# A Hybrid Multi-Step Probability Selection Particle Swarm Optimization with Dynamic Chaotic Inertial Weight and Acceleration Coefficients for Numerical Function Optimization

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## Abstract

**:**

## 1. Introduction

- The multi-step probability selection process can enhance the search ability of particles and avoid premature convergence, which also has a positive effect on convergence speed.
- The sine chaotic $\omega $ and symmetric tangent chaotic ${c}_{1},{c}_{2}$ enrich the swarm diversity and achieve a better balance between the exploration and exploitation ability, which offers higher convergence accuracy.

## 2. Related Theory about PSO

## 3. Hybrid Multi-Step Probability Selection Particle Swarm Optimization with Sine Chaotic Inertial Weight and Symmetric Tangent Chaotic Acceleration Coefficients (MPSPSO-ST)

#### 3.1. Hybrid Multi-Step Probability Selection Particle Swarm Optimization (MPSPSO)

- Calculate the objective function value $f(i=1,2,3,4)$ of each of the four particle positions.
- Due to it that the smaller $f(x)$ is, the better the fitness of the position is, so the probability of being selected should be inversely proportional to the function value of the position. From this, the probability of each position being selected is calculated as follows:$$P({x}_{i})=1-\frac{f({x}_{i})}{{\displaystyle \sum _{j=1}^{4}f({x}_{j})}}$$
- Calculate the cumulative probability of each position:$${q}_{i}={\displaystyle \sum _{j=1}^{i}P({x}_{j})}$$

- Randomly generate a uniformly distributed random number $r$ in the interval [0, 1].
- When $r$ satisfies $r<{q}_{1}$, select position ${x}_{1}^{t+1}$; when $r$ satisfies ${q}_{k-1}<r\le {q}_{k}$, select position ${x}_{k}^{t+1}$.

#### 3.2. Sine Chaotic Inertia Weight $\omega $

#### 3.3. Symmetric Tangent Chaotic Acceleration Coefficients ${c}_{1},{c}_{2}$

## 4. Experimental Results and Discussion

**d!**local minima. The parameter

**m**defines the steepness of the valleys and ridges and a larger

**m**leads to a more difficult search. ${f}_{11}$(Griewank) has many widespread local minima, which causes more difficulties in searching for the global optimum. ${f}_{13}$(levy) and ${f}_{19}$(Zakharov) are both widely used in the optimization field because they are classical. Dim, Range and

**f**represent the space dimensions of solution, the range of function variation and the minimum value of the function, that is, the optimal solutions, respectively.

_{min}#### 4.1. Comparison of MPSPSO-ST with Standard PSO, Basic PSO and MPSPSO

#### 4.2. Comparison of MPSPSO-ST with CPSO, PSO-NDAC, AIWCPSO, DE, MFO and SCA

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

ID | Test Function | Dim | Range | f_{min} | Type |
---|---|---|---|---|---|

${f}_{1}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | 30 | [−100, 100] | 0 | Unimodal |

${f}_{2}$ | $f(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}+random[0,1]}$ | 30 | [−1.28, 1.28] | 0 | Unimodal |

${f}_{3}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{({\displaystyle \sum _{j=1}^{i}{x}_{j}})}^{2}}$ | 30 | [−100, 100] | 0 | Unimodal |

${f}_{4}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{\left|{x}_{i}\right|}^{i+1}}$ | 30 | [−1, 1] | 0 | Unimodal |

${f}_{5}$ | $f(x)={({x}_{1}-1)}^{2}+{\displaystyle \sum _{i=2}^{n}i{(2{x}_{i}^{2}-{x}_{i-1})}^{2}}$ | 30 | [−10, 10] | 0 | Unimodal |

${f}_{6}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{([{x}_{i}+0.5])}^{2}}$ | 30 | [−100, 100] | 0 | Unimodal |

${f}_{7}$ | $f(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{2}}$ | 30 | [−10, 10] | 0 | Unimodal |

${f}_{8}$ | $f(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}}$ | 30 | [−1.28, 1.28] | 0 | Unimodal |

${f}_{9}$ | $f(x)=1-\mathrm{cos}(2\pi \sqrt{{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}})+0.1\sqrt{{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}}$ | 30 | [−100, 100] | 0 | Multimodal |

${f}_{10}$ | $f(x)=-{{\displaystyle \sum _{i=1}^{n}\mathrm{sin}({x}_{i})\cdot (\mathrm{sin}(\frac{i{x}_{i}^{2}}{\pi}))}}^{2m}$$,m=10$ | 30 | [0, $\pi $] | −4.687 | Multimodal |

${f}_{11}$ | $f(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}})+1}}$ | 30 | [−600, 600] | 0 | Multimodal |

${f}_{12}$ | $f(x)=\frac{\pi}{n}\{10{\mathrm{sin}}^{2}(\pi {y}_{1})+{\displaystyle \sum _{i=1}^{n-1}{({y}_{i}-1)}^{2}}$$[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1})]+{({y}_{n}-1)}^{2}\}+$$\sum _{i=1}^{n}u({x}_{i},10,100,4)},{y}_{i}=1+\frac{{x}_{i}+1}{4$$u({x}_{i},a,k,m)=\{\begin{array}{c}k{({x}_{i}-a)}^{m},{x}_{i}>a\\ 0,-a\le {x}_{i}\le a\\ k{(-{x}_{i}-a)}^{m},{x}_{i}<-a\end{array}$ | 30 | [−50, 50] | 0 | Multimodal |

${f}_{13}$ | $f(x)=0.1{\mathrm{sin}}^{2}(3\pi {x}_{1})+{\displaystyle \sum _{i=1}^{n-1}{({x}_{i}-1)}^{2}+}$${\mathrm{sin}}^{2}(3\pi {x}_{i+1})+{({x}_{n}-1)}^{2}(1+{\mathrm{sin}}^{2}(3\pi {x}_{n}))$ | 30 | [−5, 5] | 0 | Multimodal |

