4}$ | 3.97 × ${10}^{-1}$ | ||||

F15 | GWO | 3.97 × ${10}^{-1}$ | $\mathbf{4}.\mathbf{31}\times {\mathbf{10}}^{-\mathbf{7}}$ | 3.97 × ${10}^{-1}$ | |||

SSA | 3.97 × ${10}^{-1}$ | 2.35 × ${10}^{-5}$ | 3.97 × ${10}^{-1}$ | ||||

PGL-SSA | 3.97 × ${10}^{-1}$ | 1.31 × ${10}^{-5}$ | 3.97 × ${10}^{-1}$ | ||||

Dim = 2 | |||||||

PSO | 3.00 × ${10}^{0}$ | 4.59 × ${10}^{-4}$ | 3.00 × ${10}^{0}$ | ||||

GWO | 3.00 × ${10}^{0}$ | $\mathbf{7}.\mathbf{20}\times {\mathbf{10}}^{-\mathbf{6}}$ | 3.00 × ${10}^{0}$ | ||||

F16 | SSA | 3.00 × ${10}^{0}$ | 2.50 × ${10}^{-3}$ | 3.00 × ${10}^{0}$ | |||

PGL-SSA | 3.00 × ${10}^{0}$ | 1.00 × ${10}^{-3}$ | 3.00 × ${10}^{0}$ | ||||

Dim = 3 | |||||||

PSO | $-3.85\times {10}^{0}$ | 2.16 × ${10}^{-3}$ | $-3.86\times {10}^{0}$ | ||||

F17 | GWO | $-3.86\times {10}^{0}$ | $\mathbf{3}.\mathbf{24}\times {\mathbf{10}}^{-\mathbf{5}}$ | $-3.86\times {10}^{0}$ | |||

SSA | $-3.85\times {10}^{0}$ | 1.99 × ${10}^{-3}$ | $-3.86\times {10}^{0}$ | ||||

PGL-SSA | $-3.86\times {10}^{0}$ | 8.08 × ${10}^{-4}$ | $-3.86\times {10}^{0}$ | ||||

Dim = 6 | |||||||

PSO | $-3.02\times {10}^{0}$ | 1.45 × ${10}^{-1}$ | $-3.20\times {10}^{0}$ | ||||

F18 | GWO | $-3.29\times {10}^{0}$ | 4.76 × ${10}^{-2}$ | $-\mathbf{3}.\mathbf{32}\times {\mathbf{10}}^{\mathbf{0}}$ | |||

SSA | $-3.27\times {10}^{0}$ | 3.87 × ${10}^{-2}$ | $-3.31\times {10}^{0}$ | ||||

PGL-SSA | $-3.29\times {10}^{0}$ | $\mathbf{1}.\mathbf{78}\times {\mathbf{10}}^{-\mathbf{2}}$ | $-3.31\times {10}^{0}$ | ||||

Dim = 4 | |||||||

PSO | $-1.01\times {10}^{1}$ | $2.01\times {10}^{-3}$ | $-1.01\times {10}^{1}$ | ||||

F19 | GWO | $-1.01\times {10}^{1}$ | 2.51 × ${10}^{-4}$ | $-1.01\times {10}^{1}$ | |||

SSA | $-1.01\times {10}^{1}$ | 2.53 × ${10}^{-3}$ | $-1.01\times {10}^{1}$ | ||||

PGL-SSA | $-1.01\times {10}^{1}$ | $\mathbf{8}.\mathbf{42}\times {\mathbf{10}}^{-\mathbf{5}}$ | $-1.01\times {10}^{1}$ | ||||

Dim = 4 | |||||||

PSO | $-1.03\times {10}^{1}$ | 2.50 × ${10}^{-3}$ | $-1.03\times {10}^{1}$ | ||||

F20 | GWO | $-\mathbf{1}.\mathbf{04}\times {\mathbf{10}}^{\mathbf{1}}$ | 2.52 × ${10}^{-4}$ | $-\mathbf{1}.\mathbf{04}\times {\mathbf{10}}^{\mathbf{1}}$ | |||

SSA | $-1.03\times {10}^{1}$ | 7.49 × ${10}^{-3}$ | $-1.04\times {10}^{1}$ | ||||

PGL-SSA | $-\mathbf{1}.\mathbf{04}\times {\mathbf{10}}^{\mathbf{1}}$ | $\mathbf{1}.\mathbf{25}\times {\mathbf{10}}^{-\mathbf{4}}$ | $-\mathbf{1}.\mathbf{04}\times {\mathbf{10}}^{\mathbf{1}}$ | ||||

