Journal of Institute of Control, Robotics and Systems (2014) 20(8): ISSN: eissn:

Journal of Institute of Control, Robotics and Systems (0) 0(8):8-87 http://dx.doi.org/0/j.icros.0..009 ISSN:97- eissn:- On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution *,, (ByungSeok Lee,*, Joon Hwa Lee, and Moon-Beom Heo ), School of Electrical and Computer Engineering, University of Seoul, Satellite Navigation Team, Korea Aerospace Research Institute Abstract: This paper deals with the performance comparison of a PSO algorithm inspired in the process of simulating the behavior pattern of the organisms. The PSO algorithm finds the optimal solution (fitness value) of the objective function based on a stochastic process. Generally, the stochastic process, a random function, is used with the expression related to the velocity included in the PSO algorithm. In this case, the random function of the normal distribution (Gaussian) or uniform distribution are mainly used as the random function in a PSO algorithm. However, in this paper, because the probability distribution which is various with shape parameters can be expressed, the performance comparison of a PSO algorithm using the beta probability distribution function, that is a random function which has a high degree of freedom, is introduced. For performance comparison, functions (Rastrigin, Rosenbrock, Schwefel) were selected among the benchmark Set. And the convergence property was compared and analyzed using PSO-FIW to find the optimal solution. Keywords: PSO (Particle Swarm Optimization), PSO-FIW (PSO with Fixed Inertia Weight), beta probability distribution, benchmark set I. 서론 99 Kennedy Eberhart (PSO: Particle Swarm Optimization) PSO [-7]. PSO (AL: Artificial Life) (Swarm theory) (EC: Evolutionary Com- putation)., (GA: Genetic Algorithms) (EP: Evolutionary Programming) [,]. AL PSO (social behavior), (herd), (bird flock), (fish school) []. GA EC PSO (SI: Swarm Intelligence) * Corresponding Author Manuscript received December, 0 / revised April, 0 / accepted June, 0 : (byungseok@uos.ac.kr) : (joonhwa@uos.ac.kr) : (hmb@kari.re.kr). GA agent individual point (particle) [,]., particle solution. GA population PSO Swarm. PSO. PSO (convergence speed) (local search) (global search), PSO., PSO (normal distribution) (uniform distribution) (quasi-random or pseudo random) (low-discrepancy sequences) Sobol, Faure, Halton, Van der Corput [8,9], [0] Gamma. [0], (shape parameter), Copyright ICROS 0

On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 8 8, PSO Normal, Uniform., 00 ( 00 ). II PSO. III PSO (Inertia Weight) PSO (, PSO-IW) PSO-IW (convergence rate) (Constriction Coefficient) PSO (, PSO-CC). IV,. V, VI., PSO. II. PSO 알고리즘. PSO 알고리즘연산식 PSO, [,]. PSO (solution) (particle) (position) (velocity). () () PSO. () () (), PSO (). ().,. (acceleration constant) (weighting value). ()., Cognitive Component Nostalgia Swarm ( ) ( ), Social Component ( ). PSO. Fig.. Pseudo code of PSO algorithm. ( ).,, [-].. PSO 알고리즘의사코드 PSO () (). (fitness value) (objective function)., PSO best position Swarm (personal best position, ).,, personal best position (global best position, )., personal best position pbest lbest, global best position gbest. (pseudo code). III. 계수및제한성이개선된 PSO 알고리즘. 관성하중계수를갖는 PSO 알고리즘 PSO () exploitation exploration. Reynolds (bird flocks) (Collision Avoidance, Velocity Matching, Flock Centering) [7] ( ) (neighbor) swarm.

8 이병석, 이준화, 허문범 exploration exploitation () inertia weight PSO PSO-IW Shi Eberhart []. (), (diversity),. []. ( 遞增 ). 랜덤조절방법 (random adjustments). () 0.7 Normal(Gaussian) distribution. () (), () () cognitive social component [,].,.. 선형감소방법 () (linear decreasing). 0.9, 0.. () [,]. (). 비선형감소방법 (nonlinear decreasing) (7)-(9) [,]. (7) (8) (9) (7)., (8) (relative improvement).. 퍼지적응방법 (fuzzy adaptive)..,. LOW, MEDIUM, HIGH []. 관성체증방법 (increasing inertia) 0. 0.9 [].. 제한계수를갖는 PSO 알고리즘 () (0). (0) () () []. () (),. ().. 베타확률분포 IV. 베타확률분포와벤치마크함수 ( ),. ( ) (RV: Random Variable).,., () PDF (shape parameter)., (beta function). ()