${f}_{14}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{({10}^{6})}^{\frac{i-1}{n-1}}{x}_{i}^{2}}$ | 30 | [−100, 100] | 0 | Multimodal |

${f}_{15}$ | $f(x)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\mathrm{sin}({x}_{i})+0.1{x}_{i}\right|}$ | 30 | [−10, 10] | 0 | Multimodal |

${f}_{16}$ | $f(x)=[1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+$$3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2})]\times $$[30+(2{x}_{1}-3{x}_{2})\times (18-32{x}_{1}+12{x}_{1}^{2}+$$48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2})]$ | 2 | [−2, 2] | 3 | Multimodal |

${f}_{17}$ | $f(x)={(\frac{1}{500}+{\displaystyle \sum _{j=1}^{25}\frac{1}{j+{\displaystyle \sum _{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}}})}^{-1}$ | 2 | [−65.536, 65.536] | 0.998 | Multimodal |

${f}_{18}$ | $f(x)={\displaystyle \sum _{i=1}^{11}{[{a}_{i}-\frac{{x}_{1}({b}_{i}^{2}+{b}_{i}{x}_{2})}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}]}^{2}}$ | 4 | [−5, 5] | 0 | Multimodal |

${f}_{19}$ | $f(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}+{({\displaystyle \sum _{i=1}^{n}0.5i{x}_{i}})}^{2}}+{({\displaystyle \sum _{i=1}^{n}0.5i{x}_{i}})}^{4}$ | 6 | [−5, 10] | 0 | Multimodal |

${f}_{20}$ | $f(x)=-{\displaystyle \sum _{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle \sum _{j=1}^{6}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2}})}$ | 6 | [0, 1] | −3.32 | Multimodal |

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**Figure 4.**Pseudo-code of the multi-step probability selection particle swarm optimization with sine chaotic inertial weight and symmetric tangent chaotic acceleration coefficients (MPSPSO-ST) algorithm.

**Figure 5.**The convergence curves of four algorithms for the functions: (

**a**) ${f}_{1}$; (

**b**) ${f}_{4}$; (

**c**) ${f}_{5}$; (

**d**) ${f}_{8}$.

**Figure 6.**The convergence curves of four algorithms for the functions: (

**a**) ${f}_{10}$; (

**b**) ${f}_{11}$; (

**c**) ${f}_{13}$; (

**d**) ${f}_{15}$;

**(e**) ${f}_{17}$; (

**f**) ${f}_{18}$.

**Figure 7.**The convergence curves of four algorithms for the functions: (

**a**) ${f}_{19}$; (

**b**) ${f}_{20}$.

**Figure 8.**The convergence curves of seven algorithms for the functions: (

**a**) ${f}_{1}$; (

**b**) ${f}_{3}$; (

**c**) ${f}_{4}$; (

**d**) ${f}_{6}$.

**Figure 9.**The convergence curves of seven algorithms for the functions: (

**a**) ${f}_{7}$; (

**b**) ${f}_{8}$; (

**c**) ${f}_{10}$; (

**d**) ${f}_{11}$; (

**e**) ${f}_{14}$; (

**f**) ${f}_{15}$.

**Figure 10.**The convergence curves of seven algorithms for the functions: (

**a**) ${f}_{17}$; (

**b**) ${f}_{18}$; (

**c**) ${f}_{19}$; (

**d**) ${f}_{20}$.

Algorithm | Population Size | Iteration | Run Times | Parameter Settings |
---|---|---|---|---|

Standard PSO | 40 | 500 | 20 | ${c}_{1}={c}_{2}=2,\omega =0.9~0.4,{V}_{\mathrm{max}}=6$ |

Basic PSO | 40 | 500 | 20 | ${c}_{1}={c}_{2}=2,\omega =1,{V}_{\mathrm{max}}=6$ |

MPSPSO | 40 | 500 | 20 | ${c}_{1}={c}_{2}=2,\omega =0.9~0.4,{V}_{\mathrm{max}}=6$ |

MPSPSO-ST | 40 | 500 | 20 | ${c}_{1}=-\partial \times {m}^{2}\times \mathrm{tan}[\frac{\pi}{8}\times (1+{m}^{2})]+\theta +\rho \times z,$${c}_{2}=-\partial \times (1-m{)}^{2}\times \mathrm{tan}[\frac{\pi}{8}\times (1+(1-m{)}^{2})]+\theta +\rho \times z,$${\omega}_{t+1}=\varphi \times \mathrm{sin}(\pi {\omega}_{t})+\tau ,{V}_{\mathrm{max}}=6$ |

**Table 2.**Experimental results for MPSPSO-ST, Standard PSO and Basic PSO. MPSPSO on 20 classical test functions.

Function | Algorithm | The Best | The Worst | Mean | S.d. |
---|---|---|---|---|---|

${f}_{1}$ | Standard PSO | 4.5678 × 10^{−4} | 2.1867 × 10^{−2} | 5.0366 × 10^{−3} | 2.0787 × 10^{−2} |

Basic PSO | 9.0029 × 10^{1} | 1.7626 × 10^{2} | 1.3408 × 10^{2} | 8.0903 × 10^{1} | |

MPSPSO | 1.2001 × 10^{−8} | 9.9103 × 10^{−7} | 2.6957 × 10^{−7} | 1.1904 × 10^{−6} | |

MPSPSO-ST | 5.6834 × 10^{−16} | 1.9825 × 10^{−13} | 2.0811 × 10^{−14} | 1.9440 × 10^{−13} | |

${f}_{2}$ | Standard PSO | 0.1209 | 0.7743 | 0.2986 | 0.6429 |

Basic PSO | 56.7585 | 142.5399 | 102.9728 | 104.4540 | |

MPSPSO | 0.0306 | 0.1479 | 0.0640 | 0.1217 | |

MPSPSO-ST | 0.0191 | 0.0732 | 0.0396 | 0.0644 | |

${f}_{3}$ | Standard PSO | 4.3416 × 10^{1} | 1.1851 × 10^{2} | 7.6565 × 10^{1} | 9.3955 × 10^{1} |