Dim = 4 | |||||||

PSO | $-1.05\times {10}^{1}$ | 1.46 × ${10}^{-2}$ | $-1.05\times {10}^{1}$ | ||||

F21 | GWO | $-1.05\times {10}^{1}$ | 2.44 × ${10}^{-4}$ | $-1.05\times {10}^{1}$ | |||

SSA | $-1.05\times {10}^{1}$ | 5.26 × ${10}^{-3}$ | $-1.05\times {10}^{1}$ | ||||

PGL-SSA | $-1.05\times {10}^{1}$ | $\mathbf{8}.\mathbf{73}\times {\mathbf{10}}^{-\mathbf{5}}$ | $-1.05\times {10}^{1}$ |

Function | Algorithm | ||
---|---|---|---|

PGL-SSA/SSA | PGL-SSA/GWO | PGL-SSA/PSO | |

F1 | 4.34 × ${10}^{-4}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F2 | 4.17 × ${10}^{-5}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F3 | 7.91 × ${10}^{-5}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F4 | 1.81 × ${10}^{-4}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F5 | 5.78 × ${10}^{-5}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F6 | 1.04 × ${10}^{-10}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F7 | 3.35 × ${10}^{-2}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F8 | N/A | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F9 | 1.75 × ${10}^{-2}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F10 | 2.74 × ${10}^{-4}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F11 | 1.47 × ${10}^{-2}$ | 2.87 × ${10}^{-11}$ | 2.87 × ${10}^{-11}$ |

F12 | 1.76 × ${10}^{-3}$ | N/A | 2.44 × ${10}^{-3}$ |

F13 | N/A | 4.64 × ${10}^{-1}$ | 9.02 × ${10}^{-3}$ |

F14 | 4.95 × ${10}^{-2}$ | 2.75 × ${10}^{-1}$ | 4.95 × ${10}^{-2}$ |

F15 | N/A | 3.74 × ${10}^{-1}$ | 4.95 × ${10}^{-2}$ |

F16 | 4.95 × ${10}^{-2}$ | 2.75 × ${10}^{-1}$ | 5.12 × ${10}^{-1}$ |

F17 | 4.95 × ${10}^{-2}$ | 3.71 × ${10}^{-1}$ | 4.95 × ${10}^{-2}$ |

F18 | 4.95 × ${10}^{-2}$ | 1.94 × ${10}^{-1}$ | 4.95 × ${10}^{-2}$ |

F19 | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ |

F20 | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ |

F21 | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ | 4.95 × ${10}^{-2}$ |

Function | Algorithm | Dim | Mean | Std | Best |
---|---|---|---|---|---|

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 7.24 × ${10}^{-77}$ | 3.43 × ${10}^{-76}$ | 9.12 × ${10}^{-82}$ | |

F1 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 0.0 | 3.84 × ${10}^{-201}$ | |

PGL-SSA | 30 | 1.72 × ${10}^{-222}$ | 0.0 | 0.0 | |

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 2.43 × ${10}^{-40}$ | 3.74 × ${10}^{-40}$ | 4.77 × ${10}^{-51}$ | |

F2 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 4.74 × ${10}^{-103}$ | 2.34 × ${10}^{-103}$ | |

PGL-SSA | 30 | 4.79 × ${10}^{-263}$ | 0.0 | 0.0 | |

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 4.47 × ${10}^{-62}$ | 2.84 × ${10}^{-61}$ | 7.12 × ${10}^{-74}$ | |

F3 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 8.34 × ${10}^{-121}$ | 4.37 × ${10}^{-121}$ | |

PGL-SSA | 30 | 0.0 | 0.0 | 0.0 | |

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 7.64 × ${10}^{-35}$ | 4.37 × ${10}^{-34}$ | 4.47 × ${10}^{-49}$ | |

F4 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 4.64 × ${10}^{-102}$ | 7.27 × ${10}^{-104}$ | |

