# Effects of Random Values for Particle Swarm Optimization Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Standard and Modified Particle Swarm Optimization Algorithms

#### 2.1. Standard Particle Swarm Optimization Algorithm

#### 2.2. Modifications for Particle Swarm Optimization Algorithm

#### 2.2.1. Constant or Random Inertia Weight Strategies

#### 2.2.2. Time Varying Inertia Weight Strategies

#### 2.2.3. Adaptive Inertia Weight Strategies

## 3. Particle Swarm Optimization Algorithm with Different Types of Random Values

#### 3.1. Random Values with Uniform Distribution in the Range of [0, 1]

#### 3.2. Random Values with Uniform Distribution in the Range of [−1, 1]

#### 3.3. Random Values with Gauss Distribution

## 4. Experiments and Analysis

#### 4.1. Experimental Setup

#### 4.2. Experimental Results and Comparisons

#### 4.3. Application and Analysis

#### 4.3.1. Application in Engineering Problem

_{1}), thickness of the head (x

_{2}), the inner radius (x

_{3}), and the length of the cylindrical section of the vessel (x

_{4}). The highly constrained problem of pressure vessel design can be expressed as,

_{1}and x

_{2}are integer multipliers of 0.0625. x

_{3}and x

_{4}are continuous variables in the ranges of 40 ≤ x

_{3}≤ 80 and 20 ≤ x

_{4}≤ 60. In this study, the standard PSO and LDIW-PSO algorithms with different types of random values are utilized to solve this engineering problem.

#### 4.3.2. Analysis

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**The mean best fitness of standard PSO algorithm with different types of random values for benchmark functions: (

**a**) Sphere1; (

**b**) Sphere2; (

**c**) Rastrigin; (

**d**) Rosenbrock; (

**e**) Griewank; (

**f**) Ackley; (

**g**) Levy and Montalvo 2; (

**h**) Sinusoidal; (

**i**) Rotated Expanded Scaffer; (

**j**) Alpine; (

**k**) Moved axis parallel hyper-ellipsoid; (

**l**) Schwefel. (The solid, dash, short dash and short dash dot lines represent the random values generated by uniform distribution in the ranges of [0, 1] and [−1, 1], Gauss distribution, and 0.5, respectively; the black, red and blue lines represent the space dimensions 10, 30, and 100, respectively).

**Figure 6.**The mean best fitness of LDIW-PSO algorithm with different types of random values for benchmark functions: (

**a**) Sphere1; (

**b**) Sphere2; (

**c**) Rastrigin; (

**d**) Rosenbrock; (

**e**) Griewank; (

**f**) Ackley; (

**g**) Levy and Montalvo 2; (

**h**) Sinusoidal; (

**i**) Rotated Expanded Scaffer; (

**j**) Alpine; (

**k**) Moved axis parallel hyper-ellipsoid; (

**l**) Schwefel. (The solid, dash, short dash and short dash dot lines represent the random values generated by uniform distribution in the ranges of [0, 1] and [−1, 1], Gauss distribution, and 0.5, respectively; the black, red and blue lines represent the space dimensions 10, 30, and 100, respectively).

Function Name | Test Function | Search Space | The Range of Particle Velocity | The Best Solution | The Best Result |
---|---|---|---|---|---|

Sphere1 | ${f}_{1}(\mathrm{x})={\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}$ | ${\left[-100,100\right]}^{D}$ | $\left[-100,100\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Sphere2 | ${f}_{2}(\mathrm{x})={\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}$ | ${\left[-10,190\right]}^{D}$ | $\left[-10,10\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Rastrigin | ${f}_{3}(x)={\displaystyle \sum _{i=1}^{D}\left({x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10\right)}$ | ${\left[-5.12,5.12\right]}^{D}$ | $\left[-5.12,5.12\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Rosenbrock | ${f}_{4}(x)={\displaystyle \sum _{i=1}^{D-1}\left(100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right)}$ | ${\left[-30,30\right]}^{D}$ | $\left[-30,30\right]$ | $\left[1,\cdots ,1\right]$ | 0 |

