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.

Full text

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 non-
optimum 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
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(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:
),....,( 21 k
xxxfy = (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.
εββββ
∑∑∑ ===
++++= k
ijiij
k
iiii
k
iii xxxxy
11
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.
EbXY += (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.
()
YXXXb T
1
T−
= (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 consistent with
optimum performance, then the experimenter must determine
a set of adjustments to the process variables that will move the
process toward the optimum. This phase of RSM makes
considerable use of the first-order model and an optimization
technique called the method of steepest ascent (descent).
Phase 2: Phase 2 begins when the process is near the
optimum. At this point the experimenter usually wants a
model that will accurately approximate the true response
function within a relatively small region around the optimum.
Because the true response surface usually exhibits curvature
near the optimum, a second-order model (or perhaps some
higher-order polynomial) should be used. Once an appropriate
approximating model has been obtained, this model may be
analyzed to determine the optimum conditions for the process.
This sequential experimental process is usually performed
within some region of the independent variable space called
the operability region or experimentation region or region of
interest.
IV. MULTI-RESPONSE PROBLEM OVERVIEWS
Optimization of the multi-response problem is a challenge
to optimize output responses all together. Among the
simultaneous optimization methods, most of the authors used
the approaches that combine all the different response
requirements into one composite requirement. Hence, the
compromise solution is obtained in a much simpler way. A
simple weighting method was found in Ilhan et al. [10], as
applied in an electrochemical grinding (ECG) process. Zadeh
[11] normalized each response and then gave a simple weight
for each response. The discussion regarding the assignments
of weights can be found in [12].
Myers and Carter [13] proposed an algorithm for obtaining
the optimal solutions of the dual-response surface system
(DRSM). Their method assumed that the DRSM includes a
primary response and a constraint response which both of
them can be fitted as a quadratic model.
Lee-Ing Tong et al. [14] used the signal to noise (SN) ratio
and system sensitivity are used to assess the performance of
each response. They performed principal component analysis
(PCA) on SN values and system sensitivity values to obtain a
set of uncorrelated principle components, which are linear
combinations of the original responses. Additionally, they
used of variation mode chart to interpret the variation mode
(or principal component variation) resulting from PCA. They
suggested that based on engineering requirements, engineers
can determine the optimization direction for each principal
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component using the variation mode chart. Finally, technique
for order preference by similarity to ideal solution (TOPSIS)
applied to derive the overall performance index (OPI) for
multiple responses. The optimal factor/level combination can
be determined with the maximum OPI value and therefore,
simultaneously reduces the quality variation and brings the
mean to the target value.
Onur Koksoy and Tankut Yalcinoz [15] presented a
methodology for analyzing several quality characteristics
simultaneously using the mean square error (MSE) criterion
when data are collected from a combined array. They
proposed a genetic algorithm based on arithmetic crossover
for the multi-response problem in conjunction with a
composite objective function based on the individual MSE
functions of each response.
Lee-Ing Tong et al. [16] proposed procedure used the
desirability function and dual-response-surface method to
optimize the multi-response problems in a dynamic system.
They established a regression model to obtain the sensitivity
and quality variation for each experimental run and the
desirability function is used to obtain a total measurement for
the multiple responses. Next, the dual-response-surface
method was used to obtain a set of possible optimal factor–
level combinations. The optimal factor–level setting proposed
to maximize total desirability.
Liao and Chen [17] proposed data envelopment analysis
ranking (DEAR) approach to optimize multi-response
problem. The author states that Taguchi method can only be
used to optimize single response problems and PCA, although
considered to solve multi-response problem, itself has
shortcomings. The new approach is capable of decreasing
uncertainty caused by engineering judgment in the Taguchi
method and overcoming the shortcomings of PCA.
In order to overcome the single response optimization
problem of Taguchi method, Liao [18] proposed an effective
procedure called PCR-TOPSIS that is based on process