PGL-SSA | 30 | 0.0 | 0.0 | 0.0 | |

GCSSA | 30 | 4.14 × ${10}^{-9}$ | 2.65 × ${10}^{-9}$ | 1.14 × ${10}^{-13}$ | |

CSSOA | 30 | 4.44 × ${10}^{-5}$ | 6.24 × ${10}^{-5}$ | 5.33 × ${10}^{-8}$ | |

F5 | SFSSA | 30 | 4.34 × ${10}^{-8}$ | 2.62 × ${10}^{-8}$ | 3.26 × ${10}^{-11}$ |

CLSSA | 30 | 0.0 | 3.47 × ${10}^{-8}$ | 4.34 × ${10}^{-9}$ | |

PGL-SSA | 30 | 2.14 × ${10}^{-9}$ | 2.22 × ${10}^{-9}$ | 5.56 × ${10}^{-13}$ | |

GCSSA | 30 | 7.33 × ${10}^{0}$ | 1.47 × ${10}^{0}$ | 4.33 × ${10}^{-2}$ | |

CSSOA | 30 | $-2.77\times {10}^{3}$ | 4.74 × ${10}^{2}$ | 4.34 × ${10}^{0}$ | |

F6 | SFSSA | 30 | 1.47 × ${10}^{1}$ | 4.84 × ${10}^{1}$ | 2.33 × ${10}^{-1}$ |

CLSSA | 30 | $-9.47\times {10}^{3}$ | 8.47 × ${10}^{2}$ | $-8.04\times {10}^{3}$ | |