Basic PSO | 2.7751 × 10^{2} | 8.5694 × 10^{2} | 4.8829 × 10^{2} | 5.5224 × 10^{2} | |

MPSPSO | 1.8069 × 10^{0} | 1.4074 × 10^{1} | 5.7546 × 10^{0} | 1.5430 × 10^{1} | |

MPSPSO-ST | 7.9038 × 10^{−3} | 2.3292 × 10^{−1} | 7.0004 × 10^{−2} | 2.5658 × 10^{−1} | |

${f}_{4}$ | Standard PSO | 1.9400 × 10^{−7} | 4.0724 × 10^{−3} | 4.4258 × 10^{−4} | 4.1605 × 10^{−3} |

Basic PSO | 7.3786 × 10^{−2} | 1.2261 × 10^{0} | 6.5420 × 10^{−1} | 1.3869 × 10^{0} | |

MPSPSO | 4.6486 × 10^{−21} | 1.7877 × 10^{−16} | 1.2995 × 10^{−17} | 1.7277 × 10^{−16} | |

MPSPSO-ST | 1.8112 × 10^{−49} | 2.6982 × 10^{−40} | 1.9660 × 10^{−41} | 2.6120 × 10^{−40} | |

${f}_{5}$ | Standard PSO | 1.02591 | 12.0564 | 4.08932 | 12.80900 |

Basic PSO | 5.2064 × 10^{4} | 1.2287 × 10^{5} | 8.5260 × 10^{4} | 9.1677 × 10^{4} | |

MPSPSO | 0.66672 | 3.14810 | 1.26960 | 3.71450 | |

MPSPSO-ST | 0.66667 | 2.03490 | 0.79497 | 1.70870 | |

${f}_{6}$ | Standard PSO | 2.5432 × 10^{−3} | 2.4399 × 10^{−2} | 7.9220 × 10^{−3} | 2.4535 × 10^{−2} |

Basic PSO | 8.2531 × 10^{1} | 1.6702 × 10^{2} | 1.3159 × 10^{2} | 8.8401 × 10^{1} | |

MPSPSO | 1.5414 × 10^{−8} | 1.0729 × 10^{−6} | 2.3913 × 10^{−7} | 1.1474 × 10^{−6} | |

MPSPSO-ST | 6.1306 × 10^{−16} | 5.1809 × 10^{−13} | 5.0002 × 10^{−14} | 5.2461 × 10^{−13} | |

${f}_{7}$ | Standard PSO | 3.1183 × 10^{−2} | 2.5205 × 10^{−1} | 1.0527 × 10^{−1} | 2.7090 × 10^{−1} |

Basic PSO | 1.3802 × 10^{3} | 2.3578 × 10^{3} | 1.8650 × 10^{3} | 1.2075 × 10^{3} | |

MPSPSO | 5.7551 × 10^{−7} | 7.9424 × 10^{−5} | 8.8537 × 10^{−6} | 7.6769 × 10^{−5} | |

MPSPSO-ST | 5.7456 × 10^{−16} | 7.9810 × 10^{−14} | 2.7045 × 10^{−14} | 1.0692 × 10^{−13} | |

${f}_{8}$ | Standard PSO | 4.7811 × 10^{−4} | 4.7318 × 10^{−2} | 7.0241 × 10^{−3} | 4.8574 × 10^{−2} |

Basic PSO | 6.0663 × 10^{1} | 1.3860 × 10^{2} | 1.0412 × 10^{2} | 1.0270 × 10^{2} | |

MPSPSO | 2.9684 × 10^{−13} | 3.2834 × 10^{−10} | 5.1454 × 10^{−11} | 3.7774 × 10^{−10} | |

MPSPSO-ST | 7.5201 × 10^{−31} | 5.7745 × 10^{−26} | 7.5066 × 10^{−27} | 6.5486 × 10^{−26} | |

${f}_{9}$ | Standard PSO | 0.29987 | 0.49987 | 0.43487 | 0.25593 |

Basic PSO | 1.10070 | 1.5999 | 1.34680 | 0.57279 | |

MPSPSO | 0.29987 | 0.49987 | 0.42487 | 0.27839 | |

MPSPSO-ST | 0.29987 | 0.49987 | 0.36518 | 0.25683 | |

${f}_{10}$ | Standard PSO | −1.7207 × 10^{−9} | −9.1788 × 10^{−10} | −1.2473 × 10^{−9} | 9.7966 × 10^{−10} |

Basic PSO | −9.3338 × 10^{−10} | −6.8522 × 10^{−10} | −8.0715 × 10^{−10} | 3.4338 × 10^{−10} | |

MPSPSO | −2.3176 × 10^{−9} | −1.1251 × 10^{−9} | −1.8146 × 10^{−9} | 1.4152 × 10^{−9} | |

MPSPSO-ST | −3.0052 × 10^{−9} | −2.4403 × 10^{−9} | −2.7811 × 10^{−9} | 7.4879 × 10^{−10} | |

${f}_{11}$ | Standard PSO | 1.4317 × 10^{−5} | 3.2392 × 10^{−2} | 9.4530 × 10^{−3} | 3.6992 × 10^{−2} |

Basic PSO | 1.0250 × 10^{0} | 1.0392 × 10^{0} | 1.0335 × 10^{0} | 1.9501 × 10^{−2} | |

MPSPSO | 2.0319 × 10^{−8} | 3.6910 × 10^{−2} | 1.0222 × 10^{−2} | 4.3936 × 10^{−2} | |

MPSPSO-ST | 3.9968 × 10^{−15} | 8.0685 × 10^{−2} | 2.0720 × 10^{−2} | 1.1602 × 10^{−1} | |

${f}_{12}$ | Standard PSO | 1.1716 × 10^{−5} | 1.0372 × 10^{−1} | 5.2801 × 10^{−3} | 1.0100 × 10^{−1} |

