A Quick Search Dynamic VectorEvaluated Particle Swarm Optimization Algorithm Based on Fitness Distance
Abstract
:1. Introduction
2. Related Work
2.1. DMOP
2.2. Basic PSO
2.3. Basic DVEPSO
3. Quick Search DVEPSO Based on Fitness Distance (DVEPSO/FD)
3.1. System Composition
3.2. Module Design
3.2.1. Repository Update Mechanism Based on Fitness Distance
3.2.2. Quick Search Mechanism
3.2.3. Other Structures
 (1)
 Information sharing mechanism
 (2)
 Environmental monitoring and response mechanism
3.3. The PseudoCode of the Algorithm
4. Experiments and Results
4.1. Standard Benchmarks
4.2. Performance Metrics
 (1)
 Accuracy
 (2)
 Stability
4.3. Parameters’ Settings
4.4. Experiments
4.5. Results
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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PseudoCode 



$\hspace{1em}\hspace{1em}\mathrm{Randomly}\mathrm{select}\mathrm{some}\mathrm{particles},\mathrm{calculate}\mathrm{the}\mathrm{fitness}{f}_{m}^{r}\left({x}_{i}\right)$ 
$\hspace{1em}\hspace{1em}\mathbf{If}{f}_{m}^{r}\left({x}_{i}\right){f}_{m}^{r1}\left({x}_{i}\right){\alpha}_{T}$ 


Else 



End if 
$\hspace{1em}\hspace{1em}\mathrm{Calculate}\mathrm{all}\mathrm{the}\mathrm{fitness}\mathrm{distances}S{\left(i\right)}_{fd}^{m}=\frac{S{\left(i+1\right)}_{m}S{\left(i1\right)}_{m}}{{f}_{m}^{max}{f}_{m}^{min}}$, $\mathrm{steamline}\mathrm{the}\mathrm{repository}\mathrm{to}\mathrm{form}\mathrm{elite}\mathrm{repository}\mathrm{and}\mathrm{update}\mathrm{the}Re$If $Re$ out of range 
Limited elite repository based on crowding distance 
End if 
End for 
POF  POS  

No Change  Change  
No change  Type IV Problem changes  Type I FDA1; FDA4 
Change  Type III FDA2; DMOP1  Type II FDA3; FDA5; DMOP2 
Benchmarks  Definition  Benchmarks  Definition 

