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# Setting up a Response surface test (RSM) lang: en_US

Response Surface Methodology is a statistical test setup with more factors on different levels combined in one experiment. It is used when analyzing complex problems with a multiple of influence factors in once including interactions. This is done by using test arrays. A RSM is advanced DOE with specially designed arrays for calculating interactions and quadratic responses.

When comparing a standard DOE with a RSM the DOE gives a 2D image of the output of 1 factor, a RSM produces a 3D image of the output of 2 factors in one image.
 Standard DOE 2D RSM 3D

## Why use a RSM test setup

1. To find optimal settings
2. To find more robust settings
3. Understand how factors interact with each other

## What is a Factor

A factor is a input for a experiment that can change the output when variating. Its like a dimmer (a factor) of a lamp when turning the knob the brightness of the lamp changes.

## What is important for a successful RSM?

1. The output can be measured, in continuous a scale (Ratio or Interval)!
2. The influencing factors are known
3. Important Factors can be controlled (variated on a desired level or fixed on a constant level)
4. Keep the RSM simple as possible
5. DO the Confirmation Run!

## The Level of the factor

The level of a factor is the input setting of test. For the lamp dimmer the setting of the dimmer is the level (0%, 25%, 50% ect).
But the bigness of a lamp can have more influences from different factors with it own levels:
 Factor Levels Scale Power 1, 2, 9, 30, 40 60 100Watt Ratio Setting on the dimmer 0%…100% Ratio Input current 0...230V Ratio Color of glass Clear, White, Silver, Green, Red Ordinal Type of lamp Light bulb, LED, TL Ordinal Armature Silver reflector, White reflector, No reflector Ordinal Shape of lamp Ball, Cone, Candle Nominal

### Settings of levels

1. Try to chose realistic values for the levels (not impractical high or low)
2. Avoid impossible combinations of the levels with other factors in the experiment
3. The Factor must have a continuous scale
• Color of glass , Type of lamp, Armature, Shape of lamp are not suitable!
4. The
Center
point is the average of the min and max value of the levels
 Good distribution of the Levels Wrong distribution of the levels

## The test Arrays

For a RSM there are various types of Arrays each with its own pros and cons. See table below.
 Compared Box-Behnken design Central Composite design Extreme combinations No Yes Size of matrix Smaller Bigger Amount of levels 3 5 No Circumscribed points The Circumscribed point especially for the bigger arrays are far from the normal setting

### RMS Arrays compared with Full factorial and Orthogonal

The arrays designed for RSM are Rotatable around their Center point and symmetrical, this is not the case with Orthogonal Array and Factorial Array. Rotatable is desired for the quadratic fitting of the model.
 Compared Full Factorial Array Orthogonal Array Rotatable No No Symmetrical Yes No
A Full Factorial Array can be used for a RSM but compared with a CCC more test runs are needed (Full 27 , CCC 20, BB 15) and the quadratic fitting is poor.

### Array selection

As you can see in a Box-Behnken design there are less data point needed compared with a Central Composite design.
With a Box-Behnken design every factor is having 3 levels compared to Central Composite 5 with 2 circumscribed points with a bigger distance.
 Factors Box-Behnken Central Composite Test runs Test runs Distance Circumscribed 2 13 1.414 3 15 20 1.682 4 26 30 2.000 5 45 52 2.378 6 54 91 2.828

Box-Behnken compared with Central Composite design is missing the corner points, it will never occur that all the factors are high or low at the same time (No extreme combinations).
 Box-Behnken design Central Composite design No extreme points Extreme points

#### Modifications of the Central Composite Array

There are 2 modifications of the standard CCC array to eliminate the
point outside
the maximum setting.
 Standard CCC CCI CCF Circumscribed Inscribed Face centered Best fitting The extreme settings are extrapolated Poor Quadratic fitting • 5 Levels • 2 Levels outside the investigated limits • Rotatable • 5 Levels • All levels are within the investigated limits • Rotatable • 3 Levels • All levels are within the investigated limits • Not Rotatable

### Select array

#### Use coded array!

Use the coded array and don't change the coded values for the actual levels!
 Coded array Do not use a uncoded array! To see the not coded axles put the actual data in the Response Surface table
##### Example Coded and not Coded
As visible in the below graphs in the non coded result influence of the quadratic factors is gone. For this a zero crossing is needed and the minimum and maximum of all the factors must be the same.
 Coded result Data file Not coded result Data file Coded result with Non coded in table

### Building and testing the samples

Now build the samples according the array.

Some important points
• Always build and test the complete array.
• When adding repetitions it is preferred is not to test replicates on a row (samples with the same setting) but first finish the first replicate then the next. This is to randomize the order to prevent drift over time in the result.
• When testing try not to test all factors on the first level than on the second level but try to randomize.
• Do not add an extra test in the array except Center points this will create an unbalanced array, and can lead to wrong results.

The test array will be added in the Factorial table.
Now the test results put in the input table

See Data file.

### Analyzing the result

Select the Response surface box (DOE => Response surface) for the statistical analysis.

In this window you can select the factors to include in the Response surface. Develve will calculate the coefficient of the selected factors and if it is significant. See here for the formula of the calculation.

#### Colors of the cells

• Green
Not significant
• Yellow
Significant
Select all the factors with Check All.

De select all insignificant factors.

### Response Surface Graph

Display the Response Surface Graph by clicking on Graph.

By clicking on the graph the calculated result will pop-up with coded input value.
 single response 2D graph 3D response graph
By right click on the graph one of the responses can be displayed in 2 or 3D.

### Confirmation Run

After defining the optimum settings build samples according these settings to confirm the result.