Griewank | ${f}_{5}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}}+1$ | ${\left[-600,600\right]}^{D}$ | $\left[-600,600\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Ackley | $\begin{array}{ll}{f}_{6}(x)=& -20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{30}{\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}}\right)\\ & -\mathrm{exp}\left(\frac{1}{D}{\displaystyle \sum _{i=1}^{D}\mathrm{cos}(2\pi {x}_{i})}\right)+20+e\end{array}$ | ${\left[-32,32\right]}^{D}$ | $\left[-32,32\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Levy and Montalvo 2 | $\begin{array}{ll}{f}_{7}(x)=& 0.1({\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum _{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i+1}\right)\right]}\\ & +{\left({x}_{D}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{D}\right)\right])\end{array}$ | ${\left[-5,5\right]}^{D}$ | $\left[-5,5\right]$ | $\left[1,\cdots ,1\right]$ | 0 |

Sinsolidal | $\begin{array}{ll}{f}_{8}(x)=& -[2.5{\prod}_{i=1}^{D}\mathrm{sin}\left({x}_{i}-\frac{\pi}{6}\right)\\ & +{\prod}_{i=1}^{D}\mathrm{sin}\left(5\left({x}_{i}-\frac{\pi}{6}\right)\right)]\end{array}$ | ${\left[0,\pi \right]}^{D}$ | $\left[-\pi ,\pi \right]$ | $\left[\frac{2}{3}\pi ,\cdots ,\frac{2}{3}\pi \right]$ | −3.5 |

Rotated Expanded Scaffer | $\begin{array}{ll}F(x,y)& =0.5+\frac{{\mathrm{sin}}^{2}\left(\sqrt{{x}^{2}+{y}^{2}}\right)-0.5}{{\left(1+0.001\left({x}^{2}+{y}^{2}\right)\right)}^{2}}\\ {f}_{9}\left(x\right)& =F\left({x}_{1},{x}_{2}\right)+F\left({x}_{2},{x}_{3}\right)+\cdots \\ & +F\left({x}_{D-1},{x}_{D}\right)+F\left({x}_{D},{x}_{1}\right)\end{array}$ | ${\left[-100,100\right]}^{D}$ | $\left[-100,100\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Alpine | ${f}_{1}(\mathrm{x})={\displaystyle \sum _{i=1}^{D}\left|{x}_{i}\cdot \mathrm{sin}{x}_{i}+0.1{x}_{i}\right|}$ | ${\left[-9,7\right]}^{D}$ | $\left[-7,7\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Moved axis parallel hyper-ellipsoid | ${f}_{1}(\mathrm{x})={\displaystyle \sum _{i=1}^{D}5i\cdot {x}_{i}^{2}}$ | ${\left[-10,30\right]}^{D}$ | $\left[-10,10\right]$ | $\left[0,\cdots ,0\right]$ | 0 |

Schwefel | ${f}_{1}(\mathrm{x})={\displaystyle \sum _{i=1}^{D}\left[-{x}_{i}\cdot \mathrm{sin}(\sqrt{\left|{x}_{i}\right|})\right]}$ | ${\left[-300,500\right]}^{D}$ | $\left[-300,300\right]$ | $\left[\begin{array}{c}420.9687\\ \vdots \\ 420.9687\end{array}\right]$ | $-D\cdot 418.9829$ |

**Table 2.**Comparisons of standard particle swarm optimization (PSO) algorithm with different types of random values for benchmark functions.