PGL-SSA | 30 | 5.04 × ${10}^{-1}$ | 3.92 × ${10}^{-1}$ | 1.50 × ${10}^{-3}$ | |

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 0.0 | 0.0 | 0.0 | |

F7 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 0.0 | 0.0 | |

PGL-SSA | 30 | 0.0 | 0.0 | 0.0 | |

GCSSA | 30 | 7.84 × ${10}^{-16}$ | 0.0 | 6.47 × ${10}^{-16}$ | |

CSSOA | 30 | 9.43 × ${10}^{-16}$ | 0.0 | 9.04 × ${10}^{-13}$ | |

F8 | SFSSA | 30 | 8.88 × ${10}^{-16}$ | 0.0 | 8.88 × ${10}^{-16}$ |

CLSSA | 30 | 8.96 × ${10}^{-16}$ | 0.0 | 8.96 × ${10}^{-16}$ | |

PGL-SSA | 30 | 4.44 × ${10}^{-16}$ | 0.0 | 4.44 × ${10}^{-16}$ | |

GCSSA | 30 | 0.0 | 0.0 | 0.0 | |

CSSOA | 30 | 0.0 | 0.0 | 0.0 | |

F9 | SFSSA | 30 | 0.0 | 0.0 | 0.0 |

CLSSA | 30 | 0.0 | 0.0 | 0.0 | |

PGL-SSA | 30 | 0.0 | 0.0 | 0.0 | |

GCSSA | 30 | 4.84 × ${10}^{-7}$ | 4.78 × ${10}^{-8}$ | 6.43 × ${10}^{-9}$ | |

CSSOA | 30 | 2.44 × ${10}^{-5}$ | 4.37 × ${10}^{-5}$ | 7.33 × ${10}^{-7}$ | |

F10 | SFSSA | 30 | 4.37 × ${10}^{-8}$ | 3.24 × ${10}^{-8}$ | 8.34 × ${10}^{-12}$ |

CLSSA | 30 | 2.47 × ${10}^{-27}$ | 1.43 × ${10}^{-9}$ | 3.74 × ${10}^{-10}$ | |

PGL-SSA | 30 | 3.83 × ${10}^{-9}$ | 4.40 × ${10}^{-9}$ | 9.87 × ${10}^{-14}$ | |

GCSSA | 30 | 6.34 × ${10}^{-6}$ | 2.04 × ${10}^{-7}$ | 7.44 × ${10}^{-9}$ | |

CSSOA | 30 | 3.47 × ${10}^{-5}$ | 2.02 × ${10}^{-5}$ | 4.62 × ${10}^{-7}$ | |

F11 | SFSSA | 30 | 4.84 × ${10}^{-7}$ | 2.73 × ${10}^{-7}$ | 6.22 × ${10}^{-9}$ |

CLSSA | 30 | 2.48 × ${10}^{-20}$ | 2.42 × ${10}^{-8}$ | 3.77 × ${10}^{-9}$ | |

PGL-SSA | 30 | 2.39 × ${10}^{-8}$ | 2.07 × ${10}^{-8}$ | 5.29 × ${10}^{-10}$ | |

GCSSA | 2 | 2.42 × ${10}^{0}$ | 7.62 × ${10}^{-1}$ | 1.73 × ${10}^{0}$ | |

CSSOA | 2 | 1.73 × ${10}^{0}$ | 4.34 × ${10}^{-1}$ | 1.44 × ${10}^{0}$ | |

F12 | SFSSA | 2 | 1.02 × ${10}^{0}$ | 7.42 × ${10}^{-4}$ | 1.02 × ${10}^{0}$ |

CLSSA | 2 | 1.02 × ${10}^{0}$ | 2.43 × ${10}^{0}$ | 1.46 × ${10}^{0}$ | |

PGL-SSA | 2 | 9.98 × ${10}^{-1}$ | 4.35 × ${10}^{-12}$ | 9.98 × ${10}^{-1}$ | |

GCSSA | 4 | 3.42 × ${10}^{-4}$ | 4.21 × ${10}^{-5}$ | 2.77 × ${10}^{-4}$ | |

CSSOA | 4 | 3.21 × ${10}^{-4}$ | 4.44 × ${10}^{-6}$ | 3.13 × ${10}^{-4}$ | |

F13 | SFSSA | 4 | 3.13 × ${10}^{-4}$ | 4.88 × ${10}^{-6}$ | 3.13 × ${10}^{-4}$ |

CLSSA | 4 | 3.12 × ${10}^{-4}$ | 2.47 × ${10}^{-7}$ | 3.12 × ${10}^{-4}$ | |

PGL-SSA | 4 | 3.07 × ${10}^{-4}$ | 6.55 × ${10}^{-6}$ | 3.07 × ${10}^{-4}$ | |

GCSSA | 2 | $-1.03\times {10}^{0}$ | 3.77 × ${10}^{-4}$ | $-1.33\times {10}^{0}$ | |

CSSOA | 2 | $-1.03\times {10}^{0}$ | 3.41 × ${10}^{-4}$ | $-1.03\times {10}^{0}$ | |

F14 | SFSSA | 2 | $-1.03\times {10}^{0}$ | 4.47 × ${10}^{-7}$ | $-1.03\times {10}^{0}$ |

CLSSA | 2 | $-1.2\times {10}^{0}$ | 5.37 × ${10}^{-16}$ | $-1.02\times {10}^{0}$ | |

PGL-SSA | 2 | $-1.03\times {10}^{0}$ | 2.10 × ${10}^{-5}$ | $-1.03\times {10}^{0}$ | |

GCSSA | 2 | 3.97 × ${10}^{-1}$ | 1.77 × ${10}^{-4}$ | 3.97 × ${10}^{-1}$ | |

CSSOA | 2 | 4.01 × ${10}^{-1}$ | 5.42 × ${10}^{-5}$ | 4.01 × ${10}^{-1}$ | |

F15 | SFSSA | 2 | 3.99 × ${10}^{-1}$ | 1.77 × ${10}^{-5}$ | 3.99 × ${10}^{-1}$ |

CLSSA | 2 | 4.03 × ${10}^{-1}$ | 0.0 | 4.03 × ${10}^{-1}$ | |

PGL-SSA | 2 | 3.97 × ${10}^{-1}$ | 1.31 × ${10}^{-5}$ | 3.97 × ${10}^{-1}$ | |

GCSSA | 2 | 3.00 × ${10}^{0}$ | 1.27 × ${10}^{-3}$ | 3.00 × ${10}^{0}$ | |

CSSOA | 2 | 3.00 × ${10}^{0}$ | 1.00 × ${10}^{-3}$ | 3.00 × ${10}^{0}$ | |

F16 | SFSSA | 2 | 3.01 × ${10}^{0}$ | 3.77 × ${10}^{-15}$ | 3.01 × ${10}^{0}$ |

CLSSA | 2 | 3.02 × ${10}^{0}$ | 4.84 × ${10}^{-15}$ | 3.02 × ${10}^{0}$ | |

PGL-SSA | 2 | 3.00 × ${10}^{0}$ | 1.00 × ${10}^{-3}$ | 3.00 × ${10}^{0}$ | |

GCSSA | 3 | $-3.88\times {10}^{0}$ | 6.24 × ${10}^{-3}$ | $-3.88\times {10}^{0}$ | |

CSSOA | 3 | $-3.84\times {10}^{0}$ | 7.37 × ${10}^{-3}$ | $-3.84\times {10}^{0}$ | |

F17 | SFSSA | 3 | $-3.86\times {10}^{0}$ | 2.44 × ${10}^{-15}$ | $-3.86\times {10}^{0}$ |

CLSSA | 3 | $-3.94\times {10}^{0}$ | 2.74 × ${10}^{-15}$ | $-3.94\times {10}^{0}$ | |