베타확률분포를이용한입자떼최적화알고리즘의성능비교 87 Probability Density Function of Distribution a=b=. a=b=.8 a=b=. a=b=.0 a=b=0... Table. Selected benchmark functions. Func. Name & Formula Rastrigin function cos Min. Value 0 0 0 0. 0. 0. 0. 0. 0.7 0.8 0.9 R. V. f f Rosenbrock function Schwefel function sin 0 0. #. Fig.. Probability density function of beta distribution #... Probability Density Function of Distribution a=., b=. a=, b=0.8 a=.8, b=. a=.0, b=. a=, b= a=., b=. 0 0 0. 0. 0. 0. 0. 0.7 0.8 0.9 R. V.. #. Fig.. Probability density function of beta distribution #. (). 0, (). () -.,.. 벤치마크함수... (dimensionality) (non-linearity) (multi-modal),, [8-0]., domain MATLAB surf. Min. Value. (a) Rastrigin function (b) Rosenbrock function (c) Schwefel function..... Fig.. Shape of the benchmark functions.

88 ByungSeok Lee, Joon Hwa Lee, and Moon-Beom Heo. PSO. Table. Principle setting value for PSO algorithm test using Func. benchmark functions. Search Space [-0, 0] f [-0, 0] f [-00, 00] V. 시뮬레이션결과 MATLAB m-file. particle, PSO particle Swarm. PSO PSO-IW., 0.7 (fixed) (, PSO-FIW),.. 00 00 [ ]. PSO-FIW ()~(). ~ f 0 ()~()., particle. PSO. () () ()~(),, particle, domain, search space (upper bound) (lower bound). Dim. 0 0 0 0 0 0 Swarm Size Iteration # per Algorithm 00 00 00 [ ],, PSO Gussian, Uniform,, MATLAB randn, rand, random. (, ), (, ), (.8,.), (.,.8), (0., 0.), (.,.), (.8,.8), (.,.). [ ] Gaussian Uniform., f, f 0 0 PSO-FIW 00 ( 00 ).,, 8., a b pdf, a b pdf.,, Gaussian PSO-FIW Uniform ( )., Uniform PSO-FIW,, f 0, 0 (.8,.), (0., 0.) Unifrom., f 0, (, ), (.8,.), (0., 0.), (.,.), (.,.) Uniform, f 0, (.8,.), (0., 0.) Uniform. VI. 결론 (EC) PSO. PSO (MSC: Monte Carlo Simulation), (SA: Simulated Annealing) Gaussian Uniform Gaussian Uniform PSO. 8 (, ), (.,.8), (.8,.8) Gaussian Uniform, f, f 0, 0,

On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 89.. Table. Performance Comparison according to the Func. probability distribution and the Decrement compared with the Gaussian and Uniform probability distribution against the Optimum probability distribution. Dim. Distribution Compared with Gaussian Compared with Uniform 0 > > 9.9% 70.7% 0 > > 87.000% 9.9979% f 0 > > 99.9998% 99.7% 0 > > 99.990% 99.99% f 0 > >.0% 8.789% 0 > >.00% 8.7% : (, ), : (.,.8), : (.8,.8). ( ) a, b.,., MATLAB Gaussian Uniform,. 부록,, Min. Value Mean Standard Deviation 00 ( 00 ).. PSO-FIW #. Table. PSO-FIW algorithms performance comparison according to the probability distribution #.., Uniform, Gaussian, %., (.,.) 0 f Uniform 0., (, ), (.,.8). particle [8-0] (low-dicrepancy). 0.,, particle, Gaussian, Uniform particle particle particle,.,, Probability Distribution Gaussian Uniform Func. Dim. Min. Value Mean 0.0 0 Min. Value Standard Deviation 0.8 0. 0 9.8709 0.99 0 f.90 0 0.80 0 7.8 0 0.0 0 f 89.8 0.09 0.88 0 0.80 7.7079 0 7. 0.09 0 0.0 0 f. 0 0.09 0 7.97 0 0.799 0 f 0 0.0 0 8 0

80 이병석, 이준화, 허문범. PSO-FIW #. Table. PSO-FIW algorithms performance comparison accord- ing to the probability distribution #.. PSO-FIW #. Table. PSO-FIW algorithms performance comparison accord- ing to the probability distribution #. Probability Distribution Func. Dim. Min. Value Mean Min. Value Standard Deviation Probability Distribution Func. Dim. Min. Value Mean Min. Value Standard Deviation 0 9.70 0.899 0. 0.70 0.0 0 09 0.0 0 78 0 (a=, b=) 0.08 0 f 89.980 0.0 0. 0 (a=0., b=0.) 0 8 0 f.80 0 0. 0.700 0 0.00 0 f. 0 0.0 0.99 0 0.878 0 f 8 0 0.08 0.9 0 0.890. 0.90 0 7.089 0.8977.90 0. 0 8.078 (a=, b=) f 0.99 89 0.7 0.87 0 (a=., b=.) 0. 0 f.87 0 0.00 0.99 0 0.8 0 f.87 0 0 7. 0.8087 0 0.80 0 f.0 0 0.0 0.99 0 0.78 0.7 0 88 0 09 0 0 9. 8.08 0.9 0.7 (a=.8, b=.) 0.90 0 f.9 0 0.99 0 7.8 0 (a=.8, b=.8) f 0 0.8890.08 0.8 0.09 0 0.0 0 f.7 0 0.08 0.9 0 0.0 0 f.78 0 0 9.90 0.77 0 0 9.7 0.8 0. 0.08 0 0.8990 7.807 0.78 0.99 (a=., b=.8) f 0.0.9 0. 0.90 0 (a=., b=.) f 0 8.79.79 0.977 0.0 0 0.89 0 f.0 0 0 7.8 0.00 0 0.879 0 f.907 0 0 9.907 0.0 0