capability ratio (PCR) theory and on the theory of order
preference by similarity to the ideal solution (TOPSIS) to
optimize multi-response problems.
Hsu [19] presents an integrated optimization approach
based on neural networks, exponential desirability functions.
Fung and Kang [20] used Taguchi method and PCA to
optimize the given process. Initially Taguchi method was used
followed by PCA to correspond to multi-response cases, for
transforming the correlated friction properties to a set of
uncorrelated components and evaluating the principal
components. The appropriate number of the principle
components, and the influence of the number on the optimum
process condition, was subsequently studied by extracting
more than one principal component and integrating it into a
comprehensive index.
Jiju Antony et al. [21] used artificial inteligent tool (neuro-
fuzzy model) and Taguchi method of experimental design to
tackle problems involving multiple responses optimization.
They proposed a single crisp performance index called Multi-
Response Statistics (MRS) as a combined response indicator
of several responses. MRS is computed for every run by
applying neuro-fuzzy model. ANOVA is carried out on the
MRS values to identify the key factors/interactions having
significant effect on the overall process. Finally, optimal
setting of the control factors is decided by selecting the level
having highest value of MRS.
V. DESIRABILITY FUNCTION
The desirability function was originally developed by
Harrington [22] to simultaneously optimize the multiple
responses and was later modified by Derringer and Suich [23]
to improve its practicality. The desirability function approach
is one of the most frequently used multi-response optimization
techniques in practice. The desirability lies between 0 and 1
and it represents the closeness of a response to its ideal value.
If a response falls within the unacceptable intervals, the
desirability is 0, and if a response falls within the ideal
intervals or the response reaches its ideal value, the
desirability is 1. Meanwhile, when a response falls within the
tolerance intervals but not the ideal interval, or when it fails to
reach its ideal value, the desirability lies between 0 and 1. The
more closely the response approaches the ideal intervals or
ideal values, the closer the desirability is to 1. According to
the objective properties of a desirability function, the
desirability function can be categorized into the nominal-the-
best (NB) response, the larger-the-better (LB) response and
the smaller-the-better (SB) response. Interested persons can
follow the expressed relevant desirability functions in [101].
The proposed desirability function transforms each response to
a corresponding desirability value between 0 and 1. All the
desirability can be combined to form a composite desirability
function which converts a multi-response problem into a
single-response one. The desirability function is a scale-
invariant index which enables quality characteristics to be
compared to various units. In such method the plant manager
can easily determine the optimal parameters among a group of
solutions.
Kun-Lin Hsieh et al. [24] believed that when desirability
values lies more close to 0 or 1 may lead to a bad model’s
additive. To solve this problem, they referred to Taguchi
suggestion in using the Omega (Ω) transformation which is
employed to transfer the data into an additive mode. Ω
transformation’s philosophy is to simultaneously maximize
the average of the system and minimize the variation via S/N.
This transformation transfer the desirability data lying in [0,1]
to the range of . This transformation can resolve
the problem by summing up the control factor’s effect when
the data lie outside the interval [0,1].
VI. SPECIAL CASE: DUAL-RESPONSE SURFACE METHOD
In practical cases, there are many situations where the
researchers encounter to multi-responses. In such cases
surveying two or more response variables are critical.
Over the last few years in many manufacturing
organizations, multiple response optimization problems were
resolved using the past experience and engineering judgment,
which leads to increase in uncertainty during the decision-
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199International Scholarly and Scientific Research & Innovation 3(3) 2009 scholar.waset.org/1999.7/14692
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making process.
Myers and Carter [25] proposed an algorithm for obtaining
the optimal solutions of the dual-response surface method
(DRSM). Their method assumed that the DRSM includes a
primary response, p
y and a constraint response, s
y . Both
p
y and s
ycan be respectively fitted as a quadratic model as
follows:
s
k
ijiij
k
iiii
k
iiis
p
k
ijiij
k
iiii
k
iiip
xxxxy
xxxxy
εββββ
εββββ
∑∑∑
∑∑∑
===
===
++++=
++++=
11
2
1
0
11
2
1
0
(4)
where the
β
’s and
γ
’s represent the unknown coefficients,
and
P
ε
and s
ε
denote the random errors, respectively. The
random errors are assumed to possess a normal distribution with
mean 0 and variance 2
σ
.
The DRSM attempts to obtain a set of X, which can
optimize p
y