Function | r_{1}, r_{2} | $\mathit{U}\left[0,1\right]$ | 0.5 | $\mathit{U}\left[-1,1\right]$ | $\mathit{G}(0,1)$ |
---|---|---|---|---|---|

Dimension | 10/30/100 | 10/30/100 | 10/30/100 | 10/30/100 | |

Sphere1 | Average solution | 0/19.90/4203.68 | 0.74/1390.12/34,069.52 | 0/0/0 | 0/0/0 |

Standard deviation | 0/12.37/4400.60 | 1.62/575.44/10,343.18 | 0/0/0 | 0/0/0 | |

The worst solution | 0/48.87/13,080.62 | 7.90/2730.92/53,024.46 | 0/0/0 | 0/0/0 | |

The best solution | 0/4.13/303.05 | 0/443.08/17,579.98 | 0/0/0 | 0/0/0 | |

Sphere2 | Average solution | 0/1726.67/9386.74 | 130.34/5076.33/56,870.86 | 0/0/183.33 | 0/0/220 |

Standard deviation | 0/466.04/6716.49 | 180.26/9114.11/75,788.47 | 0/0/159.92 | 0/0/174.99 | |

The worst solution | 0/2600/44,700 | 600/38,700/323,027.84 | 0/0/600 | 0/0/600 | |

The best solution | 0/900.00/6400.80 | 0.01/2319.72/9236.02 | 0/0/0 | 0/0/0 | |

Rastrigin | Average solution | 1.03/52.83/357.43 | 17.01/116.61/810.69 | 0/0/0 | 0/0/0 |

Standard deviation | 1.60/21.87/81.84 | 9.70/33.22/76.28 | 0/0/0 | 0/0/0 | |

The worst solution | 4.97/100.77/499.92 | 42.81/215.88/932.59 | 0/0/0 | 0/0/0 | |

The best solution | 0/3.62/186.01 | 4.98/62.26/670.46 | 0/0/0 | 0/0/0 | |

Rosenbrock | Average solution | 0.40/462.38/308,982.29 | 250.36/63,215/13,665,740.29 | 1.93/28.86/98.92 | 0.98/28.90/98.94 |

Standard deviation | 1.22/292.83/182,460.39 | 663.05/59,859/6,367,792.97 | 2.86/0.08/0.05 | 2.43/0.07/0.03 | |

The worst solution | 3.99/1383.17/874,057.21 | 3515.43/260,355.55/27,517,029.65 | 8.93/28.96/98.98 | 8.75/28.97/98.98 | |

The best solution | 0/165.84/36,093.44 | 5.86/1614.69/5,391,184.05 | 0.00/28.67/98.77 | 1.63 × 10^{−6}/28.69/98.86 | |

Griewank | Average solution | 0.19/1.13/17.27 | 0.22/12.88/289.00 | 0/0/0 | 0/0/0 |

Standard deviation | 0.15/0.15/9.06 | 0.12/5.64/84.70 | 0/0/0 | 0/0/0 | |

The worst solution | 0.51/1.51/39.44 | 0.59/26.48/519.72 | 0/0/0 | 0/0/0 | |

The best solution | 0/0.90/3.45 | 0.08/3.34/123.68 | 0/0/0 | 0/0/0 | |

Ackley | Average solution | 1.46/6.07/11.12 | 1.64/9.00/15.63 | 0/0/0 | 0/0/0 |

Standard deviation | 1.22/1.50/2.26 | 1.02/1.25/1.10 | 0/0/0 | 0/0/0 | |

The worst solution | 3.22/8.90/15.17 | 3.58/12.06/17.46 | 0/0/0 | 0/0/0 | |

The best solution | 0/2.73/3.44 | 0.02/6.98/13.44 | 0/0/0 | 0/0/0 | |

Levy and Montalvo 2 | Average solution | 0/0.14/12.83 | 0.01/1.31/20.28 | 0/0.23/8.63 | 0/1.34/8.78 |

Standard deviation | 0/0.17/2.31 | 0.01/0.54/4.57 | 0/0.46/0.48 | 0/0.64/0.44 | |

The worst solution | 0/0.81/17.84 | 0.06/2.78/34.78 | 0/1.60/9.42 | 0/2.49/9.57 | |

The best solution | 0/0.01/9.09 | 0.00/0.28/14.28 | 0/0/7.38 | 0/0/7.89 | |

Sinsolidal | Average solution | −3.43/−1.02/−0.11 | −3.43/−0.41/0 | −3.50/−1.59/−0.09 | −3.18/−1/0 |

Standard deviation | 0.29/1.14/0.31 | 0.14/0.46/0 | 0/1.56/0.32 | 0.74/1.18/0 | |

