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Response Surface Methodologyand MINITAB
Presented by
K.A.SUNDARARAMAN,
Assistant Professor,
Department of Mechanical Engineering,SSM Institute of Engineering and Technology, Dindigul
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ORGANIZATION
What IS RSM?
Where RSM can be employed?
First order and second order models Steps in RSM
Case Analysis
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What is RSM (contd,...)
For example, in the case of the optimization of the calcinationof cement, the engineer wants to find the levels of
temperature (x1) and time (x2) that maximize the early age
strength (y) of the cement. The early age strength is a function
of the levels of temperature and time, as follows:
y = f (x1, x2) +
where represents the noise or error observed in the
response y. The surface represented by f(x1, x2) is called a
response surface.
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Where RSM can be employed?
Analytical models possesses inherent assumptions that are
not possible to be employed for designing a particular procees
or product.
Less number of experimental runs are sufficient to achieve
meaning conclusions.
In situations, where the analytical model are not possible to
developed or employed.
An efficient and cost-effective way to model and analyse the
relationship between the parameters and the response.
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First order andsecond order models
The response is well modeled by a linear function of the
independednt variables, then the approximating function is
the first order model,
Y = 0+ 1x1+ 2x2+....+ nxn+
There is curvature in the system, then a polynomial of higher
degree such as second order model must be used.
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Second order models
In situations where the first order models are inadequate to fit the
curvature in true response surface and often give lack-of-fit.
Most of the practical applications are non linear in nature.
There is considerable practical experience indicating that second
order models work well in solving response surface problems.
Second order model is very flexible that can take on a wide variety
of functional forms, so it will often work well as an approximation
to the true response surface.
Second order model is easy to fit and estimate the parameters
using the least square method.
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Steps in RSM
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Steps in RSM (contd,...)
1. Defining the parameters and their ranges, the response and
the event
Knowledge and experience are essential to choose the
parameters that are varied in the experiments.
2. Preliminary experiments
Insufficient and excessive preliminary experimentation cause
failure or troubles in the task.
The amount of time required to perform the preliminary
experiments varies depending on the process being studied
and the number and complexity of the issues.
Preliminary experiments are performed to identify the
influencing parameters and their potential ranges where the
design parameters significantly influence the response.
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Steps in RSM (contd,...)3. Experimental design
Central composite design
CCD are first-order (2N) designs augmented by additional
centre and axial points to allow estimation of the tuning
parameters of a second-order model.
The simplest of the central composite designs can be used to
fit a second order model to a response with two factors.
The design consists of a full factorial design augmented by a
few runs at the center point (such a design is shown in figure
(a) given below).
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Steps in RSM (contd,...)
A central composite design is obtained when runs at points
(-1,0), ( 1,0), (0,-1) and (0,1) are added to this design. These
points are referred to as axial points or star points and
represent runs where all but one of the factors are set at their
mid-levels. The number of axial points in a central composite
design having k factors is 2k.
Fig(b) shows the two factor central composite design with =1
The distance of the axial points from the center point is
denoted by and is specified in terms of coded values.
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Steps in RSM (contd,...)
Fig(c) shows the two factor central composite design with =1.414
It can be noted that when >1, each factor is run at five levels
(-,-1, 0, 1, and ) instead of the three levels of -1, 0, and 1.
The reason for running central composite designs with >1 is
to have a rotatable design.
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Steps in RSM (contd,...)
4. Regression Analysis , Model developement and Model Adequacy
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Model Adequacy
ANOVA and F-ratio test , Regression statistics to justify the
goodness of fit of the developed models
Normal probability plot, Plot of the residuals versus the fittedvalue, Residuals against the observation order to test the
linearity, normality, constant variance and independence of
the residuals
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Model Adequacy (contd,...)
F test for the lack of fit
The calculated value of Fratio for the lack-of-fit must be
lesser than its standard value for a desired level of confidencelevel.
F test for the model
The calculated value of Fratio for the model must be greater
than its standard value for a desired level of confidence level.
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Model Adequacy (contd,...)
Regression statistics
The coefficient of determination R2 is a measure of the
amount of reduction in the variability of response obtainedusing the regression variables in the model.
The value of R2equals to 1 if the model exactly matches.
R2and R2adjshould not differ much.
R2 and R2adj differ dramatically, there is a good chance that
non-significant terms have been included in the model.
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Model Adequacy (contd,...)
The residuals are the deviations of the observed value of the
dependent variable from the predicted values. The ANOVA used in
analysing response on the dependent variable make certain
assumptions about the distributions of residual values on the
dependent variable. The residuals
(1) are normally distributed (normality),
(2) exhibit constant variance,
(3) have a mean of zero (linearity),
(4) are independent from each other.
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Model Adequacy (contd,...)
Normal probability plot
To assess the assumption that the residuals are normally
distributed. The normality assumption is satisfied if the normal
probability plot of the residuals forms a straight line
Plot of the residuals versus the fitted value
The two assumptions linearity and constant variances can be
checked
The residuals vary randomly around zero and the plot of the
residuals versus the fitted values should not have obvious pattern
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Model Adequacy (contd,...)
Residuals against the observation order
The independence of the residuals is checked
To have runs of positive and negative residuals and a pattern
should not exist in this plot
The assumption associated with the independence is not
violated
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CONFIRMATION EXPERIMENTS
Confirmation experiments are to be conducted for
intermediate values of the process variables and the results
are compared against the results of prediction model.
The error was calculated by the equation as follows:
where Yp is the predicted value and Yexp is the experimental
value.
If the percentage error calculated is small and the predicted
values from the models are in good accord with the
experimental values.
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