Basic PSO | 4.1217 × 10^{0} | 6.0335 × 10^{0} | 5.2257 × 10^{0} | 2.4996 × 10^{0} | |

MPSPSO | 1.4159 × 10^{−9} | 1.0367 × 10^{−1} | 1.5550 × 10^{−2} | 1.6555 × 10^{−1} | |

MPSPSO-ST | 9.1102 × 10^{−17} | 3.9616 × 10^{0} | 8.3068 × 10^{−1} | 4.6458 × 10^{0} | |

${f}_{13}$ | Standard PSO | 3.4293 | 6.0435 | 4.5710 | 3.6773 |

Basic PSO | 124.6619 | 182.6454 | 150.8625 | 74.4685 | |

MPSPSO | 2.3077 | 5.9331 | 3.7067 | 3.5467 | |

MPSPSO-ST | 2.4172 | 11.536 | 5.3201 | 11.0171 | |

${f}_{14}$ | Standard PSO | 7.409 × 10^{−6} | 4.3215 × 10^{−4} | 8.4032 × 10^{−5} | 4.1803 × 10^{−4} |

Basic PSO | 1.6260 × 10^{0} | 4.1097 × 10^{0} | 2.4549 × 10^{0} | 2.8544 × 10^{0} | |

MPSPSO | 6.8133 × 10^{−11} | 2.3492 × 10^{−8} | 2.4795 × 10^{−9} | 2.4440 × 10^{−8} | |

MPSPSO-ST | 7.6029 × 10^{−17} | 2.8983 × 10^{−3} | 1.6939 × 10^{−4} | 2.8349 × 10^{−3} | |

${f}_{15}$ | Standard PSO | 2.5207 × 10^{−2} | 2.0130 × 10^{−1} | 7.7555 × 10^{−2} | 2.3124 × 10^{−1} |

Basic PSO | 2.7387 × 10^{1} | 3.6274 × 10^{1} | 3.1328 × 10^{1} | 1.0927 × 10^{1} | |

MPSPSO | 4.5570 × 10^{−5} | 4.0095 × 10^{−3} | 1.1036 × 10^{−3} | 4.9777 × 10^{−3} | |

MPSPSO-ST | 6.5547 × 10^{−8} | 3.0083 × 10^{−4} | 3.6657 × 10^{−5} | 3.3579 × 10^{−4} | |

${f}_{16}$ | Standard PSO | 3.0000 | 3.0000 | 3.0000 | 6.7349 × 10^{−15} |

Basic PSO | 3.0099 | 3.4032 | 3.1322 | 5.5096 × 10^{−1} | |

MPSPSO | 3.0000 | 3.0000 | 3.0000 | 1.9357 × 10^{−15} | |

MPSPSO-ST | 3.0000 | 3.0000 | 3.0000 | 0 | |

${f}_{17}$ | Standard PSO | 0.9980 | 7.8740 | 2.6270 | 9.4631 |

Basic PSO | 0.9980 | 7.8740 | 2.0372 | 6.5904 | |

MPSPSO | 0.9980 | 2.9821 | 1.4449 | 2.9697 | |

MPSPSO-ST | 0.9980 | 0.9980 | 0.9980 | 0 | |

${f}_{18}$ | Standard PSO | 4.3426 × 10^{−4} | 1.1096 × 10^{−3} | 8.6577 × 10^{−4} | 7.7543 × 10^{−4} |

Basic PSO | 5.8557 × 10^{−4} | 1.9988 × 10^{−3} | 1.2394 × 10^{−3} | 1.6428 × 10^{−3} | |

MPSPSO | 3.0749 × 10^{−4} | 1.0349 × 10^{−3} | 6.5378 × 10^{−4} | 1.4993 × 10^{−3} | |

MPSPSO-ST | 3.0749 × 10^{−4} | 1.0383 × 10^{−3} | 4.0910 × 10^{−4} | 1.0838 × 10^{−3} | |

${f}_{19}$ | Standard PSO | 1.8802 × 10^{−17} | 5.3438 × 10^{−15} | 9.9794 × 10^{−16} | 7.0982 × 10^{−15} |

Basic PSO | 1.6320 × 10^{0} | 4.5516 × 10^{0} | 2.8228 × 10^{0} | 3.6775 × 10^{0} | |

MPSPSO | 8.1831 × 10^{−38} | 3.5528 × 10^{−33} | 2.2073 × 10^{−34} | 3.4617 × 10^{−33} | |

MPSPSO-ST | 3.2993 × 10^{−58} | 2.8784 × 10^{−55} | 5.5092 × 10^{−56} | 3.2355 × 10^{−55} | |

${f}_{20}$ | Standard PSO | −3.3220 | −3.2031 | −3.2744 | 0.2605 |

Basic PSO | −3.1587 | −2.5910 | −2.9131 | 0.8005 | |

MPSPSO | −3.3220 | −3.2031 | −3.2566 | 0.2645 | |

MPSPSO-ST | −3.3220 | −3.2031 | −3.2982 | 0.2126 |

**Table 3.**Parameter settings for MPSPSO-ST and particle swarm optimization with the nonlinear dynamic acceleration coefficients (PSO-NDAC), chaos particle swarm optimization (CPSO), AIWCPSO, moth-flame optimization (MFO), sine cosine algorithm (SCA) and differential evolution (DE).

