Statistical Process Optimization Through Multi-Response Surface Methodology

Abstract—In recent years, response surface methodology (RSM) has

brought many attentions of many quality engineers in different

industries. Most of the published literature on robust design

methodology is basically concerned with optimization of a single

response or quality characteristic which is often most critical to

consumers. For most products, however, quality is multidimensional,

so it is common to observe multiple responses in an experimental

situation. Through this paper interested person will be familiarize

with this methodology via surveying of the most cited technical

papers.

It is believed that the proposed procedure in this study can resolve

a complex parameter design problem with more than two responses.

It can be applied to those areas where there are large data sets and a

number of responses are to be optimized simultaneously. In addition,

the proposed procedure is relatively simple and can be implemented

easily by using ready-made standard statistical packages.

Keywords—Multi-Response Surface Methodology (MRSM),

Design of Experiments (DOE), Process modeling, Quality

improvement; Robust Design.

I. INTRODUCTION

ESPONSE Surface Methodology (RSM) is a well known

up to date approach for constructing approximation

models based on either physical experiments, computer

experiments (simulations) (Box et al., [1] ; Montgomery, [2])

and experimented observations. RSM, invented by Box and

Wilson, is a collection of mathematical and statistical

techniques for empirical model building. By careful design of

experiments, the objective is to optimize a response (output

variable) which is influenced by several independent variables

(input variables). An experiment is a series of tests, called

runs, in which changes are prepared in the input variables in

order to recognize the reasons for changes in the output

response (Montgomery & Runger [3]). RSM involves two

basic concepts:

(1) The choice of the approximate model, and

(2) The plan of experiments where the response has to be

evaluated.

The performance of a manufactured product often

characterize by a group of responses. These responses in

general are correlated and measured via a different

measurement scale. Consequently, a decision-maker must

resolve the parameter selection problem to optimize each

response. This problem is regarded as a multi-response

optimization problem, subject to different response

requirements. Most of the common methods are incomplete in

such a way that a response variable is selected as the primary

R. Eslami Farsani is with Islamic Azad University, Tehran South Branch.

one and is optimized by adhering to the other constraints set

by the criteria. Many heuristic methodologies have been

developed to resolve the multi-response problem. Cornell and

Khuri [4] surveyed the multi-response problem using a

response surface method. Tai et al. [5] assigned a weight for

each response to resolve the problem. Pignatiello [6] utilized a

squared deviation-from-target and a variance to form an

expected loss function for optimizing a multiple response

problem. Layne [7] presented a procedure capable of

simultaneously considering three functions: weighted loss

function, desirability function, and distance function. While

providing a multi-response example in which Taguchi

methods are used, Byrne and Taguchi [8] discussed an

example involving a connector and a tube.

Logothetis and Haigh [9] also discussed a manufacturing

process differentiated by five responses. In doing so, they

selected one of the five response variables as primary and

optimized the objective function sequentially while ignoring

possible correlations among the responses. Optimizing the

process with respect to any single response leads to nonoptimum

values for the remaining characteristics.

II. RESPONSE SURFACE METHODOLOGY

Often engineering experimenters wish to find the conditions

under which a certain process attains the optimal results. That

is, they want to determine the levels of the design parameters

at which the response reaches its optimum. The optimum

could be either a maximum or a minimum of a function of the

design parameters. One of methodologies for obtaining the

optimum is response surface technique.

Response surface methodology is a collection of statistical

and mathematical methods that are useful for the modeling

and analyzing engineering problems. In this technique, the

main objective is to optimize the response surface that is

influenced by various process parameters. Response surface

methodology also quantifies the relationship between the

controllable input parameters and the obtained response

surfaces.

The design procedure of response surface methodology is as

follows:

(i) Designing of a series of experiments for adequate and

reliable measurement of the response of interest.

(ii) Developing a mathematical model of the second order

response surface with the best fittings.

(iii) Finding the optimal set of experimental parameters

that produce a maximum or minimum value of

response.

S. Raissi, and R- Eslami Farsani ∗

Statistical Process Optimization

Through Multi-Response Surface Methodology

R

World Academy of Science, Engineering and Technology 51 2009

267

(iv) Representing the direct and interactive effects of

process parameters through two and three dimensional

plots.

If all variables are assumed to be measurable, the response

surface can be expressed as follows:

y = f (x1, x2 ,....xk ) (1)

The goal is to optimize the response variable y . It is

assumed that the independent variables are continuous and

controllable by experiments with negligible errors. It is

required to find a suitable approximation for the true

functional relationship between independent variables and the

response surface. Usually a second-order model is utilized in

response surface methodology.

β Σβ Σβ Σβ ε

= = =

= + + + +

k

i

ij i j

k

i

ii i

k

i

y i xi x x x

1 1

2

1

0 (2)

where ε is a random error. The β coefficients, which

should be determined in the second-order model, are obtained

by the least square method. In general (2) can be written in

matrix form.

Y = bX+E (3)

where Y is defined to be a matrix of measured values, X to

be a matrix of independent variables. The matrixes b and E

consist of coefficients and errors, respectively. The solution of

(3) can be obtained by the matrix approach.

b (XTX)−1XTY

= (4)

where XT is the transpose of the matrix X and (XTX)-1 is the

inverse of the matrix XTX.

The mathematical models were evaluated for each response

by means of multiple linear regression analysis. As said

previous, modeling was started with a quadratic model

including linear, squared and interaction terms. The significant

terms in the model were found by analysis of variance

(ANOVA) for each response. Significance was judged by

determining the probability level that the F-statistic calculated

from the data is less than 5%. The model adequacies were

checked by R2, adjusted-R2, predicted-R2 and prediction error

sum of squares (PRESS). A good model will have a large

predicted R2, and a low PRESS. After model fitting was

performed, residual analysis was conducted to validate the

assumptions used in the ANOVA. This analysis included

calculating case statistics to identify outliers and examining

diagnostic plots such as normal probability plots and residual

plots.

Maximization and minimization of the polynomials thus

fitted was usually performed by desirability function method,

and mapping of the fitted responses was achieved using

computer software such as Design Expert.

III. THE SEQUENTIAL NATURE OF THE RESPONSE SURFACE

METHODOLOGY

Most applications of RSM are sequential in nature and can

be carried out based on the following phases.

Phase 0: At first some ideas are generated concerning

which factors or variables are likely to be important in

response surface study. It is usually called a screening

experiment. The objective of factor screening is to reduce the

list of candidate variables to a relatively few so that

subsequent experiments will be more efficient and require

fewer runs or tests. The purpose of this phase is the

identification of the important independent variables.

Phase 1: The experimenter’s objective is to determine if the

current settings of the independent variables result in a value

of the response that is near the optimum. If the current settings

or levels of the independent variables are not

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