The worst solution | −2.12/−0.01/0 | −2.93/−0.01/0 | −3.50/−0.01/0 | −0.87/−0.01/0 | |

The best solution | −3.50/−3.45/−1.66 | −3.50/−1.47/0 | −3.50/−3.50/−1.75 | −3.50/−3.50/0 | |

Rotated Expanded Scaffer | Average solution | 1.27/6.77/26.10 | 2.58/10.92/43.13 | 0/0/0 | 0/0/0 |

Standard deviation | 0.74/1.96/5.54 | 0.47/1.18/1.59 | 0/0/0 | 0/0/0 | |

The worst solution | 2.60/9.24/34.49 | 3.56/13.11/45.81 | 0/0/0 | 0/0/0 | |

The best solution | 0/1.92/13.45 | 1.73/8.63/39.45 | 0/0/0 | 0/0/0 | |

Alpine | Average solution | 0/2.67/39.15 | 0.12/6.36/57.50 | 0/2.14/27.04 | 0/3.36/30.12 |

Standard deviation | 0/1.70/8.56 | 0.34/1.81/8.48 | 0/1.17/3.96 | 0/1.19/3.65 | |

The worst solution | 0/7.30/62.51 | 1.81/10.83/75.846 | 0/5.71/34.30 | 0/6.04/38.17 | |

The best solution | 0/0.27/17.71 | 0/2.94/44.30 | 0/0/16.87 | 0/1/23.11 | |

Moved axis parallel hyper-ellipsoid | Average solution | 0/976.19/102,196.49 | 10.66/14,535.57/714,363.54 | 0/0/56,727.35 | 0/0/44,270.31 |

Standard deviation | 0/2276.36/46,700.85 | 21.97/6358.06/125,266.58 | 0/0/54,460.61 | 0/0/70,284.76 | |

The worst solution | 0/8532.57/196,643.04 | 105.74/33,237.39/967,098.52 | 0/0/195,537.54 | 0/0/235,721.53 | |

The best solution | 0/7.13/21,301.13 | 0.01/5625.85/484,347.13 | 0/0/0 | 0/0/0 | |

Schwefel | Average solution | −3472.55/−8400.88/−25,222.99 | −3072.44/−6957.38/−17,338.78 | −3772.47/−10,596.14/−31,640.44 | −3772.47/−10,651.80/−30,094.87 |

Standard deviation | 301.97/862.61/2717.54 | 402.88/941.95/2511.55 | 229.58/745.71/2489.84 | 233.82/635.86/1790.82 | |

The worst solution | −2865.61/−6560.66/−20,423.87 | −2151.57/−4665.91/−12,121.78 | −3355.12/−9101.35/−25,385.62 | −3235.87/−9101.35/−26,155.70 | |

The best solution | −4070.58/−9946.11/−31,017.18 | −3733.40/−8648.21/−21,643.45 | −4189.83/−11,854.02/−37,370.87 | −4189.83/−11,854.01/−32,695.87 |

**Table 3.**Comparisons of LDIW-PSO algorithm with different types of random values for benchmark functions.

Function | r_{1}, r_{2} | $\mathit{U}\left[0,1\right]$ | 0.5 | $\mathit{U}\left[-1,1\right]$ | $\mathit{G}(0,1)$ |
---|---|---|---|---|---|

Dimension | 10/30/100 | 10/30/100 | 10/30/100 | 10/30/100 | |

Sphere1 | Average solution | 0/0/1666.67 | 0/937.89/62,668.95 | 0/0/0 | 0/0/0 |

Standard deviation | 0/0/4611.33 | 0.01/1954.53/19,933.81 | 0/0/0 | 0/0/0 | |

The worst solution | 0/0/20,000 | 0.04/8616.07/110,110.64 | 0/0/0 | 0/0/0 | |

The best solution | 0/0/0 | 0/62.99/35,629.65 | 0/0/0 | 0/0/0 | |

Sphere2 | Average solution | 56.67/2033.33 /46,393.33 | 151.13/3905.81/49,939.96 | 0/0/293.33 | 0/0/236.6667 |