Algorithm | Population Size | Iteration | run times | Parameter Settings |
---|---|---|---|---|

PSO-NDAC | 40 | 500 | 20 | $\begin{array}{l}{c}_{1}=-2\times {m}^{2}+2.5,{c}_{2}=0.5\times (1-m{)}^{2}+2.5\times m\\ (m=\frac{t}{{t}_{\mathrm{max}}}),\omega =0.9~0.4,{V}_{\mathrm{max}}=6\end{array}$ |

CPSO | 40 | 500 | 20 | ${c}_{1}={c}_{2}=2,\omega =0.9~0.4,\mu =4,{V}_{\mathrm{max}}=6$ |

AIWCPSO | 40 | 500 | 20 | ${c}_{1}={c}_{2}=2,\omega =0.9~0.4,{V}_{\mathrm{max}}=6$ |

MFO | 40 | 500 | 20 | t is random number in the range[−2, 1] |

SCA | 40 | 500 | 20 | ${r}_{1}=4~0$, ${r}_{2}$ is a random number in the range [0, 2π], ${r}_{3}$ is a random number in the range [0, 2], ${r}_{4}$ is a random number in the range [0, 1] |

DE | 40 | 500 | 20 | F=0.3, CR=0.5 |

MPSPSO-ST | 40 | 500 | 20 | ${c}_{1}=-\partial \times {m}^{2}\times \mathrm{tan}[\frac{\pi}{8}\times (1+{m}^{2})]+\theta +\rho \times z,$${c}_{2}=-\partial \times (1-m{)}^{2}\times \mathrm{tan}[\frac{\pi}{8}\times (1+(1-m{)}^{2})]+\theta +\rho \times z,$${\omega}_{t+1}=\varphi \times \mathrm{sin}(\pi {\omega}_{t})+\tau ,{V}_{\mathrm{max}}=6$ |

**Table 4.**Experimental results for MPSPSO-ST and PSO-NDAC, CPSO, AIWCPSO, MFO, SCA and DE on 20 classical test functions.

Function | Algorithm | The Best | The Worst | Mean | S.D. |
---|---|---|---|---|---|