FDA1  $\begin{array}{l}{f}_{1}({X}_{I})={x}_{1}\\ {f}_{2}({X}_{II})=g\cdot h\\ g({X}_{II})=1+{\displaystyle {\sum}_{i=2}^{m}{({x}_{i}G(t))}^{2}}\\ h({f}_{1},g)=1\sqrt{\frac{{f}_{1}}{g}}\\ G(t)=\mathrm{sin}(0.5\pi \cdot t)\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ where\left{X}_{II}\right=9,{X}_{I}\in \left[0,1\right],{X}_{II}\in \left[1,1\right]\end{array}$  FDA2  $\begin{array}{l}{f}_{1}({X}_{I})={x}_{1}\\ {f}_{2}({X}_{II})=g\cdot h\\ g({X}_{II})=1+{\displaystyle {\sum}_{i\in {X}_{II}}^{m}{({x}_{i})}^{2}}\\ h({X}_{III},{f}_{1},g)=1{(\frac{{f}_{1}}{g})}^{H(t)+{{\displaystyle {\sum}_{{x}_{i}\in {X}_{III}}({x}_{i}H(t))}}^{2}}\\ H(t)=0.75+0.7\cdot \mathrm{sin}(0.5\pi \cdot t)\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ where\left{X}_{II}\right=\left{X}_{III}\right=15,{X}_{I}\in \left[0,1\right],{X}_{II},{X}_{III}\in \left[0,1\right]\end{array}$ 
FDA3  $\begin{array}{l}{f}_{1}({X}_{I})={\displaystyle {\sum}_{{x}_{i}\in {X}_{I}}{x}_{i}{}^{F(t)}}\\ {f}_{2}({X}_{II})=g\cdot h\\ g({X}_{II})=1+G(t)+{\displaystyle {\sum}_{{x}_{i}\in {X}_{II}}{({x}_{i}G(t))}^{2}}\\ h({f}_{1},g)=1\sqrt{\frac{{f}_{1}}{g}}\\ G(t)=\left\mathrm{sin}(0.5\pi \cdot t)\right\\ F(t)={10}^{2\mathrm{sin}(0.5\pi \cdot t)}\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ where\left{X}_{I}\right=5,\left{X}_{II}\right=25,{X}_{I}\in \left[0,1\right],{X}_{II}\in \left[1,1\right]\end{array}$  FDA 4  $\begin{array}{l}{f}_{1}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{cos}(\frac{{x}_{i}\pi}{2})}\\ {f}_{k}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}(\mathrm{cos}(\frac{{x}_{i}\pi}{2})})\mathrm{sin}(\frac{{x}_{Mk+1}\pi}{2})\\ {f}_{M}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{sin}(\frac{{x}_{1}\pi}{2})}\\ g({X}_{II})={\displaystyle {\sum}_{{x}_{i}\in {X}_{II}}{({x}_{i}G(t))}^{2}}\\ G(t)=\left\mathrm{sin}(0.5\pi \cdot t)\right\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ whereX\in \left[0,1\right]\end{array}$ 
FDA5  $\begin{array}{l}{f}_{1}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{cos}(\frac{{y}_{i}\pi}{2})}\\ {f}_{k}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}(\mathrm{cos}(\frac{{y}_{i}\pi}{2})})\mathrm{sin}(\frac{{y}_{Mk+1}\pi}{2})\\ {f}_{M}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{sin}(\frac{{y}_{1}\pi}{2})}\\ g({X}_{II})=G(t)+{\displaystyle {\sum}_{{x}_{i}\in {X}_{II}}{({y}_{i}G(t))}^{2}}\\ {y}_{i}={x}_{i}{}^{F(t)}\\ G(t)=\left\mathrm{sin}(0.5\pi \cdot t)\right\\ F(t)=1+100{\mathrm{sin}}^{4}(0.5\pi \cdot t)\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ whereX\in \left[0,1\right]\end{array}$  FDA5iso  $\begin{array}{l}{f}_{1}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{cos}(\frac{{y}_{i}\pi}{2})}\\ {f}_{k}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}(\mathrm{cos}(\frac{{y}_{i}\pi}{2})})\mathrm{sin}(\frac{{y}_{Mk+1}\pi}{2})\\ {f}_{M}(X)=(1+g({X}_{II})){\displaystyle {\prod}_{i=1}^{M1}\mathrm{sin}(\frac{{y}_{1}\pi}{2})}\\ g({X}_{II})={\displaystyle {\sum}_{{x}_{i}\in {X}_{II}}{({y}_{i}G(t))}^{2}}\\ {y}_{i}={x}_{i}{}^{F(t)}\\ G(t)=\left\mathrm{sin}(0.5\pi \cdot t)\right\\ F(t)=1+100{\mathrm{sin}}^{4}(0.5\pi \cdot t)\\ t=\frac{1}{{n}_{t}}\lfloor \frac{\tau}{{\tau}_{t}}\rfloor \\ whereX\in \left[0,1\right],{X}_{II}=({x}_{M},\dots ,{x}_{n})\end{array}$ 
DMOP1  $\begin{array}{l}{f}_{1}({x}_{1})={x}_{1}\\ {f}_{2}({x}_{2},\dots ,{x}_{m})=g\cdot h\\ g({x}_{2},\dots ,{x}_{m})=1+9\cdot {{\displaystyle {\sum}_{i=2}^{m}{x}_{i}}}^{2}\\ h({f}_{1},g)=1{(\frac{{f}_{1}}{g})}^{H(t)}\\ H(t)=0.75\cdot \mathrm{sin}(0.5\pi \cdot t)+1.25\\ wherem=10,{x}_{i}\in \left[0,1\right]\end{array}$  DMOP2  $\begin{array}{l}{f}_{1}({x}_{1})={x}_{1}\\ {f}_{2}({x}_{2},\dots ,{x}_{m})=g\cdot h\\ g({x}_{2},\dots ,{x}_{m})=1+{\displaystyle {\sum}_{i=2}^{m}({x}_{i}}G(t){)}^{2}\\ h({f}_{1},g)=1{(\frac{{f}_{1}}{g})}^{H(t)}\\ H(t)=0.75\cdot \mathrm{sin}(0.5\pi \cdot t)+1.25\\ G(t)=\mathrm{sin}(0.5\pi \cdot t)\\ wherem=10,{x}_{i}\in \left[0,1\right]\end{array}$ 
Parameters  ${n}_{t}$  ${\tau}_{t}$  ${w}_{0}$  ${c}_{10}$  ${c}_{20}$  $R$ 
Values  15 (FDA2: 2.5)  100  0.72 (NonQS stage)  1.49 (NonQS stage)  1.49 (NonQS stage)  1000 
Benchmarks  Accuracy  Stability  Runtime  