Standard deviation | 67.89/256.41/35,977.84 | 122.87/6591.85/40,171.84 | 0/0/228.84 | 0/0/225.1181 | |

The worst solution | 200/2600/189,200 | 500.38/38,800/153,700 | 0/0/900 | 0/0/700 | |

The best solution | 0/1600/8600 | 0.03/2361.29/9103.90 | 0/0/0 | 0/0/0 | |

Rastrigin | Average solution | 0/18.90 /130.24 | 26.30/142.34/865.88 | 0/0/0 | 0/0/0 |

Standard deviation | 0/22.91/83.07 | 11.70/31.11/82.15 | 0/0/0 | 0/0/0 | |

The worst solution | 0/82.72/335.59 | 48.75/193.01/1096.66 | 0/0/0 | 0/0/0 | |

The best solution | 0/0/28.92 | 7.97/67.63/737.39 | 0/0/0 | 0/0/0 | |

Rosenbrock | Average solution | 0.27/26.04/98.15 | 379.22/20,307.62/33,690,571.22 | 0.47/28.81/98.87 | 2.39/28.90/98.93 |

Standard deviation | 1.01/0.54/0.08 | 845.50/34,257.18/41,367,235.20 | 1.54/0.09/0.04 | 3.33/0.06/0.04 | |

The worst solution | 3.99/27.30/98.24 | 3032.11/107,129.05/231,274,237.32 | 7.48/28.94/98.94 | 8.95/28.98/98.99 | |

The best solution | 0/25.04/97.89 | 4.96/1698.53/6,545,833.08 | 0/28.59/98.77 | 1.29 × 10^{−8}/28.76/98.79 | |

Griewank | Average solution | 0/0/28.04 | 0.19/5.28/478.06 | 0/0/0 | 0/0/0 |

Standard deviation | 0/0/42.54 | 0.18/3.19/155.18 | 0/0/0 | 0/0/0 | |

The worst solution | 0/0/90.93 | 0.84/14.45/941.53 | 0/0/0 | 0/0/0 | |

The best solution | 0/0/0 | 0.05/1.70/203.77 | 0/0/0 | 0/0/0 | |

Ackley | Average solution | 0/0/9.64 | 0.80/8.25/18.27 | 0/0/0 | 0/0/0 |

Standard deviation | 0/0/6.53 | 0.85/2.90/0.79 | 0/0/0 | 0/0/0 | |

The worst solution | 0/0/19.97 | 2.81/16.67/19.44 | 0/0/0 | 0/0/0 | |

The best solution | 0/0/0 | 0/4.40/16.92 | 0/0/0 | 0/0/0 | |

Levy and Montalvo 2 | Average solution | 0/0/10.48 | 0/3.35/34.54 | 0/0.36/8.28 | 0/1.08/8.64 |

Standard deviation | 0/0/3.20 | 0/1.84/8.01 | 0/0.50/0.62 | 0/0.68/0.37 | |

The worst solution | 0/0/17.73 | 0/7.62/49.10 | 0/1.51/9.37 | 0/2.20/9.32 | |

The best solution | 0/0/5.40 | 0/0.70/18.90 | 0/0/6.31 | 0/0/7.70 | |

Sinsolidal | Average solution | −3.38/−1/−0.03 | −3.07/−0.12/0 | −3.27/−0.72/−0.05 | −3.21/−0.54/0 |

Standard deviation | 0.44/1.13/0.09 | 0.53/0.16/0 | 0.61/1.02/0.18 | 0.66/0.80/0 | |

The worst solution | −1.75/0/0 | −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 | −3.50/−3.50/−0.87 | −3.50/−3.50/0 | |

Rotated Expanded Scaffer | Average solution | 0.07/1.44/8.33 | 3/11.74/43.49 | 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 | |