${f}_{1}$ | PSO-NDAC | 5.1300 × 10^{−8} | 3.3867 × 10^{−5} | 2.6932 × 10^{−6} | 3.2888 × 10^{−5} |

CPSO | 2.5381 × 10^{−1} | 3.3147 × 10^{1} | 7.9537 × 10^{0} | 4.6559 × 10^{1} | |

AIWCPSO | 1.6492 × 10^{−6} | 2.8529 × 10^{−5} | 8.0015 × 10^{−6} | 2.8947 × 10^{−5} | |

MFO | 5.3151 × 10^{−1} | 1.0000 × 10^{4} | 5.0147 × 10^{2} | 9.7458 × 10^{3} | |

SCA | 1.2638 × 10^{−15} | 2.2184 × 10^{2} | 1.1287 × 10^{1} | 2.1604 × 10^{2} | |

DE | 6.1421 × 10^{−3} | 8.6630 × 10^{1} | 1.3950 × 10^{1} | 9.8238 × 10^{1} | |

MPSPSO-ST | 8.7871 × 10^{−16} | 2.5463 × 10^{−13} | 8.1821 × 10^{−14} | 3.3863 × 10^{−13} | |

${f}_{2}$ | PSO-NDAC | 0.02018 | 0.09354 | 0.05649 | 0.08375 |

CPSO | 0.05965 | 2.81610 | 1.25650 | 3.50140 | |

AIWCPSO | 0.03554 | 0.08707 | 0.05680 | 0.05934 | |

MFO | 0.05698 | 29.70930 | 3.39012 | 31.62590 | |

SCA | 0.00815 | 0.31364 | 0.06917 | 0.37656 | |

DE | 0.03722 | 0.20296 | 0.10343 | 0.19898 | |

MPSPSO-ST | 0.01611 | 0.05752 | 0.03274 | 0.04851 | |

${f}_{3}$ | PSO-NDAC | 8.8027 × 10^{0} | 8.9787 × 10^{1} | 2.7801 × 10^{1} | 8.7341 × 10^{1} |

CPSO | 2.4701 × 10^{1} | 1.0485 × 10^{2} | 6.0563 × 10^{1} | 9.0587 × 10^{1} | |

AIWCPSO | 1.9160 × 10^{1} | 1.1344 × 10^{2} | 4.5443 × 10^{1} | 9.0026 × 10^{1} | |

MFO | 1.7315 × 10^{3} | 5.3692 × 10^{4} | 1.9764 × 10^{4} | 5.0273 × 10^{4} | |

SCA | 2.1712 × 10^{0} | 1.2644 × 10^{4} | 1.9731 × 10^{3} | 1.8014 × 10^{4} | |

DE | 5.8248 × 10^{3} | 2.5593 × 10^{4} | 1.2039 × 10^{4} | 2.2101 × 10^{4} | |

MPSPSO-ST | 4.1427 × 10^{−2} | 5.9152 × 10^{−1} | 2.1145 × 10^{−1} | 7.7382 × 10^{−1} | |

${f}_{4}$ | PSO-NDAC | 4.0015 × 10^{−25} | 2.6450 × 10^{−19} | 3.5505 × 10^{−20} | 3.5317 × 10^{−19} |

CPSO | 2.5551 × 10^{−4} | 8.0566 × 10^{−3} | 2.2690 × 10^{−3} | 8.7752 × 10^{−3} | |

AIWCPSO | 3.0713 × 10^{−18} | 3.1036 × 10^{−12} | 2.2484 × 10^{−13} | 3.1787 × 10^{−12} | |

MFO | 8.7526 × 10^{−14} | 2.4125 × 10^{−7} | 1.2382 × 10^{−8} | 2.3484 × 10^{−7} | |

SCA | 1.7091 × 10^{−56} | 1.3950 × 10^{−3} | 6.9753 × 10^{−5} | 1.3597 × 10^{−3} | |

DE | 1.0221 × 10^{−18} | 3.9932 × 10^{−6} | 2.0004 × 10^{−7} | 3.8918 × 10^{−6} | |

MPSPSO-ST | 1.8781 × 10^{−49} | 6.6562 × 10^{−41} | 1.1225 × 10^{−41} | 9.4197 × 10^{−41} | |

${f}_{5}$ | PSO-NDAC | 6.6668 × 10^{−1} | 6.5562 × 10^{0} | 1.6663 × 10^{0} | 7.0782 × 10^{0} |

CPSO | 1.1582 × 10^{0} | 3.4957 × 10^{4} | 1.0897 × 10^{4} | 5.4315 × 10^{4} | |

AIWCPSO | 6.6749 × 10^{−1} | 4.7360 × 10^{0} | 1.3134 × 10^{0} | 5.3422 × 10^{0} | |

MFO | 1.9549 × 10^{0} | 7.2585 × 10^{4} | 1.0531 × 10^{4} | 1.1194 × 10^{5} | |

SCA | 6.6713 × 10^{−1} | 7.4345 × 10^{1} | 4.5179 × 10^{0} | 7.1707 × 10^{1} | |

DE | 3.0014 × 10^{0} | 1.5603 × 10^{3} | 1.3819 × 10^{2} | 1.6222 × 10^{3} | |

MPSPSO-ST | 6.6667 × 10^{−1} | 3.5021 × 10^{0} | 1.0923 × 10^{0} | 3.7675 × 10^{0} | |

${f}_{6}$ | PSO-NDAC | 1.7100 × 10^{−8} | 1.4269 × 10^{−5} | 2.1761 × 10^{−6} | 1.4714 × 10^{−5} |

CPSO | 1.4766 × 10^{0} | 3.7322 × 10^{1} | 1.1397 × 10^{1} | 3.7236 × 10^{1} | |

AIWCPSO | 1.2828 × 10^{−6} | 1.7912 × 10^{−5} | 6.8441 × 10^{−6} | 2.1667 × 10^{−5} | |

MFO | 5.0435 × 10^{−1} | 9.9013 × 10^{3} | 1.0440 × 10^{3} | 1.3243 × 10^{4} | |

SCA | 4.3620 × 10^{0} | 1.4545 × 10^{1} | 5.5499 × 10^{0} | 9.8103 × 10^{0} | |

DE | 6.2442 × 10^{−10} | 1.5620 × 10^{2} | 2.3732 × 10^{1} | 1.9729 × 10^{2} | |

MPSPSO-ST | 1.0678 × 10^{−15} | 4.8348 × 10^{−13} | 7.7464 × 10^{−14} | 5.0284 × 10^{−13} | |

${f}_{7}$ | PSO-NDAC | 4.4619 × 10^{−7} | 6.0106 × 10^{−5} | 8.5332 × 10^{−6} | 6.1709 × 10^{−5} |

CPSO | 5.7821 × 10^{1} | 1.1138 × 10^{3} | 5.0476 × 10^{2} | 1.3492 × 10^{3} | |

AIWCPSO | 3.5964 × 10^{−6} | 7.4185 × 10^{−5} | 2.1497 × 10^{−5} | 8.3628 × 10^{−5} | |

MFO | 3.0001 × 10^{−2} | 1.5001 × 10^{3} | 4.5025 × 10^{2} | 2.0993 × 10^{3} | |

SCA | 1.8083 × 10^{−22} | 1.9105 × 10^{1} | 1.0553 × 10^{0} | 1.8618 × 10^{1} | |

DE | 7.3789 × 10^{−5} | 1.4108 × 10^{1} | 1.3408 × 10^{0} | 1.4138 × 10^{1} | |

