# Effects of Random Values for Particle Swarm Optimization Algorithm
## 4 Experiments and Analysis
### Experimental Results and Comparisons
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & \(r_{\rm L}\)**\(r_{2}\)** & **U(0,1)** & **0.5** & **U(\(-1.1\))** & **G(0,1)** \\ \cline{2-6}  & **Dimension** & **10/30/100** & **10/30/100** & **10/30/100** & **10/30/100** \\ \hline \multirow{4}{*}{Alpine} & Average solution & 0.2/67.9/1.95 & 0.12/6.36/57.50 & 0/21.42/20.04 & 0/3.36/30.12 \\  & Standard deviation & 0.1/70.7/8.56 & 0.34/1.81/8.48 & 0/1.17  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \(r_{1},r_{2}\) & \(U(0,1)\) & \(0.5\) & \(U[-1,1]\) & \(G(0,1)\) \\ \cline{2-5}  & **Dimension** & **10/30/100** & **10/30/100** & **10/30/100** & **10/30/100** \\ \hline \multirow{4}{*}{Grievank} & Average solution & 0/0/28.04 & 0.19/53/25/475.06 & 0.0/0/0 & 0.0/0 \\  & Standard deviation & 0/0/42.54 & 0.18/31/15.18 & 0.0/0/0 & 0.0/0 \\  & The worst solution & 0/0/09.93 & 0.84/14.45/941.53 & 0.0/0/0 & 0.0/0 \\  & The best solution & 0/0/0 & 0.05/1.70/203.77 & 0.0/0/0 & 0.0/0 \\ \hline \multirow{4}{*}{Ackley} & Average solution & 0/0/0.94 & 0.80/25/18.27 & 0.0/0 & 0.0/0 \\  & Standard deviation & 0/0.633 & 0.88/29.04/0.99 & 0.0/0 & 0.0/0 \\  & The worst solution & 0/0/19.97 & 2.81/16.77/19.44 & 0.0/0 & 0.0/0 \\  & The best solution & 0/0/0 & 0.44/16.92 & 0.0/0 & 0.0/0 \\ \hline \multirow{4}{*}{Levy and Montalvo 2} & Average solution & 0/0/10.48 & 0.33/34.54 & 0.0/0.36/8.28 & 0.1/0.8/8.64 \\  & Standard deviation & 0/0/32.20 & 0.18/8.81 & 0.0/0.50/0.62 & 0.0/0.68/0.37 \\  & The worst solution & 0/17.73 & 0.72/69.40 & 0.11/51.97 & 0.22/0.92 \\  & The best solution & 0/0/5.40 & 0/0.70/18.90 & 0.0/6.31 & 0.0/0.7/7.0 \\ \hline \multirow{4}{*}{Sinusoidal} & Average solution & -3.38/1.7/-0.03 & -3.07/-0.12/0 & -3.27/-0.07/-0.05 & -3.21/-0.54/0.54 \\  & Standard deviation & 0.41/13/00 & 0.53/16/0 & 0.61/12/0.18 & 0.66/0.80/0.0 \\  & The worst solution & -175.0/0 & -1.75/0.0 & -1.75/0.0 & -1.75/0.0 \\  & The best solution & -3.50/-3.50/-0.44 & -3.50/-0.70/0.30 & -3.50/-0.87 & -3.50/-0.50/0.50 \\ \hline \multirow{4}{*}{Rotated Expanded Scifer} & Average solution & 0/07/14.83 & 3.12/14.73/4.99 & 0.0/0 & 0.0/0 \\  & Standard deviation & 0.25/1.49/3.53 & 0.47/1.03/1.02 & 0.0/0/0 & 0.0/0 \\  & The worst solution & 1/58/14/149 & 3.70/12/47.46 & 0.0/0/0 & 0.0/0 \\  & The best solution & 0/0/0 & 1.42/8.57/41.56 & 0.0/0/0 & 0.0/0 \\ \hline \multirow{4}{*}{Alpine} & Average solution & 0/091/16.46 & 0.31/8.41/73.22 & 0.2/45/27.24 & 0.2/293/30.70 \\  & Standard deviation & 0/14.74/76 & 0.73/95/13.39 & 0.1/23/6.95 & 0.1/14/4.19 \\  & The worst solution & 0/8.46/34.94 & 2.85/15.94/69.89 & 0.5/0.32/36.40 & 0.16/37.34 \\  & The best solution & 0/0.02/571 & 0/108/48.66 & 0.0/0 & 0.1/22.49 \\ \hline \multirow{4}{*}{Moved axis parallel} & Average solution & 0/13833/1320196.49 & 20.64/987.67/14.34/3.54 & 0.625/95.76/57.23 & 0.13/236.74/42.703 \\  & Standard deviation & 0/2215/48/46700.85 & 0.81/598/155.266.58 & 0.86/158.446.61 & 0.72/133.70/224.876 \\ \cline{1-1}  & The worst solution & 0/07/00/1966434 & 500/24327.28/096798.82 & 0.9/88/153.37 & 0.99/882.65/327.53 \\ \cline{1-1}  & The worst solution & 0/07/23113 & 0/221227.48/497.13 & 0.0/0.0/0 & 0.0/0 \\ \hline \multirow{4}{*}{Schwed} & Average solution & \(-3764.52/-982337926-256045\) & \(-3134.09/-74016.16\) \(-18304.47\) \(-376452.52/-10418.27/97\) \(-30589.24\) \(-3732.73/-10492.65\) \(-29986.84\) \(-29374.36\) \(-2936.84\) \(-2936. Figure 5: _Cont._  
Figure 5: The mean best fitness of standard PSO algorithm with different types of random values for benchmark functions: (**a**) Sphere1; (**b**) Sphere2; (**c**) [PERSON]; (**d**) [PERSON]; (**e**) [PERSON]; (**f**) Ackley; (**g**) Levy and Montalvo 2; (**h**) Sinusoidal; (**i**) Rotated Expanded Scaffer; (**j**) Alpine; (**k**) Moved axis parallel hyper-ellipsoid; (**l**) Schwefel.