# Improved particle swarm optimization algorithm using design of experiment and data mining techniques

- First Online:

- Received:
- Revised:
- Accepted:

DOI: 10.1007/s00158-015-1271-7

- Cite this article as:
- Liu, Z., Zhu, P., Chen, W. et al. Struct Multidisc Optim (2015) 52: 813. doi:10.1007/s00158-015-1271-7

## Abstract

Particle swarm optimization (PSO) is a relatively new global optimization algorithm. Benefitting from its simple concept, fast convergence speed and strong ability of optimization, it has gained much attention in recent years. However, PSO suffers from premature convergence problem because of the quick loss of diversity in solution search. In order to improve the optimization capability of PSO, design of experiment method, which spreads the initial particles across a design domain, and data mining technique, which is used to identify the promising optimization regions, are studied in this research to initialize the particle swarm. From the test results, the modified PSO algorithm initialized by OLHD (Optimal Latin Hypercube Design) technique successfully enhances the efficiency of the basic version but has no obvious advantage compared with other modified PSO algorithms. An extension algorithm, namely OLCPSO (Optimal Latin hypercube design and Classification and Regression tree techniques for improving basic PSO), is developed by consciously distributing more particles into potential optimal regions. The proposed method is tested and validated by benchmark functions in contrast with the basic PSO algorithm and five PSO variants. It is found from the test studies that the OLCPSO algorithm successfully enhances the efficiency of the basic PSO and possesses competitive optimization ability and algorithm stability in contrast to the existing initialization PSO methods.

### Keywords

Particle swarm optimization Design of experiment Data mining Optimization search Global optimization Algorithm stability### Nomenclature

- PSO
Particle Swarm Optimization

- OLHD
Optimal Latin Hypercube Design

- OLCPSO
Optimal Latin hypercube design and Classification and regression tree techniques for improving basic PSO

- DOE
Design of experiments

- CART
Classification and Regression Tree

- ESE
Enhanced Stochastic Evolutionary

- GPSO
Particle Swarm Optimization initialized by Gaussian distribution

- EPSO
Particle Swarm Optimization initialized by Exponential distribution

- LNPSO
Particle Swarm Optimization initialized by Lognormal distribution

- VC-PSO
Particle Swarm Optimization initialized by Vander Corput sequence

- SO-PSO
Particle Swarm Optimization initialized by Sobol sequence

- D
Dimension

**X**The position vector of a particle

**V**The velocity vector of a particle

**P**_{i}The best previously visited position vector

**P**_{g}The global best position vector of the swarm

*t*Generation number

*c*_{1}Cognitive scaling parameter

*c*_{2}Social scaling parameter

*w*Inertia weight

*r*_{1}Random numbers

*r*_{2}Random numbers

**y**Vector of classes

*M*Number of response

*N*Numbers of design variable

**xc**Matrix of design variable

*xc*_{j}^{R}Best splitting value of variable

*xc*_{j}*ϕ*_{p}Optimal criterion for design of experiment

*d*_{i}Euclidean distance

*J*_{i}Number of pairs separated by

*d*_{i}in the design*s*Number of distinct distance values

*p*A positive integer

*T*_{h}Threshold of the cooling and warming schedule

**X**_{E}Current design of DOE

**X**_{try}Best design based on ESE

*S*A uniform random number between 0 and 1

- P
Particle number

*s*_{f}Sample ratio factor

*t*_{c}Supplement coefficient

*L*Length of variable’s range

*P*_{s}Particle number in sub-region