베타확률분포를이용한입자떼최적화알고리즘의성능비교 8 x 0 80 Algo no. 0 8 Algo no. 0 000 Algo no. 0 70 700 0 00 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 7 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 900 700 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 0 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 x 0 00 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 7 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, Uniform, Dimension = 0 f, Uniform, Dimension = 0 f, Uniform, Dimension = 0 000 Algo no. 0 Algo no. 0 x 0 7 Algo no. 0 Algo no. 0 x 0. Algo no. 0 Algo no. 0 900 700 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00...8.....08 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00.0 00 0 00 0 00 0 00 0 00 0 00 0 00.0 0 00 0 00 0 00, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 00 x 0 7 x 0 00 Algo no. 0 Algo no. 0. Algo no. 0 Algo no. 0. Algo no. 0 Algo no. 0 00 000 00 00 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00....08.0.0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00.0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, Uniform, Dimension = 0 f, Uniform, Dimension = 0 f, Uniform, Dimension = 0. PSO-FIW #( = 0, 0). Fig.. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0, 0). Gaussian Uniform, f, f 0 0 PSO-FIW. 00 00. 00 0 0. Uniform Gaussian.

8 ByungSeok Lee, Joon Hwa Lee, and Moon-Beom Heo x 0 Algo no. 0 Algo no. 0 00 Algo no. 0 700 00 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 000 00 900 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 00 Algo no. 0 Algo no. 0 Algo no. 0 x 0. Algo no. 0 Algo no. 0 Algo no. 0 00 00 Algo no. 0 Algo no. 0 Algo no. 0 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00.. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 000 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 00 000 900 700 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 0 8 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 0 00 0 00 0 00 0 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 000 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 00 8 x 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 000 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 0 8 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 000 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0. PSO-FIW #( = 0). Fig.. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). a, b,, f, f 0 0 PSO-FIW. (, ) (.,.8).

On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 8 00 00 000 00 00 00 000 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 07 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0...0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 x 0 7 00 000 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 000 000 0000 900 9000 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 700 7000 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 000 00 000 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 7... Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0...08.0.0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00.0 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 x 0 7 000 00 000 00 000 00 000 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 8 7 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 000 000 0000 900 9000 0 700 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 7. PSO-FIW #( = 0). Fig. 7. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). 7 0 0,, f, f PSO-FIW. (, ) (.,.8).

8 이병석, 이준화, 허문범 900 700 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 8 7 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 00 00 00 000 900 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 0 00 0 00 0 00 0 00 0 00 0 00 700 0 00 0 00 0 00, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 x 0 00 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 7 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 700 00 00 00 00 00 00 000 900 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 700 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 x 0 00 Algo no. 0. Algo no. 0 Algo no. 0 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 0 00 0 00 0 00 0 00 0 00 0 00 00 0 00 0 00 0 00, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 x 0 0 00 0 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00... Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 00 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 8. PSO-FIW #( = 0). Fig. 8. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). 8 a, b.,, f, f 0 0 PSO-FIW. 8 (.,.) 0 f,., 0 f (0., 0.), (.,.), (.8,.8).

베타확률분포를이용한입자떼최적화알고리즘의성능비교 8 00 00 00 000 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 7. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0....08.0.0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00.0 000 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 000 00 000 00 000 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 7. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0...0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 x 0 7 x 0 000 00 00 00 000 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00...0 0.9 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 0.9 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 x 0 x 0 000 900 700 00 00 00 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00....0 0.9 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 00 0.9 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 9. PSO-FIW #( = 0). Fig. 9. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). 9 8 0 0,, f, f PSO-FIW. 9 (.,.) f. (0.) 0 f (.8,.8).

