Setting up a Response surface test (RSM)

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 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.

Add data

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.