PGL-SSA | 3 | $-3.86\times {10}^{0}$ | 8.08 × ${10}^{-4}$ | $-3.86\times {10}^{0}$ | |

GCSSA | 6 | $-3.28\times {10}^{0}$ | 7.34 × ${10}^{-1}$ | $-3.28\times {10}^{0}$ | |

CSSOA | 6 | $-3.34\times {10}^{0}$ | 2.37 × ${10}^{-2}$ | $-3.34\times {10}^{0}$ | |

F18 | SFSSA | 6 | $-3.33\times {10}^{0}$ | 1.84 × ${10}^{-2}$ | $-3.33\times {10}^{0}$ |

CLSSA | 6 | $-3.33\times {10}^{0}$ | 5.74 × ${10}^{-2}$ | $-3.33\times {10}^{0}$ | |

PGL-SSA | 6 | $-3.29\times {10}^{0}$ | 1.78 × ${10}^{-2}$ | $-3.31\times {10}^{0}$ | |

GCSSA | 4 | $-1.01\times {10}^{1}$ | 7.44 × ${10}^{-4}$ | -1.01 × ${10}^{1}$ | |

CSSOA | 4 | $-1.01\times {10}^{1}$ | 4.47 × ${10}^{-4}$ | -1.01 × ${10}^{1}$ | |

F19 | SFSSA | 4 | $-9.73\times {10}^{0}$ | 8.74 × ${10}^{-1}$ | $-1.01\times {10}^{1}$ |

CLSSA | 4 | $-1.01\times {10}^{1}$ | 5.44 × ${10}^{-8}$ | $-1.01\times {10}^{1}$ | |

PGL-SSA | 4 | $-1.01\times {10}^{1}$ | 8.42 × ${10}^{-5}$ | $-1.01\times {10}^{1}$ | |

GCSSA | 4 | $-1.04\times {10}^{1}$ | $-1.47$×${10}^{-4}$ | $-1.04\times {10}^{1}$ | |

CSSOA | 4 | $-1.04\times {10}^{1}$ | 7.33 × ${10}^{-4}$ | $-1.04\times {10}^{1}$ | |

F20 | SFSSA | 4 | $-1.04\times {10}^{1}$ | 4.43 × ${10}^{-10}$ | $-1.04\times {10}^{1}$ |

CLSSA | 4 | $-1.04\times {10}^{1}$ | 1.13 × ${10}^{-6}$ | $-1.04\times {10}^{1}$ | |

PGL-SSA | 4 | $-1.04\times {10}^{1}$ | 1.25 × ${10}^{-4}$ | $-1.04\times {10}^{1}$ | |

GCSSA | 4 | $-1.05\times {10}^{1}$ | 8.66 × ${10}^{-5}$ | $-1.05\times {10}^{1}$ | |

CSSOA | 4 | $-1.05\times {10}^{1}$ | 4.37 × ${10}^{-5}$ | $-1.05\times {10}^{1}$ | |

F21 | SFSSA | 4 | $-1.05\times {10}^{1}$ | 4.17 × ${10}^{-4}$ | $-1.05\times {10}^{1}$ |

CLSSA | 4 | $-1.05\times {10}^{1}$ | 5.64 × ${10}^{-8}$ | $-1.05\times {10}^{1}$ | |

PGL-SSA | 4 | $-1.05\times {10}^{1}$ | 8.73 × ${10}^{-5}$ | $-1.05\times {10}^{1}$ |

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**MDPI and ACS Style**

Liu, X.; Bai, Y.; Yu, C.; Yang, H.; Gao, H.; Wang, J.; Chang, Q.; Wen, X.
Multi-Strategy Improved Sparrow Search Algorithm and Application. *Math. Comput. Appl.* **2022**, *27*, 96.
https://doi.org/10.3390/mca27060096

**AMA Style**

Liu X, Bai Y, Yu C, Yang H, Gao H, Wang J, Chang Q, Wen X.
Multi-Strategy Improved Sparrow Search Algorithm and Application. *Mathematical and Computational Applications*. 2022; 27(6):96.
https://doi.org/10.3390/mca27060096

**Chicago/Turabian Style**

Liu, Xiangdong, Yan Bai, Cunhui Yu, Hailong Yang, Haoning Gao, Jing Wang, Qing Chang, and Xiaodong Wen.
2022. "Multi-Strategy Improved Sparrow Search Algorithm and Application" *Mathematical and Computational Applications* 27, no. 6: 96.
https://doi.org/10.3390/mca27060096