MPSPSO-ST | 4.6508 × 10^{−16} | 1.1855 × 10^{−12} | 1.8800 × 10^{−14} | 1.5397 × 10^{−12} | |

${f}_{8}$ | PSO-NDAC | 7.8413 × 10^{−14} | 1.4895 × 10^{−9} | 1.4322 × 10^{−10} | 1.6302 × 10^{−9} |

CPSO | 1.2651 × 10^{−2} | 2.7074 × 10^{0} | 8.0086 × 10^{−1} | 3.2157 × 10^{0} | |

AIWCPSO | 6.6641 × 10^{−11} | 1.6291 × 10^{−7} | 1.1648 × 10^{−8} | 1.6120 × 10^{−7} | |

MFO | 6.5346 × 10^{−6} | 1.3422 × 10^{1} | 1.3437 × 10^{0} | 1.3419 × 10^{1} | |

SCA | 6.8560 × 10^{−28} | 1.4863 × 10^{0} | 7.8368 × 10^{−2} | 1.4465 × 10^{0} | |

DE | 2.9936 × 10^{−7} | 3.3894 × 10^{−2} | 2.9902 × 10^{−3} | 3.2996 × 10^{−2} | |

MPSPSO-ST | 2.2820 × 10^{−31} | 2.5041 × 10^{−26} | 3.3096 × 10^{−27} | 2.8730 × 10^{−26} | |

${f}_{9}$ | PSO-NDAC | 0.29987 | 0.59987 | 0.41987 | 0.36332 |

CPSO | 0.20245 | 0.89988 | 0.59128 | 0.87943 | |

AIWCPSO | 0.19987 | 0.59987 | 0.41994 | 0.38963 | |

MFO | 1.29987 | 12.19990 | 5.56991 | 17.25210 | |

SCA | 0.09988 | 4.55470 | 0.61124 | 4.18660 | |

DE | 0.29987 | 1.39990 | 0.50236 | 1.23470 | |

MPSPSO-ST | 0.29987 | 0.49987 | 0.38987 | 0.27928 | |

${f}_{10}$ | PSO-NDAC | −2.9410 × 10^{−9} | −1.8034 × 10^{−9} | −2.3857 × 10^{−9} | 1.1856 × 10^{−9} |

CPSO | −1.1181 × 10^{−9} | −8.6693 × 10^{−10} | −9.9583 × 10^{−10} | 2.7155 × 10^{−10} | |

AIWCPSO | −2.8340 × 10^{−9} | −1.6520 × 10^{−9} | −2.4220 × 10^{−9} | 1.1758 × 10^{−9} | |

MFO | −2.6658 × 10^{−9} | −2.0384 × 10^{−9} | −2.3715 × 10^{−9} | 6.6030 × 10^{−10} | |

SCA | −7.1394 × 10^{−10} | −1.1393 × 10^{−10} | −2.6959 × 10^{−10} | 7.6044 × 10^{−10} | |

DE | −2.1077 × 10^{−9} | −1.6630 × 10^{−9} | −1.8151 × 10^{−9} | 4.4156 × 10^{−10} | |

MPSPSO-ST | −3.0193 × 10^{−9} | −2.3895 × 10^{−9} | −2.7955 × 10^{−9} | 6.9722 × 10^{−10} | |

${f}_{11}$ | PSO-NDAC | 1.8235 × 10^{−8} | 1.1349 × 10^{0} | 5.8798 × 10^{−1} | 2.1714 × 10^{0} |

CPSO | 1.2241 × 10^{−2} | 7.0711 × 10^{−1} | 2.4554 × 10^{−1} | 1.0575 × 10^{0} | |

AIWCPSO | 4.8976 × 10^{−6} | 2.7095 × 10^{−2} | 8.7549 × 10^{−3} | 3.7409 × 10^{−2} | |

MFO | 5.0123 × 10^{−1} | 9.1002 × 10^{1} | 9.9449 × 10^{0} | 1.2071 × 10^{2} | |

SCA | 7.8826 × 10^{−15} | 2.5680 × 10^{0} | 4.5262 × 10^{−1} | 2.7232 × 10^{0} | |

DE | 9.1420 × 10^{−4} | 2.0907 × 10^{0} | 6.3608 × 10^{−1} | 2.9071 × 10^{0} | |

MPSPSO-ST | 6.6613 × 10^{−16} | 9.3347 × 10^{−2} | 1.9044 × 10^{−2} | 9.7708 × 10^{−2} | |

${f}_{12}$ | PSO-NDAC | 5.7978 × 10^{−11} | 1.0370 × 10^{−1} | 1.0369 × 10^{−2} | 1.3911 × 10^{−1} |

CPSO | 2.9995 × 10^{−1} | 2.0577 × 10^{0} | 7.9385 × 10^{−1} | 2.0733 × 10^{0} | |

AIWCPSO | 5.5656 × 10^{−8} | 3.1096 × 10^{−1} | 5.1833 × 10^{−2} | 3.7374 × 10^{−1} | |

MFO | 2.0323 × 10^{0} | 1.4052 × 10^{1} | 7.0648 × 10^{0} | 1.6366 × 10^{1} | |

SCA | 5.4712 × 10^{−1} | 2.0995 × 10^{1} | 1.8558 × 10^{0} | 1.9818 × 10^{1} | |

DE | 1.4196 × 10^{−1} | 3.7376 × 10^{4} | 1.8701 × 10^{3} | 3.6428 × 10^{4} | |

MPSPSO-ST | 6.2512 × 10^{−17} | 3.9495 × 10^{0} | 1.1213 × 10^{0} | 5.1966 × 10^{0} | |

${f}_{13}$ | PSO-NDAC | 2.3083 | 5.7135 | 3.5316 | 3.5264 |

CPSO | 10.3682 | 56.6080 | 24.7796 | 48.5283 | |

AIWCPSO | 1.6613 | 4.6190 | 3.0787 | 4.1172 | |

MFO | 2.8600 | 56.0777 | 18.1767 | 66.0427 | |

SCA | 26.0094 | 34.1472 | 29.7011 | 10.6788 | |

DE | 0.0038 | 0.8927 | 0.4323 | 1.0218 | |

MPSPSO-ST | 2.4172 | 8.6693 | 5.3316 | 9.3605 | |

${f}_{14}$ | PSO-NDAC | 1.1023 × 10^{−9} | 2.4166 × 10^{−6} | 2.7192 × 10^{−7} | 2.7198 × 10^{−6} |

CPSO | 3.6970 × 10^{−2} | 1.1233 × 10^{0} | 3.4541 × 10^{−1} | 1.1816 × 10^{0} | |

AIWCPSO | 2.6544 × 10^{−8} | 2.3484 × 10^{−6} | 4.0181 × 10^{−7} | 2.6397 × 10^{−6} | |

MFO | 6.7555 × 10^{−1} | 1.1573 × 10^{2} | 2.4571 × 10^{1} | 1.2484 × 10^{2} | |

SCA | 8.2348 × 10^{−24} | 5.5803 × 10^{−2} | 3.0040 × 10^{−3} | 5.4329 × 10^{−2} | |

DE | 1.7097 × 10^{−4} | 6.8415 × 10^{−1} | 5.6675 × 10^{−2} | 6.5169 × 10^{−1} | |

MPSPSO-ST | 1.7084 × 10^{−17} | 1.1430 × 10^{−4} | 1.1812 × 10^{−5} | 1.4598 × 10^{−4} | |

