Response Surface Methodology and MINITAB

8/10/2019 Response Surface Methodology and MINITAB

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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).
http://reliawiki.org/index.php/File:Doe9.10.png


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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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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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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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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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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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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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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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Response Surface Methodology and MINITAB

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