UROP writing backup

8/7/2019 UROP writing backup

1/5

ITERATIVE DESIGN OF EXPERIMENT: IMPLICATION OF GENETIC

ALGORITHM TO KRIGING MODELLING METHOD.

Hoang K.H 1. and Laksh S. 2

Department of Chemical and Biomolecular Engineering, National University of Singapore

Engineering Drive 4, Singapore 117576

ABSTRACT

A MATLAB program was written in this work for the purpose of design of experiment for batchexperiments. We present the program that based on previous data, iteratively searching for anew set of experimental points, updating the model and searching for the optimal points. The

stochastic method genetic algorithm is used to solve the multidimensional unconstrained non-linear of maximizing likelihood function posed by Kriging method. Genetic algorithm is alsoused as a searching tool for the maximal of 'expected improvement' which is the criterion to

select additional sample points. The program is tested and shows promising result.

INTRODUCTION

In any manufacturing industry, where processes are highly complex, it has been more difficult todevelop models for global optimization. Under that condition, it is desirable to have a design of experiment (DOE) scheme that can gather most unbiased data from least number of experiments

conducted, which can eventually lead to confirmation of optimal operating condition. Tovisualize the relation between input and output, either time-consuming hard-to-buildmathematical model or experimental-based statistical model will be used. By far, the most

popular statistical model is the traditional response methodology (RSM), which employs thesecond or third order polynomial and least square regression technique. While this methodology

provided an easy estimation, it has rather limited capability to solve non-linear systemaccurately. Higher order polynomial can be used to solve the problem, but require high number of experiment to estimate all the coefficients. Recently, Kriging model, which has been appliedin geostatistics, has gained popularity in this field, by providing a flexible structure that not only

can be fitted accurately into problems that seems complex and unnatural, but also providinguncertainty for regions. The limited of this technique is the difficulty of getting the right

program to fit the model and additional effort to use the model compare to normal polynomialmodel. LecturerIn this work, we adopted the framework suggested by Jones D.R. (A Taxonomyof global optimization methods based on Response surfaces,2001) and implemented thegenetic algorithm as optimization solver to develop into iterative method.

____________________________________________________

1Student , 2Senior Lecturer


2/5

ITERATIVE METHOD

Kriging modelling

Kriging models origin in mining geostatistical application involving spatially and temporallycorrelated data. Kriging combines a global model plus local deviation:

(1)

With f(x) is known polynomial, or in this program, is a constant, while is realization of astochastic process with mean zeros, variance and non zero covariance. The matrix of covarianceis defined as

(2)

where R is the correlation matrix where R (x i,,x j) is the correlation of between any 2 sample x i, x j.There are several correlation functions, which is chosen by user. In this work, we use theGaussian correlation function in the form of

(3)

The parameter of this model are ( ) which is estimated by solving the maximization of

likelihood function:

(4)

Where both R and are function of . The predicted value of on x * and deviation are

(5)

) (6)

(7)

While any value of will create interpolative Kriging model, best fit model is created bysolving multidimensional unconstrained nonlinear problem of maximizing likelihood function.

Expected improvement

In this study, we adopted the concept of expected improvement (EI). Instead of directlysearching for the optimal, EI will increase as the predicted response is smaller than the bestvalue obtained and/or the variance is large enough. Thus, for the next iteration, we use genetic


3/5

algorithm to search for next experiment point that have highest EI value. If all the EI value isclose to zeros, the iteration may be terminated as there is nearly no improvement can be found.Improvement of point x * is

(8)

Where ybest is the minimum value has been found. Since we defined y * is Gaussian distributed,

with mean and variance , the expected improvement is calculated as

(9)

Where (.) is the standard normal cumulative distribution function and is the standardnormal density function.

Genetic Algorithm

Genetic algorithms (GAs) are stochastic global search and optimization methods that mimic themetaphor of natural biological evolution. GAs operate on a population of potential solutionsapplying the principle of survival of the fittest to produce successively better approximations toa solution. At each generation of a GA, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and reproducingthem using operators borrowed from natural genetics. This process leads to the evolution of

populations of individuals that are better suited to their environment than the individuals fromwhich they were created, just as in natural adaptation.

Iterative method

No

Ye


4/5

Figure 1: Flow chart for iterative design of experiment

The method starts with initial set of data, which can be previous experiments, or process

database, or a new set of experiment. The data collected is process by Kriging modelling andgenetic algorithm to search for best fitted , thus, the best model. Due to the nature of geneticalgorithm and depend on the level of non-linearity of the responses, the modelling process can

be done several times to get the best possible results from likelihood function. From theestimated parameter, Kriging model will be constructed, and genetic algorithm acts as EI searchengine to find the next best point. Again, as the nature of genetic algorithm, this process can bedone several times, so the best results can be found and further distilled through fuzzy c-meansto the desired number of experiments.

CASE STUDY

The test function used for the method is Himmeblaus function:

This function is a multi optimal function ( 4 minimal and 1 local maxima). In the test case, theinitial data set is 5 2 designs, which may be used for normal response surface methodology. Thenext best EI points are located by GA, and then distilled by fuzzy c-means to 1 to 4 points. Theiteration continue to 6 th batch, with 40 points tested where next best EI were 10 -6

CONCLUSION

In this work, a methodology for design of experiment and search for global optimization wasdemonstrated. We adopted Kriging modelling and Genetic Algorithm to provide the unbiased

program that has the capability of generating fairly accurate model, of searching for the nextexperiment with criteria of best expected improvement to further explore the regions andsearch for optimal. The case study implied that the proposed method is effective, and still, thereare improvement to be done for this method.

REFERENCES

A.J. Chipperfield, P. J. Fleming, The MATLAB Genetic Algorithm Toolbox, retrieve frommathwork.comJones, D.R, A Taxonomy of Global Optimization Methods Based on Response Surfaces inJournal of Global Optimization,2001Jones, D.R, Schonlau,M, Welch,W.J, Efficient Global Optimization of Expensive Black-Box

Functions in Journal of Global Optimization,1998Junghui Chen, David Shan Hill Wong,Shi-Shang Jang, Seng-Lu Yang Product and Process

Development Using Artificial Neural-Network Model and Information Analysis in AIChEJournal, April 1998Zhang,G., Olsen,M.M.,Block,D.E, New Experimental Design Method for Highly


5/5

Nonlinear and Dimensional Processes , in WileyInterscience, June 2007

Documents

UROP writing backup