${f}_{15}$ | PSO-NDAC | 5.6249 × 10^{−4} | 3.7831 × 10^{−1} | 2.7575 × 10^{−2} | 3.6192 × 10^{−1} |

CPSO | 2.9421 × 10^{0} | 1.7870 × 10^{1} | 1.0966 × 10^{1} | 2.0972 × 10^{1} | |

AIWCPSO | 3.3407 × 10^{−4} | 3.0759 × 10^{−3} | 8.7949 × 10^{−4} | 3.3632 × 10^{−3} | |

MFO | 1.9292 × 10^{−2} | 2.2203 × 10^{1} | 6.5223 × 10^{0} | 2.8871 × 10^{1} | |

SCA | 4.9150 × 10^{−5} | 7.1893 × 10^{−1} | 3.7656 × 10^{−2} | 6.9966 × 10^{−1} | |

DE | 5.4837 × 10^{−7} | 3.3238 × 10^{−2} | 4.0703 × 10^{−3} | 3.2331 × 10^{−2} | |

MPSPSO-ST | 1.0274 × 10^{−8} | 9.8253 × 10^{−4} | 1.7602 × 10^{−4} | 1.4203 × 10^{−3} | |

${f}_{16}$ | PSO-NDAC | 3.0000 | 3.0000 | 3.0000 | 1.8841 × 10^{−15} |

CPSO | 3.0008 | 3.4160 | 3.0888 | 5.3477 × 10^{−1} | |

AIWCPSO | 3.0000 | 3.0000 | 3.0000 | 1.9357 × 10^{−15} | |

MFO | 3.0000 | 3.0000 | 3.0000 | 8.2725 × 10^{−15} | |

SCA | 3.0000 | 3.0010 | 3.0002 | 1.0726 × 10^{−3} | |

DE | 3.0000 | 3.0000 | 3.0000 | 1.8310 × 10^{−15} | |

MPSPSO-ST | 3.0000 | 3.0000 | 3.0000 | 0 | |

${f}_{17}$ | PSO-NDAC | 0.99800 | 1.99200 | 1.09740 | 1.33360 |

CPSO | 4.14576 | 28.82710 | 13.89500 | 20.24470 | |

AIWCPSO | 0.99800 | 5.92880 | 1.78910 | 6.63320 | |

MFO | 0.99800 | 5.92880 | 1.59210 | 5.30030 | |

SCA | 0.99801 | 2.98210 | 1.09930 | 1.93180 | |

DE | 0.99801 | 10.76320 | 2.08246 | 11.49000 | |

MPSPSO-ST | 0.99800 | 0.99800 | 0.99800 | 0 | |

${f}_{18}$ | PSO-NDAC | 3.0749 × 10^{−4} | 1.0028 × 10^{−3} | 5.9124 × 10^{−4} | 1.1285 × 10^{−3} |

CPSO | 6.6030 × 10^{−4} | 4.0282 × 10^{−2} | 6.4056 × 10^{−3} | 5.0951 × 10^{−2} | |

AIWCPSO | 3.0749 × 10^{−4} | 1.5941 × 10^{−3} | 7.4898 × 10^{−4} | 1.5257 × 10^{−3} | |

MFO | 3.7221 × 10^{−4} | 1.6554 × 10^{−3} | 9.5160 × 10^{−4} | 1.7842 × 10^{−3} | |

SCA | 8.1546 × 10^{−4} | 1.6696 × 10^{−3} | 1.3903 × 10^{−3} | 9.2161 × 10^{−4} | |

DE | 3.1525 × 10^{−4} | 4.6017 × 10^{−3} | 1.1767 × 10^{−3} | 3.9121 × 10^{−3} | |

MPSPSO-ST | 3.0749 × 10^{−4} | 1.0371 × 10^{−3} | 3.8036 × 10^{−4} | 9.7776 × 10^{−4} | |

${f}_{19}$ | PSO-NDAC | 1.5069 × 10^{−33} | 7.4158 × 10^{−30} | 6.4473 × 10^{−31} | 7.2887 × 10^{−30} |

CPSO | 2.1384 × 10^{−2} | 2.2297 × 10^{1} | 5.7052 × 10^{0} | 3.8606 × 10^{1} | |

AIWCPSO | 2.2815 × 10^{−30} | 1.4904 × 10^{−27} | 2.6907 × 10^{−28} | 1.8522 × 10^{−27} | |

MFO | 3.3701 × 10^{−23} | 1.3478 × 10^{−19} | 3.2070 × 10^{−20} | 2.0313 × 10^{−19} | |

SCA | 3.3833 × 10^{−48} | 5.0261 × 10^{−12} | 2.5131 × 10^{−13} | 4.8989 × 10^{−12} | |

DE | 8.7979 × 10^{−27} | 8.2996 × 10^{−2} | 5.9547 × 10^{−3} | 8.4550 × 10^{−2} | |

MPSPSO-ST | 3.3480 × 10^{−58} | 4.6387 × 10^{−55} | 3.2719 × 10^{−56} | 4.4551 × 10^{−55} | |

${f}_{20}$ | PSO-NDAC | −3.3220 | −3.2031 | −3.2804 | 0.2536 |

CPSO | −3.2242 | −2.6097 | −3.0177 | 0.6943 | |

AIWCPSO | −3.3220 | −3.2031 | −3.2744 | 0.2605 | |

MFO | −3.3220 | −3.1376 | −3.2351 | 0.2619 | |

SCA | −3.0134 | −2.9619 | −2.9892 | 0.0738 | |

DE | −3.3220 | −3.2030 | −3.2799 | 0.2521 | |

MPSPSO-ST | −3.3220 | −3.2031 | −3.2982 | 0.2127 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Du, Y.; Xu, F.
A Hybrid Multi-Step Probability Selection Particle Swarm Optimization with Dynamic Chaotic Inertial Weight and Acceleration Coefficients for Numerical Function Optimization. *Symmetry* **2020**, *12*, 922.
https://doi.org/10.3390/sym12060922

**AMA Style**

Du Y, Xu F.
A Hybrid Multi-Step Probability Selection Particle Swarm Optimization with Dynamic Chaotic Inertial Weight and Acceleration Coefficients for Numerical Function Optimization. *Symmetry*. 2020; 12(6):922.
https://doi.org/10.3390/sym12060922

**Chicago/Turabian Style**

Du, Yuji, and Fanfan Xu.
2020. "A Hybrid Multi-Step Probability Selection Particle Swarm Optimization with Dynamic Chaotic Inertial Weight and Acceleration Coefficients for Numerical Function Optimization" *Symmetry* 12, no. 6: 922.
https://doi.org/10.3390/sym12060922