DVEPSO  DVEPSO/FD  DVEPSO  DVEPSO/FD  DVEPSO  DVEPSO/FD  
Mean  0.4292  0.4236  0.0223  0.0209  
FDA1  Std  0.0015  0.0004  0.0012  0.0007  111.1841  222.6885 
Best  0.4308  0.4239  0.0237  0.0213  
Mean  0.5621  0.5712  0.0399  0.0323  
FDA2  Std  0.0088  0.0065  0.0010  0.0021  139.2537  141.3722 
Best  0.5722  0.5741  0.0411  0.0338  
Mean  0.6846  0.6849  0.0412  0.0290  
FDA3  Std  0.0012  0.0043  0.0054  0.0009  119.9075  125.3159 
Best  0.6859  0.6916  0.0472  0.0297  
Mean  0.2423  0.2482  0.0284  0.0297  
FDA4  Std  0.0012  0.0023  0.0008  0.0013  161.0242  675.8527 
Best  0.2432  0.2499  0.0293  0.0310  
Mean  0.2269  0.2342  0.0256  0.0249  
FDA5  Std  0.0015  0.0023  0.0010  0.0003  168.8640  492.0685 
Best  0.2284  0.2368  0.0263  0.0251  
Mean  0.6194  0.6228  0.0204  0.0165  
DMOP1  Std  0.0054  0.0030  0.0020  0.0046  159.8208  1213.6000 
Best  0.6255  0.6296  0.0227  0.0218  
Mean  0.5680  0.5691  0.0136  0.0139  
DMOP2  Std  0.0005  0.0011  0.0003  0.0006  118.3666  163.2108 
Best  0.5683  0.5694  0.0139  0.0146 
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Wang, S.; Ma, D.; Wu, M. A Quick Search Dynamic VectorEvaluated Particle Swarm Optimization Algorithm Based on Fitness Distance. Mathematics 2022, 10, 1587. https://doi.org/10.3390/math10091587
Wang S, Ma D, Wu M. A Quick Search Dynamic VectorEvaluated Particle Swarm Optimization Algorithm Based on Fitness Distance. Mathematics. 2022; 10(9):1587. https://doi.org/10.3390/math10091587
Chicago/Turabian StyleWang, Suyu, Dengcheng Ma, and Miao Wu. 2022. "A Quick Search Dynamic VectorEvaluated Particle Swarm Optimization Algorithm Based on Fitness Distance" Mathematics 10, no. 9: 1587. https://doi.org/10.3390/math10091587