subjected to the constraint cys=

, where C is a
constant.
The desirability function simultaneously optimize the
multiple responses and was later modified by Derringer and
Such [23] to improve its practicality. The desirability function
approach is one of the most frequently used multi-response
optimization techniques in practice. The desirability lies
between 0 and 1 and it represents the closeness of a response
to its ideal value. If a response falls within the unacceptable
intervals, the desirability is 0, and if a response falls within the
ideal intervals or the response reaches its ideal value, the
desirability is 1.
Meanwhile, when a response falls within the tolerance
intervals but not the ideal interval, or when it fails to reach its
ideal value, the desirability lies between 0 and 1. The more
closely the response approaches the ideal intervals or ideal
values, the closer the desirability is to 1. According to the
objective properties of a desirability function, the desirability
function can be categorized into three forms, nominal-the-best
(NB), larger-the-better (LB) and smaller-the-better (SB).
The total desirability is defined as a geometric mean of the
individual desirability:
()
k
k
dddD 1
21 ...×××= (5)
where D is the total desirability and di is the ith desirability, i
= 1, 2, . . . , k. If all of the quality characteristics reach their
ideal values, the desirability di is 1 for all i. Consequently, the
total desirability is also 1. If any one of the responses does not
reach its ideal value, the desirability di is below 1 for that
response and the total desirability is below 1. If any one of the
responses cannot meet the quality requirements, the
desirability di is 0 for that response. Total desirability will then
be 0. The desirability function is a scale-invariant index which
enables quality characteristics to be compared to various units.
Therefore, the desirability function is an effective means of
simultaneously optimizing a multi-response problem.
VII. MODEL ADEQUACY CHECKING
To verify the derived mathematical model of each response,
model adequacy is always necessary to:
1. Examine the fitted model to ensure that it provides an
adequate approximation to the true system;
2. Verify that none of the least squares regression
assumptions are violated. There are several techniques for
checking model adequacy.
Residual Analysis: The residuals from the least squares fit,
defined by iii yye

−
=
, i = 1, 2,…, n, play an important role
in judging model adequacy. Many response surface analysts
prefer to work with scaled residuals, in contrast to the ordinary
least squares residuals. These scaled residuals often convey
more information than do the ordinary residuals.
The standardizing process scales the residuals by dividing
them by their average standard deviation. In some data sets,
residuals may have standard deviations that differ greatly.
There is some other way of scaling that takes this into account.
Let’s consider this.
The vector of fitted values i
y

corresponding to the
observed values i
yis
HyyXXXXXby 1=== −TT )(

(6)
The n x n matrix TT XXXXH 1−
=)( is usually called
the hat matrix because it maps the vector of observed values
into a vector of fitted values. The hat matrix and its properties
play a central role in regression analysis.
Since iii yye

−
=
, there are several other useful ways to
express the vector of residuals H)y(IHyyXbye −
=
−
=
−
=
(7)
The “prediction error sum of squares” (PRESS) proposed in
[26, 27], provides a useful residual scaling
2
11
∑
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=n
iii
ih
e
PRESS
(8)
From [27], it is easy to see that the PRESS residual is just
the ordinary residual weighted according to the diagonal
elements of the hat matrix ii
h. Generally, a large difference
between the ordinary residual and the PRESS residual will
indicate a point where the model fits the data well, but a model
built without that point predicts poorly.
VIII. CONCLUSION
The RSM is one of the design of experiments (DOE)
methods used to approximate an unknown function for which
only a few values are computed. The RSM stems from science
disciplines in which physical experiments are performed to
study the unknown relation between a set of variables and the
system output, or response, for which only a few experiment
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values are acquired. These relations are then modeled using a
mathematical model, called response surface.
There are many situations where the quality engineers
encounter to several correlated responses simultaneously. In
such cases decision making on optimum set of parameters is a
complicated mathematical problem. In this paper an analysis
of the most cited methods proposed and the.
Through this paper, readers could be familiar to multi-
response optimization problem via the most cited methods.
The residual analysis method and the prediction error sum of
squares (PRESS) proposed for evaluating the capability of the
designed models. Researcher could follow standard
optimization techniques such as the differentiation, the
operation research method to set their process in optimum
conditions.
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International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:3, 2009
201International Scholarly and Scientific Research & Innovation 3(3) 2009 scholar.waset.org/1999.7/14692
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