The worst solution | 1/5.98/14.94 | 3.70/12.97/46.10 | 0/0/0 | 0/0/0 | |

The best solution | 0/0/0 | 1.42/8.57/41.56 | 0/0/0 | 0/0/0 | |

Alpine | Average solution | 0/0.91/16.46 | 0.31/8.41/73.22 | 0/2.45/27.24 | 0/2.93/30.70 |

Standard deviation | 0/1.94/7.65 | 0.76/3.95/13.39 | 0/1.23/6.95 | 0/1.44/4.19 | |

The worst solution | 0/8.46/34.94 | 2.85/15.54/96.89 | 0/5.02/36.40 | 0/6.16/37.74 | |

The best solution | 0/0.02/5.71 | 0/1.08/48.66 | 0/0/0 | 0/1/22.49 | |

Moved axis parallel hyper-ellipsoid | Average solution | 0/1383.33/102,196.49 | 20.64/9487.68/714,363.54 | 0/265.79/56,727.35 | 0/132.76/44,270.31 |

Standard deviation | 0/2215.48/46,700.85 | 90.81/5798.05/125,266.58 | 0/865.18/54,460.61 | 0/727.13/70,284.76 | |

The worst solution | 0/7000/196,643.04 | 500/24,372.80/967,098.52 | 0/3982.80/195,537.54 | 0/3982.65/235,721.53 | |

The best solution | 0/0/21,301.13 | 0/2221.47/484,347.13 | 0/0/0 | 0/0/0 | |

Schwefel | Average solution | −3764.52/−9823.92/−26,540.45 | −3134.09/−7401.16/−18,304.47 | −3764.52/−10,431.89/−30,589.24 | −3732.73/−10,492.05/−29,986.84 |

Standard deviation | 142.42/1344.37/3360.98 | 450.61/794.06/2000.45 | 243.94/662.12/2767.91 | 226.00/698.81/2431.96 | |

The worst solution | −3474.36/−6944.90/−20,227.96 | −2057.33/−5583.79/−14,272.03 | −3235.87/−8743.62/−25,650.73 | −3235.87/−8624.37/−25,743.44 | |

The best solution | −3951.34/−11,854.02/−31,935.70 | −3832.10/−9056.25/−22,371.46 | −4189.83/−11,496.29/−37,956.42 | −4189.83/−11,948.07/−34,477.94 |

**Table 4.**Comparisons of standard PSO and LDIW-PSO algorithms with different types of random values for pressure vessel design.

Type | r_{1}, r_{2} | $\mathit{U}\left[0,1\right]$ | 0.5 | $\mathit{U}\left[-1,1\right]$ | $\mathit{G}(0,1)$ |
---|---|---|---|---|---|

SPSO | Average solution | 5975.93 | 5975.93 | 5975.93 | 5975.94 |

Standard deviation | 0.00 | 0.00 | 0.01 | 0.01 | |

The worst solution | 5975.93 | 5975.93 | 5975.96 | 5975.99 | |

The best solution | 5975.93 | 5975.93 | 5975.93 | 5975.93 | |

LDIW-PSO | Average solution | 5975.93 | 5975.93 | 6001.34 | 6026.76 |

Standard deviation | 0.00 | 0.00 | 139.18 | 193.41 | |

The worst solution | 5975.93 | 5975.93 | 6738.24 | 6738.26 | |

The best solution | 5975.93 | 5975.93 | 5975.93 | 5975.93 |

© 2018 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**

Dai, H.-P.; Chen, D.-D.; Zheng, Z.-S.
Effects of Random Values for Particle Swarm Optimization Algorithm. *Algorithms* **2018**, *11*, 23.
https://doi.org/10.3390/a11020023

**AMA Style**

Dai H-P, Chen D-D, Zheng Z-S.
Effects of Random Values for Particle Swarm Optimization Algorithm. *Algorithms*. 2018; 11(2):23.
https://doi.org/10.3390/a11020023

**Chicago/Turabian Style**

Dai, Hou-Ping, Dong-Dong Chen, and Zhou-Shun Zheng.
2018. "Effects of Random Values for Particle Swarm Optimization Algorithm" *Algorithms* 11, no. 2: 23.
https://doi.org/10.3390/a11020023