8 ByungSeok Lee, Joon Hwa Lee, and Moon-Beom Heo REFERENCES [] R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proc. of th International Symposium on Micro Machine and Human Science, pp. 9-, Oct. 99. [] J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. of IEEE International Conf. on Neural Networks, vol., pp. 9-98, Nov./Dec. 99. [] J. Sun, C.-H. Lai, and X.-J. Wu, Particle Swarm Optimisation : Classical and Quantum Perspectives, CRC Press, Taylor & Francis Group, London, 0. [] A. P. Engelbrecht, Fundamentals of Computational Swarm Intelligence, John Wiley & Sons Ltd., Chippenhan, Wiltshire, England, 00. [] Y. Shi and R. Eberhart, A modified particle swarm optimizer, Proc. of IEEE International Conf. on Evolutionary Computation, pp. 9-7, May 998. [] M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, vol., no., pp. 8-7, Feb. 00. [7] C. Reynolds, Flocks, herds, and schools : a distributed behavioral model, Proc. of SIGGRAPH '87 Conf. on Computer Graphics, pp. -, Jul. 987. [8] H. H. Rosenbrock, An automatic method for finding the greast or least of a function, Computer Journal, pp. 7-8, 90. [9] L. A. Rastrigin, External Control System, Theoretical Foundations of Engineering Cybernetics Series, Nauka, Moscow, Russia, 97. [0] H. P. Schwefel, Numerical Optimization of Computer Models, John Wiley & Sons, Chichester, U.K., 98. [] Y. Shi and R. C. Eberhart, Empirical study of particle swarm optimization, Proc. of the Congress on Evolutionary Computation, vol., pp. 9-90, 999. [] Y. Zheng, L.-H. Ma, L. Zhang, and J. Qian, On the convergence analysis and parameter selection in particle swarm optimization, Proc. of IEEE International Conf. on Machine Learning and Cybernetics, vol., pp. 80-807, Nov. 00. [] S. H. Park, H. T. Kim, and K. T. Kim, Improved autofocusing of stepped-fequency ISAR images using new form of particle swarm optimisation, IET Journals & Magazines on Electronics Letters, vol., no. 0, pp. 0-0, Sep. 009. [] P. Zhang, P. Wei, and H.-Y. Yu, Biogeography-based optimisation search algorithm for block matching motion estimiation, IET Journals & Magazines on Image Processing, vol., no. 7, pp. 0-0, Oct. 0. [] A. Kusiak and Z. Zhang, Adaptive control of a wind turbine with data mining and swarm intelligence, IEEE Transactions on Sustainable Energy, vol., no., pp. 8-, Jan. 0. [] B. Yang, Y. Chen, and Z. Zhao, Survey on applications of particle swarm optimization in electric power systems, Proc. of IEEE International Conf. on Control and Automation, pp. 8-8, May 007. [7] Wang Xin, Li Ran, Wang Yanghua, Peng Yong, and Qin Bin, Self-tuning PID controller with variable parameters based on particle swarm optimization, Proc. of IEEE International Conf. on Intelligent System Design and Engineering Applications, pp. -7, Jan. 0. [8] Nguyen Quang Uy, N. X. Hoai, RI. Mckay, and P. M. Tuan, Initialising PSO with randomised low-discrepancy sequences: the comparative results, Proc. of IEEE Congress on Evolutionary Computation, pp. 98-99, Sep. 007. [9] M. Pant, R. Thangaraj, C. Grosan, and A. Abraham, Improved particle swarm optimization with low-discrepancy sequences, Proc. of IEEE Congress on Evolutionary Computation, pp. 0-08, Jun. 008. [0] R. Thangaraj, M. Pant, and K. Deep, Initializing PSO with probability distributions and low-discrepancy sequences : the comparative results, Proc. of World Congress on Nature & Biologically Inspired Computing, pp. -, Dec. 009. [] J. Peng, Y. Chen, and R. C. Eberhart, Battery pack state of charge estimator design using computational intelligence approaches, Proc. of the Annual Battery Conference on Applications and Advances, pp. 7-77, 000. [] T. Peram, K. Veeramachaneni, and C. K. Mohan, Fitness-distance-ratio based particle swarm optimization, Proc. of the IEEE Swarm Intelligence Symposium, pp. 7-8, Apr. 00. [] Y. Shi and R. C. Eberhart, Fuzzy adaptive particle swarm optimization, Proc. of the IEEE Congress on Evolutionary Computation, vol., pp. 0-0, May 00. [] Y. Zheng, L. Ma, L. Zhang, and J. Qian, Empirical study of particle swarm optimizer with increasing inertia weight, Proc. of the IEEE Congress on Evolutionary Computation, vol., pp. -, Dec. 00. 이병석 00. 00 7 ~00 9. 009. 0. 0 ~.,,, (GNSS).

On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 87 이준화 987. 989. 99 8. 99 7 ~99 9. 99 0 ~99 California Institute of Technology(Caltech). 99 ~.. 허문범 99. 997 Illinois Institute of Technology. 00 Illinois Institute of Technology. 00 0 ~. GNSS,,.